0$ .","The model problem (REF )-() arises from many scientific applications such as fluid flow in porous media.","Mostly importantly, this model problem has served, and still serves, the scientific computing community as a testbed in the search and design of new and efficient computational algorithms for partial differential equations.","The classical Galerkin finite element method (see, e.g., [10], [28], [16]) is particularly a numerical technique originated from the study of elliptic problems closed related to (REF )-() or its variations.","In the last three decades, various finite element methods using discontinuous trial and test functions, including discontinuous Galerkin (DG) methods and weak Galerkin (WG) methods, have been developed for numerical solutions of partial differential equations.","These developments were often tested over testbed problems such as (REF )-() before they were generalized or applied to more complex problems in science and engineering.","The DG method, also known as the interior penalty method in different contexts, was originated in early 70s of the last century for a numerical study of model problems such as (REF )-(); see, e.g., [3], [14], [25], [38] for early incubations and [1], [13], [17], [27] for a detailed discussion and recent developments.","The weak Galerkin finite element method is a recently developed discretization framework for partial differential equations [36], [37], [24], [34].","With new concepts referred to as weak differential operators (e.g., weak gradient, weak curl, weak Laplacian etc.)","and weak continuity through the use of various stabilizers, the method allows the use of totally discontinuous functions and provides stable numerical schemes that are parameter-independent or free of locking [33].","For the convection-diffusion-reaction equation (REF )-(), the recent work in the context of weak Galerkin includes the algorithm developed and analyzed in [9], the one in [20] for singularly perturbed problems, and an earlier one in [39].","The WG finite element method has been rapidly developed and applied to several different types of problems, including second order elliptic problems, the Stokes and Navier-Stokes equations, the biharmonic and elasticity equations, div-curl systems and the Maxwell's equations, etc.","The latest development of the WG methods is the prime-dual formulation for problems that are either nonsymmetric or do not have variational forms friendly for numerical use.","Details on the new developments can be found in [30] for second order elliptic equations in nondivergence form, [31] for the Fokker-Planck equation, and [32] for elliptic Cauchy problems.","The typical WG method for the model problem (REF )-() seeks weak finite element approximations $u_h=\\lbrace u_0, u_b\\rbrace $ satisfying $u_b|_{\\partial \\Omega } = Q_b g$ and $S(u_h, v)+ (\\alpha \\nabla _w u_h, \\nabla _w v) + ({\\beta }\\cdot \\nabla _w u_h, v_0) + (c u_0, v_0) = (f,v_0)$ for all test functions $v=\\lbrace v_0, v_b\\rbrace $ satisfying $v_b|_{\\partial \\Omega }=0$ , where $Q_bg$ is an interpolation of the Dirichlet boundary data, $\\nabla _w$ is the discrete weak gradient operator, and $S(\\cdot ,\\cdot )$ is a properly selected stabilizer that gives weak continuities for the numerical solutions.","The numerical solution $u_h$ consists of two components: the approximation $u_0$ on each element and the approximation $u_b$ on the boundary of each element.","To reduce the computational complexity, some hybridized formulations have been introduced in [22], [29] for the method when applied to the diffusion equation and the biharmonic equation through the elimination of the degrees of freedom associated with the unknown function $u_0$ locally on each element.","In the superconvergence study for WG [18] on rectangular elements, this hybridized formulation was further simplified in the description of the numerical algorithm, yielding a simplified weak Galerkin (SWG) finite element scheme for the diffusion equation.","In our further investigation of the SWG to the convection-diffusion-reaction equation (REF ), we came to the conclusion that SWG represents a new discretization scheme that is different from the usual WG through a simple elimination of the unknown $u_0$ .","As a result, we believe that a systematic study of the SWG for the convection-diffusion-reaction problem (REF )-() should be conducted for its stability and convergence.","This paper is in response to this observation and shall provide a mathematical theory for the stability and the convergence of the simplified weak Galerkin finite element method for the model problem (REF )-().","We believe that the result of this paper can be extended to other types of modeling equations.","The paper is organized as follows: In Section , we shall describe the simplified weak Galerkin finite element method for (REF )-() on general polygonal partitions.","In Section , we shall present a computational formula for the element stiffness matrices and the element load vectors from SWG.","In Section , we provide a mathematical theory for the stability and well-posedness of the SWG scheme.","Sections and are devoted to a discussion of the error estimates in a discrete $H^1$ and the $L^2$ norm for the numerical solutions.","Finally, in Section , we present some numerical results to demonstrate the efficiency and accuracy of the SWG method.","Throughout the rest of the paper, we assume $d=2$ and shall use the standard notations for Sobolev spaces and norms [10], [16].","For any open set $D\\subset \\mathbb {R}^{2}$ , $\\Vert \\cdot \\Vert _{s,D}$ and $(\\cdot ,\\cdot )_{s,D}$ denote the norm and inner-product in the Sobolev space $H^s(D)$ consisting of square integrable partial derivatives up to order $s$ .","When $s=0$ or $D=\\Omega $ , we shall drop the corresponding subscripts in the norm and inner-product notation."],["Algorithm on Polymesh","Assume that the domain is of polygonal type and is partitioned into non-overlap polygons ${\\mathcal {T}}_h=\\lbrace T\\rbrace $ that are shape regular.","For each $T\\in {\\mathcal {T}}_h$ , denote by $h_T$ its diameter and by $N$ the number of edges.","For each edge $e_i, \\ i=1,\\ldots , N$ , denote by $M_i$ the midpoints and ${\\bf n}_i$ the outward normal direction of $e_i$ (see Fig.","REF for an illuatration).","The meshsize of ${\\mathcal {T}}_h$ is defined as $h=\\max _{T\\in {\\mathcal {T}}_h} h_T$ .","Let $v_b$ be a piecewise constant function defined on the boundary of $T$ , i.e., $v_b|_{e_i} =v_{b,i},$ with $v_{b,i}$ being a constant.","We define the weak gradient of $v_b$ on $T$ by: $\\nabla _w v_b:=\\displaystyle \\frac{1}{|T|}\\sum _{i=1}^N v_{b,i}|e_i|\\bf {n_i},$ where $|e_i|$ is the length of the edge $e_i$ and $|T|$ is the area of the element $T$ .","It is not hard to see that the weak gradient $\\nabla _w v_b$ satisfies the following equation: $(\\nabla _w v_b, \\mathbf {\\phi })_T=\\langle v_b, \\mathbf {\\phi }\\cdot {\\bf n}\\rangle _{\\partial T}$ for all constant vector $\\mathbf {\\phi }$ .","Here and in what follows of the paper, $\\langle \\cdot ,\\cdot \\rangle _{\\partial T}$ stands for the usual inner product in $L^2({\\partial T})$ .","Denote by $W(T)$ the space of piecewise constant functions on ${\\partial T}$ .","The global finite element space $W({\\mathcal {T}}_h)$ is constructed by patching together all the local elements $W(T)$ through single values on interior edges.","The subspace of $W_h({\\mathcal {T}}_h)$ consisting of functions with vanishing boundary value is denoted as $W_h^0({\\mathcal {T}}_h)$ .","We use the conventional notation of ${P}_j(T)$ for the space of polynomials of degree $j\\ge 0$ on $T$ .","For each $v_b\\in W(T)$ , we associate it with a linear extension in $T$ , denoted as ${\\mathfrak {s}}(v_b)\\in {P}_1 (T)$ , satisfying $\\sum _{i=1}^{N}({\\mathfrak {s}}(v_b)(M_i) -v_{b,i})\\phi (M_i)|e_i|=0,\\quad \\forall \\; \\phi \\in {P}_1(T).$ It is easy to see that ${\\mathfrak {s}}(u_b)$ is well defined by (REF ), and its computation is local and straightforward.","In fact, ${\\mathfrak {s}}(u_b)$ can be viewed as an extension of $u_b$ from $\\partial T$ to $T$ through a least-squares fitting.","Figure: An illustrative polygonal element.On each element $T \\in {\\mathcal {T}}_h $ , we introduce the following bilinear forms: $a_T(u_b,v_b)&:= & (\\alpha \\nabla _w u_b, \\nabla _w v_b)_T, \\\\b_T(u_b,v_b)&:= & ({\\beta }\\cdot \\nabla _w u_b, {\\mathfrak {s}}(v_b))_T,\\\\c_T(u_b,v_b)&:= & (c{\\mathfrak {s}}(u_b), {\\mathfrak {s}}(v_b))_T.$ For simplicity, we set $\\mathcal {B}_T(u_b,v_b):=a_T(u_b,v_b) + b_T(u_b,v_b) + c_T(u_b,v_b)$ for $u_b, v_b \\in W(T)$ .","We further introduce the stabilizer $\\begin{split}S_T(u_b,v_b):= & h^{-1}\\sum _{i=1}^N ({\\mathfrak {s}}(u_b)(M_i)-u_{b,i})({\\mathfrak {s}}(v_b)(M_i)-v_{b,i})|e_i|\\\\= & h^{-1}\\langle Q_b{\\mathfrak {s}}(u_b)-u_{b},Q_b{\\mathfrak {s}}(v_b)-v_{b}\\rangle _{\\partial T},\\end{split}$ where $Q_b$ is the $L^2$ projection operator onto $W(T)$ ; namely $Q_b u$ is the average of $u$ on each edge.","In particular, $Q_b(g)$ is well-defined and takes the average of the Dirichlet data on each boundary edge.","SWG Algorithm 2.1 The simplified weak Galerkin (SWG) scheme for the elliptic equation (REF )-() seeks $u_b \\in W_h({\\mathcal {T}}_h)$ satisfying $u_b = Q_b(g)$ on $\\partial \\Omega $ and ${\\mathcal {A}}(u_b, v_b) =(f, {\\mathfrak {s}}(v_b))\\qquad \\forall v_b \\in W_h^0({\\mathcal {T}}_h),$ where ${\\mathcal {A}}(u_b,v_b):=\\kappa S(u_b,v_b)+ {\\mathcal {B}}(u_b, v_b)$ , $S(u_b,v_b) &=& \\sum _{T\\in {\\mathcal {T}}_h} S_T(u_b,v_b),\\\\{\\mathcal {B}}(u_b,v_b)&=& \\sum _{T\\in {\\mathcal {T}}_h} \\mathcal {B}_T(u_b, v_b)$ are bilinear forms in $W_h({\\mathcal {T}}_h)$ and $(f,{\\mathfrak {s}}(v_b)):=\\sum _{T\\in {\\mathcal {T}}_h} (f, {\\mathfrak {s}}(v_b))_T$ is a linear form in $W_h({\\mathcal {T}}_h)$ ."],["Element Stiffness Matrices","The simplified weak Galerkin finite element method (REF ) is user-friendly in computer implementation.","In this section, we present a formula for the computation of the element stiffness matrices and the element load vector on general polygonal elements.","Let $T\\in {\\mathcal {T}}_h$ be a polygonal element of $N$ sides.","Denote by $X_{u_b}$ the vector representation of $u_b$ given by $(u_{b,1},u_{b,2},\\ldots ,u_{b,N})^T$ .","Then, the element stiffness matrix and the element load vector for the SWG scheme (REF ) are given in a block matrix form as follows: $(\\kappa h^{-1}A^T + B + R + C)X_{u_b} \\cong F,$ where the block components in (REF ) are given by: (1) $A:=\\lbrace a_{i,j}\\rbrace _{i,j=1}^N = E- EM(M^TEM)^{-1}M^TE$ , (2) $B:=\\lbrace b_{i,j}\\rbrace _{i,j=1}^N$ , with $b_{i,j} = (\\alpha \\mathbf {n}_i,\\mathbf {n}_j)_{T}\\displaystyle \\frac{|e_i||e_j|}{|T|^2}$ , (3) $R:=\\lbrace r_{ij}\\rbrace _{i,j=1}^N$ , with $r_{ij}= \\frac{|e_j|}{|T|}\\int _T {\\beta }\\cdot \\mathbf {n}_j \\zeta _i dT$ , (4) $C:=\\lbrace c_{ij}\\rbrace _{i,j=1}^N$ , with $c_{ij}= \\int _T c\\zeta _j\\zeta _i dT$ , (5) $F:=\\lbrace f_{i}\\rbrace _{i=1}^N$ , with $f_{i} = \\int _{T}f(x,y) \\zeta _i(x,y) dT$ , (6) $D:=\\lbrace d_{j,i}\\rbrace _{3\\times N}=(M^TEM)^{-1}M^TE$ and $\\zeta _i = d_{1,i} + d_{2,i} (x-x_T) + d_{3,i} (y-y_T)$ , (7) $M$ and $E$ are given by $M=\\begin{bmatrix}1 & x_{1} - x_T & y_{1} - y_T\\\\1 & x_{2} - x_T & y_{2} - y_T\\\\\\vdots & \\vdots & \\vdots \\\\1 & x_{N} - x_T & y_{N} - y_T\\\\\\end{bmatrix}_{N\\times 3},\\;E=\\begin{bmatrix}|e_1| & & & \\\\& |e_2| & & \\\\& & \\ddots & \\\\& & & |e_N| \\\\\\end{bmatrix}_{N\\times N}.$ Here $M_T=(x_T,y_T)$ is any point on the plane (e.g., the center of $T$ as a specific case), $(x_i, y_i)$ is the midpoint of $e_i$ , $|e_i|$ is the length of edge $e_i$ , $\\mathbf {n}_i$ is the unit outward normal vector on $e_i$ , and $|T|$ is the area of the element $T$ .","From (REF ), the element stiffness matrix on $T\\in {\\mathcal {T}}_h$ consists of two sub-matrices corresponding to the following forms: $S_T(u_b,v_b)\\ \\mbox{and } {\\mathcal {B}}_T(u_b, v_b).$ The bilinear form ${\\mathcal {B}}_T(\\cdot ,\\cdot )$ is composed of three bilinear forms given by (REF ).","The rest of this section is devoted to a computation of the element stiffness matrices for each of the bilinear forms involved."],["The stiffness matrix for $S_T(\\cdot ,\\cdot )$","For the element stiffness matrix corresponding to $S_T(u_b,v_b)$ , the key is to compute ${\\mathfrak {s}}(u_b)$ and ${\\mathfrak {s}}(v_b)$ which can be accomplished through its definition (REF ); readers are referred to [21] for a detailed derivation.","Specifically, let $M_T=(x_T,y_T)$ be the center of T (or any point on the plane), the extension ${\\mathfrak {s}}(u_b)$ can be represented as follows: ${\\mathfrak {s}}(u_b)=\\gamma _0 + \\gamma _1 (x-x_T) + \\gamma _2 (y-y_T),$ where $\\begin{bmatrix}\\gamma _0 \\\\\\gamma _1 \\\\\\gamma _2 \\\\\\end{bmatrix}=(M^TEM)^{-1}M^TE\\begin{bmatrix}u_{b,1} \\\\u_{b,2} \\\\\\vdots \\\\u_{b,N} \\\\\\end{bmatrix}.$ From ${\\mathfrak {s}}(u_b)=\\gamma _0 + \\gamma _1 (x-x_T) + \\gamma _2 (y-y_T)$ and (REF ), we have $\\begin{bmatrix}{\\mathfrak {s}}(u_{b})(M_1) \\\\{\\mathfrak {s}}(u_{b})(M_2) \\\\\\vdots \\\\{\\mathfrak {s}}(u_{b})(M_N) \\\\\\end{bmatrix}=M\\begin{bmatrix}\\gamma _0\\\\\\gamma _1\\\\\\gamma _2\\\\\\end{bmatrix}=M(M^TEM)^{-1}M^TE\\begin{bmatrix}u_{b,1} \\\\u_{b,2} \\\\\\vdots \\\\u_{b,N} \\\\\\end{bmatrix}.$ Let $v_b\\in W(T)$ be the basis function corresponding to the edge $e_j$ of $T$ : $v_b=\\left\\lbrace \\begin{array}{lllll}1, \\qquad \\text{ on } e_j,\\\\0, \\qquad \\text{ otherwise}.\\\\\\end{array}\\right.$ Then the coefficient $(\\tilde{\\gamma }_0,\\tilde{\\gamma }_1,\\tilde{\\gamma }_2)^T$ for ${\\mathfrak {s}}(v_b)$ is given by $\\begin{bmatrix}\\tilde{\\gamma }_0 \\\\\\tilde{\\gamma }_1 \\\\\\tilde{\\gamma }_2 \\\\\\end{bmatrix}=(M^TEM)^{-1}M^TE\\begin{bmatrix}v_{b,1}\\\\\\vdots \\\\v_{b,j}\\\\\\vdots \\\\v_{b,N}\\\\\\end{bmatrix}=(M^TEM)^{-1}M^TE\\begin{bmatrix}0\\\\\\vdots \\\\1\\\\\\vdots \\\\0\\\\\\end{bmatrix}\\triangleq \\begin{bmatrix}d_{1,j}\\\\d_{2,j}\\\\d_{3,j}\\\\\\end{bmatrix}.$ It follows that $\\begin{split}S_T(u_b,v_b)=&h^{-1}\\sum _{i=1}^N({\\mathfrak {s}}(u_b)(M_i)-u_{b,i})({\\mathfrak {s}}(v_b)(M_i)-v_{b,i})|e_i|\\\\=&h^{-1}\\sum _{i=1}^N (u_{b,i}-{\\mathfrak {s}}(u_b)(M_i))v_{b,i}|e_i|\\\\=&h^{-1}\\left((I_N-M(M^TEM)^{-1}M^TE)\\begin{bmatrix}u_{b,1} \\\\u_{b,2} \\\\\\vdots \\\\u_{b,N} \\\\\\end{bmatrix}\\right)_j |e_j| \\\\=&h^{-1}\\sum _{i=1}^N a_{j,i}u_{b,i},\\end{split}$ where $I_N$ is the identity matrix of size $N\\times N$ ."],["The stiffness matrix for $a_T(\\cdot ,\\cdot )$","For a computation of the element stiffness matrix corresponding to the bilinear form $a_T(u_b,v_b)=(\\alpha \\nabla _w u_b,\\nabla _w v_b )_T$ , we have from the weak gradient formula (REF ) that $(\\alpha \\nabla _w u_b,\\nabla _w v_b )_T&=&(\\alpha \\displaystyle \\frac{1}{|T|}\\sum _{j=1}^N u_{b,j}\\mathbf {n}_j|e_j|,\\displaystyle \\frac{1}{|T|}\\sum _{i=1}^N v_{b,i}\\mathbf {n}_i|e_i|)_{T}\\\\&=&\\sum _{i,j=1}^N(\\alpha \\displaystyle \\frac{1}{|T|} u_{b,j}\\mathbf {n}_j|e_j|,\\displaystyle \\frac{1}{|T|} v_{b,i}\\mathbf {n}_i|e_i|)_{T} \\nonumber \\\\&=&\\sum _{i,j=1}^N \\frac{|e_j||e_i|}{|T|^2}(\\alpha \\mathbf {n}_j,\\mathbf {n}_i)_{T}u_{b,j}v_{b,i},\\nonumber \\\\&=&\\sum _{i,j=1}^N b_{i,j}u_{b,j}v_{b,i},\\nonumber $ which leads to the block matrix $B$ in the element stiffness matrix."],["The stiffness matrix for $b_T(\\cdot ,\\cdot )$","Recall that the bilinear form $b_T(\\cdot ,\\cdot )$ is given by $b_T(u_b, v_b)=({\\beta }\\cdot \\nabla _w u_b, {\\mathfrak {s}}(v_b))_T.$ Note that the extension ${\\mathfrak {s}}(v_b)$ has the following representation: ${\\mathfrak {s}}(v_b)=\\gamma _0 + \\gamma _1 (x-x_T) + \\gamma _2 (y-y_T),$ where $\\begin{bmatrix}\\gamma _0 \\\\\\gamma _1 \\\\\\gamma _2 \\\\\\end{bmatrix}=(M^TEM)^{-1}M^TE\\begin{bmatrix}v_{b,1} \\\\v_{b,2} \\\\\\vdots \\\\v_{b,N} \\\\\\end{bmatrix}.$ Thus, with $D=(M^TEM)^{-1}M^TE$ , we have from the weak gradient formula (REF ) that $\\begin{split}& ({\\beta }\\cdot \\nabla _w u_b,{\\mathfrak {s}}(v_b))_T \\\\=&\\displaystyle \\frac{1}{|T|}\\sum _{i,j=1}^N ({\\beta }\\cdot \\mathbf {n}_j, d_{1,i} + d_{2,i} (x-x_T) + d_{3,i} (y-y_T))_T |e_j| u_{b,j} v_{b,i}\\\\=&\\displaystyle \\frac{1}{|T|}\\sum _{i,j=1}^N \\int _T {\\beta }\\cdot \\mathbf {n}_j (d_{1,i} + d_{2,i} (x-x_T) + d_{3,i} (y-y_T))dT |e_j| u_{b,j} v_{b,i}.\\end{split}$ For simplicity, we introduce the following functions: $\\zeta _i(x,y) = d_{1,i} + d_{2,i} (x-x_T) + d_{3,i} (y-y_T),\\qquad i=1,\\ldots , N.$ Then, the equation (REF ) indicates that the element stiffness matrix corresponding to the bilinear form $b_T(\\cdot ,\\cdot )$ is given by $R=\\lbrace r_{ij}\\rbrace _{N\\times N},\\ \\ r_{ij}= \\frac{|e_j|}{|T|}\\int _T {\\beta }\\cdot \\mathbf {n}_j \\zeta _i dT.$"],["The stiffness matrix for $c_T(\\cdot ,\\cdot )$","Recall that the bilinear form $c_T(\\cdot ,\\cdot )$ is given by $c_T(u_b, v_b)=(c {\\mathfrak {s}}(u_b), {\\mathfrak {s}}(v_b))_T.$ Thus, the element stiffness matrix corresponding to $c_T(\\cdot ,\\cdot )$ has the following formula: $C=\\lbrace c_{ij}\\rbrace _{N\\times N},\\quad c_{ij} = \\int _T c(x,y) \\zeta _j\\zeta _i dT,$ where $\\zeta _i$ is the function defined in (REF )."],["The element load vector","Finally, the element load vector can be obtained from $(f, {\\mathfrak {s}}(v_b))_T& =& \\int _{T}f {\\mathfrak {s}}(v_b)dT \\\\& =& \\int _{T}f(x,y) (d_{1,i} + d_{2,i}(x-x_T) + d_{3,i}(y-y_T)) dT\\\\& =& \\int _{T}f(x,y) \\zeta _i(x,y) dT$ for $i=1,\\ldots , N$ ."],["Stability and Well-Posedness","The SWG scheme (REF ) can be derived from the classical weak Galerkin finite element method [36], [24], [37] by eliminating the degrees of freedom associated with the interior of each element when ${\\beta }=0$ and $c=0$ .","But for the general case of ${\\beta }$ and $c$ , the SWG finite element method (REF ) is different from the weak Galerkin schemes in existing literature.","It is thus necessary to provide a mathematical theory for the stability and well-posedness of the numerical scheme (REF ).","Let ${\\mathcal {T}}_h$ be a shape-regular polygonal partition of the domain $\\Omega $ .","There exists a constant $C$ such that $\\Vert \\nabla {\\mathfrak {s}}(v_b)\\Vert _{T}^2 & \\le & C\\left(\\Vert \\nabla _w v_b\\Vert ^2_T + h^{-1}\\Vert v_b-Q_b{\\mathfrak {s}}(v_b)\\Vert ^2_{{\\partial T}}\\right),\\\\\\Vert v_b-{\\mathfrak {s}}(v_b)\\Vert _{0,{\\partial T}}^2&\\le & C h \\left(\\Vert \\nabla _w v_b\\Vert ^2_T + h^{-1} \\Vert v_b-Q_b{\\mathfrak {s}}(v_b)\\Vert ^2_{\\partial T}\\right).$ Moreover, the following Poincaré-type estimate holds true: $\\Vert {\\mathfrak {s}}(v_b)\\Vert ^2 & \\le & C\\left(\\Vert \\nabla _w v_b\\Vert ^2_T + h^{-1}\\Vert v_b-Q_b{\\mathfrak {s}}(v_b)\\Vert ^2_{{\\partial T}}\\right).$ From the formula (REF ) for the weak gradient, we have for any constant vector $\\mathbf {\\phi }$ that $\\begin{split}(\\nabla _w v_b, \\mathbf {\\phi })_T=&\\langle v_b, \\mathbf {\\phi }\\cdot \\mathbf {n}\\rangle _{\\partial T}\\\\=& \\langle v_b- {\\mathfrak {s}}(v_b), \\mathbf {\\phi }\\cdot \\mathbf {n}\\rangle _{\\partial T}+ \\langle {\\mathfrak {s}}(v_b), \\mathbf {\\phi }\\cdot \\mathbf {n}\\rangle _{\\partial T}\\\\=& \\langle v_b- Q_b{\\mathfrak {s}}(v_b), \\mathbf {\\phi }\\cdot \\mathbf {n}\\rangle _{\\partial T}+ (\\nabla {\\mathfrak {s}}(v_b), \\mathbf {\\phi })_T,\\end{split}$ which gives $(\\nabla {\\mathfrak {s}}(v_b), \\mathbf {\\phi })_T = (\\nabla _w v_b, \\mathbf {\\phi })_T - \\langle v_b- Q_b{\\mathfrak {s}}(v_b), \\mathbf {\\phi }\\cdot \\mathbf {n}\\rangle _{\\partial T}.$ Hence, by letting $\\mathbf {\\phi } = \\nabla {\\mathfrak {s}}(v_b)$ we arrive at $\\Vert \\nabla {\\mathfrak {s}}(v_b)\\Vert _{T}^2 \\le C\\left(\\Vert \\nabla _w v_b\\Vert ^2_T + h^{-1}\\Vert v_b-Q_b{\\mathfrak {s}}(v_b)\\Vert ^2_{{\\partial T}}\\right),$ which verifies (REF ).","Next, from the usual error estimate for the $L^2$ projection operator $Q_b$ and the estimate (REF ), we have $\\begin{split}\\Vert {\\mathfrak {s}}(v_b) - Q_b{\\mathfrak {s}}(v_b)\\Vert ^2_{{\\partial T}} \\le & C h^2 \\Vert \\nabla {\\mathfrak {s}}(v_b)\\Vert _{\\partial T}^2 \\\\\\le & C h \\Vert \\nabla {\\mathfrak {s}}(v_b)\\Vert ^2_T \\\\\\le & C\\left(h\\Vert \\nabla _w v_b\\Vert ^2_T + \\Vert v_b-Q_b{\\mathfrak {s}}(v_b)\\Vert ^2_{\\partial T}\\right).\\end{split}$ It follows that $\\begin{split}\\Vert v_b-{\\mathfrak {s}}(v_b)\\Vert _{0,{\\partial T}}\\le & \\Vert v_b-Q_b{\\mathfrak {s}}(v_b)\\Vert _{0,{\\partial T}} + \\Vert {\\mathfrak {s}}(v_b)-Q_b{\\mathfrak {s}}(v_b)\\Vert _{0,{\\partial T}}\\\\\\le & C \\left(h\\Vert \\nabla _w v_b\\Vert ^2_T + \\Vert v_b-Q_b{\\mathfrak {s}}(v_b)\\Vert ^2_{\\partial T}\\right)^{1/2},\\end{split}$ which verifies the estimate ().","To derive the inequality (REF ), we note the following discrete Poincaré inequality: $\\Vert {\\mathfrak {s}}(v_b)\\Vert ^2 \\le C\\sum _{T\\in {\\mathcal {T}}_h} \\left( \\Vert \\nabla {\\mathfrak {s}}(v_b)\\Vert _T^2 + h_T^{-1}\\Vert {\\mathfrak {s}}(v_b) - v_b\\Vert _{\\partial T}^2\\right).$ Combining the above estimate with (REF ) and (REF ) gives rise to the desired inequality (REF ).","This completes the proof of the lemma.","On each element $T\\in {\\mathcal {T}}_h$ , the following identity holds true: $\\begin{split}b_T(v_b,v_b) =&\\frac{1}{2} \\langle v_b, v_b {\\beta }\\cdot {\\bf n}\\rangle _{\\partial T}- \\frac{1}{2} ((\\nabla \\cdot {\\beta }) {\\mathfrak {s}}(v_b), {\\mathfrak {s}}(v_b))_T \\\\& - \\frac{1}{2} \\langle v_b-{\\mathfrak {s}}(v_b), (v_b-{{\\mathfrak {s}}(v_b)){\\beta }}\\cdot {\\bf n}\\rangle _{\\partial T}\\\\& + \\langle v_b-{\\mathfrak {s}}(v_b), \\overline{{\\mathfrak {s}}(v_b){\\beta }}\\cdot {\\bf n}- {{\\mathfrak {s}}(v_b){\\beta }}\\cdot {\\bf n}\\rangle _{\\partial T},\\end{split}$ where $\\overline{{\\mathfrak {s}}(v_b){\\beta }}$ is the average of ${\\mathfrak {s}}(v_b){\\beta }$ on the element $T$ .","From the formula (REF ), we have $\\begin{split}b_T(v_b,v_b) =& ({\\beta }\\cdot \\nabla _w v_b, {\\mathfrak {s}}(v_b))_T \\\\=& (\\nabla _w v_b, {{\\mathfrak {s}}(v_b){\\beta }})_T\\\\=& (\\nabla _w v_b, \\overline{{\\mathfrak {s}}(v_b){\\beta }})_T\\\\=& \\langle v_b, \\overline{{\\mathfrak {s}}(v_b){\\beta }}\\cdot {\\bf n}\\rangle _{\\partial T}\\\\=& \\langle v_b-{\\mathfrak {s}}(v_b), \\overline{{\\mathfrak {s}}(v_b){\\beta }}\\cdot {\\bf n}\\rangle _{\\partial T}+ \\langle {\\mathfrak {s}}(v_b), \\overline{{\\mathfrak {s}}(v_b){\\beta }}\\cdot {\\bf n}\\rangle _{\\partial T}.\\end{split}$ Note that $\\begin{split}\\langle {\\mathfrak {s}}(v_b), \\overline{{\\mathfrak {s}}(v_b){\\beta }}\\cdot {\\bf n}\\rangle _{\\partial T}&= (\\nabla {\\mathfrak {s}}(v_b), \\overline{{\\mathfrak {s}}(v_b){\\beta }})_T\\\\&= (\\nabla {\\mathfrak {s}}(v_b), {\\mathfrak {s}}(v_b){\\beta })_T\\\\&= \\frac{1}{2} \\langle {\\mathfrak {s}}(v_b), {\\mathfrak {s}}(v_b) {\\beta }\\cdot {\\bf n}\\rangle _{\\partial T}- \\frac{1}{2} ((\\nabla \\cdot {\\beta }) {\\mathfrak {s}}(v_b), {\\mathfrak {s}}(v_b))_T.\\end{split}$ Substituting the above identity into (REF ) yields $\\begin{split}b_T(v_b,v_b) =& \\langle v_b-{\\mathfrak {s}}(v_b), \\overline{{\\mathfrak {s}}(v_b){\\beta }}\\cdot {\\bf n}\\rangle _{\\partial T}+ \\frac{1}{2} \\langle {\\mathfrak {s}}(v_b), {\\mathfrak {s}}(v_b) {\\beta }\\cdot {\\bf n}\\rangle _{\\partial T}\\\\& \\ \\ - \\frac{1}{2} ((\\nabla \\cdot {\\beta }) {\\mathfrak {s}}(v_b), {\\mathfrak {s}}(v_b))_T\\\\=& \\langle v_b-{\\mathfrak {s}}(v_b), \\overline{{\\mathfrak {s}}(v_b){\\beta }}\\cdot {\\bf n}- {{\\mathfrak {s}}(v_b){\\beta }}\\cdot {\\bf n}\\rangle _{\\partial T}+ \\langle v_b,{{\\mathfrak {s}}(v_b){\\beta }}\\cdot {\\bf n}\\rangle _{\\partial T}\\\\& \\ \\ - \\frac{1}{2} \\langle {\\mathfrak {s}}(v_b), {\\mathfrak {s}}(v_b) {\\beta }\\cdot {\\bf n}\\rangle _{\\partial T}- \\frac{1}{2} ((\\nabla \\cdot {\\beta }) {\\mathfrak {s}}(v_b), {\\mathfrak {s}}(v_b))_T\\\\=& \\langle v_b-{\\mathfrak {s}}(v_b), \\overline{{\\mathfrak {s}}(v_b){\\beta }}\\cdot {\\bf n}- {{\\mathfrak {s}}(v_b){\\beta }}\\cdot {\\bf n}\\rangle _{\\partial T}\\\\& \\ - \\frac{1}{2} \\langle v_b-{\\mathfrak {s}}(v_b), (v_b-{{\\mathfrak {s}}(v_b)){\\beta }}\\cdot {\\bf n}\\rangle _{\\partial T}\\\\& \\ + \\frac{1}{2} \\langle v_b, v_b {\\beta }\\cdot {\\bf n}\\rangle _{\\partial T}- \\frac{1}{2} ((\\nabla \\cdot {\\beta }) {\\mathfrak {s}}(v_b), {\\mathfrak {s}}(v_b))_T,\\end{split}$ which leads to the identify (REF ).","In the finite element space $W_h({\\mathcal {T}}_h)$ , we introduce the following semi-norm: ${|\\!|\\!|} v_b{|\\!|\\!|}^2: = \\sum _{T\\in {\\mathcal {T}}_h} \\left( \\kappa S_T(v_b, v_b) + a_T(v_b, v_b) \\right)$ We claim that ${|\\!|\\!|}\\cdot {|\\!|\\!|}$ defines a norm in the closed subspace $W_h^0({\\mathcal {T}}_h)$ .","It suffices to show that $v_b \\equiv 0$ for any $v_b\\in W_h^0({\\mathcal {T}}_h)$ satisfying ${|\\!|\\!|} v_b {|\\!|\\!|} =0$ .","In fact, if ${|\\!|\\!|} v_b {|\\!|\\!|} =0$ , then from (REF ) we have $\\kappa \\sum _{T}S_T(v_b,v_b)+\\sum _{T}(\\alpha \\nabla _w v_b,\\nabla _w v_b)_T =0.$ It follows that on each element $T\\in {\\mathcal {T}}_h$ $\\nabla _w v_b=0,\\quad (v_b - {\\mathfrak {s}}(v_b))(M_i)=0$ for $i=1,\\ldots , N$ .","Thus, $\\displaystyle \\nabla {\\mathfrak {s}}(v_b)=\\frac{1}{|T|}\\sum _{i=1}^N {\\mathfrak {s}}(v_b)(M_i)|e_i|\\mathbf {n}_i=\\frac{1}{|T|}\\sum _{i=1}^N v_{b,i}|e_i|\\mathbf {n}_i=\\nabla _w v_b=0,$ so that ${\\mathfrak {s}}(v_b)$ has constant value on each element $T\\in {\\mathcal {T}}_h$ .","By using (REF ) we see that $v_b={\\mathfrak {s}}(v_b)=const$ on each edge, which, together with the fact that $v_b=0$ on $\\partial \\Omega $ , leads to $v_b\\equiv 0$ in $\\Omega $ .","For the model problem (REF ), assume that ${\\beta }\\in W^{1,\\infty }(\\Omega )$ and the condition (REF ) is satisfied.","Then, the bilinear form $\\kappa S(\\cdot ,\\cdot )+{\\mathcal {B}}(\\cdot ,\\cdot )$ is bounded and coercive in the finite element space $W_h^0({\\mathcal {T}}_h)$ ; i.e., there exist constants $M$ and $\\Lambda >0$ such that $|\\kappa S(v_b,w_b)+{\\mathcal {B}}(v_b, w_b)| & \\le & M {|\\!|\\!|} v_b{|\\!|\\!|} {|\\!|\\!|} w_b{|\\!|\\!|} \\qquad \\forall v_b, w_b \\in W_h^0({\\mathcal {T}}_h),\\\\\\kappa S(v_b,v_b)+{\\mathcal {B}}(v_b, v_b) & \\ge & \\Lambda {|\\!|\\!|} v_b{|\\!|\\!|}^2\\qquad \\forall v_b\\in W_h^0({\\mathcal {T}}_h),$ provided that the meshsize $h$ of ${\\mathcal {T}}_h$ is sufficiently small.","Recall that for any $v_b\\in W_h^0({\\mathcal {T}}_h)$ we have $\\begin{split}{\\mathcal {B}}(v_b, w_b) & \\ = \\sum _{T\\in {\\mathcal {T}}_h} \\left(a_T(v_b,w_b)+ b_T(v_b, w_b) + c_T(v_b,w_b)\\right),\\\\S(v_b,w_b) & \\ = \\sum _{T\\in {\\mathcal {T}}_h} S_T(v_b,w_b).\\end{split}$ The boundedness estimate (REF ) is then straightforward from the usual Cauchy-Schwarz and the inequality (REF ).","We shall focus on the derivation of the coercivity inequality () in the rest of the proof.","In comparison with (REF ), the key to the coercivity inequality () is to derive an estimate of the following type: $\\sum _{T\\in {\\mathcal {T}}_h} \\left(b_T(v_b, v_b) + c_T(v_b,v_b)\\right)\\ge \\eta - \\varepsilon (h) {|\\!|\\!|} v_b{|\\!|\\!|}^2,$ where $\\eta \\ge 0$ and $\\varepsilon (h)$ is a parameter satisfying $\\varepsilon (h) \\rightarrow 0$ as $h\\rightarrow 0$ .","If (REF ) indeed holds true, then we have from (REF ) that $\\begin{split}\\kappa S(v_b,v_b)+{\\mathcal {B}}(v_b, v_b) \\ge & {|\\!|\\!|} v_b {|\\!|\\!|}^2 + \\eta - \\varepsilon (h) {|\\!|\\!|} v_b{|\\!|\\!|}^2\\\\\\ge & (1- \\varepsilon (h)) {|\\!|\\!|} v_b{|\\!|\\!|}^2,\\end{split}$ which implies the coercivity () for sufficiently small $h$ .","It remains to derive the estimate (REF ).","To this end, we sum up the identify in Lemma to obtain $\\begin{split}\\sum _{T\\in {\\mathcal {T}}_h} b_T(v_b,v_b)= & -\\frac{1}{2} \\sum _{T\\in {\\mathcal {T}}_h} ( \\nabla \\cdot {\\beta }{\\mathfrak {s}}(v_b), {\\mathfrak {s}}(v_b))_T \\\\& - \\frac{1}{2} \\sum _{T\\in {\\mathcal {T}}_h}\\langle v_b-{\\mathfrak {s}}(v_b), (v_b-{{\\mathfrak {s}}(v_b)){\\beta }}\\cdot {\\bf n}\\rangle _{\\partial T}\\\\& + \\sum _{T\\in {\\mathcal {T}}_h}\\langle v_b-{\\mathfrak {s}}(v_b), \\overline{{\\mathfrak {s}}(v_b){\\beta }}\\cdot {\\bf n}- {{\\mathfrak {s}}(v_b){\\beta }}\\cdot {\\bf n}\\rangle _{\\partial T},\\end{split}$ where we have used the fact that $\\sum _{T\\in {\\mathcal {T}}_h} \\langle v_b, v_b {\\beta }\\cdot {\\bf n}\\rangle _{\\partial T}=0$ .","Thus, $\\begin{split}\\sum _{T\\in {\\mathcal {T}}_h} (b_T(v_b,v_b) + c_T(v_b,v_b))= & \\sum _{T\\in {\\mathcal {T}}_h} ( (c-\\frac{1}{2} \\nabla \\cdot {\\beta }) {\\mathfrak {s}}(v_b), {\\mathfrak {s}}(v_b))_T \\\\& - \\frac{1}{2} \\sum _{T\\in {\\mathcal {T}}_h}\\langle v_b-{\\mathfrak {s}}(v_b), (v_b-{{\\mathfrak {s}}(v_b)){\\beta }}\\cdot {\\bf n}\\rangle _{\\partial T}\\\\& + \\sum _{T\\in {\\mathcal {T}}_h}\\langle v_b-{\\mathfrak {s}}(v_b), \\overline{{\\mathfrak {s}}(v_b){\\beta }}\\cdot {\\bf n}- {{\\mathfrak {s}}(v_b){\\beta }}\\cdot {\\bf n}\\rangle _{\\partial T}.\\end{split}$ Next, from () we have $\\begin{split}\\left|\\sum _{T\\in {\\mathcal {T}}_h}\\langle v_b-{\\mathfrak {s}}(v_b), (v_b-{{\\mathfrak {s}}(v_b)){\\beta }}\\cdot {\\bf n}\\rangle _{\\partial T}\\right| & \\le C \\left( h\\Vert \\nabla _w v_b\\Vert ^2_T + \\Vert v_b - Q_b {\\mathfrak {s}}(v_b)\\Vert _{\\partial T}^2\\right) \\\\& \\le C h \\sum _{T\\in {\\mathcal {T}}_h} \\left( a_T(v_b,v_b) + S_T(v_b,v_b)\\right)\\\\&\\le C h {|\\!|\\!|} v_b{|\\!|\\!|}^2.\\end{split}$ As to the last term in (REF ), we have $\\begin{split}& \\left| \\sum _{T\\in {\\mathcal {T}}_h}\\langle v_b-{\\mathfrak {s}}(v_b), \\overline{{\\mathfrak {s}}(v_b){\\beta }}\\cdot {\\bf n}- {{\\mathfrak {s}}(v_b){\\beta }}\\cdot {\\bf n}\\rangle _{\\partial T}\\right| \\\\\\le & \\sum _{T\\in {\\mathcal {T}}_h} \\Vert v_b-{\\mathfrak {s}}(v_b)\\Vert _{\\partial T}\\Vert \\overline{{\\mathfrak {s}}(v_b){\\beta }} - {{\\mathfrak {s}}(v_b){\\beta }}\\Vert _{\\partial T}\\\\\\le & C h^{\\frac{1}{2}} \\sum _{T\\in {\\mathcal {T}}_h} \\Vert v_b-{\\mathfrak {s}}(v_b)\\Vert _{\\partial T}\\left(\\Vert {\\mathfrak {s}}(v_b)\\Vert _T + \\Vert \\nabla {\\mathfrak {s}}(v_b)\\Vert _T\\right)\\\\\\le & C h \\left(\\sum _{T\\in {\\mathcal {T}}_h} h^{-1}\\Vert v_b-{\\mathfrak {s}}(v_b)\\Vert _{\\partial T}^2\\right)^{\\frac{1}{2}} \\left(\\sum _{T\\in {\\mathcal {T}}_h} \\left(\\Vert {\\mathfrak {s}}(v_b)\\Vert _T^2 + \\Vert \\nabla {\\mathfrak {s}}(v_b)\\Vert _T^2\\right)\\right)^{\\frac{1}{2}}\\end{split}$ Combining the estimates (REF ), (), and (REF ) with (REF ) yields $\\left| \\sum _{T\\in {\\mathcal {T}}_h}\\langle v_b-{\\mathfrak {s}}(v_b), \\overline{{\\mathfrak {s}}(v_b){\\beta }}\\cdot {\\bf n}- {{\\mathfrak {s}}(v_b){\\beta }}\\cdot {\\bf n}\\rangle _{\\partial T}\\right| \\le C h {|\\!|\\!|} v_b{|\\!|\\!|}^2.$ Now by substituting (REF ) and (REF ) into (REF ) we obtain the inequality (REF ) with $\\eta = ( (c-\\frac{1}{2} \\nabla \\cdot {\\beta }) {\\mathfrak {s}}(v_b), {\\mathfrak {s}}(v_b))\\ge 0$ and $\\varepsilon (h) = Ch$ .","This completes the proof of the lemma.","The following is a direct application of Lemma .","Under the assumptions of Lemma , there exists a small, but fixed number $h_0>0$ , such that the numerical scheme (REF ) has one and only one solution $u_b\\in W_h({\\mathcal {T}}_h)$ for sufficiently fine finite element partitions ${\\mathcal {T}}_h$ satisfying $h\\le h_0$ .","It suffices to show that the homogeneous problem has only the trivial solution.","To this end, let $u_b \\in W_h^0({\\mathcal {T}}_h)$ , be the solution of scheme (REF ) with homogeneous data $f=0$ and $g=0$ .","By taking $v_b = u_b$ in (REF ) we obtain $\\kappa S(u_b,u_b)+{\\mathcal {B}}(u_b,u_b)=0,$ which, from the coercivity inequality (), gives $\\Lambda {|\\!|\\!|} u_b{|\\!|\\!|}^2 \\le \\kappa S(u_b,u_b) + {\\mathcal {B}}(u_b,u_b)=0$ , and hence $u_b\\equiv 0$ for sufficiently small $h$ ."],["Error Estimates in $H^1$","Let $u$ be the exact solution of the model problem (REF )-() and $u_b \\in W_h^0({\\mathcal {T}}_h)$ be the numerical approximation arising from the SWG scheme (REF ).","Let $Q_b u$ be the $L^2$ projection of $u$ in the space $W_h^0({\\mathcal {T}}_h)$ .","The error function refers to the difference between the $L^2$ projection and the SWG approximation: $e_b := Q_b u - u_b,$ The goal of this section is to establish an estimate for the error function $e_b$ in a discrete Sobolev norm.","Let us first state an error equation which plays an important role in the convergence analysis of the SWG scheme.","Assume that the coefficient $\\alpha $ of the model problem (REF )-() has piecewise constant values with respect to the finite element partition ${\\mathcal {T}}_h$ .","Then the following equation holds true $\\kappa S(e_b, v_b) + {\\mathcal {B}}(e_b,v_b)= \\ell _u(v_b) \\quad \\forall v_b \\in W_h^0({\\mathcal {T}}_h),$ where $\\ell _u(\\cdot )$ is a linear functional given by $\\begin{split}\\ell _u(v_b):=&\\sum _{T\\in {\\mathcal {T}}_h} \\langle \\alpha \\frac{\\partial u}{\\partial \\mathbf {n}}-\\alpha Q_0(\\nabla u)\\cdot \\mathbf {n},{\\mathfrak {s}}(v_b)-v_b\\rangle _{\\partial T}+\\kappa S(Q_bu,v_b)\\\\& + ((Q_0-I) \\nabla u, {\\mathfrak {s}}(v_b){\\beta }) + (c({\\mathfrak {s}}(Q_bu)-u), {\\mathfrak {s}}(v_b)),\\end{split}$ where $Q_0(\\nabla u)$ is the $L^2$ projection of $\\nabla u$ in the space $[P_0({\\mathcal {T}}_h)]^2$ , and $\\mathbf {n}$ is the outward normal vector on ${\\partial T}$ .","We first consider the weak gradient of $Q_b u$ , for any constant vector $\\mathbf {\\phi }$ , we have $(\\nabla _w Q_b u,\\mathbf {\\phi })_{T}&=& \\langle Q_b u, \\mathbf {\\phi }\\cdot \\mathbf {n} \\rangle _{\\partial T}= \\langle u, \\mathbf {\\phi }\\cdot \\mathbf {n} \\rangle _{\\partial T}\\\\&=& (\\nabla u, \\mathbf {\\phi })_{T}= (Q_0(\\nabla u), \\mathbf {\\phi })_{T},$ which implies $\\nabla _w Q_b u \\equiv Q_0(\\nabla u)$ .","Thus, for any $v_b \\in W_h^0(T_h)$ , we have $\\begin{split}&(\\alpha \\nabla _w Q_b u,\\nabla _w v_b)= \\sum _{T}(\\alpha Q_0(\\nabla u),\\nabla _w v_b)_T \\\\= & \\sum _{T} \\langle \\alpha Q_0(\\nabla u)\\cdot \\mathbf {n},v_b\\rangle _{\\partial T}\\\\=& \\sum _{T} \\langle \\alpha Q_0(\\nabla u)\\cdot \\mathbf {n},v_b\\rangle _{\\partial T}-\\langle \\alpha Q_0(\\nabla u)\\cdot \\mathbf {n},{\\mathfrak {s}}(v_b)\\rangle _{\\partial T} + \\langle \\alpha Q_0(\\nabla u)\\cdot \\mathbf {n},{\\mathfrak {s}}(v_b)\\rangle _{\\partial T}\\\\=& \\sum _{T} \\langle \\alpha Q_0(\\nabla u)\\cdot \\mathbf {n},v_b-{\\mathfrak {s}}(v_b)\\rangle _{\\partial T} +(\\alpha \\nabla u,\\nabla {\\mathfrak {s}}(v_b))_{T} \\\\=& \\sum _{T}\\langle \\alpha Q_0(\\nabla u)\\cdot \\mathbf {n},v_b-{\\mathfrak {s}}(v_b)\\rangle _{\\partial T} +(-\\nabla \\cdot (\\alpha \\nabla u) ,{\\mathfrak {s}}(v_b))_{T} + \\langle \\alpha \\frac{\\partial u}{\\partial \\mathbf {n}},{\\mathfrak {s}}(v_b)\\rangle _{\\partial T}\\\\=& (-\\nabla \\cdot (\\alpha \\nabla u) ,{\\mathfrak {s}}(v_b))+ \\sum _{T}\\langle \\alpha \\frac{\\partial u}{\\partial \\mathbf {n}}-\\alpha Q_0(\\nabla u)\\cdot \\mathbf {n},{\\mathfrak {s}}(v_b)-v_b\\rangle _{\\partial T}.\\end{split}$ Next, from $\\nabla _w (Q_bu) = Q_0(\\nabla u)$ , we have $\\begin{split}\\sum _T ({\\beta }\\cdot \\nabla _w (Q_bu), {\\mathfrak {s}}(v_b))_T = & \\sum _T ({\\beta }\\cdot (Q_0 \\nabla u), {\\mathfrak {s}}(v_b))_T \\\\= & \\sum _T ({\\beta }\\cdot \\nabla u, {\\mathfrak {s}}(v_b))_T + \\sum _T ((Q_0-I) \\nabla u, {\\mathfrak {s}}(v_b){\\beta })_T,\\end{split}$ and $\\sum _T (c{\\mathfrak {s}}(Q_bu), {\\mathfrak {s}}(v_b))_T = (cu, {\\mathfrak {s}}(v_b)) + (c({\\mathfrak {s}}(Q_bu)-u), {\\mathfrak {s}}(v_b)).$ The sum of (REF ), (REF ), and (REF ) gives rise to $\\begin{split}{\\mathcal {B}}(Q_b u, v_b) = &\\ (f, {\\mathfrak {s}}(v_b)) + \\sum _{T}\\langle \\alpha \\frac{\\partial u}{\\partial \\mathbf {n}}-\\alpha Q_0(\\nabla u)\\cdot \\mathbf {n},{\\mathfrak {s}}(v_b)-v_b\\rangle _{\\partial T} \\\\& \\ + \\sum _T ((Q_0-I) \\nabla u, {\\mathfrak {s}}(v_b){\\beta })_T + (c({\\mathfrak {s}}(Q_bu)-u), {\\mathfrak {s}}(v_b)),\\end{split}$ which, combined with $ (f, {\\mathfrak {s}}(v_b) = \\kappa S(u_b, v_b) + {\\mathcal {B}}(u_b,v_b)$ , leads to $\\begin{split}{\\mathcal {B}}(Q_b u - u_b, v_b) = &\\ \\kappa S(u_b,v_b) + \\sum _{T}\\langle \\alpha \\frac{\\partial u}{\\partial \\mathbf {n}}-\\alpha Q_0(\\nabla u)\\cdot \\mathbf {n},{\\mathfrak {s}}(v_b)-v_b\\rangle _{\\partial T} \\\\& \\ + \\sum _T ((Q_0-I) \\nabla u, {\\mathfrak {s}}(v_b){\\beta })_T + (c({\\mathfrak {s}}(Q_bu)-u), {\\mathfrak {s}}(v_b)),\\end{split}$ and $\\begin{split}&\\kappa S(Q_bu-u_b,v_b)+{\\mathcal {B}}(Q_b u - u_b, v_b) \\\\= &\\ \\kappa S(Q_bu,v_b) + \\sum _{T}\\langle \\alpha \\frac{\\partial u}{\\partial \\mathbf {n}}-\\alpha Q_0(\\nabla u)\\cdot \\mathbf {n},{\\mathfrak {s}}(v_b)-v_b\\rangle _{\\partial T} \\\\& \\ + \\sum _T ((Q_0-I) \\nabla u, {\\mathfrak {s}}(v_b){\\beta })_T + (c({\\mathfrak {s}}(Q_bu)-u), {\\mathfrak {s}}(v_b)).\\end{split}$ This completes the proof of the lemma.","Remark 5.1 It should be pointed out that Lemma can be extended to the case when $\\alpha $ is in $L^\\infty (\\Omega )$ and piecewise smooth with respect to the finite element partition ${\\mathcal {T}}_h$ .","Detailed analysis can be established by following the approach presented in [35].","The following result is concerned with the error estimate for the SWG numerical solutions in a discrete $H^1$ norm.","Let $u\\in H^2(\\Omega )$ be the exact solution of (REF )-() and $u_b\\in W_h({\\mathcal {T}}_h)$ be the approximate solution arising from the numerical scheme (REF ).","Assume ${\\beta }\\in C^1(\\bar{\\Omega })$ and that (REF ) is satisfied.","Then, the following error estimate holds true $\\kappa S(e_b ,e_b) +(\\alpha \\nabla _w e_b, \\nabla _w e_b )\\le Ch^2\\Vert u\\Vert _2^2,$ provided that the meshsize $h$ is sufficiently small.","Consequently, we have $\\Vert \\nabla _w u_b- \\nabla u\\Vert _0 \\le Ch \\Vert u\\Vert _2, $ The proof is based on the error equation (REF ) through a thorough analysis for the linear functional $\\ell _u(\\cdot )$ given in (REF ).","For the first term on the righ-hand side of (REF ), from the usual Cauchy-Schwarz inequality we have $\\begin{split}&\\ |\\langle \\alpha \\frac{\\partial u}{\\partial \\mathbf {n}}-\\alpha Q_0(\\nabla u)\\cdot \\mathbf {n},{\\mathfrak {s}}(v_b)-v_b\\rangle _{\\partial T}| \\\\\\le & \\ \\Vert \\alpha \\frac{\\partial u}{\\partial \\mathbf {n}}-\\alpha Q_0(\\nabla u)\\cdot \\mathbf {n}\\Vert _{0,{\\partial T}} \\Vert {\\mathfrak {s}}(v_b)-v_b\\Vert _{0,{\\partial T}}\\\\\\le & \\ \\Vert \\alpha \\Vert _{\\infty } \\Vert \\nabla u- Q_0(\\nabla u)\\Vert _{0,{\\partial T}} \\Vert {\\mathfrak {s}}(v_b)-v_b\\Vert _{0,{\\partial T}}.\\end{split}$ Now using the estimate () in the above inequality and then summing over all the element $T\\in {\\mathcal {T}}_h$ we arrive at the following: $\\begin{split}&\\sum _{T\\in {\\mathcal {T}}_h}|\\langle \\alpha \\frac{\\partial u}{\\partial \\mathbf {n}}-\\alpha Q_0(\\nabla u)\\cdot \\mathbf {n},{\\mathfrak {s}}(v_b)-v_b\\rangle _{\\partial T}| \\\\\\le & C \\Vert \\alpha \\Vert _{\\infty } \\sum _{T\\in {\\mathcal {T}}_h} \\Vert \\nabla u- Q_0(\\nabla u)\\Vert _{\\partial T}\\left(h\\Vert \\nabla _w v_b\\Vert ^2_T +\\Vert v_b-Q_b{\\mathfrak {s}}(v_b)\\Vert ^2_{\\partial T}\\right)^{\\frac{1}{2}}\\\\\\le & C\\Vert \\alpha \\Vert _{\\infty }\\left(\\Vert \\nabla u -Q_0(\\nabla u)\\Vert ^2_{0} + h^2\\Vert \\nabla ^2 u \\Vert _{0}^2\\right)^{\\frac{1}{2}}\\left( \\Vert \\nabla _w v_b\\Vert ^2+ \\kappa S(v_b, v_b)\\right)^{\\frac{1}{2}}\\\\\\le & C h \\Vert u\\Vert _2 {|\\!|\\!|} v_b {|\\!|\\!|}.\\end{split}$ As to the second term on the right hand side of (REF ), we have $\\begin{split}|S(Q_bu,v_b)| =&\\sum _{T}h^{-1}\\langle Q_bu-Q_b{\\mathfrak {s}}(Q_bu),v_b-Q_b {\\mathfrak {s}}(v_b)\\rangle _{\\partial T} \\\\=&\\sum _{T}h^{-1}\\langle Q_bu,v_b-Q_b {\\mathfrak {s}}(v_b)\\rangle _{\\partial T} \\\\=&\\sum _{T}h^{-1}\\langle Q_bu -Q_b(Q_1 u),v_b-Q_b {\\mathfrak {s}}(v_b)\\rangle _{\\partial T}\\\\=&\\sum _{T}h^{-1}\\langle u -Q_1 u,v_b-Q_b {\\mathfrak {s}}(v_b)\\rangle _{\\partial T}\\\\\\le &\\left(\\sum _{T} h^{-1} \\int _{\\partial T}|u-Q_1u|^2ds \\right)^{\\frac{1}{2}}S(v_b, v_b)^{\\frac{1}{2}}\\\\\\le & C\\left(h^{-2}\\Vert u-Q_1 u\\Vert ^2 + \\Vert u-Q_1 u\\Vert _1^2 \\right)^{\\frac{1}{2}}S(v_b, v_b)^{\\frac{1}{2}}\\\\\\le & C h \\Vert u\\Vert _2 {|\\!|\\!|} v_b{|\\!|\\!|}.\\end{split}$ The third term on the right hand side of (REF ) can be bounded by using the usual error estimate for $L^2$ projections as follows: $\\begin{split}|((Q_0-I) \\nabla u, {\\mathfrak {s}}(v_b){\\beta })| = & |((Q_0-I) \\nabla u, (Q_0-I) ({\\mathfrak {s}}(v_b){\\beta }))| \\\\\\le & \\Vert (Q_0-I) \\nabla u\\Vert \\ \\Vert (Q_0-I) ({\\mathfrak {s}}(v_b){\\beta })\\Vert \\\\\\le & C h^2 \\Vert \\nabla ^2 u\\Vert \\left( \\Vert \\nabla {\\mathfrak {s}}(v_b)\\Vert + \\Vert {\\mathfrak {s}}(v_b)\\Vert \\right) \\\\\\le & C h^2 \\Vert \\nabla ^2 u\\Vert {|\\!|\\!|} v_b{|\\!|\\!|},\\end{split}$ where we have used the estimates (REF ) and (REF ) in the last line.","The last term on the right hand side of (REF ) can be estimated as follows: $\\begin{split}|(c({\\mathfrak {s}}(Q_bu)-u), {\\mathfrak {s}}(v_b))|\\le & \\Vert c\\Vert _\\infty \\Vert {\\mathfrak {s}}(Q_bu)-u\\Vert \\Vert {\\mathfrak {s}}(v_b)\\Vert \\\\\\le & C (\\Vert {\\mathfrak {s}}(Q_bu)-Q_1 u\\Vert + \\Vert Q_1u - u\\Vert ) \\Vert {\\mathfrak {s}}(v_b)\\Vert \\\\\\le & C ( \\Vert {\\mathfrak {s}}(Q_bu)-{\\mathfrak {s}}(Q_1 u)\\Vert + \\Vert Q_1u - u\\Vert ) \\Vert {\\mathfrak {s}}(v_b)\\Vert \\\\\\le & C ( \\Vert {\\mathfrak {s}}(Q_bu-Q_1 u)\\Vert + \\Vert Q_1u - u\\Vert ) \\Vert {\\mathfrak {s}}(v_b)\\Vert \\\\\\le & C h^2 \\Vert u\\Vert _2 {|\\!|\\!|} v_b{|\\!|\\!|}.\\end{split}$ Substituting the estimates (REF )-(REF ) into the error equation (REF ) yields $\\kappa S(e_b, v_b) + {\\mathcal {B}}(e_b, v_b) \\le C h \\Vert u\\Vert _2 {|\\!|\\!|} v_b{|\\!|\\!|},$ which, together with the coercivity (), leads to $\\Lambda {|\\!|\\!|} e_b{|\\!|\\!|}^2 \\le C h \\Vert u\\Vert _2 {|\\!|\\!|} e_b{|\\!|\\!|}.$ The last inequality implies the error estimate (REF ).","Finally, from the triangle inequality and the error estimate (REF ), we obtain $\\begin{split}\\Vert \\nabla _w u_b - \\nabla u\\Vert \\le & \\Vert \\nabla _w (u_b - Q_b u)\\Vert + \\Vert \\nabla _w (Q_b u) - \\nabla u\\Vert \\\\= & \\Vert \\nabla _w e_b\\Vert + \\Vert Q_0 (\\nabla u) - \\nabla u\\Vert \\\\\\le & C h \\Vert u\\Vert _2,\\end{split}$ which gives rise to (REF ).","This completes the proof of the theorem."],["Error Estimates in $L^2$","We use the usual duality argument to derive an error estimate in $L^2$ for the numerical solutions arising from (REF ).","The analysis to be presented is a modified version of those developed in [36], [24], [35].","Consider the following auxiliary problem that seeks $\\Phi \\in H_0^1(\\Omega )$ such that $-\\nabla \\cdot (\\alpha \\nabla \\Phi ) - \\nabla \\cdot ({\\beta }\\Phi ) + c\\Phi &=&\\chi \\quad {\\rm in}\\ \\Omega \\\\\\Phi &=&0\\quad {\\rm on}\\ \\partial \\Omega , $ where $\\chi \\in L^2(\\Omega )$ .","Assume that the solution of the problem (REF )-() exists and has the $H^2$ -regularity: $\\Vert \\Phi \\Vert _2 \\le C \\Vert \\chi \\Vert ,$ where $C$ is a constant depending only on the domain and the coefficients $\\alpha , {\\beta }$ , and $c$ .","Let $u\\in H^2(\\Omega )$ be the exact solution of (REF )-() and $u_b\\in W_h({\\mathcal {T}}_h)$ be the approximate solution arising from the numerical scheme (REF ).","Assume ${\\beta }\\in C^1(\\bar{\\Omega })$ and the conditions (REF ) and (REF ) are satisfied.","Then, the following $L^2$ error estimate holds true $\\Vert u - {\\mathfrak {s}}(u_b)\\Vert \\le Ch^2\\Vert u\\Vert _2,$ provided that the meshsize $h$ is sufficiently small.","On each element $T\\in {\\mathcal {T}}_h$ , we test (REF ) against the linear function ${\\mathfrak {s}}(e_b)$ to obtain $(\\chi , {\\mathfrak {s}}(e_b))_T &=& (\\alpha \\nabla \\Phi , \\nabla {\\mathfrak {s}}(e_b))_T + ({\\beta }\\Phi , \\nabla {\\mathfrak {s}}(e_b))_T + (c\\Phi , {\\mathfrak {s}}(e_b))_T \\\\&& - \\langle \\alpha \\nabla \\Phi \\cdot {\\bf n}, {\\mathfrak {s}}(e_b)\\rangle _{\\partial T}-\\langle {\\beta }\\cdot {\\bf n}\\Phi , {\\mathfrak {s}}(e_b)\\rangle _{\\partial T}\\\\&=& (\\alpha Q_0(\\nabla \\Phi ), \\nabla {\\mathfrak {s}}(e_b))_T + (Q_0({\\beta }\\Phi ), \\nabla {\\mathfrak {s}}(e_b))_T + (c\\Phi , {\\mathfrak {s}}(e_b))_T \\\\&& - \\langle \\alpha \\nabla \\Phi \\cdot {\\bf n}, {\\mathfrak {s}}(e_b)\\rangle _{\\partial T}-\\langle {\\beta }\\cdot {\\bf n}\\Phi , {\\mathfrak {s}}(e_b)\\rangle _{\\partial T}\\\\&=& (\\alpha Q_0(\\nabla \\Phi ), \\nabla _w e_b)_T + (Q_0({\\beta }\\Phi ), \\nabla _w e_b)_T + (c\\Phi , {\\mathfrak {s}}(e_b))_T \\\\&& - \\langle \\alpha \\nabla \\Phi \\cdot {\\bf n}, {\\mathfrak {s}}(e_b)\\rangle _{\\partial T}-\\langle {\\beta }\\cdot {\\bf n}\\Phi , {\\mathfrak {s}}(e_b)\\rangle _{\\partial T}\\\\&& - \\langle \\alpha Q_0(\\nabla \\Phi )\\cdot {\\bf n}, e_b-{\\mathfrak {s}}(e_b)\\rangle _{\\partial T}- \\langle Q_0({\\beta }\\Phi )\\cdot {\\bf n}, e_b-{\\mathfrak {s}}(e_b)\\rangle _{\\partial T}$ By using $Q_0(\\nabla \\Phi ) = \\nabla _w (Q_b\\Phi )$ and $(Q_0({\\beta }\\Phi ), \\nabla _w e_b)_T = ({\\beta }\\cdot \\nabla _w e_b, \\Phi )_T$ in the above equation, we have from summing over all $T\\in {\\mathcal {T}}_h$ that $\\begin{split}(\\chi , {\\mathfrak {s}}(e_b))=& (\\alpha \\nabla _w e_b, \\nabla _w (Q_b\\Phi )) + ({\\beta }\\cdot \\nabla _w e_b, \\Phi ) + (c{\\mathfrak {s}}(e_b), \\Phi ) \\\\& - \\sum _T \\langle \\alpha \\nabla \\Phi \\cdot {\\bf n}- \\alpha Q_0(\\nabla \\Phi )\\cdot {\\bf n}, {\\mathfrak {s}}(e_b)-e_b\\rangle _{\\partial T}\\\\& -\\langle {\\beta }\\Phi \\cdot {\\bf n}- Q_0({\\beta }\\Phi )\\cdot {\\bf n}, {\\mathfrak {s}}(e_b)-e_b\\rangle _{\\partial T}.\\end{split}$ The last two terms on the right-hand side of (REF ) can be bounded by $C h \\Vert \\Phi \\Vert _2 {|\\!|\\!|} e_b{|\\!|\\!|}$ through the Cauchy-Schwarz inequality.","Thus, we have $\\begin{split}|(\\chi , {\\mathfrak {s}}(e_b))|\\le & |(\\alpha \\nabla _w e_b, \\nabla _w (Q_b\\Phi )) + ({\\beta }\\cdot \\nabla _w e_b, \\Phi ) + (c{\\mathfrak {s}}(e_b), \\Phi )| \\\\& + Ch\\Vert \\Phi \\Vert _2 {|\\!|\\!|} e_b{|\\!|\\!|}\\\\\\le & |(\\alpha \\nabla _w e_b, \\nabla _w (Q_b\\Phi )) + ({\\beta }\\cdot \\nabla _w e_b, {\\mathfrak {s}}(Q_b\\Phi )) + (c{\\mathfrak {s}}(e_b), {\\mathfrak {s}}(Q_b\\Phi ))| \\\\& + Ch\\Vert \\Phi \\Vert _2 {|\\!|\\!|} e_b{|\\!|\\!|},\\end{split}$ where have also used $\\Vert \\Phi -{\\mathfrak {s}}(Q_b\\Phi )\\Vert \\le C h^2\\Vert \\Phi \\Vert _2$ .","Now, recall that $(\\alpha \\nabla _w e_b, \\nabla _w (Q_b\\Phi )) + ({\\beta }\\cdot \\nabla _w e_b, {\\mathfrak {s}}(Q_b\\Phi )) + (c{\\mathfrak {s}}(e_b), {\\mathfrak {s}}(Q_b\\Phi ))= {\\mathcal {B}}(e_b, Q_b\\Phi ),$ and from the error equation (REF ), we have $\\begin{split}{\\mathcal {B}}(e_b, Q_b\\Phi ) = & \\ell _u(Q_b\\Phi ) - \\kappa S(e_b, Q_b\\Phi )\\\\=&\\sum _{T\\in {\\mathcal {T}}_h} \\langle \\alpha \\frac{\\partial u}{\\partial \\mathbf {n}}-\\alpha Q_0(\\nabla u)\\cdot \\mathbf {n},{\\mathfrak {s}}(Q_b\\Phi )-Q_b\\Phi \\rangle _{\\partial T}+\\kappa S(u_b,Q_b\\Phi )\\\\& + ((Q_0-I) \\nabla u, {\\mathfrak {s}}(Q_b\\Phi ){\\beta }) + (c({\\mathfrak {s}}(Q_bu)-u), {\\mathfrak {s}}(Q_b\\Phi )),\\end{split}$ The last two terms on the right-hand side of (REF ) have the following estimate: $\\left|((Q_0-I) \\nabla u, {\\mathfrak {s}}(Q_b\\Phi ){\\beta }) + (c({\\mathfrak {s}}(Q_bu)-u), {\\mathfrak {s}}(Q_b\\Phi ))\\right| \\le C h^2 \\Vert u\\Vert _2 \\Vert \\Phi \\Vert _1.$ The second term, $\\kappa S(u_b,Q_b\\Phi )$ , can be dealt with as follows: $\\begin{split}\\kappa S(u_b,Q_b\\Phi ) = & \\kappa h^{-1} \\sum _T \\langle u_b-Q_b{\\mathfrak {s}}(u_b), Q_b\\Phi - Q_b{\\mathfrak {s}}(Q_b\\Phi )\\rangle _{\\partial T}\\\\= & \\kappa h^{-1} \\sum _T \\langle u_b-Q_b{\\mathfrak {s}}(u_b), \\Phi - {\\mathfrak {s}}(Q_b\\Phi )\\rangle _{\\partial T}\\\\\\le & \\kappa h^{-1} \\sum _T \\Vert u_b-Q_b{\\mathfrak {s}}(u_b)\\Vert _{\\partial T}\\Vert \\Phi - {\\mathfrak {s}}(Q_b\\Phi )\\Vert _{\\partial T}\\\\\\le & C h ({|\\!|\\!|} e_b{|\\!|\\!|} + h\\Vert u\\Vert _2)\\Vert \\Phi \\Vert _2.\\end{split}$ As to the first term, we note from the definition of $Q_b$ and $\\Phi |_{\\partial \\Omega }=0$ that $\\begin{split}\\sum _{T\\in {\\mathcal {T}}_h} \\langle \\alpha \\frac{\\partial u}{\\partial \\mathbf {n}}-\\alpha Q_0(\\nabla u)\\cdot \\mathbf {n},\\Phi -Q_b\\Phi \\rangle _{\\partial T}= \\sum _{T\\in {\\mathcal {T}}_h} \\langle \\alpha \\frac{\\partial u}{\\partial \\mathbf {n}},\\Phi -Q_b\\Phi \\rangle _{\\partial T}= 0.\\end{split}$ Thus, we have $\\begin{split}&\\sum _{T\\in {\\mathcal {T}}_h} \\langle \\alpha \\frac{\\partial u}{\\partial \\mathbf {n}}-\\alpha Q_0(\\nabla u)\\cdot \\mathbf {n},{\\mathfrak {s}}(Q_b\\Phi )-Q_b\\Phi \\rangle _{\\partial T}\\\\= & \\sum _{T\\in {\\mathcal {T}}_h} \\langle \\alpha \\frac{\\partial u}{\\partial \\mathbf {n}}-\\alpha Q_0(\\nabla u)\\cdot \\mathbf {n},{\\mathfrak {s}}(Q_b\\Phi )-\\Phi \\rangle _{\\partial T}\\\\\\le & C h^2 \\Vert u\\Vert _2\\Vert \\Phi \\Vert _2.\\end{split}$ Substituting (REF ), (REF ), and (REF ) into (REF ) yields the following estimate: $|{\\mathcal {B}}(e_b, Q_b\\Phi )| \\le C(h^2 \\Vert u\\Vert _2 + h{|\\!|\\!|} e_b{|\\!|\\!|}) \\Vert \\Phi \\Vert _2,$ which, together with (REF ), leads to $|(\\chi , {\\mathfrak {s}}(e_b))| \\le C(h^2 \\Vert u\\Vert _2 + h{|\\!|\\!|} e_b{|\\!|\\!|}) \\Vert \\Phi \\Vert _2 \\le C(h^2 \\Vert u\\Vert _2 + h{|\\!|\\!|} e_b{|\\!|\\!|}) \\Vert \\chi \\Vert ,$ where the regularity assumption (REF ) has been employed in the last inequality.","Next, from (REF ) and the $H^1$ error estimate (REF ) in Theorem , we have $|(\\chi , {\\mathfrak {s}}(e_b))| \\le Ch^2 \\Vert u\\Vert _2 \\Vert \\chi \\Vert ,$ which leads to $\\Vert {\\mathfrak {s}}(e_b\\Vert \\le C h^2 \\Vert u\\Vert _2.$ Finally, we arrive at $\\Vert u-{\\mathfrak {s}}(u_b)\\Vert \\le \\Vert u-{\\mathfrak {s}}(Q_bu)\\Vert + \\Vert {\\mathfrak {s}}(e_b)\\Vert \\le C h^2\\Vert u\\Vert _2,$ which completes the proof of the theorem."],["Numerical Experiments","The goal of this section is to numerically verify the error estimates developed in the previous sections for the numerical scheme (REF ).","The following metrics are employed to measure the magnitude of the error function: $&&\\text{Discrete $L^2$-norm: }\\\\&&\\Vert u_b-u \\Vert _{0}=h\\left(\\sum _{i=1}^{n+1}\\sum _{j=1}^{n}|u_{i-\\frac{1}{2},j}-u(x_{i-\\frac{1}{2}},y_j)|^2 + \\sum _{i=1}^{n}\\sum _{j=1}^{n+1}|u_{i,j-\\frac{1}{2}}-u(x_{i},y_{j-\\frac{1}{2}})|^2\\right)^{1/2},\\\\&&\\text{Discrete $H^1$-norm: }\\\\&&\\Vert u_b-u\\Vert _1= h \\left(\\sum _{i=1}^{n}\\sum _{j=1}^{n} \\left|\\frac{u_{i+\\frac{1}{2},j}-u_{i-\\frac{1}{2},j}}{h} - \\frac{\\partial u}{\\partial x} (x_{i},y_{j})\\right|^2 \\right.","\\\\&&\\qquad \\qquad \\qquad \\left.","+ \\sum _{i=1}^{n}\\sum _{j=1}^{n} \\left|\\frac{u_{i,j+\\frac{1}{2}}-u_{i,j-\\frac{1}{2}}}{h} - \\frac{\\partial u}{\\partial y} (x_{i},y_{j})\\right|^2 \\right)^{1/2} ,\\\\$ Our numerical experiments are conducted for the model problem (REF )-() on polygonal domains.","The following set of test cases are considered: $\\left\\lbrace \\begin{split}&u=xy, \\\\&\\alpha =\\begin{bmatrix}1 &0\\\\0 & 1\\end{bmatrix},\\quad \\mathbf {\\beta }=\\begin{bmatrix}1\\\\1\\end{bmatrix},\\quad c=1;\\end{split}\\right.$ $\\left\\lbrace \\begin{split}&u= 3x^2+2xy, \\\\&\\alpha =\\begin{bmatrix}2 &0\\\\0 & 1\\end{bmatrix},\\quad \\mathbf {\\beta }=\\begin{bmatrix}1\\\\1\\end{bmatrix},\\quad c=1;\\end{split}\\right.$ $\\left\\lbrace \\begin{split}&u=\\sin (\\pi x)\\sin (\\pi y) + x^2-y^2, \\\\&\\alpha =\\begin{bmatrix}1 &0\\\\0 & 1\\end{bmatrix},\\quad \\mathbf {\\beta }=\\begin{bmatrix}1\\\\2\\end{bmatrix},\\quad c=1;\\end{split}\\right.$ $\\left\\lbrace \\begin{split}&u=\\sin (\\pi x)\\sin (\\pi y), \\\\&\\alpha =\\begin{bmatrix}xy+1 &0\\\\0 & 3xy\\end{bmatrix},\\quad \\mathbf {\\beta }=\\begin{bmatrix}x^3y+xy+1\\\\3x^2y+xy+2\\end{bmatrix},\\quad c=x^4y^2+xy+1;\\end{split}\\right.$ The right-hand side function $f$ and the Dirichlet boundary data $g$ are chosen to match the exact solution $u=u(x,y)$ for each test case.","Table: Error and convergence performance of the SWG scheme () with κ=4.0\\kappa =4.0 and uniform square partitions on the unit square domain Ω=(0,1) 2 \\Omega =(0,1)^2.Table REF shows the performance of the SWG scheme for each of the above test problems with the stabilizer parameter $\\kappa =4$ on uniform square partitions.","The results indicate that the numerical approximation is in the machine accuracy for the test problem (REF ) where the exact solution is a bilinear function.","For the other three test problems, the numerical solutions have the optimal rate of convergence $r=2$ in the discrete $L^2$ norm and a superconvergence of order ${\\mathcal {O}}(h^2)$ in the discrete $H^1$ norm.","The numerical results are consistent with the theoretical prediction in the discrete $L^2$ norm, but they outperform the theory in the discrete $H^1$ norm.","It should be pointed out that the superconvergence theory in [18] was developed for the diffusion equation only; but a slight modification of the analysis there will yield a superconvergence of order ${\\mathcal {O}}(h^2)$ for the SWG solutions of the full convection-diffusion equation (REF )-().","Table: Error and convergence performance of the SWG scheme () for the test case () with different values of κ\\kappa on uniform square partitions for Ω=(0,1) 2 \\Omega =(0,1)^2."],["On the influence of the stabilizer parameter","The goal of this subsection is to test the influence of the stabilizer parameter $\\kappa $ on the numerical solutions.","This part of the numerical experiment considers only the test cases (REF ) and (REF ) with the following six values of $\\kappa =0.01, \\ 0.1, \\ 1.0, \\ 4.0, \\ 6.0, \\ 20.0$ .","The case of $\\kappa =0$ is not a viable choice , as it was not covered in the convergence theory.","In fact, our computation does not suggest any convergence of the scheme when $\\kappa =0$ .","Tables REF -REF illustrate the numerical performance of the SWG scheme with different values of the stabilizer parameter $\\kappa $ .","Note that, for both test cases, the rate of convergence deteriorates as $\\kappa $ gets small (e.g.","$\\kappa =0.01$ ), particulary on coarse finite element partitions, but the rate of convergence begins to improve when the meshsize $h$ gets small.","Optimal rate of convergence and the supercovergence of order ${\\mathcal {O}}(h^2)$ are clearly shown in the tables when $\\kappa $ is away from 0 (e.g., $\\kappa \\ge 0.1$ ).","The stability and accuracy of the SWG scheme is insensitive to the value of $\\kappa $ as long as it stays away from 0.","Table: Error and convergence performance of the SWG scheme () for the test case () with different values of κ\\kappa on uniform square partitions for Ω=(0,1) 2 \\Omega =(0,1)^2."],["SWG with general polygonal partitions","The SWG scheme was applied to the test problem (REF ) with general polygonal partitions.","Table REF shows the error and convergence performance of the scheme on four types of polygonal partitions.","The stabilization parameter was set as $\\kappa =4$ in all these tests.","Optimal order of convergence in the discrete $L^2$ norm can be observed for each polygonal partition, but the superconvergence in the discrete $H^1$ norm was only seen for rectangular partitions.","The table shows a numerical rate of convergence of $r=1$ in the discrete $H^1$ norm for three other type of partitions.","The result is clearly in consistency with the error estimate developed in Section .","Fig.","REF illustrates the contour plots of the numerical solutions on different type of polygonal partitions.","It also shows the shape of the polygonal elements in our computation.","Table: Error and convergence performance of the SWG scheme () for the test problem () on general polygonal partitions for Ω=(0,1) 2 \\Omega =(0,1)^2, with κ=4\\kappa = 4.Figure: Comparison of numerical solutions obtained from SWG and the exact solution for the test problem () on various polygonal partitions of h=1/8h=1/8 and κ=1\\kappa =1."],["Numerical results on a non-convex domain","The SWG scheme with the stabilization parameter $\\kappa =4$ was applied to the test problem (REF ) on the L-shaped domain $\\Omega :=(-1,1)\\times (-1,1)/ (0,1)\\times (-1,0)$ partitioned into triangles or rectangles.","The corresponding numerical results are summarized in Table REF , which shows a convergence of order ${\\mathcal {O}}(h^2)$ in the $L^2$ norm for both the triangular and rectangular partitions.","A superconvergence of order ${\\mathcal {O}}(h^2)$ was observed in the discrete $H^1$ norm on rectangular partitions, while the optimal order of convergence with $r=1$ is confirmed numerically on triangular partitions.","It should be pointed out that the $H^2$ -regularity assumption (REF ) is not valid for non-convex polygonal domains so that the optimal order of error estimate (REF ) is not known theoretically on the L-shaped domain.","The numerical results therefore outperform the theory in the usual $L^2$ norm.","Table: Error and convergence performance of the SWG scheme () for test case () on Lshape domain, κ=4\\kappa = 4.Figure: Comparison of numerical solution obtained from SWG and the exact solution for test case () on L-shaped domain with h=1/8h=1/8."]],"string":"[\n [\n \"A Simplified Weak Galerkin Finite Element Method: Algorithm and Error\\n Estimates\"\n ],\n [\n \"Abstract In this article a simplified weak Galerkin finite element method is developed for the Dirichlet boundary value problem of convection-diffusion-reaction equations.\",\n \"The simplified weak Galerkin method utilizes only the degrees of freedom on the boundary of each element and, hence, has significantly reduced computational complexity over the regular weak Galerkin finite element method.\",\n \"A stability and some optimal order error estimates in the $H^1$ and $L^2$ norms are established for the corresponding numerical solutions.\",\n \"Numerical results are presented to verify the theory error estimates and a superconvergence phenomena on rectangular partitions.\"\n ],\n [\n \"Introduction\",\n \"This paper is concerned with the development of a simplified formulation for the weak Galerkin finite element method for second order elliptic equations.\",\n \"For simplicity, consider the model problem that seeks an unknown function $u=u(x)$ satisfying $-\\\\nabla \\\\cdot (\\\\alpha \\\\nabla u) + {\\\\beta }\\\\cdot \\\\nabla u + cu&=&f\\\\quad {\\\\rm in}\\\\ \\\\Omega \\\\\\\\u&=&g\\\\quad {\\\\rm on}\\\\ \\\\partial \\\\Omega $ where $\\\\Omega $ is a bounded polytopal domain in $\\\\mathbb {R}^d \\\\;(d\\\\ge 2)$ with boundary $\\\\partial \\\\Omega $ , $\\\\alpha =\\\\alpha (x)$ is the diffusion coefficient, ${\\\\beta }={\\\\beta }(x)$ is the convection, and $c=c(x)$ is the reaction coefficient in relevant applications.\",\n \"We assume that $\\\\alpha $ is sufficient smooth, ${\\\\beta }\\\\in [W^{1,\\\\infty }(\\\\Omega )]^d$ , and $c$ is piecewise smooth with respect to a partition of the domain.\",\n \"For well-posedness of the problem (REF )-(), we assume $f=f(x)\\\\in L^2(\\\\Omega )$ , $g=g(x)\\\\in H^{\\\\frac{1}{2}}(\\\\partial \\\\Omega )$ , and $c-\\\\frac{1}{2}\\\\nabla \\\\cdot {\\\\beta }\\\\ge 0,\\\\qquad \\\\alpha (x) \\\\ge \\\\alpha _0 \\\\qquad \\\\forall x \\\\in \\\\Omega $ for a constant $\\\\alpha _0>0$ .\",\n \"The model problem (REF )-() arises from many scientific applications such as fluid flow in porous media.\",\n \"Mostly importantly, this model problem has served, and still serves, the scientific computing community as a testbed in the search and design of new and efficient computational algorithms for partial differential equations.\",\n \"The classical Galerkin finite element method (see, e.g., [10], [28], [16]) is particularly a numerical technique originated from the study of elliptic problems closed related to (REF )-() or its variations.\",\n \"In the last three decades, various finite element methods using discontinuous trial and test functions, including discontinuous Galerkin (DG) methods and weak Galerkin (WG) methods, have been developed for numerical solutions of partial differential equations.\",\n \"These developments were often tested over testbed problems such as (REF )-() before they were generalized or applied to more complex problems in science and engineering.\",\n \"The DG method, also known as the interior penalty method in different contexts, was originated in early 70s of the last century for a numerical study of model problems such as (REF )-(); see, e.g., [3], [14], [25], [38] for early incubations and [1], [13], [17], [27] for a detailed discussion and recent developments.\",\n \"The weak Galerkin finite element method is a recently developed discretization framework for partial differential equations [36], [37], [24], [34].\",\n \"With new concepts referred to as weak differential operators (e.g., weak gradient, weak curl, weak Laplacian etc.)\",\n \"and weak continuity through the use of various stabilizers, the method allows the use of totally discontinuous functions and provides stable numerical schemes that are parameter-independent or free of locking [33].\",\n \"For the convection-diffusion-reaction equation (REF )-(), the recent work in the context of weak Galerkin includes the algorithm developed and analyzed in [9], the one in [20] for singularly perturbed problems, and an earlier one in [39].\",\n \"The WG finite element method has been rapidly developed and applied to several different types of problems, including second order elliptic problems, the Stokes and Navier-Stokes equations, the biharmonic and elasticity equations, div-curl systems and the Maxwell's equations, etc.\",\n \"The latest development of the WG methods is the prime-dual formulation for problems that are either nonsymmetric or do not have variational forms friendly for numerical use.\",\n \"Details on the new developments can be found in [30] for second order elliptic equations in nondivergence form, [31] for the Fokker-Planck equation, and [32] for elliptic Cauchy problems.\",\n \"The typical WG method for the model problem (REF )-() seeks weak finite element approximations $u_h=\\\\lbrace u_0, u_b\\\\rbrace $ satisfying $u_b|_{\\\\partial \\\\Omega } = Q_b g$ and $S(u_h, v)+ (\\\\alpha \\\\nabla _w u_h, \\\\nabla _w v) + ({\\\\beta }\\\\cdot \\\\nabla _w u_h, v_0) + (c u_0, v_0) = (f,v_0)$ for all test functions $v=\\\\lbrace v_0, v_b\\\\rbrace $ satisfying $v_b|_{\\\\partial \\\\Omega }=0$ , where $Q_bg$ is an interpolation of the Dirichlet boundary data, $\\\\nabla _w$ is the discrete weak gradient operator, and $S(\\\\cdot ,\\\\cdot )$ is a properly selected stabilizer that gives weak continuities for the numerical solutions.\",\n \"The numerical solution $u_h$ consists of two components: the approximation $u_0$ on each element and the approximation $u_b$ on the boundary of each element.\",\n \"To reduce the computational complexity, some hybridized formulations have been introduced in [22], [29] for the method when applied to the diffusion equation and the biharmonic equation through the elimination of the degrees of freedom associated with the unknown function $u_0$ locally on each element.\",\n \"In the superconvergence study for WG [18] on rectangular elements, this hybridized formulation was further simplified in the description of the numerical algorithm, yielding a simplified weak Galerkin (SWG) finite element scheme for the diffusion equation.\",\n \"In our further investigation of the SWG to the convection-diffusion-reaction equation (REF ), we came to the conclusion that SWG represents a new discretization scheme that is different from the usual WG through a simple elimination of the unknown $u_0$ .\",\n \"As a result, we believe that a systematic study of the SWG for the convection-diffusion-reaction problem (REF )-() should be conducted for its stability and convergence.\",\n \"This paper is in response to this observation and shall provide a mathematical theory for the stability and the convergence of the simplified weak Galerkin finite element method for the model problem (REF )-().\",\n \"We believe that the result of this paper can be extended to other types of modeling equations.\",\n \"The paper is organized as follows: In Section , we shall describe the simplified weak Galerkin finite element method for (REF )-() on general polygonal partitions.\",\n \"In Section , we shall present a computational formula for the element stiffness matrices and the element load vectors from SWG.\",\n \"In Section , we provide a mathematical theory for the stability and well-posedness of the SWG scheme.\",\n \"Sections and are devoted to a discussion of the error estimates in a discrete $H^1$ and the $L^2$ norm for the numerical solutions.\",\n \"Finally, in Section , we present some numerical results to demonstrate the efficiency and accuracy of the SWG method.\",\n \"Throughout the rest of the paper, we assume $d=2$ and shall use the standard notations for Sobolev spaces and norms [10], [16].\",\n \"For any open set $D\\\\subset \\\\mathbb {R}^{2}$ , $\\\\Vert \\\\cdot \\\\Vert _{s,D}$ and $(\\\\cdot ,\\\\cdot )_{s,D}$ denote the norm and inner-product in the Sobolev space $H^s(D)$ consisting of square integrable partial derivatives up to order $s$ .\",\n \"When $s=0$ or $D=\\\\Omega $ , we shall drop the corresponding subscripts in the norm and inner-product notation.\"\n ],\n [\n \"Algorithm on Polymesh\",\n \"Assume that the domain is of polygonal type and is partitioned into non-overlap polygons ${\\\\mathcal {T}}_h=\\\\lbrace T\\\\rbrace $ that are shape regular.\",\n \"For each $T\\\\in {\\\\mathcal {T}}_h$ , denote by $h_T$ its diameter and by $N$ the number of edges.\",\n \"For each edge $e_i, \\\\ i=1,\\\\ldots , N$ , denote by $M_i$ the midpoints and ${\\\\bf n}_i$ the outward normal direction of $e_i$ (see Fig.\",\n \"REF for an illuatration).\",\n \"The meshsize of ${\\\\mathcal {T}}_h$ is defined as $h=\\\\max _{T\\\\in {\\\\mathcal {T}}_h} h_T$ .\",\n \"Let $v_b$ be a piecewise constant function defined on the boundary of $T$ , i.e., $v_b|_{e_i} =v_{b,i},$ with $v_{b,i}$ being a constant.\",\n \"We define the weak gradient of $v_b$ on $T$ by: $\\\\nabla _w v_b:=\\\\displaystyle \\\\frac{1}{|T|}\\\\sum _{i=1}^N v_{b,i}|e_i|\\\\bf {n_i},$ where $|e_i|$ is the length of the edge $e_i$ and $|T|$ is the area of the element $T$ .\",\n \"It is not hard to see that the weak gradient $\\\\nabla _w v_b$ satisfies the following equation: $(\\\\nabla _w v_b, \\\\mathbf {\\\\phi })_T=\\\\langle v_b, \\\\mathbf {\\\\phi }\\\\cdot {\\\\bf n}\\\\rangle _{\\\\partial T}$ for all constant vector $\\\\mathbf {\\\\phi }$ .\",\n \"Here and in what follows of the paper, $\\\\langle \\\\cdot ,\\\\cdot \\\\rangle _{\\\\partial T}$ stands for the usual inner product in $L^2({\\\\partial T})$ .\",\n \"Denote by $W(T)$ the space of piecewise constant functions on ${\\\\partial T}$ .\",\n \"The global finite element space $W({\\\\mathcal {T}}_h)$ is constructed by patching together all the local elements $W(T)$ through single values on interior edges.\",\n \"The subspace of $W_h({\\\\mathcal {T}}_h)$ consisting of functions with vanishing boundary value is denoted as $W_h^0({\\\\mathcal {T}}_h)$ .\",\n \"We use the conventional notation of ${P}_j(T)$ for the space of polynomials of degree $j\\\\ge 0$ on $T$ .\",\n \"For each $v_b\\\\in W(T)$ , we associate it with a linear extension in $T$ , denoted as ${\\\\mathfrak {s}}(v_b)\\\\in {P}_1 (T)$ , satisfying $\\\\sum _{i=1}^{N}({\\\\mathfrak {s}}(v_b)(M_i) -v_{b,i})\\\\phi (M_i)|e_i|=0,\\\\quad \\\\forall \\\\; \\\\phi \\\\in {P}_1(T).$ It is easy to see that ${\\\\mathfrak {s}}(u_b)$ is well defined by (REF ), and its computation is local and straightforward.\",\n \"In fact, ${\\\\mathfrak {s}}(u_b)$ can be viewed as an extension of $u_b$ from $\\\\partial T$ to $T$ through a least-squares fitting.\",\n \"Figure: An illustrative polygonal element.On each element $T \\\\in {\\\\mathcal {T}}_h $ , we introduce the following bilinear forms: $a_T(u_b,v_b)&:= & (\\\\alpha \\\\nabla _w u_b, \\\\nabla _w v_b)_T, \\\\\\\\b_T(u_b,v_b)&:= & ({\\\\beta }\\\\cdot \\\\nabla _w u_b, {\\\\mathfrak {s}}(v_b))_T,\\\\\\\\c_T(u_b,v_b)&:= & (c{\\\\mathfrak {s}}(u_b), {\\\\mathfrak {s}}(v_b))_T.$ For simplicity, we set $\\\\mathcal {B}_T(u_b,v_b):=a_T(u_b,v_b) + b_T(u_b,v_b) + c_T(u_b,v_b)$ for $u_b, v_b \\\\in W(T)$ .\",\n \"We further introduce the stabilizer $\\\\begin{split}S_T(u_b,v_b):= & h^{-1}\\\\sum _{i=1}^N ({\\\\mathfrak {s}}(u_b)(M_i)-u_{b,i})({\\\\mathfrak {s}}(v_b)(M_i)-v_{b,i})|e_i|\\\\\\\\= & h^{-1}\\\\langle Q_b{\\\\mathfrak {s}}(u_b)-u_{b},Q_b{\\\\mathfrak {s}}(v_b)-v_{b}\\\\rangle _{\\\\partial T},\\\\end{split}$ where $Q_b$ is the $L^2$ projection operator onto $W(T)$ ; namely $Q_b u$ is the average of $u$ on each edge.\",\n \"In particular, $Q_b(g)$ is well-defined and takes the average of the Dirichlet data on each boundary edge.\",\n \"SWG Algorithm 2.1 The simplified weak Galerkin (SWG) scheme for the elliptic equation (REF )-() seeks $u_b \\\\in W_h({\\\\mathcal {T}}_h)$ satisfying $u_b = Q_b(g)$ on $\\\\partial \\\\Omega $ and ${\\\\mathcal {A}}(u_b, v_b) =(f, {\\\\mathfrak {s}}(v_b))\\\\qquad \\\\forall v_b \\\\in W_h^0({\\\\mathcal {T}}_h),$ where ${\\\\mathcal {A}}(u_b,v_b):=\\\\kappa S(u_b,v_b)+ {\\\\mathcal {B}}(u_b, v_b)$ , $S(u_b,v_b) &=& \\\\sum _{T\\\\in {\\\\mathcal {T}}_h} S_T(u_b,v_b),\\\\\\\\{\\\\mathcal {B}}(u_b,v_b)&=& \\\\sum _{T\\\\in {\\\\mathcal {T}}_h} \\\\mathcal {B}_T(u_b, v_b)$ are bilinear forms in $W_h({\\\\mathcal {T}}_h)$ and $(f,{\\\\mathfrak {s}}(v_b)):=\\\\sum _{T\\\\in {\\\\mathcal {T}}_h} (f, {\\\\mathfrak {s}}(v_b))_T$ is a linear form in $W_h({\\\\mathcal {T}}_h)$ .\"\n ],\n [\n \"Element Stiffness Matrices\",\n \"The simplified weak Galerkin finite element method (REF ) is user-friendly in computer implementation.\",\n \"In this section, we present a formula for the computation of the element stiffness matrices and the element load vector on general polygonal elements.\",\n \"Let $T\\\\in {\\\\mathcal {T}}_h$ be a polygonal element of $N$ sides.\",\n \"Denote by $X_{u_b}$ the vector representation of $u_b$ given by $(u_{b,1},u_{b,2},\\\\ldots ,u_{b,N})^T$ .\",\n \"Then, the element stiffness matrix and the element load vector for the SWG scheme (REF ) are given in a block matrix form as follows: $(\\\\kappa h^{-1}A^T + B + R + C)X_{u_b} \\\\cong F,$ where the block components in (REF ) are given by: (1) $A:=\\\\lbrace a_{i,j}\\\\rbrace _{i,j=1}^N = E- EM(M^TEM)^{-1}M^TE$ , (2) $B:=\\\\lbrace b_{i,j}\\\\rbrace _{i,j=1}^N$ , with $b_{i,j} = (\\\\alpha \\\\mathbf {n}_i,\\\\mathbf {n}_j)_{T}\\\\displaystyle \\\\frac{|e_i||e_j|}{|T|^2}$ , (3) $R:=\\\\lbrace r_{ij}\\\\rbrace _{i,j=1}^N$ , with $r_{ij}= \\\\frac{|e_j|}{|T|}\\\\int _T {\\\\beta }\\\\cdot \\\\mathbf {n}_j \\\\zeta _i dT$ , (4) $C:=\\\\lbrace c_{ij}\\\\rbrace _{i,j=1}^N$ , with $c_{ij}= \\\\int _T c\\\\zeta _j\\\\zeta _i dT$ , (5) $F:=\\\\lbrace f_{i}\\\\rbrace _{i=1}^N$ , with $f_{i} = \\\\int _{T}f(x,y) \\\\zeta _i(x,y) dT$ , (6) $D:=\\\\lbrace d_{j,i}\\\\rbrace _{3\\\\times N}=(M^TEM)^{-1}M^TE$ and $\\\\zeta _i = d_{1,i} + d_{2,i} (x-x_T) + d_{3,i} (y-y_T)$ , (7) $M$ and $E$ are given by $M=\\\\begin{bmatrix}1 & x_{1} - x_T & y_{1} - y_T\\\\\\\\1 & x_{2} - x_T & y_{2} - y_T\\\\\\\\\\\\vdots & \\\\vdots & \\\\vdots \\\\\\\\1 & x_{N} - x_T & y_{N} - y_T\\\\\\\\\\\\end{bmatrix}_{N\\\\times 3},\\\\;E=\\\\begin{bmatrix}|e_1| & & & \\\\\\\\& |e_2| & & \\\\\\\\& & \\\\ddots & \\\\\\\\& & & |e_N| \\\\\\\\\\\\end{bmatrix}_{N\\\\times N}.$ Here $M_T=(x_T,y_T)$ is any point on the plane (e.g., the center of $T$ as a specific case), $(x_i, y_i)$ is the midpoint of $e_i$ , $|e_i|$ is the length of edge $e_i$ , $\\\\mathbf {n}_i$ is the unit outward normal vector on $e_i$ , and $|T|$ is the area of the element $T$ .\",\n \"From (REF ), the element stiffness matrix on $T\\\\in {\\\\mathcal {T}}_h$ consists of two sub-matrices corresponding to the following forms: $S_T(u_b,v_b)\\\\ \\\\mbox{and } {\\\\mathcal {B}}_T(u_b, v_b).$ The bilinear form ${\\\\mathcal {B}}_T(\\\\cdot ,\\\\cdot )$ is composed of three bilinear forms given by (REF ).\",\n \"The rest of this section is devoted to a computation of the element stiffness matrices for each of the bilinear forms involved.\"\n ],\n [\n \"The stiffness matrix for $S_T(\\\\cdot ,\\\\cdot )$\",\n \"For the element stiffness matrix corresponding to $S_T(u_b,v_b)$ , the key is to compute ${\\\\mathfrak {s}}(u_b)$ and ${\\\\mathfrak {s}}(v_b)$ which can be accomplished through its definition (REF ); readers are referred to [21] for a detailed derivation.\",\n \"Specifically, let $M_T=(x_T,y_T)$ be the center of T (or any point on the plane), the extension ${\\\\mathfrak {s}}(u_b)$ can be represented as follows: ${\\\\mathfrak {s}}(u_b)=\\\\gamma _0 + \\\\gamma _1 (x-x_T) + \\\\gamma _2 (y-y_T),$ where $\\\\begin{bmatrix}\\\\gamma _0 \\\\\\\\\\\\gamma _1 \\\\\\\\\\\\gamma _2 \\\\\\\\\\\\end{bmatrix}=(M^TEM)^{-1}M^TE\\\\begin{bmatrix}u_{b,1} \\\\\\\\u_{b,2} \\\\\\\\\\\\vdots \\\\\\\\u_{b,N} \\\\\\\\\\\\end{bmatrix}.$ From ${\\\\mathfrak {s}}(u_b)=\\\\gamma _0 + \\\\gamma _1 (x-x_T) + \\\\gamma _2 (y-y_T)$ and (REF ), we have $\\\\begin{bmatrix}{\\\\mathfrak {s}}(u_{b})(M_1) \\\\\\\\{\\\\mathfrak {s}}(u_{b})(M_2) \\\\\\\\\\\\vdots \\\\\\\\{\\\\mathfrak {s}}(u_{b})(M_N) \\\\\\\\\\\\end{bmatrix}=M\\\\begin{bmatrix}\\\\gamma _0\\\\\\\\\\\\gamma _1\\\\\\\\\\\\gamma _2\\\\\\\\\\\\end{bmatrix}=M(M^TEM)^{-1}M^TE\\\\begin{bmatrix}u_{b,1} \\\\\\\\u_{b,2} \\\\\\\\\\\\vdots \\\\\\\\u_{b,N} \\\\\\\\\\\\end{bmatrix}.$ Let $v_b\\\\in W(T)$ be the basis function corresponding to the edge $e_j$ of $T$ : $v_b=\\\\left\\\\lbrace \\\\begin{array}{lllll}1, \\\\qquad \\\\text{ on } e_j,\\\\\\\\0, \\\\qquad \\\\text{ otherwise}.\\\\\\\\\\\\end{array}\\\\right.$ Then the coefficient $(\\\\tilde{\\\\gamma }_0,\\\\tilde{\\\\gamma }_1,\\\\tilde{\\\\gamma }_2)^T$ for ${\\\\mathfrak {s}}(v_b)$ is given by $\\\\begin{bmatrix}\\\\tilde{\\\\gamma }_0 \\\\\\\\\\\\tilde{\\\\gamma }_1 \\\\\\\\\\\\tilde{\\\\gamma }_2 \\\\\\\\\\\\end{bmatrix}=(M^TEM)^{-1}M^TE\\\\begin{bmatrix}v_{b,1}\\\\\\\\\\\\vdots \\\\\\\\v_{b,j}\\\\\\\\\\\\vdots \\\\\\\\v_{b,N}\\\\\\\\\\\\end{bmatrix}=(M^TEM)^{-1}M^TE\\\\begin{bmatrix}0\\\\\\\\\\\\vdots \\\\\\\\1\\\\\\\\\\\\vdots \\\\\\\\0\\\\\\\\\\\\end{bmatrix}\\\\triangleq \\\\begin{bmatrix}d_{1,j}\\\\\\\\d_{2,j}\\\\\\\\d_{3,j}\\\\\\\\\\\\end{bmatrix}.$ It follows that $\\\\begin{split}S_T(u_b,v_b)=&h^{-1}\\\\sum _{i=1}^N({\\\\mathfrak {s}}(u_b)(M_i)-u_{b,i})({\\\\mathfrak {s}}(v_b)(M_i)-v_{b,i})|e_i|\\\\\\\\=&h^{-1}\\\\sum _{i=1}^N (u_{b,i}-{\\\\mathfrak {s}}(u_b)(M_i))v_{b,i}|e_i|\\\\\\\\=&h^{-1}\\\\left((I_N-M(M^TEM)^{-1}M^TE)\\\\begin{bmatrix}u_{b,1} \\\\\\\\u_{b,2} \\\\\\\\\\\\vdots \\\\\\\\u_{b,N} \\\\\\\\\\\\end{bmatrix}\\\\right)_j |e_j| \\\\\\\\=&h^{-1}\\\\sum _{i=1}^N a_{j,i}u_{b,i},\\\\end{split}$ where $I_N$ is the identity matrix of size $N\\\\times N$ .\"\n ],\n [\n \"The stiffness matrix for $a_T(\\\\cdot ,\\\\cdot )$\",\n \"For a computation of the element stiffness matrix corresponding to the bilinear form $a_T(u_b,v_b)=(\\\\alpha \\\\nabla _w u_b,\\\\nabla _w v_b )_T$ , we have from the weak gradient formula (REF ) that $(\\\\alpha \\\\nabla _w u_b,\\\\nabla _w v_b )_T&=&(\\\\alpha \\\\displaystyle \\\\frac{1}{|T|}\\\\sum _{j=1}^N u_{b,j}\\\\mathbf {n}_j|e_j|,\\\\displaystyle \\\\frac{1}{|T|}\\\\sum _{i=1}^N v_{b,i}\\\\mathbf {n}_i|e_i|)_{T}\\\\\\\\&=&\\\\sum _{i,j=1}^N(\\\\alpha \\\\displaystyle \\\\frac{1}{|T|} u_{b,j}\\\\mathbf {n}_j|e_j|,\\\\displaystyle \\\\frac{1}{|T|} v_{b,i}\\\\mathbf {n}_i|e_i|)_{T} \\\\nonumber \\\\\\\\&=&\\\\sum _{i,j=1}^N \\\\frac{|e_j||e_i|}{|T|^2}(\\\\alpha \\\\mathbf {n}_j,\\\\mathbf {n}_i)_{T}u_{b,j}v_{b,i},\\\\nonumber \\\\\\\\&=&\\\\sum _{i,j=1}^N b_{i,j}u_{b,j}v_{b,i},\\\\nonumber $ which leads to the block matrix $B$ in the element stiffness matrix.\"\n ],\n [\n \"The stiffness matrix for $b_T(\\\\cdot ,\\\\cdot )$\",\n \"Recall that the bilinear form $b_T(\\\\cdot ,\\\\cdot )$ is given by $b_T(u_b, v_b)=({\\\\beta }\\\\cdot \\\\nabla _w u_b, {\\\\mathfrak {s}}(v_b))_T.$ Note that the extension ${\\\\mathfrak {s}}(v_b)$ has the following representation: ${\\\\mathfrak {s}}(v_b)=\\\\gamma _0 + \\\\gamma _1 (x-x_T) + \\\\gamma _2 (y-y_T),$ where $\\\\begin{bmatrix}\\\\gamma _0 \\\\\\\\\\\\gamma _1 \\\\\\\\\\\\gamma _2 \\\\\\\\\\\\end{bmatrix}=(M^TEM)^{-1}M^TE\\\\begin{bmatrix}v_{b,1} \\\\\\\\v_{b,2} \\\\\\\\\\\\vdots \\\\\\\\v_{b,N} \\\\\\\\\\\\end{bmatrix}.$ Thus, with $D=(M^TEM)^{-1}M^TE$ , we have from the weak gradient formula (REF ) that $\\\\begin{split}& ({\\\\beta }\\\\cdot \\\\nabla _w u_b,{\\\\mathfrak {s}}(v_b))_T \\\\\\\\=&\\\\displaystyle \\\\frac{1}{|T|}\\\\sum _{i,j=1}^N ({\\\\beta }\\\\cdot \\\\mathbf {n}_j, d_{1,i} + d_{2,i} (x-x_T) + d_{3,i} (y-y_T))_T |e_j| u_{b,j} v_{b,i}\\\\\\\\=&\\\\displaystyle \\\\frac{1}{|T|}\\\\sum _{i,j=1}^N \\\\int _T {\\\\beta }\\\\cdot \\\\mathbf {n}_j (d_{1,i} + d_{2,i} (x-x_T) + d_{3,i} (y-y_T))dT |e_j| u_{b,j} v_{b,i}.\\\\end{split}$ For simplicity, we introduce the following functions: $\\\\zeta _i(x,y) = d_{1,i} + d_{2,i} (x-x_T) + d_{3,i} (y-y_T),\\\\qquad i=1,\\\\ldots , N.$ Then, the equation (REF ) indicates that the element stiffness matrix corresponding to the bilinear form $b_T(\\\\cdot ,\\\\cdot )$ is given by $R=\\\\lbrace r_{ij}\\\\rbrace _{N\\\\times N},\\\\ \\\\ r_{ij}= \\\\frac{|e_j|}{|T|}\\\\int _T {\\\\beta }\\\\cdot \\\\mathbf {n}_j \\\\zeta _i dT.$\"\n ],\n [\n \"The stiffness matrix for $c_T(\\\\cdot ,\\\\cdot )$\",\n \"Recall that the bilinear form $c_T(\\\\cdot ,\\\\cdot )$ is given by $c_T(u_b, v_b)=(c {\\\\mathfrak {s}}(u_b), {\\\\mathfrak {s}}(v_b))_T.$ Thus, the element stiffness matrix corresponding to $c_T(\\\\cdot ,\\\\cdot )$ has the following formula: $C=\\\\lbrace c_{ij}\\\\rbrace _{N\\\\times N},\\\\quad c_{ij} = \\\\int _T c(x,y) \\\\zeta _j\\\\zeta _i dT,$ where $\\\\zeta _i$ is the function defined in (REF ).\"\n ],\n [\n \"The element load vector\",\n \"Finally, the element load vector can be obtained from $(f, {\\\\mathfrak {s}}(v_b))_T& =& \\\\int _{T}f {\\\\mathfrak {s}}(v_b)dT \\\\\\\\& =& \\\\int _{T}f(x,y) (d_{1,i} + d_{2,i}(x-x_T) + d_{3,i}(y-y_T)) dT\\\\\\\\& =& \\\\int _{T}f(x,y) \\\\zeta _i(x,y) dT$ for $i=1,\\\\ldots , N$ .\"\n ],\n [\n \"Stability and Well-Posedness\",\n \"The SWG scheme (REF ) can be derived from the classical weak Galerkin finite element method [36], [24], [37] by eliminating the degrees of freedom associated with the interior of each element when ${\\\\beta }=0$ and $c=0$ .\",\n \"But for the general case of ${\\\\beta }$ and $c$ , the SWG finite element method (REF ) is different from the weak Galerkin schemes in existing literature.\",\n \"It is thus necessary to provide a mathematical theory for the stability and well-posedness of the numerical scheme (REF ).\",\n \"Let ${\\\\mathcal {T}}_h$ be a shape-regular polygonal partition of the domain $\\\\Omega $ .\",\n \"There exists a constant $C$ such that $\\\\Vert \\\\nabla {\\\\mathfrak {s}}(v_b)\\\\Vert _{T}^2 & \\\\le & C\\\\left(\\\\Vert \\\\nabla _w v_b\\\\Vert ^2_T + h^{-1}\\\\Vert v_b-Q_b{\\\\mathfrak {s}}(v_b)\\\\Vert ^2_{{\\\\partial T}}\\\\right),\\\\\\\\\\\\Vert v_b-{\\\\mathfrak {s}}(v_b)\\\\Vert _{0,{\\\\partial T}}^2&\\\\le & C h \\\\left(\\\\Vert \\\\nabla _w v_b\\\\Vert ^2_T + h^{-1} \\\\Vert v_b-Q_b{\\\\mathfrak {s}}(v_b)\\\\Vert ^2_{\\\\partial T}\\\\right).$ Moreover, the following Poincaré-type estimate holds true: $\\\\Vert {\\\\mathfrak {s}}(v_b)\\\\Vert ^2 & \\\\le & C\\\\left(\\\\Vert \\\\nabla _w v_b\\\\Vert ^2_T + h^{-1}\\\\Vert v_b-Q_b{\\\\mathfrak {s}}(v_b)\\\\Vert ^2_{{\\\\partial T}}\\\\right).$ From the formula (REF ) for the weak gradient, we have for any constant vector $\\\\mathbf {\\\\phi }$ that $\\\\begin{split}(\\\\nabla _w v_b, \\\\mathbf {\\\\phi })_T=&\\\\langle v_b, \\\\mathbf {\\\\phi }\\\\cdot \\\\mathbf {n}\\\\rangle _{\\\\partial T}\\\\\\\\=& \\\\langle v_b- {\\\\mathfrak {s}}(v_b), \\\\mathbf {\\\\phi }\\\\cdot \\\\mathbf {n}\\\\rangle _{\\\\partial T}+ \\\\langle {\\\\mathfrak {s}}(v_b), \\\\mathbf {\\\\phi }\\\\cdot \\\\mathbf {n}\\\\rangle _{\\\\partial T}\\\\\\\\=& \\\\langle v_b- Q_b{\\\\mathfrak {s}}(v_b), \\\\mathbf {\\\\phi }\\\\cdot \\\\mathbf {n}\\\\rangle _{\\\\partial T}+ (\\\\nabla {\\\\mathfrak {s}}(v_b), \\\\mathbf {\\\\phi })_T,\\\\end{split}$ which gives $(\\\\nabla {\\\\mathfrak {s}}(v_b), \\\\mathbf {\\\\phi })_T = (\\\\nabla _w v_b, \\\\mathbf {\\\\phi })_T - \\\\langle v_b- Q_b{\\\\mathfrak {s}}(v_b), \\\\mathbf {\\\\phi }\\\\cdot \\\\mathbf {n}\\\\rangle _{\\\\partial T}.$ Hence, by letting $\\\\mathbf {\\\\phi } = \\\\nabla {\\\\mathfrak {s}}(v_b)$ we arrive at $\\\\Vert \\\\nabla {\\\\mathfrak {s}}(v_b)\\\\Vert _{T}^2 \\\\le C\\\\left(\\\\Vert \\\\nabla _w v_b\\\\Vert ^2_T + h^{-1}\\\\Vert v_b-Q_b{\\\\mathfrak {s}}(v_b)\\\\Vert ^2_{{\\\\partial T}}\\\\right),$ which verifies (REF ).\",\n \"Next, from the usual error estimate for the $L^2$ projection operator $Q_b$ and the estimate (REF ), we have $\\\\begin{split}\\\\Vert {\\\\mathfrak {s}}(v_b) - Q_b{\\\\mathfrak {s}}(v_b)\\\\Vert ^2_{{\\\\partial T}} \\\\le & C h^2 \\\\Vert \\\\nabla {\\\\mathfrak {s}}(v_b)\\\\Vert _{\\\\partial T}^2 \\\\\\\\\\\\le & C h \\\\Vert \\\\nabla {\\\\mathfrak {s}}(v_b)\\\\Vert ^2_T \\\\\\\\\\\\le & C\\\\left(h\\\\Vert \\\\nabla _w v_b\\\\Vert ^2_T + \\\\Vert v_b-Q_b{\\\\mathfrak {s}}(v_b)\\\\Vert ^2_{\\\\partial T}\\\\right).\\\\end{split}$ It follows that $\\\\begin{split}\\\\Vert v_b-{\\\\mathfrak {s}}(v_b)\\\\Vert _{0,{\\\\partial T}}\\\\le & \\\\Vert v_b-Q_b{\\\\mathfrak {s}}(v_b)\\\\Vert _{0,{\\\\partial T}} + \\\\Vert {\\\\mathfrak {s}}(v_b)-Q_b{\\\\mathfrak {s}}(v_b)\\\\Vert _{0,{\\\\partial T}}\\\\\\\\\\\\le & C \\\\left(h\\\\Vert \\\\nabla _w v_b\\\\Vert ^2_T + \\\\Vert v_b-Q_b{\\\\mathfrak {s}}(v_b)\\\\Vert ^2_{\\\\partial T}\\\\right)^{1/2},\\\\end{split}$ which verifies the estimate ().\",\n \"To derive the inequality (REF ), we note the following discrete Poincaré inequality: $\\\\Vert {\\\\mathfrak {s}}(v_b)\\\\Vert ^2 \\\\le C\\\\sum _{T\\\\in {\\\\mathcal {T}}_h} \\\\left( \\\\Vert \\\\nabla {\\\\mathfrak {s}}(v_b)\\\\Vert _T^2 + h_T^{-1}\\\\Vert {\\\\mathfrak {s}}(v_b) - v_b\\\\Vert _{\\\\partial T}^2\\\\right).$ Combining the above estimate with (REF ) and (REF ) gives rise to the desired inequality (REF ).\",\n \"This completes the proof of the lemma.\",\n \"On each element $T\\\\in {\\\\mathcal {T}}_h$ , the following identity holds true: $\\\\begin{split}b_T(v_b,v_b) =&\\\\frac{1}{2} \\\\langle v_b, v_b {\\\\beta }\\\\cdot {\\\\bf n}\\\\rangle _{\\\\partial T}- \\\\frac{1}{2} ((\\\\nabla \\\\cdot {\\\\beta }) {\\\\mathfrak {s}}(v_b), {\\\\mathfrak {s}}(v_b))_T \\\\\\\\& - \\\\frac{1}{2} \\\\langle v_b-{\\\\mathfrak {s}}(v_b), (v_b-{{\\\\mathfrak {s}}(v_b)){\\\\beta }}\\\\cdot {\\\\bf n}\\\\rangle _{\\\\partial T}\\\\\\\\& + \\\\langle v_b-{\\\\mathfrak {s}}(v_b), \\\\overline{{\\\\mathfrak {s}}(v_b){\\\\beta }}\\\\cdot {\\\\bf n}- {{\\\\mathfrak {s}}(v_b){\\\\beta }}\\\\cdot {\\\\bf n}\\\\rangle _{\\\\partial T},\\\\end{split}$ where $\\\\overline{{\\\\mathfrak {s}}(v_b){\\\\beta }}$ is the average of ${\\\\mathfrak {s}}(v_b){\\\\beta }$ on the element $T$ .\",\n \"From the formula (REF ), we have $\\\\begin{split}b_T(v_b,v_b) =& ({\\\\beta }\\\\cdot \\\\nabla _w v_b, {\\\\mathfrak {s}}(v_b))_T \\\\\\\\=& (\\\\nabla _w v_b, {{\\\\mathfrak {s}}(v_b){\\\\beta }})_T\\\\\\\\=& (\\\\nabla _w v_b, \\\\overline{{\\\\mathfrak {s}}(v_b){\\\\beta }})_T\\\\\\\\=& \\\\langle v_b, \\\\overline{{\\\\mathfrak {s}}(v_b){\\\\beta }}\\\\cdot {\\\\bf n}\\\\rangle _{\\\\partial T}\\\\\\\\=& \\\\langle v_b-{\\\\mathfrak {s}}(v_b), \\\\overline{{\\\\mathfrak {s}}(v_b){\\\\beta }}\\\\cdot {\\\\bf n}\\\\rangle _{\\\\partial T}+ \\\\langle {\\\\mathfrak {s}}(v_b), \\\\overline{{\\\\mathfrak {s}}(v_b){\\\\beta }}\\\\cdot {\\\\bf n}\\\\rangle _{\\\\partial T}.\\\\end{split}$ Note that $\\\\begin{split}\\\\langle {\\\\mathfrak {s}}(v_b), \\\\overline{{\\\\mathfrak {s}}(v_b){\\\\beta }}\\\\cdot {\\\\bf n}\\\\rangle _{\\\\partial T}&= (\\\\nabla {\\\\mathfrak {s}}(v_b), \\\\overline{{\\\\mathfrak {s}}(v_b){\\\\beta }})_T\\\\\\\\&= (\\\\nabla {\\\\mathfrak {s}}(v_b), {\\\\mathfrak {s}}(v_b){\\\\beta })_T\\\\\\\\&= \\\\frac{1}{2} \\\\langle {\\\\mathfrak {s}}(v_b), {\\\\mathfrak {s}}(v_b) {\\\\beta }\\\\cdot {\\\\bf n}\\\\rangle _{\\\\partial T}- \\\\frac{1}{2} ((\\\\nabla \\\\cdot {\\\\beta }) {\\\\mathfrak {s}}(v_b), {\\\\mathfrak {s}}(v_b))_T.\\\\end{split}$ Substituting the above identity into (REF ) yields $\\\\begin{split}b_T(v_b,v_b) =& \\\\langle v_b-{\\\\mathfrak {s}}(v_b), \\\\overline{{\\\\mathfrak {s}}(v_b){\\\\beta }}\\\\cdot {\\\\bf n}\\\\rangle _{\\\\partial T}+ \\\\frac{1}{2} \\\\langle {\\\\mathfrak {s}}(v_b), {\\\\mathfrak {s}}(v_b) {\\\\beta }\\\\cdot {\\\\bf n}\\\\rangle _{\\\\partial T}\\\\\\\\& \\\\ \\\\ - \\\\frac{1}{2} ((\\\\nabla \\\\cdot {\\\\beta }) {\\\\mathfrak {s}}(v_b), {\\\\mathfrak {s}}(v_b))_T\\\\\\\\=& \\\\langle v_b-{\\\\mathfrak {s}}(v_b), \\\\overline{{\\\\mathfrak {s}}(v_b){\\\\beta }}\\\\cdot {\\\\bf n}- {{\\\\mathfrak {s}}(v_b){\\\\beta }}\\\\cdot {\\\\bf n}\\\\rangle _{\\\\partial T}+ \\\\langle v_b,{{\\\\mathfrak {s}}(v_b){\\\\beta }}\\\\cdot {\\\\bf n}\\\\rangle _{\\\\partial T}\\\\\\\\& \\\\ \\\\ - \\\\frac{1}{2} \\\\langle {\\\\mathfrak {s}}(v_b), {\\\\mathfrak {s}}(v_b) {\\\\beta }\\\\cdot {\\\\bf n}\\\\rangle _{\\\\partial T}- \\\\frac{1}{2} ((\\\\nabla \\\\cdot {\\\\beta }) {\\\\mathfrak {s}}(v_b), {\\\\mathfrak {s}}(v_b))_T\\\\\\\\=& \\\\langle v_b-{\\\\mathfrak {s}}(v_b), \\\\overline{{\\\\mathfrak {s}}(v_b){\\\\beta }}\\\\cdot {\\\\bf n}- {{\\\\mathfrak {s}}(v_b){\\\\beta }}\\\\cdot {\\\\bf n}\\\\rangle _{\\\\partial T}\\\\\\\\& \\\\ - \\\\frac{1}{2} \\\\langle v_b-{\\\\mathfrak {s}}(v_b), (v_b-{{\\\\mathfrak {s}}(v_b)){\\\\beta }}\\\\cdot {\\\\bf n}\\\\rangle _{\\\\partial T}\\\\\\\\& \\\\ + \\\\frac{1}{2} \\\\langle v_b, v_b {\\\\beta }\\\\cdot {\\\\bf n}\\\\rangle _{\\\\partial T}- \\\\frac{1}{2} ((\\\\nabla \\\\cdot {\\\\beta }) {\\\\mathfrak {s}}(v_b), {\\\\mathfrak {s}}(v_b))_T,\\\\end{split}$ which leads to the identify (REF ).\",\n \"In the finite element space $W_h({\\\\mathcal {T}}_h)$ , we introduce the following semi-norm: ${|\\\\!|\\\\!|} v_b{|\\\\!|\\\\!|}^2: = \\\\sum _{T\\\\in {\\\\mathcal {T}}_h} \\\\left( \\\\kappa S_T(v_b, v_b) + a_T(v_b, v_b) \\\\right)$ We claim that ${|\\\\!|\\\\!|}\\\\cdot {|\\\\!|\\\\!|}$ defines a norm in the closed subspace $W_h^0({\\\\mathcal {T}}_h)$ .\",\n \"It suffices to show that $v_b \\\\equiv 0$ for any $v_b\\\\in W_h^0({\\\\mathcal {T}}_h)$ satisfying ${|\\\\!|\\\\!|} v_b {|\\\\!|\\\\!|} =0$ .\",\n \"In fact, if ${|\\\\!|\\\\!|} v_b {|\\\\!|\\\\!|} =0$ , then from (REF ) we have $\\\\kappa \\\\sum _{T}S_T(v_b,v_b)+\\\\sum _{T}(\\\\alpha \\\\nabla _w v_b,\\\\nabla _w v_b)_T =0.$ It follows that on each element $T\\\\in {\\\\mathcal {T}}_h$ $\\\\nabla _w v_b=0,\\\\quad (v_b - {\\\\mathfrak {s}}(v_b))(M_i)=0$ for $i=1,\\\\ldots , N$ .\",\n \"Thus, $\\\\displaystyle \\\\nabla {\\\\mathfrak {s}}(v_b)=\\\\frac{1}{|T|}\\\\sum _{i=1}^N {\\\\mathfrak {s}}(v_b)(M_i)|e_i|\\\\mathbf {n}_i=\\\\frac{1}{|T|}\\\\sum _{i=1}^N v_{b,i}|e_i|\\\\mathbf {n}_i=\\\\nabla _w v_b=0,$ so that ${\\\\mathfrak {s}}(v_b)$ has constant value on each element $T\\\\in {\\\\mathcal {T}}_h$ .\",\n \"By using (REF ) we see that $v_b={\\\\mathfrak {s}}(v_b)=const$ on each edge, which, together with the fact that $v_b=0$ on $\\\\partial \\\\Omega $ , leads to $v_b\\\\equiv 0$ in $\\\\Omega $ .\",\n \"For the model problem (REF ), assume that ${\\\\beta }\\\\in W^{1,\\\\infty }(\\\\Omega )$ and the condition (REF ) is satisfied.\",\n \"Then, the bilinear form $\\\\kappa S(\\\\cdot ,\\\\cdot )+{\\\\mathcal {B}}(\\\\cdot ,\\\\cdot )$ is bounded and coercive in the finite element space $W_h^0({\\\\mathcal {T}}_h)$ ; i.e., there exist constants $M$ and $\\\\Lambda >0$ such that $|\\\\kappa S(v_b,w_b)+{\\\\mathcal {B}}(v_b, w_b)| & \\\\le & M {|\\\\!|\\\\!|} v_b{|\\\\!|\\\\!|} {|\\\\!|\\\\!|} w_b{|\\\\!|\\\\!|} \\\\qquad \\\\forall v_b, w_b \\\\in W_h^0({\\\\mathcal {T}}_h),\\\\\\\\\\\\kappa S(v_b,v_b)+{\\\\mathcal {B}}(v_b, v_b) & \\\\ge & \\\\Lambda {|\\\\!|\\\\!|} v_b{|\\\\!|\\\\!|}^2\\\\qquad \\\\forall v_b\\\\in W_h^0({\\\\mathcal {T}}_h),$ provided that the meshsize $h$ of ${\\\\mathcal {T}}_h$ is sufficiently small.\",\n \"Recall that for any $v_b\\\\in W_h^0({\\\\mathcal {T}}_h)$ we have $\\\\begin{split}{\\\\mathcal {B}}(v_b, w_b) & \\\\ = \\\\sum _{T\\\\in {\\\\mathcal {T}}_h} \\\\left(a_T(v_b,w_b)+ b_T(v_b, w_b) + c_T(v_b,w_b)\\\\right),\\\\\\\\S(v_b,w_b) & \\\\ = \\\\sum _{T\\\\in {\\\\mathcal {T}}_h} S_T(v_b,w_b).\\\\end{split}$ The boundedness estimate (REF ) is then straightforward from the usual Cauchy-Schwarz and the inequality (REF ).\",\n \"We shall focus on the derivation of the coercivity inequality () in the rest of the proof.\",\n \"In comparison with (REF ), the key to the coercivity inequality () is to derive an estimate of the following type: $\\\\sum _{T\\\\in {\\\\mathcal {T}}_h} \\\\left(b_T(v_b, v_b) + c_T(v_b,v_b)\\\\right)\\\\ge \\\\eta - \\\\varepsilon (h) {|\\\\!|\\\\!|} v_b{|\\\\!|\\\\!|}^2,$ where $\\\\eta \\\\ge 0$ and $\\\\varepsilon (h)$ is a parameter satisfying $\\\\varepsilon (h) \\\\rightarrow 0$ as $h\\\\rightarrow 0$ .\",\n \"If (REF ) indeed holds true, then we have from (REF ) that $\\\\begin{split}\\\\kappa S(v_b,v_b)+{\\\\mathcal {B}}(v_b, v_b) \\\\ge & {|\\\\!|\\\\!|} v_b {|\\\\!|\\\\!|}^2 + \\\\eta - \\\\varepsilon (h) {|\\\\!|\\\\!|} v_b{|\\\\!|\\\\!|}^2\\\\\\\\\\\\ge & (1- \\\\varepsilon (h)) {|\\\\!|\\\\!|} v_b{|\\\\!|\\\\!|}^2,\\\\end{split}$ which implies the coercivity () for sufficiently small $h$ .\",\n \"It remains to derive the estimate (REF ).\",\n \"To this end, we sum up the identify in Lemma to obtain $\\\\begin{split}\\\\sum _{T\\\\in {\\\\mathcal {T}}_h} b_T(v_b,v_b)= & -\\\\frac{1}{2} \\\\sum _{T\\\\in {\\\\mathcal {T}}_h} ( \\\\nabla \\\\cdot {\\\\beta }{\\\\mathfrak {s}}(v_b), {\\\\mathfrak {s}}(v_b))_T \\\\\\\\& - \\\\frac{1}{2} \\\\sum _{T\\\\in {\\\\mathcal {T}}_h}\\\\langle v_b-{\\\\mathfrak {s}}(v_b), (v_b-{{\\\\mathfrak {s}}(v_b)){\\\\beta }}\\\\cdot {\\\\bf n}\\\\rangle _{\\\\partial T}\\\\\\\\& + \\\\sum _{T\\\\in {\\\\mathcal {T}}_h}\\\\langle v_b-{\\\\mathfrak {s}}(v_b), \\\\overline{{\\\\mathfrak {s}}(v_b){\\\\beta }}\\\\cdot {\\\\bf n}- {{\\\\mathfrak {s}}(v_b){\\\\beta }}\\\\cdot {\\\\bf n}\\\\rangle _{\\\\partial T},\\\\end{split}$ where we have used the fact that $\\\\sum _{T\\\\in {\\\\mathcal {T}}_h} \\\\langle v_b, v_b {\\\\beta }\\\\cdot {\\\\bf n}\\\\rangle _{\\\\partial T}=0$ .\",\n \"Thus, $\\\\begin{split}\\\\sum _{T\\\\in {\\\\mathcal {T}}_h} (b_T(v_b,v_b) + c_T(v_b,v_b))= & \\\\sum _{T\\\\in {\\\\mathcal {T}}_h} ( (c-\\\\frac{1}{2} \\\\nabla \\\\cdot {\\\\beta }) {\\\\mathfrak {s}}(v_b), {\\\\mathfrak {s}}(v_b))_T \\\\\\\\& - \\\\frac{1}{2} \\\\sum _{T\\\\in {\\\\mathcal {T}}_h}\\\\langle v_b-{\\\\mathfrak {s}}(v_b), (v_b-{{\\\\mathfrak {s}}(v_b)){\\\\beta }}\\\\cdot {\\\\bf n}\\\\rangle _{\\\\partial T}\\\\\\\\& + \\\\sum _{T\\\\in {\\\\mathcal {T}}_h}\\\\langle v_b-{\\\\mathfrak {s}}(v_b), \\\\overline{{\\\\mathfrak {s}}(v_b){\\\\beta }}\\\\cdot {\\\\bf n}- {{\\\\mathfrak {s}}(v_b){\\\\beta }}\\\\cdot {\\\\bf n}\\\\rangle _{\\\\partial T}.\\\\end{split}$ Next, from () we have $\\\\begin{split}\\\\left|\\\\sum _{T\\\\in {\\\\mathcal {T}}_h}\\\\langle v_b-{\\\\mathfrak {s}}(v_b), (v_b-{{\\\\mathfrak {s}}(v_b)){\\\\beta }}\\\\cdot {\\\\bf n}\\\\rangle _{\\\\partial T}\\\\right| & \\\\le C \\\\left( h\\\\Vert \\\\nabla _w v_b\\\\Vert ^2_T + \\\\Vert v_b - Q_b {\\\\mathfrak {s}}(v_b)\\\\Vert _{\\\\partial T}^2\\\\right) \\\\\\\\& \\\\le C h \\\\sum _{T\\\\in {\\\\mathcal {T}}_h} \\\\left( a_T(v_b,v_b) + S_T(v_b,v_b)\\\\right)\\\\\\\\&\\\\le C h {|\\\\!|\\\\!|} v_b{|\\\\!|\\\\!|}^2.\\\\end{split}$ As to the last term in (REF ), we have $\\\\begin{split}& \\\\left| \\\\sum _{T\\\\in {\\\\mathcal {T}}_h}\\\\langle v_b-{\\\\mathfrak {s}}(v_b), \\\\overline{{\\\\mathfrak {s}}(v_b){\\\\beta }}\\\\cdot {\\\\bf n}- {{\\\\mathfrak {s}}(v_b){\\\\beta }}\\\\cdot {\\\\bf n}\\\\rangle _{\\\\partial T}\\\\right| \\\\\\\\\\\\le & \\\\sum _{T\\\\in {\\\\mathcal {T}}_h} \\\\Vert v_b-{\\\\mathfrak {s}}(v_b)\\\\Vert _{\\\\partial T}\\\\Vert \\\\overline{{\\\\mathfrak {s}}(v_b){\\\\beta }} - {{\\\\mathfrak {s}}(v_b){\\\\beta }}\\\\Vert _{\\\\partial T}\\\\\\\\\\\\le & C h^{\\\\frac{1}{2}} \\\\sum _{T\\\\in {\\\\mathcal {T}}_h} \\\\Vert v_b-{\\\\mathfrak {s}}(v_b)\\\\Vert _{\\\\partial T}\\\\left(\\\\Vert {\\\\mathfrak {s}}(v_b)\\\\Vert _T + \\\\Vert \\\\nabla {\\\\mathfrak {s}}(v_b)\\\\Vert _T\\\\right)\\\\\\\\\\\\le & C h \\\\left(\\\\sum _{T\\\\in {\\\\mathcal {T}}_h} h^{-1}\\\\Vert v_b-{\\\\mathfrak {s}}(v_b)\\\\Vert _{\\\\partial T}^2\\\\right)^{\\\\frac{1}{2}} \\\\left(\\\\sum _{T\\\\in {\\\\mathcal {T}}_h} \\\\left(\\\\Vert {\\\\mathfrak {s}}(v_b)\\\\Vert _T^2 + \\\\Vert \\\\nabla {\\\\mathfrak {s}}(v_b)\\\\Vert _T^2\\\\right)\\\\right)^{\\\\frac{1}{2}}\\\\end{split}$ Combining the estimates (REF ), (), and (REF ) with (REF ) yields $\\\\left| \\\\sum _{T\\\\in {\\\\mathcal {T}}_h}\\\\langle v_b-{\\\\mathfrak {s}}(v_b), \\\\overline{{\\\\mathfrak {s}}(v_b){\\\\beta }}\\\\cdot {\\\\bf n}- {{\\\\mathfrak {s}}(v_b){\\\\beta }}\\\\cdot {\\\\bf n}\\\\rangle _{\\\\partial T}\\\\right| \\\\le C h {|\\\\!|\\\\!|} v_b{|\\\\!|\\\\!|}^2.$ Now by substituting (REF ) and (REF ) into (REF ) we obtain the inequality (REF ) with $\\\\eta = ( (c-\\\\frac{1}{2} \\\\nabla \\\\cdot {\\\\beta }) {\\\\mathfrak {s}}(v_b), {\\\\mathfrak {s}}(v_b))\\\\ge 0$ and $\\\\varepsilon (h) = Ch$ .\",\n \"This completes the proof of the lemma.\",\n \"The following is a direct application of Lemma .\",\n \"Under the assumptions of Lemma , there exists a small, but fixed number $h_0>0$ , such that the numerical scheme (REF ) has one and only one solution $u_b\\\\in W_h({\\\\mathcal {T}}_h)$ for sufficiently fine finite element partitions ${\\\\mathcal {T}}_h$ satisfying $h\\\\le h_0$ .\",\n \"It suffices to show that the homogeneous problem has only the trivial solution.\",\n \"To this end, let $u_b \\\\in W_h^0({\\\\mathcal {T}}_h)$ , be the solution of scheme (REF ) with homogeneous data $f=0$ and $g=0$ .\",\n \"By taking $v_b = u_b$ in (REF ) we obtain $\\\\kappa S(u_b,u_b)+{\\\\mathcal {B}}(u_b,u_b)=0,$ which, from the coercivity inequality (), gives $\\\\Lambda {|\\\\!|\\\\!|} u_b{|\\\\!|\\\\!|}^2 \\\\le \\\\kappa S(u_b,u_b) + {\\\\mathcal {B}}(u_b,u_b)=0$ , and hence $u_b\\\\equiv 0$ for sufficiently small $h$ .\"\n ],\n [\n \"Error Estimates in $H^1$\",\n \"Let $u$ be the exact solution of the model problem (REF )-() and $u_b \\\\in W_h^0({\\\\mathcal {T}}_h)$ be the numerical approximation arising from the SWG scheme (REF ).\",\n \"Let $Q_b u$ be the $L^2$ projection of $u$ in the space $W_h^0({\\\\mathcal {T}}_h)$ .\",\n \"The error function refers to the difference between the $L^2$ projection and the SWG approximation: $e_b := Q_b u - u_b,$ The goal of this section is to establish an estimate for the error function $e_b$ in a discrete Sobolev norm.\",\n \"Let us first state an error equation which plays an important role in the convergence analysis of the SWG scheme.\",\n \"Assume that the coefficient $\\\\alpha $ of the model problem (REF )-() has piecewise constant values with respect to the finite element partition ${\\\\mathcal {T}}_h$ .\",\n \"Then the following equation holds true $\\\\kappa S(e_b, v_b) + {\\\\mathcal {B}}(e_b,v_b)= \\\\ell _u(v_b) \\\\quad \\\\forall v_b \\\\in W_h^0({\\\\mathcal {T}}_h),$ where $\\\\ell _u(\\\\cdot )$ is a linear functional given by $\\\\begin{split}\\\\ell _u(v_b):=&\\\\sum _{T\\\\in {\\\\mathcal {T}}_h} \\\\langle \\\\alpha \\\\frac{\\\\partial u}{\\\\partial \\\\mathbf {n}}-\\\\alpha Q_0(\\\\nabla u)\\\\cdot \\\\mathbf {n},{\\\\mathfrak {s}}(v_b)-v_b\\\\rangle _{\\\\partial T}+\\\\kappa S(Q_bu,v_b)\\\\\\\\& + ((Q_0-I) \\\\nabla u, {\\\\mathfrak {s}}(v_b){\\\\beta }) + (c({\\\\mathfrak {s}}(Q_bu)-u), {\\\\mathfrak {s}}(v_b)),\\\\end{split}$ where $Q_0(\\\\nabla u)$ is the $L^2$ projection of $\\\\nabla u$ in the space $[P_0({\\\\mathcal {T}}_h)]^2$ , and $\\\\mathbf {n}$ is the outward normal vector on ${\\\\partial T}$ .\",\n \"We first consider the weak gradient of $Q_b u$ , for any constant vector $\\\\mathbf {\\\\phi }$ , we have $(\\\\nabla _w Q_b u,\\\\mathbf {\\\\phi })_{T}&=& \\\\langle Q_b u, \\\\mathbf {\\\\phi }\\\\cdot \\\\mathbf {n} \\\\rangle _{\\\\partial T}= \\\\langle u, \\\\mathbf {\\\\phi }\\\\cdot \\\\mathbf {n} \\\\rangle _{\\\\partial T}\\\\\\\\&=& (\\\\nabla u, \\\\mathbf {\\\\phi })_{T}= (Q_0(\\\\nabla u), \\\\mathbf {\\\\phi })_{T},$ which implies $\\\\nabla _w Q_b u \\\\equiv Q_0(\\\\nabla u)$ .\",\n \"Thus, for any $v_b \\\\in W_h^0(T_h)$ , we have $\\\\begin{split}&(\\\\alpha \\\\nabla _w Q_b u,\\\\nabla _w v_b)= \\\\sum _{T}(\\\\alpha Q_0(\\\\nabla u),\\\\nabla _w v_b)_T \\\\\\\\= & \\\\sum _{T} \\\\langle \\\\alpha Q_0(\\\\nabla u)\\\\cdot \\\\mathbf {n},v_b\\\\rangle _{\\\\partial T}\\\\\\\\=& \\\\sum _{T} \\\\langle \\\\alpha Q_0(\\\\nabla u)\\\\cdot \\\\mathbf {n},v_b\\\\rangle _{\\\\partial T}-\\\\langle \\\\alpha Q_0(\\\\nabla u)\\\\cdot \\\\mathbf {n},{\\\\mathfrak {s}}(v_b)\\\\rangle _{\\\\partial T} + \\\\langle \\\\alpha Q_0(\\\\nabla u)\\\\cdot \\\\mathbf {n},{\\\\mathfrak {s}}(v_b)\\\\rangle _{\\\\partial T}\\\\\\\\=& \\\\sum _{T} \\\\langle \\\\alpha Q_0(\\\\nabla u)\\\\cdot \\\\mathbf {n},v_b-{\\\\mathfrak {s}}(v_b)\\\\rangle _{\\\\partial T} +(\\\\alpha \\\\nabla u,\\\\nabla {\\\\mathfrak {s}}(v_b))_{T} \\\\\\\\=& \\\\sum _{T}\\\\langle \\\\alpha Q_0(\\\\nabla u)\\\\cdot \\\\mathbf {n},v_b-{\\\\mathfrak {s}}(v_b)\\\\rangle _{\\\\partial T} +(-\\\\nabla \\\\cdot (\\\\alpha \\\\nabla u) ,{\\\\mathfrak {s}}(v_b))_{T} + \\\\langle \\\\alpha \\\\frac{\\\\partial u}{\\\\partial \\\\mathbf {n}},{\\\\mathfrak {s}}(v_b)\\\\rangle _{\\\\partial T}\\\\\\\\=& (-\\\\nabla \\\\cdot (\\\\alpha \\\\nabla u) ,{\\\\mathfrak {s}}(v_b))+ \\\\sum _{T}\\\\langle \\\\alpha \\\\frac{\\\\partial u}{\\\\partial \\\\mathbf {n}}-\\\\alpha Q_0(\\\\nabla u)\\\\cdot \\\\mathbf {n},{\\\\mathfrak {s}}(v_b)-v_b\\\\rangle _{\\\\partial T}.\\\\end{split}$ Next, from $\\\\nabla _w (Q_bu) = Q_0(\\\\nabla u)$ , we have $\\\\begin{split}\\\\sum _T ({\\\\beta }\\\\cdot \\\\nabla _w (Q_bu), {\\\\mathfrak {s}}(v_b))_T = & \\\\sum _T ({\\\\beta }\\\\cdot (Q_0 \\\\nabla u), {\\\\mathfrak {s}}(v_b))_T \\\\\\\\= & \\\\sum _T ({\\\\beta }\\\\cdot \\\\nabla u, {\\\\mathfrak {s}}(v_b))_T + \\\\sum _T ((Q_0-I) \\\\nabla u, {\\\\mathfrak {s}}(v_b){\\\\beta })_T,\\\\end{split}$ and $\\\\sum _T (c{\\\\mathfrak {s}}(Q_bu), {\\\\mathfrak {s}}(v_b))_T = (cu, {\\\\mathfrak {s}}(v_b)) + (c({\\\\mathfrak {s}}(Q_bu)-u), {\\\\mathfrak {s}}(v_b)).$ The sum of (REF ), (REF ), and (REF ) gives rise to $\\\\begin{split}{\\\\mathcal {B}}(Q_b u, v_b) = &\\\\ (f, {\\\\mathfrak {s}}(v_b)) + \\\\sum _{T}\\\\langle \\\\alpha \\\\frac{\\\\partial u}{\\\\partial \\\\mathbf {n}}-\\\\alpha Q_0(\\\\nabla u)\\\\cdot \\\\mathbf {n},{\\\\mathfrak {s}}(v_b)-v_b\\\\rangle _{\\\\partial T} \\\\\\\\& \\\\ + \\\\sum _T ((Q_0-I) \\\\nabla u, {\\\\mathfrak {s}}(v_b){\\\\beta })_T + (c({\\\\mathfrak {s}}(Q_bu)-u), {\\\\mathfrak {s}}(v_b)),\\\\end{split}$ which, combined with $ (f, {\\\\mathfrak {s}}(v_b) = \\\\kappa S(u_b, v_b) + {\\\\mathcal {B}}(u_b,v_b)$ , leads to $\\\\begin{split}{\\\\mathcal {B}}(Q_b u - u_b, v_b) = &\\\\ \\\\kappa S(u_b,v_b) + \\\\sum _{T}\\\\langle \\\\alpha \\\\frac{\\\\partial u}{\\\\partial \\\\mathbf {n}}-\\\\alpha Q_0(\\\\nabla u)\\\\cdot \\\\mathbf {n},{\\\\mathfrak {s}}(v_b)-v_b\\\\rangle _{\\\\partial T} \\\\\\\\& \\\\ + \\\\sum _T ((Q_0-I) \\\\nabla u, {\\\\mathfrak {s}}(v_b){\\\\beta })_T + (c({\\\\mathfrak {s}}(Q_bu)-u), {\\\\mathfrak {s}}(v_b)),\\\\end{split}$ and $\\\\begin{split}&\\\\kappa S(Q_bu-u_b,v_b)+{\\\\mathcal {B}}(Q_b u - u_b, v_b) \\\\\\\\= &\\\\ \\\\kappa S(Q_bu,v_b) + \\\\sum _{T}\\\\langle \\\\alpha \\\\frac{\\\\partial u}{\\\\partial \\\\mathbf {n}}-\\\\alpha Q_0(\\\\nabla u)\\\\cdot \\\\mathbf {n},{\\\\mathfrak {s}}(v_b)-v_b\\\\rangle _{\\\\partial T} \\\\\\\\& \\\\ + \\\\sum _T ((Q_0-I) \\\\nabla u, {\\\\mathfrak {s}}(v_b){\\\\beta })_T + (c({\\\\mathfrak {s}}(Q_bu)-u), {\\\\mathfrak {s}}(v_b)).\\\\end{split}$ This completes the proof of the lemma.\",\n \"Remark 5.1 It should be pointed out that Lemma can be extended to the case when $\\\\alpha $ is in $L^\\\\infty (\\\\Omega )$ and piecewise smooth with respect to the finite element partition ${\\\\mathcal {T}}_h$ .\",\n \"Detailed analysis can be established by following the approach presented in [35].\",\n \"The following result is concerned with the error estimate for the SWG numerical solutions in a discrete $H^1$ norm.\",\n \"Let $u\\\\in H^2(\\\\Omega )$ be the exact solution of (REF )-() and $u_b\\\\in W_h({\\\\mathcal {T}}_h)$ be the approximate solution arising from the numerical scheme (REF ).\",\n \"Assume ${\\\\beta }\\\\in C^1(\\\\bar{\\\\Omega })$ and that (REF ) is satisfied.\",\n \"Then, the following error estimate holds true $\\\\kappa S(e_b ,e_b) +(\\\\alpha \\\\nabla _w e_b, \\\\nabla _w e_b )\\\\le Ch^2\\\\Vert u\\\\Vert _2^2,$ provided that the meshsize $h$ is sufficiently small.\",\n \"Consequently, we have $\\\\Vert \\\\nabla _w u_b- \\\\nabla u\\\\Vert _0 \\\\le Ch \\\\Vert u\\\\Vert _2, $ The proof is based on the error equation (REF ) through a thorough analysis for the linear functional $\\\\ell _u(\\\\cdot )$ given in (REF ).\",\n \"For the first term on the righ-hand side of (REF ), from the usual Cauchy-Schwarz inequality we have $\\\\begin{split}&\\\\ |\\\\langle \\\\alpha \\\\frac{\\\\partial u}{\\\\partial \\\\mathbf {n}}-\\\\alpha Q_0(\\\\nabla u)\\\\cdot \\\\mathbf {n},{\\\\mathfrak {s}}(v_b)-v_b\\\\rangle _{\\\\partial T}| \\\\\\\\\\\\le & \\\\ \\\\Vert \\\\alpha \\\\frac{\\\\partial u}{\\\\partial \\\\mathbf {n}}-\\\\alpha Q_0(\\\\nabla u)\\\\cdot \\\\mathbf {n}\\\\Vert _{0,{\\\\partial T}} \\\\Vert {\\\\mathfrak {s}}(v_b)-v_b\\\\Vert _{0,{\\\\partial T}}\\\\\\\\\\\\le & \\\\ \\\\Vert \\\\alpha \\\\Vert _{\\\\infty } \\\\Vert \\\\nabla u- Q_0(\\\\nabla u)\\\\Vert _{0,{\\\\partial T}} \\\\Vert {\\\\mathfrak {s}}(v_b)-v_b\\\\Vert _{0,{\\\\partial T}}.\\\\end{split}$ Now using the estimate () in the above inequality and then summing over all the element $T\\\\in {\\\\mathcal {T}}_h$ we arrive at the following: $\\\\begin{split}&\\\\sum _{T\\\\in {\\\\mathcal {T}}_h}|\\\\langle \\\\alpha \\\\frac{\\\\partial u}{\\\\partial \\\\mathbf {n}}-\\\\alpha Q_0(\\\\nabla u)\\\\cdot \\\\mathbf {n},{\\\\mathfrak {s}}(v_b)-v_b\\\\rangle _{\\\\partial T}| \\\\\\\\\\\\le & C \\\\Vert \\\\alpha \\\\Vert _{\\\\infty } \\\\sum _{T\\\\in {\\\\mathcal {T}}_h} \\\\Vert \\\\nabla u- Q_0(\\\\nabla u)\\\\Vert _{\\\\partial T}\\\\left(h\\\\Vert \\\\nabla _w v_b\\\\Vert ^2_T +\\\\Vert v_b-Q_b{\\\\mathfrak {s}}(v_b)\\\\Vert ^2_{\\\\partial T}\\\\right)^{\\\\frac{1}{2}}\\\\\\\\\\\\le & C\\\\Vert \\\\alpha \\\\Vert _{\\\\infty }\\\\left(\\\\Vert \\\\nabla u -Q_0(\\\\nabla u)\\\\Vert ^2_{0} + h^2\\\\Vert \\\\nabla ^2 u \\\\Vert _{0}^2\\\\right)^{\\\\frac{1}{2}}\\\\left( \\\\Vert \\\\nabla _w v_b\\\\Vert ^2+ \\\\kappa S(v_b, v_b)\\\\right)^{\\\\frac{1}{2}}\\\\\\\\\\\\le & C h \\\\Vert u\\\\Vert _2 {|\\\\!|\\\\!|} v_b {|\\\\!|\\\\!|}.\\\\end{split}$ As to the second term on the right hand side of (REF ), we have $\\\\begin{split}|S(Q_bu,v_b)| =&\\\\sum _{T}h^{-1}\\\\langle Q_bu-Q_b{\\\\mathfrak {s}}(Q_bu),v_b-Q_b {\\\\mathfrak {s}}(v_b)\\\\rangle _{\\\\partial T} \\\\\\\\=&\\\\sum _{T}h^{-1}\\\\langle Q_bu,v_b-Q_b {\\\\mathfrak {s}}(v_b)\\\\rangle _{\\\\partial T} \\\\\\\\=&\\\\sum _{T}h^{-1}\\\\langle Q_bu -Q_b(Q_1 u),v_b-Q_b {\\\\mathfrak {s}}(v_b)\\\\rangle _{\\\\partial T}\\\\\\\\=&\\\\sum _{T}h^{-1}\\\\langle u -Q_1 u,v_b-Q_b {\\\\mathfrak {s}}(v_b)\\\\rangle _{\\\\partial T}\\\\\\\\\\\\le &\\\\left(\\\\sum _{T} h^{-1} \\\\int _{\\\\partial T}|u-Q_1u|^2ds \\\\right)^{\\\\frac{1}{2}}S(v_b, v_b)^{\\\\frac{1}{2}}\\\\\\\\\\\\le & C\\\\left(h^{-2}\\\\Vert u-Q_1 u\\\\Vert ^2 + \\\\Vert u-Q_1 u\\\\Vert _1^2 \\\\right)^{\\\\frac{1}{2}}S(v_b, v_b)^{\\\\frac{1}{2}}\\\\\\\\\\\\le & C h \\\\Vert u\\\\Vert _2 {|\\\\!|\\\\!|} v_b{|\\\\!|\\\\!|}.\\\\end{split}$ The third term on the right hand side of (REF ) can be bounded by using the usual error estimate for $L^2$ projections as follows: $\\\\begin{split}|((Q_0-I) \\\\nabla u, {\\\\mathfrak {s}}(v_b){\\\\beta })| = & |((Q_0-I) \\\\nabla u, (Q_0-I) ({\\\\mathfrak {s}}(v_b){\\\\beta }))| \\\\\\\\\\\\le & \\\\Vert (Q_0-I) \\\\nabla u\\\\Vert \\\\ \\\\Vert (Q_0-I) ({\\\\mathfrak {s}}(v_b){\\\\beta })\\\\Vert \\\\\\\\\\\\le & C h^2 \\\\Vert \\\\nabla ^2 u\\\\Vert \\\\left( \\\\Vert \\\\nabla {\\\\mathfrak {s}}(v_b)\\\\Vert + \\\\Vert {\\\\mathfrak {s}}(v_b)\\\\Vert \\\\right) \\\\\\\\\\\\le & C h^2 \\\\Vert \\\\nabla ^2 u\\\\Vert {|\\\\!|\\\\!|} v_b{|\\\\!|\\\\!|},\\\\end{split}$ where we have used the estimates (REF ) and (REF ) in the last line.\",\n \"The last term on the right hand side of (REF ) can be estimated as follows: $\\\\begin{split}|(c({\\\\mathfrak {s}}(Q_bu)-u), {\\\\mathfrak {s}}(v_b))|\\\\le & \\\\Vert c\\\\Vert _\\\\infty \\\\Vert {\\\\mathfrak {s}}(Q_bu)-u\\\\Vert \\\\Vert {\\\\mathfrak {s}}(v_b)\\\\Vert \\\\\\\\\\\\le & C (\\\\Vert {\\\\mathfrak {s}}(Q_bu)-Q_1 u\\\\Vert + \\\\Vert Q_1u - u\\\\Vert ) \\\\Vert {\\\\mathfrak {s}}(v_b)\\\\Vert \\\\\\\\\\\\le & C ( \\\\Vert {\\\\mathfrak {s}}(Q_bu)-{\\\\mathfrak {s}}(Q_1 u)\\\\Vert + \\\\Vert Q_1u - u\\\\Vert ) \\\\Vert {\\\\mathfrak {s}}(v_b)\\\\Vert \\\\\\\\\\\\le & C ( \\\\Vert {\\\\mathfrak {s}}(Q_bu-Q_1 u)\\\\Vert + \\\\Vert Q_1u - u\\\\Vert ) \\\\Vert {\\\\mathfrak {s}}(v_b)\\\\Vert \\\\\\\\\\\\le & C h^2 \\\\Vert u\\\\Vert _2 {|\\\\!|\\\\!|} v_b{|\\\\!|\\\\!|}.\\\\end{split}$ Substituting the estimates (REF )-(REF ) into the error equation (REF ) yields $\\\\kappa S(e_b, v_b) + {\\\\mathcal {B}}(e_b, v_b) \\\\le C h \\\\Vert u\\\\Vert _2 {|\\\\!|\\\\!|} v_b{|\\\\!|\\\\!|},$ which, together with the coercivity (), leads to $\\\\Lambda {|\\\\!|\\\\!|} e_b{|\\\\!|\\\\!|}^2 \\\\le C h \\\\Vert u\\\\Vert _2 {|\\\\!|\\\\!|} e_b{|\\\\!|\\\\!|}.$ The last inequality implies the error estimate (REF ).\",\n \"Finally, from the triangle inequality and the error estimate (REF ), we obtain $\\\\begin{split}\\\\Vert \\\\nabla _w u_b - \\\\nabla u\\\\Vert \\\\le & \\\\Vert \\\\nabla _w (u_b - Q_b u)\\\\Vert + \\\\Vert \\\\nabla _w (Q_b u) - \\\\nabla u\\\\Vert \\\\\\\\= & \\\\Vert \\\\nabla _w e_b\\\\Vert + \\\\Vert Q_0 (\\\\nabla u) - \\\\nabla u\\\\Vert \\\\\\\\\\\\le & C h \\\\Vert u\\\\Vert _2,\\\\end{split}$ which gives rise to (REF ).\",\n \"This completes the proof of the theorem.\"\n ],\n [\n \"Error Estimates in $L^2$\",\n \"We use the usual duality argument to derive an error estimate in $L^2$ for the numerical solutions arising from (REF ).\",\n \"The analysis to be presented is a modified version of those developed in [36], [24], [35].\",\n \"Consider the following auxiliary problem that seeks $\\\\Phi \\\\in H_0^1(\\\\Omega )$ such that $-\\\\nabla \\\\cdot (\\\\alpha \\\\nabla \\\\Phi ) - \\\\nabla \\\\cdot ({\\\\beta }\\\\Phi ) + c\\\\Phi &=&\\\\chi \\\\quad {\\\\rm in}\\\\ \\\\Omega \\\\\\\\\\\\Phi &=&0\\\\quad {\\\\rm on}\\\\ \\\\partial \\\\Omega , $ where $\\\\chi \\\\in L^2(\\\\Omega )$ .\",\n \"Assume that the solution of the problem (REF )-() exists and has the $H^2$ -regularity: $\\\\Vert \\\\Phi \\\\Vert _2 \\\\le C \\\\Vert \\\\chi \\\\Vert ,$ where $C$ is a constant depending only on the domain and the coefficients $\\\\alpha , {\\\\beta }$ , and $c$ .\",\n \"Let $u\\\\in H^2(\\\\Omega )$ be the exact solution of (REF )-() and $u_b\\\\in W_h({\\\\mathcal {T}}_h)$ be the approximate solution arising from the numerical scheme (REF ).\",\n \"Assume ${\\\\beta }\\\\in C^1(\\\\bar{\\\\Omega })$ and the conditions (REF ) and (REF ) are satisfied.\",\n \"Then, the following $L^2$ error estimate holds true $\\\\Vert u - {\\\\mathfrak {s}}(u_b)\\\\Vert \\\\le Ch^2\\\\Vert u\\\\Vert _2,$ provided that the meshsize $h$ is sufficiently small.\",\n \"On each element $T\\\\in {\\\\mathcal {T}}_h$ , we test (REF ) against the linear function ${\\\\mathfrak {s}}(e_b)$ to obtain $(\\\\chi , {\\\\mathfrak {s}}(e_b))_T &=& (\\\\alpha \\\\nabla \\\\Phi , \\\\nabla {\\\\mathfrak {s}}(e_b))_T + ({\\\\beta }\\\\Phi , \\\\nabla {\\\\mathfrak {s}}(e_b))_T + (c\\\\Phi , {\\\\mathfrak {s}}(e_b))_T \\\\\\\\&& - \\\\langle \\\\alpha \\\\nabla \\\\Phi \\\\cdot {\\\\bf n}, {\\\\mathfrak {s}}(e_b)\\\\rangle _{\\\\partial T}-\\\\langle {\\\\beta }\\\\cdot {\\\\bf n}\\\\Phi , {\\\\mathfrak {s}}(e_b)\\\\rangle _{\\\\partial T}\\\\\\\\&=& (\\\\alpha Q_0(\\\\nabla \\\\Phi ), \\\\nabla {\\\\mathfrak {s}}(e_b))_T + (Q_0({\\\\beta }\\\\Phi ), \\\\nabla {\\\\mathfrak {s}}(e_b))_T + (c\\\\Phi , {\\\\mathfrak {s}}(e_b))_T \\\\\\\\&& - \\\\langle \\\\alpha \\\\nabla \\\\Phi \\\\cdot {\\\\bf n}, {\\\\mathfrak {s}}(e_b)\\\\rangle _{\\\\partial T}-\\\\langle {\\\\beta }\\\\cdot {\\\\bf n}\\\\Phi , {\\\\mathfrak {s}}(e_b)\\\\rangle _{\\\\partial T}\\\\\\\\&=& (\\\\alpha Q_0(\\\\nabla \\\\Phi ), \\\\nabla _w e_b)_T + (Q_0({\\\\beta }\\\\Phi ), \\\\nabla _w e_b)_T + (c\\\\Phi , {\\\\mathfrak {s}}(e_b))_T \\\\\\\\&& - \\\\langle \\\\alpha \\\\nabla \\\\Phi \\\\cdot {\\\\bf n}, {\\\\mathfrak {s}}(e_b)\\\\rangle _{\\\\partial T}-\\\\langle {\\\\beta }\\\\cdot {\\\\bf n}\\\\Phi , {\\\\mathfrak {s}}(e_b)\\\\rangle _{\\\\partial T}\\\\\\\\&& - \\\\langle \\\\alpha Q_0(\\\\nabla \\\\Phi )\\\\cdot {\\\\bf n}, e_b-{\\\\mathfrak {s}}(e_b)\\\\rangle _{\\\\partial T}- \\\\langle Q_0({\\\\beta }\\\\Phi )\\\\cdot {\\\\bf n}, e_b-{\\\\mathfrak {s}}(e_b)\\\\rangle _{\\\\partial T}$ By using $Q_0(\\\\nabla \\\\Phi ) = \\\\nabla _w (Q_b\\\\Phi )$ and $(Q_0({\\\\beta }\\\\Phi ), \\\\nabla _w e_b)_T = ({\\\\beta }\\\\cdot \\\\nabla _w e_b, \\\\Phi )_T$ in the above equation, we have from summing over all $T\\\\in {\\\\mathcal {T}}_h$ that $\\\\begin{split}(\\\\chi , {\\\\mathfrak {s}}(e_b))=& (\\\\alpha \\\\nabla _w e_b, \\\\nabla _w (Q_b\\\\Phi )) + ({\\\\beta }\\\\cdot \\\\nabla _w e_b, \\\\Phi ) + (c{\\\\mathfrak {s}}(e_b), \\\\Phi ) \\\\\\\\& - \\\\sum _T \\\\langle \\\\alpha \\\\nabla \\\\Phi \\\\cdot {\\\\bf n}- \\\\alpha Q_0(\\\\nabla \\\\Phi )\\\\cdot {\\\\bf n}, {\\\\mathfrak {s}}(e_b)-e_b\\\\rangle _{\\\\partial T}\\\\\\\\& -\\\\langle {\\\\beta }\\\\Phi \\\\cdot {\\\\bf n}- Q_0({\\\\beta }\\\\Phi )\\\\cdot {\\\\bf n}, {\\\\mathfrak {s}}(e_b)-e_b\\\\rangle _{\\\\partial T}.\\\\end{split}$ The last two terms on the right-hand side of (REF ) can be bounded by $C h \\\\Vert \\\\Phi \\\\Vert _2 {|\\\\!|\\\\!|} e_b{|\\\\!|\\\\!|}$ through the Cauchy-Schwarz inequality.\",\n \"Thus, we have $\\\\begin{split}|(\\\\chi , {\\\\mathfrak {s}}(e_b))|\\\\le & |(\\\\alpha \\\\nabla _w e_b, \\\\nabla _w (Q_b\\\\Phi )) + ({\\\\beta }\\\\cdot \\\\nabla _w e_b, \\\\Phi ) + (c{\\\\mathfrak {s}}(e_b), \\\\Phi )| \\\\\\\\& + Ch\\\\Vert \\\\Phi \\\\Vert _2 {|\\\\!|\\\\!|} e_b{|\\\\!|\\\\!|}\\\\\\\\\\\\le & |(\\\\alpha \\\\nabla _w e_b, \\\\nabla _w (Q_b\\\\Phi )) + ({\\\\beta }\\\\cdot \\\\nabla _w e_b, {\\\\mathfrak {s}}(Q_b\\\\Phi )) + (c{\\\\mathfrak {s}}(e_b), {\\\\mathfrak {s}}(Q_b\\\\Phi ))| \\\\\\\\& + Ch\\\\Vert \\\\Phi \\\\Vert _2 {|\\\\!|\\\\!|} e_b{|\\\\!|\\\\!|},\\\\end{split}$ where have also used $\\\\Vert \\\\Phi -{\\\\mathfrak {s}}(Q_b\\\\Phi )\\\\Vert \\\\le C h^2\\\\Vert \\\\Phi \\\\Vert _2$ .\",\n \"Now, recall that $(\\\\alpha \\\\nabla _w e_b, \\\\nabla _w (Q_b\\\\Phi )) + ({\\\\beta }\\\\cdot \\\\nabla _w e_b, {\\\\mathfrak {s}}(Q_b\\\\Phi )) + (c{\\\\mathfrak {s}}(e_b), {\\\\mathfrak {s}}(Q_b\\\\Phi ))= {\\\\mathcal {B}}(e_b, Q_b\\\\Phi ),$ and from the error equation (REF ), we have $\\\\begin{split}{\\\\mathcal {B}}(e_b, Q_b\\\\Phi ) = & \\\\ell _u(Q_b\\\\Phi ) - \\\\kappa S(e_b, Q_b\\\\Phi )\\\\\\\\=&\\\\sum _{T\\\\in {\\\\mathcal {T}}_h} \\\\langle \\\\alpha \\\\frac{\\\\partial u}{\\\\partial \\\\mathbf {n}}-\\\\alpha Q_0(\\\\nabla u)\\\\cdot \\\\mathbf {n},{\\\\mathfrak {s}}(Q_b\\\\Phi )-Q_b\\\\Phi \\\\rangle _{\\\\partial T}+\\\\kappa S(u_b,Q_b\\\\Phi )\\\\\\\\& + ((Q_0-I) \\\\nabla u, {\\\\mathfrak {s}}(Q_b\\\\Phi ){\\\\beta }) + (c({\\\\mathfrak {s}}(Q_bu)-u), {\\\\mathfrak {s}}(Q_b\\\\Phi )),\\\\end{split}$ The last two terms on the right-hand side of (REF ) have the following estimate: $\\\\left|((Q_0-I) \\\\nabla u, {\\\\mathfrak {s}}(Q_b\\\\Phi ){\\\\beta }) + (c({\\\\mathfrak {s}}(Q_bu)-u), {\\\\mathfrak {s}}(Q_b\\\\Phi ))\\\\right| \\\\le C h^2 \\\\Vert u\\\\Vert _2 \\\\Vert \\\\Phi \\\\Vert _1.$ The second term, $\\\\kappa S(u_b,Q_b\\\\Phi )$ , can be dealt with as follows: $\\\\begin{split}\\\\kappa S(u_b,Q_b\\\\Phi ) = & \\\\kappa h^{-1} \\\\sum _T \\\\langle u_b-Q_b{\\\\mathfrak {s}}(u_b), Q_b\\\\Phi - Q_b{\\\\mathfrak {s}}(Q_b\\\\Phi )\\\\rangle _{\\\\partial T}\\\\\\\\= & \\\\kappa h^{-1} \\\\sum _T \\\\langle u_b-Q_b{\\\\mathfrak {s}}(u_b), \\\\Phi - {\\\\mathfrak {s}}(Q_b\\\\Phi )\\\\rangle _{\\\\partial T}\\\\\\\\\\\\le & \\\\kappa h^{-1} \\\\sum _T \\\\Vert u_b-Q_b{\\\\mathfrak {s}}(u_b)\\\\Vert _{\\\\partial T}\\\\Vert \\\\Phi - {\\\\mathfrak {s}}(Q_b\\\\Phi )\\\\Vert _{\\\\partial T}\\\\\\\\\\\\le & C h ({|\\\\!|\\\\!|} e_b{|\\\\!|\\\\!|} + h\\\\Vert u\\\\Vert _2)\\\\Vert \\\\Phi \\\\Vert _2.\\\\end{split}$ As to the first term, we note from the definition of $Q_b$ and $\\\\Phi |_{\\\\partial \\\\Omega }=0$ that $\\\\begin{split}\\\\sum _{T\\\\in {\\\\mathcal {T}}_h} \\\\langle \\\\alpha \\\\frac{\\\\partial u}{\\\\partial \\\\mathbf {n}}-\\\\alpha Q_0(\\\\nabla u)\\\\cdot \\\\mathbf {n},\\\\Phi -Q_b\\\\Phi \\\\rangle _{\\\\partial T}= \\\\sum _{T\\\\in {\\\\mathcal {T}}_h} \\\\langle \\\\alpha \\\\frac{\\\\partial u}{\\\\partial \\\\mathbf {n}},\\\\Phi -Q_b\\\\Phi \\\\rangle _{\\\\partial T}= 0.\\\\end{split}$ Thus, we have $\\\\begin{split}&\\\\sum _{T\\\\in {\\\\mathcal {T}}_h} \\\\langle \\\\alpha \\\\frac{\\\\partial u}{\\\\partial \\\\mathbf {n}}-\\\\alpha Q_0(\\\\nabla u)\\\\cdot \\\\mathbf {n},{\\\\mathfrak {s}}(Q_b\\\\Phi )-Q_b\\\\Phi \\\\rangle _{\\\\partial T}\\\\\\\\= & \\\\sum _{T\\\\in {\\\\mathcal {T}}_h} \\\\langle \\\\alpha \\\\frac{\\\\partial u}{\\\\partial \\\\mathbf {n}}-\\\\alpha Q_0(\\\\nabla u)\\\\cdot \\\\mathbf {n},{\\\\mathfrak {s}}(Q_b\\\\Phi )-\\\\Phi \\\\rangle _{\\\\partial T}\\\\\\\\\\\\le & C h^2 \\\\Vert u\\\\Vert _2\\\\Vert \\\\Phi \\\\Vert _2.\\\\end{split}$ Substituting (REF ), (REF ), and (REF ) into (REF ) yields the following estimate: $|{\\\\mathcal {B}}(e_b, Q_b\\\\Phi )| \\\\le C(h^2 \\\\Vert u\\\\Vert _2 + h{|\\\\!|\\\\!|} e_b{|\\\\!|\\\\!|}) \\\\Vert \\\\Phi \\\\Vert _2,$ which, together with (REF ), leads to $|(\\\\chi , {\\\\mathfrak {s}}(e_b))| \\\\le C(h^2 \\\\Vert u\\\\Vert _2 + h{|\\\\!|\\\\!|} e_b{|\\\\!|\\\\!|}) \\\\Vert \\\\Phi \\\\Vert _2 \\\\le C(h^2 \\\\Vert u\\\\Vert _2 + h{|\\\\!|\\\\!|} e_b{|\\\\!|\\\\!|}) \\\\Vert \\\\chi \\\\Vert ,$ where the regularity assumption (REF ) has been employed in the last inequality.\",\n \"Next, from (REF ) and the $H^1$ error estimate (REF ) in Theorem , we have $|(\\\\chi , {\\\\mathfrak {s}}(e_b))| \\\\le Ch^2 \\\\Vert u\\\\Vert _2 \\\\Vert \\\\chi \\\\Vert ,$ which leads to $\\\\Vert {\\\\mathfrak {s}}(e_b\\\\Vert \\\\le C h^2 \\\\Vert u\\\\Vert _2.$ Finally, we arrive at $\\\\Vert u-{\\\\mathfrak {s}}(u_b)\\\\Vert \\\\le \\\\Vert u-{\\\\mathfrak {s}}(Q_bu)\\\\Vert + \\\\Vert {\\\\mathfrak {s}}(e_b)\\\\Vert \\\\le C h^2\\\\Vert u\\\\Vert _2,$ which completes the proof of the theorem.\"\n ],\n [\n \"Numerical Experiments\",\n \"The goal of this section is to numerically verify the error estimates developed in the previous sections for the numerical scheme (REF ).\",\n \"The following metrics are employed to measure the magnitude of the error function: $&&\\\\text{Discrete $L^2$-norm: }\\\\\\\\&&\\\\Vert u_b-u \\\\Vert _{0}=h\\\\left(\\\\sum _{i=1}^{n+1}\\\\sum _{j=1}^{n}|u_{i-\\\\frac{1}{2},j}-u(x_{i-\\\\frac{1}{2}},y_j)|^2 + \\\\sum _{i=1}^{n}\\\\sum _{j=1}^{n+1}|u_{i,j-\\\\frac{1}{2}}-u(x_{i},y_{j-\\\\frac{1}{2}})|^2\\\\right)^{1/2},\\\\\\\\&&\\\\text{Discrete $H^1$-norm: }\\\\\\\\&&\\\\Vert u_b-u\\\\Vert _1= h \\\\left(\\\\sum _{i=1}^{n}\\\\sum _{j=1}^{n} \\\\left|\\\\frac{u_{i+\\\\frac{1}{2},j}-u_{i-\\\\frac{1}{2},j}}{h} - \\\\frac{\\\\partial u}{\\\\partial x} (x_{i},y_{j})\\\\right|^2 \\\\right.\",\n \"\\\\\\\\&&\\\\qquad \\\\qquad \\\\qquad \\\\left.\",\n \"+ \\\\sum _{i=1}^{n}\\\\sum _{j=1}^{n} \\\\left|\\\\frac{u_{i,j+\\\\frac{1}{2}}-u_{i,j-\\\\frac{1}{2}}}{h} - \\\\frac{\\\\partial u}{\\\\partial y} (x_{i},y_{j})\\\\right|^2 \\\\right)^{1/2} ,\\\\\\\\$ Our numerical experiments are conducted for the model problem (REF )-() on polygonal domains.\",\n \"The following set of test cases are considered: $\\\\left\\\\lbrace \\\\begin{split}&u=xy, \\\\\\\\&\\\\alpha =\\\\begin{bmatrix}1 &0\\\\\\\\0 & 1\\\\end{bmatrix},\\\\quad \\\\mathbf {\\\\beta }=\\\\begin{bmatrix}1\\\\\\\\1\\\\end{bmatrix},\\\\quad c=1;\\\\end{split}\\\\right.$ $\\\\left\\\\lbrace \\\\begin{split}&u= 3x^2+2xy, \\\\\\\\&\\\\alpha =\\\\begin{bmatrix}2 &0\\\\\\\\0 & 1\\\\end{bmatrix},\\\\quad \\\\mathbf {\\\\beta }=\\\\begin{bmatrix}1\\\\\\\\1\\\\end{bmatrix},\\\\quad c=1;\\\\end{split}\\\\right.$ $\\\\left\\\\lbrace \\\\begin{split}&u=\\\\sin (\\\\pi x)\\\\sin (\\\\pi y) + x^2-y^2, \\\\\\\\&\\\\alpha =\\\\begin{bmatrix}1 &0\\\\\\\\0 & 1\\\\end{bmatrix},\\\\quad \\\\mathbf {\\\\beta }=\\\\begin{bmatrix}1\\\\\\\\2\\\\end{bmatrix},\\\\quad c=1;\\\\end{split}\\\\right.$ $\\\\left\\\\lbrace \\\\begin{split}&u=\\\\sin (\\\\pi x)\\\\sin (\\\\pi y), \\\\\\\\&\\\\alpha =\\\\begin{bmatrix}xy+1 &0\\\\\\\\0 & 3xy\\\\end{bmatrix},\\\\quad \\\\mathbf {\\\\beta }=\\\\begin{bmatrix}x^3y+xy+1\\\\\\\\3x^2y+xy+2\\\\end{bmatrix},\\\\quad c=x^4y^2+xy+1;\\\\end{split}\\\\right.$ The right-hand side function $f$ and the Dirichlet boundary data $g$ are chosen to match the exact solution $u=u(x,y)$ for each test case.\",\n \"Table: Error and convergence performance of the SWG scheme () with κ=4.0\\\\kappa =4.0 and uniform square partitions on the unit square domain Ω=(0,1) 2 \\\\Omega =(0,1)^2.Table REF shows the performance of the SWG scheme for each of the above test problems with the stabilizer parameter $\\\\kappa =4$ on uniform square partitions.\",\n \"The results indicate that the numerical approximation is in the machine accuracy for the test problem (REF ) where the exact solution is a bilinear function.\",\n \"For the other three test problems, the numerical solutions have the optimal rate of convergence $r=2$ in the discrete $L^2$ norm and a superconvergence of order ${\\\\mathcal {O}}(h^2)$ in the discrete $H^1$ norm.\",\n \"The numerical results are consistent with the theoretical prediction in the discrete $L^2$ norm, but they outperform the theory in the discrete $H^1$ norm.\",\n \"It should be pointed out that the superconvergence theory in [18] was developed for the diffusion equation only; but a slight modification of the analysis there will yield a superconvergence of order ${\\\\mathcal {O}}(h^2)$ for the SWG solutions of the full convection-diffusion equation (REF )-().\",\n \"Table: Error and convergence performance of the SWG scheme () for the test case () with different values of κ\\\\kappa on uniform square partitions for Ω=(0,1) 2 \\\\Omega =(0,1)^2.\"\n ],\n [\n \"On the influence of the stabilizer parameter\",\n \"The goal of this subsection is to test the influence of the stabilizer parameter $\\\\kappa $ on the numerical solutions.\",\n \"This part of the numerical experiment considers only the test cases (REF ) and (REF ) with the following six values of $\\\\kappa =0.01, \\\\ 0.1, \\\\ 1.0, \\\\ 4.0, \\\\ 6.0, \\\\ 20.0$ .\",\n \"The case of $\\\\kappa =0$ is not a viable choice , as it was not covered in the convergence theory.\",\n \"In fact, our computation does not suggest any convergence of the scheme when $\\\\kappa =0$ .\",\n \"Tables REF -REF illustrate the numerical performance of the SWG scheme with different values of the stabilizer parameter $\\\\kappa $ .\",\n \"Note that, for both test cases, the rate of convergence deteriorates as $\\\\kappa $ gets small (e.g.\",\n \"$\\\\kappa =0.01$ ), particulary on coarse finite element partitions, but the rate of convergence begins to improve when the meshsize $h$ gets small.\",\n \"Optimal rate of convergence and the supercovergence of order ${\\\\mathcal {O}}(h^2)$ are clearly shown in the tables when $\\\\kappa $ is away from 0 (e.g., $\\\\kappa \\\\ge 0.1$ ).\",\n \"The stability and accuracy of the SWG scheme is insensitive to the value of $\\\\kappa $ as long as it stays away from 0.\",\n \"Table: Error and convergence performance of the SWG scheme () for the test case () with different values of κ\\\\kappa on uniform square partitions for Ω=(0,1) 2 \\\\Omega =(0,1)^2.\"\n ],\n [\n \"SWG with general polygonal partitions\",\n \"The SWG scheme was applied to the test problem (REF ) with general polygonal partitions.\",\n \"Table REF shows the error and convergence performance of the scheme on four types of polygonal partitions.\",\n \"The stabilization parameter was set as $\\\\kappa =4$ in all these tests.\",\n \"Optimal order of convergence in the discrete $L^2$ norm can be observed for each polygonal partition, but the superconvergence in the discrete $H^1$ norm was only seen for rectangular partitions.\",\n \"The table shows a numerical rate of convergence of $r=1$ in the discrete $H^1$ norm for three other type of partitions.\",\n \"The result is clearly in consistency with the error estimate developed in Section .\",\n \"Fig.\",\n \"REF illustrates the contour plots of the numerical solutions on different type of polygonal partitions.\",\n \"It also shows the shape of the polygonal elements in our computation.\",\n \"Table: Error and convergence performance of the SWG scheme () for the test problem () on general polygonal partitions for Ω=(0,1) 2 \\\\Omega =(0,1)^2, with κ=4\\\\kappa = 4.Figure: Comparison of numerical solutions obtained from SWG and the exact solution for the test problem () on various polygonal partitions of h=1/8h=1/8 and κ=1\\\\kappa =1.\"\n ],\n [\n \"Numerical results on a non-convex domain\",\n \"The SWG scheme with the stabilization parameter $\\\\kappa =4$ was applied to the test problem (REF ) on the L-shaped domain $\\\\Omega :=(-1,1)\\\\times (-1,1)/ (0,1)\\\\times (-1,0)$ partitioned into triangles or rectangles.\",\n \"The corresponding numerical results are summarized in Table REF , which shows a convergence of order ${\\\\mathcal {O}}(h^2)$ in the $L^2$ norm for both the triangular and rectangular partitions.\",\n \"A superconvergence of order ${\\\\mathcal {O}}(h^2)$ was observed in the discrete $H^1$ norm on rectangular partitions, while the optimal order of convergence with $r=1$ is confirmed numerically on triangular partitions.\",\n \"It should be pointed out that the $H^2$ -regularity assumption (REF ) is not valid for non-convex polygonal domains so that the optimal order of error estimate (REF ) is not known theoretically on the L-shaped domain.\",\n \"The numerical results therefore outperform the theory in the usual $L^2$ norm.\",\n \"Table: Error and convergence performance of the SWG scheme () for test case () on Lshape domain, κ=4\\\\kappa = 4.Figure: Comparison of numerical solution obtained from SWG and the exact solution for test case () on L-shaped domain with h=1/8h=1/8.\"\n ]\n]","isConversational":false},"id":{"kind":"string","value":"1808.08667"}}},{"rowIdx":194,"cells":{"text":{"kind":"list like","value":[["Symmetries and Mass Degeneracies in the Scalar Sector"],["Abstract We explore some aspects of models with two and three SU(2) scalar doublets that lead to mass degeneracies among some of the physical scalars.","In Higgs sectors with two scalar doublets, the exact degeneracy of scalar masses, without an artificial fine-tuning of the scalar potential parameters, is possible only in the case of the inert doublet model (IDM), where the scalar potential respects a global U(1) symmetry that is not broken by the vacuum.","In the case of three doublets, we introduce and analyze the replicated inert doublet model, which possesses two inert doublets of scalars.","We then generalize this model to obtain a scalar potential, first proposed by Ivanov and Silva, with a CP4 symmetry that guarantees the existence of pairwise degenerate scalar states among two pairs of neutral scalars and two pairs of charged scalars.","Here, CP4 is a generalized CP symmetry with the property that $({\\rm CP}4)^n$ is the identity operator only for integer $n$ values that are multiples of 4.","The form of the CP4-symmetric scalar potential is simplest when expressed in the Higgs basis, where the neutral scalar field vacuum expectation value resides entirely in one of the scalar doublet fields.","The symmetries of the model permit a term in the scalar potential with a complex coefficient that cannot be removed by any redefinition of the scalar fields within the class of Higgs bases (in which case, we say that no real Higgs basis exists).","A striking feature of the CP4-symmetric model is that it preserves CP even in the absence of a real Higgs basis, as illustrated by the cancellation of the contributions to the CP violating form factors of the effective ZZZ and ZWW vertices."],["Introduction","After the initial discovery of the Higgs boson in 2012[1], [2], certain anomalies in the Higgs data (which have since disappeared) motivated the exploration of the possibility that the 125 GeV Higgs signal was comprised of two nearly mass-degenerate scalar states[3], [4], [5], [6], [7], [8], [9], [10], [11].","Although the present Higgs data is consistent with the Standard Model[12], [13], [14], one cannot yet rule out the presence of a mass degenerate scalar state at 125 GeV[15].","In this work, we consider the implications of a mass degeneracy among two (or more) scalar states of an extended Higgs sector.","Such a mass degeneracy can be either accidental or the result of a symmetry.","A trivial example of such a phenomenon arises in any doublet extended Higgs model.","All such models possess a mass degenerate state, namely the charged Higgs boson, $H^\\pm $ .","Indeed, $H^+$ and $H^-$ are mass-degenerate due to the U(1)$_{\\rm EM}$ gauge symmetry.","Moreover, the $H^+$ and $H^-$ are distinguishable by their electric charge, which can be experimentally probed using photons.","Suppose that this probe were unavailable (or equivalently, suppose one could turn off electromagnetism).","In this case, would it be possible for an experiment to reveal the existence of a mass-degenerate scalar?","In this very simple example, one could not physically distinguish (on an event by event basis) between the two degenerate states that comprise the charged Higgs scalar.","Nevertheless, there would in principle be observables that are sensitive to the number of mass-degenerate scalar states present.","For example, in the CP-conserving two Higgs doublet model, the decay rate for the decay of a heavy CP-even neutral scalar, $H\\rightarrow H^+ H^-$ (if kinematically allowed) is proportional to the number of degrees of freedom in the final state.","If we express the charged Higgs field as a linear combination of real scalar fields, $H^\\pm =(\\phi _1\\pm i\\phi _2)/\\sqrt{2}$ , then the decay rate for $H\\rightarrow H^+ H^-$ is the (incoherent) sum of the decay rates for $H\\rightarrow \\phi _1\\phi _1$ and $\\phi _2\\phi _2$ .","These two rates are identical, and the sum yields a multiplicity factor of 2.","This multiplicity factor provides the experimental signal for mass-degenerate scalars.Note that the decay rate for $Z\\rightarrow H^+ H^-$ is equal to the decay rate for $Z\\rightarrow \\phi _1\\phi _2$ .","In this case, the off-diagonal nature of the $Z\\phi _1\\phi _2$ coupling implies that no multiplicity factor is present.","Nevertheless, one can still infer the existence of mass-degenerate states, since the decays $Z\\rightarrow \\phi _1\\phi _1, \\phi _2\\phi _2$ are forbidden by Bose statistics.","Apart from the trivial mass degeneracy of $H^\\pm $ , we would like to explore in this paper the possibility of exactly mass-degenerate neutral scalars and/or mass-degenerate charged Higgs pairs in extended Higgs sectors.","In each case, the critical questions to ask are: (i) is the origin of the exact mass degeneracy natural?","and (ii) how can the mass degenerate scalars be distinguished experimentally?","Exact mass degeneracies are natural if they are a consequence of an unbroken symmetry.","In particular, accidental mass degeneracies require an artificial fine-tuning of the scalar potential parameters, and in this sense we shall call them unnatural (and in our view not especially interesting).","If mass-degenerate states are present, it is of interest to determine how to probe them experimentally.","In some cases, one can identify the presence of mass degenerate states on an event by event basis.","In other cases, the only signal of the mass degeneracy is a measurable multiplicity factor that can be determined when averaging over initial state degeneracies and summing over final state degeneracies.For example, a quark of a given flavor is a mass degenerate state due to its three possible colors.","Although the color of a quark cannot be identified experimentally, the presence of the color degree of freedom can be experimentally verified by the color multiplicity factor (most famously exhibited in the observed cross section for $e^+e^-$ annihilation into quark-antiquark pairs.)","Our focus in this paper is extended multi-doublet Higgs sectors with mass-degenerate scalar states.","It is particularly instructive to discuss mass degeneracy in the scalar sector starting from the so-called Higgs basis.","This corresponds to a subset of all possible scalar field parameterizations in which only one Higgs doublet, denoted by $H_1$ , acquires a non-zero positive vacuum expectation value (vev), while all the other scalar fields of the Higgs basis ($H_2$ , $H_3,\\ldots H_n$ ) have zero vev [16], [17], [18], [19], [20].","The neutral and charged Goldstone bosons reside entirely in $H_1$ , as this is the only doublet that possesses a non-zero vev, together with a neutral Higgs field that, in the absence of mixing with the neutral fields of the other Higgs doublets, behaves like the Standard Model (SM) Higgs boson.","In this sense, this subset of scalar bases may be viewed as Standard Model aligned bases.","That is, the Higgs basis is actually a family of basis choices, since one is always free to perform an arbitrary U($n-1$ ) transformation among $H_2$ , $H_3,\\ldots H_n$ while preserving the vev of $H_1$ [21].","There is no loss of generality in choosing any particular scalar basis as a starting point.","In order to identify the physical neutral scalars one must diagonalize the corresponding scalar squared-mass matrix.","As expected, the end result is independent of the initial choice of basis for the scalar fields.","In section , we study under what circumstances there is an exact mass degeneracy in the familiar two-Higgs-doublet model (2HDM) [22], [23].","In this model there are three physical neutral fields and one charged field, so we only consider potential mass degeneracies among the neutral fields.One can also examine mass degeneracies between the charged Higgs boson and one of the neutral Higgs bosons.","For example, in a custodial symmetric 2HDM, the CP-odd Higgs scalar is degenerate with the charged Higgs boson [24], [25], [26], [27].","However, custodial symmetry is not an exact symmetry of the full electroweak Lagrangian.","Thus, given a custodial symmetric 2HDM scalar potential, any potential mass degeneracies between neutral and charged scalars is at best approximate.","We do not consider such mass degeneracies further in this work.","We begin our 2HDM analysis by studying possible mass degeneracies among the neutral scalar states of the inert doublet model (IDM) [28], [29].","The scalar potential of this model possesses a discrete $\\mathbb {Z}_2$ symmetry that is unbroken by the vacuum.","In this case, the CP-even neutral component of $H_1$ in the Higgs basis is a mass eigenstate whose tree-level couplings are precisely those of the SM Higgs boson.","The real and imaginary parts of the neutral component of $H_2$ are odd under the discrete $\\mathbb {Z}_2$ symmetry, and have opposite signs under CP.","We will denote these two neutral states by $H$ and $A$ , although there is no way to identify which of these two states is CP-even and which is CP-odd.Equivalently, one can propose two different definitions of CP (called, say, CPa and CPb), such that $H$ is CPa-even and $A$ is CPa-odd, and vice versa for CPb.","Either definition can be consistently used to define the CP symmetry of the bosonic sector of the IDM.","It is possible that $h$ is degenerate in mass with either $H$ or $A$ , but such mass degeneracies are accidental in nature since neither case can arise due to a symmetry.","Moreover, these mass-degenerate states are physically distinguishable, since $h$ is even whereas $H$ and $A$ are odd under the $\\mathbb {Z}_2$ symmetry.","In contrast, an exact mass degeneracy of $H$ and $A$ can arise if the $\\mathbb {Z}_2$ symmetry of the scalar potential is promoted to a continuous U(1) symmetry.","In our terminology, this mass degeneracy of $H$ and $A$ is natural.","Nevertheless, the two mass-degenerate states can still be physically distinguished due to the coupling of these states to $W^\\pm H^\\mp $ .","One can now extend the above analysis to an arbitrary 2HDM.","One can show that with one exception, all 2HDM mass degeneracies are accidental.","The one exceptional case of a natural mass degeneracy is precisely the case of $m_H=m_A$ in the IDM.","This conclusion can also be obtained by considering all possible symmetries of the 2HDM scalar potential.","Among these symmetries, we can identify those that can potentially guarantee the mass degeneracy of scalar states.","By examining the consequences of these symmetries, we again confirm that the only possible neutral scalar mass degeneracy in the 2HDM arises in the IDM as previously noted.","In section we consider possible mass-degeneracies in the three Higgs doublet model.","Using the previous 2HDM analysis of mass degeneracies of the IDM, we construct a three Higgs doublet model (3HDM) generalization of the IDM, which we call the replicated inert doublet model (RIDM).","In this model, two of the three Higgs doublets are inert, and four mass-degenerate scalar pairs exist (two involving the charged scalar states from the inert doublets and two involving the neutral scalar states from the inert doublets).","We can explicitly identify the symmetries that are responsible for these mass degeneracies.","We then investigate the possibility of adding new terms to the scalar potential that partially break these symmetries while preserving the mass degeneracies.","In this way, we arrive at a model first proposed by Ivanov and Silva [30].","The Ivanov and Silva scalar potential possesses a discrete subgroup of the continuous symmetries that govern the RIDM, that maintains the mass degeneracies of the RIDM.","This discrete subgroup is the generalized CP symmetry, CP4, which has the property that $({\\rm CP}4)^n$ is the identity operator only for integer $n$ values that are multiples of 4.","The CP4 symmetry is distinguished from the conventional CP symmetry (denoted henceforth by CP2), which has the property that $(\\rm {CP}2)^2$ is the identity operator.","Some properties of specialized 3HDMs have also been analyzed recently in Ref.[31].","One of the most notable properties of the Ivanov-Silva (IS) model is that one can write down the most general CP4-invariant scalar potential with three Higgs doublets, which has the feature that at least one of the coefficients of the quartic terms of the scalar potential must be complex (with a nonvanishing imaginary part).","Indeed, as demonstrated explicitly in Appendix A, one cannot redefine the scalar fields within the family of Higgs bases such that all the coefficients of the scalar potential are real.","In this case, we say that no real Higgs basis exists.","This means that CP2 is not a symmetry of the IS scalar potential and vacuum.","In section , we identify the existence of a physical observable that is present if no CP2 symmetry exists that commutes with the CP4 symmetry of the IS model.However, this leaves open the possibility of the existence of a CP2 symmetry that does not commute with the CP4 symmetry [32]; in this case, a real Higgs basis exists and a conventional CP symmetry can be defined.","As an example, we focus on $Z$ decay into four inert neutral scalars (with some details relegated to Appendix B).","Nevertheless, the CP4 invariance guarantees that all CP-violating observables involving the Higgs/gauge boson sector of the theory must be absent.","For example, we provide an instructive analysis in section that shows how the CP4 symmetry of the IS model with no real Higgs basis ensures the cancellation of contributions to the CP-violating form factors of the effective $ZZZ$ and $ZW^+ W^-$ vertices up to three-loop order.","Finally, we state our conclusions in section ."],["2HDM mass degeneracies","Consider the 2HDM, consisting of two hypercharge-one, doublet scalar fields, $\\Phi _1$ and $\\Phi _2$ .","The most general gauge-invariant renormalizable scalar potential is $\\mathcal {V}&=& m_{11}^2 \\Phi _1^\\dagger \\Phi _1+ m_{22}^2 \\Phi _2^\\dagger \\Phi _2 -[m_{12}^2\\Phi _1^\\dagger \\Phi _2+{\\rm h.c.}]+\\tfrac{1}{2}\\lambda _1(\\Phi _1^\\dagger \\Phi _1)^2+\\tfrac{1}{2}\\lambda _2(\\Phi _2^\\dagger \\Phi _2)^2+\\lambda _3(\\Phi _1^\\dagger \\Phi _1)(\\Phi _2^\\dagger \\Phi _2)\\nonumber \\\\&&\\quad +\\lambda _4( \\Phi _1^\\dagger \\Phi _2)(\\Phi _2^\\dagger \\Phi _1)+\\left\\lbrace \\tfrac{1}{2}\\lambda _5 (\\Phi _1^\\dagger \\Phi _2)^2 +\\big [\\lambda _6 (\\Phi _1^\\dagger \\Phi _1) +\\lambda _7 (\\Phi _2^\\dagger \\Phi _2)\\big ] \\Phi _1^\\dagger \\Phi _2+{\\rm h.c.}\\right\\rbrace \\,.$ We shall assume that the minimum of the scalar potential is electric charge conserving, in which case only the neutral scalar fields possess a nonzero vacuum expectation value (vev), $\\langle \\Phi _i^0 \\rangle =v_i/\\sqrt{2}$ , where the $v_i$ are potentially complex.","The Fermi constant, $G_F$ fixes the value of $v^2\\equiv |v_1|^2+|v_2|^2=(\\sqrt{2}G_F)^{-1}\\simeq (246~{\\rm GeV})^2\\,.$ Employing a new scalar field basis consisting of two orthonormal linear combinations of $\\Phi _1$ and $\\Phi _2$ does not modify the physical predictions of the model.","One convenient choice is the Higgs basis, in which the redefined doublet fields (denoted below by $H_1$ and $H_2$ ) have the property that $H_1$ has a non-zero vev whereas $H_2$ has a zero vev [16], [17], [18].","In particular, we define new Higgs doublet fields: $H_1=\\begin{pmatrix}H_1^+\\\\ H_1^0\\end{pmatrix}\\equiv \\frac{1}{v}(v_1^* \\Phi _1+v_2^*\\Phi _2)\\,,\\qquad \\quad H_2=\\begin{pmatrix} H_2^+\\\\ H_2^0\\end{pmatrix}\\equiv \\frac{1}{v}(-v_2 \\Phi _1+v_1\\Phi _2)\\,.$ It follows that $\\langle H_1^0 \\rangle =v$ and $\\langle H_2^0 \\rangle =0$ .","The Higgs basis is uniquely defined up to an overall rephasing, $H_2\\rightarrow e^{i\\chi } H_2$ (which does not alter the fact that $\\langle H_2^0 \\rangle =0$ ).","In the Higgs basis, the scalar potential of Eq.","(REF ) is denoted as [19], [20]: $\\mathcal {V}&=& Y_1 H_1^\\dagger H_1+ Y_2 H_2^\\dagger H_2 +[Y_3H_1^\\dagger H_2+{\\rm h.c.}]+\\tfrac{1}{2}Z_1(H_1^\\dagger H_1)^2+\\tfrac{1}{2}Z_2(H_2^\\dagger H_2)^2+Z_3(H_1^\\dagger H_1)(H_2^\\dagger H_2)\\nonumber \\\\&&\\quad +Z_4( H_1^\\dagger H_2)(H_2^\\dagger H_1)+\\left\\lbrace \\tfrac{1}{2}Z_5 (H_1^\\dagger H_2)^2 +\\big [Z_6 (H_1^\\dagger H_1) +Z_7 (H_2^\\dagger H_2)\\big ] H_1^\\dagger H_2+{\\rm h.c.}\\right\\rbrace \\,,$ where $Y_1$ , $Y_2$ and $Z_1,\\ldots ,Z_4$ are real parameters, whereas $Y_3$ , $Z_5$ , $Z_6$ and $Z_7$ are potentially complex parameters.","Imposing the scalar potential minimum conditions yields, $Y_1=-\\tfrac{1}{2}Z_1 v^2\\,,\\qquad \\quad Y_3=-\\tfrac{1}{2}Z_6 v^2\\,.$"],["Mass degeneracies of the inert doublet model (IDM)","We wish to study the consequences of a 2HDM in which two or three of the neutral Higgs scalars are degenerate in mass.","For simplicity, we shall first specialize to the inert 2HDM (the so-called IDM)[28], [29] in which there is an exact discrete $\\mathbb {Z}_2$ symmetry that is preserved by the vacuum, under which all particles of the SM and one of the two Higgs doublet fields (which contains the observed Higgs boson) are even and the second Higgs doublet field is odd under the multiplicative discrete symmetry.","In particular, the discrete symmetry of the IDM is manifest in the Higgs basis, where we identify $H_1$ as even and $H_2$ as odd under the $\\mathbb {Z}_2$ symmetry.","It then follows that $Y_3=Z_6=Z_7=0$ .","The IDM scalar potential is CP-conserving since one can eliminate the phase of $Z_5$ (the only remaining potentially complex scalar potential parameter) by appropriately rephasing the Higgs basis field $H_2$ .","The Higgs basis fields are $H_1=\\begin{pmatrix} G^+ \\\\ \\frac{1}{\\sqrt{2}}\\bigl [v+h+iG^0\\bigr ] \\end{pmatrix}\\,,\\qquad \\quad H_2=\\begin{pmatrix} H^+ \\\\ \\frac{1}{\\sqrt{2}}\\bigl [H+iA\\bigr ]\\end{pmatrix}\\,,$ where $G^\\pm $ and $G^0$ are the Goldstone bosons that provide the longitudinal degrees of freedom of the massive $W^\\pm $ and $Z^0$ gauge bosons.","The physical mass spectrum of the IDM is given by, $&&m_h^2=Z_1 v^2\\,,\\qquad \\qquad \\qquad \\qquad \\quad m^2_{H^\\pm }=Y_2+\\tfrac{1}{2}Z_3 v^2\\,,\\\\&&m_{A}^2=m_{H^\\pm }^2+\\tfrac{1}{2}(Z_4- Z_5)v^2\\,,\\qquad \\,\\, m_H^2=m_A^2+Z_5 v^2\\,.$ For completeness, we exhibit the Higgs couplings of the IDM in the unitary gauge below (where the Goldstone fields are set to zero).","First, the interactions of the Higgs bosons and the gauge bosons are governed by,The photon field $A_\\mu $ should not be confused with the scalar field $A$ .","${L}_{VVH}&=&\\left(gm_W W_\\mu ^+W^{\\mu \\,-}+\\frac{g}{2c_W} m_ZZ_\\mu Z^\\mu \\right)h\\,,\\\\[8pt]{L}_{VVHH}&=&\\left[\\tfrac{1}{4}g^2 W_\\mu ^+W^{\\mu \\,-}+\\frac{g^2}{8c_W^2}Z_\\mu Z^\\mu \\right](h^2+H^2+A^2) \\nonumber \\\\&& +\\left[\\tfrac{1}{2}g^2 W_\\mu ^+ W^{\\mu \\,-}+e^2A_\\mu A^\\mu +\\frac{g^2}{c_W^2}\\left(\\tfrac{1}{2}-s_W^2\\right)^2Z_\\mu Z^\\mu +\\frac{2ge}{c_W}\\left(\\tfrac{1}{2}-s_W^2\\right)A_\\mu Z^\\mu \\right]H^+H^-\\nonumber \\\\&& +\\biggl \\lbrace \\left(\\tfrac{1}{2}eg A^\\mu W_\\mu ^+-\\frac{g^2s_W^2}{2c_W}Z^\\mu W_\\mu ^+\\right)H^-(H+iA) +{\\rm h.c.}\\biggr \\rbrace \\,, \\\\[8pt]{L}_{VHH}&=&\\frac{g}{2c_W}\\,Z^\\mu A\\,\\!\\!\\mathrel {\\leftrightarrow \\hspace{-8.50006pt}\\partial }$ H -12g[iW+ H-$\\leftrightarrow $$\\partial $$\\scriptstyle \\,\\mu $ (H+iA) +h.c.]","+[ieA+igcW(12-sW2) Z]H+$\\leftrightarrow $$\\partial $ H-, where $s_W\\equiv \\sin \\theta _W$ , $c_W\\equiv \\cos \\theta _W$ .","The trilinear and quadrilinear Higgs self-interactions are governed by ${L}_{3h}&=&-\\tfrac{1}{2}v\\bigl [Z_1 h^3+(Z_3+Z_4)h(H^2+A^2)+Z_5h(H^2-A^2)\\bigr ]-vZ_3hH^+H^-\\,, \\\\[6pt]{L}_{4h}&=&-\\tfrac{1}{8}\\bigl [Z_1 h^4+Z_2(H^2+A^2)^2+2(Z_3+Z_4)h^2(H^2+A^2)+2Z_5 h^2(H^2-A^2)\\bigr ]\\nonumber \\\\[6pt]&&-\\tfrac{1}{2}H^+ H^-\\bigl [Z_2(H^2+A^2+H^+ H^-)+Z_3 h^2\\bigr ].","$ The tree-level couplings of $h$ to SM particles obtained above are precisely those of the SM Higgs boson, corresponding to the exact Higgs alignment limit [33], [34], [35], [36], [37], [38], [39], [40], [41] (as expected in light of $Z_6=0$ ).","Moreover, an examination of the above couplings implies that $h$ is CP-even (to be identified with the SM Higgs boson) and $H$ and $A$ have opposite CP-quantum numbers (one is odd and the other is even) based on the $ZAH$ coupling.Under the rephasing, $H_2\\rightarrow iH_2$ , we note that $Z_5\\rightarrow -Z_5$ , $H\\rightarrow -A$ and $A\\rightarrow H$ .","One can check that the masses of $H$ and $A$ and their couplings are invariant under this rephasing.","Note that the CP is not uniquely defined by the IDM interactions, since two candidate definitions of CP exist (called CPa and CPb in footnote REF ), where $H$ is CPa-even and $A$ is CPa-odd, and vice versa for CPb.","Either definition of CP can be used consistently in exploring the phenomenology of the IDM.","Finally, we note that under the $\\mathbb {Z}_2$ symmetry of the IDM, the quarks and leptons can be chosen to be even.","Consequently, the tree-level couplings of $h$ to fermion pairs are identical to those of the SM Higgs boson, whereas $H$ , $A$ and $H^\\pm $ do not couple to the SM fermions.","We now examine the possibility of mass degeneracies in the IDM.","First, consider the case of $m_h=m_H$ or $m_h=m_A$ .","In this case, it is possible to physically distinguish between $h$ and its mass-degenerate partner due to their opposite $\\mathbb {Z}_2$ quantum numbers.","For example, since all SM bosons and fermions are even under the $\\mathbb {Z}_2$ symmetry, it follows that the gluon-gluon (via a top quark loop), $WW$ and $ZZ$ fusion processes can only produce $h$ whereas Drell-Yan production (via virtual $s$ -channel $Z$ exchange) can only produce $H$ in association with $A$ .","Hence, despite the mass degeneracy, the two mass-degenerate scalars are physically distinguishable.","Note that the mass degeneracy of $h$ and its scalar partner is not radiatively stable.","For example, if $h$ and $H$ are mass degenerate states, then the one-loop contributions to the $hh$ two-point function (such as $ZZ$ and $WW$ intermediate states) differ from the corresponding contributions to the $HH$ two-point function (e.g.","the $AZ$ intermediate state).","Indeed, the tree-level condition for the mass degeneracy of $h$ and $H$ , $Z_1v^2=Y_2+\\tfrac{1}{2}Z_{345}v^2,$ where $Z_{345}\\equiv Z_3+Z_4+Z_5$ , is unnatural; i.e., Eq.","(REF ) is not the result of some symmetry.The scenario where $m_h=m_H=m_A$ is a special case of the $h$ –$H$ mass degeneracy.","In the triply mass-degenerate scenario, $h$ is also distinguished from $\\phi ^\\pm $ by its U(1) charge, which is zero.","For example, there is no coupling of $ZW^\\pm H^\\mp h$ in contrast to the $Z W^\\pm H^\\mp \\phi ^\\pm $ couplings exhibited in Eq.","(REF ).","Second, consider the case of $m_H=m_A$ , which corresponds to $Z_5=0$ .","In this case, the IDM scalar potential possesses a continuous U(1) symmetry, which is not spontaneously broken by the vacuum.This case has also been noted in Ref. [41].","It is this symmetry that is responsible for the mass degenerate states $H$ and $A$ .","One can now define eigenstates of U(1) charge, $\\phi ^\\pm =\\frac{1}{\\sqrt{2}}\\bigl [H\\pm iA\\bigr ]\\,.$ The relevant interaction terms of $\\phi ^{\\pm }$ are ${L}_{\\rm int}&=&\\left[\\tfrac{1}{2}g^2 W_\\mu ^+W^{\\mu \\,-}+\\frac{g^2}{4c_W^2}Z_\\mu Z^\\mu \\right]\\phi ^+\\phi ^- +\\frac{ig}{2c_W}Z^\\mu \\phi ^-\\!\\!\\mathrel {\\leftrightarrow \\hspace{-8.50006pt}\\partial }$ + -g2[iW+ H-$\\leftrightarrow $$\\partial $$\\scriptstyle \\,\\mu $ + +h.c.]","+eg2 (AW+H-+ + AW-H+-) -g2sW22cW(ZW+H-+ + ZW-H+-) - v(Z3+Z4)h+–12[Z2(+-)2+(Z3+Z4)h2+-]-Z2 H+ H-+- .","Although $\\phi ^\\pm $ are mass degenerate states, they can be distinguished.","For example, Drell-Yan production via a virtual $s$ -channel $W^+$ exchange can produce $H^+$ in association with $\\phi ^-$ , whereas virtual $s$ -channel $W^-$ exchange can produce $H^-$ in association with $\\phi ^+$ .","Thus, the sign of the charged Higgs boson reveals the U(1)-charge of the produced neutral scalar.","The origin of this correlation lies in the fact that, by construction, $H^+$ and $\\phi ^+$ both reside in $H_2$ , whereas $H^-$ and $\\phi ^-$ both reside in $H_2^\\dagger $ ."],["2HDM mass degeneracies beyond the IDM","Although the IDM is a rather special case among all possible 2HDMs, the conclusions concerning mass degeneracies are robust.","Allowing for the most general 2HDM scalar potential, the Higgs sector is CP-violating if the relative phases among $Z_5$ , $Z_6$ and $Z_7$ cannot be removed by rephasing the Higgs basis field $H_2$ .","It is convenient to introduce three invariant quantities[42], [18], [20], whose imaginary parts are given by,Basis-invariant expressions for the $J_i$ are given in Ref.[20].","$\\operatorname{Im}J_1=\\operatorname{Im}(Z_6^* Z_7)\\,,\\qquad \\operatorname{Im}J_2=\\operatorname{Im}(Z_5^* Z_6^2)\\,,\\qquad \\operatorname{Im}J_3=\\operatorname{Im}\\bigl [Z_5^*(Z_6+Z_7)^2\\bigr ]\\,.$ The Higgs sector of the 2HDM is CP-violating unless $\\operatorname{Im}J_1=\\operatorname{Im}J_2=\\operatorname{Im}J_3=0$ .","The origin of the CP-violation can either be explicit or spontaneous [43], [44].To test for explicit CP-violation, one must employ invariant quantities that are independent of the scalar field vacuum expectation values [45], [46], [47].","Note that the neutral scalar squared-mass matrix does not involve the Higgs basis parameter $Z_7$ .","In particular, $Z_7$ only enters in the Higgs boson cubic and quartic self-couplings.","Hence, if $\\operatorname{Im}J_2=\\operatorname{Im}(Z_5^* Z_6^2)=0$ , then the neutral Higgs mass eigenstates behave like eigenstates of CP in their tree-level interactions with the gauge bosons and fermions (independently of the values of $\\operatorname{Im}J_1$ and $\\operatorname{Im}J_3$ ).","Moreover, the neutral scalar squared-mass matrix breaks up into a block diagonal form consisting of a $2\\times 2$ block (whose diagonalization yields the two CP-even neutral scalars) and a $1\\times 1$ block (which yields the CP-odd neutral scalar).","Consider the possibility of mass degeneracies among the neutral scalars of the most general 2HDM.","We now recall a remarkable tree-level relation of the CP-violating 2HDM[48], [42], [49], $ \\operatorname{Im}J_2&={\\rm Im}(Z_5^* Z_6^2)=\\frac{2 s_{13}c_{13}^2 s_{12}c_{12}}{v^6}(m_2^2-m_1^2)(m_3^2-m_1^2)(m_3^2-m_2^2)\\,,$ where the $m_i$ ($i=1,2,3$ ) are the masses of the three neutral Higgs bosons of the 2HDM, $s_{12}\\equiv \\sin \\theta _{12}$ , $c_{12}\\equiv \\cos \\theta _{12}$ , etc., and $\\theta _{12}$ and $\\theta _{13}$ are invariant mixing angles that are associated with the diagonalization of the neutral Higgs squared-mass matrix in the Higgs basis.Details on the definition of the mixing angles and their relations to the Higgs basis scalar potential parameters can be found in Ref.[49].","However, we will not need any of these details for the present argument.","In Ref.","[50], the three CP-odd invariants $\\operatorname{Im}J_i$ have been expressed in terms of the neutral scalar masses and the couplings of the neutral Higgs mass eigenstates to charged pairs, $e_i$ $(H_iW^+W^-)$ and $q_i$ $(H_iH^+H^-)$ .","In particular,In Ref.","[42], the invariant defined in Eq.","(REF ) is called $J_1$ .","$ \\operatorname{Im}J_2=\\frac{2e_1 e_2 e_3}{v^9}(m_1^2-m_2^2)(m_2^2-m_3^2)(m_3^2-m_1^2).$ If any two of the three neutral Higgs bosons are mass-degenerate, then either Eq.","(REF ) or (REF ) implies that $\\operatorname{Im}J_2=0$ , and the corresponding neutral scalar mass-eigenstates will behave as states of definite CP in their interactions with gauge bosons and fermions.","Nevertheless, if $\\operatorname{Im}J_1\\ne 0$ and/or $\\operatorname{Im}J_3\\ne 0$ (which would imply that $\\operatorname{Im}Z_7\\ne 0$ in the Higgs basis where $Z_5$ and $Z_6$ are simultaneously real), then CP-violating Higgs self-couplings must be present.","Moreover, radiative corrections will generate a non-zero $\\operatorname{Im}J_2$ and yield neutral Higgs states of indefinite CP.","That is, if CP-violation in the scalar sector is present, the tree-level relation $\\operatorname{Im}J_2=0$ can only be realized via an artificial fine-tuning of the parameters.","Nevertheless, one can consider the implications of a tree-level mass degeneracy among the neutral Higgs scalars of the 2HDM.","The above discussion illustrates the power of using scalar basis invariant conditions to analyze the CP properties of multi-Higgs models [45], [46], [51], [52], [53].","In light of Eq.","(REF ), if a tree-level mass degeneracy among the neutral Higgs scalars is present, then it is possible to rephase the Higgs basis field $H_2$ such that $Z_5$ and $Z_6$ are simultaneously real.","Thus, in the analysis that follows, we shall analyze the most general 2HDM scalar potential assuming that $Z_5$ and $Z_6$ are real parameters.","The squared-masses of the charged Higgs boson, $H^\\pm $ , and the CP-odd Higgs boson, $A$ are given by,In the notation employed in Eqs.","(REF )–(), $h$ and $H$ [$A$ ] refer to the neutral scalars that behave as CP-even [odd] mass eigenstates in their tree-level interactions with the gauge bosons and fermions.","Indeed, CP-violating interactions are present in Eqs.","(REF ) and () if $\\operatorname{Im}Z_7\\ne 0$ .","$m_{H^\\pm }^2=Y_2+\\tfrac{1}{2}Z_3 v^2\\,,\\qquad \\quad m_A^2=m_{H^\\pm }^2+\\tfrac{1}{2}(Z_4-Z_5)v^2\\,.$ The squared-masses of the CP-even Higgs bosons, $h$ and $H$ are the eigenvalues of the $2\\times 2$ matrix, $\\mathcal {M}_H^2=\\begin{pmatrix} Z_1 v^2 & \\quad Z_6 v^2 \\\\ Z_6 v^2 & \\quad m_A^2+Z_5 v^2\\end{pmatrix}.$ That is, $m^2_{H,h}=\\frac{1}{2}\\biggl \\lbrace m_A^2+(Z_1+Z_5)v^2\\pm \\sqrt{[m_A^2-(Z_1-Z_5)v^2]^2+4Z_6^2 v^4\\,}\\biggr \\rbrace \\,.$ Mass degenerate states arise if one of the following two quantities is zero, $Z_5(m_A^2-Z_1 v^2)+Z_6^2 v^2=0\\quad \\text{or}\\quad \\bigl [m_A^2-(Z_1-Z_5)v^2\\bigr ]^2+4Z_6^2 v^4=0\\,,$ where $m_A^2$ is given by Eq.","(REF ).","The case of $m_h=m_H$ arises when the second condition given in Eq.","(REF ) is satisfied.","It then follows that $m_A^2=(Z_1-Z_5)v^2$ and $Z_6=0$ , and the latter then yields the IDM mass spectrum.","As in the case of the IDM, the mass degeneracy of $h$ and $H$ requires a fine tuning of the parameters shown in Eq.","(REF ).","In principle, it is possible that $Z_7\\ne 0$ , but in this case, $Z_6=0$ is not a natural condition since the $\\mathbb {Z}_2$ symmetry of the IDM is not present.","Nevertheless, even if one accepts the two fine tuned conditions needed in this scenario, the arguments presented above Eq.","(REF ) still apply.","Namely, $Z_6=0$ corresponds to the exact alignment limit (at tree-level), in which case the tree-level interactions of the Higgs bosons and gauge bosons are still the same as those of the IDM [cf. Eqs.","(REF )–(REF )], whereas the tree-level trilinear and quadrilinear Higgs self-interactions given in Eqs.","(REF ) and () are modified by the addition of the following terms, $\\delta {L}_{3h}&=&-\\tfrac{1}{4} v \\bigl [Z_7(H+iA)+Z_7^*(H-iA)\\bigr ](HH+AA+2H^+ H^-)\\,, \\\\[5pt]\\delta {L}_{4h}&=&-\\tfrac{1}{4}\\bigl [Z_7(H+iA)+Z_7^*(H-iA)\\bigr ](HH+AA+2H^+ H^-)h\\,,$ after rephasing the Higgs basis field $H_2$ such that $Z_5$ is real.","The cases $m_h=m_A$ or $m_H=m_A$ arise when the first condition given in Eq.","(REF ) is satisfied.","This condition also requires a fine-tuning of the parameters.","Moreover, approximate Higgs alignment (as suggested by the LHC Higgs data) is not achieved unless $m_A^2\\gg Z_1 v^2$ or $|Z_6|\\ll 1$ .","Nevertheless, the physical distinction of the mass degenerate states is due to the CP quantum numbers of the neutral scalar states (which are preserved in the tree-level Higgs interactions with gauge bosons and with fermions).","One can therefore distinguish between the corresponding production mechanisms of the degenerate scalars that are mediated by gauge boson fusion or Drell-Yan production via $s$ -channel gauge boson exchange.","Finally, we consider the triply mass-degenerate case of $m_h=m_H=m_A$ .","In this case, both conditions given in Eq.","(REF ) must be satisfied, which yields $Z_5=Z_6=0$ and $m_A^2=Z_1 v^2$ .","This leaves $Z_7$ as the only potentially complex parameter of the scalar potential in the Higgs basis.","Thus, one is free to rephase the Higgs basis field $H_2$ such that $Z_7$ is real, and we conclude that the Higgs scalar potential and vacuum must be CP-conserving.","However, as long as $Z_7\\ne 0$ , the triply mass-degenerate case is unnatural, since the $\\mathbb {Z}_2$ symmetry of the IDM is not present."],["Natural 2HDM mass degeneracies: a symmetry based approach","In sections REF and REF , we derived the conditions that yield mass degeneracies among the neutral scalars of the 2HDM by brute force.","Namely, we obtained explicit expressions for the neutral scalar masses and then derived the corresponding relations among Higgs basis parameters for which mass degeneracies were present.","We then checked whether any of these relations were a consequence of a symmetry, and if yes we concluded that the corresponding mass degeneracy was natural.","In this section, we will obtain the same result by considering all possible symmetries of the 2HDM scalar potential.","Since the complete list of such symmetries is known [54], [55], [56], [57], [58], [59], we can be sure that our catalog of natural mass degeneracies of the 2HDM is complete.","We shall make use of the classification of symmetries presented in Ref.","[56], which identifies three possible Higgs family symmetries, $\\mathbb {Z}_2$ , U(1) and SO(3), and three classes of generalized CP-symmetries, denoted by GCP1, GCP2 and GCP3, respectively, as summarized in Table .In Ref.","[56] the three classes of generalized CP transformations are denoted by CP1, CP2 and CP3 respectively.","This nomenclature for the generalized CP-symmetries is awkward, in light of the notation that will be employed in section .","To avoid confusion, we have appended the letter G (for “general”) in denoting the three classes of generalized CP transformations of the 2HDM.","In the GCP transformation laws of Table , we have introduced the conjugation symbol ${}^{\\scriptscriptstyle {\\bigstar }}$ , which when applied to an SU(2) multiplet of scalar fields is defined by $\\Phi ^{\\scriptscriptstyle {\\bigstar }}\\equiv \\bigl [\\Phi ^\\dagger \\bigr ]^{\\scriptstyle T}$$,where the dagger refers both to hermitian conjugation of the quantum field operator when acting on the Hilbert space, and to complex conjugate transpose when acting on an SU(2) multiplet of fields.\\vspace{-0.72229pt}\\begin{table}[hb!","]\\begin{tabular}{|cccc|}\\hline \\rule {0pt}{}symmetry & \\hspace{36.135pt} transformation law & \\hspace{50.58878pt} \\phantom{transformation law} & \\\\[2pt]\\hline \\rule {0pt}{}\\mathbb {Z}_2 & \\Phi _1\\rightarrow \\Phi _1 & \\Phi _2\\rightarrow -\\Phi _2 &\\\\U(1) & \\Phi _1\\rightarrow \\Phi _1 & \\Phi _2\\rightarrow e^{2i\\theta }\\Phi _2 &\\\\SO(3) & \\Phi _a\\rightarrow U_{ab}\\Phi _b & U\\in {\\rm U}(2)/{\\rm U}(1)_{\\rm Y}& (\\text{for $a$, $b=1,2$} )\\\\GCP1 & \\Phi _1\\rightarrow \\Phi _1^{\\scriptscriptstyle {\\bigstar }} & \\Phi _2\\rightarrow \\Phi _2^{\\scriptscriptstyle {\\bigstar }} & \\\\GCP2 & \\Phi _1\\rightarrow \\Phi _2^{\\scriptscriptstyle {\\bigstar }} & \\Phi _2\\rightarrow -\\Phi _1^{\\scriptscriptstyle {\\bigstar }} & \\\\GCP3 & \\Phi _1 \\rightarrow \\Phi _1^{\\scriptscriptstyle {\\bigstar }} \\cos \\theta +\\Phi _2^{\\scriptscriptstyle {\\bigstar }} \\sin \\theta &\\Phi _2 \\rightarrow -\\Phi _1^{\\scriptscriptstyle {\\bigstar }}\\sin \\theta +\\Phi _2^{\\scriptscriptstyle {\\bigstar }}\\cos \\theta & \\text{(for $0<\\theta <\\tfrac{1}{2}\\pi $)} \\\\[2pt]\\hline \\hline \\rule {0pt}{}\\Pi _2& \\Phi _1\\rightarrow \\Phi _2 & \\Phi _2\\rightarrow \\Phi _1 & \\\\[2pt]\\hline \\end{tabular}\\caption {\\small Possible symmetries of the 2HDM scalar potential that are respected by the SU(2)\\times U(1)_{\\rm Y} gauge kinetic terms of the scalar fields.","The corresponding symmetry transformation laws are given in a basis where the symmetry is manifest.","Note that a scalar potential that is invariant under the mirror discretesymmetry, \\Pi _2, is also invariant under the \\mathbb {Z}_2 in another scalar field basis~\\cite {Davidson:2005cw}.","}\\end{table}$ Table: Impact of the symmetries defined in Table on the coefficients of the 2HDM scalar potential [cf. Eq.","()] in abasis where the symmetry is manifest.","A short dash indicates the absence of a constraint.","A scalar potential that is simultaneously invariant under ℤ 2 \\mathbb {Z}_2 and Π 2 \\Pi _2 isalso invariant under GCP2 in another scalar field basis , .","Likewise, a scalar potential that is simultaneously invariant under U(1) and Π 2 \\Pi _2 is also invariant under GCP3 in another scalar field basis .","The symbol ⊕\\oplus is being used above to indicate that two symmetries are enforced simultaneously within the same scalar field basis.We shall not consider the seven additional accidental symmetries of the 2HDM scalar potential identified in Refs.","[58], [59], that utilized mixed Higgs family and generalized CP transformations that leave the SU(2) gauge kinetic terms of the scalar fields invariant.","An example of such a symmetry is the well known custodial symmetry that is respected by the 2HDM scalar potential when $m_{H^\\pm }=m_A$ [24], [25], [26], [27].","However, this class of symmetries is violated by the U(1)$_{\\rm Y}$ gauge kinetic term of the scalar potential (as well as by the Yukawa couplings that are responsible for mass differences between up and down-type fermions).","Hence, any exact mass degeneracies arising from these seven accidental symmetries will be spoiled, in the absence of an artificial fine tuning of the Higgs scalar potential parameters.In cases of accidental symmetries, i.e.","symmetries of the scalar potential that are not respected by the full theory, the would-be mass degeneracies are only approximate, with calculable mass splittings.","The possibility of such approximate mass degeneracies, although technically natural, is not the subject of this paper.","Possible natural mass degeneracy of the 2HDM must be the consequence of one of the symmetries listed in Table .","Starting from a generic scalar potential given by Eq.","(REF ), if the scalar potential respects one of the symmetries listed in Table , then a scalar basis is picked out in which the symmetry is manifest.","In this basis, the coefficients of the scalar potential are constrained according to Table REF .It can be shown that for each of the symmetries listed in Table REF , a scalar field basis exists in which all scalar potential parameters and the neutral scalar field vacuum expectation values are simultaneously real, in which case CP (as defined by GCP1 in Table ) is conserved by the scalar sector Lagrangian and vacuum.","It is straightforward to check that the possible discrete symmetries of the 2HDM, namely $\\mathbb {Z}_2$ , GCP1, GCP2 (or equivalently, $\\mathbb {Z}_2\\oplus \\Pi _2$ ), do not yield scalar potentials that lead to scalar mass degeneracies.","Thus, we henceforth focus on U(1), SO(3) and GCP3 (and the related U(1)$\\oplus \\Pi _2$ symmetry).","Given a 2HDM scalar potential with a Peccei-Quinn [U(1)$_{\\rm PQ}$ ] symmetry [60] (or equivalently the U(1) transformation specified in Table In Ref.","[60], a U(1)$_{\\rm PQ}$ transformation of the 2HDM scalar fields is given by $\\Phi _1\\rightarrow e^{-i\\theta }\\Phi _1$ and $\\Phi _2\\rightarrow e^{i\\theta }\\Phi _2$ .","The U(1) transformation specified in Table corresponds to combining the U(1)$_{\\rm PQ}$ transformation with a hypercharge U(1)$_{\\rm Y}$ transformation, $\\Phi _i\\rightarrow e^{i\\theta }\\Phi _i$ (for $i=1,2$ ).)","that is spontaneously broken by the vacuum, the scalar sector will contain a massless CP-odd (Goldstone) scalar [61], [62].","In such cases, no mass degeneracy is present (without further constraints on the scalar potential parameters).","However, if the U(1) symmetry is manifestly realized in the Higgs basis, then the U(1) symmetry is unbroken by the vacuum, resulting in a mass degeneracy between the two neutral scalars residing in the Higgs basis field $H_2$ .","Indeed, this has already been shown in Section REF [see text above Eq.","(REF )], in the case of the mass degeneracy, $m_H=m_A$ , of the IDM with $Z_5=0$ .","In the case of a 2HDM scalar potential with an SO(3) symmetry, the form of the scalar potential is invariant with respect to all possible changes of the scalar basis.","Hence, it follows that the scalar potential parameters in the Higgs basis satisfy $Y_1=Y_2$ , $Z_1=Z_2=Z_3+Z_4$ and $Y_3=Z_5=Z_6=Z_7=0$ .","Using Eqs.","(REF ), (REF ) and (), it follows that $m_H=m_A=0$ .","The presence of two massless (Goldstone) scalars is a consequence of the spontaneous breaking of the SO(3) global symmetry by the vacuum.","This scalar mass degeneracy is a special case of the mass degeneracy in the case of the IDM with $Z_5=0$ , in which additional constraints among the Higgs basis parameters result in the pair of massless scalar states.","Finally, let us consider the case of a 2HDM scalar potential with a GCP3 symmetry.","Suppose that the GCP3 symmetry is manifestly realized in a basis where $\\langle \\Phi _1^0 \\rangle =\\frac{v_1}{\\sqrt{2}}\\,,\\qquad \\quad \\langle \\Phi _2^0 \\rangle =\\frac{v_2}{\\sqrt{2}}e^{i\\xi }\\,,$ where $v_1$ and $v_2$ are positive.","We define $\\tan \\beta \\equiv v_2/v_1$ (so that $0<\\beta <\\tfrac{1}{2}\\pi $ ).","Then, in light of the constraints on the GCP3 scalar potential parameters given in Table REF , the scalar potential minimum conditions yield (e.g., see eqs.","(3)–(5) of Ref.","[63]), $m_{11}^2 & = & -\\tfrac{1}{2}v^2\\bigl [\\lambda _1-2\\lambda _5\\sin ^2\\beta \\sin ^2\\xi \\bigr ]\\,, \\\\m_{22}^2 & = & -\\tfrac{1}{2}v^2\\bigl [\\lambda _1-2\\lambda _5\\cos ^2\\beta \\sin ^2\\xi \\bigr ]\\,, \\\\m_{12}^2\\sin \\xi &=& v^2\\lambda _5 \\sin \\beta \\cos \\beta \\sin \\xi \\cos \\xi \\,.$ We can assume that $\\lambda _5\\ne 0$ , since otherwise we would be dealing with an SO(3)-symmetric scalar potential.","Setting $m_{11}^2=m_{22}^2$ and $m_{12}^2=0$ then yields two conditions, $\\sin ^2 \\xi \\cos 2\\beta =0\\,,\\qquad \\quad \\sin \\xi \\cos \\xi \\sin 2\\beta =0\\,.$ Hence, there are two classes of vacua, $\\sin \\xi =0$ and $\\beta $ arbitrary ($0<\\beta <\\tfrac{1}{2}\\pi $ ) , $\\cos \\xi =0$ and $\\cos 2\\beta =0$ .","We now can calculate the parameters of the GCP3 scalar potential in the Higgs basis (e.g., see eqs.","(11)–(20) of Ref.","[63]) in the two Cases A and B defined above, $Y_1=Y_2, \\quad Y_3=Z_6=Z_7=0, \\quad Z_1=Z_2=\\lambda _1, \\qquad \\quad Z_3+Z_4=\\lambda _1-\\lambda _5,\\quad Z_5=\\lambda _5$ , $Y_1=Y_2, \\quad Y_3=Z_6=Z_7=0, \\quad Z_1=Z_2=\\lambda _1-\\lambda _5, \\quad Z_3+Z_4=\\lambda _1+\\lambda _5,\\quad Z_5=0$ .","In particular, $Z_1=Z_2=Z_3+Z_4+Z_5$ in Case A, and $Z_1=Z_2\\ne Z_3+Z_4$ (and $Z_5=0$ ) in Case B.","We can now make use of Eqs.","(REF ) and () [along with Eq.","(REF ) to eliminate $Y_2$ by virtue of $Y_1=Y_2$ ] to compute the neutral scalar mass spectrum in the two cases,Note that the positivity of the squared masses are consistent with the conditions on the 2HDM scalar potential parameters first obtained in Ref. [54].","$m_h^2=Z_1 v^2,\\quad m_H^2=0,\\quad m_A^2=(Z_3+Z_4-Z_1)v^2,\\quad m_{H^\\pm }^2=\\tfrac{1}{2}(Z_3-Z_1)v^2$ , $m_h^2=Z_1 v^2,\\quad m^2_H=m^2_A=\\tfrac{1}{2}(Z_3+Z_4-Z_1) v^2,\\qquad \\,\\,\\, m_{H^\\pm }^2=\\tfrac{1}{2}(Z_3-Z_1)v^2$ .","Note that Cases A and B correspond to degenerate vacua, since in both cases the value of the scalar potential (in the Higgs basis) at its minimum is $V_{\\rm min}=-\\tfrac{1}{8} Z_1 v^4=-\\tfrac{1}{8} v^2m_h^2$ .","In light of Table REF , we can identify case A as corresponding to realizing a GCP3 symmetry in the Higgs basis,This means that we can relax the restrictions of $\\beta \\ne 0, \\tfrac{1}{2}\\pi $ in defining Case A.","Note that for $\\beta =\\tfrac{1}{2}\\pi $ one can simply interchange the definitions of $\\Phi _1$ and $\\Phi _2$ to recover the Higgs basis result (corresponding to $\\beta =0$ ).","and case B corresponding to realizing a U(1)$\\oplus \\Pi _2$ symmetry in the Higgs basis.This result is not surprising given that U(1)$\\oplus \\Pi _2$ is equivalent to GCP3 in another scalar field basis.","In particular, case A exhibits a massless Goldstone boson corresponding to the spontaneous breaking of GCP3 by the vacuum.","In contrast, in case B, the GCP3 symmetry possesses a continuous U(1) subgroup, which is unbroken by the vacuum, that protects the mass degeneracy, $m_H=m_A$ .","Indeed, this case is again a special case of the IDM with $Z_5=0$ , where the additional constraints, $Y_1=Y_2$ and $Z_1=Z_2$ are imposed.","As a check, it is instructive to evaluate the consequences of a 2HDM scalar potential with a U(1)$\\oplus \\Pi _2$ symmetry that is manifestly realized in the $\\Phi _1$ –$\\Phi _2$ basis.","In the following, we employ primed coefficients, $\\lambda _i^\\prime $ to distinguish this case from the one above where the GCP3 symmetry is manifestly realized in the $\\Phi _1$ –$\\Phi _2$ basis.","Using the results of Table REF , the scalar potential minimum conditions yield, $m_{11}^2&=& -\\tfrac{1}{2}v^2\\bigl [\\lambda ^\\prime _1\\cos ^2\\beta +(\\lambda ^\\prime _3+\\lambda ^\\prime _4)\\sin ^2\\beta \\bigr ]\\,,\\\\m_{22}^2&=& -\\tfrac{1}{2}v^2\\bigl [\\lambda ^\\prime _1\\sin ^2\\beta +(\\lambda ^\\prime _3+\\lambda ^\\prime _4)\\cos ^2\\beta \\bigr ]\\,,$ under the assumption that $0<\\beta <\\tfrac{1}{2}\\pi $ .","We can assume that $\\lambda ^\\prime _1\\ne \\lambda ^\\prime _3+\\lambda ^\\prime _4$ , since otherwise we would be dealing with an SO(3)-symmetric scalar potential.","Setting $m_{11}^2=m_{22}^2$ then yields $\\cos 2\\beta =0$ , with $\\xi $ arbitrary.","Without loss of generality, one can set $\\xi =0$ , since the scalar potential is unchanged under a rephasing of $\\Phi _2$ .","As before, we can now compute the scalar potential parameters in the Higgs basis, $&&Y_1=Y_2,\\qquad Y_3=Z_6=Z_7=0,\\qquad Z_1=Z_2=\\tfrac{1}{2}(\\lambda ^\\prime _1+\\lambda ^\\prime _3+\\lambda ^\\prime _4),\\nonumber \\\\&& Z_3+Z_4=\\lambda ^\\prime _1,\\qquad Z_5=\\tfrac{1}{2}(\\lambda ^\\prime _1-\\lambda ^\\prime _3-\\lambda ^\\prime _4)\\,.","$ In particular, note that $Z_1=Z_2=Z_3+Z_4-Z_5$ .","Using Eqs.","(REF ), (REF ) and (), we obtain $m_h^2=Z_1 v^2\\,,\\quad m_H^2=(Z_3+Z_4-Z_1)v^2\\,,\\quad m_A^2=0\\,,\\quad m_{H^\\pm }^2=\\tfrac{1}{2}(Z_3-Z_1)v^2\\,,$ which is the same mass spectrum as Case A of the GCP3-symmetric scalar potential with $H$ and $A$ interchanged.","This result can be understood by noting that Eq.","(REF ) takes the standard form of the GCP3-symmetric scalar potential in the Higgs basis after rephasing the Higgs basis field $H_2\\rightarrow iH_2$ , which interchanges $H$ and $A$ and transforms $Z_5\\rightarrow -Z_5$ .","Finally, the case of $\\beta =0$ or $\\beta =\\tfrac{1}{2}\\pi $ must be treated separately and corresponds to a manifest realization of the U(1)$\\oplus \\Pi _2$ symmetry in the Higgs basis.","This vacuum is degenerate with the one considered above, since in both cases, $V_{\\rm min}=-\\tfrac{1}{8} Z_1 v^4=-\\tfrac{1}{8} v^2m_h^2$ .","Indeed, this latter case corresponds to Case B of the GCP3-symmetric scalar potential treated above, where the neutral scalar mass spectrum exhibits a mass degeneracy, $m_H=m_A$ .","In summary, massless scalar (Goldstone boson) states $A$ , $H$ or ($A$ , $H$ ) exist in the 2HDM with a scalar potential that exhibits, respectively, a U(1), GCP3 or SO(3) symmetry manifestly realized in a generic $\\Phi _1$ –$\\Phi _2$ basis, which agrees with the results of Table 2 of Ref. [59].","Nevertheless, in the special cases where U(1) or U(1)$\\oplus \\Pi _2$ are manifestly realized in the Higgs basis (the latter corresponding to the Case B solution of the GCP3-symmetric scalar potential), the corresponding U(1) subgroups of theses symmetries are not spontaneously broken by the vacuum, and the neutral scalar mass spectrum exhibits a mass degeneracy, $m_H=m_A$ .","In the case of the SO(3)-symmetric scalar potential, this mass degeneracy is realized by a pair of massless Goldstone boson states.","Thus, we conclude that mass-degenerate neutral scalars can arise naturally in the 2HDM only in the case of the IDM with $Z_5=0$ .","All other cases of mass-degenerate scalars require an artificial fine-tuning of the scalar potential parameters, in agreement with the analysis of section REF .","Furthermore, this conclusion is unaffected by the interactions of the scalars with the vector bosons.","Indeed, the Higgs boson–gauge boson interactions, ${L}_{\\rm int}$ , given by Eq.","(REF ) show that the global U(1) symmetry responsible for the mass degeneracy of $H$ and $A$ is an exact symmetry of ${L}_{\\rm int}$ .","Finally, as previously noted, the Higgs basis field $H_2$ of the IDM is odd whereas all other scalar, fermion and vector fields are even under the discrete $\\mathbb {Z}_2$ symmetry.","This can be achieved by employing Type-I Yukawa couplings [64] where fermions couple only to the Higgs basis field $H_1$ .","In this case, the global U(1) symmetry of the IDM scalar potential with $Z_5=0$ will also be respected by the Yukawa interactions.","However, a GCP3 [or equivalently U(1)$\\oplus \\Pi _2$ ] or SO(3) symmetry of the IDM scalar potential will be explicitly broken by the Yukawa interactions.","Hence, the U(1)-symmetric IDM is the only 2HDM for which an exact mass degeneracy of $H$ and $A$ can be preserved."],["3HDM mass degeneracies and the Ivanov Silva model","In extended Higgs sectors with more than two scalar doublets, it is now possible to have mass-degenerate charged Higgs pairs as well as mass-degenerate neutral scalars [65].","In this section, we explore new phenomena associated with mass degenerate scalars that arises for the first time in the three-Higgs doublet model (3HDM).","As a warmup exercise, we return to the IDM and add a second inert doublet and consider possible mass degeneracies among the scalar fields of the two inert doublets.","We then perturb the resulting model to obtain a version of the 3HDM that is equivalent to a model first introduced by Ivanov and Silva[30]."],["The replicated inert doublet model (RIDM)","The IDM introduced in section REF can be generalized by introducing additional inert scalar doublets.","In this section, we consider a 3HDM that consists of two inert hypercharge-one electroweak doublets, in which the inert doublets contain mass-degenerate scalar states.","The resulting models shall be called the replicated inert doublet model (RIDM).","As in the case of the IDM, we work in the Higgs basis in which the first Higgs doublet field $H_1$ contains the SM Higgs boson.","The RIDM consists of $H_1$ , with $\\langle H_1^0 \\rangle =v/\\sqrt{2},$ and two inert doublet fields $H_2$ and $H_3$ , with $\\langle H_2 \\rangle =\\langle H_3 \\rangle =0$ , and a scalar potential given by, $\\mathcal {V}_{\\rm RIDM}&=& Y_1 H_1^\\dagger H_1+ Y_2 \\left(H_2^\\dagger H_2 +H_3^\\dagger H_3\\right)+\\tfrac{1}{2}Z_1(H_1^\\dagger H_1)^2+\\tfrac{1}{2}Z_2(H_2^\\dagger H_2+H_3^\\dagger H_3)^2\\nonumber \\\\&&+Z_3(H_1^\\dagger H_1)\\left(H_2^\\dagger H_2+H_3^\\dagger H_3\\right)+Z_4\\left[( H_1^\\dagger H_2)(H_2^\\dagger H_1)+( H_1^\\dagger H_3)(H_3^\\dagger H_1)\\right]\\nonumber \\\\&&+\\tfrac{1}{2}Z_5\\left\\lbrace (H_1^\\dagger H_2)^2 +(H_2^\\dagger H_1)^2+(H_1^\\dagger H_3)^2 +(H_3^\\dagger H_1)^2\\right\\rbrace \\,.$ Without loss of generality, we have chosen $Z_5$ real and non-negative in Eq.","(REF ), which is always possible by an appropriate rephasing of the scalar fields $H_2$ and $H_3$ .","Hence, if follows that the bosonic sector of the RIDM is CP-conserving.","The charged and neutral components of the Higgs basis doublet fields of the RIDM are also mass eigenstate fields, $H_1=\\begin{pmatrix} G^+ \\\\ \\frac{1}{\\sqrt{2}}\\bigl [v+h_{\\rm SM}+iG^0\\bigr ] \\end{pmatrix},\\quad H_2=\\begin{pmatrix} H^+ \\\\ \\frac{1}{\\sqrt{2}}\\bigl [H+iA\\bigr ]\\end{pmatrix},\\quad H_3=\\begin{pmatrix} h^+ \\\\ \\frac{1}{\\sqrt{2}}\\bigl [h+ia\\bigr ]\\end{pmatrix},$ with a minor change of notation from the IDM.","The corresponding squared masses of the neutral and charged scalars are given by, $m^2_{H^\\pm }&=&m^2_{h^\\pm }=Y_2+\\tfrac{1}{2}Z_3 v^2\\,,\\qquad m^2_H=m^2_h=Y_2+\\tfrac{1}{2}(Z_3+Z_4+Z_5)v^2\\,,\\nonumber \\\\m^2_A&=&m^2_a=Y_2+\\tfrac{1}{2}(Z_3+Z_4-Z_5)v^2\\,.$ By assumption, $Z_5\\ge 0$ , in which case $m_H=m_h\\ge m_A=m_a$ .In particular, note that if $Z_5=0$ then there is an enhanced mass degeneracy in which $m_H=m_h=m_A=m_a$ .","Thus, the RIDM possesses four mass-degenerate scalar pairs: $(H^\\pm , h^\\pm )$ , $(H,h)$ and $(A,a)$ .","These mass degeneracies can be understood as a consequence of a continuous global Higgs flavor symmetry (where Higgs flavor corresponds to the multiplicity of Higgs doublets).","In order to explicitly exhibit the relevant symmetries, it is convenient to focus on the neutral scalar states of the doublet fields $H_2$ and $H_3$ , denoted henceforth by the complex fields, $H^0\\equiv \\frac{H+iA}{\\sqrt{2}}\\,,\\qquad \\quad h^0\\equiv \\frac{h+ia}{\\sqrt{2}}\\,,$ respectively.","Let us first focus on the kinetic energy terms and the terms in Eq.","(REF ) in the absence of the term proportional to $Z_5$ .","Then, one can check that the neutral complex scalar fields $H^0$ and $h^0$ appear only in the combination $H^{0\\,\\dagger }H^0+h^{0\\,\\dagger }h^0=\\tfrac{1}{2}(H^2+h^2+A^2+a^2)$ .","Thus, excluding $Z_5$ , the scalar Lagrangian possesses an O(4) global symmetry, that is responsible for four mass-degenerate neutral scalar states.","It is instructive to see how this symmetry arises when employing the complex basis $\\varphi _i=\\lbrace H^0,h^0\\rbrace $ (for $i=1,2$ ).","Noting that $\\varphi ^{\\dagger \\,i}\\varphi _i=H^{0\\,\\dagger }H^0+h^{0\\,\\dagger }h^0$ (the sum over the repeated index $i$ is implicit), it is clear that the scalar Lagrangian (in the absence of $Z_5$ ) is invariant under a U(2) global symmetry, $\\varphi _i\\rightarrow U_i{}^j\\varphi _j$ , with $U\\in {\\rm U(2)}$ .","However, the corresponding symmetry group is in fact larger than U(2).","Working in the complex basis, it is straightforward to verify that the quantity $\\varphi ^{\\dagger \\,i}\\varphi _i$ is invariant with respect to $\\varphi _i\\rightarrow U_i{}^j\\varphi _j+(V^*)^i{}_j\\varphi ^{\\dagger \\,j}\\,,$ where $U$ and $V$ are complex $2\\times 2$ matrices (which are not in general unitary), provided that the following two conditions are satisfied: $&&(i)~~~~(U^\\dagger U+V^\\dagger V)_i{}^j=\\delta _i{}^j\\,,\\\\&&(ii)~~~V^T U~\\hbox{\\rm is an antisymmetric matrix}\\,.$ One can now check that Eq.","(REF ) corresponds to an O(4) symmetry transformation.","More explicitly, the $4\\times 4$ matrix, $\\mathcal {Q}=\\left(\\begin{array}{cc}\\operatorname{Re}(U+V) &\\,\\, -\\operatorname{Im}(U+V) \\\\\\operatorname{Im}(U-V) & \\phantom{-}\\,\\, \\operatorname{Re}(U-V) \\end{array}\\right)$ is an orthogonal matrix if and only if $U$ and $V$ satisfy Eqs.","(REF ) and ().","Indeed, one can check that in light of Eqs.","(REF ) and (), the global symmetry specified by Eq.","(REF ) is governed by 6 continuous parameters as expected for an O(4) transformation.","Two special cases of Eq.","(REF ) are noteworthy.","First, if $V=0$ , then $U$ is unitary and we recover the U(2) global symmetry mentioned previously.","Second, if $U=0$ then Eq.","(REF ) corresponds to a generalized CP transformation [cf. Eq.","(REF )].The unified treatment of Higgs family transformations and generalized CP transformations has been advocated previously in Ref. [58].","A related discussion emphasizing the promotion of the U(2) basis transformation to an enlarged group of O(4) transformations appears in Ref. [66].","Both symmetries are present in the scalar Lagrangian if the $Z_5$ coupling is neglected, and either one would be sufficient to guarantee the mass degeneracy of $H$ , $h$ , $A$ and $a$ .","In the absence of the $Z_5$ coupling, the full O(4) global symmetry is respected by the pure scalar Lagrangian.","However, when we include the coupling of the scalar doublets to the gauge bosons, one must replace the ordinary derivative, $\\partial _\\mu $ , with the SU(2)$\\times $ U(1) gauge covariant derivative, $D_\\mu $ , in the scalar kinetic energy term.","The resulting coupling of the scalars to the vector bosons partially breaks the O(4) symmetry.","Employing the complex basis, it is easy to check that the symmetry transformation specified by Eq.","(REF ) is unbroken if and only if either $U=0$ or $V=0$ , namely the two special cases just highlighted above.This result is not surprising given that Eq.","(REF ) transforms the scalar field into a linear combination of two fields of opposite hypercharge unless either $U=0$ or $V=0$ .","That is, the kinetic energy term $(D^\\mu \\varphi )^{i\\,\\dagger } (D_\\mu \\varphi )_i$ is invariant under a U(2) symmetry (corresponding to $V=0$ ) and under the generalized CP symmetry (corresponding to $U=0$ ).","Mathematically, the unbroken global symmetry that remains is the semi-direct product U(2)$\\rtimes \\mathbb {Z}_2$ .This symmetry is a generalization of the U(1) symmetry (and the associated CP symmetry) of the IDM with $Z_5=0$ treated in section REF , and provides the motivation for our choice of the RIDM scalar potential given in Eq.","(REF ).","We now examine the consequence of including the term of Eq.","(REF ) proportional to $Z_5$ .","Focusing again on the neutral complex scalar fields $H^0$ and $h^0$ [cf. Eq.","(REF )], we see that a new combination of fields arises, $\\varphi _i\\varphi _i+{\\rm h.c.}=(H^0)^2+(H^{0\\,\\dagger })^2+(h^0)^2+(h^{0\\,\\dagger })^2$ .","This term is invariant with respect to Eq.","(REF ) provided that the conditions specified in Eqs.","(REF ) and () are replaced by 1.25 $&&(i^\\prime )~~~~(U^T U+V^T V)_i{}^j=\\delta _i{}^j\\,,\\\\&&(ii^\\prime )~~~V^\\dagger U~\\hbox{\\rm is an antihermitian matrix}\\,.$ The conditions specified by Eqs.","(REF ) and () are compatible with those of Eqs.","(REF ) and () if $U$ and $V$ are real matrices.","Consequently, $\\mathcal {Q}$ specified in Eq.","(REF ) is now a block diagonal orthogonal $4\\times 4 $ matrix, $\\mathcal {Q}=\\left(\\begin{array}{cc}U+V &\\,\\, 0 \\\\0 & \\phantom{-}\\,\\, U-V \\end{array}\\right)\\,,$ where $(U\\pm V)^T(U\\pm V)={1}_{2\\times 2}$ as a consequence of Eqs.","() and (REF ).","That is, the scalar Lagrangian is invariant under a global O(2)$\\times $ O(2) symmetry, which explains the presence of the mass-degenerate scalars $(H,h)$ and $(A,a)$ , respectively.","The breaking of the four-fold mass degeneracy to the two mass-degenerate pairs is due to the scalar potential term proportional to $Z_5$ , as is evident from Eq.","(REF ).","Finally, after promoting the derivative to the gauge covariant derivative in the scalar kinetic energy term, the remaining symmetry is O(2)$\\rtimes \\mathbb {Z}_2$ .","For completeness, we note that the degeneracy of the charged Higgs scalars $(H^\\pm ,h^\\pm )$ is governed by the full O(4) symmetry, which is broken down to U(2)$\\rtimes \\mathbb {Z}_2$ after promoting the derivatives of the scalar kinetic energy term to gauge covariant derivatives.","This is easily seen by noting that in the unitary gauge (in which the Goldstone fields do not explicitly appear), the physical charged scalar fields do not appear in the scalar potential term proportional to $Z_5$ .","Finally, if $Z_4=Z_5=0$ , we can make use of the vertical SU(2) global symmetry (which when gauged corresponds to the SU(2) electroweak gauge group) to conclude that all eight charged and neutral inert scalars are mass-degenerate.","Next, we examine all the bosonic couplings of the RIDM in the unitary gauge (where the Goldstone fields are set to zero).","The Higgs boson interactions with the gauge bosons and the Higgs boson self couplings of the RIDM are listed below.","${L}_{VVH}&=&\\left(gm_W W_\\mu ^+W^{\\mu \\,-}+\\frac{g}{2c_W} m_ZZ_\\mu Z^\\mu \\right)h_{\\rm SM}\\,,\\\\[8pt]{L}_{VVHH}&=&\\left[\\tfrac{1}{4}g^2 W_\\mu ^+W^{\\mu \\,-}+\\frac{g^2}{8c_W^2}Z_\\mu Z^\\mu \\right](h_{\\rm SM}^2+H^2+h^2+A^2+a^2) \\nonumber \\\\&& +\\left[\\tfrac{1}{2}g^2 W_\\mu ^+ W^{\\mu \\,-}+e^2A_\\mu A^\\mu +\\frac{g^2}{c_W^2}\\left(\\tfrac{1}{2}-s_W^2\\right)^2Z_\\mu Z^\\mu +\\frac{2ge}{c_W}\\left(\\tfrac{1}{2}-s_W^2\\right)A_\\mu Z^\\mu \\right](H^+H^- +h^+ h^-)\\nonumber \\\\&& +\\biggl \\lbrace \\left(\\tfrac{1}{2}eg A^\\mu W_\\mu ^+-\\frac{g^2s_W^2}{2c_W}Z^\\mu W_\\mu ^+\\right)\\bigl [H^-(H+iA)+h^-(h+ia)\\bigr ] +{\\rm h.c.}\\biggr \\rbrace \\,, \\\\{L}_{VHH}&=&\\frac{g}{2c_W}\\,Z^\\mu ( A\\,\\!\\!\\mathrel {\\leftrightarrow \\hspace{-8.50006pt}\\partial }$ H+a $\\leftrightarrow $$\\partial $ h) -12g{iW+[ H-$\\leftrightarrow $$\\partial $$\\scriptstyle \\,\\mu $ (H+iA)+h-$\\leftrightarrow $$\\partial $$\\scriptstyle \\,\\mu $ (h+ia)] +h.c.}","+[ieA+igcW(12-sW2) Z](H+$\\leftrightarrow $$\\partial $ H-+h+$\\leftrightarrow $$\\partial $ h-), In the RIDM, there is no experimental measurement that can physically distinguish the degenerate scalars, $(H^\\pm ,h^\\pm )$ , $(H,h)$ and $(A,a)$ .","However, a multiplicity factor will appear after summing over final mass-degenerate states, e.g., $Z\\rightarrow HA$ , $ha$ doubles the rate into a pair of neutral scalars."],["An alternative basis choice for the RIDM","So far, our discussion has employed the $\\lbrace H_1,H_2,H_3\\rbrace $ basis of doublet scalar fields.","This is one choice among a family of Higgs bases defined such that $\\langle H_1^0 \\rangle =v/\\sqrt{2}$ and $\\langle H_2^0 \\rangle =\\langle H_3^0 \\rangle =0$ .","Indeed, the Higgs basis is unique only up to an arbitrary U(2) transformation of the doublet fields $H_2$ and $H_3$ .","In the following, we shall denote the $\\lbrace H_1,H_2,H_3\\rbrace $ basis as the $H23$ -basis, since the scalar potential of Eq.","(REF ) provides a simple 3HDM extension of the inert 2HDM.","It will prove useful to consider another choice of scalar field basis that is related to the $H23$ -basis as follows,Further details are provided in Appendix REF .","$&&\\mathcal {R}\\equiv \\frac{1}{\\sqrt{2}}\\bigl (H_2+iH_3\\bigr )=\\begin{pmatrix} R^\\dagger \\\\ \\frac{1}{\\sqrt{2}}\\bigl (P+iQ^\\dagger \\bigr )\\end{pmatrix}\\,,\\nonumber \\\\[15pt]&&\\mathcal {S}\\equiv \\frac{1}{\\sqrt{2}}\\bigl (H_2-iH_3\\bigr )=\\begin{pmatrix} S^\\dagger \\\\ \\frac{1}{\\sqrt{2}}\\bigl (P^\\dagger +iQ\\bigr )\\end{pmatrix}\\,.$ This defines the $\\lbrace H_1,\\mathcal {R},\\mathcal {S}\\rbrace $ basis of doublet scalar field, henceforth denoted as the $RS$ -basis.","Note that since the real neutral fields $(H,h)$ and $(A,a)$ are mass-degenerate pairs, respectively, one can combine the mass-degenerate real fields into complex fields, $P\\equiv \\frac{H+ih}{\\sqrt{2}}\\,,\\qquad Q\\equiv \\frac{A-ia}{\\sqrt{2}}\\,,$ where $M_P\\ge M_Q$ (in our convention where $Z_5\\ge 0$ ).The relative minus sign in the definition of the imaginary parts of $P$ and $Q$ has been introduced for later convenience.","The corresponding conjugate fields are $P^\\dagger \\equiv \\frac{H-ih}{\\sqrt{2}}\\,,\\qquad Q^\\dagger \\equiv \\frac{A+ia}{\\sqrt{2}}\\,,$ Likewise, since $H^\\pm $ and $h^\\pm $ are mass-degenerate charged fields, one is free to define, $R&=\\frac{H^- - ih^-}{\\sqrt{2}}, &\\quad S&=\\frac{H^- + ih^-}{\\sqrt{2}}, \\\\R^\\dagger &=\\frac{H^+ + ih^+}{\\sqrt{2}}, &\\quad S^\\dagger &=\\frac{H^+ - ih^+}{\\sqrt{2}},$ where $R$ and $S$ are negatively charged mass-degenerate scalars and the corresponding conjugate fields, $R^\\dagger $ and $S^\\dagger $ , are positively charged mass-degenerate scalars.","In the $RS$ -basis, the scalar potential is given by $\\!\\!\\!\\!\\!\\!\\mathcal {V}_{\\rm RIDM-RS}&=& Y_1 H_1^\\dagger H_1+ Y_2 \\bigl (\\mathcal {R}^\\dagger \\mathcal {R} +\\mathcal {S}^\\dagger \\mathcal {S}\\bigr )+\\tfrac{1}{2}Z_1(H_1^\\dagger H_1)^2+\\tfrac{1}{2}{\\bar{Z}}_2(\\mathcal {R}^\\dagger \\mathcal {R}+\\mathcal {S}^\\dagger \\mathcal {S})^2+Z_3(H_1^\\dagger H_1)(\\mathcal {R}^\\dagger \\mathcal {R}+\\mathcal {S}^\\dagger \\mathcal {S}) \\nonumber \\\\&& +Z_4\\bigl [( H_1^\\dagger \\mathcal {R})(\\mathcal {R}^\\dagger H_1)+( H_1^\\dagger \\mathcal {S})(\\mathcal {S}^\\dagger H_1)\\bigr ]+ \\bar{Z}^\\prime _5\\bigl [(H_1^\\dagger \\mathcal {R})(H_1^\\dagger \\mathcal {S}) +(\\mathcal {R}^\\dagger H_1)(\\mathcal {S}^\\dagger H_1)\\bigr ]\\,,$ where $\\bar{Z}_2=Z_2$ and $\\bar{Z}_5^\\prime =Z_5$ .The reason for introducing the notation $\\bar{Z}_2$ and $\\bar{Z}^\\prime _5$ in Eq.","(REF ) is clarified in section REF .","One can then rewrite the RIDM couplings given in Eqs.","(REF )–() in terms of the neutral scalar fields $P$ and $Q$ and the charged scalar fields $R$ and $S$ (and the corresponding conjugated fields), $&&\\hspace{-14.45377pt}{L}_{VVHH}=\\left[\\tfrac{1}{4} g^2 W_\\mu ^+W^{\\mu \\,-}+\\frac{g^2}{8c_W^2}Z_\\mu Z^\\mu \\right]\\bigl (h^2_{\\rm SM}+2|P|^2+2|Q|^2\\bigr ) \\nonumber \\\\&& +\\left[\\tfrac{1}{2}g^2 W_\\mu ^+ W^{\\mu \\,-}+e^2A_\\mu A^\\mu +\\frac{g^2}{c_W^2}\\left(\\tfrac{1}{2}-s_W^2\\right)^2Z_\\mu Z^\\mu +\\frac{2ge}{c_W}\\left(\\tfrac{1}{2}-s_W^2\\right)A_\\mu Z^\\mu \\right](R^\\dagger R+S^\\dagger S)\\nonumber \\\\&&+\\biggl \\lbrace \\left(\\tfrac{1}{2}eg A^\\mu W_\\mu ^+-\\frac{g^2s_W^2}{2c_W}Z^\\mu W_\\mu ^+\\right)\\bigl [R(P+iQ^\\dagger )+S(P^\\dagger +iQ)\\bigr ] +{\\rm h.c.}\\biggr \\rbrace ,\\\\&&\\hspace{-14.45377pt}{L}_{VHH}=\\frac{g}{2c_W}\\,Z^\\mu ( Q\\,\\!\\!\\mathrel {\\leftrightarrow \\hspace{-8.50006pt}\\partial }$ P+Q $\\leftrightarrow $$\\partial $ P) +[ieA+igcW(12-sW2) Z] (R$\\leftrightarrow $$\\partial $ R+S$\\leftrightarrow $$\\partial $ S) -12g{iW+[R $\\leftrightarrow $$\\partial $ (P+iQ) +S $\\leftrightarrow $$\\partial $ (P+iQ)] +h.c.}","L3h=-12vZ1 hSM3-v[(Z3+Z4)hSM(|P|2+|Q|2)+Z5hSM(|P|2-|Q|2)] -vZ3hSM(RR+SS) , line .","L4h=-18 Z1 hSM4 -12Z2(|P|2+|Q|2)(|P|2+|Q|2+2RR + 2SS)-12(Z3+Z4) hSM2(|P|2+|Q|2) -12Z5 hSM2 (|P|2-|Q|2)-12[Z2(RR+SS) +Z3 h2SM] (RR+SS) ."],["Mass degeneracies beyond the RIDM","In this section, we add additional terms to the RIDM scalar potential while preserving the mass degeneracies of the model.","Naively, one can add to the RIDM scalar potential any gauge invariant quartic term involving the doublet fields $H_2$ and $H_3$ without upsetting the mass degeneracies of Eq.","(REF ).","However, the resulting tree-level mass degeneracies will be unnatural unless they are a consequence of a symmetry.","The simplest possible modification of the RIDM is to remove the $(H^\\dagger _2 H_2)(H^\\dagger _3 H_3)$ term entirely from the scalar potential.","That is, we can define a RIDM$^\\prime $ scalar potential as, $\\mathcal {V}_{\\rm RIDM^{\\prime }}= \\mathcal {V}_{\\rm RIDM}-Z_2(H^\\dagger _2 H_2)(H^\\dagger _3 H_3)\\,.$ Note that the term in $\\mathcal {V}_{\\rm RIDM^{\\prime }}$ that is proportional to $Z_2$ is now given by $\\tfrac{1}{2}\\bigl [(H_2^\\dagger H_2)^2+(H_3^\\dagger H_3)^2\\bigr ]$ .","Indeed, one can argue that Eq.","(REF ) provides the simplest 3HDM generalization of the IDM.","In the case of the RIDM$^\\prime $ , the tree-level mass degeneracies are no longer a consequence of a continuous symmetry, which is now explicitly broken by the presence of the explicit term in Eq.","(REF ) that is proportional to $(H^\\dagger _2 H_2)(H^\\dagger _3 H_3)$ .","Indeed, this term is invariant only under a discrete subgroup of O(2)$\\times $ O(2) [which is the symmetry group of the RIDM scalar Lagrangian as discussed in section REF ].","In the notation of Eqs.","(REF ) and (REF ), consider the following two discrete subgroups of the O(2)$\\times $ O(2) symmetry group, $&& (1)~~~U=g\\,,\\qquad \\quad V=0\\,, \\\\&& (2)~~~U=0\\,,\\qquad \\quad V=g\\,,$ where $g$ is a $2\\times 2$ matrix that acts on the Higgs basis fields $H_2$ and $H_3$ regarded as a two dimensional vector.","Then, the term $(H^\\dagger _2 H_2)(H^\\dagger _3 H_3)$ is invariant under the two discrete subgroups above if $g\\in D_4\\mathchoice{\\cong }{\\cong }{\\vbox {{ \\scriptstyle \\hfil # \\hfil \\crcr \\sim \\crcr = \\crcr }}}{\\cong }\\lbrace {1},-{1}, R,-R,S,-S,Z,-Z\\rbrace $ , where 1 is the $2\\times 2$ identity matrix and $R=\\begin{pmatrix} 1 &\\,\\,\\,\\phantom{-}0 \\\\ 0 &\\,\\,\\, \\phantom{-}1\\end{pmatrix}\\,,\\qquad \\quad S=\\begin{pmatrix} 1 &\\,\\,\\, \\phantom{-}0 \\\\ 0 &\\,\\,\\,-1\\end{pmatrix}\\,,\\qquad \\quad Z=\\begin{pmatrix} 0 &\\,\\,\\, -1 \\\\ 1 &\\,\\,\\, \\phantom{-}0\\end{pmatrix}\\,.$ We recognize $D_4$ as the dihedral group of order eight, which is the symmetry group of the square [67].","Both discrete subgroups [Eqs.","(REF ) and ()] are isomorphic to $D_4$ .","Following the discussion below Eq.","(REF ), we conclude that the RIDM$^\\prime $ scalar Lagrangian is invariant under a discrete $D_4\\times D_4$ symmetry, which is responsible for the presence of the mass-degenerate scalars $(H,h)$ and $(A,a)$ , respectively.","Finally, after promoting the derivative to the gauge covariant derivative in the scalar kinetic energy term, the remaining symmetry is $D_4\\rtimes \\mathbb {Z}_2$ .","A comprehensive treatment of natural scalar mass degeneracies in the 3HDM would require a complete classification of 3HDM scalar potential symmetries, along the lines of the 2HDM analysis given in section REF .A complete catalog of all possible finite symmetry groups of the 3HDM is known (as well as some additional partial results); however the complete classification of all possible symmetries of the 3HDM remains an open problem [68], [69].","In this paper, we shall ask a less ambitious question: can one break the discrete symmetry identified above further while still naturally maintaining the mass-degenerate states of the RIDM.","The answer turns out to be affirmative.","This investigation led us to a particular 3HDM originally introduced by Ivanov and Silva [30] for other reasons that will be reviewed below.","The Ivanov-Silva (IS) model was constructed to exhibit a number of curious properties [30], [66], which appear to rely on the existence of degenerate states in the scalar spectrum.","In particular, the IS scalar potential does not respect the conventional CP symmetry, $H_i\\rightarrow H_i^{\\scriptscriptstyle {\\bigstar }}$ , where the latter satisfies $({\\rm CP})^2={1}$ , but instead respects a generalized CP symmetry of the form $H_i\\rightarrow X_{ij}H_j^{\\scriptscriptstyle {\\bigstar }}$ for some unitary matrix $X$ .","In particular, the generalized CP symmetry of the IS scalar potential, denoted by CP4, is of order 4, signifying that $({\\rm CP4})^4={1}$ and $({\\rm CP4})^2\\ne {1}$ .","Moreover, no Higgs basis of scalar fields exists in which all the parameters of the IS scalar potential are simultaneously real.","As noted in section REF , this property is in stark contrast with the 2HDM in which the existence of any generalized CP symmetry implies that the 2HDM scalar potential automatically respects the conventional CP symmetry, i.e.","a basis of scalar fields exists such that the corresponding 2HDM scalar potential parameters are real [56], [57].","In Appendix A, we demonstrate that starting from the IS scalar potential, one can perform a basis change in order to obtain a more convenient form of the scalar potential.","By making an appropriate U(2) transformation to define the Higgs basis fields, $H_2$ and $H_3$ , we find that the IS scalar potential takes on the following form in the $H23$ -basis, $\\mathcal {V}_{\\rm IS}= \\mathcal {V}_{\\rm RIDM}+Z_3^\\prime (H_2^\\dagger H_2)(H_3^\\dagger H_3)+Z_4^\\prime (H_2^\\dagger H_3)(H_3^\\dagger H_2) +\\bigl [Z_8 (H_2^\\dagger H_3)^2+Z_9(H_2^\\dagger H_3)(H_2^\\dagger H_2-H_3^\\dagger H_3)+{\\rm h.c.}\\bigr ]\\,,$ where $\\mathcal {V}_{\\rm RIDM}$ is given in Eq.","(REF ).","In general, $Z_8$ and $Z_9$ are complex parameters.As shown in Appendix REF , one can perform an SO(2) rotation to redefine the fields $H_2$ and $H_3$ to remove the complex phase from either $Z_8$ or $Z_9$ .","We shall continue to use Eq.","(REF ) to express the Higgs basis fields in terms of mass-eigenstate fields.","Since none of the extra terms in Eq.","(REF ) involve the Higgs basis field $H_1$ , the tree-level mass relations of Eq.","(REF ) are not modified.","We now argue that the mass-degeneracies of $(H^\\pm , h^\\pm )$ , $(H,h)$ and $(A,a)$ are stable due to the presence of a symmetry.","The O(2)$\\times $ O(2) symmetry of the RIDM (prior to gauging the scalar kinetic energy terms) that is responsible for the mass degeneracies among the neutral Higgs mass eigenstates is broken by the new terms beyond $\\mathcal {V}_{\\rm RIDM}$ contained in Eq.","(REF ).","Indeed, after the extra terms are included, no unbroken continuous subgroup of O(2)$\\times $ O(2) remains.","In the notation of Eqs.","(REF ) and (REF ), consider the following two discrete subgroups of the O(2)$\\times $ O(2) symmetry group, $&& (1)~~~U=Z\\,,\\qquad \\quad V=0\\,, \\\\&& (2)~~~U=0\\,,\\qquad \\quad V=Z\\,,$ where $Z$ is given by Eq.","(REF ).","The $2\\times 2$ matrix $Z$ acts on the Higgs basis fields $H_2$ and $H_3$ .","Both discrete subgroups [Eqs.","(REF ) and ()] are isomorphic to $\\mathbb {Z}_4=\\bigl \\lbrace {1},-{1},Z,-Z\\bigr \\rbrace $ .","Note that $Z^2=-{1}$ , where 1 is the $2\\times 2$ identity matrix.In this case, gauging the scalar kinetic energy terms does not reduce the symmetry group further.","Consider first the discrete symmetry defined in Eq.","(REF ).","The fields $H_2$ and $H_3$ are odd under $-{1}$ , which simply identifies the two inert doublets.","The elements $Z$ (and $-Z$ ) act non-trivially on the inert doublets.","However, Eq.","(REF ) is invariant with respect to $\\begin{pmatrix} H_2\\\\ H_3\\end{pmatrix}\\rightarrow \\begin{pmatrix} 0 & -1\\\\ 1 & \\phantom{-}0\\end{pmatrix}\\begin{pmatrix} H_2\\\\ H_3\\end{pmatrix}\\,,$ if and only if $Z_8$ and $Z_9$ are both real.","In the model of IS where there is an unremovable complex phase in the scalar potential, only the subgroup $\\mathbb {Z}_2=\\lbrace {1},-{1}\\rbrace $ of $\\mathbb {Z}_4$ survives.","In particular, the residual symmetry in this case is not sufficient to explain the mass degeneracies of the IS model.","The discrete symmetry defined in Eq.","() is a generalized CP symmetry.","In particular, the IS scalar potential is invariant under $H_i\\rightarrow X_{ij}H_j^{\\scriptscriptstyle {\\bigstar }}\\,, \\qquad \\quad \\text{where $X=\\begin{pmatrix} 1 &\\phantom{-}0 & \\phantom{-}0 \\\\ 0 & \\phantom{-}0 & -1 \\\\ 0 & \\phantom{-}1 & \\phantom{-}0\\end{pmatrix}$,}$ This symmetry, which is also isomorphic to $\\mathbb {Z}_4$ , is the CP4 symmetry advertised above.","Moreover, this discrete symmetry is sufficient to explain the mass degeneracies of the IS model (in the case of an unremovable complex phase in the IS scalar potential).","It is instructive to consider the Higgs couplings of the IS model.","Only the quartic Higgs couplings of the RIDM are modified as follows, $\\delta {L}_{4h}&=&-\\tfrac{1}{4}(Z_3^\\prime +Z_4^\\prime )\\bigl [(H^2+A^2)(h^2+a^2)+4H^+ H^- h^+ h^-\\bigr ]-\\tfrac{1}{2}Z_3^\\prime \\bigl [(H^2+A^2)h^+ h^-+(h^2+a^2)H^+ H^-\\bigr ]\\nonumber \\\\&& -\\tfrac{1}{2}Z^{\\prime }_4\\bigl [\\bigl (Hh+Aa+i(Ha-hA)\\bigr )H^+ h^- + \\bigl (Hh+Aa-i(Ha-hA)\\bigr )h^+ H^-\\bigr ] \\nonumber \\\\&& -\\tfrac{1}{4} Z_8\\bigl [Hh+Aa+i(Ha-hA)+2h^+ H^-\\bigr ]^2-\\tfrac{1}{4} Z^*_8\\bigl [Hh+Aa-i(Ha-hA)+2H^+ h^-\\bigr ]^2 \\nonumber \\\\&& -\\tfrac{1}{4} Z_9\\bigl (H^2+A^2-h^2-a^2+2H^+ H^- -2h^+ h^-\\bigr )\\bigl [Hh+Aa+i(Ha-hA)+2h^+ H^-\\bigr ]\\nonumber \\\\&& -\\tfrac{1}{4} Z^*_9\\bigl (H^2+A^2-h^2-a^2+2H^+ H^- -2h^+ h^-\\bigr )\\bigl [Hh+Aa-i(Ha-hA)+2H^+ h^-\\bigr ]\\,.$ It is convenient to re-express the neutral scalar fields appearing in Eq.","(REF ) in terms of the complex neutral fields $P$ and $Q$ and their conjugates introduced in Eqs.","(REF ) and (REF ), and the charged fields $R$ and $S$ and their conjugates defined in Eqs.","(REF ) and ().","Note that the fields $P$ , $Q$ and the corresponding conjugate fields $P^\\dagger $ and $Q^\\dagger $ are each eigenstates of CP4.This means that each of the four states, $P$ , $Q$ , $P^\\dagger $ and $Q^\\dagger $ , are CP4-self conjugate (they are their own antiparticles).","Moreover, $P$ and the corresponding conjugate state $P^\\dagger $ are mass-degenerate, but are otherwise unrelated fields (and similarly for $Q$ and $Q^\\dagger $ ).","In particular, under a CP4 transformation, $P\\rightarrow iP$ , $Q\\rightarrow iQ$ , $P^\\dagger \\rightarrow -iP^\\dagger $ , and $Q^\\dagger \\rightarrow -iQ^\\dagger $ .","Likewise, under a CP4 transformation, $R\\rightarrow -iS^\\dagger $ , $R^\\dagger \\rightarrow iS$ , $S\\rightarrow iR^\\dagger $ and $S^\\dagger \\rightarrow -iR$ .","Note that these transformation properties are consistent with the requirement that $({\\rm CP}4)^4={1}$ .","We can evaluate the four-scalar interaction Lagrangian directly in the $RS$ -basis.","We first must rewrite Eq.","(REF ) in the RS-basis, $\\mathcal {V}_{\\rm IS-RS}= \\mathcal {V}_{\\rm RIDM-RS}+\\bar{Z}_3^\\prime (\\mathcal {R}^\\dagger \\mathcal {R})(\\mathcal {S}^\\dagger \\mathcal {S})+\\bar{Z}_4^\\prime (\\mathcal {R}^\\dagger \\mathcal {S})(\\mathcal {S}^\\dagger \\mathcal {R}) +\\bigl [\\bar{Z}_8 (\\mathcal {R}^\\dagger \\mathcal {S})^2+\\bar{Z}_9(\\mathcal {R}^\\dagger \\mathcal {S})(\\mathcal {R}^\\dagger \\mathcal {R}-\\mathcal {S}^\\dagger \\mathcal {S})+{\\rm h.c.}\\bigr ]\\,,$ where $\\mathcal {V}_{\\rm RIDM-RS}$ is given by Eq.","(REF ).","The relations between the unbarred and barred parameters are derived in Appendix REF , $\\bar{Z}_2&=&Z_2+\\tfrac{1}{2}(Z_3^\\prime +Z_4^\\prime -2\\operatorname{Re}Z_8)\\,,\\qquad \\qquad \\quad \\,\\,\\,\\,\\bar{Z}_3^\\prime =-Z_4^\\prime +2\\operatorname{Re}Z_8\\,,\\\\\\bar{Z}_4^\\prime &=&\\tfrac{1}{2}(Z_4^\\prime -Z_3^\\prime +2\\operatorname{Re}Z_8)\\,,\\qquad \\qquad \\qquad \\qquad \\bar{Z}_5^\\prime =Z_5\\,,\\\\\\bar{Z}_8&=&-\\tfrac{1}{4}(Z_3^\\prime +Z_4^\\prime +2\\operatorname{Re}Z_8)+i\\operatorname{Re}Z_9\\,,\\qquad \\quad \\bar{Z}_9= \\operatorname{Im}Z_9+i\\operatorname{Im}Z_8\\,.","$ The quartic interactions given in Eq.","(REF ) are then modified by employing the new definition of $\\bar{Z_2}$ given in Eq.","(REF ) and adding the following terms, We now consider the possible effects of the Yukawa interactions.","It is remarkable that it is possible to construct a CP4-invariant Yukawa interaction Lagrangian where the fermions transform nontrivially under a CP4 transformation [66], [70], [71].","In such a model, the mass degeneracies identified above that are a consequence of the CP4 symmetry are of course maintained.","Alternatively, if the fermions couple exclusively to the Higgs basis field $H_1$ (as in the case of the IDM), then the Yukawa interactions are invariant with respect to the $\\mathbb {Z}_2$ discrete symmetry defined below Eq.","(REF ),Note that this $\\mathbb {Z}_2$ symmetry is isomorphic to $(\\rm {CP}4)^2$ , which remains an exact symmetry of the model.","under which the inert doublet fields, $H_2$ and $H_3$ , are odd and all other fields of the model ($H_1$ , gauge bosons and fermions) are even.","However, the CP4 symmetry is no longer a symmetry of the complete model.","That is, if we define the CP4 transformation to be the conventional CP transformation when acting on the fermions and gauge fields, then the CP4 symmetry of the model will be violated by the presence of the unremovable CP-violating phase in the CKM mixing matrix.","Nevertheless, it is not clear whether this violation is sufficient to remove the scalar mass degeneracies of the IS model that were protected by the (now accidental) CP4 symmetry of the scalar potential.","This is an open question that we hope to revisit in a future work.","Finally, it is instructive to note that the scalar mass degeneracies of the CP4-invariant 3HDM is just the simplest example of a larger class of multi-Higgs models with degenerate scalars that are a consequence of a generalized CP symmetry.","In Ref.","[72], Ivanov and Laletin demonstrate how to construct $N$ Higgs doublet models with a generalized CP symmetry of order $2k$ (denoted by CP$2k$ ) with positive integer $k$ .","Nontrivial cases arise only for $2k=2^p$ with integer $p\\ge 1$ .","The simplest nontrivial models of this type (CP8 and CP16) require at least $N=5$ Higgs doublets.","Such models necessarily have mass-degenerate neutral scalars and mass-degenerate charged Higgs pairs.","A further exploration of models of this type is beyond the scope of this work."],["An observable distinction between CP2 and CP4","The distinction between the IS scalar potential in the $H23$ -basis with $Z_8$ and $Z_9$ real or complex is physical.In making this assertion, we have implicitly assumed that $Z_5\\ne 0$ .","The case of $Z_5=0$ , which is special due to the enhanced mass degeneracy noted in footnote REF , will be treated at the end of this section.","To demonstrate this assertion, we focus on the neutral scalar self-interactions in $\\delta {L}_{4h}$ that are linear in the fields $P$ or $Q$ (or their complex conjugates), $\\delta {L}_{4h}\\ni \\tfrac{1}{2}\\operatorname{Im}Z_8\\bigl [(PQ-P^\\dagger Q^\\dagger )(P^2-Q^2-P^{\\dagger \\,2}+Q^{\\dagger \\,2})\\bigr ]+\\tfrac{1}{2}i\\operatorname{Im}Z_9\\bigl [(PQ-P^\\dagger Q^\\dagger )(P^2+Q^2+P^{\\dagger \\,2}+Q^{\\dagger \\,2})\\bigr ]\\,,$ where we have used Eq.","() to re-express $\\bar{Z}_9$ [which appears in Eq.","(REF )] in terms of the $H23$ -basis parameters, $\\operatorname{Im}Z_8$ and $\\operatorname{Im}Z_9$ .","Self-interaction terms of this type are absent if $Z_8$ and $Z_9$ are both real.","Hence, the presence of these terms signals a CP4-symmetric IS scalar potential that does not respect the conventional CP symmetry, $H_i\\rightarrow H_i^{\\scriptscriptstyle {\\bigstar }}$ .","Here we provide two specific examples.","First, Eq.","(REF ) shows the existence of a $ZPQ$ interaction, which would permit the decay $Z\\rightarrow PQ, P^*Q^*$ , if kinematically available.","Since $M_Q\\le M_P$ , let us further suppose that $M_Q<\\tfrac{1}{4} m_Z<M_P$ .","In this case, the $P$ and $P^*$ would be virtual.","One possible decay of the virtual $P$ or $P^*$ makes use of the existence of the four-scalar interaction given in Eq.","(REF ).","If this interaction is present, the decay $Z\\rightarrow QQQQ^*$ , $Q^* Q^* Q^* Q$ is allowed and provides unambiguous evidence that either $Z_8$ and/or $Z_9$ possesses a nonzero imaginary part.","A second example makes use of the $W^+H^-P$ , $W^+h^-P$ , $W^+H^-Q$ , and $W^+h^-Q$ interactions of Eq.","(REF ).","In this case, we can consider the decay of a charged $W$ into a charged Higgs boson and $P$ (or $P^*$ ).","We can now make use of Eq.","(REF ) to decay the virtual $P$ or $P^*$ into $QQQ$ , $QQQ^*$ , $QQ^*Q^*$ , $Q^* Q^* Q^*$ .","Note that in each of the two cases above, there are multiple four-scalar final states involving mass-degenerate scalars.","In computing the experimentally observed rates, one must compute the squared amplitude for each of the possible final states, and then multiply the final result by a multiplicity factor that counts the number of possible final states.","In contrast, suppose that Eq.","(REF ) were a symmetry of the IS scalar potential.","In this case, the corresponding transformation properties of the scalar fields are, $P\\rightarrow iP$ , $Q\\rightarrow -iQ$ , $P^\\dagger \\rightarrow -iP^\\dagger $ , $Q^\\dagger \\rightarrow iQ^\\dagger $ , $H^{\\pm }\\rightarrow -h^\\pm $ , and $h^\\pm \\rightarrow H^\\pm $ .","One would then immediately conclude that $Z_8=Z_8^*$ and $Z_9=Z_9^*$ , as expected.","In particular, Eq.","(REF ) is not invariant under Eq.","(REF ), and thus the four scalar decay modes listed above would necessarily be absent.","As an exercise, we have evaluated the decay rate for $Z\\rightarrow QQQQ^*$ , $QQ^* Q^* Q^*$ , in an approximation where $M_Q=0$ and $M_P\\gg m_Z$ .","The computation is presented in Appendix .","The end result is $\\frac{\\Gamma (Z\\rightarrow QQQQ^*,QQ^* Q^* Q^*)}{\\Gamma (Z\\rightarrow \\nu \\bar{\\nu })}=\\frac{(\\operatorname{Im}Z_8)^2+(\\operatorname{Im}Z_9)^2}{3\\cdot 5\\cdot 2^{8}\\, \\pi ^4 }\\left(\\frac{m_Z}{M_P}\\right)^4\\,.$ This result implies that the quantity $(\\operatorname{Im}Z_8)^2+(\\operatorname{Im}Z_9)^2$ must be a physical quantity, and hence invariant with respect to scalar basis changes that are consistent with the form of the IS scalar potential given by Eq.","(REF ) in the $H23$ -basis.","However, the family of Higgs bases is larger than the set of scalar field bases in which the IS scalar potential has the form of Eq.","(REF ), as discussed in Appendix REF .","In special cases, it is possible that there exists a real Higgs basis even if $(\\operatorname{Im}Z_8)^2+(\\operatorname{Im}Z_9)^2\\ne 0$ .","In such cases, one can transform the fields ($H_2, H_3)\\rightarrow (\\bar{H}_2,\\bar{H}_3)$ , where $\\bar{H}_i\\rightarrow \\bar{H_i}^{\\scriptscriptstyle {\\bigstar }}$ is a symmetry of the Lagrangian; i.e., the model exhibits a CP2 symmetry.The notation CP2 derives from the property, $(\\text{CP2})^2={1}$ .","In the IS model, the existence of a nonzero decay rate for $Z\\rightarrow QQQQ^*$ , $QQ^* Q^* Q^*$ implies that no CP2 symmetry that commutes with the CP4 symmetry is present.We say that the CP2 symmetry commutes with CP4 if the application of these two symmetry transformations on the scalar fields does not depend on the order in which the transformations are applied.","For further details see Appendix B of Ref. [32].","However, this leaves open the possibility of a CP2 symmetry that does not commute with CP4.","In Appendix REF , we provide two examples in which $(\\operatorname{Im}Z_8)^2+(\\operatorname{Im}Z_9)^2\\ne 0$ in the $H23$ basis, but nevertheless a real Higgs basis exists: (i) $\\operatorname{Im}Z_8\\ne 0$ and $Z_9=0$ and (ii) $\\operatorname{Im}Z_8=0$ , $\\operatorname{Re}Z_9=0$ and $\\operatorname{Im}Z_9\\ne 0$ .","In both these examples, the CP2 symmetry that exists does not commute with the CP4 symmetry, even though the decay rate for $Z\\rightarrow QQQQ^*$ , $QQ^* Q^* Q^*$ is nonzero.","In light of the results of Appendix REF , this is a generic feature of a noncommuting CP2 symmetry in the IS model.","Equivalently, the nonexistence or existence of the decay $Z\\rightarrow QQQQ^*$ , $QQ^* Q^* Q^*$ is a physical distinction between the 3HDM with a CP4 symmetric IS scalar potential that either preserves or does not preserve a commuting CP2 symmetry.","Finally, we return to the special case of $Z_5=0$ (cf.","footnote REF ).","In Appendix REF , we have demonstrated explicitly that if $Z_5\\ne 0$ , then there exists a ratio of two basis-invariant quantities, which when evaluated in the $H23$ -basis yields $(\\operatorname{Im}Z_8)^2+(\\operatorname{Im}Z_9)^2$ .","Moreover, if $Z_5=0$ , then it is possible to change the basis of scalar fields of the IS model, in which the form of the IS potential is still given by Eq.","(REF ) but $\\operatorname{Im}Z_8=\\operatorname{Im}Z_9=0$ .","This result appears to be in contradiction to the result of Eq.","(REF ).","The resolution of this apparent paradox can be obtained by noting that if $Z_5=0$ , then $M_P=M_Q$ .","Since Eq.","(REF ) was derived under the assumption that $M_Q=0$ and $M_P\\gg m_Z$ , Eq.","(REF ) no longer applies if $Z_5=0$ .","But, more importantly, if $Z_5=0$ (so that $M_P=M_Q$ ), then the decay $Z\\rightarrow QQQQ^*,QQ^* Q^* Q^*$ is no longer an experimental observable, since one must also include four scalar decays involving $P$ and $P^*$ .","The possible four-body final states involve all possible combinations of $P$ and $Q$ scalars, such that either one or three of the final state scalars are complex-conjugated.","Some of the vertices that contribute to these final states are present even if $\\operatorname{Im}Z_8=\\operatorname{Im}Z_9=0$ .","For example, there is a four-scalar $|P|^2 |Q|^2$ interaction that contributes to $Z\\rightarrow QQPP^*$ .","One must compute the squared amplitude for each possible final state and then add the amplitudes incoherently to obtain the final experimentally observable decay rate.","This decay rate will involve a complicated combination of the IS potential coefficients, which will correspond to the appropriate invariant quantity in the case of $Z_5=0$ .","Thus, the possibility of finding a new basis for the IS potential in which $\\operatorname{Im}Z_8=\\operatorname{Im}Z_9=0$ when $Z_5=0$ is no longer paradoxical."],["The $ZZZ$ and {{formula:a1ad6645-e9b6-4fb4-b6f1-a07eb57f0b31}} vertices","In the CP-violating 2HDM, CP violation may manifest itself at loop level in the effective $ZZZ$ and $ZWW$ vertices.","In that model, one finds that CP-violating form factors can be described in terms of the three invariants introduced in Eq.","(REF ) [73].","In section , we noted the existence of a physical observable that could distinguish between the CP4-conserving IS models in which a CP2 symmetry that commutes with CP4 is either present or absent.","However, from a spacetime viewpoint, this physical observable was CP-even.","This raises the question as to whether any observable can exist in a CP4-invariant theory that is CP2-odd.","The answer to this question is no.","For example, there is no way to distinguish between CP2 and CP4 on the level of the form factors themselves.","Thus, if the theory respects at least one generalized CP symmetry, then all CP-violating form factors must be absent.","It is instructive to check the cancellation of contributions to the CP-violating form factors of the effective $ZZZ$ and $ZWW$ vertices in a CP4-conserving, CP2-violating theory (neglecting any effects from the Higgs-fermion Yukawa interactions).","The general $ZZZ$ vertex function (with all $Z$ bosons off-shell) can be expressed in terms of 14 different Lorentz structures [74], [75], [76], [77], [78], all preserving parity.","Some of these vanish when one or more $Z$ are on-shell.","Let us characterize them by momenta and Lorentz indices ($p_1,\\mu $ ), ($p_2,\\alpha $ ) and ($p_3,\\beta $ ), and let $Z_1$ be off-shell while $Z_2$ and $Z_3$ are on-shell.","Furthermore, we assume that $Z_1$ couples to a pair of leptons such as $e^+e^-$ , and terms proportional to the lepton mass will be neglected.","Denoting $\\ell \\equiv p_2-p_3 \\equiv 2p_2-p_1$ , the $ZZZ$ vertex structure reduces to the form [76] $-i\\Gamma _{ZZZ}^{\\alpha \\beta \\mu }=\\frac{p_1^2-M_Z^2}{M_Z^2}\\left[f_4^Z(p_1^{\\alpha }g^{\\mu \\beta }+p_1^{\\beta }g^{\\mu \\alpha })+f_5^Z\\epsilon ^{\\mu \\alpha \\beta \\rho }\\ell _\\rho \\right]\\,.$ The dimensionless form factor $f_4^Z$ violates CP while $f_5^Z$ conserves CP.","Figure: A typical pair of Feynman diagrams for Z→ZZZ\\rightarrow ZZ at two-loop order.For example, consider the case of the 2HDM.","At the one-loop level, CP violating effects yield a non-zero contribution to the $ZZZ$ vertex function, $f_4$ , that is proportional to $\\operatorname{Im}J_2$ of Eq.","(REF ) [73].","Thus, only one of the three invariants of Eq.","(REF ) contributes.","Indeed, in light of Eq.","(REF ), it follows that a non-zero $\\operatorname{Im}J_2$ requires all three neutral Higgs bosons to be non-degenerate in mass, and the $Z$ boson couples to all three non-diagonal neutral Higgs pairs.","In order to understand how the IS model conserves CP (while not respecting CP2), it is instructive to see how the CP-violating effects cancel at loop level in the effective $ZZZ$ (and $ZWW$ ) vertices.","In order to do this we have employed the software package FeynArts[79] and written a FeynArts model file containing all the bosonic couplings of the IS-model.","We have automated the construction of the diagrams contributing to the effective $ZZZ$ -vertex and evaluated their amplitude (the loop integrals are kept unevaluated in symbolic form).","We are only interested in those contributions to each diagram that contain $\\operatorname{Im}Z_8$ and/or $\\operatorname{Im}Z_9$ , since such contributions could be a signal of CP violation.","At the one-loop level there are no diagrams containing $\\operatorname{Im}Z_8$ and/or $\\operatorname{Im}Z_9$ .","Such contributions can only arise from a four-point scalar vertex.","This means that this four-point vertex must be “internal\"; i.e., none of the external $Z$ -fields can be part of this vertex.","None of the $ZZZ$ one-loop topologies can accommodate this.","Diagrams containing $\\operatorname{Im}Z_8$ and/or $\\operatorname{Im}Z_9$ first appear at two-loop order.","But even if there are individual diagrams with this type of contribution, the sum of the contributions is zero when we add the amplitudes for all the individual diagrams within each topology.","A pair of cancelling diagrams are shown in Fig.","REF .","The same happens for diagrams at three-loop order.","Repeating this exercise for the $ZWW$ vertex we find the same result.","Hence, there are no contributions at one-, two- or three-loop order containing $\\operatorname{Im}Z_8$ and/or $\\operatorname{Im}Z_9$ after adding the amplitudes for all the individual diagrams within each topology.","The arguments presented at the beginning of this section imply that this cancellation persists to all orders in perturbation theory."],["Conclusions","In this work we discussed the interplay between symmetries and natural mass degeneracies in the scalar sector.","Some cases of scalar mass degeneracy are accidental, i.e.","they are not the result of an exact symmetry and therefore can be implemented only by an artificial fine tuning of the scalar potential parameters.","The Higgs basis [16], [17], [18], in which the neutral scalar field vacuum expectation value resides entirely in one of the scalar doublet fields, is especially suitable for our study.","We began by examining the two Higgs doublet model (2HDM), with particular attention given to the special case of the inert doublet model (IDM), which possesses an unbroken $\\mathbb {Z}_2$ symmetry under which one “inert” scalar doublet is odd, and all other fields of the model are even.","In all cases in which the 2HDM exhibited scalar mass degeneracies (whether natural or accidental), the mass degenerate states can be experimentally distinguished from each other.","Moreover, with one exception, we found that all 2HDM mass degeneracies are accidental.","The one exceptional case of 2HDM scalars that can be naturally degenerate in mass are the two neutral scalar states of the inert doublet of the IDM.","This result was also confirmed by examining all possible symmetries of the 2HDM scalar potential and analyzing which of these symmetries can guarantee the presence of mass degenerate scalar states.","For models with three Higgs doublets, the analysis of the general case becomes significantly more elaborate.","We focused first on a 3HDM generalization of the IDM with mass degenerate scalars, which we denoted as the replicated IDM (RIDM), where the two doublets $H_2$ and $H_3$ are invariant under two separate unbroken $\\mathbb {Z}_2$ symmetries and the model is CP conserving.","In this framework $H_2$ and $H_3$ are composed of mass eigenstate fields, that do not mix with the SM like Higgs boson, forming four mass degenerate pairs.","Furthermore, each mass degenerate pair picks one field from each one of these doublets.","We also identified the symmetry obeyed by the neutral mass eigenstates themselves, which is responsible for the twofold mass degeneracies.","In the absence of $Z_5$ (which appears in the RIDM scalar potential) there are four mass degenerate neutral scalars and the symmetry of the scalar potential consists of an O$(4)$ global symmetry.","Introducing in the potential the term proportional to $Z_5$ , partially breaks the O$(4)$ symmetry down to an O(2)$\\times $ O(2) symmetry and the fourfold mass degeneracy is lifted, leaving a pairwise mass degeneracy.","The mass degeneracy of the two charged physical fields is governed by the full O(4) symmetry.","In the case of $Z_4 = Z_5 =0 $ there is further enhancement of the symmetry and all eight physical scalars contained in $H_2$ and $H_3$ are mass degenerate.","It is instructive to examine the Higgs boson interactions with the gauge bosons as well as the Higgs self couplings of the RIDM, since in the RIDM the components of $H_2$ and $H_3$ are already states with well defined masses.","We are then led to the conclusion that there is no experimental measurement that can physically distinguish the mass degenerate scalars of the RIDM on an event by event basis.","Nevertheless, multiplicity factors due to the production of different scalar states of the same mass do appear in physical observables and signal the existence of the mass degeneracy.","Starting with the RIDM, one can consider perturbations in which the mass degeneracies persist and yet remain natural.","By reducing the RIDM symmetries responsible for the mass degeneracies to the smallest discrete subgroup that maintains the mass degenerate scalar states, we are led to a model that is equivalent to a particular 3HDM that was originally proposed by Ivanov and Silva (IS) [30].","The IS model exhibits very special properties.","The original form of the IS scalar potential is given in Appendix REF and is the most general potential respecting the symmetry given by Eq.","(REF ).","We have rewritten the IS potential in the notation of Eq.","(REF ) where the symmetry is now given by Eq.","(REF ).","In particular, the scalar mass terms (and the corresponding mass degeneracies) are the same as in the RIDM; only the quartic couplings of the physical scalar states differ.","One must apply the symmetry given by Eq.","(REF ) [denoted by CP4] four times in order to obtain the identity transformation.","This is to be contrasted with the conventional CP symmetry transformation (denoted by CP2) whose square is the identity.","On the other hand, if we apply the CP4 transformation while at the same time transforming the space coordinates from $x$ into $-x$ , the end result can be identified as a generalized CP transformation.","This is a very unusual type of CP transformation since applying it twice does not yield the identity transformation.","However, identifying CP4 with a CP transformation is possible because from the spacetime point of view the transformation remains of order two, as it should.","Likewise, one can define a generalized time reversal operator with properties analogous to CP4 while transforming the time coordinate from $t$ to $-t$ .","Consequently, there is no contradiction with the CPT theorem, which remains intact.","A very interesting feature of the IS scalar potential is that the symmetry requires some of its coefficients to be complex (in a particular Higgs basis).","Moreover, for generic choices of the scalar potential parameters, there is no scalar basis transformation within the family of Higgs bases, of the form given by Eqs.","(REF )–(REF ), that can transform the scalar potential into a new potential with only real coefficients.","This is a surprising result in light of the statement that the IS potential is CP-conserving.","The IS model conserves CP independently of the existence or nonexistence of a real Higgs basis, although in the case where no real Higgs basis exists, the IS model is only invariant with respect to the generalized CP symmetry, CP4 (whereas CP2 is not a symmetry of the IS scalar potential).","Nevertheless, any CP-violating observable of the IS model must vanish.","For example, the contributions to the CP-violating form factors of the effective $ZZZ$ and $ZWW$ vertices generated in the IS model must exactly cancel.","As a check of this statement, we confirmed this cancellation up to three-loop order in the IS model with no real Higgs basis.","We identified a physical quartic scalar interaction made up of an odd number of mass-degenerate neutral scalar states (e.g., $P^3 Q$ and $Q^3 P$ ) that is consistent with the CP4 symmetry, but would vanish if the IS scalar potential exhibits a CP2 symmetry that commutes with CP4.","This leaves open the possibility of the existence of a CP2 symmetry that does not commute with CP4.","However, we were unable to find an observable quantity of the IS model that can distinguish between the presence or absence of a noncommuting CP2 symmetry.","Finally, we stress that the possibility of a scalar potential and vacuum that is invariant with respect to a generalized CP symmetry without the existence of a real basis appears to be inexorably connected with the existence of mass-degenerate scalar states.","We strongly suspect that this connection, which has been demonstrated in this paper for the IS model, is applicable more generally to any multi-Higgs doublet model.","If true, then the existence of a generalized CP symmetry in the absence of mass degenerate scalars necessarily implies the presence of a conventional CP symmetry; i.e., the existence of a real basis of scalar fields in which the CP symmetry corresponds simply to conjugation of the scalar fields."],["Acknowledgments","We are very grateful to Igor Ivanov for many fruitful and enlightening conversations.","We also thank Gustavo C. Branco, Nick Mavromatos, Palash B. Pal, Apostolos Pilaftsis, and Graham Ross with whom several aspects of this work were discussed.","H.E.H.","and P.O.","acknowledge the Galileo Galilei Institute for Theoretical Physics, where this work was initiated.","H.E.H., P.O.","and M.N.R.","also appreciate the hospitality of the CERN Theory group where some of this work was carried out, and M.N.R also acknowledges partial support from CERN.","H.E.H.","is supported in part by the U.S. Department of Energy grant number DE-SC0010107, and in part by the grant H2020-MSCA-RISE-2014 No.","645722 (NonMinimalHiggs).","P.O.","is supported by the Research Council of Norway.","The work of M.N.R.","was partially supported by Fundação para a Ciência e a Tecnologia (FCT, Portugal) through the projects CFTP-FCT Unit 777 (UID/FIS/00777/2013), CERN/FIS-PAR/0004/2017 and PTDC/FIS-PAR/29436/2017 which are partially funded through POCTI (FEDER), COMPETE, QREN and EU.","H.E.H.","and M.N.R.","benefited from discussions that took place at the University of Warsaw during visits supported by the HARMONIA project of the National Science Centre, Poland, under contract UMO-2015/18/M/ST2/00518 (2016–2019).","M.N.R.","and P.O.","also thank the University of Bergen and CFTP/IST/University of Lisbon, where collaboration visits took place."],["The Ivanov Silva model revisited","Consider the most general 2HDM with a scalar potential as specified in Eq.","(REF ).","Including the kinetic energy terms with SU(2)$\\times $ U(1) gauge covariant derivatives, the 2HDM [after electroweak symmetry breaking under the assumption that the vacuum preserves U(1)$_{\\rm EM}$ ] consists of a model of two scalar doublets coupled to the gauge bosons, $W^\\pm $ , $Z$ and $\\gamma $ .","We shall ignore the couplings of the bosonic sector of the 2HDM to the fermions of the SM in the following discussion.","We now ask the following question.","Does the bosonic Lagrangian conserve CP?","For CP to be conserved, two conditions must be verified.","First, the scalar potential must exhibit explicit CP conservation.","Second, the vacuum must conserve CP.","If the former is true but the latter is false, we say that CP is spontaneously broken.","However, in this discussion, we are interested in whether both explicit and spontaneous CP violation are absent.","In the 2HDM, the answer to this question is simple.","We first transform to the Higgs basis and examine the scalar potential given in Eq.","(REF ).","The Higgs basis is unique up to a possible rephasing of the Higgs basis field, $H_2\\rightarrow e^{i\\chi }H_2$ .","Then, CP is conserved if and only if there exists a choice of $\\chi $ such that all Higgs basis scalar potential parameters are real.","In the discussion above, we have not specified in detail how the scalar fields transform under a CP transformation.","Starting from the generic $\\Phi _1$ –$\\Phi _2$ basis employed in writing Eq.","(REF ), the conventional CP transformation corresponds to conjugation, $\\Phi _i^{\\rm CP}= \\Phi _i^{\\scriptscriptstyle {\\bigstar }}$ .","However, this is a basis-dependent statement.","Indeed, one is always free to change the basis, $\\Phi _i^\\prime =U_{ij}\\Phi _j$ , where $U\\in $ U(2).","In the new basis, $\\Phi ^{\\prime \\,CP}_i=X_{ij}\\Phi ^{\\prime \\,CP}_j$ , where $X=UU^T$ is a symmetric unitary matrix.","More generally, we can consider the generalized CP transformation, $\\Phi _i^{\\rm CP}(x,t)= X_{ij}\\Phi _j^{\\scriptscriptstyle {\\bigstar }}(-x,t)\\,,$ where $X\\in $ U(2).Note that it is not consistent to simply define the CP transformation of a multi-Higgs doublet model without including the matrix $X$ in Eq.","(REF ), since the form of the CP transformation depends on the choice of the scalar basis, as noted above.","Consequently, some authors prefer to call this transformation a general CP transformation rather than generalized CP transformation.","If $X$ is both unitary and symmetric, then one can find a basis in which the CP transformation is simply conjugation.As shown in Appendix D of Ref.","[80] [see the Lemma below eq.","(D.3.1)], for any symmetric unitary matrix $X$ , there exists a unitary matrix $U$ such that $X=UU^T$ .","In Ref.","[56], it is shown that in the 2HDM there are three possible classes of generalized CP transformations (GCPs): (i) $X$ is unitary and symmetric; (ii) $X$ is unitary and antisymmetric; and (iii) $X$ is unitary but is neither symmetric nor antisymmetric.","Clearly, no basis change can convert a GCP transformation of types (ii) or (iii) into the transformation of the field into its conjugate.","Nevertheless, as shown in Ref.","[56], any 2HDM scalar potential that is invariant under GCP transformations of types (ii) or (iii) is also separately invariant under a GCP transformation of type (i).","Do the above results generalize to arbitrary Higgs sectors?","In particular, consider an extended Higgs sector with $N$ hypercharge-one, complex doublets (denoted henceforth as the NHDM).","To address the question of CP invariance, we transform to the so-called charged Higgs basis defined in Ref. [81].","If the scalar fields of the charged Higgs basis are denoted by $H_i$ ($i=1,\\ldots ,n$ ), then $\\langle H_1^0 \\rangle =v/\\sqrt{2}$ , $\\langle H_j^0 \\rangle =0$ for $j=2,3,\\ldots ,n$ , and the fields $H_j^\\pm $ (for $j=2,3,\\ldots ,n$ ) are the physical, mass-eigenstate charged Higgs fields.","Note that for $N=2$ , the Higgs basis and the charged Higgs basis coincide.","For $N\\ge 3$ , consider first the case in which the physical charged Higgs bosons are mass non-degenerate.","In this case, the charged Higgs basis is uniquely defined up to a possible rephasing, $H_j\\rightarrow e^{i\\chi _j}H_j$ .","In this case, CP is conserved if and only if there exist a choice of the $\\chi _j$ such that all charged Higgs basis scalar potential parameters are real.","This generalizes the result of the 2HDM quoted above.","If there exist mass degeneracies among the physical charged Higgs fields, then one must re-evaluate the conditions for CP invariance.","To simplify the discussion, we focus on the case of $N=3$ , in which the two physical charged Higgs bosons are mass degenerate.","In this case, the charged Higgs basis is unique up to a U(2) transformation of the charged Higgs basis fields $H_2$ and $H_3$ .","Ivanov and Silva[30] constructed a 3HDM whose scalar potential and vacuum are invariant under a generalized CP transformation such that $({\\rm GCP})^2\\ne {1}$ , where 1 is the identity operator.","Moreover, some of the scalar potential parameters of the charged Higgs basis of the Ivanov–Silva (IS) scalar potential are complex, and no U(2) transformation of the charged Higgs basis fields $H_2$ and $H_3$ can be performed to remove all the complex phases.","Hence, the IS scalar potential is not invariant under a separate GCP transformation that is equivalent to conjugation in another basis, in contrast to the corresponding 2HDM result.","Ivanov and Silva denote the GCP transformation of the IS scalar potential by CP4, since it has the property that $({\\rm CP}4)^4={1}$ and $({\\rm CP}4)^2\\ne {1}$ .","Indeed, one consequence of the CP4 symmetry of the scalar potential and the vacuum is the mass degeneracy of the physical charged Higgs bosons, as well as two additional mass degeneracies among pairs of neutral Higgs bosons.","In this Appendix, we consider the 3HDM scalar potential of the Ivanov and Silva model and examine some of its properties."],["The IS scalar potential","Consider the 3HDM consisting of three hypercharge-one, complex doublet fields, $\\phi _i$ ($i=1,2,3$ ).","In the Higgs basis, the form of the scalar potential proposed initially by Ivanov and Silva (IS) in Ref.","[30] is fixed by imposing the following generalized CP symmetry, $ \\phi _i \\rightarrow W_{ij}\\phi _j^{\\scriptscriptstyle {\\bigstar }}, \\quad W=\\begin{pmatrix}1 & \\phantom{-}0 & \\phantom{-}0 \\\\0 & \\phantom{-}0 & \\phantom{-}i \\\\0 & -i & \\phantom{-}0\\end{pmatrix},$ which has the property that applying it four times yields the identity operator.","This is the CP4 symmetry transformation noted above.","The resulting IS scalar potential is given by $V=V_0+V_1\\,,$ with $V_0&=-m_{11}^2(\\phi _1^\\dagger \\phi _1)-m_{22}^2(\\phi _2^\\dagger \\phi _2+\\phi _3^\\dagger \\phi _3)+\\lambda _1(\\phi _1^\\dagger \\phi _1)^2+\\lambda _2[(\\phi _2^\\dagger \\phi _2)^2+(\\phi _3^\\dagger \\phi _3)^2]+\\lambda _3^\\prime (\\phi _2^\\dagger \\phi _2)(\\phi _3^\\dagger \\phi _3)\\nonumber \\\\& \\qquad +\\lambda _3(\\phi _1^\\dagger \\phi _1)[(\\phi _2^\\dagger \\phi _2)+(\\phi _3^\\dagger \\phi _3)]+\\lambda _4^\\prime (\\phi _2^\\dagger \\phi _3)(\\phi _3^\\dagger \\phi _2)+\\lambda _4[(\\phi _1^\\dagger \\phi _2)(\\phi _2^\\dagger \\phi _1)+(\\phi _1^\\dagger \\phi _3)(\\phi _3^\\dagger \\phi _1)],\\\\[6pt]V_1&=\\lambda _5(\\phi _3^\\dagger \\phi _1)(\\phi _2^\\dagger \\phi _1)+\\tfrac{1}{2}\\lambda _6[(\\phi _2^\\dagger \\phi _1)^2-(\\phi _1^\\dagger \\phi _3)^2]+\\lambda _8(\\phi _2^\\dagger \\phi _3)^2+\\lambda _9(\\phi _2^\\dagger \\phi _3)[(\\phi _2^\\dagger \\phi _2)-(\\phi _3^\\dagger \\phi _3)] +{\\rm h.c.}$ The hermiticity of the scalar potential implies that the coefficients of $V_0$ are real.","In contrast, the coefficients of $V_1$ are potentially complex.","However, having imposed the CP4 symmetry given by Eq.","(REF ), we see that $\\lambda _5$ is real.","Under the CP4 symmetry specified in Eq.","(REF ), the gauge-invariant bilinear quantities, $B_{ij}\\equiv \\phi _i^\\dagger \\phi _j$ , transform as follows: $B_{11} &\\rightarrow B_{11}, \\\\B_{22}&\\rightarrow B_{33} &\\quad B_{33}&\\rightarrow B_{22}, \\\\B_{12}&\\rightarrow i B_{31}, &\\quad B_{21}&\\rightarrow -i B_{13}, \\\\B_{13}&\\rightarrow -i B_{21}, &\\quad B_{31}&\\rightarrow i B_{12}, \\\\B_{23}&\\rightarrow -B_{23}, &\\quad B_{32}&\\rightarrow -B_{32}.$ It follows that $V$ given by Eqs.","(REF )–(REF ), with $\\lambda _6$ , $\\lambda _8$ and $\\lambda _9$ complex and all other scalar potential parameters real, is the most general 3HDM potential that is invariant under the CP4 transformation given in (REF ).","Without loss of generality, one can furthermore assume that $\\lambda _6$ is real after an appropriate rephasing of the scalar fields $\\phi _2$ and $\\phi _3$ .","At this stage, we have not yet found the minimum of the scalar potential and determined whether the CP4 symmetry is respected by the vacuum.","There exist a range of scalar potential parameters in which the vacuum preserves U(1)$_{\\rm EM}$ , in which case one can decompose the scalar doublets as, $ \\phi _i=\\left(\\begin{array}{c}\\varphi _i^+\\\\ (v_i+\\eta _i+i \\chi _i)/\\sqrt{2}\\end{array}\\right), \\quad i=1,2,3.$ In particular, the vacuum conserves CP4 if the minimum of the scalar potential corresponds to $(v_1,v_2,v_3)=(v,0,0)$[30].","Indeed, there exists a range of scalar potential parameters for which this corresponds to the global minimum, in which case the value of $m_{11}^2$ is fixed by the scalar potential minimum condition to be $m_{11}^2=\\lambda _1 v^2.$ In this case, the scalar field basis employed in Eqs.","(REF ) and (REF ) is the Higgs basis, with the freedom to perform U(2) transformations on $\\lbrace \\phi _2,\\phi _3\\rbrace $ .","We shall take advantage of this freedom in the next two subsections.","It is now straightforward to determine the scalar mass spectrum of the IS model.","Since we are in the Higgs basis, we can immediately identify the Goldstone bosons, $\\varphi _1^\\pm =G^\\pm $ and $\\chi _1=G^0$ .","Moreover, $\\eta _1$ is a neutral mass-eigenstate with mass $m^2_{\\eta _1}=2\\lambda _1 v^2$ , whose tree-level couplings to the gauge bosons and to itself are precisely those of the SM Higgs boson (corresponding to the exact alignment limit).","Indeed, this is analogous to the IDM in which $\\phi _1$ is equivalent to the hypercharge-one, complex scalar doublet of the SM and $\\phi _2$ and $\\phi _3$ are inert doublets.","The two physical charged Higgs fields, $\\varphi _2^\\pm $ and $\\varphi _3^\\pm $ , are mass-degenerate, $ m_{\\varphi _2^\\pm ,\\varphi _3^\\pm }^2=\\tfrac{1}{2}\\lambda _3 v^2 -m_{22}^2.$ The neutral scalar spectrum consist of the SM-like Higgs boson $\\eta _1$ and a pair of mass degenerate neutral scalars made up of linear combinations of the $\\eta _{2,3}$ and $\\chi _{2,3}$ , with masses given by[30], $M^2= a+\\sqrt{b^2+c^2}\\,,\\qquad \\quad m^2= a-\\sqrt{b^2+c^2}\\,,$ where $a=\\tfrac{1}{2}(\\lambda _3+\\lambda _4)v^2-m_{22}^2, \\qquad b=\\tfrac{1}{2}\\lambda _6 v^2, \\qquad c=\\tfrac{1}{2}\\lambda _5 v^2.$"],["A simpler form for the IS scalar potential","Given the IS scalar potential in the Higgs basis, we still have the freedom to perform a U(2) transformation on $\\lbrace \\phi _2,\\phi _3\\rbrace $ .","It is possible to remove the $\\lambda _5$ term in Eq.","(REF ) by the following basis transformation, $\\bar{\\phi }_i=U_{ij}\\phi _j,$ where $U=\\begin{pmatrix}1 & 0 & 0 \\\\0 & \\phantom{-}\\cos \\theta & -\\sin \\theta \\\\0 & \\phantom{-}\\sin \\theta & \\phantom{-}\\cos \\theta \\end{pmatrix}\\,,$ with $0\\le \\theta \\le \\pi $ .","With respect to the new basis, the CP4 transformation specified in Eq.","(REF ) is given by, $\\bar{\\phi }_i\\rightarrow V_{ij}\\bar{\\phi }^{\\scriptscriptstyle {\\bigstar }}_j\\,, \\qquad \\text{where $V=UWU^T$}.$ Using the form for $U$ given in Eq.","(REF ), it follows that $V=W$ .","Thus, in this new basis, the IS symmetry takes the same form as in the original basis.","When the scalar potential is expressed in terms of the fields $\\bar{\\phi }_i$ , the resulting scalar potential parameters will be denoted by $\\bar{m}^2_{ii}$ and $\\bar{\\lambda }_i$ .","It is straightforward to obtain expressions for $\\bar{m}_{11}^2$ , $\\bar{m}_{22}^2$ and the $\\bar{\\lambda }_i$ in terms of the scalar potential parameters defined in Eq.","(REF ).","In particular, $\\bar{m}_{11}^2=m_{11}^2$ , $\\bar{m}_{22}^2=m_{22}^2$ , and $\\bar{\\lambda }_i=\\lambda _i$ for $i=1,3$ and 4.","Next, we note that the CP4 symmetry does not mandate that $\\bar{\\lambda }_6$ is real.","However, it is straightforward to check that $\\operatorname{Im}\\bar{\\lambda }_6=\\operatorname{Im}\\lambda _6$ .","Having previously chosen $\\lambda _6$ real (after an appropriate rephasing of $\\phi _2$ and $\\phi _3$ ), it follows that $\\bar{\\lambda }_6$ is also real.","The remaining transformed coefficients are given by, $\\bar{\\lambda }_2&= \\lambda _2-\\tfrac{1}{2}\\sin ^2 2\\theta [\\lambda _2-\\operatorname{Re}\\lambda _8-\\tfrac{1}{2}(\\lambda _3^\\prime +\\lambda _4^\\prime )]-\\sin 2\\theta \\cos 2\\theta \\operatorname{Re}\\lambda _9\\,, \\\\\\bar{\\lambda }^\\prime _3&=\\lambda ^\\prime _3+\\sin ^2 2\\theta [\\lambda _2-\\operatorname{Re}\\lambda _8-\\tfrac{1}{2}(\\lambda _3^\\prime +\\lambda _4^\\prime )]+2\\sin 2\\theta \\cos 2\\theta \\operatorname{Re}\\lambda _9\\,, \\\\\\bar{\\lambda }^\\prime _4&=\\lambda ^\\prime _4+\\sin ^2 2\\theta [\\lambda _2-\\operatorname{Re}\\lambda _8-\\tfrac{1}{2}(\\lambda _3^\\prime +\\lambda _4^\\prime )]+2\\sin 2\\theta \\cos 2\\theta \\operatorname{Re}\\lambda _9\\,, \\\\\\bar{\\lambda }_5&=\\lambda _5\\cos 2\\theta +\\lambda _6\\sin 2\\theta \\,,\\\\\\bar{\\lambda }_6&=\\lambda _6\\cos 2\\theta -\\lambda _5\\sin 2\\theta \\,,\\\\\\operatorname{Re}\\bar{\\lambda }_8&=\\operatorname{Re}\\lambda _8+\\tfrac{1}{2}\\sin ^2 2\\theta [\\lambda _2-\\operatorname{Re}\\lambda _8-\\tfrac{1}{2}(\\lambda _3^\\prime +\\lambda _4^\\prime )]+\\sin 2\\theta \\cos 2\\theta \\operatorname{Re}\\lambda _9\\,,\\\\\\operatorname{Re}\\bar{\\lambda }_9&=(1-2\\sin ^2 2\\theta )\\operatorname{Re}\\lambda _9+\\sin 2\\theta \\cos 2\\theta [\\lambda _2-\\operatorname{Re}\\lambda _8-\\tfrac{1}{2}(\\lambda _3^\\prime +\\lambda _4^\\prime )]\\,,\\\\\\operatorname{Im}\\bar{\\lambda }_8&=\\cos 2\\theta \\operatorname{Im}\\lambda _8+\\sin 2\\theta \\operatorname{Im}\\lambda _9\\,,\\\\\\operatorname{Im}\\bar{\\lambda }_9&=\\cos 2\\theta \\operatorname{Im}\\lambda _9-\\sin 2\\theta \\operatorname{Im}\\lambda _8\\,.$ One can now choose the angle $\\theta $ such that $\\bar{\\lambda }_5=0$ .Note that a different choice of $\\tan 2\\theta $ could have been made to set either $\\bar{\\lambda }_6=0$ , $\\operatorname{Im}\\bar{\\lambda }_8=0$ or $\\operatorname{Im}\\bar{\\lambda }_9=0$ .","That is, one can always perform a change of Higgs basis to remove one degree of freedom from the coefficients of the IS scalar potential.","This yields $\\tan 2\\theta =-\\lambda _5/\\lambda _6$ .","Then, $\\sin 2\\theta $ and $\\cos 2\\theta $ are determined up to an overall sign.","Introducing the following notation, $\\lambda _{56}\\equiv \\sqrt{\\lambda _5^2+\\lambda _6^2}\\,,$ we choose the angle $\\theta $ such that, $\\sin 2\\theta =\\frac{\\lambda _5}{\\lambda _{56}}\\,,\\qquad \\quad \\cos 2\\theta =-\\,\\frac{\\lambda _6}{\\lambda _{56}}\\,.$ Thus, the $\\lambda _5$ -term in Eq.","(REF ) is actually redundant.A similar simplification was presented recently in Ref. [70].","Inserting the results of Eq.","(REF ) back into Eq.","(REF ) yields $\\bar{\\lambda }_5=0$ and, $\\bar{\\lambda }_2&=\\frac{\\lambda _5 \\left[\\lambda _5 \\left(\\lambda ^\\prime _3+\\lambda ^\\prime _4+2 \\operatorname{Re}\\lambda _8\\right)+4 \\lambda _6 \\operatorname{Re}\\lambda _9\\right]+2 \\lambda _2 \\left(\\lambda _5^2+2 \\lambda _6^2\\right)}{4 \\lambda ^2_{56}},\\\\\\bar{\\lambda }^\\prime _3&=\\frac{\\lambda _5^2 \\left(\\lambda ^\\prime _3-\\lambda ^\\prime _4-2 \\operatorname{Re}\\lambda _8\\right)+2 \\lambda _6^2 \\lambda ^\\prime _3-4 \\lambda _5 \\lambda _6 \\operatorname{Re}\\lambda _9+2 \\lambda _2 \\lambda _5^2}{2\\lambda ^2_{56}},\\\\\\bar{\\lambda }^\\prime _4&=\\frac{\\lambda _5^2 \\left(-\\lambda ^\\prime _3+\\lambda ^\\prime _4-2 \\operatorname{Re}\\lambda _8\\right)+2 \\lambda _6^2 \\lambda ^\\prime _4-4 \\lambda _5 \\lambda _6 \\operatorname{Re}\\lambda _9+2 \\lambda _2 \\lambda _5^2}{2 \\lambda ^2_{56}},\\\\\\bar{\\lambda }_6&=-\\lambda _{56},\\\\\\operatorname{Re}\\bar{\\lambda }_8&=\\frac{-\\lambda _5^2 \\left(\\lambda ^\\prime _3+\\lambda ^\\prime _4-2 \\operatorname{Re}\\lambda _8\\right)-4 \\lambda _5 \\lambda _6 \\operatorname{Re}\\lambda _9+4 \\lambda _6^2 \\operatorname{Re}\\lambda _8+2 \\lambda _2 \\lambda _5^2}{4 \\lambda ^2_{56}},\\\\\\operatorname{Re}\\bar{\\lambda }_9&=\\frac{\\lambda _5\\lambda _6 \\left(\\lambda ^\\prime _3+\\lambda ^\\prime _4+2 \\operatorname{Re}\\lambda _8\\right)-2 \\lambda _5^2 \\operatorname{Re}\\lambda _9+2 \\lambda _6^2 \\operatorname{Re}\\lambda _9-2 \\lambda _2 \\lambda _5 \\lambda _6}{2 \\lambda ^2_{56}},\\\\\\operatorname{Im}\\bar{\\lambda }_8&=\\frac{\\lambda _5 \\operatorname{Im}\\lambda _9-\\lambda _6 \\operatorname{Im}\\lambda _8}{\\lambda _{56}},\\\\\\operatorname{Im}\\bar{\\lambda }_9&=\\frac{-\\lambda _5 \\operatorname{Im}\\lambda _8-\\lambda _6 \\operatorname{Im}\\lambda _9}{\\lambda _{56}}.$ An additional feature of the IS scalar potential with $\\bar{\\lambda }_5=0$ is that the real and imaginary parts of the neutral fields $\\bar{\\phi }_2^0$ and $\\bar{\\phi }_3^0$ are mass eigenstates.","That is, the neutral squared-mass matrices are already diagonal in the $\\lbrace \\bar{\\phi }_1,\\bar{\\phi }_2,\\bar{\\phi }_3\\rbrace $ basis.","In particular, the lightest of the two mass-degenerate states lives in the imaginary part of $\\bar{\\phi }_2$ and in the real part of $\\bar{\\phi }_3$ .","The heaviest of the two mass-degenerate neutral states lives in the real part of $\\bar{\\phi }_2$ and in the imaginary part of $\\bar{\\phi }_3$ .","It is convenient to make an additional field redefinition, $\\bar{\\phi }_3\\rightarrow i\\bar{\\phi }_3$ .","The effect of this modification is to modify $\\bar{V}_1$ by flipping the sign of $(\\bar{\\phi }^\\dagger _1\\bar{\\phi }_3)^2$ in the term proportional to $\\bar{\\lambda }_6$ and to transform $\\bar{\\lambda }_8\\rightarrow -\\bar{\\lambda }_8$ and $\\bar{\\lambda }_9\\rightarrow -i\\bar{\\lambda }_9$ .","To make contact with the $H23$ -basis employed in Eq.","(REF ), we define, $H_1=\\bar{\\phi }_1\\,,\\qquad H_2=\\bar{\\phi }_2\\,,\\qquad H_3=i\\bar{\\phi }_3\\,,$ corresponding to a basis change, $H_i\\rightarrow \\widetilde{U}_{ij}\\bar{\\phi }_j$ , with $\\widetilde{U}={\\rm diag}(1\\,,\\,1\\,,\\,i)$ .","Note that the heaviest mass degenerate neutral fields now reside in the real part of the neutral components of $H_2$ and $H_3$ , and the lightest mass degenerate neutral fields reside in the imaginary part of the neutral components of $H_2$ and $H_3$ .","When expressed in the $H23$ -basis, the IS scalar potential is given by, $\\mathcal {V}_{\\rm IS}= \\mathcal {V}_{\\rm RIDM}+Z_3^\\prime (H_2^\\dagger H_2)(H_3^\\dagger H_3)+Z_4^\\prime (H_2^\\dagger H_3)(H_3^\\dagger H_2) +\\bigl [Z_8 (H_2^\\dagger H_3)^2+Z_9(H_2^\\dagger H_3)(H_2^\\dagger H_2-H_3^\\dagger H_3)+{\\rm h.c.}\\bigr ]\\,,$ where $\\mathcal {V}_{\\rm RIDM}$ is given by Eq.","(REF ), with $Z_8$ and $Z_9$ potentially complex and all other scalar potential parameters real.","Eq.","(REF ) is the version of the IS scalar potential employed in section REF .","To make contact with the previous notation used above, we note that $Y_1&=& -m_{11}^2\\,,\\qquad Y_2=-m_{22}^2\\,,\\qquad Z_1=2\\lambda _1\\,,\\qquad Z_2=2\\bar{\\lambda }_2\\,,\\qquad Z_3=\\lambda _3\\,,\\qquad Z_4=\\lambda _4 \\nonumber \\\\Z_3^\\prime &=&\\bar{\\lambda }_3^\\prime -2\\bar{\\lambda }_2\\,,\\qquad Z_4^\\prime =\\bar{\\lambda }_4^\\prime \\,,\\qquad Z_5=\\bar{\\lambda }_6\\,,\\qquad Z_8=-\\bar{\\lambda }_8\\,,\\qquad Z_9=-i\\bar{\\lambda }_9\\,.$ The corresponding CP4 symmetry transformation now takes the form $H_i\\rightarrow X_{ij}H_j^{\\scriptscriptstyle {\\bigstar }}\\,,\\qquad \\text{where $X=\\widetilde{U}W\\widetilde{U}^T=\\begin{pmatrix} 1 & \\phantom{-}0 & \\phantom{-}0 \\\\ 0 & \\phantom{-}0 & -1 \\\\ 0 & \\phantom{-}1 & \\phantom{-}0\\end{pmatrix}$}\\,,$ as indicated in Eq.","(REF )."],["Non-existence of a real Higgs basis","Consider the IS scalar potential [cf. Eq.","(REF ) with $\\mathcal {V}_{\\rm RIDM}$ given by Eq.","(REF )] expressed in terms of the Higgs basis of scalar doublet fields, $\\lbrace H_1,H_2,H_3\\rbrace $ , where $\\langle H_1^0 \\rangle \\ne 0$ and the vevs of the other two doublet fields vanish.","The coefficients $Z_8$ and $Z_9$ are potentially complex and all other scalar potential parameters are real.","Recall that the Higgs basis is unique only up to an arbitrary U(2) transformation of $\\lbrace H_2, H_3\\rbrace $ .","Is it possible to transform to a new Higgs basis in which all the IS scalar potential parameters are real?","Such a Higgs basis, if it exists, is called a real Higgs basis.","The most general basis transformation that preserves the general class of Higgs bases is given (in block diagonal form) by, $\\begin{pmatrix} \\bar{H}_1 \\\\ \\bar{H}_{23}\\end{pmatrix}=\\begin{pmatrix} 1 & \\,\\,0 \\\\ 0 &\\,\\, \\widetilde{V}\\end{pmatrix} \\begin{pmatrix} H_1 \\\\ H_{23}\\end{pmatrix}\\,,$ where $H_{23}\\equiv \\begin{pmatrix} H_2\\\\ H_3\\end{pmatrix}\\,,\\qquad \\quad \\bar{H}_{23}\\equiv \\begin{pmatrix} \\bar{H}_2\\\\ \\bar{H}_3\\end{pmatrix}\\,,$ and $\\widetilde{V}$ is the most general U(2) matrix, $\\widetilde{V}=e^{i\\psi /2}\\begin{pmatrix} e^{i\\alpha }\\cos \\phi & -e^{-i\\beta }\\sin \\phi \\\\ e^{i\\beta }\\sin \\phi & \\phantom{-}e^{-i\\alpha }\\cos \\phi \\end{pmatrix}\\,,$ where $0\\le \\phi <\\pi $ , $-\\pi <\\psi \\le \\pi $ , $0\\le \\alpha \\le \\pi $ and $0\\le \\beta \\le \\pi $ .","Applying Eq.","(REF ) to the IS scalar potential given in Eq.","(REF ) yields $\\mathcal {V}_{\\rm IS}&=&Y_1 \\bar{H}_1^\\dagger \\bar{H}_1+ Y_2 \\left( \\bar{H}_2^\\dagger \\bar{H}_2 + \\bar{H}_3^\\dagger \\bar{H}_3\\right)+\\tfrac{1}{2}Z_1( \\bar{H}_1^\\dagger \\bar{H}_1)^2+\\tfrac{1}{2}\\bar{Z}_2( \\bar{H}_2^\\dagger \\bar{H}_2+ \\bar{H}_3^\\dagger \\bar{H}_3)^2 \\\\&&+Z_3(\\bar{H}_1^\\dagger \\bar{H}_1)( \\bar{H}_2^\\dagger \\bar{H}_2+ \\bar{H}_3^\\dagger \\bar{H}_3)+Z_4\\bigl [( \\bar{H}_1^\\dagger \\bar{H}_2)( \\bar{H}_2^\\dagger \\bar{H}_1)+( \\bar{H}_1^\\dagger \\bar{H}_3)( \\bar{H}_3^\\dagger \\bar{H}_1)\\bigr ]\\nonumber \\\\&&+\\bar{Z}_3^\\prime ( \\bar{H}_2^\\dagger \\bar{H}_2)( \\bar{H}_3^\\dagger \\bar{H}_3)+\\bar{Z}_4^\\prime ( \\bar{H}_2^\\dagger \\bar{H}_3)( \\bar{H}_3^\\dagger \\bar{H}_2)+i\\bar{Z}_5^\\prime \\bigl [e^{i\\psi } (\\bar{H}_3^\\dagger \\bar{H}_1)( \\bar{H}_2^\\dagger \\bar{H}_1)-e^{-i\\psi }( \\bar{H}_1^\\dagger \\bar{H}_2)( \\bar{H}_1^\\dagger \\bar{H}_3)\\bigr ]\\,,\\nonumber \\\\&&+\\bigl \\lbrace \\tfrac{1}{2}\\bar{Z}_5\\bigl [e^{i\\psi } (\\bar{H}_2^\\dagger \\bar{H}_1)^2 +e^{-i\\psi }( \\bar{H}_1^\\dagger \\bar{H}_3)^2\\bigr ]+\\bar{Z}_8 ( \\bar{H}_2^\\dagger \\bar{H}_3)^2+\\bar{Z}_9( \\bar{H}_2^\\dagger \\bar{H}_3)(\\bar{H}_2^\\dagger \\bar{H}_2- \\bar{H}_3^\\dagger \\bar{H}_3)+{\\rm h.c.} \\bigr \\rbrace \\,.\\nonumber $ The coefficients $Y_1$ , $Y_2$ , $Z_1$ , $Z_3$ and $Z_4$ are unmodified, whereas, $&& \\bar{Z}_2=Z_2+\\tfrac{1}{2}\\sin ^2 2\\phi \\bigl (Z_3^\\prime +Z_4^\\prime +Z_8 e^{2i\\xi }+Z^*_8 e^{-2i\\xi }\\bigr )-\\sin 2\\phi \\cos 2\\phi \\bigl (Z_9e^{i\\xi }+Z^*_9e^{-i\\xi }\\bigr ), \\\\&&\\bar{Z}_3^\\prime =Z_3^\\prime -\\sin ^2 2\\phi \\bigl (Z_3^\\prime +Z_4^\\prime +Z_8 e^{2i\\xi }+Z^*_8 e^{-2i\\xi }\\bigr )+2\\sin 2\\phi \\cos 2\\phi \\bigl (Z_9e^{i\\xi }+Z^*_9e^{-i\\xi }\\bigr ), \\\\&&\\bar{Z}_4^\\prime =Z_4^\\prime -\\tfrac{1}{2}\\sin ^2 2\\phi \\bigl (Z_3^\\prime +Z_4^\\prime +Z_8 e^{2i\\xi }+Z^*_8 e^{-2i\\xi }\\bigr )+\\sin 2\\phi \\cos 2\\phi \\bigl (Z_9e^{i\\xi }+Z^*_9e^{-i\\xi }\\bigr ), \\\\&&\\bar{Z}_5^\\prime =Z_5\\sin 2\\phi \\sin \\xi \\,, \\\\&&\\bar{Z}_5=e^{i\\chi }Z_5\\bigl (e^{i\\xi }\\cos ^2\\phi +e^{-i\\xi }\\sin ^2\\phi \\bigr )\\,, \\\\&&\\bar{Z}_8= e^{2i\\chi }\\bigl \\lbrace -\\tfrac{1}{4} \\sin ^2 2\\phi \\bigl (Z_3^\\prime +Z_4^\\prime \\bigr )+e^{2i\\xi }\\cos ^4\\phi \\,Z_8+e^{-2i\\xi }\\sin ^4\\phi \\, Z_8^* \\nonumber \\\\&& \\qquad \\qquad \\qquad +\\sin 2\\phi \\bigl [e^{i\\xi }\\cos ^2\\phi \\,Z_9-e^{-i\\xi }\\sin ^2\\phi \\, Z_9^*\\bigr ]\\bigr \\rbrace \\,, \\\\&&\\bar{Z}_9=e^{i\\chi }\\bigl \\lbrace -\\tfrac{1}{2}\\sin 2\\phi \\cos 2\\phi (Z_3^\\prime +Z_4^\\prime ) -\\sin 2\\phi \\bigl [e^{2i\\xi }\\cos ^2\\phi \\,Z_8-e^{-2i\\xi } \\sin ^2\\phi \\,Z_8^*\\bigr ] \\nonumber \\\\&& \\qquad \\qquad \\qquad +\\tfrac{1}{2}e^{i\\xi }(\\cos 4\\phi +\\cos 2\\phi )Z_9+\\tfrac{1}{2}e^{-i\\xi }(\\cos 4\\phi -\\cos 2\\phi )Z_9^*\\bigr \\rbrace \\,,$ where $\\xi \\equiv \\alpha +\\beta \\,,\\qquad \\quad \\chi \\equiv \\alpha -\\beta \\,.$ By definition of the $H23$ -basis, $Z_5$ is real and $Z_5^\\prime =0$ [the latter is a consequence of the absence of a term in Eq.","(REF ) that involves $(H_3^\\dagger H_1)(H_2^\\dagger H_1)$ and its hermitian conjugate].","After employing a generic U(2) basis change [Eq.","(REF )], a nonzero $\\bar{Z}_5^\\prime $ and a complex $\\bar{Z}_5$ are generated [cf. Eqs.","() and ()], such that $Z_5^2=|\\bar{Z}_5|^2+\\bar{Z}_5^{\\prime \\,2}\\,.$ It is instructive to examine the form of the CP4 transformation in the $\\lbrace \\bar{H}_1$ , $\\bar{H}_2$ , $\\bar{H_3}\\rbrace $ basis.","Starting from Eq.","(REF ) and transforming $H_i\\rightarrow \\bar{H}_i= \\widetilde{V}_{ij}H_j$ ($i,j=2,3$ ), it follows that the CP4 transformation of the barred fields is given by, $\\bar{H}_i\\rightarrow \\bar{X}_{ij}\\bar{H}_j^{\\scriptscriptstyle {\\bigstar }}\\,,\\qquad \\text{where $\\bar{X}=VW{V}^T$}\\,,$ where the $3\\times 3$ matrices $\\bar{X}$ , ${V}$ and $W$ in block form are given by $\\bar{X}=\\begin{pmatrix} 1 & 0 \\\\ 0 & \\widetilde{X}\\end{pmatrix}\\,,\\qquad {V}=\\begin{pmatrix} 1 & 0 \\\\ 0 & \\widetilde{V}\\end{pmatrix}\\,,\\qquad W=\\begin{pmatrix} 1 & 0 \\\\ 0 & \\epsilon \\end{pmatrix}\\,,$ and $\\epsilon \\equiv \\left({\\begin{matrix} 0 & -1 \\\\ 1 & \\phantom{-}0\\end{matrix}}\\right)$ .","For any $\\widetilde{V}\\in $ U(2), we have $\\widetilde{X}=\\widetilde{V}\\epsilon \\widetilde{V}^T=e^{i\\psi }\\epsilon \\,,$ after taking the determinant of Eq.","(REF ) and noting that $\\det \\widetilde{V}=e^{i\\psi }$ .","Indeed, if we impose invariance of the scalar potential under CP4 in the $\\lbrace \\bar{H}_1$ , $\\bar{H}_2$ , $\\bar{H_3}\\rbrace $ basis, then the IS scalar potential must have the form given by Eq.","(REF ), with $\\bar{Z}_8$ and $\\bar{Z}_9$ potentially complex and all other scalar potential coefficients [excluding factors of $i$ or $e^{\\pm i\\psi }$ that explicitly appear in Eq.","(REF )] real.","It is possible to choose a basis in which all but one of the scalar potential parameters are real.","This can be achieved by choosing $\\psi =\\alpha =\\beta =0$ in Eq.","(REF ).","In this case, Eqs.","() and () yield $\\bar{Z}_5^\\prime =0$ , $\\bar{Z}_5=Z_5$ and $\\operatorname{Im}\\bar{Z}_8&=&\\cos 2\\phi \\operatorname{Im}Z_8+\\sin 2\\phi \\operatorname{Im}Z_9\\,,\\nonumber \\\\\\operatorname{Im}\\bar{Z}_9&=&\\cos 2\\phi \\operatorname{Im}Z_9-\\sin 2\\phi \\operatorname{Im}Z_8\\,.$ Indeed, there is a choice of $\\phi $ in Eq.","(REF ) such that $\\operatorname{Im}\\bar{Z}_9=0$ (and another choice of $\\phi $ such that $\\operatorname{Im}\\bar{Z}_8=0$ ).","Thus, by a series of basis changes, we have reduced the number of independent parameters in the IS scalar potential from 14 to 12."],["Transforming to a Higgs basis where $Z_8$ and {{formula:67350f71-efd2-4657-aecf-2e42710893ea}} are real","We now examine whether a choice of $\\psi $ , $\\chi $ , $\\xi $ and $\\phi $ exists such that $i\\bar{Z}_5^\\prime e^{\\pm i\\psi }$ , $\\bar{Z}_5 e^{\\pm i\\psi }$ , $\\bar{Z}_8$ and $\\bar{Z}_9$ are all real.","To begin, we first show that a Higgs basis exists in which $\\bar{Z}_8$ and $\\bar{Z}_9$ are both real.","Here, we follow the analysis given in Appendix C of Ref. [46].","By assumption, $Z_9$ is real and $Z_8=|Z_8|e^{i\\theta _8}$ (where $\\theta _8$ is not an integer multiple of $\\pi $ so that $\\operatorname{Im}Z_8\\ne 0$ ).","Setting $\\operatorname{Im}\\bar{Z_8}=\\operatorname{Im}\\bar{Z_9}=0$ in Eqs.","() and () yields, $\\operatorname{Im}\\bar{Z}_8=f_a\\cos 2\\chi -f_b\\sin 2\\chi =0\\,,\\qquad \\quad \\operatorname{Im}\\bar{Z}_9=f_c\\cos \\chi -f_d\\sin \\chi =0\\,,\\ $ where $f_a&=& |Z_8|\\cos 2\\phi \\sin (2\\xi +\\theta _8)+Z_9\\sin 2\\phi \\sin \\xi \\,, \\\\f_b&=&\\tfrac{1}{4}(Z_3^\\prime +Z_4^\\prime )\\sin ^2 2\\phi -|Z_8|(1-\\tfrac{1}{2}\\sin ^2 2\\phi )\\cos (2\\xi +\\theta _8)-Z_9\\sin 2\\phi \\cos 2\\phi \\cos \\xi \\,,\\\\f_c&=&-|Z_8|\\sin 2\\phi \\sin (2\\xi +\\theta _8)+Z_9\\cos 2\\phi \\sin \\xi \\,,\\\\f_d&=&\\tfrac{1}{2}(Z_3^\\prime +Z_4^\\prime ) \\sin 2\\phi \\cos 2\\phi +|Z_8|\\sin 2\\phi \\cos 2\\phi \\cos (2\\xi +\\theta _8)-Z_9\\cos 4\\phi \\cos \\xi \\,.$ Assuming that $f_a\\ne 0$ and $f_c\\ne 0$ , Eq.","(REF ) implies that $\\cot \\chi =\\frac{f_d}{f_c}\\,,\\qquad \\quad \\cot 2\\chi =\\frac{f_b}{f_a}\\,.$ Employing the trigonometric identity, $\\cot 2\\chi =(\\cot ^2\\chi -1)/(2\\cot \\chi )$ , we end up with, $G(\\phi ,\\xi )\\equiv f_a(f_d^2-f_c^2)-2f_b f_c f_d=0\\,.$ Note that the above condition is independent of the angle $\\psi $ .","Inserting the results of Eqs.","(REF )–() into Eq.","(REF ) leads to a very complicated expression.","However, it is quite easy to check that $G(0,\\xi )=-G(\\tfrac{1}{2}\\pi ,\\xi )=Z_9^2\\operatorname{Im}Z_8\\,.$ As a consequence of Eq.","(REF ), for any choice of $\\xi $ , there must exist a value of $\\phi $ between 0 and $\\tfrac{1}{2}\\pi $ such that $G(\\phi ,\\xi )=0$ .","Plugging these values of $\\phi $ and $\\xi $ back into Eqs.","(REF )–(), we can then use Eq.","(REF ) to determine $\\chi $ .","Thus, we have shown that for any choice of $\\xi $ and $\\psi $ , there must exist a corresponding $\\phi $ and $\\chi $ (whose values depend on the choice of $\\xi $ ) such that $\\operatorname{Im}\\bar{Z}_8=\\operatorname{Im}\\bar{Z}_9=0$ .Although we have reached this conclusion under the assumption that $f_c$ and $f_a$ are nonzero, it is straightforward to modify the analysis if either $f_a=0$ and/or $f_c=0$ .","If $f_a=f_b=f_c=f_d=0$ , then Eq.","(REF ) immediately yields $\\operatorname{Im}\\bar{Z}_8=\\operatorname{Im}\\bar{Z}_9=0$ .","If at least one of the quantities $f_a$ , $f_b$ , $f_c$ and $f_d$ is nonzero, then $\\chi $ can be determined from one of the two expressions in Eq.","(REF )."],["Does a Higgs basis exist were all scalar potential parameters are real?","Having found a Higgs basis with real $\\bar{Z}_8$ and $\\bar{Z}_9$ for an arbitrary choice of $\\xi $ and $\\psi $ (where the parameters $\\phi $ and $\\chi $ have been determined), we now examine whether it is also possible to choose particular values of $\\xi $ and $\\psi $ such that $i\\bar{Z}_5^\\prime e^{\\pm i\\psi }$ and $\\bar{Z}_5 e^{\\pm i\\psi }$ are both real.","If this were possible, then one would have succeeded in finding a U(2) transformation, $\\bar{H}_i = \\widetilde{V}_{ij} H_j$ ($i,j=2,3$ ) such that all the coefficients of the IS scalar potential are real.","For example, if $Z_5=0$ , then it follows from Eqs.","() and () that $\\bar{Z}_5^\\prime =\\bar{Z}_5=0$ , in which case all the coefficients of the IS scalar potential, when expressed in terms of the barred scalar doublet fields, are real.","Thus a real Higgs basis exists when $Z_5=0$ .","It therefore follows that if $Z_5=0$ , then the IS scalar potential must possess a CP2 symmetry of the form, $H_i\\rightarrow Y_{ij}H_j^{\\scriptscriptstyle {\\bigstar }}$ , where $Y$ is a symmetric unitary matrix, which in block diagonal form [cf. Eq.","(REF )] is given by, $Y=\\begin{pmatrix} 1 & \\,\\,\\,0 \\\\ 0 &\\,\\,\\, \\widetilde{Y}\\end{pmatrix}\\,,\\qquad \\quad \\text{with $\\widetilde{Y}\\equiv (\\tilde{V}^T\\tilde{V})^*$},$ and $\\tilde{V}$ [given by Eq.","(REF )] is the unitary matrix that transforms the $H23$ basis into a real Higgs basis.","Suppose one performs a CP4 transformation [Eq.","(REF )] followed by a CP2 transformation, $H_i\\rightarrow Y_{ij}H_j^{\\scriptscriptstyle {\\bigstar }}$ , and compares the result obtained by performing these two transformations in the opposite order.","The results of applying CP4 followed by CP2 as compared to CP2 followed by CP4 are equivalent to the Higgs family transformations, $YX^*$ and $XY^*$ , respectively [32].","Using Eqs.","(REF ), (REF ) and (REF ), it follows that $XY^*=e^{2i\\psi }YX^*\\,.$ That is, the CP2 and CP4 transformations commute if and only if $\\det Y= e^{2i\\psi }=1$ .","For example, the $H23$ basis is a real Higgs basis in the trivial case where $\\operatorname{Im}Z_8=\\operatorname{Im}Z_9=0$ , independently of the value of $Z_5$ .","In this case, the corresponding CP2 symmetry, $H_i\\rightarrow Y_{ij}H_j^{\\scriptscriptstyle {\\bigstar }}$ , with $Y={1}$ , commutes with the CP4 symmetry of the IS potential.","If $Z_5=0$ , the real Higgs basis obtained above is independent of $\\psi $ .","In this case, it is convenient to choose $\\psi =0$ in defining the CP2 transformation.","It then follows from Eq.","(REF ) that the CP2 and CP4 transformations commute if $Z_5=0$ and either $\\operatorname{Im}Z_8$ and/or $\\operatorname{Im}Z_9$ is nonzero.","Two other special cases, first pointed out in Appendix B of Ref.","[32], are noteworthy.","First, suppose that $Z_9=0$ .","In this case, the choice of $\\psi =\\chi =\\tfrac{1}{2}\\pi $ , $\\xi =0$ and $\\phi =\\tfrac{1}{4}\\pi $ inserted into Eqs.","()–() will yield a real Higgs basis, with $\\bar{Z}_5^\\prime =0$ , $e^{\\pm i\\psi }\\bar{Z}_5=\\mp Z_5$ , $\\bar{Z_8}=\\tfrac{1}{4}(Z_3^\\prime +Z_4^\\prime -2\\operatorname{Re}Z_8)$ and $\\bar{Z}_9=\\operatorname{Im}Z_8$ .","In light of Eqs.","(REF ) and (REF ), the barred and unbarred scalar fields are related by $\\bar{H}_2=\\frac{i}{\\sqrt{2}}(H_2-H_3)\\,,\\qquad \\quad \\bar{H}_3=\\frac{1}{\\sqrt{2}}(H_2+H_3)\\,.$ In the $H23$ basis, we can identify the corresponding CP2 transformation as $H_i\\rightarrow Y_{ij}H_j^{\\scriptscriptstyle {\\bigstar }}$ , with $Y=\\begin{pmatrix} 1 & \\,\\,\\, 0 &\\,\\,\\, 0 \\\\ 0 &\\,\\,\\, 0 &\\,\\,\\, 1 \\\\ 0 &\\,\\,\\, 1 &\\,\\,\\, 0\\end{pmatrix}\\,.$ Since $XY^*\\ne YX^*$ , it follows that the CP2 and CP4 transformations do not commute.","Second, suppose that $\\operatorname{Im}Z_8=0$ , $\\operatorname{Re}Z_9=0$ and $\\operatorname{Im}Z_9\\ne 0$ .","In this case, we simply choose $\\bar{H}_2=H_2$ and $\\bar{H_3}=iH_3$ , corresponding to $\\psi =\\tfrac{1}{2}\\pi $ , $\\chi =\\xi =-\\tfrac{1}{4}\\pi $ and $\\phi =0$ in Eq.","(REF ).","A real Higgs basis is then achieved with $\\bar{Z}_5^\\prime =0$ , $e^{\\pm i\\psi }\\bar{Z}_5=\\pm Z_5$ , $\\bar{Z}_8=-\\operatorname{Re}Z_8$ , and $\\bar{Z}_9=\\operatorname{Im}Z_9$ .","In the $H23$ basis, we can identify the corresponding CP2 transformation as $H_i\\rightarrow Y_{ij}H_j^{\\scriptscriptstyle {\\bigstar }}$ , with $Y=\\begin{pmatrix} 1 & \\phantom{-}0 &\\phantom{-}0 \\\\ 0 & \\phantom{-}1 &\\phantom{-}0 \\\\ 0 & \\phantom{-}0 & -1\\end{pmatrix}\\,.$ Once again, the CP2 and CP4 transformations do not commute.","It should be noted that the last two cases are related by a simple basis transformation.","Namely, starting from an $H23$ basis with $\\operatorname{Im}Z_8=0$ and $\\operatorname{Re}Z_9=0$ and employing $\\chi =\\psi =\\xi =0$ and $\\phi =\\tfrac{1}{4}\\pi $ in Eqs.","()–() yields $\\bar{Z}_5^\\prime =0$ , $\\bar{Z}_5=Z_5$ , $\\operatorname{Im}\\bar{Z}_8\\ne 0$ and $\\bar{Z}_9=0$ , thereby reducing to the previous case above.","We now consider the IS scalar potential with generic parameters (excluding the special cases considered above) and investigate whether a real Higgs basis exists.","In particular, consider the $H23$ basis with $Z_5$ , $Z_9\\ne 0$ .","As noted above, we can assume without loss of generality that $Z_5$ and $Z_9$ are real.The case of $\\operatorname{Im}Z_8=0$ and $\\operatorname{Re}Z_9=0$ is thus eliminated from consideration, since it is related by a scalar basis transformation to the case of $Z_9=0$ as noted above.","We examine two different cases: Case 1: $\\psi \\ne \\pm \\tfrac{1}{2}\\pi $ Case 2: $\\psi =\\pm \\tfrac{1}{2}\\pi $ In case 1, a real Higgs basis would require $\\bar{Z}_5^\\prime =0$ .","Since $\\sin 2\\phi \\ne 0$ [in light of Eq.","(REF )], it follows that $\\sin \\xi =0$ , in which case $e^{\\pm i\\psi }\\bar{Z}_5=\\pm e^{i(\\chi \\pm \\psi )}Z_5$ is real if and only if $\\sin (\\chi \\pm \\psi )=0$ .","This equation must be satisfied for both sign choices, which yields $\\sin \\chi \\cos \\psi =\\cos \\chi \\sin \\psi =0$ .","For generic values of the parameters, Eqs.","(REF )–(REF ) imply that $\\sin \\chi \\ne 0$ and $\\cos \\chi \\ne 0$ .","Thus, in general no value of $\\psi $ exists such that $\\sin (\\chi \\pm \\psi )=0$ holds for both sign choices.","That is, case 1 cannot yield a real Higgs basis for a generic choice of the IS scalar potential parameters.","In case 2, $i\\bar{Z}_5^\\prime e^{\\pm i\\psi }$ is real for all choices of $\\xi $ and one must check whether there exists a $\\xi $ that yields a real value of $i\\bar{Z}_5=ie^{i\\chi }Z_5(e^{i\\xi }\\cos ^2\\phi +e^{-i\\xi }\\sin ^2\\phi $ ).","The condition that $i\\bar{Z}_5$ is real is equivalent to $\\operatorname{Re}\\bigl [e^{i(\\chi +\\xi )}\\cos ^2\\phi +e^{i(\\chi -\\xi )}\\sin ^2\\phi \\bigr ]=0\\,,$ which can be simplified to the condition, $\\cot \\chi =\\cos 2\\phi \\tan \\xi \\,.$ It follows that either $\\chi \\pm \\xi $ are both half odd integer multiples of $\\tfrac{1}{2}\\pi $ or $\\cos 2\\phi =\\cot \\chi \\cot \\xi $ .","If $\\chi \\pm \\xi $ are both half odd integer multiples of $\\tfrac{1}{2}\\pi $ , then either $\\chi $ is a half odd integer of $\\tfrac{1}{2}\\pi $ and $\\xi $ is an integer multiple of $\\pi $ or vice versa.","If $\\chi $ is a half odd integer multiple of $\\tfrac{1}{2}\\pi $ and $\\xi $ is an integer multiple of $\\pi $ , then Eq.","(REF ) yields $f_a=f_d=0$ .","However, these latter two equations cannot be simultaneously satisfied if $Z_9\\ne 0$ .","Similarly, if $\\chi $ is an integer multiple of $\\pi $ and $\\xi $ is a half odd integer multiple of $\\tfrac{1}{2}\\pi $ , then Eq.","(REF ) yields $f_a=f_c=0$ which cannot be simultaneously satisfied if $\\operatorname{Im}Z_8$ and $Z_9$ are nonzero.","Thus, if $\\chi \\pm \\xi $ are both half odd integer multiples of $\\tfrac{1}{2}\\pi $ , then no real Higgs basis exists for generic values of the IS scalar potential parameters.","Finally, we examine the possibility that $i\\bar{Z}_5$ is real due to $\\cos 2\\phi =\\cot \\chi \\cot \\xi $ .","We can also assume that $\\xi $ is not an integer multiple of $\\tfrac{1}{2}\\pi $ , as this case was already treated above.","In order that $\\operatorname{Im}\\bar{Z}_8=\\operatorname{Im}\\bar{Z}_9=0$ , one must satisfy $\\cot \\chi =f_d/f_c$ and $G(\\phi ,\\xi )=0$ , under the assumption of $f_c\\ne 0$ .","In this case we can satisfy $\\operatorname{Im}\\bar{Z}_9=0$ if $\\phi =\\phi _\\xi $ , where $\\cos 2\\phi _\\xi =\\left(\\frac{f_d}{f_c}\\right)_{\\phi =\\phi _\\xi }\\cot \\xi \\,.$ Using Eqs.","() and (), one can employ Eq.","(REF ) to obtain a quadratic equation for $\\cot 2\\phi _\\xi $ , whose solution is given by $\\cot 2\\phi _\\xi &=&\\frac{1}{2Z_9}\\biggl [\\tfrac{1}{2}(Z_3^\\prime +Z_4^\\prime )\\cos \\xi +|Z_8|\\cos (\\xi +\\theta _8) \\nonumber \\\\&&\\qquad \\qquad \\quad \\pm \\sqrt{[\\tfrac{1}{2}(Z_3^\\prime +Z_4^\\prime )\\cos \\xi +|Z_8|\\cos (\\xi +\\theta _8)]^2+4Z_9^2\\cos ^2\\xi }\\biggr ]\\,.$ Eq.","(REF ) determines $\\sin 2\\phi _\\xi $ up to an overall sign.","It is convenient to choose this sign to be positive.","It is sufficient to demonstrate one example of the IS scalar potential parameters in which no real Higgs basis exists.","Thus, consider an example where $Z_8=-\\tfrac{1}{2}(Z_3^\\prime +Z_4^\\prime )+2iZ_9\\,.$ Then, $\\tfrac{1}{2}(Z_3^\\prime +Z_4^\\prime )\\cos \\xi +|Z_8|(\\cos (\\xi +\\theta _8) =-2Z_9\\sin \\xi $ .","It follows that $\\cot 2\\phi _\\xi =\\pm 1-\\sin \\xi \\,.$ We now investigate whether a value of $\\xi \\ne \\tfrac{1}{2}n\\pi $ (where $n$ is an integer) exists such that $G(\\phi _\\xi ,\\xi )=0$ .","We introduce the notation, $\\tilde{f}\\equiv f(\\phi _\\xi ,\\xi )$ , where $\\phi _\\xi $ has been determined from Eq.","(REF ).","Then, Eq.","(REF ) implies that $\\tilde{f}_d=\\tilde{f}_c\\cos 2\\phi _\\xi \\tan \\xi $ .","Inserting this result into Eq.","(REF ) yields, $G(\\phi _\\xi ,\\xi )= \\tilde{f}_c^2\\bigl [\\tilde{f}_a(\\cos ^2 2\\phi _\\xi \\tan ^2\\xi -1)-2\\tilde{f}_b \\cos 2\\phi _\\xi \\tan \\xi \\bigr ]\\,.$ An explicit calculation yields $\\tilde{f}_c=\\sin 2\\phi _\\xi \\bigl [-\\operatorname{Re}Z_8\\sin 2\\xi +(3\\sin \\xi \\mp 2)(\\sin \\xi \\pm 1)Z_9\\bigr ]\\,,$ and $\\tilde{f}_a(\\cos ^2 2\\phi _\\xi \\tan ^2\\xi -1)-2\\tilde{f}_b \\cos 2\\phi _\\xi \\tan \\xi =2Z_9\\frac{\\sin 2\\phi _\\xi }{\\cos ^2\\xi }(\\sin \\xi \\mp 1)\\,.$ Hence, we end up withWe have made use of the identity, $1+\\cot ^2 2\\phi _\\xi =1/\\sin ^2 2\\phi _\\xi $ .","As noted below Eq.","(REF ) , we have assumed that $\\sin 2\\phi _\\xi $ is positive.","$G(\\phi _\\xi ,\\xi )=\\frac{2Z_9}{\\cos ^2\\xi }\\bigl [1+(1\\mp \\sin \\xi )^2\\bigr ]^{-3/2}(\\sin \\xi \\mp 1)\\bigl [\\operatorname{Re}Z_8\\sin 2\\xi -(3\\sin \\xi \\mp 2)(\\sin \\xi \\pm 1)Z_9\\bigr ]^2\\,.$ Since the above analysis has assumed that $\\tilde{f}_c\\ne 0$ and $\\xi \\ne \\tfrac{1}{2}n\\pi $ (for integer $n$ ) it follows that $G(\\phi _\\xi ,\\xi )$ is strictly nonzero, which implies that $\\operatorname{Im}\\bar{Z}_8\\ne 0$ .By expanding to squared expression in Eq.","(REF ), one sees that the factor of $\\cos ^2\\xi $ in the denominator is canceled by terms in the numerator.","Hence, there is no singularity in the limit of $\\cos \\xi \\rightarrow 0$ ."],["Special cases","Cases where $f_c=0$ need to be treated separately.","First, we examine the case of $f_c=0$ and $f_d\\ne 0$ .","Inserting Eq.","(REF ) into Eq.","(), one can solve for $\\cot 2\\phi _\\xi $ , $\\cot 2\\phi _\\xi =\\frac{\\operatorname{Re}Z_8\\sin 2\\xi +2Z_9\\cos 2\\xi }{Z_9\\sin \\xi }\\,.$ We next impose $\\operatorname{Im}\\bar{Z}_9=0$ .","Then Eq.","(REF ) implies that $\\sin \\chi =0$ , in which case Eq.","(REF ) yields $\\cos \\xi =0$ .","Inserting the latter result back into Eq.","(REF ) yields $\\cot 2\\phi _\\xi =\\pm 2$ .","If one now attempts to impose $\\operatorname{Im}\\bar{Z}_8=0$ using Eq.","(REF ) with $\\sin \\chi =0$ , then one would conclude that $f_a=0$ .","However, one can explicitly show that $f_a\\ne 0$ by inserting Eq.","(REF ) into Eq.","() and employing $\\cot 2\\phi _\\xi =\\pm 2$ and $\\cos \\xi =0$ .","Finally, we briefly consider the case of $f_c=f_d=0$ .","Inserting Eq.","(REF ) into Eqs.","() and () yields equations for $\\tan 2\\phi $ and $\\tan 4\\phi $ respectively.","The compatibility of these two equations then fixes the value of $\\xi $ .","In this case, $\\operatorname{Im}\\bar{Z}_9=0$ is automatic, and $\\operatorname{Im}\\bar{Z}_8=0$ implies via Eq.","(REF ) that $\\cot 2\\chi =f_b/f_a$ .","At this stage, $\\chi $ , $\\xi $ and $\\phi $ are all determined prior to imposing Eq.","(REF ).","The latter is an independent condition; hence for generic values of $\\operatorname{Re}Z_8$ and $Z_9$ , it is not possible to perform a basis change such that $i\\bar{Z}_5$ , $\\bar{Z}_8$ and $\\bar{Z}_9$ are simultaneously real.","This completes all the subcases of case 2.","We conclude that if the IS scalar potential possesses at least one non-real coefficient (for generic choices of the scalar potential parameters), then no real Higgs basis exists and it is not possible to perform a U(2) transformation of the Higgs basis fields $\\lbrace H_2,H_3\\rbrace $ such that all coefficients of the scalar potential are real."],["Basis-invariant polynomial functions of the IS scalar potential parameters","In our analysis of the IS model, we have advocated the choice of a particular class of Higgs bases in which $\\bar{Z}_5^\\prime =0$ .","Nevertheless, it is instructive to show that physical observables that depend on the parameters of the IS scalar potential are independent of the choice of the scalar basis.","In this appendix, we introduce a number of basis invariant quantities and evaluate them in the $H23$ -basis.","Consider a generic basis of scalar fields, $\\lbrace \\Phi _a\\rbrace $ , where $a=1,2,3$ labels hypercharge-one, doublet fields of the 3HDM.","Basis transformations that leave invariant the form of the canonical kinetic energy terms correspond to global U(3) transformations, $\\Phi _a\\rightarrow U_{a{\\bar{b}}}\\Phi _b$ [and $\\Phi _{\\bar{a}}^\\dagger \\rightarrow \\Phi _{\\bar{b}}^\\dagger U^\\dagger _{b{\\bar{a}}}$ ], where the $3\\times 3$ unitary matrix $U$ satisfies $U^\\dagger _{b{\\bar{a}}}U_{a{\\bar{c}}}=\\delta _{b{\\bar{c}}}$ .","Here, we follow the index conventions introduced in Ref.","[20], in which replacing an unbarred index with a barred index is equivalent to hermitian conjugation.","We only allow sums over barred–unbarred index pairs, which are performed by employing the U(3)-invariant tensor $\\delta _{a{\\bar{b}}}$ .","In this notation, the 3HDM scalar potential in a generic $\\Phi _a$ -basis is given by, $\\mathcal {V}=Y_{a{\\bar{b}}}\\Phi _{\\bar{a}}^\\dagger \\Phi _b+\\tfrac{1}{2}Z_{a{\\bar{b}}c{\\bar{d}}}(\\Phi _{\\bar{a}}^\\dagger \\Phi _b)(\\Phi _{\\bar{c}}^\\dagger \\Phi _d)\\,,$ where $Z_{a{\\bar{b}}c{\\bar{d}}}=Z_{c{\\bar{d}}a{\\bar{b}}}$ .","Hermiticity of $\\mathcal {V}$ implies that $Y_{a {\\bar{b}}}= (Y_{b {\\bar{a}}})^\\ast $ and $Z_{a{\\bar{b}}c{\\bar{d}}}= (Z_{b{\\bar{a}}d{\\bar{c}}})^\\ast $ .","Minimizing the scalar potential, under the assumption that the vacuum preserves U(1)$_{\\rm EM}$ , yields the neutral Higgs vacuum expectation values, $\\langle \\Phi ^0_a \\rangle =v \\widehat{v}_a/\\sqrt{2}$ , where $v=246$ GeV and $\\widehat{v}_a$ is a vector of unit norm.","It is convenient to define the hermitian matrix[19] $V_{a{\\bar{b}}}\\equiv \\widehat{v}_a \\,\\widehat{v}_{\\bar{b}}^\\ast \\,.$ One can now construct basis-invariant quantities that depend on knowledge of the scalar potential minimum by forming products of $V_{a{\\bar{b}}}$ and $Z_{a{\\bar{b}}c{\\bar{d}}}$ such that all barred–unbarred index pairs are summed over.","We define six invariant quantities below, The invariants above can be evaluated in any basis.","In particular, in the $H23$ -basis, the only nonzero component of $V_{a{\\bar{b}}}$ is $V_{11}=1$ .","We thus obtain, $J_1&=&Z_1,\\\\J_2&=&Z_1+2Z_3,\\\\J_3&=&Z_1+2Z_4,\\\\J_4&=&Z_1^2+2 Z_3^2+2 Z_4^2+2 Z_5^2,\\\\J_5&=&Z_1^3+4 Z_5^2 Z_1+2 Z_3^3+6 Z_3 Z_4^2+2 Z_2 Z_5^2+4 Z_5^2 \\operatorname{Re}Z_8 ,\\\\J_6&=&Z_1^4+2 Z_3^4+2 Z_4^4+12 Z_3^2 Z_4^2+4 Z_5^4+2Z_5^2(3Z_1^2+2Z_1 Z_2+Z_2^2)\\nonumber \\\\&&+8 Z_5^2\\bigl [|Z_8|^2+(Z_1+Z_2)\\operatorname{Re}Z_8+ \\left(\\operatorname{Im}Z_9\\right)^2\\bigr ].$ Using the first four invariant quantities above, one can show that $Z_5$ can be expressed in terms of an invariant quantity.In the $H23$ -basis (where $Z_5^\\prime =0$ ), one expects that $Z^2_5$ can be expressed in terms of an invariant quantity in light of the mass relation, $M_P^2-M_Q^2=Z_5 v^2$ , which implies that $Z_5^2$ is a physical parameter.","In a general class of Higgs bases, the corresponding invariant quantity is $|Z_5|^2+|Z_5^\\prime |^2$ [cf. Eq.","(REF )].","In particular, $Z_5^2=-J_1^2+\\tfrac{1}{2}J_1 \\left(J_2+J_3\\right) -\\tfrac{1}{4} (J_2^2+J_3^2)+\\tfrac{1}{2}J_4\\,.$ Finally, we have discovered a remarkable invariant quantity, $\\mathcal {N}&=&32Z_5^2 J_6-16J_5^2+8J_5(3J_{21}J_{31}^2+K) -J_{31}^4(9J_{21}^2+4Z_5^2) -6KJ_{21}J_{31}^2-24Z_5^2 J_{21}^2 J_{31}^2 \\nonumber \\\\[5pt]&& -J_{21}^6 - 4Z_5^2J_{21}^4-8J_1(J_1^2+2Z_5^2)J_{21}^3 -16J_1^6-96 Z_5^2 J_1^4 -192 Z_5^4 J_1^2 -128Z_5^6\\,, $ where $J_{ij}\\equiv J_i-J_j$ , the invariant quantity $Z^2_5$ is given by Eq.","(REF ) and $K\\equiv 4J_1^3+8Z_5^2 J_1+J_{21}^3\\,.$ Plugging in the expressions for $J_1\\,,\\ldots \\,,J_6$ given above, we find $\\mathcal {N}=256 Z_5^4\\bigl [(\\operatorname{Im}Z_8)^2 + (\\operatorname{Im}Z_9)^2\\bigr ]\\,.$ It follows that if $Z_5\\ne 0$ then there exists a ratio of invariant quantities, which when evaluated in the $H23$ -basis, is equal to $(\\operatorname{Im}Z_8)^2 + (\\operatorname{Im}Z_9)^2$ .","In contrast, if $Z_5=0$ , then there is no invariant quantity that reduces in the $H23$ -basis to $(\\operatorname{Im}Z_8)^2 + (\\operatorname{Im}Z_9)^2$ .","Nevertheless, the invariant condition, $Z_5=0$ , signals the presence of four mass-degenerate neutral scalars.","The significance of the invariant $\\mathcal {N}$ is as follows.","The CP4-conserving IS model possesses a CP2 symmetry that commutes with CP4 if and only if $\\mathcal {N}=0$ .","Note that the nonvanishing of $\\mathcal {N}$ does not exclude the possibility of a CP2 symmetry that does not commute with CP4.","Two explicit examples of this phenomenon were presented in Appendix REF : (i) $Z_9=0$ and $\\operatorname{Im}Z_8\\ne 0$ ; and (ii) $\\operatorname{Im}Z_8=\\operatorname{Re}Z_9=0$ and $\\operatorname{Im}Z_9\\ne 0$ .","In both these cases, a real Higgs basis exists, and the corresponding CP2 transformation does not commute with CP4.One cannot employ Eq.","(REF ) to compute $\\mathcal {N}$ in the real Higgs basis in cases (i) and (ii), since in both cases, the real Higgs basis lies outside the set of $H23$ bases.","Nevertheless, we have checked that evaluating $\\mathcal {N}$ directly in the real Higgs basis in cases (i) and (ii) reproduces the corresponding results obtained in the $H23$ basis via Eq.","(REF ).","We also noted in Section that under the assumption that $M_P\\ne M_Q$ (or equivalently for $Z_5\\ne 0$ in the $H23$ basis), the decay rate for $Z\\rightarrow QQQQ^*$ , $QQ^* Q^* Q^*$ , if kinematically allowed, is nonzero if and only if $\\mathcal {N}\\ne 0$ .","Note that the invariant quantity $\\mathcal {N}$ constructed above has been expressed in terms of Higgs basis parameters.","This means that this invariant quantity depends on the knowledge of the vacuum, i.e.","the minimum of the scalar potential (which is needed to formally define the Higgs basis).","Given an explicitly CP4-invariant scalar potential, one could ask a slightly different question: is there an invariant quantity that can differentiate between scalar potentials that explicitly preserve or violate the CP2 symmetry, independently of the vacuum.","This question has been recently addressed and answered in Ref. [32].","However, it is not clear that such an invariant quantity can be directly related in practice to a physical observable."],["An alternative Higgs basis","In this paper, we first defined the $H23$ -basis by employing the scalar doublet fields $\\lbrace H_1,H_2,H_3\\rbrace $ , which was one particular choice among possible Higgs bases.","An arbitrary Higgs basis can be obtained by performing the U(2) basis transformation given by Eqs.","(REF ) and (REF ).","The corresponding IS scalar potential is given by Eq.","(REF ), where the barred coefficients in terms of the unbarred coefficients are given in Eqs.","(REF )–().","In section REF , we explored another Higgs basis choice, called the $RS$ -basis, which employs the scalar doublet fields, $\\lbrace H_1,\\mathcal {R},\\mathcal {S}\\rbrace $ .","The relations between the $H23$ -basis and $RS$ -basis are given by, $\\mathcal {R}\\equiv \\frac{1}{\\sqrt{2}}\\bigl (H_2+iH_3\\bigr )=\\begin{pmatrix} R^\\dagger \\\\ \\frac{1}{\\sqrt{2}}\\bigl (P+iQ^\\dagger \\bigr )\\end{pmatrix}\\,,\\quad \\qquad \\mathcal {S}\\equiv \\frac{1}{\\sqrt{2}}\\bigl (H_2-iH_3\\bigr )=\\begin{pmatrix} S^\\dagger \\\\ \\frac{1}{\\sqrt{2}}\\bigl (P^\\dagger +iQ\\bigr )\\end{pmatrix}\\,.$ Note that the form of the CP4 transformation in this basis is $\\begin{pmatrix} H_1 \\\\ \\mathcal {R} \\\\ \\mathcal {S}\\end{pmatrix}\\longrightarrow \\begin{pmatrix} 1 & \\phantom{-}0 & \\phantom{-}0 \\\\ 0 & \\phantom{-}0 & \\phantom{-}i \\\\ 0 & -i & \\phantom{-}0\\end{pmatrix}\\begin{pmatrix} H_1^\\dagger \\\\ \\mathcal {R}^\\dagger \\\\ \\mathcal {S}^\\dagger \\end{pmatrix}\\,.$ The change of basis from $\\lbrace H_1,H_2,H_3\\rbrace $ to $\\lbrace H_1,R,S\\rbrace $ corresponds to choosing $\\alpha =\\beta =\\phi =\\tfrac{1}{4}\\pi $ and $\\psi =-\\tfrac{1}{2}\\pi $ .","Inserting these results into Eqs.","(REF )–() yields, 1.25 $\\bar{Z}_2&=&Z_2+\\tfrac{1}{2}(Z_3^\\prime +Z_4^\\prime -2\\operatorname{Re}Z_8)\\,,\\\\\\bar{Z}_3^\\prime &=&-Z_4^\\prime +2\\operatorname{Re}Z_8\\,,\\\\\\bar{Z}_4^\\prime &=&\\tfrac{1}{2}(Z_4^\\prime -Z_3^\\prime +2\\operatorname{Re}Z_8)\\,,\\\\\\bar{Z}_5^\\prime &=&Z_5\\,,\\\\\\bar{Z}_5&=&0\\,,\\\\\\bar{Z}_8&=&-\\tfrac{1}{4}(Z_3^\\prime +Z_4^\\prime +2\\operatorname{Re}Z_8)+i\\operatorname{Re}Z_9\\,,\\\\\\bar{Z}_9&=& \\operatorname{Im}Z_9+i\\operatorname{Im}Z_8\\,.$ Note that in the $RS$ -basis, the invariant $\\mathcal {N}$ defined in Eq.","(REF ) is $\\mathcal {N}=256\\bar{Z}_5^{\\prime \\,4}|\\bar{Z}_9|^2\\,.$ In particular, the absence [or presence] of the $Z\\rightarrow QQQQ^*$ , $QQ^*Q^*Q^*$ decay discussed in Appendix depends on the [non-]vanishing of $\\bar{Z}_9$ .","These decays are governed by Eq.","(REF ), which when expressed in the $RS$ -basis is given by, $\\delta {L}_{4h}\\ni \\tfrac{1}{2}i(PQ-P^\\dagger Q^\\dagger )\\bigl [\\bar{Z}_9(P^{\\dagger \\,2}+Q^2)+\\bar{Z}_9^*(P^2+Q^{\\dagger \\,2})\\bigr ]\\,.$"],["$Z$ decay into four inert neutral scalars","Consider a universe (not ours) in which the electroweak theory of elementary particles at the electroweak scale consists of the IS model, with $M_{H^\\pm ,h^\\pm }<M_Q<\\tfrac{1}{4} m_Z\\ll M_P\\,.$ In this case, the decay of the $Z$ to four neutral inert scalars would be consistent with a CP4-symmetric IS scalar potential that does not possess a real scalar basis.","Experimentally, the final state would be detected via the decay $Q\\rightarrow (H^\\pm , h^\\pm )+W^{*\\,\\mp }$ , with the virtual $W^{*\\,\\mp }$ decaying to quark or lepton pairs.","In this universe, the $H^\\pm , h^\\pm $ are the lightest particles of the inert scalar sector and hence stable.","Although this is not our universe, this example provides a proof in principle of the existence of an experimental distinction between the CP4-conserving/CP2-nonconserving case and the CP4/CP2-conserving case.","Figure: Feynman diagrams for Z→QQQQ * Z\\rightarrow QQQQ^\\ast In light of Eq.","(REF ), there are four contributing tree-level Feynman diagrams to the decay amplitude $Z\\rightarrow QQQQ^\\ast $ , which are shown in Fig.","REF .","Employing the Feynman rules obtained from Eq.","(REF ) (and including the appropriate symmetry factors in obtaining the rules for the four-scalar vertex), the invariant matrix element is given by, $i\\mathcal {M}&=&\\frac{g}{2\\cos \\theta _W}\\bigl (\\operatorname{Im}Z_8 +i\\operatorname{Im}Z_9\\bigr )\\,\\varepsilon _\\lambda (p)\\cdot \\left[\\frac{p-2k_1}{(p-k_1)^2-M_P^2}+\\frac{p-2k_2}{(p-k_2)^2-M_P^2}\\right.","\\nonumber \\\\[6pt]&&\\qquad \\qquad \\qquad \\left.+\\frac{p-2k_3}{(p-k_3)^2-M_P^2}-\\frac{3(p-2k_4)}{(p-k_4)^2-M_P^2}\\right]\\,,\\nonumber $ where $p$ is the four-momentum of the $Z$ and the $k_i$ are the final state momenta (with $k_4$ the momentum of $Q^*$ ).","We then square the matrix element and average over the initial state spins, using $|\\mathcal {M}|^2_{\\rm ave}\\equiv \\tfrac{1}{3}\\sum _\\lambda |X\\cdot \\varepsilon _\\lambda (p)|^2=-\\tfrac{1}{3} X_\\mu X^*_\\nu \\left( g^{\\mu \\nu }-\\frac{p^\\mu p^\\nu }{m_Z^2}\\right)\\,.$ where $X$ is the four vector dotted into the polarization vector in the expression for $i\\mathcal {M}$ .","We shall work in the approximation that $M_P\\gg m_Z$ and $M_Q=0$ .","In this case, $X=\\frac{g}{M_P^2\\cos \\theta _W}\\bigl (\\operatorname{Im}Z_8 +i\\operatorname{Im}Z_9\\bigr )(p-4k_4)\\,,$ where we have used conservation of momentum, $p=k_1+k_2+k_3+k_4$ .","It then follows after some simplification (with $p^2=m_Z^2$ ) that, $|\\mathcal {M}|^2_{\\rm ave}=\\frac{16g^2}{3m_Z^2 M_P^4\\cos ^2\\theta _W}\\bigl [(\\operatorname{Im}Z_8)^2+ (\\operatorname{Im}Z_9)^2\\bigr ] (p\\cdot k_4)^2\\,.$ The four body decay width for $Z\\rightarrow QQQQ^*$ is given by $\\Gamma =\\frac{1}{6}\\,\\frac{(2\\pi )^{-8}}{2m_Z}\\int \\left(\\prod _{i=1}^4 \\frac{d^3 k_i}{2E_i}\\right)\\delta ^4(p-k_1-k_2-k_3-k_4)|\\mathcal {M}|^2_{\\rm ave}\\,,$ where the factor of $1/6$ is due to the three identical Qs in the final state (which means we overcount by a factor of 3!","by integrating over the full phase space).","Using the above results, we obtain, $\\Gamma =\\frac{4g^2(2\\pi )^{-8}}{9m_Z^3 M_P^4\\cos ^2\\theta _W}\\bigl [(\\operatorname{Im}Z_8)^2+ (\\operatorname{Im}Z_9)^2\\bigr ]\\int \\left(\\prod _{i=1}^4 \\frac{d^3 k_i}{2E_i}\\right)\\delta ^4(p-k_1-k_2-k_3-k_4) \\,(p\\cdot k_1)^2$ after changing integration variables $k_1\\longleftrightarrow k_4$ .","To perform the phase space integration, we follow Ref. [82].","To integrate over $d^3k_1d^3k_2$ we use, $\\int \\frac{d^3k_1}{2E_1}\\frac{d^3k_2}{E_2}\\delta ^{4}(N-k_1-k_2)\\times {\\left\\lbrace \\begin{array}{ll}1& \\text{$=\\frac{1}{2}\\pi $,}\\\\k_{1\\mu }&\\text{$=\\frac{1}{4}\\pi N_\\mu $,}\\\\k_{1\\mu }k_{1\\nu }&\\text{$=-\\frac{1}{24}\\pi (N^2g_{\\mu \\nu }-4N_\\mu N_\\nu )$,}\\\\k_{1\\mu }k_{2\\nu }&\\text{$=\\frac{1}{24}\\pi (N^2g_{\\mu \\nu }+2N_\\mu N_\\nu )$,}\\end{array}\\right.","}$ where $N$ is an arbitrary four-vector.","In the present application, $N=p-k_3-k_4$ .","After performing this integration, we have two further integrations to do over $k_3$ and $k_4$ .","It is convenient to work in the $Z$ rest frame: $p=(m_Z;0,0,0);\\ \\ k_3=E_3(1;0,0,1);\\ \\ k_4=E_4(1;\\sin \\theta ,0,\\cos \\theta )\\,.$ We introduce the following scaled kinematic variables $w\\equiv \\frac{1-\\cos \\theta }{2},\\ \\ \\ y\\equiv \\frac{2E_3}{m_Z},\\ \\ \\ z\\equiv \\frac{2E_4}{m_Z}\\,.$ Then, $\\int \\frac{d^3 k_3}{2E_3}\\,\\frac{d^3k_4}{2E_4}=\\frac{\\pi ^2 m_Z^4}{4}\\int _0^{1}\\!z\\,dz\\!","\\left\\lbrace \\int _0^{1-z}y\\,dy\\int _0^1 dw+\\!\\int _{1-z}^1 y\\,dy\\!","\\int _{(y+z-1)/yz}^1 \\!dw\\right\\rbrace \\,.$ We now evaluate the integral in Eq.","(REF ).","Using the above results, $\\int \\frac{d^3k_1}{2E_1}\\frac{d^3k_2}{E_2}\\delta ^{4}(N-k_1-k_2)\\,(p\\cdot k_1)^2&=&-\\frac{\\pi }{24}\\biggl \\lbrace m_Z^2(p-k_3-k_4)^2-4\\bigl [p\\cdot (p-k_3-k_4)\\bigr ]^2\\biggr \\rbrace \\nonumber \\\\[6pt]&=&\\frac{\\pi m_Z^4}{24}\\bigl [(2-y-z)^2-1+y+z-yzw\\bigr ]\\,,$ where $N\\equiv p-k_3-k_4$ .","In obtaining the above result, we have used [cf. Eq.","(REF )], $(p-k_3-k_4)^2&=&m_Z^2-2p\\cdot (k_3+k_4)+2k_3\\cdot k_4=m_Z^2(1-y-z+yzw)\\,, \\\\[5pt]p\\cdot (p-k_3-k_4) &=&m^2_Z\\bigl [1-\\tfrac{1}{2}(y+z)\\bigr ]\\,.$ Hence, after employing Eq.","(REF ), we end up with $&& \\int \\left(\\prod _{i=1}^4 \\frac{d^3 k_i}{2E_i}\\right)\\delta ^4(p-k_1-k_2-k_3-k_4) \\,(p\\cdot k_1)^2\\nonumber \\\\[10pt]&& \\qquad =\\frac{\\pi ^3 m_Z^8}{96}\\int _0^{1}\\!z\\,dz\\!","\\left\\lbrace \\int _0^{1-z} y\\,dy\\int _0^1 dw+\\!\\int _{1-z}^1 y\\,dy\\!","\\int _{(y+z-1)/yz}^1 \\!dw\\right\\rbrace \\bigl [(2-y-z)^2-1+y+z-yzw\\bigr ]\\nonumber \\\\[6pt]&&\\qquad \\,\\, =\\frac{\\pi ^3 m_Z^8}{1280}\\,.$ Collecting all our results, we end up with $\\Gamma =\\frac{g^2 m_Z^5\\bigl [(\\operatorname{Im}Z_8)^2+(\\operatorname{Im}Z_9)^2\\bigr ]}{3^2\\cdot 5\\cdot 2^{14}\\, \\pi ^5 M_P^4\\cos ^2\\theta _W}\\,.$ Relative to the decay rate of $Z$ into a neutrino pair, $\\Gamma (Z\\rightarrow \\nu \\bar{\\nu })= g^2 m_Z/(96\\pi \\cos ^2\\theta _W)$ , $\\frac{\\Gamma (Z\\rightarrow QQQQ^*)}{\\Gamma (Z\\rightarrow \\nu \\bar{\\nu })}=\\frac{(\\operatorname{Im}Z_8)^2+(\\operatorname{Im}Z_9)^2}{3\\cdot 5\\cdot 2^{9}\\, \\pi ^4 }\\left(\\frac{m_Z}{M_P}\\right)^4\\,.$ Finally, we note that the decay rate for $Z\\rightarrow QQ^*Q^*Q^*$ is identical to the one given above.","Since $Q$ and $Q^*$ are mass degenerate, the experimentally observable width would be a factor of 2 times the one given in Eq.","(REF ), as quoted in Eq.","(REF )."]],"string":"[\n [\n \"Symmetries and Mass Degeneracies in the Scalar Sector\"\n ],\n [\n \"Abstract We explore some aspects of models with two and three SU(2) scalar doublets that lead to mass degeneracies among some of the physical scalars.\",\n \"In Higgs sectors with two scalar doublets, the exact degeneracy of scalar masses, without an artificial fine-tuning of the scalar potential parameters, is possible only in the case of the inert doublet model (IDM), where the scalar potential respects a global U(1) symmetry that is not broken by the vacuum.\",\n \"In the case of three doublets, we introduce and analyze the replicated inert doublet model, which possesses two inert doublets of scalars.\",\n \"We then generalize this model to obtain a scalar potential, first proposed by Ivanov and Silva, with a CP4 symmetry that guarantees the existence of pairwise degenerate scalar states among two pairs of neutral scalars and two pairs of charged scalars.\",\n \"Here, CP4 is a generalized CP symmetry with the property that $({\\\\rm CP}4)^n$ is the identity operator only for integer $n$ values that are multiples of 4.\",\n \"The form of the CP4-symmetric scalar potential is simplest when expressed in the Higgs basis, where the neutral scalar field vacuum expectation value resides entirely in one of the scalar doublet fields.\",\n \"The symmetries of the model permit a term in the scalar potential with a complex coefficient that cannot be removed by any redefinition of the scalar fields within the class of Higgs bases (in which case, we say that no real Higgs basis exists).\",\n \"A striking feature of the CP4-symmetric model is that it preserves CP even in the absence of a real Higgs basis, as illustrated by the cancellation of the contributions to the CP violating form factors of the effective ZZZ and ZWW vertices.\"\n ],\n [\n \"Introduction\",\n \"After the initial discovery of the Higgs boson in 2012[1], [2], certain anomalies in the Higgs data (which have since disappeared) motivated the exploration of the possibility that the 125 GeV Higgs signal was comprised of two nearly mass-degenerate scalar states[3], [4], [5], [6], [7], [8], [9], [10], [11].\",\n \"Although the present Higgs data is consistent with the Standard Model[12], [13], [14], one cannot yet rule out the presence of a mass degenerate scalar state at 125 GeV[15].\",\n \"In this work, we consider the implications of a mass degeneracy among two (or more) scalar states of an extended Higgs sector.\",\n \"Such a mass degeneracy can be either accidental or the result of a symmetry.\",\n \"A trivial example of such a phenomenon arises in any doublet extended Higgs model.\",\n \"All such models possess a mass degenerate state, namely the charged Higgs boson, $H^\\\\pm $ .\",\n \"Indeed, $H^+$ and $H^-$ are mass-degenerate due to the U(1)$_{\\\\rm EM}$ gauge symmetry.\",\n \"Moreover, the $H^+$ and $H^-$ are distinguishable by their electric charge, which can be experimentally probed using photons.\",\n \"Suppose that this probe were unavailable (or equivalently, suppose one could turn off electromagnetism).\",\n \"In this case, would it be possible for an experiment to reveal the existence of a mass-degenerate scalar?\",\n \"In this very simple example, one could not physically distinguish (on an event by event basis) between the two degenerate states that comprise the charged Higgs scalar.\",\n \"Nevertheless, there would in principle be observables that are sensitive to the number of mass-degenerate scalar states present.\",\n \"For example, in the CP-conserving two Higgs doublet model, the decay rate for the decay of a heavy CP-even neutral scalar, $H\\\\rightarrow H^+ H^-$ (if kinematically allowed) is proportional to the number of degrees of freedom in the final state.\",\n \"If we express the charged Higgs field as a linear combination of real scalar fields, $H^\\\\pm =(\\\\phi _1\\\\pm i\\\\phi _2)/\\\\sqrt{2}$ , then the decay rate for $H\\\\rightarrow H^+ H^-$ is the (incoherent) sum of the decay rates for $H\\\\rightarrow \\\\phi _1\\\\phi _1$ and $\\\\phi _2\\\\phi _2$ .\",\n \"These two rates are identical, and the sum yields a multiplicity factor of 2.\",\n \"This multiplicity factor provides the experimental signal for mass-degenerate scalars.Note that the decay rate for $Z\\\\rightarrow H^+ H^-$ is equal to the decay rate for $Z\\\\rightarrow \\\\phi _1\\\\phi _2$ .\",\n \"In this case, the off-diagonal nature of the $Z\\\\phi _1\\\\phi _2$ coupling implies that no multiplicity factor is present.\",\n \"Nevertheless, one can still infer the existence of mass-degenerate states, since the decays $Z\\\\rightarrow \\\\phi _1\\\\phi _1, \\\\phi _2\\\\phi _2$ are forbidden by Bose statistics.\",\n \"Apart from the trivial mass degeneracy of $H^\\\\pm $ , we would like to explore in this paper the possibility of exactly mass-degenerate neutral scalars and/or mass-degenerate charged Higgs pairs in extended Higgs sectors.\",\n \"In each case, the critical questions to ask are: (i) is the origin of the exact mass degeneracy natural?\",\n \"and (ii) how can the mass degenerate scalars be distinguished experimentally?\",\n \"Exact mass degeneracies are natural if they are a consequence of an unbroken symmetry.\",\n \"In particular, accidental mass degeneracies require an artificial fine-tuning of the scalar potential parameters, and in this sense we shall call them unnatural (and in our view not especially interesting).\",\n \"If mass-degenerate states are present, it is of interest to determine how to probe them experimentally.\",\n \"In some cases, one can identify the presence of mass degenerate states on an event by event basis.\",\n \"In other cases, the only signal of the mass degeneracy is a measurable multiplicity factor that can be determined when averaging over initial state degeneracies and summing over final state degeneracies.For example, a quark of a given flavor is a mass degenerate state due to its three possible colors.\",\n \"Although the color of a quark cannot be identified experimentally, the presence of the color degree of freedom can be experimentally verified by the color multiplicity factor (most famously exhibited in the observed cross section for $e^+e^-$ annihilation into quark-antiquark pairs.)\",\n \"Our focus in this paper is extended multi-doublet Higgs sectors with mass-degenerate scalar states.\",\n \"It is particularly instructive to discuss mass degeneracy in the scalar sector starting from the so-called Higgs basis.\",\n \"This corresponds to a subset of all possible scalar field parameterizations in which only one Higgs doublet, denoted by $H_1$ , acquires a non-zero positive vacuum expectation value (vev), while all the other scalar fields of the Higgs basis ($H_2$ , $H_3,\\\\ldots H_n$ ) have zero vev [16], [17], [18], [19], [20].\",\n \"The neutral and charged Goldstone bosons reside entirely in $H_1$ , as this is the only doublet that possesses a non-zero vev, together with a neutral Higgs field that, in the absence of mixing with the neutral fields of the other Higgs doublets, behaves like the Standard Model (SM) Higgs boson.\",\n \"In this sense, this subset of scalar bases may be viewed as Standard Model aligned bases.\",\n \"That is, the Higgs basis is actually a family of basis choices, since one is always free to perform an arbitrary U($n-1$ ) transformation among $H_2$ , $H_3,\\\\ldots H_n$ while preserving the vev of $H_1$ [21].\",\n \"There is no loss of generality in choosing any particular scalar basis as a starting point.\",\n \"In order to identify the physical neutral scalars one must diagonalize the corresponding scalar squared-mass matrix.\",\n \"As expected, the end result is independent of the initial choice of basis for the scalar fields.\",\n \"In section , we study under what circumstances there is an exact mass degeneracy in the familiar two-Higgs-doublet model (2HDM) [22], [23].\",\n \"In this model there are three physical neutral fields and one charged field, so we only consider potential mass degeneracies among the neutral fields.One can also examine mass degeneracies between the charged Higgs boson and one of the neutral Higgs bosons.\",\n \"For example, in a custodial symmetric 2HDM, the CP-odd Higgs scalar is degenerate with the charged Higgs boson [24], [25], [26], [27].\",\n \"However, custodial symmetry is not an exact symmetry of the full electroweak Lagrangian.\",\n \"Thus, given a custodial symmetric 2HDM scalar potential, any potential mass degeneracies between neutral and charged scalars is at best approximate.\",\n \"We do not consider such mass degeneracies further in this work.\",\n \"We begin our 2HDM analysis by studying possible mass degeneracies among the neutral scalar states of the inert doublet model (IDM) [28], [29].\",\n \"The scalar potential of this model possesses a discrete $\\\\mathbb {Z}_2$ symmetry that is unbroken by the vacuum.\",\n \"In this case, the CP-even neutral component of $H_1$ in the Higgs basis is a mass eigenstate whose tree-level couplings are precisely those of the SM Higgs boson.\",\n \"The real and imaginary parts of the neutral component of $H_2$ are odd under the discrete $\\\\mathbb {Z}_2$ symmetry, and have opposite signs under CP.\",\n \"We will denote these two neutral states by $H$ and $A$ , although there is no way to identify which of these two states is CP-even and which is CP-odd.Equivalently, one can propose two different definitions of CP (called, say, CPa and CPb), such that $H$ is CPa-even and $A$ is CPa-odd, and vice versa for CPb.\",\n \"Either definition can be consistently used to define the CP symmetry of the bosonic sector of the IDM.\",\n \"It is possible that $h$ is degenerate in mass with either $H$ or $A$ , but such mass degeneracies are accidental in nature since neither case can arise due to a symmetry.\",\n \"Moreover, these mass-degenerate states are physically distinguishable, since $h$ is even whereas $H$ and $A$ are odd under the $\\\\mathbb {Z}_2$ symmetry.\",\n \"In contrast, an exact mass degeneracy of $H$ and $A$ can arise if the $\\\\mathbb {Z}_2$ symmetry of the scalar potential is promoted to a continuous U(1) symmetry.\",\n \"In our terminology, this mass degeneracy of $H$ and $A$ is natural.\",\n \"Nevertheless, the two mass-degenerate states can still be physically distinguished due to the coupling of these states to $W^\\\\pm H^\\\\mp $ .\",\n \"One can now extend the above analysis to an arbitrary 2HDM.\",\n \"One can show that with one exception, all 2HDM mass degeneracies are accidental.\",\n \"The one exceptional case of a natural mass degeneracy is precisely the case of $m_H=m_A$ in the IDM.\",\n \"This conclusion can also be obtained by considering all possible symmetries of the 2HDM scalar potential.\",\n \"Among these symmetries, we can identify those that can potentially guarantee the mass degeneracy of scalar states.\",\n \"By examining the consequences of these symmetries, we again confirm that the only possible neutral scalar mass degeneracy in the 2HDM arises in the IDM as previously noted.\",\n \"In section we consider possible mass-degeneracies in the three Higgs doublet model.\",\n \"Using the previous 2HDM analysis of mass degeneracies of the IDM, we construct a three Higgs doublet model (3HDM) generalization of the IDM, which we call the replicated inert doublet model (RIDM).\",\n \"In this model, two of the three Higgs doublets are inert, and four mass-degenerate scalar pairs exist (two involving the charged scalar states from the inert doublets and two involving the neutral scalar states from the inert doublets).\",\n \"We can explicitly identify the symmetries that are responsible for these mass degeneracies.\",\n \"We then investigate the possibility of adding new terms to the scalar potential that partially break these symmetries while preserving the mass degeneracies.\",\n \"In this way, we arrive at a model first proposed by Ivanov and Silva [30].\",\n \"The Ivanov and Silva scalar potential possesses a discrete subgroup of the continuous symmetries that govern the RIDM, that maintains the mass degeneracies of the RIDM.\",\n \"This discrete subgroup is the generalized CP symmetry, CP4, which has the property that $({\\\\rm CP}4)^n$ is the identity operator only for integer $n$ values that are multiples of 4.\",\n \"The CP4 symmetry is distinguished from the conventional CP symmetry (denoted henceforth by CP2), which has the property that $(\\\\rm {CP}2)^2$ is the identity operator.\",\n \"Some properties of specialized 3HDMs have also been analyzed recently in Ref.[31].\",\n \"One of the most notable properties of the Ivanov-Silva (IS) model is that one can write down the most general CP4-invariant scalar potential with three Higgs doublets, which has the feature that at least one of the coefficients of the quartic terms of the scalar potential must be complex (with a nonvanishing imaginary part).\",\n \"Indeed, as demonstrated explicitly in Appendix A, one cannot redefine the scalar fields within the family of Higgs bases such that all the coefficients of the scalar potential are real.\",\n \"In this case, we say that no real Higgs basis exists.\",\n \"This means that CP2 is not a symmetry of the IS scalar potential and vacuum.\",\n \"In section , we identify the existence of a physical observable that is present if no CP2 symmetry exists that commutes with the CP4 symmetry of the IS model.However, this leaves open the possibility of the existence of a CP2 symmetry that does not commute with the CP4 symmetry [32]; in this case, a real Higgs basis exists and a conventional CP symmetry can be defined.\",\n \"As an example, we focus on $Z$ decay into four inert neutral scalars (with some details relegated to Appendix B).\",\n \"Nevertheless, the CP4 invariance guarantees that all CP-violating observables involving the Higgs/gauge boson sector of the theory must be absent.\",\n \"For example, we provide an instructive analysis in section that shows how the CP4 symmetry of the IS model with no real Higgs basis ensures the cancellation of contributions to the CP-violating form factors of the effective $ZZZ$ and $ZW^+ W^-$ vertices up to three-loop order.\",\n \"Finally, we state our conclusions in section .\"\n ],\n [\n \"2HDM mass degeneracies\",\n \"Consider the 2HDM, consisting of two hypercharge-one, doublet scalar fields, $\\\\Phi _1$ and $\\\\Phi _2$ .\",\n \"The most general gauge-invariant renormalizable scalar potential is $\\\\mathcal {V}&=& m_{11}^2 \\\\Phi _1^\\\\dagger \\\\Phi _1+ m_{22}^2 \\\\Phi _2^\\\\dagger \\\\Phi _2 -[m_{12}^2\\\\Phi _1^\\\\dagger \\\\Phi _2+{\\\\rm h.c.}]+\\\\tfrac{1}{2}\\\\lambda _1(\\\\Phi _1^\\\\dagger \\\\Phi _1)^2+\\\\tfrac{1}{2}\\\\lambda _2(\\\\Phi _2^\\\\dagger \\\\Phi _2)^2+\\\\lambda _3(\\\\Phi _1^\\\\dagger \\\\Phi _1)(\\\\Phi _2^\\\\dagger \\\\Phi _2)\\\\nonumber \\\\\\\\&&\\\\quad +\\\\lambda _4( \\\\Phi _1^\\\\dagger \\\\Phi _2)(\\\\Phi _2^\\\\dagger \\\\Phi _1)+\\\\left\\\\lbrace \\\\tfrac{1}{2}\\\\lambda _5 (\\\\Phi _1^\\\\dagger \\\\Phi _2)^2 +\\\\big [\\\\lambda _6 (\\\\Phi _1^\\\\dagger \\\\Phi _1) +\\\\lambda _7 (\\\\Phi _2^\\\\dagger \\\\Phi _2)\\\\big ] \\\\Phi _1^\\\\dagger \\\\Phi _2+{\\\\rm h.c.}\\\\right\\\\rbrace \\\\,.$ We shall assume that the minimum of the scalar potential is electric charge conserving, in which case only the neutral scalar fields possess a nonzero vacuum expectation value (vev), $\\\\langle \\\\Phi _i^0 \\\\rangle =v_i/\\\\sqrt{2}$ , where the $v_i$ are potentially complex.\",\n \"The Fermi constant, $G_F$ fixes the value of $v^2\\\\equiv |v_1|^2+|v_2|^2=(\\\\sqrt{2}G_F)^{-1}\\\\simeq (246~{\\\\rm GeV})^2\\\\,.$ Employing a new scalar field basis consisting of two orthonormal linear combinations of $\\\\Phi _1$ and $\\\\Phi _2$ does not modify the physical predictions of the model.\",\n \"One convenient choice is the Higgs basis, in which the redefined doublet fields (denoted below by $H_1$ and $H_2$ ) have the property that $H_1$ has a non-zero vev whereas $H_2$ has a zero vev [16], [17], [18].\",\n \"In particular, we define new Higgs doublet fields: $H_1=\\\\begin{pmatrix}H_1^+\\\\\\\\ H_1^0\\\\end{pmatrix}\\\\equiv \\\\frac{1}{v}(v_1^* \\\\Phi _1+v_2^*\\\\Phi _2)\\\\,,\\\\qquad \\\\quad H_2=\\\\begin{pmatrix} H_2^+\\\\\\\\ H_2^0\\\\end{pmatrix}\\\\equiv \\\\frac{1}{v}(-v_2 \\\\Phi _1+v_1\\\\Phi _2)\\\\,.$ It follows that $\\\\langle H_1^0 \\\\rangle =v$ and $\\\\langle H_2^0 \\\\rangle =0$ .\",\n \"The Higgs basis is uniquely defined up to an overall rephasing, $H_2\\\\rightarrow e^{i\\\\chi } H_2$ (which does not alter the fact that $\\\\langle H_2^0 \\\\rangle =0$ ).\",\n \"In the Higgs basis, the scalar potential of Eq.\",\n \"(REF ) is denoted as [19], [20]: $\\\\mathcal {V}&=& Y_1 H_1^\\\\dagger H_1+ Y_2 H_2^\\\\dagger H_2 +[Y_3H_1^\\\\dagger H_2+{\\\\rm h.c.}]+\\\\tfrac{1}{2}Z_1(H_1^\\\\dagger H_1)^2+\\\\tfrac{1}{2}Z_2(H_2^\\\\dagger H_2)^2+Z_3(H_1^\\\\dagger H_1)(H_2^\\\\dagger H_2)\\\\nonumber \\\\\\\\&&\\\\quad +Z_4( H_1^\\\\dagger H_2)(H_2^\\\\dagger H_1)+\\\\left\\\\lbrace \\\\tfrac{1}{2}Z_5 (H_1^\\\\dagger H_2)^2 +\\\\big [Z_6 (H_1^\\\\dagger H_1) +Z_7 (H_2^\\\\dagger H_2)\\\\big ] H_1^\\\\dagger H_2+{\\\\rm h.c.}\\\\right\\\\rbrace \\\\,,$ where $Y_1$ , $Y_2$ and $Z_1,\\\\ldots ,Z_4$ are real parameters, whereas $Y_3$ , $Z_5$ , $Z_6$ and $Z_7$ are potentially complex parameters.\",\n \"Imposing the scalar potential minimum conditions yields, $Y_1=-\\\\tfrac{1}{2}Z_1 v^2\\\\,,\\\\qquad \\\\quad Y_3=-\\\\tfrac{1}{2}Z_6 v^2\\\\,.$\"\n ],\n [\n \"Mass degeneracies of the inert doublet model (IDM)\",\n \"We wish to study the consequences of a 2HDM in which two or three of the neutral Higgs scalars are degenerate in mass.\",\n \"For simplicity, we shall first specialize to the inert 2HDM (the so-called IDM)[28], [29] in which there is an exact discrete $\\\\mathbb {Z}_2$ symmetry that is preserved by the vacuum, under which all particles of the SM and one of the two Higgs doublet fields (which contains the observed Higgs boson) are even and the second Higgs doublet field is odd under the multiplicative discrete symmetry.\",\n \"In particular, the discrete symmetry of the IDM is manifest in the Higgs basis, where we identify $H_1$ as even and $H_2$ as odd under the $\\\\mathbb {Z}_2$ symmetry.\",\n \"It then follows that $Y_3=Z_6=Z_7=0$ .\",\n \"The IDM scalar potential is CP-conserving since one can eliminate the phase of $Z_5$ (the only remaining potentially complex scalar potential parameter) by appropriately rephasing the Higgs basis field $H_2$ .\",\n \"The Higgs basis fields are $H_1=\\\\begin{pmatrix} G^+ \\\\\\\\ \\\\frac{1}{\\\\sqrt{2}}\\\\bigl [v+h+iG^0\\\\bigr ] \\\\end{pmatrix}\\\\,,\\\\qquad \\\\quad H_2=\\\\begin{pmatrix} H^+ \\\\\\\\ \\\\frac{1}{\\\\sqrt{2}}\\\\bigl [H+iA\\\\bigr ]\\\\end{pmatrix}\\\\,,$ where $G^\\\\pm $ and $G^0$ are the Goldstone bosons that provide the longitudinal degrees of freedom of the massive $W^\\\\pm $ and $Z^0$ gauge bosons.\",\n \"The physical mass spectrum of the IDM is given by, $&&m_h^2=Z_1 v^2\\\\,,\\\\qquad \\\\qquad \\\\qquad \\\\qquad \\\\quad m^2_{H^\\\\pm }=Y_2+\\\\tfrac{1}{2}Z_3 v^2\\\\,,\\\\\\\\&&m_{A}^2=m_{H^\\\\pm }^2+\\\\tfrac{1}{2}(Z_4- Z_5)v^2\\\\,,\\\\qquad \\\\,\\\\, m_H^2=m_A^2+Z_5 v^2\\\\,.$ For completeness, we exhibit the Higgs couplings of the IDM in the unitary gauge below (where the Goldstone fields are set to zero).\",\n \"First, the interactions of the Higgs bosons and the gauge bosons are governed by,The photon field $A_\\\\mu $ should not be confused with the scalar field $A$ .\",\n \"${L}_{VVH}&=&\\\\left(gm_W W_\\\\mu ^+W^{\\\\mu \\\\,-}+\\\\frac{g}{2c_W} m_ZZ_\\\\mu Z^\\\\mu \\\\right)h\\\\,,\\\\\\\\[8pt]{L}_{VVHH}&=&\\\\left[\\\\tfrac{1}{4}g^2 W_\\\\mu ^+W^{\\\\mu \\\\,-}+\\\\frac{g^2}{8c_W^2}Z_\\\\mu Z^\\\\mu \\\\right](h^2+H^2+A^2) \\\\nonumber \\\\\\\\&& +\\\\left[\\\\tfrac{1}{2}g^2 W_\\\\mu ^+ W^{\\\\mu \\\\,-}+e^2A_\\\\mu A^\\\\mu +\\\\frac{g^2}{c_W^2}\\\\left(\\\\tfrac{1}{2}-s_W^2\\\\right)^2Z_\\\\mu Z^\\\\mu +\\\\frac{2ge}{c_W}\\\\left(\\\\tfrac{1}{2}-s_W^2\\\\right)A_\\\\mu Z^\\\\mu \\\\right]H^+H^-\\\\nonumber \\\\\\\\&& +\\\\biggl \\\\lbrace \\\\left(\\\\tfrac{1}{2}eg A^\\\\mu W_\\\\mu ^+-\\\\frac{g^2s_W^2}{2c_W}Z^\\\\mu W_\\\\mu ^+\\\\right)H^-(H+iA) +{\\\\rm h.c.}\\\\biggr \\\\rbrace \\\\,, \\\\\\\\[8pt]{L}_{VHH}&=&\\\\frac{g}{2c_W}\\\\,Z^\\\\mu A\\\\,\\\\!\\\\!\\\\mathrel {\\\\leftrightarrow \\\\hspace{-8.50006pt}\\\\partial }$ H -12g[iW+ H-$\\\\leftrightarrow $$\\\\partial $$\\\\scriptstyle \\\\,\\\\mu $ (H+iA) +h.c.]\",\n \"+[ieA+igcW(12-sW2) Z]H+$\\\\leftrightarrow $$\\\\partial $ H-, where $s_W\\\\equiv \\\\sin \\\\theta _W$ , $c_W\\\\equiv \\\\cos \\\\theta _W$ .\",\n \"The trilinear and quadrilinear Higgs self-interactions are governed by ${L}_{3h}&=&-\\\\tfrac{1}{2}v\\\\bigl [Z_1 h^3+(Z_3+Z_4)h(H^2+A^2)+Z_5h(H^2-A^2)\\\\bigr ]-vZ_3hH^+H^-\\\\,, \\\\\\\\[6pt]{L}_{4h}&=&-\\\\tfrac{1}{8}\\\\bigl [Z_1 h^4+Z_2(H^2+A^2)^2+2(Z_3+Z_4)h^2(H^2+A^2)+2Z_5 h^2(H^2-A^2)\\\\bigr ]\\\\nonumber \\\\\\\\[6pt]&&-\\\\tfrac{1}{2}H^+ H^-\\\\bigl [Z_2(H^2+A^2+H^+ H^-)+Z_3 h^2\\\\bigr ].\",\n \"$ The tree-level couplings of $h$ to SM particles obtained above are precisely those of the SM Higgs boson, corresponding to the exact Higgs alignment limit [33], [34], [35], [36], [37], [38], [39], [40], [41] (as expected in light of $Z_6=0$ ).\",\n \"Moreover, an examination of the above couplings implies that $h$ is CP-even (to be identified with the SM Higgs boson) and $H$ and $A$ have opposite CP-quantum numbers (one is odd and the other is even) based on the $ZAH$ coupling.Under the rephasing, $H_2\\\\rightarrow iH_2$ , we note that $Z_5\\\\rightarrow -Z_5$ , $H\\\\rightarrow -A$ and $A\\\\rightarrow H$ .\",\n \"One can check that the masses of $H$ and $A$ and their couplings are invariant under this rephasing.\",\n \"Note that the CP is not uniquely defined by the IDM interactions, since two candidate definitions of CP exist (called CPa and CPb in footnote REF ), where $H$ is CPa-even and $A$ is CPa-odd, and vice versa for CPb.\",\n \"Either definition of CP can be used consistently in exploring the phenomenology of the IDM.\",\n \"Finally, we note that under the $\\\\mathbb {Z}_2$ symmetry of the IDM, the quarks and leptons can be chosen to be even.\",\n \"Consequently, the tree-level couplings of $h$ to fermion pairs are identical to those of the SM Higgs boson, whereas $H$ , $A$ and $H^\\\\pm $ do not couple to the SM fermions.\",\n \"We now examine the possibility of mass degeneracies in the IDM.\",\n \"First, consider the case of $m_h=m_H$ or $m_h=m_A$ .\",\n \"In this case, it is possible to physically distinguish between $h$ and its mass-degenerate partner due to their opposite $\\\\mathbb {Z}_2$ quantum numbers.\",\n \"For example, since all SM bosons and fermions are even under the $\\\\mathbb {Z}_2$ symmetry, it follows that the gluon-gluon (via a top quark loop), $WW$ and $ZZ$ fusion processes can only produce $h$ whereas Drell-Yan production (via virtual $s$ -channel $Z$ exchange) can only produce $H$ in association with $A$ .\",\n \"Hence, despite the mass degeneracy, the two mass-degenerate scalars are physically distinguishable.\",\n \"Note that the mass degeneracy of $h$ and its scalar partner is not radiatively stable.\",\n \"For example, if $h$ and $H$ are mass degenerate states, then the one-loop contributions to the $hh$ two-point function (such as $ZZ$ and $WW$ intermediate states) differ from the corresponding contributions to the $HH$ two-point function (e.g.\",\n \"the $AZ$ intermediate state).\",\n \"Indeed, the tree-level condition for the mass degeneracy of $h$ and $H$ , $Z_1v^2=Y_2+\\\\tfrac{1}{2}Z_{345}v^2,$ where $Z_{345}\\\\equiv Z_3+Z_4+Z_5$ , is unnatural; i.e., Eq.\",\n \"(REF ) is not the result of some symmetry.The scenario where $m_h=m_H=m_A$ is a special case of the $h$ –$H$ mass degeneracy.\",\n \"In the triply mass-degenerate scenario, $h$ is also distinguished from $\\\\phi ^\\\\pm $ by its U(1) charge, which is zero.\",\n \"For example, there is no coupling of $ZW^\\\\pm H^\\\\mp h$ in contrast to the $Z W^\\\\pm H^\\\\mp \\\\phi ^\\\\pm $ couplings exhibited in Eq.\",\n \"(REF ).\",\n \"Second, consider the case of $m_H=m_A$ , which corresponds to $Z_5=0$ .\",\n \"In this case, the IDM scalar potential possesses a continuous U(1) symmetry, which is not spontaneously broken by the vacuum.This case has also been noted in Ref. [41].\",\n \"It is this symmetry that is responsible for the mass degenerate states $H$ and $A$ .\",\n \"One can now define eigenstates of U(1) charge, $\\\\phi ^\\\\pm =\\\\frac{1}{\\\\sqrt{2}}\\\\bigl [H\\\\pm iA\\\\bigr ]\\\\,.$ The relevant interaction terms of $\\\\phi ^{\\\\pm }$ are ${L}_{\\\\rm int}&=&\\\\left[\\\\tfrac{1}{2}g^2 W_\\\\mu ^+W^{\\\\mu \\\\,-}+\\\\frac{g^2}{4c_W^2}Z_\\\\mu Z^\\\\mu \\\\right]\\\\phi ^+\\\\phi ^- +\\\\frac{ig}{2c_W}Z^\\\\mu \\\\phi ^-\\\\!\\\\!\\\\mathrel {\\\\leftrightarrow \\\\hspace{-8.50006pt}\\\\partial }$ + -g2[iW+ H-$\\\\leftrightarrow $$\\\\partial $$\\\\scriptstyle \\\\,\\\\mu $ + +h.c.]\",\n \"+eg2 (AW+H-+ + AW-H+-) -g2sW22cW(ZW+H-+ + ZW-H+-) - v(Z3+Z4)h+–12[Z2(+-)2+(Z3+Z4)h2+-]-Z2 H+ H-+- .\",\n \"Although $\\\\phi ^\\\\pm $ are mass degenerate states, they can be distinguished.\",\n \"For example, Drell-Yan production via a virtual $s$ -channel $W^+$ exchange can produce $H^+$ in association with $\\\\phi ^-$ , whereas virtual $s$ -channel $W^-$ exchange can produce $H^-$ in association with $\\\\phi ^+$ .\",\n \"Thus, the sign of the charged Higgs boson reveals the U(1)-charge of the produced neutral scalar.\",\n \"The origin of this correlation lies in the fact that, by construction, $H^+$ and $\\\\phi ^+$ both reside in $H_2$ , whereas $H^-$ and $\\\\phi ^-$ both reside in $H_2^\\\\dagger $ .\"\n ],\n [\n \"2HDM mass degeneracies beyond the IDM\",\n \"Although the IDM is a rather special case among all possible 2HDMs, the conclusions concerning mass degeneracies are robust.\",\n \"Allowing for the most general 2HDM scalar potential, the Higgs sector is CP-violating if the relative phases among $Z_5$ , $Z_6$ and $Z_7$ cannot be removed by rephasing the Higgs basis field $H_2$ .\",\n \"It is convenient to introduce three invariant quantities[42], [18], [20], whose imaginary parts are given by,Basis-invariant expressions for the $J_i$ are given in Ref.[20].\",\n \"$\\\\operatorname{Im}J_1=\\\\operatorname{Im}(Z_6^* Z_7)\\\\,,\\\\qquad \\\\operatorname{Im}J_2=\\\\operatorname{Im}(Z_5^* Z_6^2)\\\\,,\\\\qquad \\\\operatorname{Im}J_3=\\\\operatorname{Im}\\\\bigl [Z_5^*(Z_6+Z_7)^2\\\\bigr ]\\\\,.$ The Higgs sector of the 2HDM is CP-violating unless $\\\\operatorname{Im}J_1=\\\\operatorname{Im}J_2=\\\\operatorname{Im}J_3=0$ .\",\n \"The origin of the CP-violation can either be explicit or spontaneous [43], [44].To test for explicit CP-violation, one must employ invariant quantities that are independent of the scalar field vacuum expectation values [45], [46], [47].\",\n \"Note that the neutral scalar squared-mass matrix does not involve the Higgs basis parameter $Z_7$ .\",\n \"In particular, $Z_7$ only enters in the Higgs boson cubic and quartic self-couplings.\",\n \"Hence, if $\\\\operatorname{Im}J_2=\\\\operatorname{Im}(Z_5^* Z_6^2)=0$ , then the neutral Higgs mass eigenstates behave like eigenstates of CP in their tree-level interactions with the gauge bosons and fermions (independently of the values of $\\\\operatorname{Im}J_1$ and $\\\\operatorname{Im}J_3$ ).\",\n \"Moreover, the neutral scalar squared-mass matrix breaks up into a block diagonal form consisting of a $2\\\\times 2$ block (whose diagonalization yields the two CP-even neutral scalars) and a $1\\\\times 1$ block (which yields the CP-odd neutral scalar).\",\n \"Consider the possibility of mass degeneracies among the neutral scalars of the most general 2HDM.\",\n \"We now recall a remarkable tree-level relation of the CP-violating 2HDM[48], [42], [49], $ \\\\operatorname{Im}J_2&={\\\\rm Im}(Z_5^* Z_6^2)=\\\\frac{2 s_{13}c_{13}^2 s_{12}c_{12}}{v^6}(m_2^2-m_1^2)(m_3^2-m_1^2)(m_3^2-m_2^2)\\\\,,$ where the $m_i$ ($i=1,2,3$ ) are the masses of the three neutral Higgs bosons of the 2HDM, $s_{12}\\\\equiv \\\\sin \\\\theta _{12}$ , $c_{12}\\\\equiv \\\\cos \\\\theta _{12}$ , etc., and $\\\\theta _{12}$ and $\\\\theta _{13}$ are invariant mixing angles that are associated with the diagonalization of the neutral Higgs squared-mass matrix in the Higgs basis.Details on the definition of the mixing angles and their relations to the Higgs basis scalar potential parameters can be found in Ref.[49].\",\n \"However, we will not need any of these details for the present argument.\",\n \"In Ref.\",\n \"[50], the three CP-odd invariants $\\\\operatorname{Im}J_i$ have been expressed in terms of the neutral scalar masses and the couplings of the neutral Higgs mass eigenstates to charged pairs, $e_i$ $(H_iW^+W^-)$ and $q_i$ $(H_iH^+H^-)$ .\",\n \"In particular,In Ref.\",\n \"[42], the invariant defined in Eq.\",\n \"(REF ) is called $J_1$ .\",\n \"$ \\\\operatorname{Im}J_2=\\\\frac{2e_1 e_2 e_3}{v^9}(m_1^2-m_2^2)(m_2^2-m_3^2)(m_3^2-m_1^2).$ If any two of the three neutral Higgs bosons are mass-degenerate, then either Eq.\",\n \"(REF ) or (REF ) implies that $\\\\operatorname{Im}J_2=0$ , and the corresponding neutral scalar mass-eigenstates will behave as states of definite CP in their interactions with gauge bosons and fermions.\",\n \"Nevertheless, if $\\\\operatorname{Im}J_1\\\\ne 0$ and/or $\\\\operatorname{Im}J_3\\\\ne 0$ (which would imply that $\\\\operatorname{Im}Z_7\\\\ne 0$ in the Higgs basis where $Z_5$ and $Z_6$ are simultaneously real), then CP-violating Higgs self-couplings must be present.\",\n \"Moreover, radiative corrections will generate a non-zero $\\\\operatorname{Im}J_2$ and yield neutral Higgs states of indefinite CP.\",\n \"That is, if CP-violation in the scalar sector is present, the tree-level relation $\\\\operatorname{Im}J_2=0$ can only be realized via an artificial fine-tuning of the parameters.\",\n \"Nevertheless, one can consider the implications of a tree-level mass degeneracy among the neutral Higgs scalars of the 2HDM.\",\n \"The above discussion illustrates the power of using scalar basis invariant conditions to analyze the CP properties of multi-Higgs models [45], [46], [51], [52], [53].\",\n \"In light of Eq.\",\n \"(REF ), if a tree-level mass degeneracy among the neutral Higgs scalars is present, then it is possible to rephase the Higgs basis field $H_2$ such that $Z_5$ and $Z_6$ are simultaneously real.\",\n \"Thus, in the analysis that follows, we shall analyze the most general 2HDM scalar potential assuming that $Z_5$ and $Z_6$ are real parameters.\",\n \"The squared-masses of the charged Higgs boson, $H^\\\\pm $ , and the CP-odd Higgs boson, $A$ are given by,In the notation employed in Eqs.\",\n \"(REF )–(), $h$ and $H$ [$A$ ] refer to the neutral scalars that behave as CP-even [odd] mass eigenstates in their tree-level interactions with the gauge bosons and fermions.\",\n \"Indeed, CP-violating interactions are present in Eqs.\",\n \"(REF ) and () if $\\\\operatorname{Im}Z_7\\\\ne 0$ .\",\n \"$m_{H^\\\\pm }^2=Y_2+\\\\tfrac{1}{2}Z_3 v^2\\\\,,\\\\qquad \\\\quad m_A^2=m_{H^\\\\pm }^2+\\\\tfrac{1}{2}(Z_4-Z_5)v^2\\\\,.$ The squared-masses of the CP-even Higgs bosons, $h$ and $H$ are the eigenvalues of the $2\\\\times 2$ matrix, $\\\\mathcal {M}_H^2=\\\\begin{pmatrix} Z_1 v^2 & \\\\quad Z_6 v^2 \\\\\\\\ Z_6 v^2 & \\\\quad m_A^2+Z_5 v^2\\\\end{pmatrix}.$ That is, $m^2_{H,h}=\\\\frac{1}{2}\\\\biggl \\\\lbrace m_A^2+(Z_1+Z_5)v^2\\\\pm \\\\sqrt{[m_A^2-(Z_1-Z_5)v^2]^2+4Z_6^2 v^4\\\\,}\\\\biggr \\\\rbrace \\\\,.$ Mass degenerate states arise if one of the following two quantities is zero, $Z_5(m_A^2-Z_1 v^2)+Z_6^2 v^2=0\\\\quad \\\\text{or}\\\\quad \\\\bigl [m_A^2-(Z_1-Z_5)v^2\\\\bigr ]^2+4Z_6^2 v^4=0\\\\,,$ where $m_A^2$ is given by Eq.\",\n \"(REF ).\",\n \"The case of $m_h=m_H$ arises when the second condition given in Eq.\",\n \"(REF ) is satisfied.\",\n \"It then follows that $m_A^2=(Z_1-Z_5)v^2$ and $Z_6=0$ , and the latter then yields the IDM mass spectrum.\",\n \"As in the case of the IDM, the mass degeneracy of $h$ and $H$ requires a fine tuning of the parameters shown in Eq.\",\n \"(REF ).\",\n \"In principle, it is possible that $Z_7\\\\ne 0$ , but in this case, $Z_6=0$ is not a natural condition since the $\\\\mathbb {Z}_2$ symmetry of the IDM is not present.\",\n \"Nevertheless, even if one accepts the two fine tuned conditions needed in this scenario, the arguments presented above Eq.\",\n \"(REF ) still apply.\",\n \"Namely, $Z_6=0$ corresponds to the exact alignment limit (at tree-level), in which case the tree-level interactions of the Higgs bosons and gauge bosons are still the same as those of the IDM [cf. Eqs.\",\n \"(REF )–(REF )], whereas the tree-level trilinear and quadrilinear Higgs self-interactions given in Eqs.\",\n \"(REF ) and () are modified by the addition of the following terms, $\\\\delta {L}_{3h}&=&-\\\\tfrac{1}{4} v \\\\bigl [Z_7(H+iA)+Z_7^*(H-iA)\\\\bigr ](HH+AA+2H^+ H^-)\\\\,, \\\\\\\\[5pt]\\\\delta {L}_{4h}&=&-\\\\tfrac{1}{4}\\\\bigl [Z_7(H+iA)+Z_7^*(H-iA)\\\\bigr ](HH+AA+2H^+ H^-)h\\\\,,$ after rephasing the Higgs basis field $H_2$ such that $Z_5$ is real.\",\n \"The cases $m_h=m_A$ or $m_H=m_A$ arise when the first condition given in Eq.\",\n \"(REF ) is satisfied.\",\n \"This condition also requires a fine-tuning of the parameters.\",\n \"Moreover, approximate Higgs alignment (as suggested by the LHC Higgs data) is not achieved unless $m_A^2\\\\gg Z_1 v^2$ or $|Z_6|\\\\ll 1$ .\",\n \"Nevertheless, the physical distinction of the mass degenerate states is due to the CP quantum numbers of the neutral scalar states (which are preserved in the tree-level Higgs interactions with gauge bosons and with fermions).\",\n \"One can therefore distinguish between the corresponding production mechanisms of the degenerate scalars that are mediated by gauge boson fusion or Drell-Yan production via $s$ -channel gauge boson exchange.\",\n \"Finally, we consider the triply mass-degenerate case of $m_h=m_H=m_A$ .\",\n \"In this case, both conditions given in Eq.\",\n \"(REF ) must be satisfied, which yields $Z_5=Z_6=0$ and $m_A^2=Z_1 v^2$ .\",\n \"This leaves $Z_7$ as the only potentially complex parameter of the scalar potential in the Higgs basis.\",\n \"Thus, one is free to rephase the Higgs basis field $H_2$ such that $Z_7$ is real, and we conclude that the Higgs scalar potential and vacuum must be CP-conserving.\",\n \"However, as long as $Z_7\\\\ne 0$ , the triply mass-degenerate case is unnatural, since the $\\\\mathbb {Z}_2$ symmetry of the IDM is not present.\"\n ],\n [\n \"Natural 2HDM mass degeneracies: a symmetry based approach\",\n \"In sections REF and REF , we derived the conditions that yield mass degeneracies among the neutral scalars of the 2HDM by brute force.\",\n \"Namely, we obtained explicit expressions for the neutral scalar masses and then derived the corresponding relations among Higgs basis parameters for which mass degeneracies were present.\",\n \"We then checked whether any of these relations were a consequence of a symmetry, and if yes we concluded that the corresponding mass degeneracy was natural.\",\n \"In this section, we will obtain the same result by considering all possible symmetries of the 2HDM scalar potential.\",\n \"Since the complete list of such symmetries is known [54], [55], [56], [57], [58], [59], we can be sure that our catalog of natural mass degeneracies of the 2HDM is complete.\",\n \"We shall make use of the classification of symmetries presented in Ref.\",\n \"[56], which identifies three possible Higgs family symmetries, $\\\\mathbb {Z}_2$ , U(1) and SO(3), and three classes of generalized CP-symmetries, denoted by GCP1, GCP2 and GCP3, respectively, as summarized in Table .In Ref.\",\n \"[56] the three classes of generalized CP transformations are denoted by CP1, CP2 and CP3 respectively.\",\n \"This nomenclature for the generalized CP-symmetries is awkward, in light of the notation that will be employed in section .\",\n \"To avoid confusion, we have appended the letter G (for “general”) in denoting the three classes of generalized CP transformations of the 2HDM.\",\n \"In the GCP transformation laws of Table , we have introduced the conjugation symbol ${}^{\\\\scriptscriptstyle {\\\\bigstar }}$ , which when applied to an SU(2) multiplet of scalar fields is defined by $\\\\Phi ^{\\\\scriptscriptstyle {\\\\bigstar }}\\\\equiv \\\\bigl [\\\\Phi ^\\\\dagger \\\\bigr ]^{\\\\scriptstyle T}$$,where the dagger refers both to hermitian conjugation of the quantum field operator when acting on the Hilbert space, and to complex conjugate transpose when acting on an SU(2) multiplet of fields.\\\\vspace{-0.72229pt}\\\\begin{table}[hb!\",\n \"]\\\\begin{tabular}{|cccc|}\\\\hline \\\\rule {0pt}{}symmetry & \\\\hspace{36.135pt} transformation law & \\\\hspace{50.58878pt} \\\\phantom{transformation law} & \\\\\\\\[2pt]\\\\hline \\\\rule {0pt}{}\\\\mathbb {Z}_2 & \\\\Phi _1\\\\rightarrow \\\\Phi _1 & \\\\Phi _2\\\\rightarrow -\\\\Phi _2 &\\\\\\\\U(1) & \\\\Phi _1\\\\rightarrow \\\\Phi _1 & \\\\Phi _2\\\\rightarrow e^{2i\\\\theta }\\\\Phi _2 &\\\\\\\\SO(3) & \\\\Phi _a\\\\rightarrow U_{ab}\\\\Phi _b & U\\\\in {\\\\rm U}(2)/{\\\\rm U}(1)_{\\\\rm Y}& (\\\\text{for $a$, $b=1,2$} )\\\\\\\\GCP1 & \\\\Phi _1\\\\rightarrow \\\\Phi _1^{\\\\scriptscriptstyle {\\\\bigstar }} & \\\\Phi _2\\\\rightarrow \\\\Phi _2^{\\\\scriptscriptstyle {\\\\bigstar }} & \\\\\\\\GCP2 & \\\\Phi _1\\\\rightarrow \\\\Phi _2^{\\\\scriptscriptstyle {\\\\bigstar }} & \\\\Phi _2\\\\rightarrow -\\\\Phi _1^{\\\\scriptscriptstyle {\\\\bigstar }} & \\\\\\\\GCP3 & \\\\Phi _1 \\\\rightarrow \\\\Phi _1^{\\\\scriptscriptstyle {\\\\bigstar }} \\\\cos \\\\theta +\\\\Phi _2^{\\\\scriptscriptstyle {\\\\bigstar }} \\\\sin \\\\theta &\\\\Phi _2 \\\\rightarrow -\\\\Phi _1^{\\\\scriptscriptstyle {\\\\bigstar }}\\\\sin \\\\theta +\\\\Phi _2^{\\\\scriptscriptstyle {\\\\bigstar }}\\\\cos \\\\theta & \\\\text{(for $0<\\\\theta <\\\\tfrac{1}{2}\\\\pi $)} \\\\\\\\[2pt]\\\\hline \\\\hline \\\\rule {0pt}{}\\\\Pi _2& \\\\Phi _1\\\\rightarrow \\\\Phi _2 & \\\\Phi _2\\\\rightarrow \\\\Phi _1 & \\\\\\\\[2pt]\\\\hline \\\\end{tabular}\\\\caption {\\\\small Possible symmetries of the 2HDM scalar potential that are respected by the SU(2)\\\\times U(1)_{\\\\rm Y} gauge kinetic terms of the scalar fields.\",\n \"The corresponding symmetry transformation laws are given in a basis where the symmetry is manifest.\",\n \"Note that a scalar potential that is invariant under the mirror discretesymmetry, \\\\Pi _2, is also invariant under the \\\\mathbb {Z}_2 in another scalar field basis~\\\\cite {Davidson:2005cw}.\",\n \"}\\\\end{table}$ Table: Impact of the symmetries defined in Table on the coefficients of the 2HDM scalar potential [cf. Eq.\",\n \"()] in abasis where the symmetry is manifest.\",\n \"A short dash indicates the absence of a constraint.\",\n \"A scalar potential that is simultaneously invariant under ℤ 2 \\\\mathbb {Z}_2 and Π 2 \\\\Pi _2 isalso invariant under GCP2 in another scalar field basis , .\",\n \"Likewise, a scalar potential that is simultaneously invariant under U(1) and Π 2 \\\\Pi _2 is also invariant under GCP3 in another scalar field basis .\",\n \"The symbol ⊕\\\\oplus is being used above to indicate that two symmetries are enforced simultaneously within the same scalar field basis.We shall not consider the seven additional accidental symmetries of the 2HDM scalar potential identified in Refs.\",\n \"[58], [59], that utilized mixed Higgs family and generalized CP transformations that leave the SU(2) gauge kinetic terms of the scalar fields invariant.\",\n \"An example of such a symmetry is the well known custodial symmetry that is respected by the 2HDM scalar potential when $m_{H^\\\\pm }=m_A$ [24], [25], [26], [27].\",\n \"However, this class of symmetries is violated by the U(1)$_{\\\\rm Y}$ gauge kinetic term of the scalar potential (as well as by the Yukawa couplings that are responsible for mass differences between up and down-type fermions).\",\n \"Hence, any exact mass degeneracies arising from these seven accidental symmetries will be spoiled, in the absence of an artificial fine tuning of the Higgs scalar potential parameters.In cases of accidental symmetries, i.e.\",\n \"symmetries of the scalar potential that are not respected by the full theory, the would-be mass degeneracies are only approximate, with calculable mass splittings.\",\n \"The possibility of such approximate mass degeneracies, although technically natural, is not the subject of this paper.\",\n \"Possible natural mass degeneracy of the 2HDM must be the consequence of one of the symmetries listed in Table .\",\n \"Starting from a generic scalar potential given by Eq.\",\n \"(REF ), if the scalar potential respects one of the symmetries listed in Table , then a scalar basis is picked out in which the symmetry is manifest.\",\n \"In this basis, the coefficients of the scalar potential are constrained according to Table REF .It can be shown that for each of the symmetries listed in Table REF , a scalar field basis exists in which all scalar potential parameters and the neutral scalar field vacuum expectation values are simultaneously real, in which case CP (as defined by GCP1 in Table ) is conserved by the scalar sector Lagrangian and vacuum.\",\n \"It is straightforward to check that the possible discrete symmetries of the 2HDM, namely $\\\\mathbb {Z}_2$ , GCP1, GCP2 (or equivalently, $\\\\mathbb {Z}_2\\\\oplus \\\\Pi _2$ ), do not yield scalar potentials that lead to scalar mass degeneracies.\",\n \"Thus, we henceforth focus on U(1), SO(3) and GCP3 (and the related U(1)$\\\\oplus \\\\Pi _2$ symmetry).\",\n \"Given a 2HDM scalar potential with a Peccei-Quinn [U(1)$_{\\\\rm PQ}$ ] symmetry [60] (or equivalently the U(1) transformation specified in Table In Ref.\",\n \"[60], a U(1)$_{\\\\rm PQ}$ transformation of the 2HDM scalar fields is given by $\\\\Phi _1\\\\rightarrow e^{-i\\\\theta }\\\\Phi _1$ and $\\\\Phi _2\\\\rightarrow e^{i\\\\theta }\\\\Phi _2$ .\",\n \"The U(1) transformation specified in Table corresponds to combining the U(1)$_{\\\\rm PQ}$ transformation with a hypercharge U(1)$_{\\\\rm Y}$ transformation, $\\\\Phi _i\\\\rightarrow e^{i\\\\theta }\\\\Phi _i$ (for $i=1,2$ ).)\",\n \"that is spontaneously broken by the vacuum, the scalar sector will contain a massless CP-odd (Goldstone) scalar [61], [62].\",\n \"In such cases, no mass degeneracy is present (without further constraints on the scalar potential parameters).\",\n \"However, if the U(1) symmetry is manifestly realized in the Higgs basis, then the U(1) symmetry is unbroken by the vacuum, resulting in a mass degeneracy between the two neutral scalars residing in the Higgs basis field $H_2$ .\",\n \"Indeed, this has already been shown in Section REF [see text above Eq.\",\n \"(REF )], in the case of the mass degeneracy, $m_H=m_A$ , of the IDM with $Z_5=0$ .\",\n \"In the case of a 2HDM scalar potential with an SO(3) symmetry, the form of the scalar potential is invariant with respect to all possible changes of the scalar basis.\",\n \"Hence, it follows that the scalar potential parameters in the Higgs basis satisfy $Y_1=Y_2$ , $Z_1=Z_2=Z_3+Z_4$ and $Y_3=Z_5=Z_6=Z_7=0$ .\",\n \"Using Eqs.\",\n \"(REF ), (REF ) and (), it follows that $m_H=m_A=0$ .\",\n \"The presence of two massless (Goldstone) scalars is a consequence of the spontaneous breaking of the SO(3) global symmetry by the vacuum.\",\n \"This scalar mass degeneracy is a special case of the mass degeneracy in the case of the IDM with $Z_5=0$ , in which additional constraints among the Higgs basis parameters result in the pair of massless scalar states.\",\n \"Finally, let us consider the case of a 2HDM scalar potential with a GCP3 symmetry.\",\n \"Suppose that the GCP3 symmetry is manifestly realized in a basis where $\\\\langle \\\\Phi _1^0 \\\\rangle =\\\\frac{v_1}{\\\\sqrt{2}}\\\\,,\\\\qquad \\\\quad \\\\langle \\\\Phi _2^0 \\\\rangle =\\\\frac{v_2}{\\\\sqrt{2}}e^{i\\\\xi }\\\\,,$ where $v_1$ and $v_2$ are positive.\",\n \"We define $\\\\tan \\\\beta \\\\equiv v_2/v_1$ (so that $0<\\\\beta <\\\\tfrac{1}{2}\\\\pi $ ).\",\n \"Then, in light of the constraints on the GCP3 scalar potential parameters given in Table REF , the scalar potential minimum conditions yield (e.g., see eqs.\",\n \"(3)–(5) of Ref.\",\n \"[63]), $m_{11}^2 & = & -\\\\tfrac{1}{2}v^2\\\\bigl [\\\\lambda _1-2\\\\lambda _5\\\\sin ^2\\\\beta \\\\sin ^2\\\\xi \\\\bigr ]\\\\,, \\\\\\\\m_{22}^2 & = & -\\\\tfrac{1}{2}v^2\\\\bigl [\\\\lambda _1-2\\\\lambda _5\\\\cos ^2\\\\beta \\\\sin ^2\\\\xi \\\\bigr ]\\\\,, \\\\\\\\m_{12}^2\\\\sin \\\\xi &=& v^2\\\\lambda _5 \\\\sin \\\\beta \\\\cos \\\\beta \\\\sin \\\\xi \\\\cos \\\\xi \\\\,.$ We can assume that $\\\\lambda _5\\\\ne 0$ , since otherwise we would be dealing with an SO(3)-symmetric scalar potential.\",\n \"Setting $m_{11}^2=m_{22}^2$ and $m_{12}^2=0$ then yields two conditions, $\\\\sin ^2 \\\\xi \\\\cos 2\\\\beta =0\\\\,,\\\\qquad \\\\quad \\\\sin \\\\xi \\\\cos \\\\xi \\\\sin 2\\\\beta =0\\\\,.$ Hence, there are two classes of vacua, $\\\\sin \\\\xi =0$ and $\\\\beta $ arbitrary ($0<\\\\beta <\\\\tfrac{1}{2}\\\\pi $ ) , $\\\\cos \\\\xi =0$ and $\\\\cos 2\\\\beta =0$ .\",\n \"We now can calculate the parameters of the GCP3 scalar potential in the Higgs basis (e.g., see eqs.\",\n \"(11)–(20) of Ref.\",\n \"[63]) in the two Cases A and B defined above, $Y_1=Y_2, \\\\quad Y_3=Z_6=Z_7=0, \\\\quad Z_1=Z_2=\\\\lambda _1, \\\\qquad \\\\quad Z_3+Z_4=\\\\lambda _1-\\\\lambda _5,\\\\quad Z_5=\\\\lambda _5$ , $Y_1=Y_2, \\\\quad Y_3=Z_6=Z_7=0, \\\\quad Z_1=Z_2=\\\\lambda _1-\\\\lambda _5, \\\\quad Z_3+Z_4=\\\\lambda _1+\\\\lambda _5,\\\\quad Z_5=0$ .\",\n \"In particular, $Z_1=Z_2=Z_3+Z_4+Z_5$ in Case A, and $Z_1=Z_2\\\\ne Z_3+Z_4$ (and $Z_5=0$ ) in Case B.\",\n \"We can now make use of Eqs.\",\n \"(REF ) and () [along with Eq.\",\n \"(REF ) to eliminate $Y_2$ by virtue of $Y_1=Y_2$ ] to compute the neutral scalar mass spectrum in the two cases,Note that the positivity of the squared masses are consistent with the conditions on the 2HDM scalar potential parameters first obtained in Ref. [54].\",\n \"$m_h^2=Z_1 v^2,\\\\quad m_H^2=0,\\\\quad m_A^2=(Z_3+Z_4-Z_1)v^2,\\\\quad m_{H^\\\\pm }^2=\\\\tfrac{1}{2}(Z_3-Z_1)v^2$ , $m_h^2=Z_1 v^2,\\\\quad m^2_H=m^2_A=\\\\tfrac{1}{2}(Z_3+Z_4-Z_1) v^2,\\\\qquad \\\\,\\\\,\\\\, m_{H^\\\\pm }^2=\\\\tfrac{1}{2}(Z_3-Z_1)v^2$ .\",\n \"Note that Cases A and B correspond to degenerate vacua, since in both cases the value of the scalar potential (in the Higgs basis) at its minimum is $V_{\\\\rm min}=-\\\\tfrac{1}{8} Z_1 v^4=-\\\\tfrac{1}{8} v^2m_h^2$ .\",\n \"In light of Table REF , we can identify case A as corresponding to realizing a GCP3 symmetry in the Higgs basis,This means that we can relax the restrictions of $\\\\beta \\\\ne 0, \\\\tfrac{1}{2}\\\\pi $ in defining Case A.\",\n \"Note that for $\\\\beta =\\\\tfrac{1}{2}\\\\pi $ one can simply interchange the definitions of $\\\\Phi _1$ and $\\\\Phi _2$ to recover the Higgs basis result (corresponding to $\\\\beta =0$ ).\",\n \"and case B corresponding to realizing a U(1)$\\\\oplus \\\\Pi _2$ symmetry in the Higgs basis.This result is not surprising given that U(1)$\\\\oplus \\\\Pi _2$ is equivalent to GCP3 in another scalar field basis.\",\n \"In particular, case A exhibits a massless Goldstone boson corresponding to the spontaneous breaking of GCP3 by the vacuum.\",\n \"In contrast, in case B, the GCP3 symmetry possesses a continuous U(1) subgroup, which is unbroken by the vacuum, that protects the mass degeneracy, $m_H=m_A$ .\",\n \"Indeed, this case is again a special case of the IDM with $Z_5=0$ , where the additional constraints, $Y_1=Y_2$ and $Z_1=Z_2$ are imposed.\",\n \"As a check, it is instructive to evaluate the consequences of a 2HDM scalar potential with a U(1)$\\\\oplus \\\\Pi _2$ symmetry that is manifestly realized in the $\\\\Phi _1$ –$\\\\Phi _2$ basis.\",\n \"In the following, we employ primed coefficients, $\\\\lambda _i^\\\\prime $ to distinguish this case from the one above where the GCP3 symmetry is manifestly realized in the $\\\\Phi _1$ –$\\\\Phi _2$ basis.\",\n \"Using the results of Table REF , the scalar potential minimum conditions yield, $m_{11}^2&=& -\\\\tfrac{1}{2}v^2\\\\bigl [\\\\lambda ^\\\\prime _1\\\\cos ^2\\\\beta +(\\\\lambda ^\\\\prime _3+\\\\lambda ^\\\\prime _4)\\\\sin ^2\\\\beta \\\\bigr ]\\\\,,\\\\\\\\m_{22}^2&=& -\\\\tfrac{1}{2}v^2\\\\bigl [\\\\lambda ^\\\\prime _1\\\\sin ^2\\\\beta +(\\\\lambda ^\\\\prime _3+\\\\lambda ^\\\\prime _4)\\\\cos ^2\\\\beta \\\\bigr ]\\\\,,$ under the assumption that $0<\\\\beta <\\\\tfrac{1}{2}\\\\pi $ .\",\n \"We can assume that $\\\\lambda ^\\\\prime _1\\\\ne \\\\lambda ^\\\\prime _3+\\\\lambda ^\\\\prime _4$ , since otherwise we would be dealing with an SO(3)-symmetric scalar potential.\",\n \"Setting $m_{11}^2=m_{22}^2$ then yields $\\\\cos 2\\\\beta =0$ , with $\\\\xi $ arbitrary.\",\n \"Without loss of generality, one can set $\\\\xi =0$ , since the scalar potential is unchanged under a rephasing of $\\\\Phi _2$ .\",\n \"As before, we can now compute the scalar potential parameters in the Higgs basis, $&&Y_1=Y_2,\\\\qquad Y_3=Z_6=Z_7=0,\\\\qquad Z_1=Z_2=\\\\tfrac{1}{2}(\\\\lambda ^\\\\prime _1+\\\\lambda ^\\\\prime _3+\\\\lambda ^\\\\prime _4),\\\\nonumber \\\\\\\\&& Z_3+Z_4=\\\\lambda ^\\\\prime _1,\\\\qquad Z_5=\\\\tfrac{1}{2}(\\\\lambda ^\\\\prime _1-\\\\lambda ^\\\\prime _3-\\\\lambda ^\\\\prime _4)\\\\,.\",\n \"$ In particular, note that $Z_1=Z_2=Z_3+Z_4-Z_5$ .\",\n \"Using Eqs.\",\n \"(REF ), (REF ) and (), we obtain $m_h^2=Z_1 v^2\\\\,,\\\\quad m_H^2=(Z_3+Z_4-Z_1)v^2\\\\,,\\\\quad m_A^2=0\\\\,,\\\\quad m_{H^\\\\pm }^2=\\\\tfrac{1}{2}(Z_3-Z_1)v^2\\\\,,$ which is the same mass spectrum as Case A of the GCP3-symmetric scalar potential with $H$ and $A$ interchanged.\",\n \"This result can be understood by noting that Eq.\",\n \"(REF ) takes the standard form of the GCP3-symmetric scalar potential in the Higgs basis after rephasing the Higgs basis field $H_2\\\\rightarrow iH_2$ , which interchanges $H$ and $A$ and transforms $Z_5\\\\rightarrow -Z_5$ .\",\n \"Finally, the case of $\\\\beta =0$ or $\\\\beta =\\\\tfrac{1}{2}\\\\pi $ must be treated separately and corresponds to a manifest realization of the U(1)$\\\\oplus \\\\Pi _2$ symmetry in the Higgs basis.\",\n \"This vacuum is degenerate with the one considered above, since in both cases, $V_{\\\\rm min}=-\\\\tfrac{1}{8} Z_1 v^4=-\\\\tfrac{1}{8} v^2m_h^2$ .\",\n \"Indeed, this latter case corresponds to Case B of the GCP3-symmetric scalar potential treated above, where the neutral scalar mass spectrum exhibits a mass degeneracy, $m_H=m_A$ .\",\n \"In summary, massless scalar (Goldstone boson) states $A$ , $H$ or ($A$ , $H$ ) exist in the 2HDM with a scalar potential that exhibits, respectively, a U(1), GCP3 or SO(3) symmetry manifestly realized in a generic $\\\\Phi _1$ –$\\\\Phi _2$ basis, which agrees with the results of Table 2 of Ref. [59].\",\n \"Nevertheless, in the special cases where U(1) or U(1)$\\\\oplus \\\\Pi _2$ are manifestly realized in the Higgs basis (the latter corresponding to the Case B solution of the GCP3-symmetric scalar potential), the corresponding U(1) subgroups of theses symmetries are not spontaneously broken by the vacuum, and the neutral scalar mass spectrum exhibits a mass degeneracy, $m_H=m_A$ .\",\n \"In the case of the SO(3)-symmetric scalar potential, this mass degeneracy is realized by a pair of massless Goldstone boson states.\",\n \"Thus, we conclude that mass-degenerate neutral scalars can arise naturally in the 2HDM only in the case of the IDM with $Z_5=0$ .\",\n \"All other cases of mass-degenerate scalars require an artificial fine-tuning of the scalar potential parameters, in agreement with the analysis of section REF .\",\n \"Furthermore, this conclusion is unaffected by the interactions of the scalars with the vector bosons.\",\n \"Indeed, the Higgs boson–gauge boson interactions, ${L}_{\\\\rm int}$ , given by Eq.\",\n \"(REF ) show that the global U(1) symmetry responsible for the mass degeneracy of $H$ and $A$ is an exact symmetry of ${L}_{\\\\rm int}$ .\",\n \"Finally, as previously noted, the Higgs basis field $H_2$ of the IDM is odd whereas all other scalar, fermion and vector fields are even under the discrete $\\\\mathbb {Z}_2$ symmetry.\",\n \"This can be achieved by employing Type-I Yukawa couplings [64] where fermions couple only to the Higgs basis field $H_1$ .\",\n \"In this case, the global U(1) symmetry of the IDM scalar potential with $Z_5=0$ will also be respected by the Yukawa interactions.\",\n \"However, a GCP3 [or equivalently U(1)$\\\\oplus \\\\Pi _2$ ] or SO(3) symmetry of the IDM scalar potential will be explicitly broken by the Yukawa interactions.\",\n \"Hence, the U(1)-symmetric IDM is the only 2HDM for which an exact mass degeneracy of $H$ and $A$ can be preserved.\"\n ],\n [\n \"3HDM mass degeneracies and the Ivanov Silva model\",\n \"In extended Higgs sectors with more than two scalar doublets, it is now possible to have mass-degenerate charged Higgs pairs as well as mass-degenerate neutral scalars [65].\",\n \"In this section, we explore new phenomena associated with mass degenerate scalars that arises for the first time in the three-Higgs doublet model (3HDM).\",\n \"As a warmup exercise, we return to the IDM and add a second inert doublet and consider possible mass degeneracies among the scalar fields of the two inert doublets.\",\n \"We then perturb the resulting model to obtain a version of the 3HDM that is equivalent to a model first introduced by Ivanov and Silva[30].\"\n ],\n [\n \"The replicated inert doublet model (RIDM)\",\n \"The IDM introduced in section REF can be generalized by introducing additional inert scalar doublets.\",\n \"In this section, we consider a 3HDM that consists of two inert hypercharge-one electroweak doublets, in which the inert doublets contain mass-degenerate scalar states.\",\n \"The resulting models shall be called the replicated inert doublet model (RIDM).\",\n \"As in the case of the IDM, we work in the Higgs basis in which the first Higgs doublet field $H_1$ contains the SM Higgs boson.\",\n \"The RIDM consists of $H_1$ , with $\\\\langle H_1^0 \\\\rangle =v/\\\\sqrt{2},$ and two inert doublet fields $H_2$ and $H_3$ , with $\\\\langle H_2 \\\\rangle =\\\\langle H_3 \\\\rangle =0$ , and a scalar potential given by, $\\\\mathcal {V}_{\\\\rm RIDM}&=& Y_1 H_1^\\\\dagger H_1+ Y_2 \\\\left(H_2^\\\\dagger H_2 +H_3^\\\\dagger H_3\\\\right)+\\\\tfrac{1}{2}Z_1(H_1^\\\\dagger H_1)^2+\\\\tfrac{1}{2}Z_2(H_2^\\\\dagger H_2+H_3^\\\\dagger H_3)^2\\\\nonumber \\\\\\\\&&+Z_3(H_1^\\\\dagger H_1)\\\\left(H_2^\\\\dagger H_2+H_3^\\\\dagger H_3\\\\right)+Z_4\\\\left[( H_1^\\\\dagger H_2)(H_2^\\\\dagger H_1)+( H_1^\\\\dagger H_3)(H_3^\\\\dagger H_1)\\\\right]\\\\nonumber \\\\\\\\&&+\\\\tfrac{1}{2}Z_5\\\\left\\\\lbrace (H_1^\\\\dagger H_2)^2 +(H_2^\\\\dagger H_1)^2+(H_1^\\\\dagger H_3)^2 +(H_3^\\\\dagger H_1)^2\\\\right\\\\rbrace \\\\,.$ Without loss of generality, we have chosen $Z_5$ real and non-negative in Eq.\",\n \"(REF ), which is always possible by an appropriate rephasing of the scalar fields $H_2$ and $H_3$ .\",\n \"Hence, if follows that the bosonic sector of the RIDM is CP-conserving.\",\n \"The charged and neutral components of the Higgs basis doublet fields of the RIDM are also mass eigenstate fields, $H_1=\\\\begin{pmatrix} G^+ \\\\\\\\ \\\\frac{1}{\\\\sqrt{2}}\\\\bigl [v+h_{\\\\rm SM}+iG^0\\\\bigr ] \\\\end{pmatrix},\\\\quad H_2=\\\\begin{pmatrix} H^+ \\\\\\\\ \\\\frac{1}{\\\\sqrt{2}}\\\\bigl [H+iA\\\\bigr ]\\\\end{pmatrix},\\\\quad H_3=\\\\begin{pmatrix} h^+ \\\\\\\\ \\\\frac{1}{\\\\sqrt{2}}\\\\bigl [h+ia\\\\bigr ]\\\\end{pmatrix},$ with a minor change of notation from the IDM.\",\n \"The corresponding squared masses of the neutral and charged scalars are given by, $m^2_{H^\\\\pm }&=&m^2_{h^\\\\pm }=Y_2+\\\\tfrac{1}{2}Z_3 v^2\\\\,,\\\\qquad m^2_H=m^2_h=Y_2+\\\\tfrac{1}{2}(Z_3+Z_4+Z_5)v^2\\\\,,\\\\nonumber \\\\\\\\m^2_A&=&m^2_a=Y_2+\\\\tfrac{1}{2}(Z_3+Z_4-Z_5)v^2\\\\,.$ By assumption, $Z_5\\\\ge 0$ , in which case $m_H=m_h\\\\ge m_A=m_a$ .In particular, note that if $Z_5=0$ then there is an enhanced mass degeneracy in which $m_H=m_h=m_A=m_a$ .\",\n \"Thus, the RIDM possesses four mass-degenerate scalar pairs: $(H^\\\\pm , h^\\\\pm )$ , $(H,h)$ and $(A,a)$ .\",\n \"These mass degeneracies can be understood as a consequence of a continuous global Higgs flavor symmetry (where Higgs flavor corresponds to the multiplicity of Higgs doublets).\",\n \"In order to explicitly exhibit the relevant symmetries, it is convenient to focus on the neutral scalar states of the doublet fields $H_2$ and $H_3$ , denoted henceforth by the complex fields, $H^0\\\\equiv \\\\frac{H+iA}{\\\\sqrt{2}}\\\\,,\\\\qquad \\\\quad h^0\\\\equiv \\\\frac{h+ia}{\\\\sqrt{2}}\\\\,,$ respectively.\",\n \"Let us first focus on the kinetic energy terms and the terms in Eq.\",\n \"(REF ) in the absence of the term proportional to $Z_5$ .\",\n \"Then, one can check that the neutral complex scalar fields $H^0$ and $h^0$ appear only in the combination $H^{0\\\\,\\\\dagger }H^0+h^{0\\\\,\\\\dagger }h^0=\\\\tfrac{1}{2}(H^2+h^2+A^2+a^2)$ .\",\n \"Thus, excluding $Z_5$ , the scalar Lagrangian possesses an O(4) global symmetry, that is responsible for four mass-degenerate neutral scalar states.\",\n \"It is instructive to see how this symmetry arises when employing the complex basis $\\\\varphi _i=\\\\lbrace H^0,h^0\\\\rbrace $ (for $i=1,2$ ).\",\n \"Noting that $\\\\varphi ^{\\\\dagger \\\\,i}\\\\varphi _i=H^{0\\\\,\\\\dagger }H^0+h^{0\\\\,\\\\dagger }h^0$ (the sum over the repeated index $i$ is implicit), it is clear that the scalar Lagrangian (in the absence of $Z_5$ ) is invariant under a U(2) global symmetry, $\\\\varphi _i\\\\rightarrow U_i{}^j\\\\varphi _j$ , with $U\\\\in {\\\\rm U(2)}$ .\",\n \"However, the corresponding symmetry group is in fact larger than U(2).\",\n \"Working in the complex basis, it is straightforward to verify that the quantity $\\\\varphi ^{\\\\dagger \\\\,i}\\\\varphi _i$ is invariant with respect to $\\\\varphi _i\\\\rightarrow U_i{}^j\\\\varphi _j+(V^*)^i{}_j\\\\varphi ^{\\\\dagger \\\\,j}\\\\,,$ where $U$ and $V$ are complex $2\\\\times 2$ matrices (which are not in general unitary), provided that the following two conditions are satisfied: $&&(i)~~~~(U^\\\\dagger U+V^\\\\dagger V)_i{}^j=\\\\delta _i{}^j\\\\,,\\\\\\\\&&(ii)~~~V^T U~\\\\hbox{\\\\rm is an antisymmetric matrix}\\\\,.$ One can now check that Eq.\",\n \"(REF ) corresponds to an O(4) symmetry transformation.\",\n \"More explicitly, the $4\\\\times 4$ matrix, $\\\\mathcal {Q}=\\\\left(\\\\begin{array}{cc}\\\\operatorname{Re}(U+V) &\\\\,\\\\, -\\\\operatorname{Im}(U+V) \\\\\\\\\\\\operatorname{Im}(U-V) & \\\\phantom{-}\\\\,\\\\, \\\\operatorname{Re}(U-V) \\\\end{array}\\\\right)$ is an orthogonal matrix if and only if $U$ and $V$ satisfy Eqs.\",\n \"(REF ) and ().\",\n \"Indeed, one can check that in light of Eqs.\",\n \"(REF ) and (), the global symmetry specified by Eq.\",\n \"(REF ) is governed by 6 continuous parameters as expected for an O(4) transformation.\",\n \"Two special cases of Eq.\",\n \"(REF ) are noteworthy.\",\n \"First, if $V=0$ , then $U$ is unitary and we recover the U(2) global symmetry mentioned previously.\",\n \"Second, if $U=0$ then Eq.\",\n \"(REF ) corresponds to a generalized CP transformation [cf. Eq.\",\n \"(REF )].The unified treatment of Higgs family transformations and generalized CP transformations has been advocated previously in Ref. [58].\",\n \"A related discussion emphasizing the promotion of the U(2) basis transformation to an enlarged group of O(4) transformations appears in Ref. [66].\",\n \"Both symmetries are present in the scalar Lagrangian if the $Z_5$ coupling is neglected, and either one would be sufficient to guarantee the mass degeneracy of $H$ , $h$ , $A$ and $a$ .\",\n \"In the absence of the $Z_5$ coupling, the full O(4) global symmetry is respected by the pure scalar Lagrangian.\",\n \"However, when we include the coupling of the scalar doublets to the gauge bosons, one must replace the ordinary derivative, $\\\\partial _\\\\mu $ , with the SU(2)$\\\\times $ U(1) gauge covariant derivative, $D_\\\\mu $ , in the scalar kinetic energy term.\",\n \"The resulting coupling of the scalars to the vector bosons partially breaks the O(4) symmetry.\",\n \"Employing the complex basis, it is easy to check that the symmetry transformation specified by Eq.\",\n \"(REF ) is unbroken if and only if either $U=0$ or $V=0$ , namely the two special cases just highlighted above.This result is not surprising given that Eq.\",\n \"(REF ) transforms the scalar field into a linear combination of two fields of opposite hypercharge unless either $U=0$ or $V=0$ .\",\n \"That is, the kinetic energy term $(D^\\\\mu \\\\varphi )^{i\\\\,\\\\dagger } (D_\\\\mu \\\\varphi )_i$ is invariant under a U(2) symmetry (corresponding to $V=0$ ) and under the generalized CP symmetry (corresponding to $U=0$ ).\",\n \"Mathematically, the unbroken global symmetry that remains is the semi-direct product U(2)$\\\\rtimes \\\\mathbb {Z}_2$ .This symmetry is a generalization of the U(1) symmetry (and the associated CP symmetry) of the IDM with $Z_5=0$ treated in section REF , and provides the motivation for our choice of the RIDM scalar potential given in Eq.\",\n \"(REF ).\",\n \"We now examine the consequence of including the term of Eq.\",\n \"(REF ) proportional to $Z_5$ .\",\n \"Focusing again on the neutral complex scalar fields $H^0$ and $h^0$ [cf. Eq.\",\n \"(REF )], we see that a new combination of fields arises, $\\\\varphi _i\\\\varphi _i+{\\\\rm h.c.}=(H^0)^2+(H^{0\\\\,\\\\dagger })^2+(h^0)^2+(h^{0\\\\,\\\\dagger })^2$ .\",\n \"This term is invariant with respect to Eq.\",\n \"(REF ) provided that the conditions specified in Eqs.\",\n \"(REF ) and () are replaced by 1.25 $&&(i^\\\\prime )~~~~(U^T U+V^T V)_i{}^j=\\\\delta _i{}^j\\\\,,\\\\\\\\&&(ii^\\\\prime )~~~V^\\\\dagger U~\\\\hbox{\\\\rm is an antihermitian matrix}\\\\,.$ The conditions specified by Eqs.\",\n \"(REF ) and () are compatible with those of Eqs.\",\n \"(REF ) and () if $U$ and $V$ are real matrices.\",\n \"Consequently, $\\\\mathcal {Q}$ specified in Eq.\",\n \"(REF ) is now a block diagonal orthogonal $4\\\\times 4 $ matrix, $\\\\mathcal {Q}=\\\\left(\\\\begin{array}{cc}U+V &\\\\,\\\\, 0 \\\\\\\\0 & \\\\phantom{-}\\\\,\\\\, U-V \\\\end{array}\\\\right)\\\\,,$ where $(U\\\\pm V)^T(U\\\\pm V)={1}_{2\\\\times 2}$ as a consequence of Eqs.\",\n \"() and (REF ).\",\n \"That is, the scalar Lagrangian is invariant under a global O(2)$\\\\times $ O(2) symmetry, which explains the presence of the mass-degenerate scalars $(H,h)$ and $(A,a)$ , respectively.\",\n \"The breaking of the four-fold mass degeneracy to the two mass-degenerate pairs is due to the scalar potential term proportional to $Z_5$ , as is evident from Eq.\",\n \"(REF ).\",\n \"Finally, after promoting the derivative to the gauge covariant derivative in the scalar kinetic energy term, the remaining symmetry is O(2)$\\\\rtimes \\\\mathbb {Z}_2$ .\",\n \"For completeness, we note that the degeneracy of the charged Higgs scalars $(H^\\\\pm ,h^\\\\pm )$ is governed by the full O(4) symmetry, which is broken down to U(2)$\\\\rtimes \\\\mathbb {Z}_2$ after promoting the derivatives of the scalar kinetic energy term to gauge covariant derivatives.\",\n \"This is easily seen by noting that in the unitary gauge (in which the Goldstone fields do not explicitly appear), the physical charged scalar fields do not appear in the scalar potential term proportional to $Z_5$ .\",\n \"Finally, if $Z_4=Z_5=0$ , we can make use of the vertical SU(2) global symmetry (which when gauged corresponds to the SU(2) electroweak gauge group) to conclude that all eight charged and neutral inert scalars are mass-degenerate.\",\n \"Next, we examine all the bosonic couplings of the RIDM in the unitary gauge (where the Goldstone fields are set to zero).\",\n \"The Higgs boson interactions with the gauge bosons and the Higgs boson self couplings of the RIDM are listed below.\",\n \"${L}_{VVH}&=&\\\\left(gm_W W_\\\\mu ^+W^{\\\\mu \\\\,-}+\\\\frac{g}{2c_W} m_ZZ_\\\\mu Z^\\\\mu \\\\right)h_{\\\\rm SM}\\\\,,\\\\\\\\[8pt]{L}_{VVHH}&=&\\\\left[\\\\tfrac{1}{4}g^2 W_\\\\mu ^+W^{\\\\mu \\\\,-}+\\\\frac{g^2}{8c_W^2}Z_\\\\mu Z^\\\\mu \\\\right](h_{\\\\rm SM}^2+H^2+h^2+A^2+a^2) \\\\nonumber \\\\\\\\&& +\\\\left[\\\\tfrac{1}{2}g^2 W_\\\\mu ^+ W^{\\\\mu \\\\,-}+e^2A_\\\\mu A^\\\\mu +\\\\frac{g^2}{c_W^2}\\\\left(\\\\tfrac{1}{2}-s_W^2\\\\right)^2Z_\\\\mu Z^\\\\mu +\\\\frac{2ge}{c_W}\\\\left(\\\\tfrac{1}{2}-s_W^2\\\\right)A_\\\\mu Z^\\\\mu \\\\right](H^+H^- +h^+ h^-)\\\\nonumber \\\\\\\\&& +\\\\biggl \\\\lbrace \\\\left(\\\\tfrac{1}{2}eg A^\\\\mu W_\\\\mu ^+-\\\\frac{g^2s_W^2}{2c_W}Z^\\\\mu W_\\\\mu ^+\\\\right)\\\\bigl [H^-(H+iA)+h^-(h+ia)\\\\bigr ] +{\\\\rm h.c.}\\\\biggr \\\\rbrace \\\\,, \\\\\\\\{L}_{VHH}&=&\\\\frac{g}{2c_W}\\\\,Z^\\\\mu ( A\\\\,\\\\!\\\\!\\\\mathrel {\\\\leftrightarrow \\\\hspace{-8.50006pt}\\\\partial }$ H+a $\\\\leftrightarrow $$\\\\partial $ h) -12g{iW+[ H-$\\\\leftrightarrow $$\\\\partial $$\\\\scriptstyle \\\\,\\\\mu $ (H+iA)+h-$\\\\leftrightarrow $$\\\\partial $$\\\\scriptstyle \\\\,\\\\mu $ (h+ia)] +h.c.}\",\n \"+[ieA+igcW(12-sW2) Z](H+$\\\\leftrightarrow $$\\\\partial $ H-+h+$\\\\leftrightarrow $$\\\\partial $ h-), In the RIDM, there is no experimental measurement that can physically distinguish the degenerate scalars, $(H^\\\\pm ,h^\\\\pm )$ , $(H,h)$ and $(A,a)$ .\",\n \"However, a multiplicity factor will appear after summing over final mass-degenerate states, e.g., $Z\\\\rightarrow HA$ , $ha$ doubles the rate into a pair of neutral scalars.\"\n ],\n [\n \"An alternative basis choice for the RIDM\",\n \"So far, our discussion has employed the $\\\\lbrace H_1,H_2,H_3\\\\rbrace $ basis of doublet scalar fields.\",\n \"This is one choice among a family of Higgs bases defined such that $\\\\langle H_1^0 \\\\rangle =v/\\\\sqrt{2}$ and $\\\\langle H_2^0 \\\\rangle =\\\\langle H_3^0 \\\\rangle =0$ .\",\n \"Indeed, the Higgs basis is unique only up to an arbitrary U(2) transformation of the doublet fields $H_2$ and $H_3$ .\",\n \"In the following, we shall denote the $\\\\lbrace H_1,H_2,H_3\\\\rbrace $ basis as the $H23$ -basis, since the scalar potential of Eq.\",\n \"(REF ) provides a simple 3HDM extension of the inert 2HDM.\",\n \"It will prove useful to consider another choice of scalar field basis that is related to the $H23$ -basis as follows,Further details are provided in Appendix REF .\",\n \"$&&\\\\mathcal {R}\\\\equiv \\\\frac{1}{\\\\sqrt{2}}\\\\bigl (H_2+iH_3\\\\bigr )=\\\\begin{pmatrix} R^\\\\dagger \\\\\\\\ \\\\frac{1}{\\\\sqrt{2}}\\\\bigl (P+iQ^\\\\dagger \\\\bigr )\\\\end{pmatrix}\\\\,,\\\\nonumber \\\\\\\\[15pt]&&\\\\mathcal {S}\\\\equiv \\\\frac{1}{\\\\sqrt{2}}\\\\bigl (H_2-iH_3\\\\bigr )=\\\\begin{pmatrix} S^\\\\dagger \\\\\\\\ \\\\frac{1}{\\\\sqrt{2}}\\\\bigl (P^\\\\dagger +iQ\\\\bigr )\\\\end{pmatrix}\\\\,.$ This defines the $\\\\lbrace H_1,\\\\mathcal {R},\\\\mathcal {S}\\\\rbrace $ basis of doublet scalar field, henceforth denoted as the $RS$ -basis.\",\n \"Note that since the real neutral fields $(H,h)$ and $(A,a)$ are mass-degenerate pairs, respectively, one can combine the mass-degenerate real fields into complex fields, $P\\\\equiv \\\\frac{H+ih}{\\\\sqrt{2}}\\\\,,\\\\qquad Q\\\\equiv \\\\frac{A-ia}{\\\\sqrt{2}}\\\\,,$ where $M_P\\\\ge M_Q$ (in our convention where $Z_5\\\\ge 0$ ).The relative minus sign in the definition of the imaginary parts of $P$ and $Q$ has been introduced for later convenience.\",\n \"The corresponding conjugate fields are $P^\\\\dagger \\\\equiv \\\\frac{H-ih}{\\\\sqrt{2}}\\\\,,\\\\qquad Q^\\\\dagger \\\\equiv \\\\frac{A+ia}{\\\\sqrt{2}}\\\\,,$ Likewise, since $H^\\\\pm $ and $h^\\\\pm $ are mass-degenerate charged fields, one is free to define, $R&=\\\\frac{H^- - ih^-}{\\\\sqrt{2}}, &\\\\quad S&=\\\\frac{H^- + ih^-}{\\\\sqrt{2}}, \\\\\\\\R^\\\\dagger &=\\\\frac{H^+ + ih^+}{\\\\sqrt{2}}, &\\\\quad S^\\\\dagger &=\\\\frac{H^+ - ih^+}{\\\\sqrt{2}},$ where $R$ and $S$ are negatively charged mass-degenerate scalars and the corresponding conjugate fields, $R^\\\\dagger $ and $S^\\\\dagger $ , are positively charged mass-degenerate scalars.\",\n \"In the $RS$ -basis, the scalar potential is given by $\\\\!\\\\!\\\\!\\\\!\\\\!\\\\!\\\\mathcal {V}_{\\\\rm RIDM-RS}&=& Y_1 H_1^\\\\dagger H_1+ Y_2 \\\\bigl (\\\\mathcal {R}^\\\\dagger \\\\mathcal {R} +\\\\mathcal {S}^\\\\dagger \\\\mathcal {S}\\\\bigr )+\\\\tfrac{1}{2}Z_1(H_1^\\\\dagger H_1)^2+\\\\tfrac{1}{2}{\\\\bar{Z}}_2(\\\\mathcal {R}^\\\\dagger \\\\mathcal {R}+\\\\mathcal {S}^\\\\dagger \\\\mathcal {S})^2+Z_3(H_1^\\\\dagger H_1)(\\\\mathcal {R}^\\\\dagger \\\\mathcal {R}+\\\\mathcal {S}^\\\\dagger \\\\mathcal {S}) \\\\nonumber \\\\\\\\&& +Z_4\\\\bigl [( H_1^\\\\dagger \\\\mathcal {R})(\\\\mathcal {R}^\\\\dagger H_1)+( H_1^\\\\dagger \\\\mathcal {S})(\\\\mathcal {S}^\\\\dagger H_1)\\\\bigr ]+ \\\\bar{Z}^\\\\prime _5\\\\bigl [(H_1^\\\\dagger \\\\mathcal {R})(H_1^\\\\dagger \\\\mathcal {S}) +(\\\\mathcal {R}^\\\\dagger H_1)(\\\\mathcal {S}^\\\\dagger H_1)\\\\bigr ]\\\\,,$ where $\\\\bar{Z}_2=Z_2$ and $\\\\bar{Z}_5^\\\\prime =Z_5$ .The reason for introducing the notation $\\\\bar{Z}_2$ and $\\\\bar{Z}^\\\\prime _5$ in Eq.\",\n \"(REF ) is clarified in section REF .\",\n \"One can then rewrite the RIDM couplings given in Eqs.\",\n \"(REF )–() in terms of the neutral scalar fields $P$ and $Q$ and the charged scalar fields $R$ and $S$ (and the corresponding conjugated fields), $&&\\\\hspace{-14.45377pt}{L}_{VVHH}=\\\\left[\\\\tfrac{1}{4} g^2 W_\\\\mu ^+W^{\\\\mu \\\\,-}+\\\\frac{g^2}{8c_W^2}Z_\\\\mu Z^\\\\mu \\\\right]\\\\bigl (h^2_{\\\\rm SM}+2|P|^2+2|Q|^2\\\\bigr ) \\\\nonumber \\\\\\\\&& +\\\\left[\\\\tfrac{1}{2}g^2 W_\\\\mu ^+ W^{\\\\mu \\\\,-}+e^2A_\\\\mu A^\\\\mu +\\\\frac{g^2}{c_W^2}\\\\left(\\\\tfrac{1}{2}-s_W^2\\\\right)^2Z_\\\\mu Z^\\\\mu +\\\\frac{2ge}{c_W}\\\\left(\\\\tfrac{1}{2}-s_W^2\\\\right)A_\\\\mu Z^\\\\mu \\\\right](R^\\\\dagger R+S^\\\\dagger S)\\\\nonumber \\\\\\\\&&+\\\\biggl \\\\lbrace \\\\left(\\\\tfrac{1}{2}eg A^\\\\mu W_\\\\mu ^+-\\\\frac{g^2s_W^2}{2c_W}Z^\\\\mu W_\\\\mu ^+\\\\right)\\\\bigl [R(P+iQ^\\\\dagger )+S(P^\\\\dagger +iQ)\\\\bigr ] +{\\\\rm h.c.}\\\\biggr \\\\rbrace ,\\\\\\\\&&\\\\hspace{-14.45377pt}{L}_{VHH}=\\\\frac{g}{2c_W}\\\\,Z^\\\\mu ( Q\\\\,\\\\!\\\\!\\\\mathrel {\\\\leftrightarrow \\\\hspace{-8.50006pt}\\\\partial }$ P+Q $\\\\leftrightarrow $$\\\\partial $ P) +[ieA+igcW(12-sW2) Z] (R$\\\\leftrightarrow $$\\\\partial $ R+S$\\\\leftrightarrow $$\\\\partial $ S) -12g{iW+[R $\\\\leftrightarrow $$\\\\partial $ (P+iQ) +S $\\\\leftrightarrow $$\\\\partial $ (P+iQ)] +h.c.}\",\n \"L3h=-12vZ1 hSM3-v[(Z3+Z4)hSM(|P|2+|Q|2)+Z5hSM(|P|2-|Q|2)] -vZ3hSM(RR+SS) , line .\",\n \"L4h=-18 Z1 hSM4 -12Z2(|P|2+|Q|2)(|P|2+|Q|2+2RR + 2SS)-12(Z3+Z4) hSM2(|P|2+|Q|2) -12Z5 hSM2 (|P|2-|Q|2)-12[Z2(RR+SS) +Z3 h2SM] (RR+SS) .\"\n ],\n [\n \"Mass degeneracies beyond the RIDM\",\n \"In this section, we add additional terms to the RIDM scalar potential while preserving the mass degeneracies of the model.\",\n \"Naively, one can add to the RIDM scalar potential any gauge invariant quartic term involving the doublet fields $H_2$ and $H_3$ without upsetting the mass degeneracies of Eq.\",\n \"(REF ).\",\n \"However, the resulting tree-level mass degeneracies will be unnatural unless they are a consequence of a symmetry.\",\n \"The simplest possible modification of the RIDM is to remove the $(H^\\\\dagger _2 H_2)(H^\\\\dagger _3 H_3)$ term entirely from the scalar potential.\",\n \"That is, we can define a RIDM$^\\\\prime $ scalar potential as, $\\\\mathcal {V}_{\\\\rm RIDM^{\\\\prime }}= \\\\mathcal {V}_{\\\\rm RIDM}-Z_2(H^\\\\dagger _2 H_2)(H^\\\\dagger _3 H_3)\\\\,.$ Note that the term in $\\\\mathcal {V}_{\\\\rm RIDM^{\\\\prime }}$ that is proportional to $Z_2$ is now given by $\\\\tfrac{1}{2}\\\\bigl [(H_2^\\\\dagger H_2)^2+(H_3^\\\\dagger H_3)^2\\\\bigr ]$ .\",\n \"Indeed, one can argue that Eq.\",\n \"(REF ) provides the simplest 3HDM generalization of the IDM.\",\n \"In the case of the RIDM$^\\\\prime $ , the tree-level mass degeneracies are no longer a consequence of a continuous symmetry, which is now explicitly broken by the presence of the explicit term in Eq.\",\n \"(REF ) that is proportional to $(H^\\\\dagger _2 H_2)(H^\\\\dagger _3 H_3)$ .\",\n \"Indeed, this term is invariant only under a discrete subgroup of O(2)$\\\\times $ O(2) [which is the symmetry group of the RIDM scalar Lagrangian as discussed in section REF ].\",\n \"In the notation of Eqs.\",\n \"(REF ) and (REF ), consider the following two discrete subgroups of the O(2)$\\\\times $ O(2) symmetry group, $&& (1)~~~U=g\\\\,,\\\\qquad \\\\quad V=0\\\\,, \\\\\\\\&& (2)~~~U=0\\\\,,\\\\qquad \\\\quad V=g\\\\,,$ where $g$ is a $2\\\\times 2$ matrix that acts on the Higgs basis fields $H_2$ and $H_3$ regarded as a two dimensional vector.\",\n \"Then, the term $(H^\\\\dagger _2 H_2)(H^\\\\dagger _3 H_3)$ is invariant under the two discrete subgroups above if $g\\\\in D_4\\\\mathchoice{\\\\cong }{\\\\cong }{\\\\vbox {{ \\\\scriptstyle \\\\hfil # \\\\hfil \\\\crcr \\\\sim \\\\crcr = \\\\crcr }}}{\\\\cong }\\\\lbrace {1},-{1}, R,-R,S,-S,Z,-Z\\\\rbrace $ , where 1 is the $2\\\\times 2$ identity matrix and $R=\\\\begin{pmatrix} 1 &\\\\,\\\\,\\\\,\\\\phantom{-}0 \\\\\\\\ 0 &\\\\,\\\\,\\\\, \\\\phantom{-}1\\\\end{pmatrix}\\\\,,\\\\qquad \\\\quad S=\\\\begin{pmatrix} 1 &\\\\,\\\\,\\\\, \\\\phantom{-}0 \\\\\\\\ 0 &\\\\,\\\\,\\\\,-1\\\\end{pmatrix}\\\\,,\\\\qquad \\\\quad Z=\\\\begin{pmatrix} 0 &\\\\,\\\\,\\\\, -1 \\\\\\\\ 1 &\\\\,\\\\,\\\\, \\\\phantom{-}0\\\\end{pmatrix}\\\\,.$ We recognize $D_4$ as the dihedral group of order eight, which is the symmetry group of the square [67].\",\n \"Both discrete subgroups [Eqs.\",\n \"(REF ) and ()] are isomorphic to $D_4$ .\",\n \"Following the discussion below Eq.\",\n \"(REF ), we conclude that the RIDM$^\\\\prime $ scalar Lagrangian is invariant under a discrete $D_4\\\\times D_4$ symmetry, which is responsible for the presence of the mass-degenerate scalars $(H,h)$ and $(A,a)$ , respectively.\",\n \"Finally, after promoting the derivative to the gauge covariant derivative in the scalar kinetic energy term, the remaining symmetry is $D_4\\\\rtimes \\\\mathbb {Z}_2$ .\",\n \"A comprehensive treatment of natural scalar mass degeneracies in the 3HDM would require a complete classification of 3HDM scalar potential symmetries, along the lines of the 2HDM analysis given in section REF .A complete catalog of all possible finite symmetry groups of the 3HDM is known (as well as some additional partial results); however the complete classification of all possible symmetries of the 3HDM remains an open problem [68], [69].\",\n \"In this paper, we shall ask a less ambitious question: can one break the discrete symmetry identified above further while still naturally maintaining the mass-degenerate states of the RIDM.\",\n \"The answer turns out to be affirmative.\",\n \"This investigation led us to a particular 3HDM originally introduced by Ivanov and Silva [30] for other reasons that will be reviewed below.\",\n \"The Ivanov-Silva (IS) model was constructed to exhibit a number of curious properties [30], [66], which appear to rely on the existence of degenerate states in the scalar spectrum.\",\n \"In particular, the IS scalar potential does not respect the conventional CP symmetry, $H_i\\\\rightarrow H_i^{\\\\scriptscriptstyle {\\\\bigstar }}$ , where the latter satisfies $({\\\\rm CP})^2={1}$ , but instead respects a generalized CP symmetry of the form $H_i\\\\rightarrow X_{ij}H_j^{\\\\scriptscriptstyle {\\\\bigstar }}$ for some unitary matrix $X$ .\",\n \"In particular, the generalized CP symmetry of the IS scalar potential, denoted by CP4, is of order 4, signifying that $({\\\\rm CP4})^4={1}$ and $({\\\\rm CP4})^2\\\\ne {1}$ .\",\n \"Moreover, no Higgs basis of scalar fields exists in which all the parameters of the IS scalar potential are simultaneously real.\",\n \"As noted in section REF , this property is in stark contrast with the 2HDM in which the existence of any generalized CP symmetry implies that the 2HDM scalar potential automatically respects the conventional CP symmetry, i.e.\",\n \"a basis of scalar fields exists such that the corresponding 2HDM scalar potential parameters are real [56], [57].\",\n \"In Appendix A, we demonstrate that starting from the IS scalar potential, one can perform a basis change in order to obtain a more convenient form of the scalar potential.\",\n \"By making an appropriate U(2) transformation to define the Higgs basis fields, $H_2$ and $H_3$ , we find that the IS scalar potential takes on the following form in the $H23$ -basis, $\\\\mathcal {V}_{\\\\rm IS}= \\\\mathcal {V}_{\\\\rm RIDM}+Z_3^\\\\prime (H_2^\\\\dagger H_2)(H_3^\\\\dagger H_3)+Z_4^\\\\prime (H_2^\\\\dagger H_3)(H_3^\\\\dagger H_2) +\\\\bigl [Z_8 (H_2^\\\\dagger H_3)^2+Z_9(H_2^\\\\dagger H_3)(H_2^\\\\dagger H_2-H_3^\\\\dagger H_3)+{\\\\rm h.c.}\\\\bigr ]\\\\,,$ where $\\\\mathcal {V}_{\\\\rm RIDM}$ is given in Eq.\",\n \"(REF ).\",\n \"In general, $Z_8$ and $Z_9$ are complex parameters.As shown in Appendix REF , one can perform an SO(2) rotation to redefine the fields $H_2$ and $H_3$ to remove the complex phase from either $Z_8$ or $Z_9$ .\",\n \"We shall continue to use Eq.\",\n \"(REF ) to express the Higgs basis fields in terms of mass-eigenstate fields.\",\n \"Since none of the extra terms in Eq.\",\n \"(REF ) involve the Higgs basis field $H_1$ , the tree-level mass relations of Eq.\",\n \"(REF ) are not modified.\",\n \"We now argue that the mass-degeneracies of $(H^\\\\pm , h^\\\\pm )$ , $(H,h)$ and $(A,a)$ are stable due to the presence of a symmetry.\",\n \"The O(2)$\\\\times $ O(2) symmetry of the RIDM (prior to gauging the scalar kinetic energy terms) that is responsible for the mass degeneracies among the neutral Higgs mass eigenstates is broken by the new terms beyond $\\\\mathcal {V}_{\\\\rm RIDM}$ contained in Eq.\",\n \"(REF ).\",\n \"Indeed, after the extra terms are included, no unbroken continuous subgroup of O(2)$\\\\times $ O(2) remains.\",\n \"In the notation of Eqs.\",\n \"(REF ) and (REF ), consider the following two discrete subgroups of the O(2)$\\\\times $ O(2) symmetry group, $&& (1)~~~U=Z\\\\,,\\\\qquad \\\\quad V=0\\\\,, \\\\\\\\&& (2)~~~U=0\\\\,,\\\\qquad \\\\quad V=Z\\\\,,$ where $Z$ is given by Eq.\",\n \"(REF ).\",\n \"The $2\\\\times 2$ matrix $Z$ acts on the Higgs basis fields $H_2$ and $H_3$ .\",\n \"Both discrete subgroups [Eqs.\",\n \"(REF ) and ()] are isomorphic to $\\\\mathbb {Z}_4=\\\\bigl \\\\lbrace {1},-{1},Z,-Z\\\\bigr \\\\rbrace $ .\",\n \"Note that $Z^2=-{1}$ , where 1 is the $2\\\\times 2$ identity matrix.In this case, gauging the scalar kinetic energy terms does not reduce the symmetry group further.\",\n \"Consider first the discrete symmetry defined in Eq.\",\n \"(REF ).\",\n \"The fields $H_2$ and $H_3$ are odd under $-{1}$ , which simply identifies the two inert doublets.\",\n \"The elements $Z$ (and $-Z$ ) act non-trivially on the inert doublets.\",\n \"However, Eq.\",\n \"(REF ) is invariant with respect to $\\\\begin{pmatrix} H_2\\\\\\\\ H_3\\\\end{pmatrix}\\\\rightarrow \\\\begin{pmatrix} 0 & -1\\\\\\\\ 1 & \\\\phantom{-}0\\\\end{pmatrix}\\\\begin{pmatrix} H_2\\\\\\\\ H_3\\\\end{pmatrix}\\\\,,$ if and only if $Z_8$ and $Z_9$ are both real.\",\n \"In the model of IS where there is an unremovable complex phase in the scalar potential, only the subgroup $\\\\mathbb {Z}_2=\\\\lbrace {1},-{1}\\\\rbrace $ of $\\\\mathbb {Z}_4$ survives.\",\n \"In particular, the residual symmetry in this case is not sufficient to explain the mass degeneracies of the IS model.\",\n \"The discrete symmetry defined in Eq.\",\n \"() is a generalized CP symmetry.\",\n \"In particular, the IS scalar potential is invariant under $H_i\\\\rightarrow X_{ij}H_j^{\\\\scriptscriptstyle {\\\\bigstar }}\\\\,, \\\\qquad \\\\quad \\\\text{where $X=\\\\begin{pmatrix} 1 &\\\\phantom{-}0 & \\\\phantom{-}0 \\\\\\\\ 0 & \\\\phantom{-}0 & -1 \\\\\\\\ 0 & \\\\phantom{-}1 & \\\\phantom{-}0\\\\end{pmatrix}$,}$ This symmetry, which is also isomorphic to $\\\\mathbb {Z}_4$ , is the CP4 symmetry advertised above.\",\n \"Moreover, this discrete symmetry is sufficient to explain the mass degeneracies of the IS model (in the case of an unremovable complex phase in the IS scalar potential).\",\n \"It is instructive to consider the Higgs couplings of the IS model.\",\n \"Only the quartic Higgs couplings of the RIDM are modified as follows, $\\\\delta {L}_{4h}&=&-\\\\tfrac{1}{4}(Z_3^\\\\prime +Z_4^\\\\prime )\\\\bigl [(H^2+A^2)(h^2+a^2)+4H^+ H^- h^+ h^-\\\\bigr ]-\\\\tfrac{1}{2}Z_3^\\\\prime \\\\bigl [(H^2+A^2)h^+ h^-+(h^2+a^2)H^+ H^-\\\\bigr ]\\\\nonumber \\\\\\\\&& -\\\\tfrac{1}{2}Z^{\\\\prime }_4\\\\bigl [\\\\bigl (Hh+Aa+i(Ha-hA)\\\\bigr )H^+ h^- + \\\\bigl (Hh+Aa-i(Ha-hA)\\\\bigr )h^+ H^-\\\\bigr ] \\\\nonumber \\\\\\\\&& -\\\\tfrac{1}{4} Z_8\\\\bigl [Hh+Aa+i(Ha-hA)+2h^+ H^-\\\\bigr ]^2-\\\\tfrac{1}{4} Z^*_8\\\\bigl [Hh+Aa-i(Ha-hA)+2H^+ h^-\\\\bigr ]^2 \\\\nonumber \\\\\\\\&& -\\\\tfrac{1}{4} Z_9\\\\bigl (H^2+A^2-h^2-a^2+2H^+ H^- -2h^+ h^-\\\\bigr )\\\\bigl [Hh+Aa+i(Ha-hA)+2h^+ H^-\\\\bigr ]\\\\nonumber \\\\\\\\&& -\\\\tfrac{1}{4} Z^*_9\\\\bigl (H^2+A^2-h^2-a^2+2H^+ H^- -2h^+ h^-\\\\bigr )\\\\bigl [Hh+Aa-i(Ha-hA)+2H^+ h^-\\\\bigr ]\\\\,.$ It is convenient to re-express the neutral scalar fields appearing in Eq.\",\n \"(REF ) in terms of the complex neutral fields $P$ and $Q$ and their conjugates introduced in Eqs.\",\n \"(REF ) and (REF ), and the charged fields $R$ and $S$ and their conjugates defined in Eqs.\",\n \"(REF ) and ().\",\n \"Note that the fields $P$ , $Q$ and the corresponding conjugate fields $P^\\\\dagger $ and $Q^\\\\dagger $ are each eigenstates of CP4.This means that each of the four states, $P$ , $Q$ , $P^\\\\dagger $ and $Q^\\\\dagger $ , are CP4-self conjugate (they are their own antiparticles).\",\n \"Moreover, $P$ and the corresponding conjugate state $P^\\\\dagger $ are mass-degenerate, but are otherwise unrelated fields (and similarly for $Q$ and $Q^\\\\dagger $ ).\",\n \"In particular, under a CP4 transformation, $P\\\\rightarrow iP$ , $Q\\\\rightarrow iQ$ , $P^\\\\dagger \\\\rightarrow -iP^\\\\dagger $ , and $Q^\\\\dagger \\\\rightarrow -iQ^\\\\dagger $ .\",\n \"Likewise, under a CP4 transformation, $R\\\\rightarrow -iS^\\\\dagger $ , $R^\\\\dagger \\\\rightarrow iS$ , $S\\\\rightarrow iR^\\\\dagger $ and $S^\\\\dagger \\\\rightarrow -iR$ .\",\n \"Note that these transformation properties are consistent with the requirement that $({\\\\rm CP}4)^4={1}$ .\",\n \"We can evaluate the four-scalar interaction Lagrangian directly in the $RS$ -basis.\",\n \"We first must rewrite Eq.\",\n \"(REF ) in the RS-basis, $\\\\mathcal {V}_{\\\\rm IS-RS}= \\\\mathcal {V}_{\\\\rm RIDM-RS}+\\\\bar{Z}_3^\\\\prime (\\\\mathcal {R}^\\\\dagger \\\\mathcal {R})(\\\\mathcal {S}^\\\\dagger \\\\mathcal {S})+\\\\bar{Z}_4^\\\\prime (\\\\mathcal {R}^\\\\dagger \\\\mathcal {S})(\\\\mathcal {S}^\\\\dagger \\\\mathcal {R}) +\\\\bigl [\\\\bar{Z}_8 (\\\\mathcal {R}^\\\\dagger \\\\mathcal {S})^2+\\\\bar{Z}_9(\\\\mathcal {R}^\\\\dagger \\\\mathcal {S})(\\\\mathcal {R}^\\\\dagger \\\\mathcal {R}-\\\\mathcal {S}^\\\\dagger \\\\mathcal {S})+{\\\\rm h.c.}\\\\bigr ]\\\\,,$ where $\\\\mathcal {V}_{\\\\rm RIDM-RS}$ is given by Eq.\",\n \"(REF ).\",\n \"The relations between the unbarred and barred parameters are derived in Appendix REF , $\\\\bar{Z}_2&=&Z_2+\\\\tfrac{1}{2}(Z_3^\\\\prime +Z_4^\\\\prime -2\\\\operatorname{Re}Z_8)\\\\,,\\\\qquad \\\\qquad \\\\quad \\\\,\\\\,\\\\,\\\\,\\\\bar{Z}_3^\\\\prime =-Z_4^\\\\prime +2\\\\operatorname{Re}Z_8\\\\,,\\\\\\\\\\\\bar{Z}_4^\\\\prime &=&\\\\tfrac{1}{2}(Z_4^\\\\prime -Z_3^\\\\prime +2\\\\operatorname{Re}Z_8)\\\\,,\\\\qquad \\\\qquad \\\\qquad \\\\qquad \\\\bar{Z}_5^\\\\prime =Z_5\\\\,,\\\\\\\\\\\\bar{Z}_8&=&-\\\\tfrac{1}{4}(Z_3^\\\\prime +Z_4^\\\\prime +2\\\\operatorname{Re}Z_8)+i\\\\operatorname{Re}Z_9\\\\,,\\\\qquad \\\\quad \\\\bar{Z}_9= \\\\operatorname{Im}Z_9+i\\\\operatorname{Im}Z_8\\\\,.\",\n \"$ The quartic interactions given in Eq.\",\n \"(REF ) are then modified by employing the new definition of $\\\\bar{Z_2}$ given in Eq.\",\n \"(REF ) and adding the following terms, We now consider the possible effects of the Yukawa interactions.\",\n \"It is remarkable that it is possible to construct a CP4-invariant Yukawa interaction Lagrangian where the fermions transform nontrivially under a CP4 transformation [66], [70], [71].\",\n \"In such a model, the mass degeneracies identified above that are a consequence of the CP4 symmetry are of course maintained.\",\n \"Alternatively, if the fermions couple exclusively to the Higgs basis field $H_1$ (as in the case of the IDM), then the Yukawa interactions are invariant with respect to the $\\\\mathbb {Z}_2$ discrete symmetry defined below Eq.\",\n \"(REF ),Note that this $\\\\mathbb {Z}_2$ symmetry is isomorphic to $(\\\\rm {CP}4)^2$ , which remains an exact symmetry of the model.\",\n \"under which the inert doublet fields, $H_2$ and $H_3$ , are odd and all other fields of the model ($H_1$ , gauge bosons and fermions) are even.\",\n \"However, the CP4 symmetry is no longer a symmetry of the complete model.\",\n \"That is, if we define the CP4 transformation to be the conventional CP transformation when acting on the fermions and gauge fields, then the CP4 symmetry of the model will be violated by the presence of the unremovable CP-violating phase in the CKM mixing matrix.\",\n \"Nevertheless, it is not clear whether this violation is sufficient to remove the scalar mass degeneracies of the IS model that were protected by the (now accidental) CP4 symmetry of the scalar potential.\",\n \"This is an open question that we hope to revisit in a future work.\",\n \"Finally, it is instructive to note that the scalar mass degeneracies of the CP4-invariant 3HDM is just the simplest example of a larger class of multi-Higgs models with degenerate scalars that are a consequence of a generalized CP symmetry.\",\n \"In Ref.\",\n \"[72], Ivanov and Laletin demonstrate how to construct $N$ Higgs doublet models with a generalized CP symmetry of order $2k$ (denoted by CP$2k$ ) with positive integer $k$ .\",\n \"Nontrivial cases arise only for $2k=2^p$ with integer $p\\\\ge 1$ .\",\n \"The simplest nontrivial models of this type (CP8 and CP16) require at least $N=5$ Higgs doublets.\",\n \"Such models necessarily have mass-degenerate neutral scalars and mass-degenerate charged Higgs pairs.\",\n \"A further exploration of models of this type is beyond the scope of this work.\"\n ],\n [\n \"An observable distinction between CP2 and CP4\",\n \"The distinction between the IS scalar potential in the $H23$ -basis with $Z_8$ and $Z_9$ real or complex is physical.In making this assertion, we have implicitly assumed that $Z_5\\\\ne 0$ .\",\n \"The case of $Z_5=0$ , which is special due to the enhanced mass degeneracy noted in footnote REF , will be treated at the end of this section.\",\n \"To demonstrate this assertion, we focus on the neutral scalar self-interactions in $\\\\delta {L}_{4h}$ that are linear in the fields $P$ or $Q$ (or their complex conjugates), $\\\\delta {L}_{4h}\\\\ni \\\\tfrac{1}{2}\\\\operatorname{Im}Z_8\\\\bigl [(PQ-P^\\\\dagger Q^\\\\dagger )(P^2-Q^2-P^{\\\\dagger \\\\,2}+Q^{\\\\dagger \\\\,2})\\\\bigr ]+\\\\tfrac{1}{2}i\\\\operatorname{Im}Z_9\\\\bigl [(PQ-P^\\\\dagger Q^\\\\dagger )(P^2+Q^2+P^{\\\\dagger \\\\,2}+Q^{\\\\dagger \\\\,2})\\\\bigr ]\\\\,,$ where we have used Eq.\",\n \"() to re-express $\\\\bar{Z}_9$ [which appears in Eq.\",\n \"(REF )] in terms of the $H23$ -basis parameters, $\\\\operatorname{Im}Z_8$ and $\\\\operatorname{Im}Z_9$ .\",\n \"Self-interaction terms of this type are absent if $Z_8$ and $Z_9$ are both real.\",\n \"Hence, the presence of these terms signals a CP4-symmetric IS scalar potential that does not respect the conventional CP symmetry, $H_i\\\\rightarrow H_i^{\\\\scriptscriptstyle {\\\\bigstar }}$ .\",\n \"Here we provide two specific examples.\",\n \"First, Eq.\",\n \"(REF ) shows the existence of a $ZPQ$ interaction, which would permit the decay $Z\\\\rightarrow PQ, P^*Q^*$ , if kinematically available.\",\n \"Since $M_Q\\\\le M_P$ , let us further suppose that $M_Q<\\\\tfrac{1}{4} m_Z<M_P$ .\",\n \"In this case, the $P$ and $P^*$ would be virtual.\",\n \"One possible decay of the virtual $P$ or $P^*$ makes use of the existence of the four-scalar interaction given in Eq.\",\n \"(REF ).\",\n \"If this interaction is present, the decay $Z\\\\rightarrow QQQQ^*$ , $Q^* Q^* Q^* Q$ is allowed and provides unambiguous evidence that either $Z_8$ and/or $Z_9$ possesses a nonzero imaginary part.\",\n \"A second example makes use of the $W^+H^-P$ , $W^+h^-P$ , $W^+H^-Q$ , and $W^+h^-Q$ interactions of Eq.\",\n \"(REF ).\",\n \"In this case, we can consider the decay of a charged $W$ into a charged Higgs boson and $P$ (or $P^*$ ).\",\n \"We can now make use of Eq.\",\n \"(REF ) to decay the virtual $P$ or $P^*$ into $QQQ$ , $QQQ^*$ , $QQ^*Q^*$ , $Q^* Q^* Q^*$ .\",\n \"Note that in each of the two cases above, there are multiple four-scalar final states involving mass-degenerate scalars.\",\n \"In computing the experimentally observed rates, one must compute the squared amplitude for each of the possible final states, and then multiply the final result by a multiplicity factor that counts the number of possible final states.\",\n \"In contrast, suppose that Eq.\",\n \"(REF ) were a symmetry of the IS scalar potential.\",\n \"In this case, the corresponding transformation properties of the scalar fields are, $P\\\\rightarrow iP$ , $Q\\\\rightarrow -iQ$ , $P^\\\\dagger \\\\rightarrow -iP^\\\\dagger $ , $Q^\\\\dagger \\\\rightarrow iQ^\\\\dagger $ , $H^{\\\\pm }\\\\rightarrow -h^\\\\pm $ , and $h^\\\\pm \\\\rightarrow H^\\\\pm $ .\",\n \"One would then immediately conclude that $Z_8=Z_8^*$ and $Z_9=Z_9^*$ , as expected.\",\n \"In particular, Eq.\",\n \"(REF ) is not invariant under Eq.\",\n \"(REF ), and thus the four scalar decay modes listed above would necessarily be absent.\",\n \"As an exercise, we have evaluated the decay rate for $Z\\\\rightarrow QQQQ^*$ , $QQ^* Q^* Q^*$ , in an approximation where $M_Q=0$ and $M_P\\\\gg m_Z$ .\",\n \"The computation is presented in Appendix .\",\n \"The end result is $\\\\frac{\\\\Gamma (Z\\\\rightarrow QQQQ^*,QQ^* Q^* Q^*)}{\\\\Gamma (Z\\\\rightarrow \\\\nu \\\\bar{\\\\nu })}=\\\\frac{(\\\\operatorname{Im}Z_8)^2+(\\\\operatorname{Im}Z_9)^2}{3\\\\cdot 5\\\\cdot 2^{8}\\\\, \\\\pi ^4 }\\\\left(\\\\frac{m_Z}{M_P}\\\\right)^4\\\\,.$ This result implies that the quantity $(\\\\operatorname{Im}Z_8)^2+(\\\\operatorname{Im}Z_9)^2$ must be a physical quantity, and hence invariant with respect to scalar basis changes that are consistent with the form of the IS scalar potential given by Eq.\",\n \"(REF ) in the $H23$ -basis.\",\n \"However, the family of Higgs bases is larger than the set of scalar field bases in which the IS scalar potential has the form of Eq.\",\n \"(REF ), as discussed in Appendix REF .\",\n \"In special cases, it is possible that there exists a real Higgs basis even if $(\\\\operatorname{Im}Z_8)^2+(\\\\operatorname{Im}Z_9)^2\\\\ne 0$ .\",\n \"In such cases, one can transform the fields ($H_2, H_3)\\\\rightarrow (\\\\bar{H}_2,\\\\bar{H}_3)$ , where $\\\\bar{H}_i\\\\rightarrow \\\\bar{H_i}^{\\\\scriptscriptstyle {\\\\bigstar }}$ is a symmetry of the Lagrangian; i.e., the model exhibits a CP2 symmetry.The notation CP2 derives from the property, $(\\\\text{CP2})^2={1}$ .\",\n \"In the IS model, the existence of a nonzero decay rate for $Z\\\\rightarrow QQQQ^*$ , $QQ^* Q^* Q^*$ implies that no CP2 symmetry that commutes with the CP4 symmetry is present.We say that the CP2 symmetry commutes with CP4 if the application of these two symmetry transformations on the scalar fields does not depend on the order in which the transformations are applied.\",\n \"For further details see Appendix B of Ref. [32].\",\n \"However, this leaves open the possibility of a CP2 symmetry that does not commute with CP4.\",\n \"In Appendix REF , we provide two examples in which $(\\\\operatorname{Im}Z_8)^2+(\\\\operatorname{Im}Z_9)^2\\\\ne 0$ in the $H23$ basis, but nevertheless a real Higgs basis exists: (i) $\\\\operatorname{Im}Z_8\\\\ne 0$ and $Z_9=0$ and (ii) $\\\\operatorname{Im}Z_8=0$ , $\\\\operatorname{Re}Z_9=0$ and $\\\\operatorname{Im}Z_9\\\\ne 0$ .\",\n \"In both these examples, the CP2 symmetry that exists does not commute with the CP4 symmetry, even though the decay rate for $Z\\\\rightarrow QQQQ^*$ , $QQ^* Q^* Q^*$ is nonzero.\",\n \"In light of the results of Appendix REF , this is a generic feature of a noncommuting CP2 symmetry in the IS model.\",\n \"Equivalently, the nonexistence or existence of the decay $Z\\\\rightarrow QQQQ^*$ , $QQ^* Q^* Q^*$ is a physical distinction between the 3HDM with a CP4 symmetric IS scalar potential that either preserves or does not preserve a commuting CP2 symmetry.\",\n \"Finally, we return to the special case of $Z_5=0$ (cf.\",\n \"footnote REF ).\",\n \"In Appendix REF , we have demonstrated explicitly that if $Z_5\\\\ne 0$ , then there exists a ratio of two basis-invariant quantities, which when evaluated in the $H23$ -basis yields $(\\\\operatorname{Im}Z_8)^2+(\\\\operatorname{Im}Z_9)^2$ .\",\n \"Moreover, if $Z_5=0$ , then it is possible to change the basis of scalar fields of the IS model, in which the form of the IS potential is still given by Eq.\",\n \"(REF ) but $\\\\operatorname{Im}Z_8=\\\\operatorname{Im}Z_9=0$ .\",\n \"This result appears to be in contradiction to the result of Eq.\",\n \"(REF ).\",\n \"The resolution of this apparent paradox can be obtained by noting that if $Z_5=0$ , then $M_P=M_Q$ .\",\n \"Since Eq.\",\n \"(REF ) was derived under the assumption that $M_Q=0$ and $M_P\\\\gg m_Z$ , Eq.\",\n \"(REF ) no longer applies if $Z_5=0$ .\",\n \"But, more importantly, if $Z_5=0$ (so that $M_P=M_Q$ ), then the decay $Z\\\\rightarrow QQQQ^*,QQ^* Q^* Q^*$ is no longer an experimental observable, since one must also include four scalar decays involving $P$ and $P^*$ .\",\n \"The possible four-body final states involve all possible combinations of $P$ and $Q$ scalars, such that either one or three of the final state scalars are complex-conjugated.\",\n \"Some of the vertices that contribute to these final states are present even if $\\\\operatorname{Im}Z_8=\\\\operatorname{Im}Z_9=0$ .\",\n \"For example, there is a four-scalar $|P|^2 |Q|^2$ interaction that contributes to $Z\\\\rightarrow QQPP^*$ .\",\n \"One must compute the squared amplitude for each possible final state and then add the amplitudes incoherently to obtain the final experimentally observable decay rate.\",\n \"This decay rate will involve a complicated combination of the IS potential coefficients, which will correspond to the appropriate invariant quantity in the case of $Z_5=0$ .\",\n \"Thus, the possibility of finding a new basis for the IS potential in which $\\\\operatorname{Im}Z_8=\\\\operatorname{Im}Z_9=0$ when $Z_5=0$ is no longer paradoxical.\"\n ],\n [\n \"The $ZZZ$ and {{formula:a1ad6645-e9b6-4fb4-b6f1-a07eb57f0b31}} vertices\",\n \"In the CP-violating 2HDM, CP violation may manifest itself at loop level in the effective $ZZZ$ and $ZWW$ vertices.\",\n \"In that model, one finds that CP-violating form factors can be described in terms of the three invariants introduced in Eq.\",\n \"(REF ) [73].\",\n \"In section , we noted the existence of a physical observable that could distinguish between the CP4-conserving IS models in which a CP2 symmetry that commutes with CP4 is either present or absent.\",\n \"However, from a spacetime viewpoint, this physical observable was CP-even.\",\n \"This raises the question as to whether any observable can exist in a CP4-invariant theory that is CP2-odd.\",\n \"The answer to this question is no.\",\n \"For example, there is no way to distinguish between CP2 and CP4 on the level of the form factors themselves.\",\n \"Thus, if the theory respects at least one generalized CP symmetry, then all CP-violating form factors must be absent.\",\n \"It is instructive to check the cancellation of contributions to the CP-violating form factors of the effective $ZZZ$ and $ZWW$ vertices in a CP4-conserving, CP2-violating theory (neglecting any effects from the Higgs-fermion Yukawa interactions).\",\n \"The general $ZZZ$ vertex function (with all $Z$ bosons off-shell) can be expressed in terms of 14 different Lorentz structures [74], [75], [76], [77], [78], all preserving parity.\",\n \"Some of these vanish when one or more $Z$ are on-shell.\",\n \"Let us characterize them by momenta and Lorentz indices ($p_1,\\\\mu $ ), ($p_2,\\\\alpha $ ) and ($p_3,\\\\beta $ ), and let $Z_1$ be off-shell while $Z_2$ and $Z_3$ are on-shell.\",\n \"Furthermore, we assume that $Z_1$ couples to a pair of leptons such as $e^+e^-$ , and terms proportional to the lepton mass will be neglected.\",\n \"Denoting $\\\\ell \\\\equiv p_2-p_3 \\\\equiv 2p_2-p_1$ , the $ZZZ$ vertex structure reduces to the form [76] $-i\\\\Gamma _{ZZZ}^{\\\\alpha \\\\beta \\\\mu }=\\\\frac{p_1^2-M_Z^2}{M_Z^2}\\\\left[f_4^Z(p_1^{\\\\alpha }g^{\\\\mu \\\\beta }+p_1^{\\\\beta }g^{\\\\mu \\\\alpha })+f_5^Z\\\\epsilon ^{\\\\mu \\\\alpha \\\\beta \\\\rho }\\\\ell _\\\\rho \\\\right]\\\\,.$ The dimensionless form factor $f_4^Z$ violates CP while $f_5^Z$ conserves CP.\",\n \"Figure: A typical pair of Feynman diagrams for Z→ZZZ\\\\rightarrow ZZ at two-loop order.For example, consider the case of the 2HDM.\",\n \"At the one-loop level, CP violating effects yield a non-zero contribution to the $ZZZ$ vertex function, $f_4$ , that is proportional to $\\\\operatorname{Im}J_2$ of Eq.\",\n \"(REF ) [73].\",\n \"Thus, only one of the three invariants of Eq.\",\n \"(REF ) contributes.\",\n \"Indeed, in light of Eq.\",\n \"(REF ), it follows that a non-zero $\\\\operatorname{Im}J_2$ requires all three neutral Higgs bosons to be non-degenerate in mass, and the $Z$ boson couples to all three non-diagonal neutral Higgs pairs.\",\n \"In order to understand how the IS model conserves CP (while not respecting CP2), it is instructive to see how the CP-violating effects cancel at loop level in the effective $ZZZ$ (and $ZWW$ ) vertices.\",\n \"In order to do this we have employed the software package FeynArts[79] and written a FeynArts model file containing all the bosonic couplings of the IS-model.\",\n \"We have automated the construction of the diagrams contributing to the effective $ZZZ$ -vertex and evaluated their amplitude (the loop integrals are kept unevaluated in symbolic form).\",\n \"We are only interested in those contributions to each diagram that contain $\\\\operatorname{Im}Z_8$ and/or $\\\\operatorname{Im}Z_9$ , since such contributions could be a signal of CP violation.\",\n \"At the one-loop level there are no diagrams containing $\\\\operatorname{Im}Z_8$ and/or $\\\\operatorname{Im}Z_9$ .\",\n \"Such contributions can only arise from a four-point scalar vertex.\",\n \"This means that this four-point vertex must be “internal\\\"; i.e., none of the external $Z$ -fields can be part of this vertex.\",\n \"None of the $ZZZ$ one-loop topologies can accommodate this.\",\n \"Diagrams containing $\\\\operatorname{Im}Z_8$ and/or $\\\\operatorname{Im}Z_9$ first appear at two-loop order.\",\n \"But even if there are individual diagrams with this type of contribution, the sum of the contributions is zero when we add the amplitudes for all the individual diagrams within each topology.\",\n \"A pair of cancelling diagrams are shown in Fig.\",\n \"REF .\",\n \"The same happens for diagrams at three-loop order.\",\n \"Repeating this exercise for the $ZWW$ vertex we find the same result.\",\n \"Hence, there are no contributions at one-, two- or three-loop order containing $\\\\operatorname{Im}Z_8$ and/or $\\\\operatorname{Im}Z_9$ after adding the amplitudes for all the individual diagrams within each topology.\",\n \"The arguments presented at the beginning of this section imply that this cancellation persists to all orders in perturbation theory.\"\n ],\n [\n \"Conclusions\",\n \"In this work we discussed the interplay between symmetries and natural mass degeneracies in the scalar sector.\",\n \"Some cases of scalar mass degeneracy are accidental, i.e.\",\n \"they are not the result of an exact symmetry and therefore can be implemented only by an artificial fine tuning of the scalar potential parameters.\",\n \"The Higgs basis [16], [17], [18], in which the neutral scalar field vacuum expectation value resides entirely in one of the scalar doublet fields, is especially suitable for our study.\",\n \"We began by examining the two Higgs doublet model (2HDM), with particular attention given to the special case of the inert doublet model (IDM), which possesses an unbroken $\\\\mathbb {Z}_2$ symmetry under which one “inert” scalar doublet is odd, and all other fields of the model are even.\",\n \"In all cases in which the 2HDM exhibited scalar mass degeneracies (whether natural or accidental), the mass degenerate states can be experimentally distinguished from each other.\",\n \"Moreover, with one exception, we found that all 2HDM mass degeneracies are accidental.\",\n \"The one exceptional case of 2HDM scalars that can be naturally degenerate in mass are the two neutral scalar states of the inert doublet of the IDM.\",\n \"This result was also confirmed by examining all possible symmetries of the 2HDM scalar potential and analyzing which of these symmetries can guarantee the presence of mass degenerate scalar states.\",\n \"For models with three Higgs doublets, the analysis of the general case becomes significantly more elaborate.\",\n \"We focused first on a 3HDM generalization of the IDM with mass degenerate scalars, which we denoted as the replicated IDM (RIDM), where the two doublets $H_2$ and $H_3$ are invariant under two separate unbroken $\\\\mathbb {Z}_2$ symmetries and the model is CP conserving.\",\n \"In this framework $H_2$ and $H_3$ are composed of mass eigenstate fields, that do not mix with the SM like Higgs boson, forming four mass degenerate pairs.\",\n \"Furthermore, each mass degenerate pair picks one field from each one of these doublets.\",\n \"We also identified the symmetry obeyed by the neutral mass eigenstates themselves, which is responsible for the twofold mass degeneracies.\",\n \"In the absence of $Z_5$ (which appears in the RIDM scalar potential) there are four mass degenerate neutral scalars and the symmetry of the scalar potential consists of an O$(4)$ global symmetry.\",\n \"Introducing in the potential the term proportional to $Z_5$ , partially breaks the O$(4)$ symmetry down to an O(2)$\\\\times $ O(2) symmetry and the fourfold mass degeneracy is lifted, leaving a pairwise mass degeneracy.\",\n \"The mass degeneracy of the two charged physical fields is governed by the full O(4) symmetry.\",\n \"In the case of $Z_4 = Z_5 =0 $ there is further enhancement of the symmetry and all eight physical scalars contained in $H_2$ and $H_3$ are mass degenerate.\",\n \"It is instructive to examine the Higgs boson interactions with the gauge bosons as well as the Higgs self couplings of the RIDM, since in the RIDM the components of $H_2$ and $H_3$ are already states with well defined masses.\",\n \"We are then led to the conclusion that there is no experimental measurement that can physically distinguish the mass degenerate scalars of the RIDM on an event by event basis.\",\n \"Nevertheless, multiplicity factors due to the production of different scalar states of the same mass do appear in physical observables and signal the existence of the mass degeneracy.\",\n \"Starting with the RIDM, one can consider perturbations in which the mass degeneracies persist and yet remain natural.\",\n \"By reducing the RIDM symmetries responsible for the mass degeneracies to the smallest discrete subgroup that maintains the mass degenerate scalar states, we are led to a model that is equivalent to a particular 3HDM that was originally proposed by Ivanov and Silva (IS) [30].\",\n \"The IS model exhibits very special properties.\",\n \"The original form of the IS scalar potential is given in Appendix REF and is the most general potential respecting the symmetry given by Eq.\",\n \"(REF ).\",\n \"We have rewritten the IS potential in the notation of Eq.\",\n \"(REF ) where the symmetry is now given by Eq.\",\n \"(REF ).\",\n \"In particular, the scalar mass terms (and the corresponding mass degeneracies) are the same as in the RIDM; only the quartic couplings of the physical scalar states differ.\",\n \"One must apply the symmetry given by Eq.\",\n \"(REF ) [denoted by CP4] four times in order to obtain the identity transformation.\",\n \"This is to be contrasted with the conventional CP symmetry transformation (denoted by CP2) whose square is the identity.\",\n \"On the other hand, if we apply the CP4 transformation while at the same time transforming the space coordinates from $x$ into $-x$ , the end result can be identified as a generalized CP transformation.\",\n \"This is a very unusual type of CP transformation since applying it twice does not yield the identity transformation.\",\n \"However, identifying CP4 with a CP transformation is possible because from the spacetime point of view the transformation remains of order two, as it should.\",\n \"Likewise, one can define a generalized time reversal operator with properties analogous to CP4 while transforming the time coordinate from $t$ to $-t$ .\",\n \"Consequently, there is no contradiction with the CPT theorem, which remains intact.\",\n \"A very interesting feature of the IS scalar potential is that the symmetry requires some of its coefficients to be complex (in a particular Higgs basis).\",\n \"Moreover, for generic choices of the scalar potential parameters, there is no scalar basis transformation within the family of Higgs bases, of the form given by Eqs.\",\n \"(REF )–(REF ), that can transform the scalar potential into a new potential with only real coefficients.\",\n \"This is a surprising result in light of the statement that the IS potential is CP-conserving.\",\n \"The IS model conserves CP independently of the existence or nonexistence of a real Higgs basis, although in the case where no real Higgs basis exists, the IS model is only invariant with respect to the generalized CP symmetry, CP4 (whereas CP2 is not a symmetry of the IS scalar potential).\",\n \"Nevertheless, any CP-violating observable of the IS model must vanish.\",\n \"For example, the contributions to the CP-violating form factors of the effective $ZZZ$ and $ZWW$ vertices generated in the IS model must exactly cancel.\",\n \"As a check of this statement, we confirmed this cancellation up to three-loop order in the IS model with no real Higgs basis.\",\n \"We identified a physical quartic scalar interaction made up of an odd number of mass-degenerate neutral scalar states (e.g., $P^3 Q$ and $Q^3 P$ ) that is consistent with the CP4 symmetry, but would vanish if the IS scalar potential exhibits a CP2 symmetry that commutes with CP4.\",\n \"This leaves open the possibility of the existence of a CP2 symmetry that does not commute with CP4.\",\n \"However, we were unable to find an observable quantity of the IS model that can distinguish between the presence or absence of a noncommuting CP2 symmetry.\",\n \"Finally, we stress that the possibility of a scalar potential and vacuum that is invariant with respect to a generalized CP symmetry without the existence of a real basis appears to be inexorably connected with the existence of mass-degenerate scalar states.\",\n \"We strongly suspect that this connection, which has been demonstrated in this paper for the IS model, is applicable more generally to any multi-Higgs doublet model.\",\n \"If true, then the existence of a generalized CP symmetry in the absence of mass degenerate scalars necessarily implies the presence of a conventional CP symmetry; i.e., the existence of a real basis of scalar fields in which the CP symmetry corresponds simply to conjugation of the scalar fields.\"\n ],\n [\n \"Acknowledgments\",\n \"We are very grateful to Igor Ivanov for many fruitful and enlightening conversations.\",\n \"We also thank Gustavo C. Branco, Nick Mavromatos, Palash B. Pal, Apostolos Pilaftsis, and Graham Ross with whom several aspects of this work were discussed.\",\n \"H.E.H.\",\n \"and P.O.\",\n \"acknowledge the Galileo Galilei Institute for Theoretical Physics, where this work was initiated.\",\n \"H.E.H., P.O.\",\n \"and M.N.R.\",\n \"also appreciate the hospitality of the CERN Theory group where some of this work was carried out, and M.N.R also acknowledges partial support from CERN.\",\n \"H.E.H.\",\n \"is supported in part by the U.S. Department of Energy grant number DE-SC0010107, and in part by the grant H2020-MSCA-RISE-2014 No.\",\n \"645722 (NonMinimalHiggs).\",\n \"P.O.\",\n \"is supported by the Research Council of Norway.\",\n \"The work of M.N.R.\",\n \"was partially supported by Fundação para a Ciência e a Tecnologia (FCT, Portugal) through the projects CFTP-FCT Unit 777 (UID/FIS/00777/2013), CERN/FIS-PAR/0004/2017 and PTDC/FIS-PAR/29436/2017 which are partially funded through POCTI (FEDER), COMPETE, QREN and EU.\",\n \"H.E.H.\",\n \"and M.N.R.\",\n \"benefited from discussions that took place at the University of Warsaw during visits supported by the HARMONIA project of the National Science Centre, Poland, under contract UMO-2015/18/M/ST2/00518 (2016–2019).\",\n \"M.N.R.\",\n \"and P.O.\",\n \"also thank the University of Bergen and CFTP/IST/University of Lisbon, where collaboration visits took place.\"\n ],\n [\n \"The Ivanov Silva model revisited\",\n \"Consider the most general 2HDM with a scalar potential as specified in Eq.\",\n \"(REF ).\",\n \"Including the kinetic energy terms with SU(2)$\\\\times $ U(1) gauge covariant derivatives, the 2HDM [after electroweak symmetry breaking under the assumption that the vacuum preserves U(1)$_{\\\\rm EM}$ ] consists of a model of two scalar doublets coupled to the gauge bosons, $W^\\\\pm $ , $Z$ and $\\\\gamma $ .\",\n \"We shall ignore the couplings of the bosonic sector of the 2HDM to the fermions of the SM in the following discussion.\",\n \"We now ask the following question.\",\n \"Does the bosonic Lagrangian conserve CP?\",\n \"For CP to be conserved, two conditions must be verified.\",\n \"First, the scalar potential must exhibit explicit CP conservation.\",\n \"Second, the vacuum must conserve CP.\",\n \"If the former is true but the latter is false, we say that CP is spontaneously broken.\",\n \"However, in this discussion, we are interested in whether both explicit and spontaneous CP violation are absent.\",\n \"In the 2HDM, the answer to this question is simple.\",\n \"We first transform to the Higgs basis and examine the scalar potential given in Eq.\",\n \"(REF ).\",\n \"The Higgs basis is unique up to a possible rephasing of the Higgs basis field, $H_2\\\\rightarrow e^{i\\\\chi }H_2$ .\",\n \"Then, CP is conserved if and only if there exists a choice of $\\\\chi $ such that all Higgs basis scalar potential parameters are real.\",\n \"In the discussion above, we have not specified in detail how the scalar fields transform under a CP transformation.\",\n \"Starting from the generic $\\\\Phi _1$ –$\\\\Phi _2$ basis employed in writing Eq.\",\n \"(REF ), the conventional CP transformation corresponds to conjugation, $\\\\Phi _i^{\\\\rm CP}= \\\\Phi _i^{\\\\scriptscriptstyle {\\\\bigstar }}$ .\",\n \"However, this is a basis-dependent statement.\",\n \"Indeed, one is always free to change the basis, $\\\\Phi _i^\\\\prime =U_{ij}\\\\Phi _j$ , where $U\\\\in $ U(2).\",\n \"In the new basis, $\\\\Phi ^{\\\\prime \\\\,CP}_i=X_{ij}\\\\Phi ^{\\\\prime \\\\,CP}_j$ , where $X=UU^T$ is a symmetric unitary matrix.\",\n \"More generally, we can consider the generalized CP transformation, $\\\\Phi _i^{\\\\rm CP}(x,t)= X_{ij}\\\\Phi _j^{\\\\scriptscriptstyle {\\\\bigstar }}(-x,t)\\\\,,$ where $X\\\\in $ U(2).Note that it is not consistent to simply define the CP transformation of a multi-Higgs doublet model without including the matrix $X$ in Eq.\",\n \"(REF ), since the form of the CP transformation depends on the choice of the scalar basis, as noted above.\",\n \"Consequently, some authors prefer to call this transformation a general CP transformation rather than generalized CP transformation.\",\n \"If $X$ is both unitary and symmetric, then one can find a basis in which the CP transformation is simply conjugation.As shown in Appendix D of Ref.\",\n \"[80] [see the Lemma below eq.\",\n \"(D.3.1)], for any symmetric unitary matrix $X$ , there exists a unitary matrix $U$ such that $X=UU^T$ .\",\n \"In Ref.\",\n \"[56], it is shown that in the 2HDM there are three possible classes of generalized CP transformations (GCPs): (i) $X$ is unitary and symmetric; (ii) $X$ is unitary and antisymmetric; and (iii) $X$ is unitary but is neither symmetric nor antisymmetric.\",\n \"Clearly, no basis change can convert a GCP transformation of types (ii) or (iii) into the transformation of the field into its conjugate.\",\n \"Nevertheless, as shown in Ref.\",\n \"[56], any 2HDM scalar potential that is invariant under GCP transformations of types (ii) or (iii) is also separately invariant under a GCP transformation of type (i).\",\n \"Do the above results generalize to arbitrary Higgs sectors?\",\n \"In particular, consider an extended Higgs sector with $N$ hypercharge-one, complex doublets (denoted henceforth as the NHDM).\",\n \"To address the question of CP invariance, we transform to the so-called charged Higgs basis defined in Ref. [81].\",\n \"If the scalar fields of the charged Higgs basis are denoted by $H_i$ ($i=1,\\\\ldots ,n$ ), then $\\\\langle H_1^0 \\\\rangle =v/\\\\sqrt{2}$ , $\\\\langle H_j^0 \\\\rangle =0$ for $j=2,3,\\\\ldots ,n$ , and the fields $H_j^\\\\pm $ (for $j=2,3,\\\\ldots ,n$ ) are the physical, mass-eigenstate charged Higgs fields.\",\n \"Note that for $N=2$ , the Higgs basis and the charged Higgs basis coincide.\",\n \"For $N\\\\ge 3$ , consider first the case in which the physical charged Higgs bosons are mass non-degenerate.\",\n \"In this case, the charged Higgs basis is uniquely defined up to a possible rephasing, $H_j\\\\rightarrow e^{i\\\\chi _j}H_j$ .\",\n \"In this case, CP is conserved if and only if there exist a choice of the $\\\\chi _j$ such that all charged Higgs basis scalar potential parameters are real.\",\n \"This generalizes the result of the 2HDM quoted above.\",\n \"If there exist mass degeneracies among the physical charged Higgs fields, then one must re-evaluate the conditions for CP invariance.\",\n \"To simplify the discussion, we focus on the case of $N=3$ , in which the two physical charged Higgs bosons are mass degenerate.\",\n \"In this case, the charged Higgs basis is unique up to a U(2) transformation of the charged Higgs basis fields $H_2$ and $H_3$ .\",\n \"Ivanov and Silva[30] constructed a 3HDM whose scalar potential and vacuum are invariant under a generalized CP transformation such that $({\\\\rm GCP})^2\\\\ne {1}$ , where 1 is the identity operator.\",\n \"Moreover, some of the scalar potential parameters of the charged Higgs basis of the Ivanov–Silva (IS) scalar potential are complex, and no U(2) transformation of the charged Higgs basis fields $H_2$ and $H_3$ can be performed to remove all the complex phases.\",\n \"Hence, the IS scalar potential is not invariant under a separate GCP transformation that is equivalent to conjugation in another basis, in contrast to the corresponding 2HDM result.\",\n \"Ivanov and Silva denote the GCP transformation of the IS scalar potential by CP4, since it has the property that $({\\\\rm CP}4)^4={1}$ and $({\\\\rm CP}4)^2\\\\ne {1}$ .\",\n \"Indeed, one consequence of the CP4 symmetry of the scalar potential and the vacuum is the mass degeneracy of the physical charged Higgs bosons, as well as two additional mass degeneracies among pairs of neutral Higgs bosons.\",\n \"In this Appendix, we consider the 3HDM scalar potential of the Ivanov and Silva model and examine some of its properties.\"\n ],\n [\n \"The IS scalar potential\",\n \"Consider the 3HDM consisting of three hypercharge-one, complex doublet fields, $\\\\phi _i$ ($i=1,2,3$ ).\",\n \"In the Higgs basis, the form of the scalar potential proposed initially by Ivanov and Silva (IS) in Ref.\",\n \"[30] is fixed by imposing the following generalized CP symmetry, $ \\\\phi _i \\\\rightarrow W_{ij}\\\\phi _j^{\\\\scriptscriptstyle {\\\\bigstar }}, \\\\quad W=\\\\begin{pmatrix}1 & \\\\phantom{-}0 & \\\\phantom{-}0 \\\\\\\\0 & \\\\phantom{-}0 & \\\\phantom{-}i \\\\\\\\0 & -i & \\\\phantom{-}0\\\\end{pmatrix},$ which has the property that applying it four times yields the identity operator.\",\n \"This is the CP4 symmetry transformation noted above.\",\n \"The resulting IS scalar potential is given by $V=V_0+V_1\\\\,,$ with $V_0&=-m_{11}^2(\\\\phi _1^\\\\dagger \\\\phi _1)-m_{22}^2(\\\\phi _2^\\\\dagger \\\\phi _2+\\\\phi _3^\\\\dagger \\\\phi _3)+\\\\lambda _1(\\\\phi _1^\\\\dagger \\\\phi _1)^2+\\\\lambda _2[(\\\\phi _2^\\\\dagger \\\\phi _2)^2+(\\\\phi _3^\\\\dagger \\\\phi _3)^2]+\\\\lambda _3^\\\\prime (\\\\phi _2^\\\\dagger \\\\phi _2)(\\\\phi _3^\\\\dagger \\\\phi _3)\\\\nonumber \\\\\\\\& \\\\qquad +\\\\lambda _3(\\\\phi _1^\\\\dagger \\\\phi _1)[(\\\\phi _2^\\\\dagger \\\\phi _2)+(\\\\phi _3^\\\\dagger \\\\phi _3)]+\\\\lambda _4^\\\\prime (\\\\phi _2^\\\\dagger \\\\phi _3)(\\\\phi _3^\\\\dagger \\\\phi _2)+\\\\lambda _4[(\\\\phi _1^\\\\dagger \\\\phi _2)(\\\\phi _2^\\\\dagger \\\\phi _1)+(\\\\phi _1^\\\\dagger \\\\phi _3)(\\\\phi _3^\\\\dagger \\\\phi _1)],\\\\\\\\[6pt]V_1&=\\\\lambda _5(\\\\phi _3^\\\\dagger \\\\phi _1)(\\\\phi _2^\\\\dagger \\\\phi _1)+\\\\tfrac{1}{2}\\\\lambda _6[(\\\\phi _2^\\\\dagger \\\\phi _1)^2-(\\\\phi _1^\\\\dagger \\\\phi _3)^2]+\\\\lambda _8(\\\\phi _2^\\\\dagger \\\\phi _3)^2+\\\\lambda _9(\\\\phi _2^\\\\dagger \\\\phi _3)[(\\\\phi _2^\\\\dagger \\\\phi _2)-(\\\\phi _3^\\\\dagger \\\\phi _3)] +{\\\\rm h.c.}$ The hermiticity of the scalar potential implies that the coefficients of $V_0$ are real.\",\n \"In contrast, the coefficients of $V_1$ are potentially complex.\",\n \"However, having imposed the CP4 symmetry given by Eq.\",\n \"(REF ), we see that $\\\\lambda _5$ is real.\",\n \"Under the CP4 symmetry specified in Eq.\",\n \"(REF ), the gauge-invariant bilinear quantities, $B_{ij}\\\\equiv \\\\phi _i^\\\\dagger \\\\phi _j$ , transform as follows: $B_{11} &\\\\rightarrow B_{11}, \\\\\\\\B_{22}&\\\\rightarrow B_{33} &\\\\quad B_{33}&\\\\rightarrow B_{22}, \\\\\\\\B_{12}&\\\\rightarrow i B_{31}, &\\\\quad B_{21}&\\\\rightarrow -i B_{13}, \\\\\\\\B_{13}&\\\\rightarrow -i B_{21}, &\\\\quad B_{31}&\\\\rightarrow i B_{12}, \\\\\\\\B_{23}&\\\\rightarrow -B_{23}, &\\\\quad B_{32}&\\\\rightarrow -B_{32}.$ It follows that $V$ given by Eqs.\",\n \"(REF )–(REF ), with $\\\\lambda _6$ , $\\\\lambda _8$ and $\\\\lambda _9$ complex and all other scalar potential parameters real, is the most general 3HDM potential that is invariant under the CP4 transformation given in (REF ).\",\n \"Without loss of generality, one can furthermore assume that $\\\\lambda _6$ is real after an appropriate rephasing of the scalar fields $\\\\phi _2$ and $\\\\phi _3$ .\",\n \"At this stage, we have not yet found the minimum of the scalar potential and determined whether the CP4 symmetry is respected by the vacuum.\",\n \"There exist a range of scalar potential parameters in which the vacuum preserves U(1)$_{\\\\rm EM}$ , in which case one can decompose the scalar doublets as, $ \\\\phi _i=\\\\left(\\\\begin{array}{c}\\\\varphi _i^+\\\\\\\\ (v_i+\\\\eta _i+i \\\\chi _i)/\\\\sqrt{2}\\\\end{array}\\\\right), \\\\quad i=1,2,3.$ In particular, the vacuum conserves CP4 if the minimum of the scalar potential corresponds to $(v_1,v_2,v_3)=(v,0,0)$[30].\",\n \"Indeed, there exists a range of scalar potential parameters for which this corresponds to the global minimum, in which case the value of $m_{11}^2$ is fixed by the scalar potential minimum condition to be $m_{11}^2=\\\\lambda _1 v^2.$ In this case, the scalar field basis employed in Eqs.\",\n \"(REF ) and (REF ) is the Higgs basis, with the freedom to perform U(2) transformations on $\\\\lbrace \\\\phi _2,\\\\phi _3\\\\rbrace $ .\",\n \"We shall take advantage of this freedom in the next two subsections.\",\n \"It is now straightforward to determine the scalar mass spectrum of the IS model.\",\n \"Since we are in the Higgs basis, we can immediately identify the Goldstone bosons, $\\\\varphi _1^\\\\pm =G^\\\\pm $ and $\\\\chi _1=G^0$ .\",\n \"Moreover, $\\\\eta _1$ is a neutral mass-eigenstate with mass $m^2_{\\\\eta _1}=2\\\\lambda _1 v^2$ , whose tree-level couplings to the gauge bosons and to itself are precisely those of the SM Higgs boson (corresponding to the exact alignment limit).\",\n \"Indeed, this is analogous to the IDM in which $\\\\phi _1$ is equivalent to the hypercharge-one, complex scalar doublet of the SM and $\\\\phi _2$ and $\\\\phi _3$ are inert doublets.\",\n \"The two physical charged Higgs fields, $\\\\varphi _2^\\\\pm $ and $\\\\varphi _3^\\\\pm $ , are mass-degenerate, $ m_{\\\\varphi _2^\\\\pm ,\\\\varphi _3^\\\\pm }^2=\\\\tfrac{1}{2}\\\\lambda _3 v^2 -m_{22}^2.$ The neutral scalar spectrum consist of the SM-like Higgs boson $\\\\eta _1$ and a pair of mass degenerate neutral scalars made up of linear combinations of the $\\\\eta _{2,3}$ and $\\\\chi _{2,3}$ , with masses given by[30], $M^2= a+\\\\sqrt{b^2+c^2}\\\\,,\\\\qquad \\\\quad m^2= a-\\\\sqrt{b^2+c^2}\\\\,,$ where $a=\\\\tfrac{1}{2}(\\\\lambda _3+\\\\lambda _4)v^2-m_{22}^2, \\\\qquad b=\\\\tfrac{1}{2}\\\\lambda _6 v^2, \\\\qquad c=\\\\tfrac{1}{2}\\\\lambda _5 v^2.$\"\n ],\n [\n \"A simpler form for the IS scalar potential\",\n \"Given the IS scalar potential in the Higgs basis, we still have the freedom to perform a U(2) transformation on $\\\\lbrace \\\\phi _2,\\\\phi _3\\\\rbrace $ .\",\n \"It is possible to remove the $\\\\lambda _5$ term in Eq.\",\n \"(REF ) by the following basis transformation, $\\\\bar{\\\\phi }_i=U_{ij}\\\\phi _j,$ where $U=\\\\begin{pmatrix}1 & 0 & 0 \\\\\\\\0 & \\\\phantom{-}\\\\cos \\\\theta & -\\\\sin \\\\theta \\\\\\\\0 & \\\\phantom{-}\\\\sin \\\\theta & \\\\phantom{-}\\\\cos \\\\theta \\\\end{pmatrix}\\\\,,$ with $0\\\\le \\\\theta \\\\le \\\\pi $ .\",\n \"With respect to the new basis, the CP4 transformation specified in Eq.\",\n \"(REF ) is given by, $\\\\bar{\\\\phi }_i\\\\rightarrow V_{ij}\\\\bar{\\\\phi }^{\\\\scriptscriptstyle {\\\\bigstar }}_j\\\\,, \\\\qquad \\\\text{where $V=UWU^T$}.$ Using the form for $U$ given in Eq.\",\n \"(REF ), it follows that $V=W$ .\",\n \"Thus, in this new basis, the IS symmetry takes the same form as in the original basis.\",\n \"When the scalar potential is expressed in terms of the fields $\\\\bar{\\\\phi }_i$ , the resulting scalar potential parameters will be denoted by $\\\\bar{m}^2_{ii}$ and $\\\\bar{\\\\lambda }_i$ .\",\n \"It is straightforward to obtain expressions for $\\\\bar{m}_{11}^2$ , $\\\\bar{m}_{22}^2$ and the $\\\\bar{\\\\lambda }_i$ in terms of the scalar potential parameters defined in Eq.\",\n \"(REF ).\",\n \"In particular, $\\\\bar{m}_{11}^2=m_{11}^2$ , $\\\\bar{m}_{22}^2=m_{22}^2$ , and $\\\\bar{\\\\lambda }_i=\\\\lambda _i$ for $i=1,3$ and 4.\",\n \"Next, we note that the CP4 symmetry does not mandate that $\\\\bar{\\\\lambda }_6$ is real.\",\n \"However, it is straightforward to check that $\\\\operatorname{Im}\\\\bar{\\\\lambda }_6=\\\\operatorname{Im}\\\\lambda _6$ .\",\n \"Having previously chosen $\\\\lambda _6$ real (after an appropriate rephasing of $\\\\phi _2$ and $\\\\phi _3$ ), it follows that $\\\\bar{\\\\lambda }_6$ is also real.\",\n \"The remaining transformed coefficients are given by, $\\\\bar{\\\\lambda }_2&= \\\\lambda _2-\\\\tfrac{1}{2}\\\\sin ^2 2\\\\theta [\\\\lambda _2-\\\\operatorname{Re}\\\\lambda _8-\\\\tfrac{1}{2}(\\\\lambda _3^\\\\prime +\\\\lambda _4^\\\\prime )]-\\\\sin 2\\\\theta \\\\cos 2\\\\theta \\\\operatorname{Re}\\\\lambda _9\\\\,, \\\\\\\\\\\\bar{\\\\lambda }^\\\\prime _3&=\\\\lambda ^\\\\prime _3+\\\\sin ^2 2\\\\theta [\\\\lambda _2-\\\\operatorname{Re}\\\\lambda _8-\\\\tfrac{1}{2}(\\\\lambda _3^\\\\prime +\\\\lambda _4^\\\\prime )]+2\\\\sin 2\\\\theta \\\\cos 2\\\\theta \\\\operatorname{Re}\\\\lambda _9\\\\,, \\\\\\\\\\\\bar{\\\\lambda }^\\\\prime _4&=\\\\lambda ^\\\\prime _4+\\\\sin ^2 2\\\\theta [\\\\lambda _2-\\\\operatorname{Re}\\\\lambda _8-\\\\tfrac{1}{2}(\\\\lambda _3^\\\\prime +\\\\lambda _4^\\\\prime )]+2\\\\sin 2\\\\theta \\\\cos 2\\\\theta \\\\operatorname{Re}\\\\lambda _9\\\\,, \\\\\\\\\\\\bar{\\\\lambda }_5&=\\\\lambda _5\\\\cos 2\\\\theta +\\\\lambda _6\\\\sin 2\\\\theta \\\\,,\\\\\\\\\\\\bar{\\\\lambda }_6&=\\\\lambda _6\\\\cos 2\\\\theta -\\\\lambda _5\\\\sin 2\\\\theta \\\\,,\\\\\\\\\\\\operatorname{Re}\\\\bar{\\\\lambda }_8&=\\\\operatorname{Re}\\\\lambda _8+\\\\tfrac{1}{2}\\\\sin ^2 2\\\\theta [\\\\lambda _2-\\\\operatorname{Re}\\\\lambda _8-\\\\tfrac{1}{2}(\\\\lambda _3^\\\\prime +\\\\lambda _4^\\\\prime )]+\\\\sin 2\\\\theta \\\\cos 2\\\\theta \\\\operatorname{Re}\\\\lambda _9\\\\,,\\\\\\\\\\\\operatorname{Re}\\\\bar{\\\\lambda }_9&=(1-2\\\\sin ^2 2\\\\theta )\\\\operatorname{Re}\\\\lambda _9+\\\\sin 2\\\\theta \\\\cos 2\\\\theta [\\\\lambda _2-\\\\operatorname{Re}\\\\lambda _8-\\\\tfrac{1}{2}(\\\\lambda _3^\\\\prime +\\\\lambda _4^\\\\prime )]\\\\,,\\\\\\\\\\\\operatorname{Im}\\\\bar{\\\\lambda }_8&=\\\\cos 2\\\\theta \\\\operatorname{Im}\\\\lambda _8+\\\\sin 2\\\\theta \\\\operatorname{Im}\\\\lambda _9\\\\,,\\\\\\\\\\\\operatorname{Im}\\\\bar{\\\\lambda }_9&=\\\\cos 2\\\\theta \\\\operatorname{Im}\\\\lambda _9-\\\\sin 2\\\\theta \\\\operatorname{Im}\\\\lambda _8\\\\,.$ One can now choose the angle $\\\\theta $ such that $\\\\bar{\\\\lambda }_5=0$ .Note that a different choice of $\\\\tan 2\\\\theta $ could have been made to set either $\\\\bar{\\\\lambda }_6=0$ , $\\\\operatorname{Im}\\\\bar{\\\\lambda }_8=0$ or $\\\\operatorname{Im}\\\\bar{\\\\lambda }_9=0$ .\",\n \"That is, one can always perform a change of Higgs basis to remove one degree of freedom from the coefficients of the IS scalar potential.\",\n \"This yields $\\\\tan 2\\\\theta =-\\\\lambda _5/\\\\lambda _6$ .\",\n \"Then, $\\\\sin 2\\\\theta $ and $\\\\cos 2\\\\theta $ are determined up to an overall sign.\",\n \"Introducing the following notation, $\\\\lambda _{56}\\\\equiv \\\\sqrt{\\\\lambda _5^2+\\\\lambda _6^2}\\\\,,$ we choose the angle $\\\\theta $ such that, $\\\\sin 2\\\\theta =\\\\frac{\\\\lambda _5}{\\\\lambda _{56}}\\\\,,\\\\qquad \\\\quad \\\\cos 2\\\\theta =-\\\\,\\\\frac{\\\\lambda _6}{\\\\lambda _{56}}\\\\,.$ Thus, the $\\\\lambda _5$ -term in Eq.\",\n \"(REF ) is actually redundant.A similar simplification was presented recently in Ref. [70].\",\n \"Inserting the results of Eq.\",\n \"(REF ) back into Eq.\",\n \"(REF ) yields $\\\\bar{\\\\lambda }_5=0$ and, $\\\\bar{\\\\lambda }_2&=\\\\frac{\\\\lambda _5 \\\\left[\\\\lambda _5 \\\\left(\\\\lambda ^\\\\prime _3+\\\\lambda ^\\\\prime _4+2 \\\\operatorname{Re}\\\\lambda _8\\\\right)+4 \\\\lambda _6 \\\\operatorname{Re}\\\\lambda _9\\\\right]+2 \\\\lambda _2 \\\\left(\\\\lambda _5^2+2 \\\\lambda _6^2\\\\right)}{4 \\\\lambda ^2_{56}},\\\\\\\\\\\\bar{\\\\lambda }^\\\\prime _3&=\\\\frac{\\\\lambda _5^2 \\\\left(\\\\lambda ^\\\\prime _3-\\\\lambda ^\\\\prime _4-2 \\\\operatorname{Re}\\\\lambda _8\\\\right)+2 \\\\lambda _6^2 \\\\lambda ^\\\\prime _3-4 \\\\lambda _5 \\\\lambda _6 \\\\operatorname{Re}\\\\lambda _9+2 \\\\lambda _2 \\\\lambda _5^2}{2\\\\lambda ^2_{56}},\\\\\\\\\\\\bar{\\\\lambda }^\\\\prime _4&=\\\\frac{\\\\lambda _5^2 \\\\left(-\\\\lambda ^\\\\prime _3+\\\\lambda ^\\\\prime _4-2 \\\\operatorname{Re}\\\\lambda _8\\\\right)+2 \\\\lambda _6^2 \\\\lambda ^\\\\prime _4-4 \\\\lambda _5 \\\\lambda _6 \\\\operatorname{Re}\\\\lambda _9+2 \\\\lambda _2 \\\\lambda _5^2}{2 \\\\lambda ^2_{56}},\\\\\\\\\\\\bar{\\\\lambda }_6&=-\\\\lambda _{56},\\\\\\\\\\\\operatorname{Re}\\\\bar{\\\\lambda }_8&=\\\\frac{-\\\\lambda _5^2 \\\\left(\\\\lambda ^\\\\prime _3+\\\\lambda ^\\\\prime _4-2 \\\\operatorname{Re}\\\\lambda _8\\\\right)-4 \\\\lambda _5 \\\\lambda _6 \\\\operatorname{Re}\\\\lambda _9+4 \\\\lambda _6^2 \\\\operatorname{Re}\\\\lambda _8+2 \\\\lambda _2 \\\\lambda _5^2}{4 \\\\lambda ^2_{56}},\\\\\\\\\\\\operatorname{Re}\\\\bar{\\\\lambda }_9&=\\\\frac{\\\\lambda _5\\\\lambda _6 \\\\left(\\\\lambda ^\\\\prime _3+\\\\lambda ^\\\\prime _4+2 \\\\operatorname{Re}\\\\lambda _8\\\\right)-2 \\\\lambda _5^2 \\\\operatorname{Re}\\\\lambda _9+2 \\\\lambda _6^2 \\\\operatorname{Re}\\\\lambda _9-2 \\\\lambda _2 \\\\lambda _5 \\\\lambda _6}{2 \\\\lambda ^2_{56}},\\\\\\\\\\\\operatorname{Im}\\\\bar{\\\\lambda }_8&=\\\\frac{\\\\lambda _5 \\\\operatorname{Im}\\\\lambda _9-\\\\lambda _6 \\\\operatorname{Im}\\\\lambda _8}{\\\\lambda _{56}},\\\\\\\\\\\\operatorname{Im}\\\\bar{\\\\lambda }_9&=\\\\frac{-\\\\lambda _5 \\\\operatorname{Im}\\\\lambda _8-\\\\lambda _6 \\\\operatorname{Im}\\\\lambda _9}{\\\\lambda _{56}}.$ An additional feature of the IS scalar potential with $\\\\bar{\\\\lambda }_5=0$ is that the real and imaginary parts of the neutral fields $\\\\bar{\\\\phi }_2^0$ and $\\\\bar{\\\\phi }_3^0$ are mass eigenstates.\",\n \"That is, the neutral squared-mass matrices are already diagonal in the $\\\\lbrace \\\\bar{\\\\phi }_1,\\\\bar{\\\\phi }_2,\\\\bar{\\\\phi }_3\\\\rbrace $ basis.\",\n \"In particular, the lightest of the two mass-degenerate states lives in the imaginary part of $\\\\bar{\\\\phi }_2$ and in the real part of $\\\\bar{\\\\phi }_3$ .\",\n \"The heaviest of the two mass-degenerate neutral states lives in the real part of $\\\\bar{\\\\phi }_2$ and in the imaginary part of $\\\\bar{\\\\phi }_3$ .\",\n \"It is convenient to make an additional field redefinition, $\\\\bar{\\\\phi }_3\\\\rightarrow i\\\\bar{\\\\phi }_3$ .\",\n \"The effect of this modification is to modify $\\\\bar{V}_1$ by flipping the sign of $(\\\\bar{\\\\phi }^\\\\dagger _1\\\\bar{\\\\phi }_3)^2$ in the term proportional to $\\\\bar{\\\\lambda }_6$ and to transform $\\\\bar{\\\\lambda }_8\\\\rightarrow -\\\\bar{\\\\lambda }_8$ and $\\\\bar{\\\\lambda }_9\\\\rightarrow -i\\\\bar{\\\\lambda }_9$ .\",\n \"To make contact with the $H23$ -basis employed in Eq.\",\n \"(REF ), we define, $H_1=\\\\bar{\\\\phi }_1\\\\,,\\\\qquad H_2=\\\\bar{\\\\phi }_2\\\\,,\\\\qquad H_3=i\\\\bar{\\\\phi }_3\\\\,,$ corresponding to a basis change, $H_i\\\\rightarrow \\\\widetilde{U}_{ij}\\\\bar{\\\\phi }_j$ , with $\\\\widetilde{U}={\\\\rm diag}(1\\\\,,\\\\,1\\\\,,\\\\,i)$ .\",\n \"Note that the heaviest mass degenerate neutral fields now reside in the real part of the neutral components of $H_2$ and $H_3$ , and the lightest mass degenerate neutral fields reside in the imaginary part of the neutral components of $H_2$ and $H_3$ .\",\n \"When expressed in the $H23$ -basis, the IS scalar potential is given by, $\\\\mathcal {V}_{\\\\rm IS}= \\\\mathcal {V}_{\\\\rm RIDM}+Z_3^\\\\prime (H_2^\\\\dagger H_2)(H_3^\\\\dagger H_3)+Z_4^\\\\prime (H_2^\\\\dagger H_3)(H_3^\\\\dagger H_2) +\\\\bigl [Z_8 (H_2^\\\\dagger H_3)^2+Z_9(H_2^\\\\dagger H_3)(H_2^\\\\dagger H_2-H_3^\\\\dagger H_3)+{\\\\rm h.c.}\\\\bigr ]\\\\,,$ where $\\\\mathcal {V}_{\\\\rm RIDM}$ is given by Eq.\",\n \"(REF ), with $Z_8$ and $Z_9$ potentially complex and all other scalar potential parameters real.\",\n \"Eq.\",\n \"(REF ) is the version of the IS scalar potential employed in section REF .\",\n \"To make contact with the previous notation used above, we note that $Y_1&=& -m_{11}^2\\\\,,\\\\qquad Y_2=-m_{22}^2\\\\,,\\\\qquad Z_1=2\\\\lambda _1\\\\,,\\\\qquad Z_2=2\\\\bar{\\\\lambda }_2\\\\,,\\\\qquad Z_3=\\\\lambda _3\\\\,,\\\\qquad Z_4=\\\\lambda _4 \\\\nonumber \\\\\\\\Z_3^\\\\prime &=&\\\\bar{\\\\lambda }_3^\\\\prime -2\\\\bar{\\\\lambda }_2\\\\,,\\\\qquad Z_4^\\\\prime =\\\\bar{\\\\lambda }_4^\\\\prime \\\\,,\\\\qquad Z_5=\\\\bar{\\\\lambda }_6\\\\,,\\\\qquad Z_8=-\\\\bar{\\\\lambda }_8\\\\,,\\\\qquad Z_9=-i\\\\bar{\\\\lambda }_9\\\\,.$ The corresponding CP4 symmetry transformation now takes the form $H_i\\\\rightarrow X_{ij}H_j^{\\\\scriptscriptstyle {\\\\bigstar }}\\\\,,\\\\qquad \\\\text{where $X=\\\\widetilde{U}W\\\\widetilde{U}^T=\\\\begin{pmatrix} 1 & \\\\phantom{-}0 & \\\\phantom{-}0 \\\\\\\\ 0 & \\\\phantom{-}0 & -1 \\\\\\\\ 0 & \\\\phantom{-}1 & \\\\phantom{-}0\\\\end{pmatrix}$}\\\\,,$ as indicated in Eq.\",\n \"(REF ).\"\n ],\n [\n \"Non-existence of a real Higgs basis\",\n \"Consider the IS scalar potential [cf. Eq.\",\n \"(REF ) with $\\\\mathcal {V}_{\\\\rm RIDM}$ given by Eq.\",\n \"(REF )] expressed in terms of the Higgs basis of scalar doublet fields, $\\\\lbrace H_1,H_2,H_3\\\\rbrace $ , where $\\\\langle H_1^0 \\\\rangle \\\\ne 0$ and the vevs of the other two doublet fields vanish.\",\n \"The coefficients $Z_8$ and $Z_9$ are potentially complex and all other scalar potential parameters are real.\",\n \"Recall that the Higgs basis is unique only up to an arbitrary U(2) transformation of $\\\\lbrace H_2, H_3\\\\rbrace $ .\",\n \"Is it possible to transform to a new Higgs basis in which all the IS scalar potential parameters are real?\",\n \"Such a Higgs basis, if it exists, is called a real Higgs basis.\",\n \"The most general basis transformation that preserves the general class of Higgs bases is given (in block diagonal form) by, $\\\\begin{pmatrix} \\\\bar{H}_1 \\\\\\\\ \\\\bar{H}_{23}\\\\end{pmatrix}=\\\\begin{pmatrix} 1 & \\\\,\\\\,0 \\\\\\\\ 0 &\\\\,\\\\, \\\\widetilde{V}\\\\end{pmatrix} \\\\begin{pmatrix} H_1 \\\\\\\\ H_{23}\\\\end{pmatrix}\\\\,,$ where $H_{23}\\\\equiv \\\\begin{pmatrix} H_2\\\\\\\\ H_3\\\\end{pmatrix}\\\\,,\\\\qquad \\\\quad \\\\bar{H}_{23}\\\\equiv \\\\begin{pmatrix} \\\\bar{H}_2\\\\\\\\ \\\\bar{H}_3\\\\end{pmatrix}\\\\,,$ and $\\\\widetilde{V}$ is the most general U(2) matrix, $\\\\widetilde{V}=e^{i\\\\psi /2}\\\\begin{pmatrix} e^{i\\\\alpha }\\\\cos \\\\phi & -e^{-i\\\\beta }\\\\sin \\\\phi \\\\\\\\ e^{i\\\\beta }\\\\sin \\\\phi & \\\\phantom{-}e^{-i\\\\alpha }\\\\cos \\\\phi \\\\end{pmatrix}\\\\,,$ where $0\\\\le \\\\phi <\\\\pi $ , $-\\\\pi <\\\\psi \\\\le \\\\pi $ , $0\\\\le \\\\alpha \\\\le \\\\pi $ and $0\\\\le \\\\beta \\\\le \\\\pi $ .\",\n \"Applying Eq.\",\n \"(REF ) to the IS scalar potential given in Eq.\",\n \"(REF ) yields $\\\\mathcal {V}_{\\\\rm IS}&=&Y_1 \\\\bar{H}_1^\\\\dagger \\\\bar{H}_1+ Y_2 \\\\left( \\\\bar{H}_2^\\\\dagger \\\\bar{H}_2 + \\\\bar{H}_3^\\\\dagger \\\\bar{H}_3\\\\right)+\\\\tfrac{1}{2}Z_1( \\\\bar{H}_1^\\\\dagger \\\\bar{H}_1)^2+\\\\tfrac{1}{2}\\\\bar{Z}_2( \\\\bar{H}_2^\\\\dagger \\\\bar{H}_2+ \\\\bar{H}_3^\\\\dagger \\\\bar{H}_3)^2 \\\\\\\\&&+Z_3(\\\\bar{H}_1^\\\\dagger \\\\bar{H}_1)( \\\\bar{H}_2^\\\\dagger \\\\bar{H}_2+ \\\\bar{H}_3^\\\\dagger \\\\bar{H}_3)+Z_4\\\\bigl [( \\\\bar{H}_1^\\\\dagger \\\\bar{H}_2)( \\\\bar{H}_2^\\\\dagger \\\\bar{H}_1)+( \\\\bar{H}_1^\\\\dagger \\\\bar{H}_3)( \\\\bar{H}_3^\\\\dagger \\\\bar{H}_1)\\\\bigr ]\\\\nonumber \\\\\\\\&&+\\\\bar{Z}_3^\\\\prime ( \\\\bar{H}_2^\\\\dagger \\\\bar{H}_2)( \\\\bar{H}_3^\\\\dagger \\\\bar{H}_3)+\\\\bar{Z}_4^\\\\prime ( \\\\bar{H}_2^\\\\dagger \\\\bar{H}_3)( \\\\bar{H}_3^\\\\dagger \\\\bar{H}_2)+i\\\\bar{Z}_5^\\\\prime \\\\bigl [e^{i\\\\psi } (\\\\bar{H}_3^\\\\dagger \\\\bar{H}_1)( \\\\bar{H}_2^\\\\dagger \\\\bar{H}_1)-e^{-i\\\\psi }( \\\\bar{H}_1^\\\\dagger \\\\bar{H}_2)( \\\\bar{H}_1^\\\\dagger \\\\bar{H}_3)\\\\bigr ]\\\\,,\\\\nonumber \\\\\\\\&&+\\\\bigl \\\\lbrace \\\\tfrac{1}{2}\\\\bar{Z}_5\\\\bigl [e^{i\\\\psi } (\\\\bar{H}_2^\\\\dagger \\\\bar{H}_1)^2 +e^{-i\\\\psi }( \\\\bar{H}_1^\\\\dagger \\\\bar{H}_3)^2\\\\bigr ]+\\\\bar{Z}_8 ( \\\\bar{H}_2^\\\\dagger \\\\bar{H}_3)^2+\\\\bar{Z}_9( \\\\bar{H}_2^\\\\dagger \\\\bar{H}_3)(\\\\bar{H}_2^\\\\dagger \\\\bar{H}_2- \\\\bar{H}_3^\\\\dagger \\\\bar{H}_3)+{\\\\rm h.c.} \\\\bigr \\\\rbrace \\\\,.\\\\nonumber $ The coefficients $Y_1$ , $Y_2$ , $Z_1$ , $Z_3$ and $Z_4$ are unmodified, whereas, $&& \\\\bar{Z}_2=Z_2+\\\\tfrac{1}{2}\\\\sin ^2 2\\\\phi \\\\bigl (Z_3^\\\\prime +Z_4^\\\\prime +Z_8 e^{2i\\\\xi }+Z^*_8 e^{-2i\\\\xi }\\\\bigr )-\\\\sin 2\\\\phi \\\\cos 2\\\\phi \\\\bigl (Z_9e^{i\\\\xi }+Z^*_9e^{-i\\\\xi }\\\\bigr ), \\\\\\\\&&\\\\bar{Z}_3^\\\\prime =Z_3^\\\\prime -\\\\sin ^2 2\\\\phi \\\\bigl (Z_3^\\\\prime +Z_4^\\\\prime +Z_8 e^{2i\\\\xi }+Z^*_8 e^{-2i\\\\xi }\\\\bigr )+2\\\\sin 2\\\\phi \\\\cos 2\\\\phi \\\\bigl (Z_9e^{i\\\\xi }+Z^*_9e^{-i\\\\xi }\\\\bigr ), \\\\\\\\&&\\\\bar{Z}_4^\\\\prime =Z_4^\\\\prime -\\\\tfrac{1}{2}\\\\sin ^2 2\\\\phi \\\\bigl (Z_3^\\\\prime +Z_4^\\\\prime +Z_8 e^{2i\\\\xi }+Z^*_8 e^{-2i\\\\xi }\\\\bigr )+\\\\sin 2\\\\phi \\\\cos 2\\\\phi \\\\bigl (Z_9e^{i\\\\xi }+Z^*_9e^{-i\\\\xi }\\\\bigr ), \\\\\\\\&&\\\\bar{Z}_5^\\\\prime =Z_5\\\\sin 2\\\\phi \\\\sin \\\\xi \\\\,, \\\\\\\\&&\\\\bar{Z}_5=e^{i\\\\chi }Z_5\\\\bigl (e^{i\\\\xi }\\\\cos ^2\\\\phi +e^{-i\\\\xi }\\\\sin ^2\\\\phi \\\\bigr )\\\\,, \\\\\\\\&&\\\\bar{Z}_8= e^{2i\\\\chi }\\\\bigl \\\\lbrace -\\\\tfrac{1}{4} \\\\sin ^2 2\\\\phi \\\\bigl (Z_3^\\\\prime +Z_4^\\\\prime \\\\bigr )+e^{2i\\\\xi }\\\\cos ^4\\\\phi \\\\,Z_8+e^{-2i\\\\xi }\\\\sin ^4\\\\phi \\\\, Z_8^* \\\\nonumber \\\\\\\\&& \\\\qquad \\\\qquad \\\\qquad +\\\\sin 2\\\\phi \\\\bigl [e^{i\\\\xi }\\\\cos ^2\\\\phi \\\\,Z_9-e^{-i\\\\xi }\\\\sin ^2\\\\phi \\\\, Z_9^*\\\\bigr ]\\\\bigr \\\\rbrace \\\\,, \\\\\\\\&&\\\\bar{Z}_9=e^{i\\\\chi }\\\\bigl \\\\lbrace -\\\\tfrac{1}{2}\\\\sin 2\\\\phi \\\\cos 2\\\\phi (Z_3^\\\\prime +Z_4^\\\\prime ) -\\\\sin 2\\\\phi \\\\bigl [e^{2i\\\\xi }\\\\cos ^2\\\\phi \\\\,Z_8-e^{-2i\\\\xi } \\\\sin ^2\\\\phi \\\\,Z_8^*\\\\bigr ] \\\\nonumber \\\\\\\\&& \\\\qquad \\\\qquad \\\\qquad +\\\\tfrac{1}{2}e^{i\\\\xi }(\\\\cos 4\\\\phi +\\\\cos 2\\\\phi )Z_9+\\\\tfrac{1}{2}e^{-i\\\\xi }(\\\\cos 4\\\\phi -\\\\cos 2\\\\phi )Z_9^*\\\\bigr \\\\rbrace \\\\,,$ where $\\\\xi \\\\equiv \\\\alpha +\\\\beta \\\\,,\\\\qquad \\\\quad \\\\chi \\\\equiv \\\\alpha -\\\\beta \\\\,.$ By definition of the $H23$ -basis, $Z_5$ is real and $Z_5^\\\\prime =0$ [the latter is a consequence of the absence of a term in Eq.\",\n \"(REF ) that involves $(H_3^\\\\dagger H_1)(H_2^\\\\dagger H_1)$ and its hermitian conjugate].\",\n \"After employing a generic U(2) basis change [Eq.\",\n \"(REF )], a nonzero $\\\\bar{Z}_5^\\\\prime $ and a complex $\\\\bar{Z}_5$ are generated [cf. Eqs.\",\n \"() and ()], such that $Z_5^2=|\\\\bar{Z}_5|^2+\\\\bar{Z}_5^{\\\\prime \\\\,2}\\\\,.$ It is instructive to examine the form of the CP4 transformation in the $\\\\lbrace \\\\bar{H}_1$ , $\\\\bar{H}_2$ , $\\\\bar{H_3}\\\\rbrace $ basis.\",\n \"Starting from Eq.\",\n \"(REF ) and transforming $H_i\\\\rightarrow \\\\bar{H}_i= \\\\widetilde{V}_{ij}H_j$ ($i,j=2,3$ ), it follows that the CP4 transformation of the barred fields is given by, $\\\\bar{H}_i\\\\rightarrow \\\\bar{X}_{ij}\\\\bar{H}_j^{\\\\scriptscriptstyle {\\\\bigstar }}\\\\,,\\\\qquad \\\\text{where $\\\\bar{X}=VW{V}^T$}\\\\,,$ where the $3\\\\times 3$ matrices $\\\\bar{X}$ , ${V}$ and $W$ in block form are given by $\\\\bar{X}=\\\\begin{pmatrix} 1 & 0 \\\\\\\\ 0 & \\\\widetilde{X}\\\\end{pmatrix}\\\\,,\\\\qquad {V}=\\\\begin{pmatrix} 1 & 0 \\\\\\\\ 0 & \\\\widetilde{V}\\\\end{pmatrix}\\\\,,\\\\qquad W=\\\\begin{pmatrix} 1 & 0 \\\\\\\\ 0 & \\\\epsilon \\\\end{pmatrix}\\\\,,$ and $\\\\epsilon \\\\equiv \\\\left({\\\\begin{matrix} 0 & -1 \\\\\\\\ 1 & \\\\phantom{-}0\\\\end{matrix}}\\\\right)$ .\",\n \"For any $\\\\widetilde{V}\\\\in $ U(2), we have $\\\\widetilde{X}=\\\\widetilde{V}\\\\epsilon \\\\widetilde{V}^T=e^{i\\\\psi }\\\\epsilon \\\\,,$ after taking the determinant of Eq.\",\n \"(REF ) and noting that $\\\\det \\\\widetilde{V}=e^{i\\\\psi }$ .\",\n \"Indeed, if we impose invariance of the scalar potential under CP4 in the $\\\\lbrace \\\\bar{H}_1$ , $\\\\bar{H}_2$ , $\\\\bar{H_3}\\\\rbrace $ basis, then the IS scalar potential must have the form given by Eq.\",\n \"(REF ), with $\\\\bar{Z}_8$ and $\\\\bar{Z}_9$ potentially complex and all other scalar potential coefficients [excluding factors of $i$ or $e^{\\\\pm i\\\\psi }$ that explicitly appear in Eq.\",\n \"(REF )] real.\",\n \"It is possible to choose a basis in which all but one of the scalar potential parameters are real.\",\n \"This can be achieved by choosing $\\\\psi =\\\\alpha =\\\\beta =0$ in Eq.\",\n \"(REF ).\",\n \"In this case, Eqs.\",\n \"() and () yield $\\\\bar{Z}_5^\\\\prime =0$ , $\\\\bar{Z}_5=Z_5$ and $\\\\operatorname{Im}\\\\bar{Z}_8&=&\\\\cos 2\\\\phi \\\\operatorname{Im}Z_8+\\\\sin 2\\\\phi \\\\operatorname{Im}Z_9\\\\,,\\\\nonumber \\\\\\\\\\\\operatorname{Im}\\\\bar{Z}_9&=&\\\\cos 2\\\\phi \\\\operatorname{Im}Z_9-\\\\sin 2\\\\phi \\\\operatorname{Im}Z_8\\\\,.$ Indeed, there is a choice of $\\\\phi $ in Eq.\",\n \"(REF ) such that $\\\\operatorname{Im}\\\\bar{Z}_9=0$ (and another choice of $\\\\phi $ such that $\\\\operatorname{Im}\\\\bar{Z}_8=0$ ).\",\n \"Thus, by a series of basis changes, we have reduced the number of independent parameters in the IS scalar potential from 14 to 12.\"\n ],\n [\n \"Transforming to a Higgs basis where $Z_8$ and {{formula:67350f71-efd2-4657-aecf-2e42710893ea}} are real\",\n \"We now examine whether a choice of $\\\\psi $ , $\\\\chi $ , $\\\\xi $ and $\\\\phi $ exists such that $i\\\\bar{Z}_5^\\\\prime e^{\\\\pm i\\\\psi }$ , $\\\\bar{Z}_5 e^{\\\\pm i\\\\psi }$ , $\\\\bar{Z}_8$ and $\\\\bar{Z}_9$ are all real.\",\n \"To begin, we first show that a Higgs basis exists in which $\\\\bar{Z}_8$ and $\\\\bar{Z}_9$ are both real.\",\n \"Here, we follow the analysis given in Appendix C of Ref. [46].\",\n \"By assumption, $Z_9$ is real and $Z_8=|Z_8|e^{i\\\\theta _8}$ (where $\\\\theta _8$ is not an integer multiple of $\\\\pi $ so that $\\\\operatorname{Im}Z_8\\\\ne 0$ ).\",\n \"Setting $\\\\operatorname{Im}\\\\bar{Z_8}=\\\\operatorname{Im}\\\\bar{Z_9}=0$ in Eqs.\",\n \"() and () yields, $\\\\operatorname{Im}\\\\bar{Z}_8=f_a\\\\cos 2\\\\chi -f_b\\\\sin 2\\\\chi =0\\\\,,\\\\qquad \\\\quad \\\\operatorname{Im}\\\\bar{Z}_9=f_c\\\\cos \\\\chi -f_d\\\\sin \\\\chi =0\\\\,,\\\\ $ where $f_a&=& |Z_8|\\\\cos 2\\\\phi \\\\sin (2\\\\xi +\\\\theta _8)+Z_9\\\\sin 2\\\\phi \\\\sin \\\\xi \\\\,, \\\\\\\\f_b&=&\\\\tfrac{1}{4}(Z_3^\\\\prime +Z_4^\\\\prime )\\\\sin ^2 2\\\\phi -|Z_8|(1-\\\\tfrac{1}{2}\\\\sin ^2 2\\\\phi )\\\\cos (2\\\\xi +\\\\theta _8)-Z_9\\\\sin 2\\\\phi \\\\cos 2\\\\phi \\\\cos \\\\xi \\\\,,\\\\\\\\f_c&=&-|Z_8|\\\\sin 2\\\\phi \\\\sin (2\\\\xi +\\\\theta _8)+Z_9\\\\cos 2\\\\phi \\\\sin \\\\xi \\\\,,\\\\\\\\f_d&=&\\\\tfrac{1}{2}(Z_3^\\\\prime +Z_4^\\\\prime ) \\\\sin 2\\\\phi \\\\cos 2\\\\phi +|Z_8|\\\\sin 2\\\\phi \\\\cos 2\\\\phi \\\\cos (2\\\\xi +\\\\theta _8)-Z_9\\\\cos 4\\\\phi \\\\cos \\\\xi \\\\,.$ Assuming that $f_a\\\\ne 0$ and $f_c\\\\ne 0$ , Eq.\",\n \"(REF ) implies that $\\\\cot \\\\chi =\\\\frac{f_d}{f_c}\\\\,,\\\\qquad \\\\quad \\\\cot 2\\\\chi =\\\\frac{f_b}{f_a}\\\\,.$ Employing the trigonometric identity, $\\\\cot 2\\\\chi =(\\\\cot ^2\\\\chi -1)/(2\\\\cot \\\\chi )$ , we end up with, $G(\\\\phi ,\\\\xi )\\\\equiv f_a(f_d^2-f_c^2)-2f_b f_c f_d=0\\\\,.$ Note that the above condition is independent of the angle $\\\\psi $ .\",\n \"Inserting the results of Eqs.\",\n \"(REF )–() into Eq.\",\n \"(REF ) leads to a very complicated expression.\",\n \"However, it is quite easy to check that $G(0,\\\\xi )=-G(\\\\tfrac{1}{2}\\\\pi ,\\\\xi )=Z_9^2\\\\operatorname{Im}Z_8\\\\,.$ As a consequence of Eq.\",\n \"(REF ), for any choice of $\\\\xi $ , there must exist a value of $\\\\phi $ between 0 and $\\\\tfrac{1}{2}\\\\pi $ such that $G(\\\\phi ,\\\\xi )=0$ .\",\n \"Plugging these values of $\\\\phi $ and $\\\\xi $ back into Eqs.\",\n \"(REF )–(), we can then use Eq.\",\n \"(REF ) to determine $\\\\chi $ .\",\n \"Thus, we have shown that for any choice of $\\\\xi $ and $\\\\psi $ , there must exist a corresponding $\\\\phi $ and $\\\\chi $ (whose values depend on the choice of $\\\\xi $ ) such that $\\\\operatorname{Im}\\\\bar{Z}_8=\\\\operatorname{Im}\\\\bar{Z}_9=0$ .Although we have reached this conclusion under the assumption that $f_c$ and $f_a$ are nonzero, it is straightforward to modify the analysis if either $f_a=0$ and/or $f_c=0$ .\",\n \"If $f_a=f_b=f_c=f_d=0$ , then Eq.\",\n \"(REF ) immediately yields $\\\\operatorname{Im}\\\\bar{Z}_8=\\\\operatorname{Im}\\\\bar{Z}_9=0$ .\",\n \"If at least one of the quantities $f_a$ , $f_b$ , $f_c$ and $f_d$ is nonzero, then $\\\\chi $ can be determined from one of the two expressions in Eq.\",\n \"(REF ).\"\n ],\n [\n \"Does a Higgs basis exist were all scalar potential parameters are real?\",\n \"Having found a Higgs basis with real $\\\\bar{Z}_8$ and $\\\\bar{Z}_9$ for an arbitrary choice of $\\\\xi $ and $\\\\psi $ (where the parameters $\\\\phi $ and $\\\\chi $ have been determined), we now examine whether it is also possible to choose particular values of $\\\\xi $ and $\\\\psi $ such that $i\\\\bar{Z}_5^\\\\prime e^{\\\\pm i\\\\psi }$ and $\\\\bar{Z}_5 e^{\\\\pm i\\\\psi }$ are both real.\",\n \"If this were possible, then one would have succeeded in finding a U(2) transformation, $\\\\bar{H}_i = \\\\widetilde{V}_{ij} H_j$ ($i,j=2,3$ ) such that all the coefficients of the IS scalar potential are real.\",\n \"For example, if $Z_5=0$ , then it follows from Eqs.\",\n \"() and () that $\\\\bar{Z}_5^\\\\prime =\\\\bar{Z}_5=0$ , in which case all the coefficients of the IS scalar potential, when expressed in terms of the barred scalar doublet fields, are real.\",\n \"Thus a real Higgs basis exists when $Z_5=0$ .\",\n \"It therefore follows that if $Z_5=0$ , then the IS scalar potential must possess a CP2 symmetry of the form, $H_i\\\\rightarrow Y_{ij}H_j^{\\\\scriptscriptstyle {\\\\bigstar }}$ , where $Y$ is a symmetric unitary matrix, which in block diagonal form [cf. Eq.\",\n \"(REF )] is given by, $Y=\\\\begin{pmatrix} 1 & \\\\,\\\\,\\\\,0 \\\\\\\\ 0 &\\\\,\\\\,\\\\, \\\\widetilde{Y}\\\\end{pmatrix}\\\\,,\\\\qquad \\\\quad \\\\text{with $\\\\widetilde{Y}\\\\equiv (\\\\tilde{V}^T\\\\tilde{V})^*$},$ and $\\\\tilde{V}$ [given by Eq.\",\n \"(REF )] is the unitary matrix that transforms the $H23$ basis into a real Higgs basis.\",\n \"Suppose one performs a CP4 transformation [Eq.\",\n \"(REF )] followed by a CP2 transformation, $H_i\\\\rightarrow Y_{ij}H_j^{\\\\scriptscriptstyle {\\\\bigstar }}$ , and compares the result obtained by performing these two transformations in the opposite order.\",\n \"The results of applying CP4 followed by CP2 as compared to CP2 followed by CP4 are equivalent to the Higgs family transformations, $YX^*$ and $XY^*$ , respectively [32].\",\n \"Using Eqs.\",\n \"(REF ), (REF ) and (REF ), it follows that $XY^*=e^{2i\\\\psi }YX^*\\\\,.$ That is, the CP2 and CP4 transformations commute if and only if $\\\\det Y= e^{2i\\\\psi }=1$ .\",\n \"For example, the $H23$ basis is a real Higgs basis in the trivial case where $\\\\operatorname{Im}Z_8=\\\\operatorname{Im}Z_9=0$ , independently of the value of $Z_5$ .\",\n \"In this case, the corresponding CP2 symmetry, $H_i\\\\rightarrow Y_{ij}H_j^{\\\\scriptscriptstyle {\\\\bigstar }}$ , with $Y={1}$ , commutes with the CP4 symmetry of the IS potential.\",\n \"If $Z_5=0$ , the real Higgs basis obtained above is independent of $\\\\psi $ .\",\n \"In this case, it is convenient to choose $\\\\psi =0$ in defining the CP2 transformation.\",\n \"It then follows from Eq.\",\n \"(REF ) that the CP2 and CP4 transformations commute if $Z_5=0$ and either $\\\\operatorname{Im}Z_8$ and/or $\\\\operatorname{Im}Z_9$ is nonzero.\",\n \"Two other special cases, first pointed out in Appendix B of Ref.\",\n \"[32], are noteworthy.\",\n \"First, suppose that $Z_9=0$ .\",\n \"In this case, the choice of $\\\\psi =\\\\chi =\\\\tfrac{1}{2}\\\\pi $ , $\\\\xi =0$ and $\\\\phi =\\\\tfrac{1}{4}\\\\pi $ inserted into Eqs.\",\n \"()–() will yield a real Higgs basis, with $\\\\bar{Z}_5^\\\\prime =0$ , $e^{\\\\pm i\\\\psi }\\\\bar{Z}_5=\\\\mp Z_5$ , $\\\\bar{Z_8}=\\\\tfrac{1}{4}(Z_3^\\\\prime +Z_4^\\\\prime -2\\\\operatorname{Re}Z_8)$ and $\\\\bar{Z}_9=\\\\operatorname{Im}Z_8$ .\",\n \"In light of Eqs.\",\n \"(REF ) and (REF ), the barred and unbarred scalar fields are related by $\\\\bar{H}_2=\\\\frac{i}{\\\\sqrt{2}}(H_2-H_3)\\\\,,\\\\qquad \\\\quad \\\\bar{H}_3=\\\\frac{1}{\\\\sqrt{2}}(H_2+H_3)\\\\,.$ In the $H23$ basis, we can identify the corresponding CP2 transformation as $H_i\\\\rightarrow Y_{ij}H_j^{\\\\scriptscriptstyle {\\\\bigstar }}$ , with $Y=\\\\begin{pmatrix} 1 & \\\\,\\\\,\\\\, 0 &\\\\,\\\\,\\\\, 0 \\\\\\\\ 0 &\\\\,\\\\,\\\\, 0 &\\\\,\\\\,\\\\, 1 \\\\\\\\ 0 &\\\\,\\\\,\\\\, 1 &\\\\,\\\\,\\\\, 0\\\\end{pmatrix}\\\\,.$ Since $XY^*\\\\ne YX^*$ , it follows that the CP2 and CP4 transformations do not commute.\",\n \"Second, suppose that $\\\\operatorname{Im}Z_8=0$ , $\\\\operatorname{Re}Z_9=0$ and $\\\\operatorname{Im}Z_9\\\\ne 0$ .\",\n \"In this case, we simply choose $\\\\bar{H}_2=H_2$ and $\\\\bar{H_3}=iH_3$ , corresponding to $\\\\psi =\\\\tfrac{1}{2}\\\\pi $ , $\\\\chi =\\\\xi =-\\\\tfrac{1}{4}\\\\pi $ and $\\\\phi =0$ in Eq.\",\n \"(REF ).\",\n \"A real Higgs basis is then achieved with $\\\\bar{Z}_5^\\\\prime =0$ , $e^{\\\\pm i\\\\psi }\\\\bar{Z}_5=\\\\pm Z_5$ , $\\\\bar{Z}_8=-\\\\operatorname{Re}Z_8$ , and $\\\\bar{Z}_9=\\\\operatorname{Im}Z_9$ .\",\n \"In the $H23$ basis, we can identify the corresponding CP2 transformation as $H_i\\\\rightarrow Y_{ij}H_j^{\\\\scriptscriptstyle {\\\\bigstar }}$ , with $Y=\\\\begin{pmatrix} 1 & \\\\phantom{-}0 &\\\\phantom{-}0 \\\\\\\\ 0 & \\\\phantom{-}1 &\\\\phantom{-}0 \\\\\\\\ 0 & \\\\phantom{-}0 & -1\\\\end{pmatrix}\\\\,.$ Once again, the CP2 and CP4 transformations do not commute.\",\n \"It should be noted that the last two cases are related by a simple basis transformation.\",\n \"Namely, starting from an $H23$ basis with $\\\\operatorname{Im}Z_8=0$ and $\\\\operatorname{Re}Z_9=0$ and employing $\\\\chi =\\\\psi =\\\\xi =0$ and $\\\\phi =\\\\tfrac{1}{4}\\\\pi $ in Eqs.\",\n \"()–() yields $\\\\bar{Z}_5^\\\\prime =0$ , $\\\\bar{Z}_5=Z_5$ , $\\\\operatorname{Im}\\\\bar{Z}_8\\\\ne 0$ and $\\\\bar{Z}_9=0$ , thereby reducing to the previous case above.\",\n \"We now consider the IS scalar potential with generic parameters (excluding the special cases considered above) and investigate whether a real Higgs basis exists.\",\n \"In particular, consider the $H23$ basis with $Z_5$ , $Z_9\\\\ne 0$ .\",\n \"As noted above, we can assume without loss of generality that $Z_5$ and $Z_9$ are real.The case of $\\\\operatorname{Im}Z_8=0$ and $\\\\operatorname{Re}Z_9=0$ is thus eliminated from consideration, since it is related by a scalar basis transformation to the case of $Z_9=0$ as noted above.\",\n \"We examine two different cases: Case 1: $\\\\psi \\\\ne \\\\pm \\\\tfrac{1}{2}\\\\pi $ Case 2: $\\\\psi =\\\\pm \\\\tfrac{1}{2}\\\\pi $ In case 1, a real Higgs basis would require $\\\\bar{Z}_5^\\\\prime =0$ .\",\n \"Since $\\\\sin 2\\\\phi \\\\ne 0$ [in light of Eq.\",\n \"(REF )], it follows that $\\\\sin \\\\xi =0$ , in which case $e^{\\\\pm i\\\\psi }\\\\bar{Z}_5=\\\\pm e^{i(\\\\chi \\\\pm \\\\psi )}Z_5$ is real if and only if $\\\\sin (\\\\chi \\\\pm \\\\psi )=0$ .\",\n \"This equation must be satisfied for both sign choices, which yields $\\\\sin \\\\chi \\\\cos \\\\psi =\\\\cos \\\\chi \\\\sin \\\\psi =0$ .\",\n \"For generic values of the parameters, Eqs.\",\n \"(REF )–(REF ) imply that $\\\\sin \\\\chi \\\\ne 0$ and $\\\\cos \\\\chi \\\\ne 0$ .\",\n \"Thus, in general no value of $\\\\psi $ exists such that $\\\\sin (\\\\chi \\\\pm \\\\psi )=0$ holds for both sign choices.\",\n \"That is, case 1 cannot yield a real Higgs basis for a generic choice of the IS scalar potential parameters.\",\n \"In case 2, $i\\\\bar{Z}_5^\\\\prime e^{\\\\pm i\\\\psi }$ is real for all choices of $\\\\xi $ and one must check whether there exists a $\\\\xi $ that yields a real value of $i\\\\bar{Z}_5=ie^{i\\\\chi }Z_5(e^{i\\\\xi }\\\\cos ^2\\\\phi +e^{-i\\\\xi }\\\\sin ^2\\\\phi $ ).\",\n \"The condition that $i\\\\bar{Z}_5$ is real is equivalent to $\\\\operatorname{Re}\\\\bigl [e^{i(\\\\chi +\\\\xi )}\\\\cos ^2\\\\phi +e^{i(\\\\chi -\\\\xi )}\\\\sin ^2\\\\phi \\\\bigr ]=0\\\\,,$ which can be simplified to the condition, $\\\\cot \\\\chi =\\\\cos 2\\\\phi \\\\tan \\\\xi \\\\,.$ It follows that either $\\\\chi \\\\pm \\\\xi $ are both half odd integer multiples of $\\\\tfrac{1}{2}\\\\pi $ or $\\\\cos 2\\\\phi =\\\\cot \\\\chi \\\\cot \\\\xi $ .\",\n \"If $\\\\chi \\\\pm \\\\xi $ are both half odd integer multiples of $\\\\tfrac{1}{2}\\\\pi $ , then either $\\\\chi $ is a half odd integer of $\\\\tfrac{1}{2}\\\\pi $ and $\\\\xi $ is an integer multiple of $\\\\pi $ or vice versa.\",\n \"If $\\\\chi $ is a half odd integer multiple of $\\\\tfrac{1}{2}\\\\pi $ and $\\\\xi $ is an integer multiple of $\\\\pi $ , then Eq.\",\n \"(REF ) yields $f_a=f_d=0$ .\",\n \"However, these latter two equations cannot be simultaneously satisfied if $Z_9\\\\ne 0$ .\",\n \"Similarly, if $\\\\chi $ is an integer multiple of $\\\\pi $ and $\\\\xi $ is a half odd integer multiple of $\\\\tfrac{1}{2}\\\\pi $ , then Eq.\",\n \"(REF ) yields $f_a=f_c=0$ which cannot be simultaneously satisfied if $\\\\operatorname{Im}Z_8$ and $Z_9$ are nonzero.\",\n \"Thus, if $\\\\chi \\\\pm \\\\xi $ are both half odd integer multiples of $\\\\tfrac{1}{2}\\\\pi $ , then no real Higgs basis exists for generic values of the IS scalar potential parameters.\",\n \"Finally, we examine the possibility that $i\\\\bar{Z}_5$ is real due to $\\\\cos 2\\\\phi =\\\\cot \\\\chi \\\\cot \\\\xi $ .\",\n \"We can also assume that $\\\\xi $ is not an integer multiple of $\\\\tfrac{1}{2}\\\\pi $ , as this case was already treated above.\",\n \"In order that $\\\\operatorname{Im}\\\\bar{Z}_8=\\\\operatorname{Im}\\\\bar{Z}_9=0$ , one must satisfy $\\\\cot \\\\chi =f_d/f_c$ and $G(\\\\phi ,\\\\xi )=0$ , under the assumption of $f_c\\\\ne 0$ .\",\n \"In this case we can satisfy $\\\\operatorname{Im}\\\\bar{Z}_9=0$ if $\\\\phi =\\\\phi _\\\\xi $ , where $\\\\cos 2\\\\phi _\\\\xi =\\\\left(\\\\frac{f_d}{f_c}\\\\right)_{\\\\phi =\\\\phi _\\\\xi }\\\\cot \\\\xi \\\\,.$ Using Eqs.\",\n \"() and (), one can employ Eq.\",\n \"(REF ) to obtain a quadratic equation for $\\\\cot 2\\\\phi _\\\\xi $ , whose solution is given by $\\\\cot 2\\\\phi _\\\\xi &=&\\\\frac{1}{2Z_9}\\\\biggl [\\\\tfrac{1}{2}(Z_3^\\\\prime +Z_4^\\\\prime )\\\\cos \\\\xi +|Z_8|\\\\cos (\\\\xi +\\\\theta _8) \\\\nonumber \\\\\\\\&&\\\\qquad \\\\qquad \\\\quad \\\\pm \\\\sqrt{[\\\\tfrac{1}{2}(Z_3^\\\\prime +Z_4^\\\\prime )\\\\cos \\\\xi +|Z_8|\\\\cos (\\\\xi +\\\\theta _8)]^2+4Z_9^2\\\\cos ^2\\\\xi }\\\\biggr ]\\\\,.$ Eq.\",\n \"(REF ) determines $\\\\sin 2\\\\phi _\\\\xi $ up to an overall sign.\",\n \"It is convenient to choose this sign to be positive.\",\n \"It is sufficient to demonstrate one example of the IS scalar potential parameters in which no real Higgs basis exists.\",\n \"Thus, consider an example where $Z_8=-\\\\tfrac{1}{2}(Z_3^\\\\prime +Z_4^\\\\prime )+2iZ_9\\\\,.$ Then, $\\\\tfrac{1}{2}(Z_3^\\\\prime +Z_4^\\\\prime )\\\\cos \\\\xi +|Z_8|(\\\\cos (\\\\xi +\\\\theta _8) =-2Z_9\\\\sin \\\\xi $ .\",\n \"It follows that $\\\\cot 2\\\\phi _\\\\xi =\\\\pm 1-\\\\sin \\\\xi \\\\,.$ We now investigate whether a value of $\\\\xi \\\\ne \\\\tfrac{1}{2}n\\\\pi $ (where $n$ is an integer) exists such that $G(\\\\phi _\\\\xi ,\\\\xi )=0$ .\",\n \"We introduce the notation, $\\\\tilde{f}\\\\equiv f(\\\\phi _\\\\xi ,\\\\xi )$ , where $\\\\phi _\\\\xi $ has been determined from Eq.\",\n \"(REF ).\",\n \"Then, Eq.\",\n \"(REF ) implies that $\\\\tilde{f}_d=\\\\tilde{f}_c\\\\cos 2\\\\phi _\\\\xi \\\\tan \\\\xi $ .\",\n \"Inserting this result into Eq.\",\n \"(REF ) yields, $G(\\\\phi _\\\\xi ,\\\\xi )= \\\\tilde{f}_c^2\\\\bigl [\\\\tilde{f}_a(\\\\cos ^2 2\\\\phi _\\\\xi \\\\tan ^2\\\\xi -1)-2\\\\tilde{f}_b \\\\cos 2\\\\phi _\\\\xi \\\\tan \\\\xi \\\\bigr ]\\\\,.$ An explicit calculation yields $\\\\tilde{f}_c=\\\\sin 2\\\\phi _\\\\xi \\\\bigl [-\\\\operatorname{Re}Z_8\\\\sin 2\\\\xi +(3\\\\sin \\\\xi \\\\mp 2)(\\\\sin \\\\xi \\\\pm 1)Z_9\\\\bigr ]\\\\,,$ and $\\\\tilde{f}_a(\\\\cos ^2 2\\\\phi _\\\\xi \\\\tan ^2\\\\xi -1)-2\\\\tilde{f}_b \\\\cos 2\\\\phi _\\\\xi \\\\tan \\\\xi =2Z_9\\\\frac{\\\\sin 2\\\\phi _\\\\xi }{\\\\cos ^2\\\\xi }(\\\\sin \\\\xi \\\\mp 1)\\\\,.$ Hence, we end up withWe have made use of the identity, $1+\\\\cot ^2 2\\\\phi _\\\\xi =1/\\\\sin ^2 2\\\\phi _\\\\xi $ .\",\n \"As noted below Eq.\",\n \"(REF ) , we have assumed that $\\\\sin 2\\\\phi _\\\\xi $ is positive.\",\n \"$G(\\\\phi _\\\\xi ,\\\\xi )=\\\\frac{2Z_9}{\\\\cos ^2\\\\xi }\\\\bigl [1+(1\\\\mp \\\\sin \\\\xi )^2\\\\bigr ]^{-3/2}(\\\\sin \\\\xi \\\\mp 1)\\\\bigl [\\\\operatorname{Re}Z_8\\\\sin 2\\\\xi -(3\\\\sin \\\\xi \\\\mp 2)(\\\\sin \\\\xi \\\\pm 1)Z_9\\\\bigr ]^2\\\\,.$ Since the above analysis has assumed that $\\\\tilde{f}_c\\\\ne 0$ and $\\\\xi \\\\ne \\\\tfrac{1}{2}n\\\\pi $ (for integer $n$ ) it follows that $G(\\\\phi _\\\\xi ,\\\\xi )$ is strictly nonzero, which implies that $\\\\operatorname{Im}\\\\bar{Z}_8\\\\ne 0$ .By expanding to squared expression in Eq.\",\n \"(REF ), one sees that the factor of $\\\\cos ^2\\\\xi $ in the denominator is canceled by terms in the numerator.\",\n \"Hence, there is no singularity in the limit of $\\\\cos \\\\xi \\\\rightarrow 0$ .\"\n ],\n [\n \"Special cases\",\n \"Cases where $f_c=0$ need to be treated separately.\",\n \"First, we examine the case of $f_c=0$ and $f_d\\\\ne 0$ .\",\n \"Inserting Eq.\",\n \"(REF ) into Eq.\",\n \"(), one can solve for $\\\\cot 2\\\\phi _\\\\xi $ , $\\\\cot 2\\\\phi _\\\\xi =\\\\frac{\\\\operatorname{Re}Z_8\\\\sin 2\\\\xi +2Z_9\\\\cos 2\\\\xi }{Z_9\\\\sin \\\\xi }\\\\,.$ We next impose $\\\\operatorname{Im}\\\\bar{Z}_9=0$ .\",\n \"Then Eq.\",\n \"(REF ) implies that $\\\\sin \\\\chi =0$ , in which case Eq.\",\n \"(REF ) yields $\\\\cos \\\\xi =0$ .\",\n \"Inserting the latter result back into Eq.\",\n \"(REF ) yields $\\\\cot 2\\\\phi _\\\\xi =\\\\pm 2$ .\",\n \"If one now attempts to impose $\\\\operatorname{Im}\\\\bar{Z}_8=0$ using Eq.\",\n \"(REF ) with $\\\\sin \\\\chi =0$ , then one would conclude that $f_a=0$ .\",\n \"However, one can explicitly show that $f_a\\\\ne 0$ by inserting Eq.\",\n \"(REF ) into Eq.\",\n \"() and employing $\\\\cot 2\\\\phi _\\\\xi =\\\\pm 2$ and $\\\\cos \\\\xi =0$ .\",\n \"Finally, we briefly consider the case of $f_c=f_d=0$ .\",\n \"Inserting Eq.\",\n \"(REF ) into Eqs.\",\n \"() and () yields equations for $\\\\tan 2\\\\phi $ and $\\\\tan 4\\\\phi $ respectively.\",\n \"The compatibility of these two equations then fixes the value of $\\\\xi $ .\",\n \"In this case, $\\\\operatorname{Im}\\\\bar{Z}_9=0$ is automatic, and $\\\\operatorname{Im}\\\\bar{Z}_8=0$ implies via Eq.\",\n \"(REF ) that $\\\\cot 2\\\\chi =f_b/f_a$ .\",\n \"At this stage, $\\\\chi $ , $\\\\xi $ and $\\\\phi $ are all determined prior to imposing Eq.\",\n \"(REF ).\",\n \"The latter is an independent condition; hence for generic values of $\\\\operatorname{Re}Z_8$ and $Z_9$ , it is not possible to perform a basis change such that $i\\\\bar{Z}_5$ , $\\\\bar{Z}_8$ and $\\\\bar{Z}_9$ are simultaneously real.\",\n \"This completes all the subcases of case 2.\",\n \"We conclude that if the IS scalar potential possesses at least one non-real coefficient (for generic choices of the scalar potential parameters), then no real Higgs basis exists and it is not possible to perform a U(2) transformation of the Higgs basis fields $\\\\lbrace H_2,H_3\\\\rbrace $ such that all coefficients of the scalar potential are real.\"\n ],\n [\n \"Basis-invariant polynomial functions of the IS scalar potential parameters\",\n \"In our analysis of the IS model, we have advocated the choice of a particular class of Higgs bases in which $\\\\bar{Z}_5^\\\\prime =0$ .\",\n \"Nevertheless, it is instructive to show that physical observables that depend on the parameters of the IS scalar potential are independent of the choice of the scalar basis.\",\n \"In this appendix, we introduce a number of basis invariant quantities and evaluate them in the $H23$ -basis.\",\n \"Consider a generic basis of scalar fields, $\\\\lbrace \\\\Phi _a\\\\rbrace $ , where $a=1,2,3$ labels hypercharge-one, doublet fields of the 3HDM.\",\n \"Basis transformations that leave invariant the form of the canonical kinetic energy terms correspond to global U(3) transformations, $\\\\Phi _a\\\\rightarrow U_{a{\\\\bar{b}}}\\\\Phi _b$ [and $\\\\Phi _{\\\\bar{a}}^\\\\dagger \\\\rightarrow \\\\Phi _{\\\\bar{b}}^\\\\dagger U^\\\\dagger _{b{\\\\bar{a}}}$ ], where the $3\\\\times 3$ unitary matrix $U$ satisfies $U^\\\\dagger _{b{\\\\bar{a}}}U_{a{\\\\bar{c}}}=\\\\delta _{b{\\\\bar{c}}}$ .\",\n \"Here, we follow the index conventions introduced in Ref.\",\n \"[20], in which replacing an unbarred index with a barred index is equivalent to hermitian conjugation.\",\n \"We only allow sums over barred–unbarred index pairs, which are performed by employing the U(3)-invariant tensor $\\\\delta _{a{\\\\bar{b}}}$ .\",\n \"In this notation, the 3HDM scalar potential in a generic $\\\\Phi _a$ -basis is given by, $\\\\mathcal {V}=Y_{a{\\\\bar{b}}}\\\\Phi _{\\\\bar{a}}^\\\\dagger \\\\Phi _b+\\\\tfrac{1}{2}Z_{a{\\\\bar{b}}c{\\\\bar{d}}}(\\\\Phi _{\\\\bar{a}}^\\\\dagger \\\\Phi _b)(\\\\Phi _{\\\\bar{c}}^\\\\dagger \\\\Phi _d)\\\\,,$ where $Z_{a{\\\\bar{b}}c{\\\\bar{d}}}=Z_{c{\\\\bar{d}}a{\\\\bar{b}}}$ .\",\n \"Hermiticity of $\\\\mathcal {V}$ implies that $Y_{a {\\\\bar{b}}}= (Y_{b {\\\\bar{a}}})^\\\\ast $ and $Z_{a{\\\\bar{b}}c{\\\\bar{d}}}= (Z_{b{\\\\bar{a}}d{\\\\bar{c}}})^\\\\ast $ .\",\n \"Minimizing the scalar potential, under the assumption that the vacuum preserves U(1)$_{\\\\rm EM}$ , yields the neutral Higgs vacuum expectation values, $\\\\langle \\\\Phi ^0_a \\\\rangle =v \\\\widehat{v}_a/\\\\sqrt{2}$ , where $v=246$ GeV and $\\\\widehat{v}_a$ is a vector of unit norm.\",\n \"It is convenient to define the hermitian matrix[19] $V_{a{\\\\bar{b}}}\\\\equiv \\\\widehat{v}_a \\\\,\\\\widehat{v}_{\\\\bar{b}}^\\\\ast \\\\,.$ One can now construct basis-invariant quantities that depend on knowledge of the scalar potential minimum by forming products of $V_{a{\\\\bar{b}}}$ and $Z_{a{\\\\bar{b}}c{\\\\bar{d}}}$ such that all barred–unbarred index pairs are summed over.\",\n \"We define six invariant quantities below, The invariants above can be evaluated in any basis.\",\n \"In particular, in the $H23$ -basis, the only nonzero component of $V_{a{\\\\bar{b}}}$ is $V_{11}=1$ .\",\n \"We thus obtain, $J_1&=&Z_1,\\\\\\\\J_2&=&Z_1+2Z_3,\\\\\\\\J_3&=&Z_1+2Z_4,\\\\\\\\J_4&=&Z_1^2+2 Z_3^2+2 Z_4^2+2 Z_5^2,\\\\\\\\J_5&=&Z_1^3+4 Z_5^2 Z_1+2 Z_3^3+6 Z_3 Z_4^2+2 Z_2 Z_5^2+4 Z_5^2 \\\\operatorname{Re}Z_8 ,\\\\\\\\J_6&=&Z_1^4+2 Z_3^4+2 Z_4^4+12 Z_3^2 Z_4^2+4 Z_5^4+2Z_5^2(3Z_1^2+2Z_1 Z_2+Z_2^2)\\\\nonumber \\\\\\\\&&+8 Z_5^2\\\\bigl [|Z_8|^2+(Z_1+Z_2)\\\\operatorname{Re}Z_8+ \\\\left(\\\\operatorname{Im}Z_9\\\\right)^2\\\\bigr ].$ Using the first four invariant quantities above, one can show that $Z_5$ can be expressed in terms of an invariant quantity.In the $H23$ -basis (where $Z_5^\\\\prime =0$ ), one expects that $Z^2_5$ can be expressed in terms of an invariant quantity in light of the mass relation, $M_P^2-M_Q^2=Z_5 v^2$ , which implies that $Z_5^2$ is a physical parameter.\",\n \"In a general class of Higgs bases, the corresponding invariant quantity is $|Z_5|^2+|Z_5^\\\\prime |^2$ [cf. Eq.\",\n \"(REF )].\",\n \"In particular, $Z_5^2=-J_1^2+\\\\tfrac{1}{2}J_1 \\\\left(J_2+J_3\\\\right) -\\\\tfrac{1}{4} (J_2^2+J_3^2)+\\\\tfrac{1}{2}J_4\\\\,.$ Finally, we have discovered a remarkable invariant quantity, $\\\\mathcal {N}&=&32Z_5^2 J_6-16J_5^2+8J_5(3J_{21}J_{31}^2+K) -J_{31}^4(9J_{21}^2+4Z_5^2) -6KJ_{21}J_{31}^2-24Z_5^2 J_{21}^2 J_{31}^2 \\\\nonumber \\\\\\\\[5pt]&& -J_{21}^6 - 4Z_5^2J_{21}^4-8J_1(J_1^2+2Z_5^2)J_{21}^3 -16J_1^6-96 Z_5^2 J_1^4 -192 Z_5^4 J_1^2 -128Z_5^6\\\\,, $ where $J_{ij}\\\\equiv J_i-J_j$ , the invariant quantity $Z^2_5$ is given by Eq.\",\n \"(REF ) and $K\\\\equiv 4J_1^3+8Z_5^2 J_1+J_{21}^3\\\\,.$ Plugging in the expressions for $J_1\\\\,,\\\\ldots \\\\,,J_6$ given above, we find $\\\\mathcal {N}=256 Z_5^4\\\\bigl [(\\\\operatorname{Im}Z_8)^2 + (\\\\operatorname{Im}Z_9)^2\\\\bigr ]\\\\,.$ It follows that if $Z_5\\\\ne 0$ then there exists a ratio of invariant quantities, which when evaluated in the $H23$ -basis, is equal to $(\\\\operatorname{Im}Z_8)^2 + (\\\\operatorname{Im}Z_9)^2$ .\",\n \"In contrast, if $Z_5=0$ , then there is no invariant quantity that reduces in the $H23$ -basis to $(\\\\operatorname{Im}Z_8)^2 + (\\\\operatorname{Im}Z_9)^2$ .\",\n \"Nevertheless, the invariant condition, $Z_5=0$ , signals the presence of four mass-degenerate neutral scalars.\",\n \"The significance of the invariant $\\\\mathcal {N}$ is as follows.\",\n \"The CP4-conserving IS model possesses a CP2 symmetry that commutes with CP4 if and only if $\\\\mathcal {N}=0$ .\",\n \"Note that the nonvanishing of $\\\\mathcal {N}$ does not exclude the possibility of a CP2 symmetry that does not commute with CP4.\",\n \"Two explicit examples of this phenomenon were presented in Appendix REF : (i) $Z_9=0$ and $\\\\operatorname{Im}Z_8\\\\ne 0$ ; and (ii) $\\\\operatorname{Im}Z_8=\\\\operatorname{Re}Z_9=0$ and $\\\\operatorname{Im}Z_9\\\\ne 0$ .\",\n \"In both these cases, a real Higgs basis exists, and the corresponding CP2 transformation does not commute with CP4.One cannot employ Eq.\",\n \"(REF ) to compute $\\\\mathcal {N}$ in the real Higgs basis in cases (i) and (ii), since in both cases, the real Higgs basis lies outside the set of $H23$ bases.\",\n \"Nevertheless, we have checked that evaluating $\\\\mathcal {N}$ directly in the real Higgs basis in cases (i) and (ii) reproduces the corresponding results obtained in the $H23$ basis via Eq.\",\n \"(REF ).\",\n \"We also noted in Section that under the assumption that $M_P\\\\ne M_Q$ (or equivalently for $Z_5\\\\ne 0$ in the $H23$ basis), the decay rate for $Z\\\\rightarrow QQQQ^*$ , $QQ^* Q^* Q^*$ , if kinematically allowed, is nonzero if and only if $\\\\mathcal {N}\\\\ne 0$ .\",\n \"Note that the invariant quantity $\\\\mathcal {N}$ constructed above has been expressed in terms of Higgs basis parameters.\",\n \"This means that this invariant quantity depends on the knowledge of the vacuum, i.e.\",\n \"the minimum of the scalar potential (which is needed to formally define the Higgs basis).\",\n \"Given an explicitly CP4-invariant scalar potential, one could ask a slightly different question: is there an invariant quantity that can differentiate between scalar potentials that explicitly preserve or violate the CP2 symmetry, independently of the vacuum.\",\n \"This question has been recently addressed and answered in Ref. [32].\",\n \"However, it is not clear that such an invariant quantity can be directly related in practice to a physical observable.\"\n ],\n [\n \"An alternative Higgs basis\",\n \"In this paper, we first defined the $H23$ -basis by employing the scalar doublet fields $\\\\lbrace H_1,H_2,H_3\\\\rbrace $ , which was one particular choice among possible Higgs bases.\",\n \"An arbitrary Higgs basis can be obtained by performing the U(2) basis transformation given by Eqs.\",\n \"(REF ) and (REF ).\",\n \"The corresponding IS scalar potential is given by Eq.\",\n \"(REF ), where the barred coefficients in terms of the unbarred coefficients are given in Eqs.\",\n \"(REF )–().\",\n \"In section REF , we explored another Higgs basis choice, called the $RS$ -basis, which employs the scalar doublet fields, $\\\\lbrace H_1,\\\\mathcal {R},\\\\mathcal {S}\\\\rbrace $ .\",\n \"The relations between the $H23$ -basis and $RS$ -basis are given by, $\\\\mathcal {R}\\\\equiv \\\\frac{1}{\\\\sqrt{2}}\\\\bigl (H_2+iH_3\\\\bigr )=\\\\begin{pmatrix} R^\\\\dagger \\\\\\\\ \\\\frac{1}{\\\\sqrt{2}}\\\\bigl (P+iQ^\\\\dagger \\\\bigr )\\\\end{pmatrix}\\\\,,\\\\quad \\\\qquad \\\\mathcal {S}\\\\equiv \\\\frac{1}{\\\\sqrt{2}}\\\\bigl (H_2-iH_3\\\\bigr )=\\\\begin{pmatrix} S^\\\\dagger \\\\\\\\ \\\\frac{1}{\\\\sqrt{2}}\\\\bigl (P^\\\\dagger +iQ\\\\bigr )\\\\end{pmatrix}\\\\,.$ Note that the form of the CP4 transformation in this basis is $\\\\begin{pmatrix} H_1 \\\\\\\\ \\\\mathcal {R} \\\\\\\\ \\\\mathcal {S}\\\\end{pmatrix}\\\\longrightarrow \\\\begin{pmatrix} 1 & \\\\phantom{-}0 & \\\\phantom{-}0 \\\\\\\\ 0 & \\\\phantom{-}0 & \\\\phantom{-}i \\\\\\\\ 0 & -i & \\\\phantom{-}0\\\\end{pmatrix}\\\\begin{pmatrix} H_1^\\\\dagger \\\\\\\\ \\\\mathcal {R}^\\\\dagger \\\\\\\\ \\\\mathcal {S}^\\\\dagger \\\\end{pmatrix}\\\\,.$ The change of basis from $\\\\lbrace H_1,H_2,H_3\\\\rbrace $ to $\\\\lbrace H_1,R,S\\\\rbrace $ corresponds to choosing $\\\\alpha =\\\\beta =\\\\phi =\\\\tfrac{1}{4}\\\\pi $ and $\\\\psi =-\\\\tfrac{1}{2}\\\\pi $ .\",\n \"Inserting these results into Eqs.\",\n \"(REF )–() yields, 1.25 $\\\\bar{Z}_2&=&Z_2+\\\\tfrac{1}{2}(Z_3^\\\\prime +Z_4^\\\\prime -2\\\\operatorname{Re}Z_8)\\\\,,\\\\\\\\\\\\bar{Z}_3^\\\\prime &=&-Z_4^\\\\prime +2\\\\operatorname{Re}Z_8\\\\,,\\\\\\\\\\\\bar{Z}_4^\\\\prime &=&\\\\tfrac{1}{2}(Z_4^\\\\prime -Z_3^\\\\prime +2\\\\operatorname{Re}Z_8)\\\\,,\\\\\\\\\\\\bar{Z}_5^\\\\prime &=&Z_5\\\\,,\\\\\\\\\\\\bar{Z}_5&=&0\\\\,,\\\\\\\\\\\\bar{Z}_8&=&-\\\\tfrac{1}{4}(Z_3^\\\\prime +Z_4^\\\\prime +2\\\\operatorname{Re}Z_8)+i\\\\operatorname{Re}Z_9\\\\,,\\\\\\\\\\\\bar{Z}_9&=& \\\\operatorname{Im}Z_9+i\\\\operatorname{Im}Z_8\\\\,.$ Note that in the $RS$ -basis, the invariant $\\\\mathcal {N}$ defined in Eq.\",\n \"(REF ) is $\\\\mathcal {N}=256\\\\bar{Z}_5^{\\\\prime \\\\,4}|\\\\bar{Z}_9|^2\\\\,.$ In particular, the absence [or presence] of the $Z\\\\rightarrow QQQQ^*$ , $QQ^*Q^*Q^*$ decay discussed in Appendix depends on the [non-]vanishing of $\\\\bar{Z}_9$ .\",\n \"These decays are governed by Eq.\",\n \"(REF ), which when expressed in the $RS$ -basis is given by, $\\\\delta {L}_{4h}\\\\ni \\\\tfrac{1}{2}i(PQ-P^\\\\dagger Q^\\\\dagger )\\\\bigl [\\\\bar{Z}_9(P^{\\\\dagger \\\\,2}+Q^2)+\\\\bar{Z}_9^*(P^2+Q^{\\\\dagger \\\\,2})\\\\bigr ]\\\\,.$\"\n ],\n [\n \"$Z$ decay into four inert neutral scalars\",\n \"Consider a universe (not ours) in which the electroweak theory of elementary particles at the electroweak scale consists of the IS model, with $M_{H^\\\\pm ,h^\\\\pm }<M_Q<\\\\tfrac{1}{4} m_Z\\\\ll M_P\\\\,.$ In this case, the decay of the $Z$ to four neutral inert scalars would be consistent with a CP4-symmetric IS scalar potential that does not possess a real scalar basis.\",\n \"Experimentally, the final state would be detected via the decay $Q\\\\rightarrow (H^\\\\pm , h^\\\\pm )+W^{*\\\\,\\\\mp }$ , with the virtual $W^{*\\\\,\\\\mp }$ decaying to quark or lepton pairs.\",\n \"In this universe, the $H^\\\\pm , h^\\\\pm $ are the lightest particles of the inert scalar sector and hence stable.\",\n \"Although this is not our universe, this example provides a proof in principle of the existence of an experimental distinction between the CP4-conserving/CP2-nonconserving case and the CP4/CP2-conserving case.\",\n \"Figure: Feynman diagrams for Z→QQQQ * Z\\\\rightarrow QQQQ^\\\\ast In light of Eq.\",\n \"(REF ), there are four contributing tree-level Feynman diagrams to the decay amplitude $Z\\\\rightarrow QQQQ^\\\\ast $ , which are shown in Fig.\",\n \"REF .\",\n \"Employing the Feynman rules obtained from Eq.\",\n \"(REF ) (and including the appropriate symmetry factors in obtaining the rules for the four-scalar vertex), the invariant matrix element is given by, $i\\\\mathcal {M}&=&\\\\frac{g}{2\\\\cos \\\\theta _W}\\\\bigl (\\\\operatorname{Im}Z_8 +i\\\\operatorname{Im}Z_9\\\\bigr )\\\\,\\\\varepsilon _\\\\lambda (p)\\\\cdot \\\\left[\\\\frac{p-2k_1}{(p-k_1)^2-M_P^2}+\\\\frac{p-2k_2}{(p-k_2)^2-M_P^2}\\\\right.\",\n \"\\\\nonumber \\\\\\\\[6pt]&&\\\\qquad \\\\qquad \\\\qquad \\\\left.+\\\\frac{p-2k_3}{(p-k_3)^2-M_P^2}-\\\\frac{3(p-2k_4)}{(p-k_4)^2-M_P^2}\\\\right]\\\\,,\\\\nonumber $ where $p$ is the four-momentum of the $Z$ and the $k_i$ are the final state momenta (with $k_4$ the momentum of $Q^*$ ).\",\n \"We then square the matrix element and average over the initial state spins, using $|\\\\mathcal {M}|^2_{\\\\rm ave}\\\\equiv \\\\tfrac{1}{3}\\\\sum _\\\\lambda |X\\\\cdot \\\\varepsilon _\\\\lambda (p)|^2=-\\\\tfrac{1}{3} X_\\\\mu X^*_\\\\nu \\\\left( g^{\\\\mu \\\\nu }-\\\\frac{p^\\\\mu p^\\\\nu }{m_Z^2}\\\\right)\\\\,.$ where $X$ is the four vector dotted into the polarization vector in the expression for $i\\\\mathcal {M}$ .\",\n \"We shall work in the approximation that $M_P\\\\gg m_Z$ and $M_Q=0$ .\",\n \"In this case, $X=\\\\frac{g}{M_P^2\\\\cos \\\\theta _W}\\\\bigl (\\\\operatorname{Im}Z_8 +i\\\\operatorname{Im}Z_9\\\\bigr )(p-4k_4)\\\\,,$ where we have used conservation of momentum, $p=k_1+k_2+k_3+k_4$ .\",\n \"It then follows after some simplification (with $p^2=m_Z^2$ ) that, $|\\\\mathcal {M}|^2_{\\\\rm ave}=\\\\frac{16g^2}{3m_Z^2 M_P^4\\\\cos ^2\\\\theta _W}\\\\bigl [(\\\\operatorname{Im}Z_8)^2+ (\\\\operatorname{Im}Z_9)^2\\\\bigr ] (p\\\\cdot k_4)^2\\\\,.$ The four body decay width for $Z\\\\rightarrow QQQQ^*$ is given by $\\\\Gamma =\\\\frac{1}{6}\\\\,\\\\frac{(2\\\\pi )^{-8}}{2m_Z}\\\\int \\\\left(\\\\prod _{i=1}^4 \\\\frac{d^3 k_i}{2E_i}\\\\right)\\\\delta ^4(p-k_1-k_2-k_3-k_4)|\\\\mathcal {M}|^2_{\\\\rm ave}\\\\,,$ where the factor of $1/6$ is due to the three identical Qs in the final state (which means we overcount by a factor of 3!\",\n \"by integrating over the full phase space).\",\n \"Using the above results, we obtain, $\\\\Gamma =\\\\frac{4g^2(2\\\\pi )^{-8}}{9m_Z^3 M_P^4\\\\cos ^2\\\\theta _W}\\\\bigl [(\\\\operatorname{Im}Z_8)^2+ (\\\\operatorname{Im}Z_9)^2\\\\bigr ]\\\\int \\\\left(\\\\prod _{i=1}^4 \\\\frac{d^3 k_i}{2E_i}\\\\right)\\\\delta ^4(p-k_1-k_2-k_3-k_4) \\\\,(p\\\\cdot k_1)^2$ after changing integration variables $k_1\\\\longleftrightarrow k_4$ .\",\n \"To perform the phase space integration, we follow Ref. [82].\",\n \"To integrate over $d^3k_1d^3k_2$ we use, $\\\\int \\\\frac{d^3k_1}{2E_1}\\\\frac{d^3k_2}{E_2}\\\\delta ^{4}(N-k_1-k_2)\\\\times {\\\\left\\\\lbrace \\\\begin{array}{ll}1& \\\\text{$=\\\\frac{1}{2}\\\\pi $,}\\\\\\\\k_{1\\\\mu }&\\\\text{$=\\\\frac{1}{4}\\\\pi N_\\\\mu $,}\\\\\\\\k_{1\\\\mu }k_{1\\\\nu }&\\\\text{$=-\\\\frac{1}{24}\\\\pi (N^2g_{\\\\mu \\\\nu }-4N_\\\\mu N_\\\\nu )$,}\\\\\\\\k_{1\\\\mu }k_{2\\\\nu }&\\\\text{$=\\\\frac{1}{24}\\\\pi (N^2g_{\\\\mu \\\\nu }+2N_\\\\mu N_\\\\nu )$,}\\\\end{array}\\\\right.\",\n \"}$ where $N$ is an arbitrary four-vector.\",\n \"In the present application, $N=p-k_3-k_4$ .\",\n \"After performing this integration, we have two further integrations to do over $k_3$ and $k_4$ .\",\n \"It is convenient to work in the $Z$ rest frame: $p=(m_Z;0,0,0);\\\\ \\\\ k_3=E_3(1;0,0,1);\\\\ \\\\ k_4=E_4(1;\\\\sin \\\\theta ,0,\\\\cos \\\\theta )\\\\,.$ We introduce the following scaled kinematic variables $w\\\\equiv \\\\frac{1-\\\\cos \\\\theta }{2},\\\\ \\\\ \\\\ y\\\\equiv \\\\frac{2E_3}{m_Z},\\\\ \\\\ \\\\ z\\\\equiv \\\\frac{2E_4}{m_Z}\\\\,.$ Then, $\\\\int \\\\frac{d^3 k_3}{2E_3}\\\\,\\\\frac{d^3k_4}{2E_4}=\\\\frac{\\\\pi ^2 m_Z^4}{4}\\\\int _0^{1}\\\\!z\\\\,dz\\\\!\",\n \"\\\\left\\\\lbrace \\\\int _0^{1-z}y\\\\,dy\\\\int _0^1 dw+\\\\!\\\\int _{1-z}^1 y\\\\,dy\\\\!\",\n \"\\\\int _{(y+z-1)/yz}^1 \\\\!dw\\\\right\\\\rbrace \\\\,.$ We now evaluate the integral in Eq.\",\n \"(REF ).\",\n \"Using the above results, $\\\\int \\\\frac{d^3k_1}{2E_1}\\\\frac{d^3k_2}{E_2}\\\\delta ^{4}(N-k_1-k_2)\\\\,(p\\\\cdot k_1)^2&=&-\\\\frac{\\\\pi }{24}\\\\biggl \\\\lbrace m_Z^2(p-k_3-k_4)^2-4\\\\bigl [p\\\\cdot (p-k_3-k_4)\\\\bigr ]^2\\\\biggr \\\\rbrace \\\\nonumber \\\\\\\\[6pt]&=&\\\\frac{\\\\pi m_Z^4}{24}\\\\bigl [(2-y-z)^2-1+y+z-yzw\\\\bigr ]\\\\,,$ where $N\\\\equiv p-k_3-k_4$ .\",\n \"In obtaining the above result, we have used [cf. Eq.\",\n \"(REF )], $(p-k_3-k_4)^2&=&m_Z^2-2p\\\\cdot (k_3+k_4)+2k_3\\\\cdot k_4=m_Z^2(1-y-z+yzw)\\\\,, \\\\\\\\[5pt]p\\\\cdot (p-k_3-k_4) &=&m^2_Z\\\\bigl [1-\\\\tfrac{1}{2}(y+z)\\\\bigr ]\\\\,.$ Hence, after employing Eq.\",\n \"(REF ), we end up with $&& \\\\int \\\\left(\\\\prod _{i=1}^4 \\\\frac{d^3 k_i}{2E_i}\\\\right)\\\\delta ^4(p-k_1-k_2-k_3-k_4) \\\\,(p\\\\cdot k_1)^2\\\\nonumber \\\\\\\\[10pt]&& \\\\qquad =\\\\frac{\\\\pi ^3 m_Z^8}{96}\\\\int _0^{1}\\\\!z\\\\,dz\\\\!\",\n \"\\\\left\\\\lbrace \\\\int _0^{1-z} y\\\\,dy\\\\int _0^1 dw+\\\\!\\\\int _{1-z}^1 y\\\\,dy\\\\!\",\n \"\\\\int _{(y+z-1)/yz}^1 \\\\!dw\\\\right\\\\rbrace \\\\bigl [(2-y-z)^2-1+y+z-yzw\\\\bigr ]\\\\nonumber \\\\\\\\[6pt]&&\\\\qquad \\\\,\\\\, =\\\\frac{\\\\pi ^3 m_Z^8}{1280}\\\\,.$ Collecting all our results, we end up with $\\\\Gamma =\\\\frac{g^2 m_Z^5\\\\bigl [(\\\\operatorname{Im}Z_8)^2+(\\\\operatorname{Im}Z_9)^2\\\\bigr ]}{3^2\\\\cdot 5\\\\cdot 2^{14}\\\\, \\\\pi ^5 M_P^4\\\\cos ^2\\\\theta _W}\\\\,.$ Relative to the decay rate of $Z$ into a neutrino pair, $\\\\Gamma (Z\\\\rightarrow \\\\nu \\\\bar{\\\\nu })= g^2 m_Z/(96\\\\pi \\\\cos ^2\\\\theta _W)$ , $\\\\frac{\\\\Gamma (Z\\\\rightarrow QQQQ^*)}{\\\\Gamma (Z\\\\rightarrow \\\\nu \\\\bar{\\\\nu })}=\\\\frac{(\\\\operatorname{Im}Z_8)^2+(\\\\operatorname{Im}Z_9)^2}{3\\\\cdot 5\\\\cdot 2^{9}\\\\, \\\\pi ^4 }\\\\left(\\\\frac{m_Z}{M_P}\\\\right)^4\\\\,.$ Finally, we note that the decay rate for $Z\\\\rightarrow QQ^*Q^*Q^*$ is identical to the one given above.\",\n \"Since $Q$ and $Q^*$ are mass degenerate, the experimentally observable width would be a factor of 2 times the one given in Eq.\",\n \"(REF ), as quoted in Eq.\",\n \"(REF ).\"\n ]\n]","isConversational":false},"id":{"kind":"string","value":"1808.08629"}}},{"rowIdx":195,"cells":{"text":{"kind":"list like","value":[["Quasistates and quasiprobabilities"],["Abstract The quasiprobability representation of quantum states addresses two main concerns, the identification of nonclassical features and the decomposition of the density operator.","While the former aspect is a main focus of current research, the latter decomposition property has been studied less frequently.","In this contribution, we introduce a method for the generalized expansion of quantum states in terms of so-called quasistates.","In contrast to the quasiprobability decomposition through nonclassical distributions and pure-state operators, our technique results in classical probabilities and nonpositive semidefinite operators, defining the notion of quasistates, that carry the information about the nonclassical characteristics of the system.","Therefore, our method presents a complementary approach to prominent quasiprobability representations.","We explore the usefulness of our technique with several examples, including applications for quantum state reconstruction and the representation of nonclassical light.","In addition, using our framework, we also demonstrate that inseparable quantum correlations can be described in terms of classical joint probabilities and tensor-product quasistates for an unambiguous identification of quantum entanglement."],["Introduction","Quantum systems can exhibit properties which have no analog in the classical realm, opening possibilities for technological innovations beyond the limitations posed by classical physics [1].","For assessing the quantumness of a state, the bisection of a system into a classically accessible domain and a genuinely quantum-mechanical sector has become one of the major challenges of current research.","A successful approach to performing such a desired separation are quasiprobability distributions, such as the Wigner-Weyl function [2], [3].","Within this framework, a state is identified to be nonclassical when the corresponding quasiprobability does not have the properties of a classical probability distribution.","Beyond this nonclassicality aspect, another purpose of quasiprobabilities is the full characterization of a quantum state, for example, using the Glauber-Sudarshan representation to describe quantized harmonic oscillators [4], [5].","As such, the description of a density operator in terms of quasiprobabilities and classical reference states, e.g., coherent states, is an equally important feature of quasiprobabilities.","However, despite having such a vital impact on the state's representation, the density operator expansion is, by far, less frequently considered.","This is at least partially a result of a missing framework which focuses on the decomposition of density operators.","Thus, the main objective of this contributions is to devise such a missing methodology that naturally leads to the useful and complementary concept of quasistates.","The notion of quasiprobabilities has become one of the main hallmarks for certifying quantum features since it allows us to study resources for practical tasks.","For instance, negative quasiprobabilities have been identified as indicators for the state's usefulness in performing quantum computation protocols [6], [7].","Moreover, negativities in quasiprobabilities can be related to a broad range of concepts of nonclassicality, such as contextuality [8] and quantum entanglement [9].","In addition, quasiprobabilities can be helpful to determine the actual probabilities of measurement outcomes [10], [11].","Furthermore, quasiprobabilities can be defined for many relevant physical systems, such as continuous-variable harmonic oscillators [4], [5] or discrete-variable angular-momentum-type degrees of freedom [12], [13].","Typically, the underlying quasiprobabilities are obtained by generalizing a classical phase space to the quantum domain; see Ref.","[14] for an early study and Ref.","[15] for a recent generalization.","For example, quasiprobabilities in finite-dimensional systems can be introduced in this manner, cf., e.g., Refs.","[16], [17], [18], [19].","With regard to the quantum state representation, this typically requires us to identify pure reference states which are compatible with the properties of classical physics [20].","Based on such classical reference states, a general quasiprobability decomposition of a quantum state can be constructed [19].","Examples relevant for quantum technologies include the formulation of entanglement quasiprobabilities [21], [22], which become negative for quantum correlated states.","Once a quasiprobability representation is constructed, it can be further generalized.","Again, pioneering applications can be found in quantum optics, where the Glauber-Sudarshan and Wigner-Weyl quasiprobabilities can be unified in terms of so-called $s$ -parametrized quasiprobabilities [23], [24], which also include the well-known Husimi-Kano distributions [25], [26].","This unification is achieved by a convolution of the quasiprobability with a Gaussian function and can be further generalized [27], [28], [29], including non-Gaussian scenarios [30], [31].","A successful application of the $s$ -parameter approach in a finite-dimensional system has been reported as well [32].","In addition, joint non-Gaussian quasiprobabilities for multiple optical modes and points in time have been introduced to study quantum correlated light [33], [34].","Although such formal aspects led to profound insights into quantum physics when compared to classical statistical theories (see, e.g., [35], [36]), the method of quasiprobabilities is also of experimental importance.","The most frequent implementation is the identification of nonclassical light in terms of negative Wigner-Weyl functions; see Refs.","[37], [38] for early and recent examples.","While the Glauber-Sudarshan distribution can be highly singular or even ambiguous [39], [40], [41], the reconstruction of this function is feasible in certain experiments as well [42].","Moreover, the non-Gaussian generalization of this distribution is always experimentally accessible, such as reported for squeezed light [43].","Even imperfect detection schemes can be employed for the reconstruction of nonclassical phase-space distribution of light [44].","Conversely, a phase-space representation of the detector can experimentally verify quantum properties of the detection device used [45].","Another approach [46], [47] enables an experimental reconstruction of phase-space distributions and entirely circumvents a detector characterization by analyzing certain data patterns [48].","Beyond quantum light, equally insightful is the characterization of matter systems using quasiprobabilities, such as demonstrated for motional states of ions [49] and large atomic ensembles [50].","It is also worth mentioning that quantum correlations in composite hybrid systems are also accessible via generalized quasiprobabilities [51], [52].","Therefore, quasiprobabilities present a highly successful and versatile approach to identifying nonclassical properties of quantum states in theory and experiment.","Still, as outlined above, another important aspect is the quantum state decomposition.","Yet, in contrast to the vast number of examples for the nonclassicality certification, studies of the decomposition properties are rarely done; a gap we aim to close in this article.","Based on the decomposition of states in terms of quasiprobabilities, we introduce a generalized expansion of quantum states in terms of quasistates.","This method is complementary to the previously known approaches in which a density operator is decomposed in terms of classical reference states and a negative quasiprobability density.","In our general method, we can find a decomposition which results in a nonnegative (i.e., classical) probability density by overcoming the usage of physical reference states in the decomposition.","This naturally establishes the concept of quasistates, which then become the relevant objects for certifying the different kinds of nonclassicality.","The usefulness of this change of perspective from quasiprobabilities to quasistates is studied in detail.","One practical application relates to the experimental reconstruction of density operators.","Moreover, as quasiprobabilities are most frequently applied in optical systems, we also perform a detailed analysis of quasistates that correspond to prominent phase-space representations of light.","Finally, we investigate how quantum entanglement can be uniquely characterized via classical joint probability distributions when employing quasistates.","Therefore, our approach offers a toolbox for the characterization and decomposition of quantum states.","This article is structured as follows.","In Sec.",", we discuss the general methodological framework.","An application to the quantum state reconstruction is developed in Sec.",".","Phase-space based quasistates for quantized light are studied in Sec.",".","In Sec.",", quantum correlations are analyzed within the proposed framework.","Finally, concluding discussions are presented in Sec.","."],["Conceptual framework","To characterize the state of a quantum system, a representation of the density operator is required.","In general, a state decomposition consists of a family of pure states, defining a set $\\mathcal {S}$ , and corresponding expansion coefficients.","Then the density operator takes the form $\\hat{\\rho }=\\int _{\\mathcal {S}} d\\psi \\, P(\\psi ) |\\psi \\rangle \\langle \\psi |,$ where $d\\psi $ indicates an integral over the volume of $\\mathcal {S}$ and $P$ is a correspondingly defined density.","In the case of a discrete decomposition, the integral may be replaced by a sum.","Note that we deliberately choose not to specify the set $\\mathcal {S}$ to keep our treatment as broadly applicable as possible.","Arguably the most well-known example of a representation (REF ) is the spectral decomposition.","In this case, $\\mathcal {S}$ is the set of eigenstates and the probability mass function $P$ returns the corresponding eigenvalues.","Other examples for a decomposition (REF ) are studied in the continuation of this work and have applications, for example, in quantum optics and quantum information.","It is worth mentioning that a general method to construct quasiprobabilities $P$ for a given set $\\mathcal {S}$ has been recently devised [19].","In that approach, it is ensured that $P$ is a classical probability distribution when the state is in the convex hull $\\mathrm {conv}\\lbrace |\\psi \\rangle \\langle \\psi |:\\psi \\in \\mathcal {S}\\rbrace $ , which is a nontrivial task as the decomposition (REF ) is typically not unique [19].","For our purpose, we may generalize the above treatment.","Specifically, a convolution kernel $K:\\mathcal {S}\\rightarrow \\mathcal {S}$ is introduced together with the kernel $K^{-1}$ for the corresponding deconvolution.","Then we get an equivalent decomposition as $\\hat{\\rho }=\\int _{\\mathcal {S}} d\\chi \\, P_K(\\chi )\\hat{\\Delta }_{K}(\\chi ).$ Therein we have a modified density $P_K$ and a modified family of operators $\\hat{\\Delta }_{K}$ , $P_K(\\chi )=&\\int _{\\mathcal {S}} d\\psi \\, P(\\psi )K(\\psi ,\\chi ),\\\\\\hat{\\Delta }_{K}(\\chi )=&\\int _{\\mathcal {S}} d\\psi ^{\\prime }\\, K^{-1}(\\chi ,\\psi ^{\\prime }) |\\psi ^{\\prime }\\rangle \\langle \\psi ^{\\prime }|.$ Using the properties that the operations used are inverse to one another, $\\delta (\\psi ,\\psi ^{\\prime })=\\int _{\\mathcal {S}} d\\chi K(\\psi ,\\chi )K^{-1}(\\chi ,\\psi ^{\\prime })$ with $\\delta $ being the Dirac distribution, one can directly see that Eqs.","(REF ) and (REF ) describe the same density operator $\\hat{\\rho }$ .","In general, the distribution $P_K$ in Eq.","(REF ) is not a probability density, even if $P$ was.","For such generalized distributions, the name quasiprobabilities was established.","As discussed in the Introduction, such quasiprobability densities cover a wide range of applications, mainly for purpose of identifying quantum features.","In close analogy to the notion of a quasiprobability, we are going to demonstrate that the operators $\\hat{\\Delta }_K$ in Eq.","() are, in general, not physical density operators.","Consequently, we refer to such operators as quasistates.","A main focus of previous research was devoted to characterizing quasiprobabilities.","Often, the idea of a decomposition of the quantum state [cf.","Eqs.","(REF ) and (REF )] in terms of such distributions was neglected.","In particular, a general characterization of the distinctive features of quasistates and their applications beyond being a mathematical tool does not exist.","For this reason, we are going to perform the missing comprehensive investigation of quasistates as defined in Eq.","() and discuss useful applications of such operators."],["State reconstruction","One interesting application of quasistates are state reconstruction protocols as we show in this section.","The following considerations are based on a recent work [11] in which the dual representation of measurement operators has been introduced.","Here we demonstrate how this method relates to quasistates and can be used for the general reconstruction of density operators."],["Dual representation","Let us briefly revisit the findings in Ref.","[11] with regards to our method.","Suppose $\\lbrace \\hat{\\Pi }(j)\\rbrace _{j\\in \\mathcal {S}}$ is a positive operator-valued measure (POVM).","As a complementary set of operators, the notion of contravariant operator-valued measured (COVM) was introduced.","This defines a set $\\lbrace \\hat{\\Gamma }(j^{\\prime })\\rbrace _{j^{\\prime }\\in \\mathcal {S}}$ that satisfies the orthonormality relation $\\mathrm {tr}\\big [\\hat{\\Gamma }(j^{\\prime })\\hat{\\Pi }(j)\\big ]=\\delta (j^{\\prime },j).$ This means that the COVM represent dual basis operators to the measured POVM.","In contrast to the POVM, COVM operators are not necessarily positive semidefinite.","However, they are of particular interest, for example, when considering imperfections in the measurement process [11].","The construction of COVM operators is based on a kernel defined via the elements $K(l,j)=\\mathrm {tr}\\big [\\hat{\\Pi }(l)\\hat{\\Pi }(j)\\big ].$ Specifically, it was shown that $\\hat{\\Gamma }(j^{\\prime })=\\sum _{l\\in \\mathcal {S}}K^{-1}(j^{\\prime },l)\\hat{\\Pi }(l),$ for the discrete case.","In the following, let us explore the relation between the COVM and quasistates.","For simplicity, we assume that the considered complex Hilbert space is finite dimensional, $\\dim _{\\mathbb {C}}\\mathcal {H}=d<\\infty $ .","The set of Hermitian operators over $\\mathcal {H}$ is a real valued space with the dimension $\\dim _{\\mathbb {R}}\\mathrm {Herm}(\\mathcal {H})=d^2$ .","Further, rather than restricting to a POVM for a single observable as done previously, we consider an informationally complete set $\\lbrace \\hat{\\Pi }(\\psi )\\rbrace _{\\psi \\in \\mathcal {S}}$ , with $\\hat{\\Pi }(\\psi )=|\\psi \\rangle \\langle \\psi |,$ which guarantees that a reconstruction of the full quantum state is possible and implies the cardinality $|\\mathcal {S}|=d^2$ .","However, this also means that not all elements of $\\mathcal {S}$ can be pairwise orthogonal vectors.","See Refs.","[53], [54], [55] for introductions to informationally complete measurements.","Because of the properties of the now studied POVM, we can expand the quantum state as given in Eq.","(REF ), $\\hat{\\rho }=\\sum _{\\psi \\in \\mathcal {S}}P(\\psi ) \\hat{\\Pi }(\\psi ).$ It is important to mention that $P(\\psi )$ , in general, does not correspond to the probability of the measurement of $\\hat{\\Pi }(\\psi )$ , i.e., $\\mathrm {tr}[\\hat{\\rho }\\hat{\\Pi }(\\psi )]\\ne P(\\psi )$ .","However, we can employ the same formalism as discussed above, cf.","Eqs.","(REF ) and (REF ), to expand the state as [11] $\\hat{\\rho }=\\sum _{\\chi \\in \\mathcal {S}}\\rho (\\chi ) \\hat{\\Gamma }(\\chi ).$ Now we can identify the coefficient with the measurement probability, $\\rho (\\psi ^{\\prime })=\\mathrm {tr}[\\hat{\\rho }\\hat{\\Pi }(\\psi ^{\\prime })]$ for all $\\psi ^{\\prime }\\in \\mathcal {S}$ , because of relation (REF ).","In other words, the expansion in Eq.","(REF ) can be used to directly reconstruct the density operator in terms of measurement outcomes and dual operators.","Let us specifically relate the found reconstruction with the generalized decomposition (REF ).","Using the definitions (REF ) and (REF ), we find that the kernel under study is given by the $d^2\\times d^2$ Gram-Schmidt matrix $K=\\left[|\\langle \\psi |\\chi \\rangle |^2\\right]_{\\psi ,\\chi \\in \\mathcal {S}}.$ Recall that information completeness implies that this matrix is a bijection, thus invertible.","Further, as a result of Eqs.","() and (REF ), the thought-after quasistates are identical to the COVM elements, $\\hat{\\Delta }_K(\\chi )=\\hat{\\Gamma }(\\chi ).$ Consequently, it also holds true that $P_K(\\chi )=\\rho (\\chi )$ ; see Eqs.","(REF ) and (REF ).","Therefore, quasistates allow for the direct reconstruction of the density operator under study.","It is worth mentioning that imperfections in any measurement, meaning that $\\hat{\\Pi }(\\psi )$ is not a pure-state projector, can be treated similarly to the thorough discussion for a single observable carried out in Ref.","[11].","Furthermore, if the POVM does not allow for a full reconstruction or is over-complete (e.g., because $|\\mathcal {S}|\\ne d^2$ ), then the state in the spanned subspace can be reconstructed or a pseudo-inversion of $K$ can be performed [11]."],["Example","To further discuss the above findings, we study the specific example of a qubit system ($d=2$ ), defined via the orthonormal Hilbert space basis $\\lbrace |0\\rangle ,|1\\rangle \\rbrace $ .","The four rank-one projection operators $|\\psi _j\\rangle \\langle \\psi _j|$ in Eq.","(REF ), also defining the set $\\mathcal {S}$ , can be chosen according to the states $\\nonumber |\\psi _1\\rangle =|1\\rangle ,|\\psi _2\\rangle =&\\frac{\\sqrt{2} |0\\rangle +|1\\rangle }{\\sqrt{3}},|\\psi _3\\rangle =\\frac{\\sqrt{2} w|0\\rangle +|1\\rangle }{\\sqrt{3}},\\\\\\text{and }|\\psi _4\\rangle =&\\frac{\\sqrt{2} w^2|0\\rangle +|1\\rangle }{\\sqrt{3}},$ where $w=\\exp [2\\pi i/3]$ ; see the top-left panel in Fig.","REF for their geometric Bloch-sphere representation.","This allows us to construct $K$ via Eq.","(REF ) and compute its inverse, $K=\\frac{1}{3}\\begin{bmatrix}3 & 1 & 1 & 1\\\\1 & 3 & 1 & 1\\\\1 & 1 & 3 & 1\\\\1 & 1 & 1 & 3\\end{bmatrix}\\!,\\,K^{-1}=\\frac{1}{4}\\begin{bmatrix}5 & -1 & -1 & -1\\\\-1 & 5 & -1 & -1\\\\-1 & -1 & 5 & -1\\\\-1 & -1 & -1 & 5\\end{bmatrix}\\!.$ This then yields the quasistates as $\\hat{\\Delta }_K(\\psi _1)=&|1\\rangle \\langle 1|-\\frac{1}{2}|0\\rangle \\langle 0|\\text{ and}\\\\\\nonumber \\hat{\\Delta }_K(\\psi _j)=&\\frac{|0\\rangle \\langle 0|+\\sqrt{2}w^{j-2}|0\\rangle \\langle 1|+\\sqrt{2}w^{\\ast (j-2)}|1\\rangle \\langle 0|}{2},$ for $j=2,3,4$ .","As we can see from the top-right panel in Fig.","REF , these quasistates do not resemble physical quantum states because of the negative eigenvalue.","Figure: (Color online)The elements of 𝒮\\mathcal {S} given in Eq.","() in a tetrahedron configuration (larger green bullets) are shown in the top-left Bloch-sphere representation.The smaller red bullet indicates the state ρ ^=|0〉〈0|\\hat{\\rho }=|0\\rangle \\langle 0| to be reconstructed.The top-right panel shows the two eigenvalues of the resulting quasistates Δ ^ K (ψ j )\\hat{\\Delta }_K(\\psi _j), being identical for j=1,...,4j=1,\\ldots ,4.For the reconstruction of the state under study, the bottom panels depict the expansion coefficients P(ψ j )P(\\psi _j) and P K (ψ j )P_K(\\psi _j) to be used with |ψ j 〉〈ψ j ||\\psi _j\\rangle \\langle \\psi _j| and Δ ^ K (ψ j )\\hat{\\Delta }_K(\\psi _j) [Eqs.","() and ()], respectively.Let us now consider the specific state $\\hat{\\rho }=|0\\rangle \\langle 0|$ for the reconstruction.","Then, the measured and, therefore, nonnegative expansion coefficients are $P_K(\\psi _j)=\\mathrm {tr}(\\hat{\\rho }|\\psi _j\\rangle \\langle \\psi _j|)=\\langle \\psi _j|\\hat{\\rho }|\\psi _j\\rangle ,$ using the projectors defined via the states in Eq.","(REF ).","In addition, the corresponding COVM elements, being the quasistates in Eq.","(REF ), result in the expansion coefficients $P(\\psi _j)=\\mathrm {tr}[\\hat{\\rho }\\hat{\\Delta }_K(\\psi _j)]$ , which resembles a quasiprobability distribution.","The bottom part in Fig.","REF shows the values of $P_K$ and $P$ for the state $\\hat{\\rho }=|0\\rangle \\langle 0|$ .","Using the nonnegative $P_K$ (see Fig.","REF ), we can now directly confirm that the density operator $\\hat{\\rho }=|0\\rangle \\langle 0|$ is expanded as described in Eq.","(REF ) while using the quasistates in Eq.","(REF ), $\\hat{\\rho }=\\sum _{j=1,\\ldots ,4}P_K(\\psi _j)\\hat{\\Delta }_K(\\psi _j)$ .","The possibly negative elements of $P$ also allow for the decomposition of the state, $\\hat{\\rho }=\\sum _{j=1,\\ldots ,4}P(\\psi _j)|\\psi _j\\rangle \\langle \\psi _j|$ .","However, $P$ does not correspond to a directly measured quantity.","Thus, quasistates provide a beneficial tool to reconstruct density operators using measured quantities.","Let us also briefly comment on the extension of our approach to continuous-variable systems, $d=\\infty $ .","In fact, already in Refs.","[56], [57], a method was introduced to expand the state of a single-mode radiation field in terms of so-called pattern functions.","Using the here developed framework, we can directly identify these pattern functions with the wave functions of the corresponding quasistates.","Beyond this reconstruction approach, we study the expansion of nonclassical states of light in the following section."],["Quantum Light","We now demonstrate the application of quasistates for the description of nonclassical quantum states.","In particular, we consider a harmonic oscillator, which can, for example, describe a quantized radiation mode.","To describe the system's nonclassicality, the prominent Glauber-Sudarshan representation has been developed [4], [5], $\\hat{\\rho }=\\int _{\\mathbb {C}}d\\alpha \\,P(\\alpha )|\\alpha \\rangle \\langle \\alpha |.$ Here this decomposition implies that we identify the set $\\mathcal {S}$ with the complex coherent amplitudes $\\alpha \\in \\mathbb {C}$ , defining the coherent states $|\\alpha \\rangle $ .","For a thorough introduction to quantum optics in phase space, see Ref.","[58].","A state is referred to as nonclassical if $P$ does not exhibit the properties of a classical probability distribution.","In the following, we begin our considerations of quasistates for nonclassical light in the Fourier picture and then proceed to prominent examples of quasiprobabilities and their dual representation via quasistates."],["Fourier representation","The characteristic function $\\Phi (\\beta )$ is the Fourier transform $F$ of a distribution, which is accessible via the kernel $F(\\alpha ,\\beta )=\\exp \\left(\\beta \\alpha ^\\ast -\\beta ^\\ast \\alpha \\right),$ i.e., in our notation, $P_{F}=\\Phi $ .","The inverse Fourier transform, defined via $F^{-1}(\\beta ,\\alpha )=F(\\alpha ,\\beta )^\\ast /\\pi ^2$ , results in the corresponding quasistates [Eq.","()], $\\hat{\\Delta }_{F}(\\beta )=\\int _{\\mathbb {C}}d\\beta \\,\\frac{\\exp \\left(\\beta ^\\ast \\alpha -\\beta \\alpha ^\\ast \\right)}{\\pi ^2}|\\alpha \\rangle \\langle \\alpha |.$ For the evaluation of the integral, it is useful to represent the coherent-state projectors in terms of normally ordered exponential of creation ($\\hat{a}^\\dag $ ) and annihilation ($\\hat{a}$ ) operators, which reads $|\\alpha \\rangle \\langle \\alpha |={:}\\exp (-[\\hat{a}-\\alpha ]^\\ast [\\hat{a}-\\alpha ]){:}$ [58].","This representation yields $\\hat{\\Delta }_{F}(\\beta )=\\frac{\\exp (-|\\beta |^2)}{\\pi }{:}\\hat{D}(-\\beta ){:},$ where $\\hat{D}(\\gamma )=\\exp (\\gamma \\hat{a}^\\dag -\\gamma ^\\ast \\hat{a})=\\exp (-|\\gamma |^2/2){:}\\hat{D}(\\gamma ){:}$ is the unitary displacement operator.","Therefore, the quasistates $\\hat{\\Delta }_{F}(\\beta )$ correspond (up to a rescaling) to unitary operators in the case that the convolution kernel is the Fourier transformation.","Note that instead of the Fourier transform, the Laplace transform can be advantageous in certain scenarios as well [59].","One application, which also connects to the previous section, is the state reconstruction via balanced homodyne detection.","For this purpose, let us recall that the characteristic function of the Wigner-Weyl distribution takes the form [58] $\\Phi (\\beta )e^{-|\\beta |^2/2}=\\langle \\hat{D}(\\beta )\\rangle =\\langle \\exp \\left(i|\\beta |\\hat{x}[\\pi /2-\\arg \\beta ]\\right)\\rangle ,$ where we used the quadrature operator $\\hat{x}[\\varphi ]=e^{i\\varphi }\\hat{a}+e^{-i\\varphi }\\hat{a}^\\dag $ .","This means for a balanced homodyne measurement, providing a data set $\\lbrace x_j[\\varphi _j]\\rbrace _{j=1,\\ldots ,N}$ of quadratures $x_j$ for the phase values $\\varphi _j$ , we can approximate $\\begin{aligned}\\hat{\\rho }=&\\int _{\\mathbb {C}}d\\beta \\,\\Phi (\\beta )\\hat{\\Delta }_{F}(\\beta )\\\\=&\\int _{-\\infty }^\\infty dr\\int _{-\\pi /2}^{\\pi /2} d\\varphi \\,\\frac{\\pi }{\\pi }e^{r^2/2}\\langle e^{ir\\hat{x}[\\varphi ]}\\rangle \\hat{\\Delta }_{F}(ire^{-i\\varphi })\\\\\\approx &\\int _{-\\infty }^\\infty dr\\, e^{r^2/2} \\frac{\\pi }{N}\\sum _{j=1}^N e^{irx_j[\\varphi _j]}\\hat{\\Delta }_{F}(ire^{-i\\varphi _j}),\\end{aligned}$ which corresponds to a sampling approach as a consequence of a quasistate representation [see specifically the first line of Eq.","(REF ) and the use of $\\hat{\\Delta }_{F}$ ].","In fact, further analyses show that this approach is indeed equivalent to established reconstruction methods bases on balanced homodyne detection and using pattern functions [56], [57], [60].","In addition, it is worth emphasizing that the phases $\\varphi $ have to be uniformly distributed in the interval $[-\\pi /2,\\pi /2]$ to ensure an optimal sampling [61].","Figure: (Color online)The QQ function [Eq.","()] of the quasistate [Eq.","()] for different parameters ss, pp, and qq.The top-left plot corresponds to a coherent (vacuum) state.The top-center plot shows the maximally singular quasistate .The top-right plot includes, in addition, squeezing.The bottom row depicts cases in which the QQ function becomes complex;the larger and corresponding smaller plots show the amplitude |Q||Q| and phase argQ\\arg Q, respectively.Note the difference in the phases when comparing the left and right plots."],["Generalized optical quasistates","Translational types of convolutions in the original space, e.g., $P_{\\kappa }(\\alpha ^{\\prime })=\\int _{\\mathbb {C}}d\\alpha \\,P(\\alpha )\\kappa (\\alpha ^{\\prime }-\\alpha )$ , take a product form in the Fourier domain.","In this scenario, we can consider the representation $\\hat{\\rho }=\\int _{\\mathbb {C}}d\\beta \\,\\left[\\Phi (\\beta )\\Omega (\\beta )\\right]\\left[\\frac{e^{-|\\beta |^2}}{\\pi \\Omega (\\beta )}{:}\\hat{D}(-\\beta ){:}\\right],$ for a kernel $\\Omega $ .","The inverse Fourier transform gives the distribution $P_{F^{-1}\\Omega F}$ as well as the quasistates $\\hat{\\Delta }_{F^{-1}\\Omega F}(\\alpha )=\\int _{\\mathbb {C}}d\\beta \\,e^{\\beta \\alpha ^\\ast -\\beta ^\\ast \\alpha }\\frac{e^{-|\\beta |^2}}{\\pi \\Omega (\\beta )}{:}\\hat{D}(-\\beta ){:}.$ Such operators have been extensively studied in connection to operator orderings and their resulting quasiprobabilities [23], [24], [27], [28], [29].","Here, however, let us analyze them based on their own merit as quasistates.","Of particular interest are the Gaussian functions $\\Omega (\\beta )=\\exp \\left(-\\frac{1-s}{2}|\\beta |^2-\\frac{p}{4}\\beta ^2-\\frac{q}{4}\\beta ^\\ast \\right),$ where $|p|=|q|$ [27], [28], [29].","In Table REF , specific examples and their connection to known quasiprobabilities are given.","Evaluating the integrals as demonstrated in Appendix , we get the quasistates in normal ordering as $\\hat{\\Delta }_{F^{-1}\\Omega F}(0)=\\frac{2\\,{:}\\exp \\left(\\frac{-2[1+s]\\hat{a}^\\dag \\hat{a} +q\\hat{a}^{\\dag 2}+p\\hat{a}^2}{(1+s)^2-pq}\\right){:}}{\\sqrt{(1+s)^2-pq}}.$ Note that because of the convolution property, we have $\\hat{\\Delta }_{F^{-1}\\Omega F}(\\alpha )=\\hat{D}(\\alpha )\\hat{\\Delta }_{F^{-1}\\Omega F}(0)\\hat{D}(\\alpha )^\\dag $ for the displaced quasistates.","In Fig.","REF , several examples of the phase-space $Q$ -function representation, $Q(\\alpha )=\\frac{\\langle \\alpha |\\hat{\\Delta }_{F^{-1}\\Omega F}(0)|\\alpha \\rangle }{\\pi },$ are shown for different parameters that define the quasistates in Eq.","(REF ).","It is also worth mentioning that the $Q$ function is the Husimi-Kano distribution.","Table: Parameters [cf.","Eq.","()] for prominent quasiprobability distributions frequently used in quantum optics.Interestingly, the considered class of quasistates, given in Eq.","(REF ), can be related to squeezed versions of thermal-like quasistates, $\\begin{aligned}\\hat{\\Delta }_{F^{-1}\\Omega F}(0)=&\\exp ([\\zeta ^\\ast \\hat{a}^2-\\zeta \\hat{a}^{\\dag 2}]/2)\\\\&\\times \\frac{1}{\\bar{n}+1}\\sum _{n\\in \\mathbb {N}}\\left(\\frac{\\bar{n}}{\\bar{n}+1}\\right)^n|n\\rangle \\langle n|\\\\&\\times \\exp ([\\zeta ^\\ast \\hat{a}^2-\\zeta \\hat{a}^{\\dag 2}]/2)^\\dag ,\\end{aligned}$ where $\\bar{n}\\in \\mathbb {C}$ generalizes the notion of a mean thermal photon number and $\\zeta $ is the squeezing parameter.","A full and exact analysis is provided in Appendix , and it provides the relations between the different parameter sets as $\\zeta =e^{i\\arg \\tau }\\mathrm {artanh}\\,|\\tau |\\quad \\text{and}\\quad \\bar{n}=\\frac{\\omega }{1-\\omega },$ with $\\omega =\\frac{r-1}{r+1},\\,|\\tau |^2=\\frac{s-r}{s+r},\\,\\text{and}\\,\\frac{\\tau }{\\tau ^\\ast }=\\frac{q}{p},$ where $r=\\sqrt{s^2-pq}$ .","Recall that we require $|q|=|p|$ .","Also note that the squeezing operation is unitary which implies that we have the eigenvalues $(1-\\omega )\\omega ^n$ , following a geometric distribution.","In the following, let us study the resulting quasistates in more detail."],["$s$ -Parameterized quasistates","Arguably the most frequently used choice of parameters is obtained for $p=q=0$ , likewise $\\zeta =\\tau =0$ .","This choice results in the $s$ -parametrized quasiprobability distributions $P_{F^{-1}\\Omega F}$ .","Here, we additionally find the quasistates $\\hat{\\Delta }_{F^{-1}\\Omega F}$ .","In particular, those quasistates have the eigenvalues $(1-\\omega )\\omega ^n=\\frac{\\bar{n}^{n}}{(\\bar{n}+1)^{n+1}}=\\frac{2}{1+s}\\left(-\\frac{1-s}{1+s}\\right)^n$ for $n\\in \\mathbb {N}$ ; see Eq.","(REF ).","For the parameters $-1<s<1$ , the eigenvalues (REF ) of the quasistates are negative for odd $n$ , certifying that these operators are only accessible in terms of our generalization of the notion of a physical state.","In the limit $s\\rightarrow 1$ , we get the eigenvalue 1 for $n=0$ and 0 for $n>0$ , defining the vacuum state.","Its $Q$ function is shown on the left in the top row of Fig.","REF , and the corresponding quasiprobability distribution is the original Glauber-Sudarshan distribution.","Furthermore, the case $s=0$ interestingly describes an operator with the maximally possible singularities a Glauber-Sudarshan distribution can exhibit [41]; see also the center plot in the top row of Fig.","REF for its $Q$ function.","There, we observe that the spread (i.e., variance) in phase space is reduced in all directions when compared to the vacuum state, further highlighting that it is a genuine quasistate that is incompatible with the uncertainty relation which holds true for any physical state.","The underlying phase-space distribution $P_{F^{-1}\\Omega F}$ is the Wigner-Weyl distribution.","Finally, the quasistates for the Husimi-Kano distribution are obtained in the limit $s\\rightarrow -1$ , and its corresponding $Q$ function is described by a singular Dirac distribution.","Beyond real-valued $s$ parameters, we can additionally consider complex values as well [58].","Then, the eigenvalues (REF ) are complex too.","The example $s=i$ is shown on the left of the bottom row in Fig.","REF .","While the amplitude $|Q|$ is identical with the vacuum state, the quasistate characteristics are clearly visible through the nontrivial phases, $\\arg Q\\ne 0$ ."],["Non-$s$ -parametrized quasistates","Moreover, we can also consider the quasistates for cases with $p,q\\ne 0$ .","As we demand $|q|=|p|$ , we have two extremal scenarios, $q=p^\\ast $ and $q=-p^\\ast $ .","In addition to the $s$ -parametrized quasiprobabilities in Table REF , we also listed the Agarwal-Wolf distributions.","They correspond to the case $q=-p^\\ast =\\mp 1$ (also, $s=0$ ).","The complex-valued $Q$ function of the resulting quasistate $\\hat{\\Delta }_{F^{-1}\\Omega F}$ is shown in the bottom-right plot of Fig.","REF .","The amplitude $|Q|$ is again compatible with the uncertainty principle bounded by the vacuum state, similarly to the previously discussed scenario of a complex $s$ parameter.","Here, however, the functional behavior of the phase $\\arg Q$ does not possess a radial symmetry.","Rather, the quasistates to the Agarwal-Wolf distributions exhibit hyperbolic isophase contours.","Note that this dismisses a sometimes held believe that Agarwal-Wolf distributions are an example of a $s$ -parametrized quasiprobability.","Finally, we can ask ourselves what happens in the case $q=p^\\ast $ .","An example is shown in the top-right plot in Fig.","REF .","Interestingly, the resulting quasistates are squeezed.","Specifically for a squeezing parameter $\\zeta $ , we can choose $s=\\frac{1+\\tanh ^2|\\zeta |}{1-\\tanh ^2|\\zeta |}\\text{ and }q=\\frac{-2e^{i\\arg \\zeta }\\tanh |\\zeta |}{1-\\tanh ^2|\\zeta |}=p^\\ast .$ This choice implies $\\bar{n}=0$ and means that the quasistates are pure squeezed states, cf.","Eqs.","(REF ) and (REF ).","Consequently, the quasiprobability density $P_{F^{-1}\\Omega F}(\\alpha )$ now describes the decomposition of the density operator in terms of displaced squeezed states with a coherent amplitude $\\alpha $ and a squeezing defined by $\\zeta $ .","This further implies that we have a generalized Glauber-Sudarshan quasiprobability distribution which is, however, a phase-space representation that expands the state in terms of squeezed states.","Thus, the case $q=p^\\ast $ significantly extends the notion of $s$ -parametrized distributions to additionally include arbitrary squeezed states."],["Quantum correlations","In a recent work [19], we generalized the concept of quasiprobability representations to more general notions of quantum coherence [62], beyond the specific example of harmonic oscillators studied in the previous section.","Among the various types of quantumness, quantum correlations between multiple degrees of freedom play an outstanding role for applications in quantum technologies [1], [63].","For this reason, let us study the entanglement of quantum systems within the framework of quasistates.","A comprehensive introduction to the theory of entanglement (likewise, inseparability) can be found in Ref.","[63]."],["Entanglement quasiprobabilities","A pure separable state in a multipartite system is described by a tensor-product vector, $|(a,b,\\ldots )\\rangle =|a\\rangle \\otimes |b\\rangle \\otimes \\cdots ,$ where $a\\in \\mathcal {S}_A$ , $b\\in \\mathcal {S}_B$ , etc.","In addition, the inclusion of classical correlations in terms of statistical mixtures, given by a classical probability density $P$ , yields the definition of a mixed separable state [64], $\\hat{\\rho }{=}\\int _{S_A{\\times } S_B{\\times }\\cdots } da\\,db\\cdots P(a,b,\\ldots )|(a,b,\\ldots )\\rangle \\langle (a,b,\\ldots )|.$ A state is defined to be inseparable (i.e., entangled) if it cannot be written in the form of Eq.","(REF ).","However, when we allow for $P$ to be a quasiprobability, even entangled states can be expanded in terms of separable states using Eq.","(REF ) [21].","A construction approach for bipartite entanglement quasiprobabilities was introduced in Ref.","[22].","As this approach can be generalized from the bipartite scenario to the multipartite case [19], we restrict ourselves to bipartite systems in the following.","The optimal decomposition of entangled states in terms of separable ones warrants that the entanglement is uniquely identified through negativities in $P(a,b)$ [22], [19].","The separable states $|(a,b)\\rangle $ needed for the decomposition are obtained from the solution of the so-called separability eigenvalue equations (SEEs), $\\hat{\\rho }_a|b\\rangle =g|b\\rangle \\text{ and }\\hat{\\rho }_b|a\\rangle =g|a\\rangle ,$ with $\\hat{\\rho }_a=\\mathrm {tr}_A[\\hat{\\rho }(|a\\rangle \\langle a|\\otimes \\hat{1}_B)]$ and $\\hat{\\rho }_b=\\mathrm {tr}_B[\\hat{\\rho }(\\hat{1}_A\\otimes |b\\rangle \\langle b|)]$ .","The SEEs approach generalizes the eigenvalue problem to composite systems while respecting the tensor-product structure of the corresponding eigenvectors and was initially introduced to construct entanglement witnesses [65], [66].","The solutions $(a,b)\\in \\mathcal {S}$ allow us to construct the distribution $P$ by solving the linear problem $G\\vec{p}=\\vec{g},$ where $G=[|\\langle (a,b)|(a^{\\prime },b^{\\prime })\\rangle |^2]_{(a,b),(a^{\\prime },b^{\\prime })\\in \\mathcal {S}}$ is the Gram-Schmidt matrix of separability eigenvectors.","Furthermore, the solution vector $\\vec{p}=[P(a,b)]_{(a,b)\\in \\mathcal {S}}$ defines the quasiprobabilities, and $\\vec{g}=[g]_{(a,b)\\in \\mathcal {S}}$ is the vector of separability eigenvalues, cf.","Eq.","(REF ).","Let us point out that the here-used Gram-Schmidt matrix also relates to the reconstruction approach in Sec.","[Eq.","(REF )] and is a result of a underlying principle of convex decomposition, discussed in greater detail in Ref.","[19].","One can always find a set $\\mathcal {S}_0$ such that $\\mathcal {S}\\subset \\mathcal {S}_0\\times \\mathcal {S}_0$ ; we then set $P(a,b)=0$ for all $(a,b)\\in \\mathcal {S}_0\\times \\mathcal {S}_0\\setminus \\mathcal {S}$ .","Using the construction to find optimal entanglement quasiprobabilities via the SEEs, we can decompose any state, be it entangled or separable [22], [19], as $\\hat{\\rho }=\\sum _{(a,b)\\in \\mathcal {S}_0\\times \\mathcal {S}_0}P(a,b)|(a,b)\\rangle \\langle (a,b)|.$ It is worth mentioning that the entanglement quasiprobability is always real valued and normalized."],["Quasistate representation","A universally applicable approach to diminish the negativities in quasiprobabilities for a single system is a convolution of the form $P_K(\\chi )=rP(\\chi )+\\frac{1-r}{|\\mathcal {S}_0|}\\sum _{\\psi \\in \\mathcal {S}_0}P(\\psi ),$ with $0< r\\le 1$ and $|\\mathcal {S}_0|$ being the cardinality of the set; see Appendix for details.","The kernel under study consequently leads to quasistates of the form $\\hat{\\Delta }_K(\\chi )=\\frac{1}{r}|\\chi \\rangle \\langle \\chi |-\\frac{1-r}{r|\\mathcal {S}_0|}\\sum _{\\psi \\in \\mathcal {S}_0}|\\psi \\rangle \\langle \\psi |.$ We emphasize that quasistates defined in this manner are both Hermitian, $\\hat{\\Delta }_K(\\chi )=\\hat{\\Delta }_K(\\chi )^\\dag $ , and normalized, $\\mathrm {tr}[\\hat{\\Delta }_K(\\chi )]=1$ .","Thus, the only distinction to true states is the fact that $\\hat{\\Delta }_K(\\chi )$ is, in general, not a positive semidefinite operator.","The convolution can be generalized to composite systems via product kernels.","Choosing, for example, the same $r$ for both systems, we can reformulate Eq.","(REF ) as $\\hat{\\rho }=\\sum _{(\\tilde{a},\\tilde{b})\\in \\mathcal {S}_0\\times \\mathcal {S}_0}P_{K\\otimes K}(\\tilde{a},\\tilde{b})\\,\\hat{\\Delta }_K(\\tilde{a})\\otimes \\hat{\\Delta }_K(\\tilde{b}).$ In particular, we have $\\begin{aligned}P_{K\\otimes K}(\\tilde{a},\\tilde{b})=&r^2P(\\tilde{a},\\tilde{b})+\\frac{(1-r)^2}{|\\mathcal {S}_0|^2}\\\\&+\\frac{r(1-r)}{|\\mathcal {S}_0|}\\left[P(\\tilde{a})+P(\\tilde{b})\\right],\\end{aligned}$ using the marginal distributions $P(\\tilde{a})=\\sum _{b\\in \\mathcal {S}_0}P(\\tilde{a},b)$ and $P(\\tilde{b})=\\sum _{a\\in \\mathcal {S}_0}P(a,\\tilde{b})$ .","Eventually, we get a nonnegative distribution $P_{K\\otimes K}$ for a sufficiently large $r$ , which is also demonstrated later.","This means that an entangled state can be decomposed according to Eq.","(REF ) in terms of a classical joint probability distribution $P_{K\\otimes K}(a,b)\\ge 0$ and tensor-product quasistates $\\hat{\\Delta }_K(a)\\otimes \\hat{\\Delta }_K(b)\\ngeq 0$ [cf.","Eq.","(REF )].","Since we can chose $r=0$ for separable states, we find that quasistates are not required in this case.","Therefore, we are able to conclude that entanglement is unambiguously identified in terms of classical joint probability distributions if and only if the tensor-product operators of the decomposition have to be unphysical quasistates."],["Example","As a proof of concept, let us consider an entangled two-qubit quantum state parametrized as $\\hat{\\rho }=\\frac{1}{4}\\left[\\hat{1}\\otimes \\hat{1}+\\sum _{j\\in \\lbrace x,y,z\\rbrace }\\rho (j)\\,\\hat{\\sigma }_j\\otimes \\hat{\\sigma }_j\\right].$ This corresponds to a physical density operator if and only if the real coefficients satisfy $[\\rho (x),\\rho (y),\\rho (z)]\\in \\mathrm {conv}\\lbrace [-1,1,1],[1,-1,1],[1,1,-1],[-1,-1,-1]\\rbrace $ , where the extremal points represent Bell states.","In Ref.","[19], we solved the SEEs [Eq.","(REF )] for this family of states.","In particular, the separability eigenvectors are tensor products of eigenstates of Pauli operators ($\\hat{\\sigma }_x$ , $\\hat{\\sigma }_y$ , and $\\hat{\\sigma }_z$ ).","This implies that we have $\\mathcal {S}_0=\\lbrace x_{+},x_{-},y_{+},y_{-},z_{+},z_{-}\\rbrace ,$ where $\\hat{\\sigma }_w|w_s\\rangle =s|w_{s}\\rangle $ for $w_s\\in \\mathcal {S}_0$ .","The exact entanglement quasiprobability reads [19] $P(a_s,b_t)=\\delta (a,b)\\left[\\frac{q}{12}+\\frac{|\\rho (a)|+st\\rho (a)}{4}\\right],$ with $a,b\\in \\lbrace x,y,z\\rbrace $ and $s,t\\in \\lbrace +1,-1\\rbrace $ , identifying the separability eigenvectors, $(a_s, b_t)\\in \\mathcal {S}_0\\times \\mathcal {S}_0$ .","Of particular importance is the parameter $q=1-|\\rho (x)|-|\\rho (y)|-|\\rho (z)|$ as it determines if the state is separable; namely, $\\hat{\\rho }$ in Eq.","(REF ) is separable if and only if $q\\ge 0$ [19].","Conversely, the negativity of the quasiprobability in the case of entanglement is given by this quantity as well; it reads $\\min _{(u, v)\\in \\mathcal {S}_0\\times \\mathcal {S}_0}P(u,v)=q/12\\ge -1/6$ , and the lower bound is attained for Bell states.","Figure: (Color online)The top panel depicts the entanglement quasiprobabilities [Eq.","()] for a Bell state with ρ(x)=ρ(y)=ρ(z)=-1\\rho (x)=\\rho (y)=\\rho (z)=-1.Each patch in the AA-BB plane corresponds to a product state |a〉〈a|⊗|b〉〈b||a\\rangle \\langle a|\\otimes |b\\rangle \\langle b|, with a,b∈𝒮 0 a,b\\in \\mathcal {S}_0 according to Eq.","().The bottom panel shows the eigenvalues of those separable-state projectors.The negativities in the quasiprobabilities PP identify the entanglement.Now we can apply the kernel in Eq.","(REF ).","Inserting the distribution in Eq.","(REF ) yields $\\begin{aligned}&P_{K\\otimes K}(a_s,b_t)\\\\= & r^2\\delta (a,b)\\left[\\frac{q}{12}+\\frac{|\\rho (a)|+st\\rho (a)}{4}\\right]+\\frac{(1-r)^2}{36}\\\\&+\\frac{r(1-r)}{6}\\left[\\frac{q}{3}+\\frac{|\\rho (a)|}{2}+\\frac{|\\rho (b)|}{2}\\right].\\end{aligned}$ After some straightforward algebra, we find that $P_{K\\otimes K}$ is always nonnegative for $0<r\\le 1/\\sqrt{7}$ .","In addition, we get tensor-product quasistates which are formed by the subsystem components $\\hat{\\Delta }_{K}(j)=\\frac{1}{r}|j\\rangle \\langle j|-\\frac{1-r}{2r}\\hat{1}$ for $j\\in \\mathcal {S}_0$ (see Appendix ).","Consequently, the eigenvalues of $\\hat{\\Delta }_{K}(j)$ are $(1+r)/(2r)$ and $-(1-r)/(2r)$ .","Figure: (Color online)The top panel depicts the classical joint probabilities, obtained using the kernel KK [see Eq.","() for r=1/7r=1/\\sqrt{7}], for the same Bell state as shown in Fig.",".The bottom panel shows the positive and negative eigenvalues of the quasistates in Eq.","() which are then required for the decomposition of the entangled state.Now the entanglement is captured by the unphysical nature of the quasistates.Let us consider a Bell state $\\hat{\\rho }=|\\psi \\rangle \\langle \\psi |$ as a specific example, where $|\\psi \\rangle =\\frac{|0\\rangle \\otimes |1\\rangle -|1\\rangle \\otimes |0\\rangle }{\\sqrt{2}},$ i.e., $\\rho (x)=\\rho (y)=\\rho (z)=-1$ .","The quasiprobability decomposition of this state is shown in Fig.","REF .","The negativities in the joint quasiprobability, $P\\ngeq 0$ , certify the entanglement and enable us to decompose the entangled state $\\hat{\\rho }$ using nonnegative tensor-product projectors, $|a\\rangle \\langle a|\\otimes |b\\rangle \\langle b|\\ge 0$ .","In contrast, in Fig.","REF , we show the result of applying the uncorrelated and nonnegative kernel $K\\otimes K$ for $r=1/\\sqrt{7}$ ; see Eq.","(REF ).","This results in a joint classical probability distribution.","However, the consequence is that the tensor-product operators for the decomposition have to be unphysical quasistates (because of the negative eigenvalues) in order to expand the entangled Bell state under study."],["Conclusions and discussion","Since the early theoretical descriptions of quantum states in phase space, quasiprobabilities have become one of the most important tools for studying quantum phenomena.","Here, we complemented this treatment by introducing the concept of quasistates.","While nonclassical states can be expanded in terms of a quasiprobability density and physical states, we demonstrated that one can equivalently use a classical, i.e., nonnegative, distribution and quasistates to perform such an expansion.","Then the quantum features of the state are carried over from the necessity of negativities in the distribution to the requirement that unphysical quasistates are needed to describe the state of a system.","Furthermore, we elaborated a number of different aspects and applications of quasistate, rendering them a useful tool to characterize quantum systems.","In simple terms, quasistates almost describe a density operator—only certain properties of a physical state do not apply.","In particular, a physical density operator is a Hermitian and positive semidefinite operator with a unit trace.","Throughout this work, we discussed several examples for which one (or multiple) of the defining properties of a density operator are violated by a quasistate.","For instance, eigenvalues can be negative, violating the positive semidefiniteness, or a Fourier representation was shown to lead to non-Hermitian quasistates.","As one practical implementation, we studied the reconstruction of the density operator.","This was based on the findings of a recent work [11], where Born's rule was reinterpreted in terms of contravariant operator-valued measures.","For example, this dual concept is useful when the positive operator-valued measure describes imperfect measurement outcomes.","Here, we proved that the contravariant operator-valued measure directly relates to the concept of quasistates and, therefore, can be considered as a special case of our general notion.","Consequently, we were able to apply quasistates for a direct quantum state reconstruction in terms of measured probabilities, such as exemplified for a two-level qubit state.","The duality between quasiprobability densities and their corresponding quasistates was studied for prominent quantum-optical phase-space distributions.","For instance, we found that the quasistates of the Wigner-Weyl distribution coincide with an operator that describes the maximal singularities the Glauber-Sudarshan distribution can have [41].","Beyond known quasiprobabilities in quantum optics, we further showed that our approach also leads to generalized phase-space distributions which expand the state in terms of squeezed states rather than coherent ones.","This, for example, can be useful to characterize non-Gaussianity, which is required for universal quantum computation and, in contrast to nonclassicality in terms of the Glauber-Sudarshan distribution, not only relies on coherent states but general squeezed states [67].","Furthermore, we characterized quantum correlations using quasistates.","In particular, we showed that entanglement can be identified with a classical joint probability distribution and tensor-product quasistates.","This result is surprising when considering that a violation of local-hidden-variable models typically excludes such classical distributions because states are (obviously) implicitly assumed to be physical in such models.","Here, we proved the necessary and sufficient condition that a density operator is separable, i.e., not entangled, if and only if both the distribution and the used states are classical; conversely, entanglement requires that at least one of them is nonclassical.","As an example, we identified the entanglement of a Bell state using either optimal quasiprobabilities [22] and tensor-product states or classical probabilities and unphysical quasistates.","Based on the general construction of quasiprobabilities for other notions of quantum coherences [19], the found results can be straightforwardly generalized to other forms of quantumness.","In conclusion, we developed the versatile and useful framework of quasistates for decomposing density operators.","Previously established concepts and methods have been demonstrated to be equivalently accessible with our technique which additionally allowed us to go beyond this state of the art.","Thus, we believe that, analogously to quasiprobabilities, the notion of quasistates has the potential to significantly contribute to the description and reconstruction of nonclassical quantum states in theory and experiment.","This work has received funding from the European Union's Horizon 2020 Research and Innovation Program under Grant Agreement No.","665148 (QCUMbER)."],["Exponential functions of bosonic operators","For our treatment of quantum light in Sec.",", some additional algebra is required.","In this appendix, we formulate the needed relations."],["General relations","We frequently apply the Gaussian integral formula $\\begin{aligned}&\\int _{\\mathbb {C}}d\\xi \\,\\exp \\left(-z|\\xi |^2-x\\xi ^2-y\\xi ^{\\ast 2}+u\\xi +v\\xi ^\\ast \\right)\\\\=&\\frac{\\pi }{\\sqrt{z^2-4xy}}\\exp \\left(\\frac{zuv-yu^2-xv^2}{z^2-4xy}\\right),\\end{aligned}$ where $\\mathrm {Re}[z\\pm (x+y)]>0$ and $\\mathrm {Re}[z^2-4xy]>0$ .","We use the branch with a nonnegative real part for the square root of a complex number.","The above relation can be extended to integrals of normally ordered expressions because under this prescription, we have an algebra of commuting operators.","In the case that operator ordering becomes relevant, let us formulate additional relations.","From the observations that $\\hat{a}^mf(\\hat{n})|n\\rangle =f(\\hat{n}+m)\\hat{a}^m|n\\rangle $ and $f(\\hat{n})\\hat{a}^{\\dag m}|n\\rangle =\\hat{a}^{\\dag m}f(\\hat{n}+m)|n\\rangle $ hold true, we conclude that $e^{x\\hat{a}^2}\\omega ^{\\hat{n}}=\\omega ^{\\hat{n}}e^{\\omega ^2x\\hat{a}^2}\\text{ and }\\omega ^{\\hat{n}}e^{y\\hat{a}^{\\dag 2}}=e^{\\omega ^2y\\hat{a}^{\\dag 2}}\\omega ^{\\hat{n}}.$ In the same manner, we get $\\exp (u\\hat{a})\\omega ^{\\hat{n}}=\\omega ^{\\hat{n}}\\exp (\\omega u\\hat{a})$ and $\\omega ^{\\hat{n}}\\exp (v\\hat{a}^\\dag )=\\exp (\\omega v\\hat{a}^\\dag )\\omega ^{\\hat{n}}$ .","Furthermore, using the above integral, $|\\alpha \\rangle \\langle \\alpha |={:}\\exp (-[\\hat{a}-\\alpha ]^\\dag [\\hat{a}-\\alpha ]){:}$ , and $\\pi \\hat{1}=\\int _{\\mathbb {C}}d\\alpha \\,|\\alpha \\rangle \\langle \\alpha |$ , we obtain $\\begin{aligned}&e^{x\\hat{a}^2}e^{y\\hat{a}^{\\dag 2}}=\\exp [x\\hat{a}^2]\\hat{1}\\exp [y\\hat{a}^{\\dag 2}]\\\\=&\\frac{1}{\\sqrt{1-4xy}}{:}\\exp \\left(\\frac{4xy\\hat{a}^\\dag \\hat{a}+x\\hat{a}^2+y\\hat{a}^{\\dag 2}}{1-4uv}\\right){:}\\\\=&e^{v\\hat{a}^{\\dag 2}/(1-4uv)}\\left(\\frac{1}{1-4uv}\\right)^{\\hat{n}+1/2}e^{u\\hat{a}^2/(1-4uv)}.\\end{aligned}$ The same approach yields $\\exp [u\\hat{a}]\\exp [y\\hat{a}^{\\dag 2}]=\\exp [y(\\hat{a}^\\dag +u)^2]\\exp [u\\hat{a}]$ and $\\exp [x\\hat{a}^2]\\exp [v\\hat{a}^\\dag ]=\\exp [v\\hat{a}^\\dag ]\\exp [x(\\hat{a}+v)^2]$ .","Finally, let us also mention the well-known formula $\\exp (u\\hat{a})\\exp (v\\hat{a}^\\dag )=\\exp (uv)\\exp (v\\hat{a}^\\dag )\\exp (u\\hat{a})$ ."],["Spectral decomposition of Gaussian quasistates","In the main part of this contribution, we consider filter functions of a Gaussian form [cf.","Eq.","(REF )].","The exact analysis for the resulting quasistates is performed here.","For zero displacement, the integral in Eq.","(REF ) yields $\\begin{aligned}\\hat{\\Delta }=&\\int _{\\mathbb {C}}d\\beta \\,\\frac{e^{-|\\beta |^2}}{\\pi \\Omega (\\beta )}e^{-\\beta \\hat{a}^\\dag }e^{\\beta ^\\ast \\hat{a}}\\\\=&\\frac{2\\,{:}\\exp \\left(\\frac{-2[1+s]\\hat{a}^\\dag \\hat{a} +q\\hat{a}^{\\dag 2}+p\\hat{a}^2}{(1+s)^2-pq}\\right){:}}{\\sqrt{(1+s)^2-pq}}.\\end{aligned}$ To characterize this quasistate, let us consider its decomposition in terms of exponential operators.","First, the squeezing operator $\\hat{S}=\\exp ([\\zeta ^\\ast \\hat{a}^2-\\zeta \\hat{a}^{\\dag 2}]/2)$ can be equivalently given as [58] $\\hat{S}=\\exp \\left(-\\frac{\\tau }{2}\\hat{a}^{\\dag 2}\\right)\\sqrt{1-|\\tau |^2}^{\\hat{n}+1/2}\\exp \\left(\\frac{\\tau ^\\ast }{2}\\hat{a}^2\\right),$ where $\\tau =e^{i\\arg \\zeta }\\tanh |\\zeta |$ .","Note that $|\\tau |<1$ , and we have infinite squeezing for $|\\tau |\\rightarrow 1$ .","In addition, we consider thermal-state-like operator $\\begin{aligned}\\hat{W}=&(1-\\omega )\\omega ^{\\hat{n}}=\\sum _{n\\in \\mathbb {N}}(1-\\omega )\\omega ^{n}|n\\rangle \\langle n|\\\\=&(1-\\omega ){:}e^{-(1-\\omega )\\hat{n}}{:}\\\\=&\\int _{\\mathbb {C}}d\\alpha \\,\\frac{1-\\omega }{\\pi \\omega }\\exp \\left(-\\frac{(1-\\omega )|\\alpha |^2}{\\omega }\\right)|\\alpha \\rangle \\langle \\alpha |.\\end{aligned}$ See Ref.","[68] for a derivation and additional considerations.","Also note that the vacuum state $\\hat{W}=|0\\rangle \\langle 0|$ is obtained in the limit $\\omega \\rightarrow 0$ .","Now, we defined an operator $\\hat{T}$ to be compared with the quasistate $\\hat{\\Delta }$ , $\\hat{T}=\\hat{S}\\hat{W}\\hat{S}^\\dag .$ Using the exchange relations in Eqs.","(REF ) and (REF ), we find after some algebra a normally ordered expression, $\\begin{aligned}\\hat{T}=&(1-\\omega )\\sqrt{\\frac{1-|\\tau |^2}{1-|\\tau |^2\\omega ^2}}\\\\&\\times \\exp \\left(-\\frac{\\tau }{2}\\frac{1-\\omega ^2}{1-|\\tau |^2\\omega ^2}\\hat{a}^{\\dag 2}\\right)\\\\&\\times {:}\\exp \\left(-\\frac{[1-\\omega ][1+\\omega |\\tau |^2]}{1-|\\tau |^2\\omega ^2}\\hat{n}\\right){:}\\\\&\\times \\exp \\left(-\\frac{\\tau ^\\ast }{2}\\frac{1-\\omega ^2}{1-|\\tau |^2\\omega ^2}\\hat{a}^2\\right).\\end{aligned}$ Finally, we can compare this expression with Eq.","(REF ), $\\hat{\\Delta }=\\hat{T}$ .","Namely, equating coefficient yields $\\omega =\\frac{r-1}{r+1},\\,|\\tau |^2=\\frac{s-r}{s+r},\\,\\text{and}\\,\\frac{\\tau }{\\tau ^\\ast }=\\frac{q}{p},$ where $r=\\sqrt{s^2-pq}$ .","This also implies $\\tau =-q/(s+r)$ and $\\tau ^\\ast =-p/(s+r)$ .","Conversely, we can deduce parameters $s$ , $p$ , and $q$ from given values of $\\omega $ and $\\tau $ , $\\begin{aligned}p=\\frac{-2(1+\\omega )\\tau ^\\ast }{(1-\\omega )(1-|\\tau |^2)},\\quad q=\\frac{-2(1+\\omega )\\tau }{(1-\\omega )(1-|\\tau |^2)},\\\\\\text{and}\\quad s+1=\\frac{2(1+\\omega |\\tau |^2)}{(1-\\omega )(1-|\\tau |^2)}.\\end{aligned}$"],["Uniform attenuation","In the main text, we consider a specific example of a kernel.","For studying its properties, let us consider the corresponding convolution $P_K(\\chi )=\\sum _{\\psi \\in \\mathcal {S}}P(\\psi )K(\\psi ,\\chi )=a\\,P(\\chi )+b\\sum _{\\psi \\in \\mathcal {S}}P(\\psi )$ for a set $\\mathcal {S}$ of states.","Note that $\\sum _{\\psi \\in \\mathcal {S}}P(\\psi )=1$ is the normalization condition.","To ensure that the result is normalized as well, $1=\\sum _{\\chi \\in \\mathcal {S}}P_K(\\chi )=a+b|\\mathcal {S}|$ , we find $b=\\frac{1-a}{|\\mathcal {S}|},$ using the cardinality $|\\mathcal {S}|$ of the set of states.","Further, when $0\\le a\\le 1$ is satisfied, the non-negativity of $P_K$ is guaranteed for any nonnegative $P$ .","The considered convolution can be formulated via the $|\\mathcal {S}|\\times |\\mathcal {S}|$ map $K=a\\,\\mathrm {id}+b\\,nn^\\mathrm {T},$ using the identity $\\mathrm {id}$ and the $|\\mathcal {S}|$ -dimensional vector $n=[1,\\ldots ,1]^\\mathrm {T}$ .","Then the inverse takes a similar form, $K^{-1}=\\frac{1}{a}\\mathrm {id}-\\frac{b/a}{a+b|\\mathcal {S}|}nn^\\mathrm {T},$ where $a+b|\\mathcal {S}|=1$ for the given $b$ .","Using the normalization, the convolution kernel $K$ corresponds to uniform addition of noise to the probabilities.","On the level of the operators, we get $\\hat{\\Delta }_K(\\chi )=\\frac{1}{a}|\\chi \\rangle \\langle \\chi |-\\frac{1-a}{a|\\mathcal {S}|}\\sum _{\\psi \\in \\mathcal {S}}|\\psi \\rangle \\langle \\psi |.$ In the continuous case, we can generalize the above relations as $P_K(\\chi )=\\int _{\\mathcal {S}} d\\psi \\,P(\\psi )\\left[a\\delta (\\psi ,\\chi )+\\frac{1-a}{|\\mathcal {S}|}\\right],$ where the volume $|\\mathcal {S}|=\\int _{\\mathcal {S}}d\\psi $ is used instead.","Furthermore, the projection operators transform as $\\hat{\\Delta }_{K}(\\chi )=\\int _{\\mathcal {S}} d\\psi \\left[\\frac{1}{a}\\delta (\\psi ,\\chi )-\\frac{1-a}{a|\\mathcal {S}|}\\right]|\\psi \\rangle \\langle \\psi |.$ As a discrete-variable example, we study a qubit system with orthogonal basis $\\lbrace |0\\rangle ,|1\\rangle \\rbrace $ .","We can identify the Pauli matrices as $\\hat{\\sigma }_z=|1\\rangle \\langle 1|-|0\\rangle \\langle 0|$ , $\\hat{\\sigma }_x=|0\\rangle \\langle 1|+|1\\rangle \\langle 0|$ , and $\\hat{\\sigma }_y=i|0\\rangle \\langle 1|-i|1\\rangle \\langle 0|$ .","The Hermitian operator basis can be completed with the operator $\\hat{1}=|0\\rangle \\langle 0|+|1\\rangle \\langle 1|$ .","The eigenvectors $|j_{\\pm 1}\\rangle $ of the Pauli matrices $\\hat{\\sigma }_j$ to the eigenvalues $\\pm 1$ for $j\\in \\lbrace z,x,y\\rbrace $ form the set $\\mathcal {S}$ .","Then we can decompose any density operator as $\\hat{\\rho }=\\sum _{j\\in \\lbrace x,y,z\\rbrace ,s\\in \\lbrace +1,-1\\rbrace }P(j_s)|j_{s}\\rangle \\langle j_{s}|.$ Conversely, the attenuated distribution reads $P_{K}(j_s)=aP(j_s)+(1-a)/6$ , and the transformed operators take the form $\\hat{\\Delta }_K(j_s)=a^{-1}|j_s\\rangle \\langle j_s|-(1-a)(2a)^{-1}\\hat{1}$ ."]],"string":"[\n [\n \"Quasistates and quasiprobabilities\"\n ],\n [\n \"Abstract The quasiprobability representation of quantum states addresses two main concerns, the identification of nonclassical features and the decomposition of the density operator.\",\n \"While the former aspect is a main focus of current research, the latter decomposition property has been studied less frequently.\",\n \"In this contribution, we introduce a method for the generalized expansion of quantum states in terms of so-called quasistates.\",\n \"In contrast to the quasiprobability decomposition through nonclassical distributions and pure-state operators, our technique results in classical probabilities and nonpositive semidefinite operators, defining the notion of quasistates, that carry the information about the nonclassical characteristics of the system.\",\n \"Therefore, our method presents a complementary approach to prominent quasiprobability representations.\",\n \"We explore the usefulness of our technique with several examples, including applications for quantum state reconstruction and the representation of nonclassical light.\",\n \"In addition, using our framework, we also demonstrate that inseparable quantum correlations can be described in terms of classical joint probabilities and tensor-product quasistates for an unambiguous identification of quantum entanglement.\"\n ],\n [\n \"Introduction\",\n \"Quantum systems can exhibit properties which have no analog in the classical realm, opening possibilities for technological innovations beyond the limitations posed by classical physics [1].\",\n \"For assessing the quantumness of a state, the bisection of a system into a classically accessible domain and a genuinely quantum-mechanical sector has become one of the major challenges of current research.\",\n \"A successful approach to performing such a desired separation are quasiprobability distributions, such as the Wigner-Weyl function [2], [3].\",\n \"Within this framework, a state is identified to be nonclassical when the corresponding quasiprobability does not have the properties of a classical probability distribution.\",\n \"Beyond this nonclassicality aspect, another purpose of quasiprobabilities is the full characterization of a quantum state, for example, using the Glauber-Sudarshan representation to describe quantized harmonic oscillators [4], [5].\",\n \"As such, the description of a density operator in terms of quasiprobabilities and classical reference states, e.g., coherent states, is an equally important feature of quasiprobabilities.\",\n \"However, despite having such a vital impact on the state's representation, the density operator expansion is, by far, less frequently considered.\",\n \"This is at least partially a result of a missing framework which focuses on the decomposition of density operators.\",\n \"Thus, the main objective of this contributions is to devise such a missing methodology that naturally leads to the useful and complementary concept of quasistates.\",\n \"The notion of quasiprobabilities has become one of the main hallmarks for certifying quantum features since it allows us to study resources for practical tasks.\",\n \"For instance, negative quasiprobabilities have been identified as indicators for the state's usefulness in performing quantum computation protocols [6], [7].\",\n \"Moreover, negativities in quasiprobabilities can be related to a broad range of concepts of nonclassicality, such as contextuality [8] and quantum entanglement [9].\",\n \"In addition, quasiprobabilities can be helpful to determine the actual probabilities of measurement outcomes [10], [11].\",\n \"Furthermore, quasiprobabilities can be defined for many relevant physical systems, such as continuous-variable harmonic oscillators [4], [5] or discrete-variable angular-momentum-type degrees of freedom [12], [13].\",\n \"Typically, the underlying quasiprobabilities are obtained by generalizing a classical phase space to the quantum domain; see Ref.\",\n \"[14] for an early study and Ref.\",\n \"[15] for a recent generalization.\",\n \"For example, quasiprobabilities in finite-dimensional systems can be introduced in this manner, cf., e.g., Refs.\",\n \"[16], [17], [18], [19].\",\n \"With regard to the quantum state representation, this typically requires us to identify pure reference states which are compatible with the properties of classical physics [20].\",\n \"Based on such classical reference states, a general quasiprobability decomposition of a quantum state can be constructed [19].\",\n \"Examples relevant for quantum technologies include the formulation of entanglement quasiprobabilities [21], [22], which become negative for quantum correlated states.\",\n \"Once a quasiprobability representation is constructed, it can be further generalized.\",\n \"Again, pioneering applications can be found in quantum optics, where the Glauber-Sudarshan and Wigner-Weyl quasiprobabilities can be unified in terms of so-called $s$ -parametrized quasiprobabilities [23], [24], which also include the well-known Husimi-Kano distributions [25], [26].\",\n \"This unification is achieved by a convolution of the quasiprobability with a Gaussian function and can be further generalized [27], [28], [29], including non-Gaussian scenarios [30], [31].\",\n \"A successful application of the $s$ -parameter approach in a finite-dimensional system has been reported as well [32].\",\n \"In addition, joint non-Gaussian quasiprobabilities for multiple optical modes and points in time have been introduced to study quantum correlated light [33], [34].\",\n \"Although such formal aspects led to profound insights into quantum physics when compared to classical statistical theories (see, e.g., [35], [36]), the method of quasiprobabilities is also of experimental importance.\",\n \"The most frequent implementation is the identification of nonclassical light in terms of negative Wigner-Weyl functions; see Refs.\",\n \"[37], [38] for early and recent examples.\",\n \"While the Glauber-Sudarshan distribution can be highly singular or even ambiguous [39], [40], [41], the reconstruction of this function is feasible in certain experiments as well [42].\",\n \"Moreover, the non-Gaussian generalization of this distribution is always experimentally accessible, such as reported for squeezed light [43].\",\n \"Even imperfect detection schemes can be employed for the reconstruction of nonclassical phase-space distribution of light [44].\",\n \"Conversely, a phase-space representation of the detector can experimentally verify quantum properties of the detection device used [45].\",\n \"Another approach [46], [47] enables an experimental reconstruction of phase-space distributions and entirely circumvents a detector characterization by analyzing certain data patterns [48].\",\n \"Beyond quantum light, equally insightful is the characterization of matter systems using quasiprobabilities, such as demonstrated for motional states of ions [49] and large atomic ensembles [50].\",\n \"It is also worth mentioning that quantum correlations in composite hybrid systems are also accessible via generalized quasiprobabilities [51], [52].\",\n \"Therefore, quasiprobabilities present a highly successful and versatile approach to identifying nonclassical properties of quantum states in theory and experiment.\",\n \"Still, as outlined above, another important aspect is the quantum state decomposition.\",\n \"Yet, in contrast to the vast number of examples for the nonclassicality certification, studies of the decomposition properties are rarely done; a gap we aim to close in this article.\",\n \"Based on the decomposition of states in terms of quasiprobabilities, we introduce a generalized expansion of quantum states in terms of quasistates.\",\n \"This method is complementary to the previously known approaches in which a density operator is decomposed in terms of classical reference states and a negative quasiprobability density.\",\n \"In our general method, we can find a decomposition which results in a nonnegative (i.e., classical) probability density by overcoming the usage of physical reference states in the decomposition.\",\n \"This naturally establishes the concept of quasistates, which then become the relevant objects for certifying the different kinds of nonclassicality.\",\n \"The usefulness of this change of perspective from quasiprobabilities to quasistates is studied in detail.\",\n \"One practical application relates to the experimental reconstruction of density operators.\",\n \"Moreover, as quasiprobabilities are most frequently applied in optical systems, we also perform a detailed analysis of quasistates that correspond to prominent phase-space representations of light.\",\n \"Finally, we investigate how quantum entanglement can be uniquely characterized via classical joint probability distributions when employing quasistates.\",\n \"Therefore, our approach offers a toolbox for the characterization and decomposition of quantum states.\",\n \"This article is structured as follows.\",\n \"In Sec.\",\n \", we discuss the general methodological framework.\",\n \"An application to the quantum state reconstruction is developed in Sec.\",\n \".\",\n \"Phase-space based quasistates for quantized light are studied in Sec.\",\n \".\",\n \"In Sec.\",\n \", quantum correlations are analyzed within the proposed framework.\",\n \"Finally, concluding discussions are presented in Sec.\",\n \".\"\n ],\n [\n \"Conceptual framework\",\n \"To characterize the state of a quantum system, a representation of the density operator is required.\",\n \"In general, a state decomposition consists of a family of pure states, defining a set $\\\\mathcal {S}$ , and corresponding expansion coefficients.\",\n \"Then the density operator takes the form $\\\\hat{\\\\rho }=\\\\int _{\\\\mathcal {S}} d\\\\psi \\\\, P(\\\\psi ) |\\\\psi \\\\rangle \\\\langle \\\\psi |,$ where $d\\\\psi $ indicates an integral over the volume of $\\\\mathcal {S}$ and $P$ is a correspondingly defined density.\",\n \"In the case of a discrete decomposition, the integral may be replaced by a sum.\",\n \"Note that we deliberately choose not to specify the set $\\\\mathcal {S}$ to keep our treatment as broadly applicable as possible.\",\n \"Arguably the most well-known example of a representation (REF ) is the spectral decomposition.\",\n \"In this case, $\\\\mathcal {S}$ is the set of eigenstates and the probability mass function $P$ returns the corresponding eigenvalues.\",\n \"Other examples for a decomposition (REF ) are studied in the continuation of this work and have applications, for example, in quantum optics and quantum information.\",\n \"It is worth mentioning that a general method to construct quasiprobabilities $P$ for a given set $\\\\mathcal {S}$ has been recently devised [19].\",\n \"In that approach, it is ensured that $P$ is a classical probability distribution when the state is in the convex hull $\\\\mathrm {conv}\\\\lbrace |\\\\psi \\\\rangle \\\\langle \\\\psi |:\\\\psi \\\\in \\\\mathcal {S}\\\\rbrace $ , which is a nontrivial task as the decomposition (REF ) is typically not unique [19].\",\n \"For our purpose, we may generalize the above treatment.\",\n \"Specifically, a convolution kernel $K:\\\\mathcal {S}\\\\rightarrow \\\\mathcal {S}$ is introduced together with the kernel $K^{-1}$ for the corresponding deconvolution.\",\n \"Then we get an equivalent decomposition as $\\\\hat{\\\\rho }=\\\\int _{\\\\mathcal {S}} d\\\\chi \\\\, P_K(\\\\chi )\\\\hat{\\\\Delta }_{K}(\\\\chi ).$ Therein we have a modified density $P_K$ and a modified family of operators $\\\\hat{\\\\Delta }_{K}$ , $P_K(\\\\chi )=&\\\\int _{\\\\mathcal {S}} d\\\\psi \\\\, P(\\\\psi )K(\\\\psi ,\\\\chi ),\\\\\\\\\\\\hat{\\\\Delta }_{K}(\\\\chi )=&\\\\int _{\\\\mathcal {S}} d\\\\psi ^{\\\\prime }\\\\, K^{-1}(\\\\chi ,\\\\psi ^{\\\\prime }) |\\\\psi ^{\\\\prime }\\\\rangle \\\\langle \\\\psi ^{\\\\prime }|.$ Using the properties that the operations used are inverse to one another, $\\\\delta (\\\\psi ,\\\\psi ^{\\\\prime })=\\\\int _{\\\\mathcal {S}} d\\\\chi K(\\\\psi ,\\\\chi )K^{-1}(\\\\chi ,\\\\psi ^{\\\\prime })$ with $\\\\delta $ being the Dirac distribution, one can directly see that Eqs.\",\n \"(REF ) and (REF ) describe the same density operator $\\\\hat{\\\\rho }$ .\",\n \"In general, the distribution $P_K$ in Eq.\",\n \"(REF ) is not a probability density, even if $P$ was.\",\n \"For such generalized distributions, the name quasiprobabilities was established.\",\n \"As discussed in the Introduction, such quasiprobability densities cover a wide range of applications, mainly for purpose of identifying quantum features.\",\n \"In close analogy to the notion of a quasiprobability, we are going to demonstrate that the operators $\\\\hat{\\\\Delta }_K$ in Eq.\",\n \"() are, in general, not physical density operators.\",\n \"Consequently, we refer to such operators as quasistates.\",\n \"A main focus of previous research was devoted to characterizing quasiprobabilities.\",\n \"Often, the idea of a decomposition of the quantum state [cf.\",\n \"Eqs.\",\n \"(REF ) and (REF )] in terms of such distributions was neglected.\",\n \"In particular, a general characterization of the distinctive features of quasistates and their applications beyond being a mathematical tool does not exist.\",\n \"For this reason, we are going to perform the missing comprehensive investigation of quasistates as defined in Eq.\",\n \"() and discuss useful applications of such operators.\"\n ],\n [\n \"State reconstruction\",\n \"One interesting application of quasistates are state reconstruction protocols as we show in this section.\",\n \"The following considerations are based on a recent work [11] in which the dual representation of measurement operators has been introduced.\",\n \"Here we demonstrate how this method relates to quasistates and can be used for the general reconstruction of density operators.\"\n ],\n [\n \"Dual representation\",\n \"Let us briefly revisit the findings in Ref.\",\n \"[11] with regards to our method.\",\n \"Suppose $\\\\lbrace \\\\hat{\\\\Pi }(j)\\\\rbrace _{j\\\\in \\\\mathcal {S}}$ is a positive operator-valued measure (POVM).\",\n \"As a complementary set of operators, the notion of contravariant operator-valued measured (COVM) was introduced.\",\n \"This defines a set $\\\\lbrace \\\\hat{\\\\Gamma }(j^{\\\\prime })\\\\rbrace _{j^{\\\\prime }\\\\in \\\\mathcal {S}}$ that satisfies the orthonormality relation $\\\\mathrm {tr}\\\\big [\\\\hat{\\\\Gamma }(j^{\\\\prime })\\\\hat{\\\\Pi }(j)\\\\big ]=\\\\delta (j^{\\\\prime },j).$ This means that the COVM represent dual basis operators to the measured POVM.\",\n \"In contrast to the POVM, COVM operators are not necessarily positive semidefinite.\",\n \"However, they are of particular interest, for example, when considering imperfections in the measurement process [11].\",\n \"The construction of COVM operators is based on a kernel defined via the elements $K(l,j)=\\\\mathrm {tr}\\\\big [\\\\hat{\\\\Pi }(l)\\\\hat{\\\\Pi }(j)\\\\big ].$ Specifically, it was shown that $\\\\hat{\\\\Gamma }(j^{\\\\prime })=\\\\sum _{l\\\\in \\\\mathcal {S}}K^{-1}(j^{\\\\prime },l)\\\\hat{\\\\Pi }(l),$ for the discrete case.\",\n \"In the following, let us explore the relation between the COVM and quasistates.\",\n \"For simplicity, we assume that the considered complex Hilbert space is finite dimensional, $\\\\dim _{\\\\mathbb {C}}\\\\mathcal {H}=d<\\\\infty $ .\",\n \"The set of Hermitian operators over $\\\\mathcal {H}$ is a real valued space with the dimension $\\\\dim _{\\\\mathbb {R}}\\\\mathrm {Herm}(\\\\mathcal {H})=d^2$ .\",\n \"Further, rather than restricting to a POVM for a single observable as done previously, we consider an informationally complete set $\\\\lbrace \\\\hat{\\\\Pi }(\\\\psi )\\\\rbrace _{\\\\psi \\\\in \\\\mathcal {S}}$ , with $\\\\hat{\\\\Pi }(\\\\psi )=|\\\\psi \\\\rangle \\\\langle \\\\psi |,$ which guarantees that a reconstruction of the full quantum state is possible and implies the cardinality $|\\\\mathcal {S}|=d^2$ .\",\n \"However, this also means that not all elements of $\\\\mathcal {S}$ can be pairwise orthogonal vectors.\",\n \"See Refs.\",\n \"[53], [54], [55] for introductions to informationally complete measurements.\",\n \"Because of the properties of the now studied POVM, we can expand the quantum state as given in Eq.\",\n \"(REF ), $\\\\hat{\\\\rho }=\\\\sum _{\\\\psi \\\\in \\\\mathcal {S}}P(\\\\psi ) \\\\hat{\\\\Pi }(\\\\psi ).$ It is important to mention that $P(\\\\psi )$ , in general, does not correspond to the probability of the measurement of $\\\\hat{\\\\Pi }(\\\\psi )$ , i.e., $\\\\mathrm {tr}[\\\\hat{\\\\rho }\\\\hat{\\\\Pi }(\\\\psi )]\\\\ne P(\\\\psi )$ .\",\n \"However, we can employ the same formalism as discussed above, cf.\",\n \"Eqs.\",\n \"(REF ) and (REF ), to expand the state as [11] $\\\\hat{\\\\rho }=\\\\sum _{\\\\chi \\\\in \\\\mathcal {S}}\\\\rho (\\\\chi ) \\\\hat{\\\\Gamma }(\\\\chi ).$ Now we can identify the coefficient with the measurement probability, $\\\\rho (\\\\psi ^{\\\\prime })=\\\\mathrm {tr}[\\\\hat{\\\\rho }\\\\hat{\\\\Pi }(\\\\psi ^{\\\\prime })]$ for all $\\\\psi ^{\\\\prime }\\\\in \\\\mathcal {S}$ , because of relation (REF ).\",\n \"In other words, the expansion in Eq.\",\n \"(REF ) can be used to directly reconstruct the density operator in terms of measurement outcomes and dual operators.\",\n \"Let us specifically relate the found reconstruction with the generalized decomposition (REF ).\",\n \"Using the definitions (REF ) and (REF ), we find that the kernel under study is given by the $d^2\\\\times d^2$ Gram-Schmidt matrix $K=\\\\left[|\\\\langle \\\\psi |\\\\chi \\\\rangle |^2\\\\right]_{\\\\psi ,\\\\chi \\\\in \\\\mathcal {S}}.$ Recall that information completeness implies that this matrix is a bijection, thus invertible.\",\n \"Further, as a result of Eqs.\",\n \"() and (REF ), the thought-after quasistates are identical to the COVM elements, $\\\\hat{\\\\Delta }_K(\\\\chi )=\\\\hat{\\\\Gamma }(\\\\chi ).$ Consequently, it also holds true that $P_K(\\\\chi )=\\\\rho (\\\\chi )$ ; see Eqs.\",\n \"(REF ) and (REF ).\",\n \"Therefore, quasistates allow for the direct reconstruction of the density operator under study.\",\n \"It is worth mentioning that imperfections in any measurement, meaning that $\\\\hat{\\\\Pi }(\\\\psi )$ is not a pure-state projector, can be treated similarly to the thorough discussion for a single observable carried out in Ref.\",\n \"[11].\",\n \"Furthermore, if the POVM does not allow for a full reconstruction or is over-complete (e.g., because $|\\\\mathcal {S}|\\\\ne d^2$ ), then the state in the spanned subspace can be reconstructed or a pseudo-inversion of $K$ can be performed [11].\"\n ],\n [\n \"Example\",\n \"To further discuss the above findings, we study the specific example of a qubit system ($d=2$ ), defined via the orthonormal Hilbert space basis $\\\\lbrace |0\\\\rangle ,|1\\\\rangle \\\\rbrace $ .\",\n \"The four rank-one projection operators $|\\\\psi _j\\\\rangle \\\\langle \\\\psi _j|$ in Eq.\",\n \"(REF ), also defining the set $\\\\mathcal {S}$ , can be chosen according to the states $\\\\nonumber |\\\\psi _1\\\\rangle =|1\\\\rangle ,|\\\\psi _2\\\\rangle =&\\\\frac{\\\\sqrt{2} |0\\\\rangle +|1\\\\rangle }{\\\\sqrt{3}},|\\\\psi _3\\\\rangle =\\\\frac{\\\\sqrt{2} w|0\\\\rangle +|1\\\\rangle }{\\\\sqrt{3}},\\\\\\\\\\\\text{and }|\\\\psi _4\\\\rangle =&\\\\frac{\\\\sqrt{2} w^2|0\\\\rangle +|1\\\\rangle }{\\\\sqrt{3}},$ where $w=\\\\exp [2\\\\pi i/3]$ ; see the top-left panel in Fig.\",\n \"REF for their geometric Bloch-sphere representation.\",\n \"This allows us to construct $K$ via Eq.\",\n \"(REF ) and compute its inverse, $K=\\\\frac{1}{3}\\\\begin{bmatrix}3 & 1 & 1 & 1\\\\\\\\1 & 3 & 1 & 1\\\\\\\\1 & 1 & 3 & 1\\\\\\\\1 & 1 & 1 & 3\\\\end{bmatrix}\\\\!,\\\\,K^{-1}=\\\\frac{1}{4}\\\\begin{bmatrix}5 & -1 & -1 & -1\\\\\\\\-1 & 5 & -1 & -1\\\\\\\\-1 & -1 & 5 & -1\\\\\\\\-1 & -1 & -1 & 5\\\\end{bmatrix}\\\\!.$ This then yields the quasistates as $\\\\hat{\\\\Delta }_K(\\\\psi _1)=&|1\\\\rangle \\\\langle 1|-\\\\frac{1}{2}|0\\\\rangle \\\\langle 0|\\\\text{ and}\\\\\\\\\\\\nonumber \\\\hat{\\\\Delta }_K(\\\\psi _j)=&\\\\frac{|0\\\\rangle \\\\langle 0|+\\\\sqrt{2}w^{j-2}|0\\\\rangle \\\\langle 1|+\\\\sqrt{2}w^{\\\\ast (j-2)}|1\\\\rangle \\\\langle 0|}{2},$ for $j=2,3,4$ .\",\n \"As we can see from the top-right panel in Fig.\",\n \"REF , these quasistates do not resemble physical quantum states because of the negative eigenvalue.\",\n \"Figure: (Color online)The elements of 𝒮\\\\mathcal {S} given in Eq.\",\n \"() in a tetrahedron configuration (larger green bullets) are shown in the top-left Bloch-sphere representation.The smaller red bullet indicates the state ρ ^=|0〉〈0|\\\\hat{\\\\rho }=|0\\\\rangle \\\\langle 0| to be reconstructed.The top-right panel shows the two eigenvalues of the resulting quasistates Δ ^ K (ψ j )\\\\hat{\\\\Delta }_K(\\\\psi _j), being identical for j=1,...,4j=1,\\\\ldots ,4.For the reconstruction of the state under study, the bottom panels depict the expansion coefficients P(ψ j )P(\\\\psi _j) and P K (ψ j )P_K(\\\\psi _j) to be used with |ψ j 〉〈ψ j ||\\\\psi _j\\\\rangle \\\\langle \\\\psi _j| and Δ ^ K (ψ j )\\\\hat{\\\\Delta }_K(\\\\psi _j) [Eqs.\",\n \"() and ()], respectively.Let us now consider the specific state $\\\\hat{\\\\rho }=|0\\\\rangle \\\\langle 0|$ for the reconstruction.\",\n \"Then, the measured and, therefore, nonnegative expansion coefficients are $P_K(\\\\psi _j)=\\\\mathrm {tr}(\\\\hat{\\\\rho }|\\\\psi _j\\\\rangle \\\\langle \\\\psi _j|)=\\\\langle \\\\psi _j|\\\\hat{\\\\rho }|\\\\psi _j\\\\rangle ,$ using the projectors defined via the states in Eq.\",\n \"(REF ).\",\n \"In addition, the corresponding COVM elements, being the quasistates in Eq.\",\n \"(REF ), result in the expansion coefficients $P(\\\\psi _j)=\\\\mathrm {tr}[\\\\hat{\\\\rho }\\\\hat{\\\\Delta }_K(\\\\psi _j)]$ , which resembles a quasiprobability distribution.\",\n \"The bottom part in Fig.\",\n \"REF shows the values of $P_K$ and $P$ for the state $\\\\hat{\\\\rho }=|0\\\\rangle \\\\langle 0|$ .\",\n \"Using the nonnegative $P_K$ (see Fig.\",\n \"REF ), we can now directly confirm that the density operator $\\\\hat{\\\\rho }=|0\\\\rangle \\\\langle 0|$ is expanded as described in Eq.\",\n \"(REF ) while using the quasistates in Eq.\",\n \"(REF ), $\\\\hat{\\\\rho }=\\\\sum _{j=1,\\\\ldots ,4}P_K(\\\\psi _j)\\\\hat{\\\\Delta }_K(\\\\psi _j)$ .\",\n \"The possibly negative elements of $P$ also allow for the decomposition of the state, $\\\\hat{\\\\rho }=\\\\sum _{j=1,\\\\ldots ,4}P(\\\\psi _j)|\\\\psi _j\\\\rangle \\\\langle \\\\psi _j|$ .\",\n \"However, $P$ does not correspond to a directly measured quantity.\",\n \"Thus, quasistates provide a beneficial tool to reconstruct density operators using measured quantities.\",\n \"Let us also briefly comment on the extension of our approach to continuous-variable systems, $d=\\\\infty $ .\",\n \"In fact, already in Refs.\",\n \"[56], [57], a method was introduced to expand the state of a single-mode radiation field in terms of so-called pattern functions.\",\n \"Using the here developed framework, we can directly identify these pattern functions with the wave functions of the corresponding quasistates.\",\n \"Beyond this reconstruction approach, we study the expansion of nonclassical states of light in the following section.\"\n ],\n [\n \"Quantum Light\",\n \"We now demonstrate the application of quasistates for the description of nonclassical quantum states.\",\n \"In particular, we consider a harmonic oscillator, which can, for example, describe a quantized radiation mode.\",\n \"To describe the system's nonclassicality, the prominent Glauber-Sudarshan representation has been developed [4], [5], $\\\\hat{\\\\rho }=\\\\int _{\\\\mathbb {C}}d\\\\alpha \\\\,P(\\\\alpha )|\\\\alpha \\\\rangle \\\\langle \\\\alpha |.$ Here this decomposition implies that we identify the set $\\\\mathcal {S}$ with the complex coherent amplitudes $\\\\alpha \\\\in \\\\mathbb {C}$ , defining the coherent states $|\\\\alpha \\\\rangle $ .\",\n \"For a thorough introduction to quantum optics in phase space, see Ref.\",\n \"[58].\",\n \"A state is referred to as nonclassical if $P$ does not exhibit the properties of a classical probability distribution.\",\n \"In the following, we begin our considerations of quasistates for nonclassical light in the Fourier picture and then proceed to prominent examples of quasiprobabilities and their dual representation via quasistates.\"\n ],\n [\n \"Fourier representation\",\n \"The characteristic function $\\\\Phi (\\\\beta )$ is the Fourier transform $F$ of a distribution, which is accessible via the kernel $F(\\\\alpha ,\\\\beta )=\\\\exp \\\\left(\\\\beta \\\\alpha ^\\\\ast -\\\\beta ^\\\\ast \\\\alpha \\\\right),$ i.e., in our notation, $P_{F}=\\\\Phi $ .\",\n \"The inverse Fourier transform, defined via $F^{-1}(\\\\beta ,\\\\alpha )=F(\\\\alpha ,\\\\beta )^\\\\ast /\\\\pi ^2$ , results in the corresponding quasistates [Eq.\",\n \"()], $\\\\hat{\\\\Delta }_{F}(\\\\beta )=\\\\int _{\\\\mathbb {C}}d\\\\beta \\\\,\\\\frac{\\\\exp \\\\left(\\\\beta ^\\\\ast \\\\alpha -\\\\beta \\\\alpha ^\\\\ast \\\\right)}{\\\\pi ^2}|\\\\alpha \\\\rangle \\\\langle \\\\alpha |.$ For the evaluation of the integral, it is useful to represent the coherent-state projectors in terms of normally ordered exponential of creation ($\\\\hat{a}^\\\\dag $ ) and annihilation ($\\\\hat{a}$ ) operators, which reads $|\\\\alpha \\\\rangle \\\\langle \\\\alpha |={:}\\\\exp (-[\\\\hat{a}-\\\\alpha ]^\\\\ast [\\\\hat{a}-\\\\alpha ]){:}$ [58].\",\n \"This representation yields $\\\\hat{\\\\Delta }_{F}(\\\\beta )=\\\\frac{\\\\exp (-|\\\\beta |^2)}{\\\\pi }{:}\\\\hat{D}(-\\\\beta ){:},$ where $\\\\hat{D}(\\\\gamma )=\\\\exp (\\\\gamma \\\\hat{a}^\\\\dag -\\\\gamma ^\\\\ast \\\\hat{a})=\\\\exp (-|\\\\gamma |^2/2){:}\\\\hat{D}(\\\\gamma ){:}$ is the unitary displacement operator.\",\n \"Therefore, the quasistates $\\\\hat{\\\\Delta }_{F}(\\\\beta )$ correspond (up to a rescaling) to unitary operators in the case that the convolution kernel is the Fourier transformation.\",\n \"Note that instead of the Fourier transform, the Laplace transform can be advantageous in certain scenarios as well [59].\",\n \"One application, which also connects to the previous section, is the state reconstruction via balanced homodyne detection.\",\n \"For this purpose, let us recall that the characteristic function of the Wigner-Weyl distribution takes the form [58] $\\\\Phi (\\\\beta )e^{-|\\\\beta |^2/2}=\\\\langle \\\\hat{D}(\\\\beta )\\\\rangle =\\\\langle \\\\exp \\\\left(i|\\\\beta |\\\\hat{x}[\\\\pi /2-\\\\arg \\\\beta ]\\\\right)\\\\rangle ,$ where we used the quadrature operator $\\\\hat{x}[\\\\varphi ]=e^{i\\\\varphi }\\\\hat{a}+e^{-i\\\\varphi }\\\\hat{a}^\\\\dag $ .\",\n \"This means for a balanced homodyne measurement, providing a data set $\\\\lbrace x_j[\\\\varphi _j]\\\\rbrace _{j=1,\\\\ldots ,N}$ of quadratures $x_j$ for the phase values $\\\\varphi _j$ , we can approximate $\\\\begin{aligned}\\\\hat{\\\\rho }=&\\\\int _{\\\\mathbb {C}}d\\\\beta \\\\,\\\\Phi (\\\\beta )\\\\hat{\\\\Delta }_{F}(\\\\beta )\\\\\\\\=&\\\\int _{-\\\\infty }^\\\\infty dr\\\\int _{-\\\\pi /2}^{\\\\pi /2} d\\\\varphi \\\\,\\\\frac{\\\\pi }{\\\\pi }e^{r^2/2}\\\\langle e^{ir\\\\hat{x}[\\\\varphi ]}\\\\rangle \\\\hat{\\\\Delta }_{F}(ire^{-i\\\\varphi })\\\\\\\\\\\\approx &\\\\int _{-\\\\infty }^\\\\infty dr\\\\, e^{r^2/2} \\\\frac{\\\\pi }{N}\\\\sum _{j=1}^N e^{irx_j[\\\\varphi _j]}\\\\hat{\\\\Delta }_{F}(ire^{-i\\\\varphi _j}),\\\\end{aligned}$ which corresponds to a sampling approach as a consequence of a quasistate representation [see specifically the first line of Eq.\",\n \"(REF ) and the use of $\\\\hat{\\\\Delta }_{F}$ ].\",\n \"In fact, further analyses show that this approach is indeed equivalent to established reconstruction methods bases on balanced homodyne detection and using pattern functions [56], [57], [60].\",\n \"In addition, it is worth emphasizing that the phases $\\\\varphi $ have to be uniformly distributed in the interval $[-\\\\pi /2,\\\\pi /2]$ to ensure an optimal sampling [61].\",\n \"Figure: (Color online)The QQ function [Eq.\",\n \"()] of the quasistate [Eq.\",\n \"()] for different parameters ss, pp, and qq.The top-left plot corresponds to a coherent (vacuum) state.The top-center plot shows the maximally singular quasistate .The top-right plot includes, in addition, squeezing.The bottom row depicts cases in which the QQ function becomes complex;the larger and corresponding smaller plots show the amplitude |Q||Q| and phase argQ\\\\arg Q, respectively.Note the difference in the phases when comparing the left and right plots.\"\n ],\n [\n \"Generalized optical quasistates\",\n \"Translational types of convolutions in the original space, e.g., $P_{\\\\kappa }(\\\\alpha ^{\\\\prime })=\\\\int _{\\\\mathbb {C}}d\\\\alpha \\\\,P(\\\\alpha )\\\\kappa (\\\\alpha ^{\\\\prime }-\\\\alpha )$ , take a product form in the Fourier domain.\",\n \"In this scenario, we can consider the representation $\\\\hat{\\\\rho }=\\\\int _{\\\\mathbb {C}}d\\\\beta \\\\,\\\\left[\\\\Phi (\\\\beta )\\\\Omega (\\\\beta )\\\\right]\\\\left[\\\\frac{e^{-|\\\\beta |^2}}{\\\\pi \\\\Omega (\\\\beta )}{:}\\\\hat{D}(-\\\\beta ){:}\\\\right],$ for a kernel $\\\\Omega $ .\",\n \"The inverse Fourier transform gives the distribution $P_{F^{-1}\\\\Omega F}$ as well as the quasistates $\\\\hat{\\\\Delta }_{F^{-1}\\\\Omega F}(\\\\alpha )=\\\\int _{\\\\mathbb {C}}d\\\\beta \\\\,e^{\\\\beta \\\\alpha ^\\\\ast -\\\\beta ^\\\\ast \\\\alpha }\\\\frac{e^{-|\\\\beta |^2}}{\\\\pi \\\\Omega (\\\\beta )}{:}\\\\hat{D}(-\\\\beta ){:}.$ Such operators have been extensively studied in connection to operator orderings and their resulting quasiprobabilities [23], [24], [27], [28], [29].\",\n \"Here, however, let us analyze them based on their own merit as quasistates.\",\n \"Of particular interest are the Gaussian functions $\\\\Omega (\\\\beta )=\\\\exp \\\\left(-\\\\frac{1-s}{2}|\\\\beta |^2-\\\\frac{p}{4}\\\\beta ^2-\\\\frac{q}{4}\\\\beta ^\\\\ast \\\\right),$ where $|p|=|q|$ [27], [28], [29].\",\n \"In Table REF , specific examples and their connection to known quasiprobabilities are given.\",\n \"Evaluating the integrals as demonstrated in Appendix , we get the quasistates in normal ordering as $\\\\hat{\\\\Delta }_{F^{-1}\\\\Omega F}(0)=\\\\frac{2\\\\,{:}\\\\exp \\\\left(\\\\frac{-2[1+s]\\\\hat{a}^\\\\dag \\\\hat{a} +q\\\\hat{a}^{\\\\dag 2}+p\\\\hat{a}^2}{(1+s)^2-pq}\\\\right){:}}{\\\\sqrt{(1+s)^2-pq}}.$ Note that because of the convolution property, we have $\\\\hat{\\\\Delta }_{F^{-1}\\\\Omega F}(\\\\alpha )=\\\\hat{D}(\\\\alpha )\\\\hat{\\\\Delta }_{F^{-1}\\\\Omega F}(0)\\\\hat{D}(\\\\alpha )^\\\\dag $ for the displaced quasistates.\",\n \"In Fig.\",\n \"REF , several examples of the phase-space $Q$ -function representation, $Q(\\\\alpha )=\\\\frac{\\\\langle \\\\alpha |\\\\hat{\\\\Delta }_{F^{-1}\\\\Omega F}(0)|\\\\alpha \\\\rangle }{\\\\pi },$ are shown for different parameters that define the quasistates in Eq.\",\n \"(REF ).\",\n \"It is also worth mentioning that the $Q$ function is the Husimi-Kano distribution.\",\n \"Table: Parameters [cf.\",\n \"Eq.\",\n \"()] for prominent quasiprobability distributions frequently used in quantum optics.Interestingly, the considered class of quasistates, given in Eq.\",\n \"(REF ), can be related to squeezed versions of thermal-like quasistates, $\\\\begin{aligned}\\\\hat{\\\\Delta }_{F^{-1}\\\\Omega F}(0)=&\\\\exp ([\\\\zeta ^\\\\ast \\\\hat{a}^2-\\\\zeta \\\\hat{a}^{\\\\dag 2}]/2)\\\\\\\\&\\\\times \\\\frac{1}{\\\\bar{n}+1}\\\\sum _{n\\\\in \\\\mathbb {N}}\\\\left(\\\\frac{\\\\bar{n}}{\\\\bar{n}+1}\\\\right)^n|n\\\\rangle \\\\langle n|\\\\\\\\&\\\\times \\\\exp ([\\\\zeta ^\\\\ast \\\\hat{a}^2-\\\\zeta \\\\hat{a}^{\\\\dag 2}]/2)^\\\\dag ,\\\\end{aligned}$ where $\\\\bar{n}\\\\in \\\\mathbb {C}$ generalizes the notion of a mean thermal photon number and $\\\\zeta $ is the squeezing parameter.\",\n \"A full and exact analysis is provided in Appendix , and it provides the relations between the different parameter sets as $\\\\zeta =e^{i\\\\arg \\\\tau }\\\\mathrm {artanh}\\\\,|\\\\tau |\\\\quad \\\\text{and}\\\\quad \\\\bar{n}=\\\\frac{\\\\omega }{1-\\\\omega },$ with $\\\\omega =\\\\frac{r-1}{r+1},\\\\,|\\\\tau |^2=\\\\frac{s-r}{s+r},\\\\,\\\\text{and}\\\\,\\\\frac{\\\\tau }{\\\\tau ^\\\\ast }=\\\\frac{q}{p},$ where $r=\\\\sqrt{s^2-pq}$ .\",\n \"Recall that we require $|q|=|p|$ .\",\n \"Also note that the squeezing operation is unitary which implies that we have the eigenvalues $(1-\\\\omega )\\\\omega ^n$ , following a geometric distribution.\",\n \"In the following, let us study the resulting quasistates in more detail.\"\n ],\n [\n \"$s$ -Parameterized quasistates\",\n \"Arguably the most frequently used choice of parameters is obtained for $p=q=0$ , likewise $\\\\zeta =\\\\tau =0$ .\",\n \"This choice results in the $s$ -parametrized quasiprobability distributions $P_{F^{-1}\\\\Omega F}$ .\",\n \"Here, we additionally find the quasistates $\\\\hat{\\\\Delta }_{F^{-1}\\\\Omega F}$ .\",\n \"In particular, those quasistates have the eigenvalues $(1-\\\\omega )\\\\omega ^n=\\\\frac{\\\\bar{n}^{n}}{(\\\\bar{n}+1)^{n+1}}=\\\\frac{2}{1+s}\\\\left(-\\\\frac{1-s}{1+s}\\\\right)^n$ for $n\\\\in \\\\mathbb {N}$ ; see Eq.\",\n \"(REF ).\",\n \"For the parameters $-1<s<1$ , the eigenvalues (REF ) of the quasistates are negative for odd $n$ , certifying that these operators are only accessible in terms of our generalization of the notion of a physical state.\",\n \"In the limit $s\\\\rightarrow 1$ , we get the eigenvalue 1 for $n=0$ and 0 for $n>0$ , defining the vacuum state.\",\n \"Its $Q$ function is shown on the left in the top row of Fig.\",\n \"REF , and the corresponding quasiprobability distribution is the original Glauber-Sudarshan distribution.\",\n \"Furthermore, the case $s=0$ interestingly describes an operator with the maximally possible singularities a Glauber-Sudarshan distribution can exhibit [41]; see also the center plot in the top row of Fig.\",\n \"REF for its $Q$ function.\",\n \"There, we observe that the spread (i.e., variance) in phase space is reduced in all directions when compared to the vacuum state, further highlighting that it is a genuine quasistate that is incompatible with the uncertainty relation which holds true for any physical state.\",\n \"The underlying phase-space distribution $P_{F^{-1}\\\\Omega F}$ is the Wigner-Weyl distribution.\",\n \"Finally, the quasistates for the Husimi-Kano distribution are obtained in the limit $s\\\\rightarrow -1$ , and its corresponding $Q$ function is described by a singular Dirac distribution.\",\n \"Beyond real-valued $s$ parameters, we can additionally consider complex values as well [58].\",\n \"Then, the eigenvalues (REF ) are complex too.\",\n \"The example $s=i$ is shown on the left of the bottom row in Fig.\",\n \"REF .\",\n \"While the amplitude $|Q|$ is identical with the vacuum state, the quasistate characteristics are clearly visible through the nontrivial phases, $\\\\arg Q\\\\ne 0$ .\"\n ],\n [\n \"Non-$s$ -parametrized quasistates\",\n \"Moreover, we can also consider the quasistates for cases with $p,q\\\\ne 0$ .\",\n \"As we demand $|q|=|p|$ , we have two extremal scenarios, $q=p^\\\\ast $ and $q=-p^\\\\ast $ .\",\n \"In addition to the $s$ -parametrized quasiprobabilities in Table REF , we also listed the Agarwal-Wolf distributions.\",\n \"They correspond to the case $q=-p^\\\\ast =\\\\mp 1$ (also, $s=0$ ).\",\n \"The complex-valued $Q$ function of the resulting quasistate $\\\\hat{\\\\Delta }_{F^{-1}\\\\Omega F}$ is shown in the bottom-right plot of Fig.\",\n \"REF .\",\n \"The amplitude $|Q|$ is again compatible with the uncertainty principle bounded by the vacuum state, similarly to the previously discussed scenario of a complex $s$ parameter.\",\n \"Here, however, the functional behavior of the phase $\\\\arg Q$ does not possess a radial symmetry.\",\n \"Rather, the quasistates to the Agarwal-Wolf distributions exhibit hyperbolic isophase contours.\",\n \"Note that this dismisses a sometimes held believe that Agarwal-Wolf distributions are an example of a $s$ -parametrized quasiprobability.\",\n \"Finally, we can ask ourselves what happens in the case $q=p^\\\\ast $ .\",\n \"An example is shown in the top-right plot in Fig.\",\n \"REF .\",\n \"Interestingly, the resulting quasistates are squeezed.\",\n \"Specifically for a squeezing parameter $\\\\zeta $ , we can choose $s=\\\\frac{1+\\\\tanh ^2|\\\\zeta |}{1-\\\\tanh ^2|\\\\zeta |}\\\\text{ and }q=\\\\frac{-2e^{i\\\\arg \\\\zeta }\\\\tanh |\\\\zeta |}{1-\\\\tanh ^2|\\\\zeta |}=p^\\\\ast .$ This choice implies $\\\\bar{n}=0$ and means that the quasistates are pure squeezed states, cf.\",\n \"Eqs.\",\n \"(REF ) and (REF ).\",\n \"Consequently, the quasiprobability density $P_{F^{-1}\\\\Omega F}(\\\\alpha )$ now describes the decomposition of the density operator in terms of displaced squeezed states with a coherent amplitude $\\\\alpha $ and a squeezing defined by $\\\\zeta $ .\",\n \"This further implies that we have a generalized Glauber-Sudarshan quasiprobability distribution which is, however, a phase-space representation that expands the state in terms of squeezed states.\",\n \"Thus, the case $q=p^\\\\ast $ significantly extends the notion of $s$ -parametrized distributions to additionally include arbitrary squeezed states.\"\n ],\n [\n \"Quantum correlations\",\n \"In a recent work [19], we generalized the concept of quasiprobability representations to more general notions of quantum coherence [62], beyond the specific example of harmonic oscillators studied in the previous section.\",\n \"Among the various types of quantumness, quantum correlations between multiple degrees of freedom play an outstanding role for applications in quantum technologies [1], [63].\",\n \"For this reason, let us study the entanglement of quantum systems within the framework of quasistates.\",\n \"A comprehensive introduction to the theory of entanglement (likewise, inseparability) can be found in Ref.\",\n \"[63].\"\n ],\n [\n \"Entanglement quasiprobabilities\",\n \"A pure separable state in a multipartite system is described by a tensor-product vector, $|(a,b,\\\\ldots )\\\\rangle =|a\\\\rangle \\\\otimes |b\\\\rangle \\\\otimes \\\\cdots ,$ where $a\\\\in \\\\mathcal {S}_A$ , $b\\\\in \\\\mathcal {S}_B$ , etc.\",\n \"In addition, the inclusion of classical correlations in terms of statistical mixtures, given by a classical probability density $P$ , yields the definition of a mixed separable state [64], $\\\\hat{\\\\rho }{=}\\\\int _{S_A{\\\\times } S_B{\\\\times }\\\\cdots } da\\\\,db\\\\cdots P(a,b,\\\\ldots )|(a,b,\\\\ldots )\\\\rangle \\\\langle (a,b,\\\\ldots )|.$ A state is defined to be inseparable (i.e., entangled) if it cannot be written in the form of Eq.\",\n \"(REF ).\",\n \"However, when we allow for $P$ to be a quasiprobability, even entangled states can be expanded in terms of separable states using Eq.\",\n \"(REF ) [21].\",\n \"A construction approach for bipartite entanglement quasiprobabilities was introduced in Ref.\",\n \"[22].\",\n \"As this approach can be generalized from the bipartite scenario to the multipartite case [19], we restrict ourselves to bipartite systems in the following.\",\n \"The optimal decomposition of entangled states in terms of separable ones warrants that the entanglement is uniquely identified through negativities in $P(a,b)$ [22], [19].\",\n \"The separable states $|(a,b)\\\\rangle $ needed for the decomposition are obtained from the solution of the so-called separability eigenvalue equations (SEEs), $\\\\hat{\\\\rho }_a|b\\\\rangle =g|b\\\\rangle \\\\text{ and }\\\\hat{\\\\rho }_b|a\\\\rangle =g|a\\\\rangle ,$ with $\\\\hat{\\\\rho }_a=\\\\mathrm {tr}_A[\\\\hat{\\\\rho }(|a\\\\rangle \\\\langle a|\\\\otimes \\\\hat{1}_B)]$ and $\\\\hat{\\\\rho }_b=\\\\mathrm {tr}_B[\\\\hat{\\\\rho }(\\\\hat{1}_A\\\\otimes |b\\\\rangle \\\\langle b|)]$ .\",\n \"The SEEs approach generalizes the eigenvalue problem to composite systems while respecting the tensor-product structure of the corresponding eigenvectors and was initially introduced to construct entanglement witnesses [65], [66].\",\n \"The solutions $(a,b)\\\\in \\\\mathcal {S}$ allow us to construct the distribution $P$ by solving the linear problem $G\\\\vec{p}=\\\\vec{g},$ where $G=[|\\\\langle (a,b)|(a^{\\\\prime },b^{\\\\prime })\\\\rangle |^2]_{(a,b),(a^{\\\\prime },b^{\\\\prime })\\\\in \\\\mathcal {S}}$ is the Gram-Schmidt matrix of separability eigenvectors.\",\n \"Furthermore, the solution vector $\\\\vec{p}=[P(a,b)]_{(a,b)\\\\in \\\\mathcal {S}}$ defines the quasiprobabilities, and $\\\\vec{g}=[g]_{(a,b)\\\\in \\\\mathcal {S}}$ is the vector of separability eigenvalues, cf.\",\n \"Eq.\",\n \"(REF ).\",\n \"Let us point out that the here-used Gram-Schmidt matrix also relates to the reconstruction approach in Sec.\",\n \"[Eq.\",\n \"(REF )] and is a result of a underlying principle of convex decomposition, discussed in greater detail in Ref.\",\n \"[19].\",\n \"One can always find a set $\\\\mathcal {S}_0$ such that $\\\\mathcal {S}\\\\subset \\\\mathcal {S}_0\\\\times \\\\mathcal {S}_0$ ; we then set $P(a,b)=0$ for all $(a,b)\\\\in \\\\mathcal {S}_0\\\\times \\\\mathcal {S}_0\\\\setminus \\\\mathcal {S}$ .\",\n \"Using the construction to find optimal entanglement quasiprobabilities via the SEEs, we can decompose any state, be it entangled or separable [22], [19], as $\\\\hat{\\\\rho }=\\\\sum _{(a,b)\\\\in \\\\mathcal {S}_0\\\\times \\\\mathcal {S}_0}P(a,b)|(a,b)\\\\rangle \\\\langle (a,b)|.$ It is worth mentioning that the entanglement quasiprobability is always real valued and normalized.\"\n ],\n [\n \"Quasistate representation\",\n \"A universally applicable approach to diminish the negativities in quasiprobabilities for a single system is a convolution of the form $P_K(\\\\chi )=rP(\\\\chi )+\\\\frac{1-r}{|\\\\mathcal {S}_0|}\\\\sum _{\\\\psi \\\\in \\\\mathcal {S}_0}P(\\\\psi ),$ with $0< r\\\\le 1$ and $|\\\\mathcal {S}_0|$ being the cardinality of the set; see Appendix for details.\",\n \"The kernel under study consequently leads to quasistates of the form $\\\\hat{\\\\Delta }_K(\\\\chi )=\\\\frac{1}{r}|\\\\chi \\\\rangle \\\\langle \\\\chi |-\\\\frac{1-r}{r|\\\\mathcal {S}_0|}\\\\sum _{\\\\psi \\\\in \\\\mathcal {S}_0}|\\\\psi \\\\rangle \\\\langle \\\\psi |.$ We emphasize that quasistates defined in this manner are both Hermitian, $\\\\hat{\\\\Delta }_K(\\\\chi )=\\\\hat{\\\\Delta }_K(\\\\chi )^\\\\dag $ , and normalized, $\\\\mathrm {tr}[\\\\hat{\\\\Delta }_K(\\\\chi )]=1$ .\",\n \"Thus, the only distinction to true states is the fact that $\\\\hat{\\\\Delta }_K(\\\\chi )$ is, in general, not a positive semidefinite operator.\",\n \"The convolution can be generalized to composite systems via product kernels.\",\n \"Choosing, for example, the same $r$ for both systems, we can reformulate Eq.\",\n \"(REF ) as $\\\\hat{\\\\rho }=\\\\sum _{(\\\\tilde{a},\\\\tilde{b})\\\\in \\\\mathcal {S}_0\\\\times \\\\mathcal {S}_0}P_{K\\\\otimes K}(\\\\tilde{a},\\\\tilde{b})\\\\,\\\\hat{\\\\Delta }_K(\\\\tilde{a})\\\\otimes \\\\hat{\\\\Delta }_K(\\\\tilde{b}).$ In particular, we have $\\\\begin{aligned}P_{K\\\\otimes K}(\\\\tilde{a},\\\\tilde{b})=&r^2P(\\\\tilde{a},\\\\tilde{b})+\\\\frac{(1-r)^2}{|\\\\mathcal {S}_0|^2}\\\\\\\\&+\\\\frac{r(1-r)}{|\\\\mathcal {S}_0|}\\\\left[P(\\\\tilde{a})+P(\\\\tilde{b})\\\\right],\\\\end{aligned}$ using the marginal distributions $P(\\\\tilde{a})=\\\\sum _{b\\\\in \\\\mathcal {S}_0}P(\\\\tilde{a},b)$ and $P(\\\\tilde{b})=\\\\sum _{a\\\\in \\\\mathcal {S}_0}P(a,\\\\tilde{b})$ .\",\n \"Eventually, we get a nonnegative distribution $P_{K\\\\otimes K}$ for a sufficiently large $r$ , which is also demonstrated later.\",\n \"This means that an entangled state can be decomposed according to Eq.\",\n \"(REF ) in terms of a classical joint probability distribution $P_{K\\\\otimes K}(a,b)\\\\ge 0$ and tensor-product quasistates $\\\\hat{\\\\Delta }_K(a)\\\\otimes \\\\hat{\\\\Delta }_K(b)\\\\ngeq 0$ [cf.\",\n \"Eq.\",\n \"(REF )].\",\n \"Since we can chose $r=0$ for separable states, we find that quasistates are not required in this case.\",\n \"Therefore, we are able to conclude that entanglement is unambiguously identified in terms of classical joint probability distributions if and only if the tensor-product operators of the decomposition have to be unphysical quasistates.\"\n ],\n [\n \"Example\",\n \"As a proof of concept, let us consider an entangled two-qubit quantum state parametrized as $\\\\hat{\\\\rho }=\\\\frac{1}{4}\\\\left[\\\\hat{1}\\\\otimes \\\\hat{1}+\\\\sum _{j\\\\in \\\\lbrace x,y,z\\\\rbrace }\\\\rho (j)\\\\,\\\\hat{\\\\sigma }_j\\\\otimes \\\\hat{\\\\sigma }_j\\\\right].$ This corresponds to a physical density operator if and only if the real coefficients satisfy $[\\\\rho (x),\\\\rho (y),\\\\rho (z)]\\\\in \\\\mathrm {conv}\\\\lbrace [-1,1,1],[1,-1,1],[1,1,-1],[-1,-1,-1]\\\\rbrace $ , where the extremal points represent Bell states.\",\n \"In Ref.\",\n \"[19], we solved the SEEs [Eq.\",\n \"(REF )] for this family of states.\",\n \"In particular, the separability eigenvectors are tensor products of eigenstates of Pauli operators ($\\\\hat{\\\\sigma }_x$ , $\\\\hat{\\\\sigma }_y$ , and $\\\\hat{\\\\sigma }_z$ ).\",\n \"This implies that we have $\\\\mathcal {S}_0=\\\\lbrace x_{+},x_{-},y_{+},y_{-},z_{+},z_{-}\\\\rbrace ,$ where $\\\\hat{\\\\sigma }_w|w_s\\\\rangle =s|w_{s}\\\\rangle $ for $w_s\\\\in \\\\mathcal {S}_0$ .\",\n \"The exact entanglement quasiprobability reads [19] $P(a_s,b_t)=\\\\delta (a,b)\\\\left[\\\\frac{q}{12}+\\\\frac{|\\\\rho (a)|+st\\\\rho (a)}{4}\\\\right],$ with $a,b\\\\in \\\\lbrace x,y,z\\\\rbrace $ and $s,t\\\\in \\\\lbrace +1,-1\\\\rbrace $ , identifying the separability eigenvectors, $(a_s, b_t)\\\\in \\\\mathcal {S}_0\\\\times \\\\mathcal {S}_0$ .\",\n \"Of particular importance is the parameter $q=1-|\\\\rho (x)|-|\\\\rho (y)|-|\\\\rho (z)|$ as it determines if the state is separable; namely, $\\\\hat{\\\\rho }$ in Eq.\",\n \"(REF ) is separable if and only if $q\\\\ge 0$ [19].\",\n \"Conversely, the negativity of the quasiprobability in the case of entanglement is given by this quantity as well; it reads $\\\\min _{(u, v)\\\\in \\\\mathcal {S}_0\\\\times \\\\mathcal {S}_0}P(u,v)=q/12\\\\ge -1/6$ , and the lower bound is attained for Bell states.\",\n \"Figure: (Color online)The top panel depicts the entanglement quasiprobabilities [Eq.\",\n \"()] for a Bell state with ρ(x)=ρ(y)=ρ(z)=-1\\\\rho (x)=\\\\rho (y)=\\\\rho (z)=-1.Each patch in the AA-BB plane corresponds to a product state |a〉〈a|⊗|b〉〈b||a\\\\rangle \\\\langle a|\\\\otimes |b\\\\rangle \\\\langle b|, with a,b∈𝒮 0 a,b\\\\in \\\\mathcal {S}_0 according to Eq.\",\n \"().The bottom panel shows the eigenvalues of those separable-state projectors.The negativities in the quasiprobabilities PP identify the entanglement.Now we can apply the kernel in Eq.\",\n \"(REF ).\",\n \"Inserting the distribution in Eq.\",\n \"(REF ) yields $\\\\begin{aligned}&P_{K\\\\otimes K}(a_s,b_t)\\\\\\\\= & r^2\\\\delta (a,b)\\\\left[\\\\frac{q}{12}+\\\\frac{|\\\\rho (a)|+st\\\\rho (a)}{4}\\\\right]+\\\\frac{(1-r)^2}{36}\\\\\\\\&+\\\\frac{r(1-r)}{6}\\\\left[\\\\frac{q}{3}+\\\\frac{|\\\\rho (a)|}{2}+\\\\frac{|\\\\rho (b)|}{2}\\\\right].\\\\end{aligned}$ After some straightforward algebra, we find that $P_{K\\\\otimes K}$ is always nonnegative for $0<r\\\\le 1/\\\\sqrt{7}$ .\",\n \"In addition, we get tensor-product quasistates which are formed by the subsystem components $\\\\hat{\\\\Delta }_{K}(j)=\\\\frac{1}{r}|j\\\\rangle \\\\langle j|-\\\\frac{1-r}{2r}\\\\hat{1}$ for $j\\\\in \\\\mathcal {S}_0$ (see Appendix ).\",\n \"Consequently, the eigenvalues of $\\\\hat{\\\\Delta }_{K}(j)$ are $(1+r)/(2r)$ and $-(1-r)/(2r)$ .\",\n \"Figure: (Color online)The top panel depicts the classical joint probabilities, obtained using the kernel KK [see Eq.\",\n \"() for r=1/7r=1/\\\\sqrt{7}], for the same Bell state as shown in Fig.\",\n \".The bottom panel shows the positive and negative eigenvalues of the quasistates in Eq.\",\n \"() which are then required for the decomposition of the entangled state.Now the entanglement is captured by the unphysical nature of the quasistates.Let us consider a Bell state $\\\\hat{\\\\rho }=|\\\\psi \\\\rangle \\\\langle \\\\psi |$ as a specific example, where $|\\\\psi \\\\rangle =\\\\frac{|0\\\\rangle \\\\otimes |1\\\\rangle -|1\\\\rangle \\\\otimes |0\\\\rangle }{\\\\sqrt{2}},$ i.e., $\\\\rho (x)=\\\\rho (y)=\\\\rho (z)=-1$ .\",\n \"The quasiprobability decomposition of this state is shown in Fig.\",\n \"REF .\",\n \"The negativities in the joint quasiprobability, $P\\\\ngeq 0$ , certify the entanglement and enable us to decompose the entangled state $\\\\hat{\\\\rho }$ using nonnegative tensor-product projectors, $|a\\\\rangle \\\\langle a|\\\\otimes |b\\\\rangle \\\\langle b|\\\\ge 0$ .\",\n \"In contrast, in Fig.\",\n \"REF , we show the result of applying the uncorrelated and nonnegative kernel $K\\\\otimes K$ for $r=1/\\\\sqrt{7}$ ; see Eq.\",\n \"(REF ).\",\n \"This results in a joint classical probability distribution.\",\n \"However, the consequence is that the tensor-product operators for the decomposition have to be unphysical quasistates (because of the negative eigenvalues) in order to expand the entangled Bell state under study.\"\n ],\n [\n \"Conclusions and discussion\",\n \"Since the early theoretical descriptions of quantum states in phase space, quasiprobabilities have become one of the most important tools for studying quantum phenomena.\",\n \"Here, we complemented this treatment by introducing the concept of quasistates.\",\n \"While nonclassical states can be expanded in terms of a quasiprobability density and physical states, we demonstrated that one can equivalently use a classical, i.e., nonnegative, distribution and quasistates to perform such an expansion.\",\n \"Then the quantum features of the state are carried over from the necessity of negativities in the distribution to the requirement that unphysical quasistates are needed to describe the state of a system.\",\n \"Furthermore, we elaborated a number of different aspects and applications of quasistate, rendering them a useful tool to characterize quantum systems.\",\n \"In simple terms, quasistates almost describe a density operator—only certain properties of a physical state do not apply.\",\n \"In particular, a physical density operator is a Hermitian and positive semidefinite operator with a unit trace.\",\n \"Throughout this work, we discussed several examples for which one (or multiple) of the defining properties of a density operator are violated by a quasistate.\",\n \"For instance, eigenvalues can be negative, violating the positive semidefiniteness, or a Fourier representation was shown to lead to non-Hermitian quasistates.\",\n \"As one practical implementation, we studied the reconstruction of the density operator.\",\n \"This was based on the findings of a recent work [11], where Born's rule was reinterpreted in terms of contravariant operator-valued measures.\",\n \"For example, this dual concept is useful when the positive operator-valued measure describes imperfect measurement outcomes.\",\n \"Here, we proved that the contravariant operator-valued measure directly relates to the concept of quasistates and, therefore, can be considered as a special case of our general notion.\",\n \"Consequently, we were able to apply quasistates for a direct quantum state reconstruction in terms of measured probabilities, such as exemplified for a two-level qubit state.\",\n \"The duality between quasiprobability densities and their corresponding quasistates was studied for prominent quantum-optical phase-space distributions.\",\n \"For instance, we found that the quasistates of the Wigner-Weyl distribution coincide with an operator that describes the maximal singularities the Glauber-Sudarshan distribution can have [41].\",\n \"Beyond known quasiprobabilities in quantum optics, we further showed that our approach also leads to generalized phase-space distributions which expand the state in terms of squeezed states rather than coherent ones.\",\n \"This, for example, can be useful to characterize non-Gaussianity, which is required for universal quantum computation and, in contrast to nonclassicality in terms of the Glauber-Sudarshan distribution, not only relies on coherent states but general squeezed states [67].\",\n \"Furthermore, we characterized quantum correlations using quasistates.\",\n \"In particular, we showed that entanglement can be identified with a classical joint probability distribution and tensor-product quasistates.\",\n \"This result is surprising when considering that a violation of local-hidden-variable models typically excludes such classical distributions because states are (obviously) implicitly assumed to be physical in such models.\",\n \"Here, we proved the necessary and sufficient condition that a density operator is separable, i.e., not entangled, if and only if both the distribution and the used states are classical; conversely, entanglement requires that at least one of them is nonclassical.\",\n \"As an example, we identified the entanglement of a Bell state using either optimal quasiprobabilities [22] and tensor-product states or classical probabilities and unphysical quasistates.\",\n \"Based on the general construction of quasiprobabilities for other notions of quantum coherences [19], the found results can be straightforwardly generalized to other forms of quantumness.\",\n \"In conclusion, we developed the versatile and useful framework of quasistates for decomposing density operators.\",\n \"Previously established concepts and methods have been demonstrated to be equivalently accessible with our technique which additionally allowed us to go beyond this state of the art.\",\n \"Thus, we believe that, analogously to quasiprobabilities, the notion of quasistates has the potential to significantly contribute to the description and reconstruction of nonclassical quantum states in theory and experiment.\",\n \"This work has received funding from the European Union's Horizon 2020 Research and Innovation Program under Grant Agreement No.\",\n \"665148 (QCUMbER).\"\n ],\n [\n \"Exponential functions of bosonic operators\",\n \"For our treatment of quantum light in Sec.\",\n \", some additional algebra is required.\",\n \"In this appendix, we formulate the needed relations.\"\n ],\n [\n \"General relations\",\n \"We frequently apply the Gaussian integral formula $\\\\begin{aligned}&\\\\int _{\\\\mathbb {C}}d\\\\xi \\\\,\\\\exp \\\\left(-z|\\\\xi |^2-x\\\\xi ^2-y\\\\xi ^{\\\\ast 2}+u\\\\xi +v\\\\xi ^\\\\ast \\\\right)\\\\\\\\=&\\\\frac{\\\\pi }{\\\\sqrt{z^2-4xy}}\\\\exp \\\\left(\\\\frac{zuv-yu^2-xv^2}{z^2-4xy}\\\\right),\\\\end{aligned}$ where $\\\\mathrm {Re}[z\\\\pm (x+y)]>0$ and $\\\\mathrm {Re}[z^2-4xy]>0$ .\",\n \"We use the branch with a nonnegative real part for the square root of a complex number.\",\n \"The above relation can be extended to integrals of normally ordered expressions because under this prescription, we have an algebra of commuting operators.\",\n \"In the case that operator ordering becomes relevant, let us formulate additional relations.\",\n \"From the observations that $\\\\hat{a}^mf(\\\\hat{n})|n\\\\rangle =f(\\\\hat{n}+m)\\\\hat{a}^m|n\\\\rangle $ and $f(\\\\hat{n})\\\\hat{a}^{\\\\dag m}|n\\\\rangle =\\\\hat{a}^{\\\\dag m}f(\\\\hat{n}+m)|n\\\\rangle $ hold true, we conclude that $e^{x\\\\hat{a}^2}\\\\omega ^{\\\\hat{n}}=\\\\omega ^{\\\\hat{n}}e^{\\\\omega ^2x\\\\hat{a}^2}\\\\text{ and }\\\\omega ^{\\\\hat{n}}e^{y\\\\hat{a}^{\\\\dag 2}}=e^{\\\\omega ^2y\\\\hat{a}^{\\\\dag 2}}\\\\omega ^{\\\\hat{n}}.$ In the same manner, we get $\\\\exp (u\\\\hat{a})\\\\omega ^{\\\\hat{n}}=\\\\omega ^{\\\\hat{n}}\\\\exp (\\\\omega u\\\\hat{a})$ and $\\\\omega ^{\\\\hat{n}}\\\\exp (v\\\\hat{a}^\\\\dag )=\\\\exp (\\\\omega v\\\\hat{a}^\\\\dag )\\\\omega ^{\\\\hat{n}}$ .\",\n \"Furthermore, using the above integral, $|\\\\alpha \\\\rangle \\\\langle \\\\alpha |={:}\\\\exp (-[\\\\hat{a}-\\\\alpha ]^\\\\dag [\\\\hat{a}-\\\\alpha ]){:}$ , and $\\\\pi \\\\hat{1}=\\\\int _{\\\\mathbb {C}}d\\\\alpha \\\\,|\\\\alpha \\\\rangle \\\\langle \\\\alpha |$ , we obtain $\\\\begin{aligned}&e^{x\\\\hat{a}^2}e^{y\\\\hat{a}^{\\\\dag 2}}=\\\\exp [x\\\\hat{a}^2]\\\\hat{1}\\\\exp [y\\\\hat{a}^{\\\\dag 2}]\\\\\\\\=&\\\\frac{1}{\\\\sqrt{1-4xy}}{:}\\\\exp \\\\left(\\\\frac{4xy\\\\hat{a}^\\\\dag \\\\hat{a}+x\\\\hat{a}^2+y\\\\hat{a}^{\\\\dag 2}}{1-4uv}\\\\right){:}\\\\\\\\=&e^{v\\\\hat{a}^{\\\\dag 2}/(1-4uv)}\\\\left(\\\\frac{1}{1-4uv}\\\\right)^{\\\\hat{n}+1/2}e^{u\\\\hat{a}^2/(1-4uv)}.\\\\end{aligned}$ The same approach yields $\\\\exp [u\\\\hat{a}]\\\\exp [y\\\\hat{a}^{\\\\dag 2}]=\\\\exp [y(\\\\hat{a}^\\\\dag +u)^2]\\\\exp [u\\\\hat{a}]$ and $\\\\exp [x\\\\hat{a}^2]\\\\exp [v\\\\hat{a}^\\\\dag ]=\\\\exp [v\\\\hat{a}^\\\\dag ]\\\\exp [x(\\\\hat{a}+v)^2]$ .\",\n \"Finally, let us also mention the well-known formula $\\\\exp (u\\\\hat{a})\\\\exp (v\\\\hat{a}^\\\\dag )=\\\\exp (uv)\\\\exp (v\\\\hat{a}^\\\\dag )\\\\exp (u\\\\hat{a})$ .\"\n ],\n [\n \"Spectral decomposition of Gaussian quasistates\",\n \"In the main part of this contribution, we consider filter functions of a Gaussian form [cf.\",\n \"Eq.\",\n \"(REF )].\",\n \"The exact analysis for the resulting quasistates is performed here.\",\n \"For zero displacement, the integral in Eq.\",\n \"(REF ) yields $\\\\begin{aligned}\\\\hat{\\\\Delta }=&\\\\int _{\\\\mathbb {C}}d\\\\beta \\\\,\\\\frac{e^{-|\\\\beta |^2}}{\\\\pi \\\\Omega (\\\\beta )}e^{-\\\\beta \\\\hat{a}^\\\\dag }e^{\\\\beta ^\\\\ast \\\\hat{a}}\\\\\\\\=&\\\\frac{2\\\\,{:}\\\\exp \\\\left(\\\\frac{-2[1+s]\\\\hat{a}^\\\\dag \\\\hat{a} +q\\\\hat{a}^{\\\\dag 2}+p\\\\hat{a}^2}{(1+s)^2-pq}\\\\right){:}}{\\\\sqrt{(1+s)^2-pq}}.\\\\end{aligned}$ To characterize this quasistate, let us consider its decomposition in terms of exponential operators.\",\n \"First, the squeezing operator $\\\\hat{S}=\\\\exp ([\\\\zeta ^\\\\ast \\\\hat{a}^2-\\\\zeta \\\\hat{a}^{\\\\dag 2}]/2)$ can be equivalently given as [58] $\\\\hat{S}=\\\\exp \\\\left(-\\\\frac{\\\\tau }{2}\\\\hat{a}^{\\\\dag 2}\\\\right)\\\\sqrt{1-|\\\\tau |^2}^{\\\\hat{n}+1/2}\\\\exp \\\\left(\\\\frac{\\\\tau ^\\\\ast }{2}\\\\hat{a}^2\\\\right),$ where $\\\\tau =e^{i\\\\arg \\\\zeta }\\\\tanh |\\\\zeta |$ .\",\n \"Note that $|\\\\tau |<1$ , and we have infinite squeezing for $|\\\\tau |\\\\rightarrow 1$ .\",\n \"In addition, we consider thermal-state-like operator $\\\\begin{aligned}\\\\hat{W}=&(1-\\\\omega )\\\\omega ^{\\\\hat{n}}=\\\\sum _{n\\\\in \\\\mathbb {N}}(1-\\\\omega )\\\\omega ^{n}|n\\\\rangle \\\\langle n|\\\\\\\\=&(1-\\\\omega ){:}e^{-(1-\\\\omega )\\\\hat{n}}{:}\\\\\\\\=&\\\\int _{\\\\mathbb {C}}d\\\\alpha \\\\,\\\\frac{1-\\\\omega }{\\\\pi \\\\omega }\\\\exp \\\\left(-\\\\frac{(1-\\\\omega )|\\\\alpha |^2}{\\\\omega }\\\\right)|\\\\alpha \\\\rangle \\\\langle \\\\alpha |.\\\\end{aligned}$ See Ref.\",\n \"[68] for a derivation and additional considerations.\",\n \"Also note that the vacuum state $\\\\hat{W}=|0\\\\rangle \\\\langle 0|$ is obtained in the limit $\\\\omega \\\\rightarrow 0$ .\",\n \"Now, we defined an operator $\\\\hat{T}$ to be compared with the quasistate $\\\\hat{\\\\Delta }$ , $\\\\hat{T}=\\\\hat{S}\\\\hat{W}\\\\hat{S}^\\\\dag .$ Using the exchange relations in Eqs.\",\n \"(REF ) and (REF ), we find after some algebra a normally ordered expression, $\\\\begin{aligned}\\\\hat{T}=&(1-\\\\omega )\\\\sqrt{\\\\frac{1-|\\\\tau |^2}{1-|\\\\tau |^2\\\\omega ^2}}\\\\\\\\&\\\\times \\\\exp \\\\left(-\\\\frac{\\\\tau }{2}\\\\frac{1-\\\\omega ^2}{1-|\\\\tau |^2\\\\omega ^2}\\\\hat{a}^{\\\\dag 2}\\\\right)\\\\\\\\&\\\\times {:}\\\\exp \\\\left(-\\\\frac{[1-\\\\omega ][1+\\\\omega |\\\\tau |^2]}{1-|\\\\tau |^2\\\\omega ^2}\\\\hat{n}\\\\right){:}\\\\\\\\&\\\\times \\\\exp \\\\left(-\\\\frac{\\\\tau ^\\\\ast }{2}\\\\frac{1-\\\\omega ^2}{1-|\\\\tau |^2\\\\omega ^2}\\\\hat{a}^2\\\\right).\\\\end{aligned}$ Finally, we can compare this expression with Eq.\",\n \"(REF ), $\\\\hat{\\\\Delta }=\\\\hat{T}$ .\",\n \"Namely, equating coefficient yields $\\\\omega =\\\\frac{r-1}{r+1},\\\\,|\\\\tau |^2=\\\\frac{s-r}{s+r},\\\\,\\\\text{and}\\\\,\\\\frac{\\\\tau }{\\\\tau ^\\\\ast }=\\\\frac{q}{p},$ where $r=\\\\sqrt{s^2-pq}$ .\",\n \"This also implies $\\\\tau =-q/(s+r)$ and $\\\\tau ^\\\\ast =-p/(s+r)$ .\",\n \"Conversely, we can deduce parameters $s$ , $p$ , and $q$ from given values of $\\\\omega $ and $\\\\tau $ , $\\\\begin{aligned}p=\\\\frac{-2(1+\\\\omega )\\\\tau ^\\\\ast }{(1-\\\\omega )(1-|\\\\tau |^2)},\\\\quad q=\\\\frac{-2(1+\\\\omega )\\\\tau }{(1-\\\\omega )(1-|\\\\tau |^2)},\\\\\\\\\\\\text{and}\\\\quad s+1=\\\\frac{2(1+\\\\omega |\\\\tau |^2)}{(1-\\\\omega )(1-|\\\\tau |^2)}.\\\\end{aligned}$\"\n ],\n [\n \"Uniform attenuation\",\n \"In the main text, we consider a specific example of a kernel.\",\n \"For studying its properties, let us consider the corresponding convolution $P_K(\\\\chi )=\\\\sum _{\\\\psi \\\\in \\\\mathcal {S}}P(\\\\psi )K(\\\\psi ,\\\\chi )=a\\\\,P(\\\\chi )+b\\\\sum _{\\\\psi \\\\in \\\\mathcal {S}}P(\\\\psi )$ for a set $\\\\mathcal {S}$ of states.\",\n \"Note that $\\\\sum _{\\\\psi \\\\in \\\\mathcal {S}}P(\\\\psi )=1$ is the normalization condition.\",\n \"To ensure that the result is normalized as well, $1=\\\\sum _{\\\\chi \\\\in \\\\mathcal {S}}P_K(\\\\chi )=a+b|\\\\mathcal {S}|$ , we find $b=\\\\frac{1-a}{|\\\\mathcal {S}|},$ using the cardinality $|\\\\mathcal {S}|$ of the set of states.\",\n \"Further, when $0\\\\le a\\\\le 1$ is satisfied, the non-negativity of $P_K$ is guaranteed for any nonnegative $P$ .\",\n \"The considered convolution can be formulated via the $|\\\\mathcal {S}|\\\\times |\\\\mathcal {S}|$ map $K=a\\\\,\\\\mathrm {id}+b\\\\,nn^\\\\mathrm {T},$ using the identity $\\\\mathrm {id}$ and the $|\\\\mathcal {S}|$ -dimensional vector $n=[1,\\\\ldots ,1]^\\\\mathrm {T}$ .\",\n \"Then the inverse takes a similar form, $K^{-1}=\\\\frac{1}{a}\\\\mathrm {id}-\\\\frac{b/a}{a+b|\\\\mathcal {S}|}nn^\\\\mathrm {T},$ where $a+b|\\\\mathcal {S}|=1$ for the given $b$ .\",\n \"Using the normalization, the convolution kernel $K$ corresponds to uniform addition of noise to the probabilities.\",\n \"On the level of the operators, we get $\\\\hat{\\\\Delta }_K(\\\\chi )=\\\\frac{1}{a}|\\\\chi \\\\rangle \\\\langle \\\\chi |-\\\\frac{1-a}{a|\\\\mathcal {S}|}\\\\sum _{\\\\psi \\\\in \\\\mathcal {S}}|\\\\psi \\\\rangle \\\\langle \\\\psi |.$ In the continuous case, we can generalize the above relations as $P_K(\\\\chi )=\\\\int _{\\\\mathcal {S}} d\\\\psi \\\\,P(\\\\psi )\\\\left[a\\\\delta (\\\\psi ,\\\\chi )+\\\\frac{1-a}{|\\\\mathcal {S}|}\\\\right],$ where the volume $|\\\\mathcal {S}|=\\\\int _{\\\\mathcal {S}}d\\\\psi $ is used instead.\",\n \"Furthermore, the projection operators transform as $\\\\hat{\\\\Delta }_{K}(\\\\chi )=\\\\int _{\\\\mathcal {S}} d\\\\psi \\\\left[\\\\frac{1}{a}\\\\delta (\\\\psi ,\\\\chi )-\\\\frac{1-a}{a|\\\\mathcal {S}|}\\\\right]|\\\\psi \\\\rangle \\\\langle \\\\psi |.$ As a discrete-variable example, we study a qubit system with orthogonal basis $\\\\lbrace |0\\\\rangle ,|1\\\\rangle \\\\rbrace $ .\",\n \"We can identify the Pauli matrices as $\\\\hat{\\\\sigma }_z=|1\\\\rangle \\\\langle 1|-|0\\\\rangle \\\\langle 0|$ , $\\\\hat{\\\\sigma }_x=|0\\\\rangle \\\\langle 1|+|1\\\\rangle \\\\langle 0|$ , and $\\\\hat{\\\\sigma }_y=i|0\\\\rangle \\\\langle 1|-i|1\\\\rangle \\\\langle 0|$ .\",\n \"The Hermitian operator basis can be completed with the operator $\\\\hat{1}=|0\\\\rangle \\\\langle 0|+|1\\\\rangle \\\\langle 1|$ .\",\n \"The eigenvectors $|j_{\\\\pm 1}\\\\rangle $ of the Pauli matrices $\\\\hat{\\\\sigma }_j$ to the eigenvalues $\\\\pm 1$ for $j\\\\in \\\\lbrace z,x,y\\\\rbrace $ form the set $\\\\mathcal {S}$ .\",\n \"Then we can decompose any density operator as $\\\\hat{\\\\rho }=\\\\sum _{j\\\\in \\\\lbrace x,y,z\\\\rbrace ,s\\\\in \\\\lbrace +1,-1\\\\rbrace }P(j_s)|j_{s}\\\\rangle \\\\langle j_{s}|.$ Conversely, the attenuated distribution reads $P_{K}(j_s)=aP(j_s)+(1-a)/6$ , and the transformed operators take the form $\\\\hat{\\\\Delta }_K(j_s)=a^{-1}|j_s\\\\rangle \\\\langle j_s|-(1-a)(2a)^{-1}\\\\hat{1}$ .\"\n ]\n]","isConversational":false},"id":{"kind":"string","value":"1808.08471"}}},{"rowIdx":196,"cells":{"text":{"kind":"list like","value":[["Invariant Sets in Quasiperiodically Forced Dynamical Systems"],["Abstract This paper addresses structures of state space in quasiperiodically forced dynamical systems.","We develop a theory of ergodic partition of state space in a class of measure-preserving and dissipative flows, which is a natural extension of the existing theory for measure-preserving maps.","The ergodic partition result is based on eigenspace at eigenvalue 0 of the associated Koopman operator, which is realized via time-averages of observables, and provides a constructive way to visualize a low-dimensional slice through a high-dimensional invariant set.","We apply the result to the systems with a finite number of attractors and show that the time-average of a continuous observable is well-defined and reveals the invariant sets, namely, a finite number of basins of attraction.","We provide a characterization of invariant sets in the quasiperiodically forced systems.","A theoretical result on uniform boundedness of the invariant sets is presented.","The series of theoretical results enables numerical analysis of invariant sets in the quasiperiodically forced systems based on the ergodic partition and time-averages.","Using this, we analyze a nonlinear model of complex power grids that represents the short-term swing instability, named the coherent swing instability.","We show that our theoretical results can be used to understand stability regions in such complex systems."],["Introduction","Methods of ergodic theory [1], [2] have been applied for solving a number of problems in science and technology such as control of mixing of fluid flows [3], analysis of traffic jams [4] and of quantum many-body models [5], performance evaluation of ecosystem models [6].","A visualization method for invariant sets of discrete-time dynamical systems (or maps) based on ergodic theory was developed in [7].","This method is based on the ergodic partition or ergodic decomposition [8] of state space in a given dynamical system, which is associated with the eigenspace at eigenvalue 1 of the Koopman operator [9].","This method partitions a (compact and metric) state space $X$ using joint level sets of time-averages of a basis of functions defined on $X$ .","This leads to an approximation of invariant sets on which the dynamics of the system are ergodic.","In an ergodic invariant set, almost all points are accessible in the sense that the initial conditions in this set thoroughly sample the set.","The method enables the visualization of low-dimensional (in practice usually two-dimensional) slices through high-dimensional invariant structures and offers a computationally tractable method for analyzing global structures of state space in discrete-time nonlinear models.","The numerical methods associated with the ergodic partition theory have been developed: see [9] and references therein.","The purpose of this paper is to extend the ergodic partition theory and exploit it to provide global analysis of state space in quasiperiodically forced dynamical systems.","Quasiperiodicity is one of the three types of commonly observed dynamics in both deterministic models and experiments: see, e.g., [10], [11].","Quasiperiodically forced systems are important objects in nonlinear dynamics and have been studied by many groups of researchers: phenomenology of chaotic attractors [12], analysis of dynamics described by the Schrödinger equations [13] and of quantum chaos [14], and dynamical systems analysis with applications to fluid mixing [15], [16].","In the present paper, we study the global structure of state space in quasiperiodically forced systems from the viewpoint of ergodic partition.","First, we develop a theory of ergodic partition of state space in measure-preserving and dissipative, continuous-time dynamical systems or flows.","This is a natural extension of the existing theory for measure-preserving maps in [7].","The ergodic partition is again based on eigenspace at eigenvalue 0 of the associated Koopman group.","Related analyses of time-periodic flows via spectral properties of linear operators are reported in [17], [18].","Since an arbitrary quasiperiodically forced system with a smooth invariant measure is transformed into an autonomous system determining a measure-preserving flow (see Section ), the ergodic partition theory is used for visualization of invariant sets for the quasiperiodically forced system.","Next, we provide a characterization of invariant sets in quasiperiodically forced systems.","In general, understanding invariant structures in the quasiperiodically forced systems is not easy because the associated portraits of state space change with time in an aperiodic manner.","Here, we introduce a notion of uniform boundedness of invariant sets.","A theoretical result on uniform boundedness of invariant sets is presented in this paper: see Theorem REF , and Corollaries REF and REF .","This clarifies a dynamical feature of invariant sets in the quasiperiodically forced systems that is directly determined by the ergodic partition and its associated visualization technique.","Also, we apply the theory developed above to analysis of a nonlinear model of complex power grids that represents the short-term swing instability, which we name the Coherent Swing Instability (CSI) [19].","Preliminary results in this paper were published in the conference proceedings [20], [21].","This paper contains detailed discussion of the theory with a new proof and a set of new numerical simulations.","The rest of this paper is organized as follows.","In Section we introduce set-up and mathematical preliminaries from dynamical systems theory.","A theory of ergodic partition in measure-preserving flows is developed in Section .","Based on the result, the basin of attraction in the dissipative case is explored in ().","In Section we provide a new characterization of invariant sets in the quasiperiodically forced dynamical systems.","The developed theory is applied in Section to two simple examples of the quasiperiodically forced system and to a nonlinear model of the CSI phenomenon of a power grid.","Conclusions of this paper are presented in Section ."],["Quasiperiodically Forced Dynamical Systems","Throughout this paper, we address the following quasiperiodically forced dynamical system that evolves on a finite-dimensional metric space $M$ : for ${m}\\in {M}$ and $t\\in \\mathbb {R}$ , $\\frac{{d}{m}}{{d} t}={g}({m},t).$ The function $g$ is assumed to be quasiperiodic on $t$ in the sense of Moser [22], that is, ${g}({m},t)={G}({m},\\mathit {\\Omega }_1t,\\mathit {\\Omega }_2t,\\ldots ,\\mathit {\\Omega }_Nt),$ where ${G}({m},\\theta _1,\\theta _2,\\ldots ,\\theta _N)$ is assumed to be a smooth vector-valued function and of period $2\\pi $ in $\\theta _1,\\theta _2,\\ldots ,\\theta _N$ .","The real numbers $\\mathit {\\Omega }_1,\\mathit {\\Omega }_2,\\ldots ,\\mathit {\\Omega }_N$ are the $N$ basic angular frequencies and assumed to be rationally independent: there exists no $N$ -dimensional integer vector $(k_1,k_2,\\ldots ,k_N)^\\top $ ($\\top $ stands for the transpose operation of real-valued vectors) in which the entries do not all vanish, satisfying the resonance relation: $\\sum ^{N}_{i=1}k_i\\mathit {\\Omega }_i=0.$ The system (REF ) is non-autonomous and transformed into an autonomous system defined on the augmented state space $X:=M\\times \\mathbb {T}^N$ , in the same manner as in [16].","By introducing the $N$ variables $\\theta _i=\\mathit {\\Omega }_it\\in \\mathbb {T}$ ($i=1,2,\\ldots ,N$ ), we have $\\frac{d{m}}{dt}={G}({m},\\theta _1,\\theta _2,\\ldots ,\\theta _N), \\qquad \\frac{d\\theta _i}{dt}=\\mathit {\\Omega }_i \\qquad i=1,2,\\ldots ,N.$ In this way, by taking trajectories of the augmented system (REF ) on the augmented state variable ${x}:=({m},\\theta _1,\\theta _2,\\ldots ,\\theta _N)\\in X$ , a flow, that is, one-parameter group of diffeomorphisms is defined for the quasiperiodically forced system (REF )."],["Measure-Preserving Flows and Partition of State Space","In Section , as one type of flows induced by (REF ), we will investigate a measure-preserving flow defined on a class of probability spaces, where ergodic theory has been developed [8], [23].","The probability space considered here corresponds to the tuple $({X},\\mathfrak {B}_{X},\\mu )$ , where ${X}$ is a compact metric space, $\\mathfrak {B}_{X}$ is the Borel $\\sigma $ -algebra of ${X}$ , and $\\mu $ is a probability measure.","A measure-preserving flow of the probability space is a one-parameter group of measure-preserving diffeomorphisms ${S}^t: {X}\\rightarrow {X}$ , $t\\in \\mathbb {R}$ such that ${S}^t$ is measure-preserving (for all $A\\in \\mathfrak {B}_{X}$ we have $\\mu ({S}^{-t}(A))=\\mu (A)$ ), ${S}^0$ the identity, and ${S}^{t_1+t_2}={S}^{t_1}\\circ {S}^{t_2}$ for $t_1,t_2\\in \\mathbb {R}$ .","The important notion which we utilize throughout this paper is the partition of state space in dynamical systems.","One motivation behind the following definitions is that we aim to locate such a partition that plays a role in the data-driven estimation of certain statistical properties of non-ergodic measure-preserving dynamical systems; we will show this later as the ergodic partition.","The definitions of partition and measurable partition from [7] are the following.","Definition 1 A partition of ${X}$ is a family $\\zeta $ of sets satisfying $A,B\\in \\zeta \\quad \\Rightarrow \\quad \\mu (A\\cap B)=0, \\quad \\mu \\left({X}\\setminus \\bigcup _{A\\in \\zeta }A\\right)=0.$ The element of $\\zeta $ containing a point ${x}\\in {X}$ is denoted by $\\zeta ({x})$ .","A partition $\\zeta $ of ${X}$ is said to be measurable if there exists a countable family $\\mathfrak {D}$ of measurable sets $\\lbrace D_i\\rbrace $ such that every $D_i$ is a union of elements of $\\zeta $ , and for any pair $A_1,A_2$ of elements of $\\zeta $ there exists $D_j\\in \\mathfrak {D}$ such that $A_1\\subset D_j$ and $A_2\\subset D_j^{\\rm c}$ , where $D_j^{\\rm c}$ stands for the complement of $D_j$ in ${X}$ .","The family $\\mathfrak {D}$ is called a basis for $\\zeta $ .","In Section we use a product operation on the set of partitions of ${X}$ .","The product operation is defined in [23] as follows.","Definition 2 If $\\zeta _n$ , $n=1,\\ldots ,N$ are partitions, we then define their product $\\vee ^{N}_{n=1}\\zeta _n$ as the partition whose elements are the sets of the form $\\cap ^{N}_{i=1}{A_i}$ , for $A_i\\in \\zeta _i$ , $i=1,\\ldots ,N$ , satisfying $\\mu \\left(\\cap ^{N}_{i=1}A_i\\right)\\ne 0$ .","For a countable sequence of partitions, the notation $\\vee ^{\\infty }_{n=1}\\zeta _n$ will be used for the $\\sigma $ -algebra generated by $\\cup ^\\infty _{n=1}\\zeta _n$ ."],["Koopman Group","Consider a flow ${S}^t: {X}\\rightarrow {X}$ ($t\\in \\mathbb {R}$ ) on a finite-dimensional space ${X}$ and denote by $\\mathcal {F}$ a space of scalar-valued functions defined on ${X}$ : $\\mathcal {F}\\ni f: {X}\\rightarrow \\mathbb {C}$ , which we call the space of observables.","In the following, we suppose that the existence and uniqueness of solutions associated with the flow hold for all $t$ .","Then, we define the one-parameter group of linear operators $U^t$ ($t\\in \\mathbb {R}$ ) as a group of composition operators with ${S}^t$ : for $f\\in \\mathcal {F}$ , $(U^tf)({x}):=f({S}^t({x}))=(f\\circ {S}^t)({x}).$ where it is known as the Koopman group [24], [9].","For spaces as $\\mathcal {F}=\\mathcal {C}^0(X)$ and $\\mathcal {L}^p(X)$ , the Koopman group is strongly continuous.","Although the original flow ${S}^t$ is possibly described by a nonlinear differential equation and evolves on the finite-dimensional space ${X}$ , the operators $U^t$ are linear but evolve on the infinite-dimensional space $\\mathcal {F}$ .","The eigenvalue $\\lambda \\in \\mathbb {C}$ and eigenfunction $\\phi _\\lambda \\in \\mathcal {F}\\setminus \\lbrace 0\\rbrace $ of the Koopman group are defined as follows: $(U^t\\phi _\\lambda )({x})=\\exp (\\lambda t)\\phi _\\lambda ({x}).$ The notion of the Koopman group and its spectral characterization will be used in the following sections."],["Ergodic Partition in Measure-Preserving Flows","For completeness, in this section we extend the theory of ergodic partition in [7] to the measure-preserving flow ${S}^t$ on the probability space $(X,\\mathfrak {B}_{X},\\mu )$ , which is introduced in Section REF .","This section consists of a few lemmas and a theorem.","Their proofs are almost identical to the proofs for maps [7] and appear in Appendices and .","We denote by $\\mathcal {L}^1_\\mu ({X})$ the space of all $\\mu $ -integrable functions on ${X}$ and by $\\mathcal {C}({X})$ the space of all real-valued continuous functions on ${X}$ endowed with the sup norm.","Roughly speaking, an ergodic partition is a partition of the state space ${X}$ into invariant sets on which the dynamics are ergodic.","The precise definition of ergodic partition is the following.","Definition 3 A measurable partition $\\zeta $ of ${X}$ is said to be ergodic under the flow ${S}^t$ if for any element $A$ of $\\zeta $ , (i) $A$ is invariant under ${S}^t$ , and (ii) there exists an invariant probability measure $\\mu _A$ on $A$ such that the restriction of ${S}^t$ to $A$ , denoted by ${S}^t|_A$ , is an ergodic diffeomorphism on $A$ , and for all $f\\in \\mathcal {L}^1_\\mu ({X})$ , $\\int _{X} f{d}\\mu =\\int _{X}\\left[\\int _{A=:\\zeta ({x})} f|_A{d}\\mu _A\\right]{d}\\mu ({x}),$ where $f|_A$ stands for the restriction of $f$ to the ergodic element $A$ , and $\\mu ({x})$ is again a probability measure on $X$ .","It is remarked that the ergodic partition is defined for the entire state space $X$ with $\\sigma $ -algebra, but it can be relaxed in context of the $\\sigma $ -algebra of invariant sets which can be parts of the state space.","Here, we denote by $f^\\ast $ the time-average of $f$ under the flow ${S}^t$ if the right-hand side of $f^\\ast ({x}):=\\lim _{T\\rightarrow \\infty }\\frac{1}{T}\\int ^T_0 f({S}^t({x})){d} t,$ exists for almost every (a.e.)","point ${x}\\in {X}$ with respect to the measure $\\mu $ .","Birkhoff's Ergodic Theorem [25], [1] shows that for all $f\\in \\mathcal {L}^1_\\mu ({X})$ , (i) $f^\\ast ({x})$ exists for a.e.","point ${x}\\in {X}$ (with respect to $\\mu $ ); (ii) $f^\\ast ({S}^t({x}))=f^\\ast ({x})$ for a.e.","point ${x}\\in {X}$ ; and (iii) $\\int _{X}f{d}\\mu =\\int _{X}f^\\ast {d}\\mu $ .","Here, we let $\\mathit {\\Sigma }$ be the set of points ${x}\\in {X}$ such that $f^\\ast ({x})$ exists for all $f\\in \\mathcal {C}({X})$ , and $\\mathit {\\Sigma }(f)$ the set of points ${x}\\in {X}$ such that $f^\\ast ({x})$ exists for a particular $f\\in \\mathcal {C}({X})$ .","The following lemma is standard (see page 129 of [23] for discrete-time dynamical systems (maps): it is obvious for flows).","Lemma 4 For a countable and dense set $S$ in $\\mathcal {C}({X})$ , $\\mathit {\\Sigma }=\\bigcap _{f\\in S}\\mathit {\\Sigma }(f).$ Now, we have a set $\\mathit {\\Sigma }$ such that the time-averages of all continuous functions on ${X}$ exist.","Then, the complimentary set $\\mathit {\\Sigma }^{\\rm c}$ is of measure zero.","This is because according to Birkhoff's Ergodic Theorem (i), each $\\lbrace \\mathit {\\Sigma }(f)\\rbrace ^{\\rm c}$ is of measure zero, and thus $\\mathit {\\Sigma }^{\\rm c}:=\\cup _{f\\in S}\\lbrace \\mathit {\\Sigma }(f)\\rbrace ^{\\rm c}$ is a countable union of sets with zero measure, which is again of measure zero.","The next lemma shows that the time-average of a continuous function induces a measurable partition on ${X}$ .","Lemma 5 Let $f$ be a continuous function on ${X}$ .","The family of level sets of $f^\\ast $ , $A_\\alpha :=\\lbrace {x} : {x}\\in \\mathit {\\Sigma }, f^\\ast ({x})=\\alpha \\rbrace ,~~~\\alpha \\in \\mathbb {R},$ is a measurable partition of ${X}$ .","We denote by $\\zeta _f$ this partition and call it the partition induced by the function $f$ .","See Appedix .","Now, we can prove the first theorem stating that $\\zeta _f$ induces the ergodic partition of ${X}$ , which holds for general measure-preserving flows including the augmented system (REF ) for the quasiperiodically forced system (REF ).","Theorem 6 Consider the measure-preserving flow ${S}^t: X\\rightarrow X$ ($t\\in \\mathbb {R}$ ) associated with $\\left.\\frac{d{S}^t({x})}{dt}\\right|_{t=0} ={F}({x}) \\quad \\textrm {for each}~{x}\\in X$ where ${F}: X\\rightarrow \\mathrm {T}X$ (tangent bundle of $X$ ) is a nonlinear vector field.","Let $\\zeta {e}$ be the product of measurable partitions of ${X}$ induced by every $f\\in S$ : $\\zeta {e}=\\bigvee _{f\\in S}\\zeta _f.$ Then, $\\zeta {e}$ is the ergodic partition of ${X}$ .","See Appendix .","By combining the above theorem and lemma 20 in [26], we have the following corollary based on a finite number of basis functions.","Corollary 7 Assume there exists a complete system of functions $\\lbrace f_i\\rbrace $ , $f_i\\in \\mathcal {C}(X)$ , $i\\in \\mathbb {N}_{>0}$ (set of all natural numbers except for 0) i.e.","finite linear combinations of $f_i$ are dense in $\\mathcal {C}(X)$ .","The ergodic partition $\\zeta {e}$ is $\\zeta {e}=\\bigvee _{i\\in \\mathbb {N}_{>0}}\\zeta _{f_i}.$ For a given function $f$ , it is obvious that each level set $A_\\alpha =\\lbrace {x} : {x}\\in \\mathit {\\Sigma }, f^\\ast ({x})=\\alpha \\rbrace $ ($\\alpha \\in \\mathbb {R}$ ) as an element of $\\zeta _f$ is invariant under ${S}^t$ .","In [7] the authors proposed to use the level sets $A_\\alpha $ , which are directly computed with the time-averaging of $f$ under ${S}^t$ , for identifying and visualizing invariant sets in discrete-time dynamical systems possessing a smooth invariant measure.","Theorem REF implies that the same computation can be used for visualization of invariant sets for the measure-preserving flow, namely, continuous-time dynamical systems with a smooth invariant measure.","Since, as shown in Section , the quasiperiodically forced system (REF ) defined on the state space $M$ is transformed to the flow defined on the augmented state space $X=M\\times \\mathbb {T}^N$ , the ergodic partition theory is applicable to visualization of invariant sets in $X$ for the original quasiperiodically forced system (REF ) possessing a smooth invariant measure.","However, in the quasiperiodically forced system (REF ) we are interested in dynamics on $M$ not on $X=M\\times \\mathbb {T}^N$ .","While for $N=1$ it is clear that every intersection of the invariant set in $M\\times \\mathbb {T}^1$ with $M$ is an invariant set of the associated Poincaré map.","The relationship in the quasiperiodically forced system (namely, $N\\ge 2$ ) is much less clear.","We make it precise in Section .","Here, let us introduce the connection of time-average $f^\\ast $ and the Koopman group.","It is obvious that for any $f\\in \\mathcal {L}^1_\\mu (X)$ its time-average $f^\\ast $ is an eigenfuction of the operators $U^t$ at eigenvalue 0: for a.e.","point ${x}\\in {X}$ with respect to $\\mu $ , $(U^tf^\\ast )({x})=f^\\ast ({S}^t({x}))=f^\\ast ({x})=\\exp (0t)f^\\ast ({x}).$ Since the eigenfunctions at 0 are invariant under the flow, we have a complete characterization of (possibly non-smooth) invariant sets."],["Basin of Attraction in Dissipative Flows","In the last of the previous section, we indicate that the time-average of an observable enables the visualization of low-dimensional slices through high-dimensional invariant sets of the quasiperiodically forced system with a smooth invariant measure.","Here, we consider a more general class of the quasiperiodically forced systems with dissipation and will point out that the use of time-average works for characterizing an important invariant set—basin of attraction.","In Section we considered the measure-preserving flow on a compact metric space.","Every continuous flow on a compact metric space preserves a measure.","This is the content of the so-called Krylov-Bogolyubov theorem [27].","However, this might be a useless statement if our intent is to study the behavior of trajectories from their time-averages, by the method used in the measure-preserving cases in Section .","For example, let us take a simple dissipative flow, $\\frac{{d} x}{{d} t}=-\\lambda x \\qquad x\\in \\mathbb {R}, \\qquad \\lambda >0.$ All of the initial conditions converge to $x=0$ as $t\\rightarrow \\infty $ along the flow $S^t(x)=\\exp (-\\lambda t)x$ .","The invariant measure clearly exists and is the Dirac measure supported at $x=0$ (to which any “initial\" measure defined on $\\mathbb {R}$ evolves).","Thus, the following statement holds: for a function $f:\\mathbb {R}\\rightarrow \\mathbb {C}$ , $f^\\ast (x)=\\int _{\\mathbb {R}}f{d}\\mu ,$ for a.e.","point $x\\in \\mathbb {R}$ with respect to the Dirac measure.","However, that excludes the whole real line except for the origin itself.","The situation feels better though.","For example, it is easy to see that the time-average of any continuous function on $\\mathbb {R}$ is just its value at 0: $f^*(x)=f(0)$ .","Note that for the measure-preserving case in Section , we basically considered integrable functions for which time-averages exist.","We can not do that in the dissipative case.","A function that is 1 everywhere on a finite interval $(\\underline{a},\\overline{a})$ containing 0, except at 0 where its value is 0, is clearly integrable (being a union of two simple functions).","Then, its time-average under the dissipative flow is identically 1, and its integral with respect to the invariant Dirac measure is 0.","It turns out that in dissipative systems, it is best for the time-average to work with continuous functions.","Adopting that idea, the whole method of ergodic partitioning can be extended to capture basins of attraction for continuous flows that are not necessarily measure-preserving.","Consider a continuous flow ${S}^t: X\\rightarrow X$ , where $X$ is not necessary compact, and label by $F$ the set on which time averages of continuous functions do not exist.","On the set $X\\setminus F$ , consider a set $C$ on which time-averages of continuous functions (or, better, a countable, separating set of continuous functions) are constant.There can be uncountably many such sets inside $X\\setminus F$ : for example, let ${S}^t$ be the identity map, mapping every point into itself.","Then, by the same construction described in Section and Appendix , there is an invariant measure $\\mu _C$ such that $f^*({x})=\\int _C f d\\mu _C$ for any continuous function $f$ , ${x}\\in C$ .","Such a measure is called a physical measure [28], provided $M$ is equipped with a measure $\\mu $ that ${S}^t$ does not preserve (but we are interested in)—say Lebesgue measure—and $C$ contains an open (and thus positive measure) set in $M$ .In light of this discussion, one might say that the notion of “ergodic measure\" should be preserved for such constructs even in dissipative systems.","Namely, many ergodic measures we obtain in measure-preserving system are certainly physically important, although their support does not contain an open set.","The tricky part is that due to Birkhoff's Ergodic Theorem, ergodic measures are associated with time averages of integrable functions.","In dissipative systems, this does not work (see the above example).","We could keep the requirement of equality of space and time averages, like in (REF ), but for continuous functions only.","Physical measures would then be ergodic measures with support that contains an open set.","Suppose that the state space of a continuous flow admits a finite number of attractors, and that the union of their basins of attraction is of full measure.","We state the result for flows (continuous-time systems) to emphasize that considerations here work for both discrete and continuous time.","Theorem 8 Let the system $\\left.\\frac{d{S}^t({x})}{dt}\\right|_{t=0}={F}({x}), \\quad {x}\\in X\\subset \\mathbb {R}^n$ with a flow ${S}^t$ have a finite number $N$ of attractors with basins of attraction $A_j, j=1,...,N$ , such that $\\mu (\\cup _j A_j)=\\mu (X)$ , where $\\mu $ is the Lebesgue measure.","Also, let the time averages of continuous functions exist everywhere on a set $X\\setminus F$ , where $\\mu (F)=0$ .","Then, the time-average $h^*({x})$ of a continuous function $h\\in \\mathcal {C}(X)$ is a piecewise constant eigenfunction of the Koopman operators $U^t$ at eigenvalue 0 defined a.e.","point with respect to $\\mu $ .","For any point ${x}\\in A_j,$ there exists a set $C$ such that $(U^t h^*)({x})&=U^t \\lim _{T\\rightarrow \\infty }\\frac{1}{T}\\int _0^{T} h({S}^\\tau ({x}))d\\tau \\nonumber \\\\&= \\int _C (U^t h) d\\mu _C\\nonumber \\\\&=\\int _C (h\\circ {S}^t) d\\mu _C\\nonumber \\\\&=\\int _C h d\\mu _C \\nonumber \\\\&=h^*({x}), \\nonumber $ where in the second line we utilized (REF ), and transitioning from line 3 to 4 we used the fact that $\\mu _C$ is invariant under ${S}^t$ .","Since $(U^t h^*)({x})=h^*({x})$ is exactly the equation which $h^*({x})$ has to satisfy in order to be an eigenfunction of $U^t$ , and the basin of attraction $A_j$ is arbitrary, by taking into account that $\\mu (\\cup _j A_j)=\\mu (X)$ the proposition is proven.","The concept of the ergodic partition in dissipative systems is applied to a more general class of systems than those treated in Theorem REF .","It provides ergodic measures on invariant sets that are not necessarily attractors—because they do not have an open set of initial conditions converging to them.","Yet, these sets are dynamically important and can not be refined without losing the equivalence of space and time averages.","More precisely, the ergodic partition $\\zeta {e}$ of $M$ under ${S}^t$ (not necessarily measure-$\\mu $ preserving) is a partition into sets $C_{\\alpha }$ , where $\\alpha $ is a member of an indexing set, such that on each set $C_{\\alpha }$ (ergodic set) there exists an ergodic measure $\\mu _{C_\\alpha }$ such that $\\mu _{C_{\\alpha }}(C_{\\alpha })=1,$ For every $f\\in \\mathcal {C}(M),f^{\\ast }({x}\\in C_{\\alpha })=\\int _{C_{\\alpha }}fd\\mu _{C_{\\alpha }},$ for a.e ${x}$ point with respect to $\\mu $ .","The last condition emphasizes the role of the measure $\\mu $ —the time averages are equal to space averages with respect to $\\mu _{C_\\alpha }$ , but their equality is almost everywhere with respect to measure $\\mu $ that might be of our interest.","In this way, we have escaped the realm of the Krylov-Bogolyubov Theorem, which, in the case when dynamics are not measure-preserving, neglects dynamics on possibly large swaths of the state space."],["Sample-Based Characterization of Invariant Sets","In this section, we provide a characterization of invariant sets in the quasiperiodically forced system (REF ).","Generally speaking, understanding invariant structures of (REF ) is not easy because the associated portrait of $M$ changes with time in an aperiodic manner.","As shown in Section , the ergodic partition theory enables one to visualize an invariant set as its low-dimensional slice in $M$ by the time-averaging technique.","Also, in () it was shown that the time-averaging technique works for dissipative case to visualize the basin of attraction.","The obtained slice here is just a sample of the invariant set at a particular initial time (in other words, an initial condition in $\\mathbb {T}^N$ ).","Because of the aperiodic nature, such a sample does not seem to provide complete information on the entire structure of invariant set in the augmented state space $M\\times \\mathbb {T}^N$ .","However, we will prove that a boundedness property of the invariant set in $M\\times \\mathbb {T}^N$ is captured by means of one sample of it in $M$ (see Theorem REF ).","In the following, in order to encompass both the cases in Section and (), we consider the original state space $M$ that is metric and not necessarily compact.","A continuous, linear skew-product flow ${S}^t$ (diffeomorphism) on the augmented state space $X:=M\\times \\mathbb {T}^N$ derived from the quasiperiodically forced system (REF ) is described below: for ${x}=({m},{\\theta })\\in M\\times \\mathbb {T}^N$ , ${S}^t({x}):=\\left({S}^t_{M}({m},{\\theta }),{S}^t_\\mathit {\\Omega }({\\theta })\\right),$ where ${S}^t_{M}: M\\times \\mathbb {T}^N\\rightarrow M$ is the continuous map defined by trajectories of the original system (REF ), and ${S}^t_\\mathit {\\Omega }: \\mathbb {T}^N\\rightarrow \\mathbb {T}^N$ the linear (continuous) flow on the torus $\\mathbb {T}^N$ .","For a fixed ${\\theta }_0\\in \\mathbb {T}^N$ , we will write the orbit on $\\mathbb {T}^N$ through ${\\theta }_0$ as $\\mathcal {O}_{\\mathbb {T}^N}({\\theta }_0):=\\lbrace {\\theta } : {\\theta }={S}^t_\\mathit {\\Omega }({\\theta }_0)\\in \\mathbb {T}^N, t\\in \\mathbb {R}\\rbrace $ .","Since the $N$ basic frequencies in the original system (REF ) are rationally independent, the following lemma is obvious.","Lemma 9 For any ${\\theta }_0\\in \\mathbb {T}^N$ , $\\mathcal {O}_{\\mathbb {T}^N}({\\theta }_0)$ is dense in $\\mathbb {T}^N$ .","Now, we investigate an invariant set in the augmented system (REF ).","Let $I\\subseteq X$ be a positively invariant set of (REF ) that is supposed to be closedIf $I$ is invariant and open, then its closure $\\mathrm {cl}(I)$ is still invariant..","It follows from Lemma REF and by closure that the following decomposition of $I$ holds: $I=\\bigcup _{\\theta \\in \\mathbb {T}^N} A_\\theta \\times \\lbrace {\\theta }\\rbrace ,$ where $A_{\\theta }$ is a closed subset of $M$ .","Let us denote by $A_{\\theta _0}$ an intersection or sample of the invariant set $I$ at ${\\theta }={\\theta }_0$ .","The next lemma provides a topological property of the intersections $A_\\theta $ .","Lemma 10 For each $\\epsilon >0$ there exists a positive constant $\\delta $ so that $|{\\theta }-{\\theta }_0|_{\\mathbb {T}^N}<\\delta $ implies $d(A_\\theta ,A_{\\theta _0})<\\epsilon $ , where $d(A_{\\theta },A_{\\theta _0})$ is the Hausdorff distance that induces a topology on the family of all closed subsets of $M$ .","Assume not.","Then, for each $\\delta >0$ there exist a sequence of times $\\lbrace t_j\\rbrace $ , where $t_j\\rightarrow \\infty $ as $j\\rightarrow \\infty $ , and a positive constant $\\epsilon _1$ such that ${\\theta }_j={S}^{t_j}_\\mathit {\\Omega }({\\theta }_0)$ satisfies $|{\\theta }_j-{\\theta }_0|_{\\mathbb {T}^N}<\\delta $ for every $j>J$ ($J$ is an integer), while $d(A_{\\theta _k},A_{\\theta _0})>\\epsilon _1$ holds for some $k>J$ .","Here, from the continuity of ${S}^t_M$ in $t$ , for each ${m}\\in A_{\\theta _0}$ and $\\epsilon >0$ , there exists a positive constant $\\delta $ such that $|{\\theta }_k-{\\theta }_0|_{\\mathbb {T}^N}<\\delta $ implies $|{m}_k-{m}|_{M}<\\epsilon $ , where ${m}_k:={S}^{t_k}_{M}({m},{\\theta }_0)\\in A_{\\theta _k}$ .","This gives us a contradiction of $d(A_{\\theta _k},A_{\\theta _0})>\\epsilon _1$ by taking $\\epsilon =\\epsilon _1$ and from the definition of Hausdorff distance.","This lemma suggests that the invariant set $I$ is topologically connected with respect to ${\\theta }$ .","One trivial example is provided by taking as $I$ the closure of a single trajectory starting at a point $({m},{\\theta }_0)$ , for which $A_{\\theta _0}$ consists of a single point.","In this case, $I$ is topologically regarded as a product set of a single point in $M$ and the torus $\\mathbb {T}^N$ .","This is rigorously stated in the next theorem.","Theorem 11 Suppose that $A_{\\theta _0}$ is bounded in $M$ .","Then, $A_{\\theta _0}\\times \\mathbb {T}^N$ is homeomorphic to $I$ .","Now, let us construct the mapping $h_{\\theta _0}: A_{\\theta _0}\\times \\mathcal {O}_{\\mathbb {T}^N}({\\theta }_0)\\rightarrow I$ in the following manner.","For each $({m},{\\theta })\\in A_{\\theta _0}\\times \\mathcal {O}_{\\mathbb {T}^N}({\\theta }_0)$ , by choosing time $\\tau _\\theta $ that satisfies ${\\theta }={S}^{\\tau _\\theta }_{\\it \\Omega }({\\theta }_0)$ , we define $h_{\\theta _0}({m},{\\theta })$ as $({S}^{\\tau _\\theta }_M({m},{\\theta }_0),{\\theta })$ .","We here prove the continuity of $h_{\\theta _0}$ .","For each $\\epsilon >0$ and ${\\theta }\\in \\mathcal {O}_{\\mathbb {T}^N}({\\theta }_0)$ , there exists a sequence of $\\lbrace t_j\\rbrace $ (where $t_j\\rightarrow \\infty $ as $j\\rightarrow \\infty $ ) such that ${\\theta }_j={S}^{t_j}_\\mathit {\\Omega }({\\theta }_0)$ satisfies $\\left|{\\theta }_j-{\\theta }\\right|_{\\mathbb {T}^N}<\\epsilon /2$ for every $j>J$ ($J$ is an integer).","Furthermore, because of the continuity of ${S}^t_{M}$ in $t$ , for each ${m}_j\\in A_{\\theta _0}$ there exists a positive constant $\\delta _j$ such that $\\left|{m}-{m}_j\\right|_M<\\delta _j$ implies $\\left|{S}_M^{\\tau _\\theta }({m},{\\theta }_0)-{S}_M^{\\tau _{\\theta _j}}({m}_j,{\\theta }_0)\\right|_M<\\epsilon /2$ .","Here, because for every $j>J$ $\\left|({m},{\\theta })-({m}_j,{\\theta }_j)\\right|_{M\\times \\mathbb {T}^N}\\le \\left|{m}-{m}_j\\right|_{M}+\\left|{\\theta }-{\\theta }_j\\right|_{\\mathbb {T}^N}< \\delta _j+\\frac{\\epsilon }{2},$ we set $\\delta :=\\delta _j+\\epsilon /2$ .","Thus, if $\\left|({m},{\\theta })-({m}_j,{\\theta }_j)\\right|_{M\\times \\mathbb {T}^N}<\\delta $ , then $\\left|h_{\\theta _0}({m},{\\theta })-h_{\\theta _0}({m}_j,{\\theta }_j)\\right|_{M\\times \\mathbb {T}^N}&\\le \\left|{S}_M^{\\tau _\\theta }({m},{\\theta }_0)-{S}_M^{\\tau _{\\theta _j}}({m}_j,{\\theta }_0)\\right|_M+\\left|{\\theta }-{\\theta }_j\\right|_{\\mathbb {T}^N}\\nonumber \\\\& <\\frac{\\epsilon }{2}+\\frac{\\epsilon }{2}=\\epsilon .\\nonumber $ This shows that $h_{\\theta _0}$ is continuous.","We next prove that $h_{\\theta _0}$ is uniformly continuous.","Assume not.","Then, for each $\\delta >0$ there exists a positive constant $\\epsilon _1$ such that $\\left|({m}^{\\prime }_j,{\\theta }^{\\prime }_j)-({m}_j,{\\theta }_j)\\right|_{M\\times \\mathbb {T}^N}<\\delta $ implies $\\left|h_{\\theta _0}({m}^{\\prime }_j,{\\theta }^{\\prime }_j)-h_{\\theta _0}({m}_j,{\\theta }_j)\\right|_{M\\times \\mathbb {T}^N}\\ge \\epsilon _1$ for some $({m}_j,{\\theta }_j) $ and $({m}^{\\prime }_j,{\\theta }^{\\prime }_j)\\in A_{\\theta _0}\\times \\mathcal {O}_{\\mathbb {T}^N}({\\theta }_0)$ .","We here set $\\delta =1/j$ .","Since it is supposed that $A_{\\theta _0}$ is bounded and closed (and $\\mathcal {O}_{\\mathbb {T}^N}({\\theta }_0)\\subset \\mathbb {T}^N$ ), the two sequences of points $\\lbrace ({m}_j,{\\theta }_j)\\rbrace $ and $\\lbrace ({m}^{\\prime }_j,{\\theta }^{\\prime }_j)\\rbrace $ have convergent subsequences, denoted as $\\lbrace ({m}_{j_k},{\\theta }_{j_k})\\rbrace $ and $\\lbrace ({m}^{\\prime }_{j_k},{\\theta }^{\\prime }_{j_k})\\rbrace $ .","Because of $\\left|({m}^{\\prime }_j,{\\theta }^{\\prime }_j)-({m}_j,{\\theta }_j)\\right|_{M\\times \\mathbb {T}^N}<1/j$ , both the subsequences have the same convergent point, denoted as $({m}^\\ast ,{\\theta }^\\ast )$ .","That is, for each $\\delta _1>0$ , there exists a positive integer $J_k$ such that both the inequalities $\\left|({m}^{\\prime }_{j_k},{\\theta }^{\\prime }_{j_k})-({m}^\\ast ,{\\theta }^\\ast )\\right|_{M\\times \\mathbb {T}^N}<\\delta _1$ and $\\left|({m}_{j_k},{\\theta }_{j_k})-({m}^\\ast ,{\\theta }^\\ast )\\right|_{M\\times \\mathbb {T}^N}<\\delta _1$ hold for every $j_k\\ge J_k$ .","Here, since $h_{\\theta _0}$ is proven to be continuous at $({m}^\\ast ,{\\theta }^\\ast )$ , for each $\\epsilon >0$ , there exists a positive constant $\\delta _2$ such that $\\left|({m},{\\theta })-({m}^\\ast ,{\\theta }^\\ast )\\right|_{M\\times \\mathbb {T}^N}<\\delta _2$ implies $\\left|h_{\\theta _0}({m},{\\theta })-h_{\\theta _0}({m}^\\ast ,{\\theta }^\\ast )\\right|_{M\\times \\mathbb {T}^N}<\\epsilon /2$ .","By taking $\\epsilon =\\epsilon _1$ and $\\delta _1=\\delta _2$ thus choosing $J_k$ appropriately, we see that both the inequalities $\\left|h_{\\theta _0}({m}^{\\prime }_{j_k},{\\theta }^{\\prime }_{j_k})-h_{\\theta _0}({m}^\\ast ,{\\theta }^\\ast )\\right|_{M\\times \\mathbb {T}^M}<\\epsilon _1/2$ and $\\left|h_{\\theta _0}({m}_{j_k},{\\theta }_{j_k})-h_{\\theta _0}({m}^\\ast ,{\\theta }^\\ast )\\right|_{M\\times \\mathbb {T}^N}<\\epsilon _1/2$ hold for every $j_k\\ge J_k$ , and $\\begin{aligned}\\left|h_{\\theta _0}({m}^{\\prime }_{j_k},{\\theta }^{\\prime }_{j_k})-h_{\\theta _0}({m}_{j_k},{\\theta }_{j_k})\\right|_{M\\times \\mathbb {T}^M}\\le &\\left|h_{\\theta _0}({m}^{\\prime }_{j_k},{\\theta }^{\\prime }_{j_k})-h_{\\theta _0}({m}^\\ast ,{\\theta }^\\ast )\\right|_{M\\times \\mathbb {T}^N} \\\\& + \\left|h_{\\theta _0}({m}^\\ast ,{\\theta }^\\ast )-h_{\\theta _0}({m}_{j_k},{\\theta }_{j_k})\\right|_{M\\times \\mathbb {T}^N} \\\\< & \\frac{\\epsilon _1}{2}+\\frac{\\epsilon _1}{2} = \\epsilon _1,\\end{aligned}$ implying the contradiction for $\\left|h_{\\theta _0}({m}^{\\prime }_{j_k},{\\theta }^{\\prime }_{j_k})-h_{\\theta _0}({m}_{j_k},{\\theta }_{j_k})\\right|_{M\\times \\mathbb {T}^N}\\ge \\epsilon _1$ .","Thus, $h_{\\theta _0}$ is uniformly continuous.","By virtue of Theorem 3.45 in page 136 of [29], the uniformly-continuous $h_{\\theta _0}: A_{\\theta _0}\\times \\mathcal {O}_{\\mathbb {T}^N}({\\theta }_0)\\rightarrow I$ is extended to the mapping $\\hat{h}_{\\theta _0}: A_{\\theta _0}\\times \\mathbb {T}^N\\rightarrow I$ that is also (uniformly) continuous.","We now prove that $\\hat{h}_{\\theta _0}$ is a bijection.","For each ${\\theta }\\in \\mathcal {O}_{\\mathbb {T}^N}({\\theta }_0)$ , the statement is obvious from the construction of $h_{\\theta _0}$ (see its dependence on ${\\theta }$ ).","It is thus enough to check the case ${\\theta }\\in \\mathbb {T}^N\\setminus \\mathcal {O}_{\\mathbb {T}^N}({\\theta }_0)$ .","For each ${\\theta }\\in \\mathbb {T}^N\\setminus \\mathcal {O}_{\\mathbb {T}^N}({\\theta }_0)$ , there exists a sequence of times $\\lbrace t_j\\rbrace $ (where $t_j\\rightarrow \\infty $ as $j\\rightarrow \\infty $ ) such that ${\\theta }_j={S}^{t_j}_\\mathit {\\Omega }({\\theta }_0)$ converges to ${\\theta }$ .","Because $\\hat{h}_{\\theta _0}$ is continuous, the sequence $\\lbrace \\hat{h}_{\\theta _0}({m},{\\theta }_j)=h_{\\theta _0}({m},{\\theta }_j)\\rbrace $ converges to $\\hat{h}_{\\theta _0}({m},{\\theta })$ , which is represented as $\\displaystyle \\left(\\lim _{t_j\\rightarrow \\infty }{S}_M^{t_j}({m},{\\theta }_0),{\\theta }\\right)$ .","Namely, the limit exists for every ${m}\\in A_{\\theta _0}$ .","Assume that $\\hat{h}_{\\theta _0}$ is not an injection.","Then, there exist ${m},{m}^{\\prime }\\in A_{\\theta _0}$ satisfying ${m}\\ne {m}^{\\prime }$ such that for each $\\epsilon >0$ , $\\left|{S}_M^{t_j}({m},{\\theta }_0)-{S}_M^{t_j}({m}^{\\prime },{\\theta }_0)\\right|_M<\\epsilon $ holds for every $j>J$ ($J$ is an integer and depends on $\\epsilon $ ).","Here, the fundamental property of uniqueness of trajectories in (REF ) (or ${S}^t$ diffeomorphsim) implies that for each $t_j$ , $\\left|{S}_M^{t_j}({m},{\\theta }_0)-{S}_M^{t_j}({m}^{\\prime },{\\theta }_0)\\right|_M\\ge \\epsilon _j$ holds ($\\epsilon _j$ is a positive constant and depends on $t_j$ ), showing the contradiction by taking $\\epsilon =\\epsilon _j$ .","Hence, $\\hat{h}_{\\theta _0}$ is an injection.","Regarding $\\hat{h}_{\\theta _0}$ surjective, it is obvious that for each ${\\theta }\\in \\mathcal {O}_{\\mathbb {T}^N}({\\theta }_0)$ , $\\hat{h}_{\\theta _0}(A_{\\theta _0},{\\theta })=h_{\\theta _0}(A_{\\theta _0},{\\theta })=\\left({S}_M^{\\tau _\\theta }(A_{\\theta _0},{\\theta }_0),{\\theta }\\right)=(A_\\theta ,{\\theta })$ .","For each ${\\theta }\\in \\mathbb {T}^N\\setminus \\mathcal {O}_{\\mathbb {T}^N}({\\theta }_0)$ we see $\\displaystyle \\hat{h}_{\\theta _0}(A_{\\theta _0},{\\theta })=\\left(\\lim _{t_j\\rightarrow \\infty }{S}_M^{t_j}(A_{\\theta _0},{\\theta }_0),{\\theta }\\right)$ , and the limit converges to $A_\\theta $ from Lemma REF .","Hence, from (REF ), $\\hat{h}_{\\theta _0}$ is a surjection.","Finally, the inverse of $\\hat{h}_{\\theta _0}$ exists according to the above construction and is continuous.","Therefore, $\\hat{h}_{\\theta _0}: A_{\\theta _0}\\times \\mathbb {T}^N\\rightarrow I$ is a homeomorphism, namely, $A_{\\theta _0}\\times \\mathbb {T}^N$ is homeomorphic to $I$ .","The following two corollaries provide a way of characterization of boundedness of $I$ by means of one sample of it.","Corollary 12 Suppose that $A_{\\theta _0}$ is bounded in $M$ .","Then, $I$ is bounded in $X$ .","Corollary 13 Suppose that a closed subset $B_{\\theta _0}$ of $A_{\\theta _0}$ is bounded in $M$ .","Then, $\\hat{h}_{\\theta _0}(B_{\\theta _0}\\times \\mathbb {T}^N)$ is a subset of $I$ and bounded in $X$ .","Corollaries REF and REF imply that by computing a slice of the invariant set in $M$ at one sample onset, it is possible to determine whether the invariant set or its subset is bounded in the augmented state space $X$ .","For a bounded invariant set $I$ in $X$ , any cross-section $I_{\\theta _0}=A_{\\theta _0}\\times \\lbrace {\\theta }_0\\rbrace $ (${\\theta }_0\\in \\mathbb {T}^N$ ) is bounded in terms of $M$ , in other words, the boundedness property of $A_{\\theta _0}$ does not depend on the choice of ${\\theta }_0$ .","In this way, we call the bounded invariant set $I$ in $X$ uniformly bounded invariant set."],["Example Studies","In this section, we provide three examples of the application of the above theoretical results.","The last example is related to practical problems on stability of power grids."],["Forced Linear Harmonic Oscillator","First, we will consider the following one-degree-of-freedom linear harmonic oscillator with quasi-periodic forcing of (multiple) $N$ frequencies: $\\frac{dm_1}{dt}=m_2, \\qquad \\frac{dm_2}{dt}=-m_1+\\sum ^{N}_{i=1}F_i\\sin (\\mathit {\\Omega }_it+\\theta _{i0}),$ or $\\frac{dm_1}{dt}=m_2, \\qquad \\frac{dm_2}{dt}=-m_1+\\sum ^{N}_{i=1}F_i\\sin \\theta _i, \\qquad \\frac{d\\theta _i}{dt}=\\mathit {\\Omega }_i \\qquad i=1,2,\\ldots ,N$ where $F_i$ ($>0$ ) ($i=1,2,\\ldots ,N$ ) are the amplitudes of periodic forces, $\\mathit {\\Omega }_i$ ($>0$ ) the angular frequencies of the forces, and $\\theta _{i0}\\in \\mathbb {T}$ the initial phases.","We assume there is no resonance: the $N$ angular frequencies $\\mathit {\\Omega }_i$ are rationally independent; and $\\mathit {\\Omega }_i\\ne 1$ ($i=1,2,\\ldots ,N$ ).","The solution of (REF ) with initial condition $(m_{10},m_{20},\\theta _{10},\\theta _{20},\\ldots ,\\theta _{N0})$ is analytically represented as follows: $\\left.\\begin{aligned}m_1(t) &=C_1\\cos t+C_2\\sin t+\\sum ^{N}_{i=1}\\frac{F_i}{1-\\mathit {\\Omega }^2_i}\\cos (\\mathit {\\Omega }_it+\\theta _{i0}), \\\\m_2(t) &=\\displaystyle \\frac{{d}}{{d} t}m_1(t), \\\\{\\vspace{14.22636pt}}\\theta _i(t) &= \\mathit {\\Omega }_it+\\theta _{i0} \\qquad i=1,2,\\ldots ,N,\\end{aligned}\\right\\rbrace $ with $C_1:=m_{10}-\\sum ^{N}_{i=1}\\frac{F_i}{1-\\mathit {\\Omega }^2_i}\\cos \\theta _{i0}, \\qquad C_2:=m_{20}+\\sum ^{N}_{i=1}\\frac{F_i\\mathit {\\Omega }_i}{1-\\mathit {\\Omega }^2_i}\\sin \\theta _{i0}.$ Now, we estimate the time-averages of an observable under the dynamics described by the linear system (REF ).","We define the following quadratic observable $f: \\mathbb {R}^2\\times \\mathbb {T}^N\\rightarrow \\mathbb {R}$ , related to the potential energy $m^2_1/2$ : $f(m_1,m_2,\\theta _1,\\theta _2,\\ldots ,\\theta _N)=m^2_1.$ Then, we calculate its time-average under the solution as $\\lim _{T\\rightarrow \\infty }\\frac{1}{T}\\int ^T_0(U^tf)(m_{10},m_{20},\\theta _{10},\\theta _{20},\\ldots ,\\theta _{N0}){d} t=\\frac{1}{2}\\left\\lbrace C^2_1+C^2_2+\\sum ^{N}_{i=1}\\frac{F^2_i}{(1-\\mathit {\\Omega }^2_i)^2}\\right\\rbrace .$ Here, as shown in (REF ), the constants $C_1$ and $C_2$ are determined by the initial condition $(m_{10},m_{20},\\theta _{10},\\theta _{20},\\ldots ,\\theta _{N0})$ .","The level set, parameterized by a real-valued constant $c\\in \\mathbb {R}$ , of the time-average in the two-dimensional initial plane $(m_{10},m_{20})$ at fixed $(\\theta _{10},\\theta _{20},\\ldots ,\\theta _{N0})$ becomes a circle with center of $\\left(\\sum ^{N}_{i=1}\\frac{F_i}{1-\\mathit {\\Omega }^2_i}\\cos \\theta _{i0},-\\sum ^{N}_{i=1}\\frac{F_i\\mathit {\\Omega }_i}{1-\\mathit {\\Omega }^2_i}\\sin \\theta _{i0}\\right),$ if the following inequality holds: $r^2:=2c-\\sum ^{N}_{i=1}\\frac{F^2_i}{(1-\\mathit {\\Omega }^2_i)^2}>0$ Thus, the constant $r$ corresponds to the radius of the circle of the level set for $c$ .","The center of the level set seems to stay close to the origin and is shifted with the choice of initial phases $(\\theta _{10},\\theta _{20},\\ldots ,\\theta _{N0})$ .","Numerical results of the level sets for $N=2$ , $\\mathit {\\Omega }_1=\\pi /3$ , $\\mathit {\\Omega }_2=11/10$ , and $F_1=F_2=0.2$ are presented in Figure REF .","Figure: Level sets of the quasiperiodically-forced one-degree-of-freedom harmonic oscillator () for different (θ 10 ,θ 20 )(\\theta _{10},\\theta _{20}):(a) (0,0)(0,0), (b) (π/2,π/2)(\\pi /2,\\pi /2), (c) (π,π)(\\pi ,\\pi ), and (d) (3π/2,3π/2)(3\\pi /2,3\\pi /2).The horizontal (or vertical) axis for each figure is x 1 ∈[-10,10]x_1\\in [-10,10] (or x 2 ∈[-10,10]x_2\\in [-10,10])."],["One-Dimensional Dissipative System","Next, we consider the following linear dissipative system with quasi-periodic forcing: $\\frac{{d} m}{{d} t}=-\\lambda m+\\sum ^{N}_{i=1}F_i\\sin (\\mathit {\\Omega }_it+\\theta _{i0}).$ or $\\frac{dm}{dt}=-\\lambda m+\\sum ^{N}_{i=1}F_i\\sin \\theta _i, \\qquad \\frac{d\\theta _i}{dt}=\\mathit {\\Omega }_i \\qquad i=1,2,\\ldots ,N$ where $\\lambda >0$ .","In the same manner as above, we suppose the $N$ angular frequencies $\\mathit {\\Omega }_i$ are rationally independent.","The solution of (REF ) with initial condition $(m_{0},\\theta _{10},\\theta _{20},\\ldots ,\\theta _{N0})$ is analytically represented as follows: $\\left.\\begin{array}{ccl}m(t) &=& \\displaystyle C^{-\\lambda t}+\\sum ^{N}_{i=1}\\frac{F_i}{\\lambda ^2+\\mathit {\\Omega }^2_i}\\lbrace \\lambda \\sin (\\mathit {\\Omega }_it+\\theta _{i0})-\\mathit {\\Omega }_i\\cos (\\mathit {\\Omega }_it+\\theta _{i0})\\rbrace ,\\\\{\\vspace{11.38109pt}}\\theta _i(t) &=& \\mathit {\\Omega }_it+\\theta _{i0}~~~(i=1,2,\\ldots ,N),\\end{array}\\right\\rbrace $ with $C:=m_0-\\sum ^{N}_{i=1}\\frac{F_i}{\\lambda ^2+\\mathit {\\Omega }^2_i}\\lbrace \\lambda \\sin \\theta _{i0}-\\mathit {\\Omega }_i\\cos \\theta _{i0}\\rbrace .$ Here, we estimate the time-averages of an observable under the dynamics described by the linear system (REF ).","We consider the quadratic observable as $f(m,\\theta _1,\\theta _2,\\ldots ,\\theta _N)=m^2.$ Then, its time-average under the solution is directly calculated as follows: $\\lim _{T\\rightarrow \\infty }\\frac{1}{T}\\int ^T_0(U^tf)(m_0,\\theta _{10},\\theta _{20},\\ldots ,\\theta _{N0}){d} t=\\frac{1}{2}\\sum ^{N}_{i=1}\\frac{F^2_i}{\\lambda ^2+\\mathit {\\Omega }^2_i}.$ Although this is trivial, since the transient term in the solution is filtered out, the value of the time-average is constant in the augmented state space $\\mathbb {R}\\times \\mathbb {T}^N$ , in other words, does not depend on the initial condition $(m_0,\\theta _{10},\\theta _{20},\\ldots ,\\theta _{N0})$ .","This implies that for any choice of sample at $(\\theta _{10},\\theta _{20},\\ldots ,\\theta _{N0})^\\top \\in \\mathbb {T}^N$ , the partition of $M=\\mathbb {R}$ based on the level set of the time-averages is just one and corresponds to $\\mathbb {R}$ itself.","Also, the above observation on the partition holds when we pick up a polynomial observable like $f(m,\\theta _1,\\theta _2,\\ldots ,\\theta _N)=\\sum ^{n}_{j=0}a_jm^j,$ with coefficients $a_j\\in \\mathbb {R}$ and finite integer $n$ .","For every continuous function defined on a closed interval in $\\mathbb {R}$ , from the Weierstrass Approximation Theorem, it can be uniformly approximated on that interval by polynomials to any degree of accuracy.","This implies that the above observation of the partition holds for every continuous function, which is an example of Theorem REF .","Thus, the augmented state space $\\mathbb {R}\\times \\mathbb {T}^N$ is the basin of attraction for the system (REF ), that is to say, the torus $\\mathbb {T}^N$ is the unique attractor for the system."],["Two-Dimensional Nonlinear Model of a Power Grid","To show the effectiveness of the theory beyond the basic examples, we apply the developed theory for analyzing the CSI phenomenon in a loop power grid.","CSI is a undesirable and emergent phenomenon of synchronous machines in a power grid, in which most of the machines coherently lose synchronism with the rest of the grid after being subjected to a finite disturbance [19].","In the case of small dissipation, this phenomenon generally does not happen upon an infinitesimally small perturbation around a steady operating state (stable equilibrium).If we have no dissipation, nonlinear stability properties are not understood in the high-dimensional cases.","However, it encompasses the situation when the grid's operating state escapes a predefined, positive-measure set around the equilibrium.","In this way, the notion of instability that we address here is non-local.","In [19], we derived a reduced-order dynamical system that described averaged dynamics of machines in a simple loop power grid and explained the non-local instability.","The reduced-order system has an quasiperiodic forcing (so it is non-autonomous), and its solutions define a measure-preserving flow.","The goal of this section is to characterize the non-local instability by analyzing invariant sets of the quasiperiodically forced system, which is introduced in (REF ) below."],["Mathematical Model","Consider short-term (zero to ten seconds) swing dynamics of a rudimentary power grid with the loop topology shown in Figure REF , where the small gray circles denote synchronous generators.","The loop part of the grid consists of $N{G}$ small, identical generators, encompassed by the dotted box, which operate in the grid and are connected to the infinite busAn ideal voltage source of constant voltage and constant frequency.","The loss-less transmission lines joining the infinite bus and a generator are much longer than those joining two generators in the loop part.","Thus, the magnitude of interaction between the infinite bus and a generator is much smaller than that between two neighboring generators on the loop part.","We call the model of Figure REF the loop power grid in the following.","Here, we assume that the lengths of transmission lines between two neighboring generators are identical.","In [19], we showed that the CSI phenomenon in the loop power grid was accurately captured by the following dynamical system defined on the cylindrical state space $\\mathbb {T}^1\\times \\mathbb {R}$ : $\\frac{d\\delta }{dt}=\\omega , \\qquad \\frac{d\\omega }{dt}=p{m}-\\frac{b}{N{G}}\\sum _{i=1}^{N{G}}\\sin \\left(\\sum _{j\\in \\mathcal {J}}e_{ij}c_j\\cos \\mathit {\\Omega }_jt+\\delta \\right)$ where $e_{ij}=\\sqrt{\\frac{2}{N{G}}}\\cos \\left(\\frac{2\\pi ij}{N{G}}+\\frac{\\pi }{4}\\right),\\qquad \\mathit {\\Omega }_j=2\\sqrt{|b{int}|}\\left|\\sin \\frac{\\pi j}{N{G}}\\right|.$ The system (REF ) represents spatially-averaged dynamics of the $N{G}$ generators in the loop power grid.","The variable $\\delta \\in \\mathbb {T}^1$ is the average of angular positions of rotors (with respect to the infinite bus) of the $N{G}$ generators, and $\\omega \\in \\mathbb {R}$ is the average of deviations of rotor speeds in the $N{G}$ generators relative to the system angular frequency.","The parameter $p{m}$ stands for the mechanical input power to a generator, $b$ for the maximum transmission power between the infinite bus and a generator, and $b{int}$ for the maximum transmission power between two neighboring generators in the loop power grid.","The constants $e_{ij}$ are the eigenfunctions of linear modal oscillations between coupled generators in the loop part, $\\mathit {\\Omega }_j$ their eigen-(angular) frequency, and $c_j$ the strengths of modal oscillations.","The finite index set $\\mathcal {J}$ determines which modes are excited in the loop part.","The system (REF ) is derived under the observation that the linear modal oscillations in the loop part act as perturbations on the spatially-averaged dynamics of the $N{G}$ generators: see [19] and references therein.","Note that (REF ) is the Hamiltonian system: $\\frac{d\\delta }{dt}=\\frac{}{\\omega }{H}(\\delta ,\\omega ,t), \\qquad \\frac{d\\omega }{dt}=-\\frac{}{\\delta }{H}(\\delta ,\\omega ,t),$ with the time-dependent Hamiltonian function ${H}(\\delta ,\\omega ,t)$ , given by ${H}(\\delta ,\\omega ,t):=\\frac{1}{2}\\omega ^2-p{m}\\delta -\\frac{b}{N{G}}\\sum ^{N{G}}_{i=1}\\cos \\left(\\sum _{j\\in \\mathcal {J}}e_{ij}c_j\\cos \\mathit {\\Omega }_jt+\\delta \\right).$ Because the flow defined here is divergence-free, i.e.","$({H}/\\delta )({d}\\delta /{d} t)+({H}/\\omega )({d}\\omega /{d} t)=0$ , the system (REF ) preserves the Liouville measure ${d}\\delta {d}\\omega $ .","Note that it does not conserve the Hamiltonian function ${H}$ , because of ${d}{H}/{d} t\\ne 0$ if $c_j\\ne 0$ .","In this way, the augmented system for (REF ), namely $\\frac{d\\delta }{dt}=\\omega , \\quad \\frac{d\\omega }{dt}=p{m}-\\frac{b}{N{G}}\\sum _{i=1}^{N{G}}\\sin \\left(\\sum _{j\\in \\mathcal {J}}e_{ij}c_j\\cos \\theta _j+\\delta \\right), \\quad \\frac{d\\theta _j}{dt}=\\mathit {\\Omega }_j \\qquad j\\in \\mathcal {J},$ defines a measure-preserving flow on $M\\times \\mathbb {T}^{|{\\cal J}|}$ with $M=\\mathbb {T}^1\\times \\mathbb {R}$ , where $|{\\cal J}|$ stands for the cardinality of ${\\cal J}$ ."],["Results and Implications","It was shown in [19] that the unbounded motion in the quasiperiodically forced system (REF ) corresponds to the CSI phenomenon.","Because of $\\delta \\in \\mathbb {T}^1$ , the unbounded motion involves the unbounded trajectory in the $\\omega $ -direction, that is, the average of deviation of rotor speeds.","We use Corollaries REF and REF to analyze invariant sets of the flow (defined by the system (REF )), in which all the generators show bounded deviation of rotor speeds in time.","The analysis is crucial to understanding the so-called stability region of the loop power grid, i.e.","how the grid's behavior depends on initial conditions representing failures in the grid.","Numerical simulations are performed for analysis of invariant sets.","To do so, we need to fix (i) a function $f$ , (ii) the subset of state space on which we identify invariant sets, and (iii) the exit time $T{ex}$ to obtain an approximation of each time-average $f^\\ast $ .","We use the function $f(\\delta )=\\sin 2\\delta $ and the grid of $401\\times 401$ of initial conditions $(\\delta ,\\omega )$ on $[1,2]\\times [-0.15,0.15]$ .","The averaging operation of a single function can be used for the identification of invariant sets.","Numerical integration of the system (REF ) is performed with the 4th-order symplectic integrator [30] with time step $h$ : see Appendix for details.","The parameter settings are the following: $p{m}=0.95, \\quad b=1, \\quad N{G}=20, \\quad b{int}=100, \\quad h=\\frac{2\\pi }{\\mathit {\\Omega }_1}\\frac{1}{N},\\quad T{ex}=\\frac{2\\pi }{\\mathit {\\Omega }_1}\\times 2000,$ where $N=8$ or 16 depending on the setting of $\\mathcal {J}$ .","The values of $p{m}$ , $b$ , $N{G}$ , and $ b{int}$ are the same as in [19].","Figure: Analysis of invariant sets of the measure-preserving flow defined by ()–I: Initial phase θ 0 =0{\\theta }_0={0} and multiple setting of 𝒥\\mathcal {J} i.e.","(a) 𝒥={1}\\mathcal {J}=\\lbrace 1\\rbrace , (b) 𝒥={1,2}\\mathcal {J}=\\lbrace 1,2\\rbrace , (c) 𝒥={1,2,3}\\mathcal {J}=\\lbrace 1,2,3\\rbrace , (d) 𝒥={1,2,3,4}\\mathcal {J}=\\lbrace 1,2,3,4\\rbrace , (e) 𝒥={1,2,3,4,5}\\mathcal {J}=\\lbrace 1,2,3,4,5\\rbrace , and (f) 𝒥={1,2,3,4,5,6}\\mathcal {J}=\\lbrace 1,2,3,4,5,6\\rbrace .The horizontal axis for each figure is δ∈[1,2]\\delta \\in [1,2], and the vertical axis is ω∈[-0.15,0.15]\\omega \\in [-0.15,0.15].For the outer dark-green regions, every trajectory starting from them is unbounded in time.Figure: Analysis of invariant sets of the measure-preserving flow defined by ()–II: 𝒥={1,2}\\mathcal {J}=\\lbrace 1,2\\rbrace and multiple setting of initial phase θ 0 =(θ 10 ,θ 20 ) ⊤ =(2πkΩ 1 /Ω 2 ,0) ⊤ {\\theta }_0=(\\theta _{10},\\theta _{20})^\\top =(2\\pi k\\mathit {\\Omega }_1/\\mathit {\\Omega }_2,0)^\\top for (a) k=0k=0 (same as Figure (b)), (b) k=1k=1, (c) k=2k=2, (d) k=3k=3, (e) k=4k=4, and (f) k=5k=5.The horizontal axis for each figure is δ∈[0,π]\\delta \\in [0,\\pi ], and the vertical axis is ω∈[-0.15,0.15]\\omega \\in [-0.15,0.15].For the outer dark-green regions, every trajectory starting from them is unbounded in time.Figure REF shows numerical results on analysis of invariant sets of the flow defined by the quasiperiodically forced system (REF ).","In this figure we change the number of excitation modes, i.e.","$\\mathcal {J}$ ; (a) $\\mathcal {J}=\\lbrace 1\\rbrace $ , (b) $\\mathcal {J}=\\lbrace 1,2\\rbrace $ , (c) $\\mathcal {J}=\\lbrace 1,2,3\\rbrace $ , (d) $\\mathcal {J}=\\lbrace 1,2,3,4\\rbrace $ , (e) $\\mathcal {J}=\\lbrace 1,2,3,4,5\\rbrace $ , and (f) $\\mathcal {J}=\\lbrace 1,2,3,4,5,6\\rbrace $ .","The amplitude $c_j$ in the figure is common and satisfies $\\displaystyle \\sqrt{\\sum _{j\\in \\mathcal {J}}c_j^2}=1.5$ , implying that the root-means-square of the forcing term does not change for any setting of $\\mathcal {J}$ .","All the initial phases ${\\theta }_0$ are set to zero.","For the outer dark-green region in each figure, every trajectory starting from it is unbounded in time.","Except for the dark-green on the bottom, the color bar attached to each figure denotes the value of time-average $f^\\ast (\\delta )$ .","The level sets of $f^\\ast (\\delta )$ are colored by the same color.","That is, the set of the same color belongs to one invariant set.","By Corollary REF , the fact that the level set is bounded in these figures implies that the associated subset of invariant set is bounded in the augmented state space $M\\times \\mathbb {T}^{|\\mathcal {J}|}$ .","In the figures (a,b,c), we see that the color plot of the level sets forms concentric rings.","However, in the figures (d,e,f), we see that the color plot does does not necessarily exhibit concentric rings and does become scattered in the bands close to the outer dark-green regions.","This implies that the structure of invariant sets is complicated in the bands.","We anticipate this results from the so-called resonance phenomenon (see, e.g., [31], [32]) as an interaction between a family of bounded oscillations in the unforced system and quasiperiodic forcing.","Figure REF shows other numerical results on analysis of invariant sets.","In this figure, we consider the forcing term with two frequencies, $\\mathcal {J}=\\lbrace 1,2\\rbrace $ , and we change the initial phases ${\\theta }_0=(\\theta _{10},\\theta _{20})^\\top =(2\\pi k\\mathit {\\Omega }_1/\\mathit {\\Omega }_2,0)$ where $k=0,1,\\ldots ,5$ .","The color plots here are conducted in the same manner as in Figure REF , and for the outer dark-green regions every trajectory starting from them is unbounded in time.","By Corollary REF , the fact that the level set with same color is bounded in these figures implies that the associated subset of invariant set is bounded in the augmented state space.","Here, under the current setting of parameters, it is conjectured that outside the outer dark-green regions (i.e., outside the computational domain of the analysis), there exists no state from which trajectory is bounded in time.","This is true in the unperturbed case because there exists one homoclinic orbit separating the bounded and unbounded trajectories inside the computational domain.","Thus, it can be inferred that the level sets discussed above are bounded in $M=\\mathbb {T}^1\\times \\mathbb {R}$ , implying by Corollary REF that the whole of the corresponding invariant sets are uniformly bounded in the augmented state space.","The intersection of uniformly bounded invariant sets for all ${\\theta }$ corresponds to the stability region of the loop power grid, in which all the generators show bounded deviation of rotor speeds in time."],["Conclusions","In this paper, we studied the ergodic partition and invariant sets of the quasiperiodically forced dynamical system (REF ).","The main theoretical contributions of this paper are twofold.","One is to provide a theory of ergodic partition of state space for smooth flows.","The theory is a natural extension of that in [7] and is applicable to measure-preserving and dissipative flows arising in various physical and engineering systems.","Examples of them include dynamical systems induced by time-dependent Hamiltonians and incompressible fluid flows with time-dependent velocity profiles.","The other is to provide a new characterization of invariant sets in the the quasiperiodically forced system (REF ), in which we introduced a concept of uniformly bounded invariant sets.","The developed theory was applied to characterize the CSI phenomenon of a rudimentary power grid.","We have speculated that the phenomenon can be characterized, in particular, the stability region corresponds to the intersection of uniformly bounded sets for all initial phases; or a sufficient condition for the phenomenon is that the operating state of the grid is placed outside of the bounded sets at a particular initial phase or time like $t=0$ ."],["Acknowledgments","Y.S.","thanks Dr. Marko $\\rm Budi\\check{s}i\\acute{c}$ for his introduction to theory and computation of ergodic partition and fruitful discussions.","The authors also appreciate the reviewers for their valuable suggestion of the manuscript.","During part of the work on this paper, Y.S.","was at the Department of Mechanical Engineering, University of California, Santa Barbara, United States, and at the Department of Electrical Engineering, Kyoto University, Japan."],["Proof of Lemma ","A continuous function $f$ on $X$ is measurable and, from the assumption that $X$ is compact for the theoretical analysis in Section , $f$ is bounded on $X$ .","Here, we note that the time-average $f^\\ast $ of the measurable function $f$ is measurable as a limit of measurable functions $f_T$ on $X$ , defined as $f_T(x):=\\frac{1}{T}\\int ^T_0 f({S}^t(x)){d}t \\qquad T>0.$ Since we consider the sets $A_\\alpha $ on $\\mathit {\\Sigma }\\subset X$ , the fact that the family of $A_\\alpha $ is a partition of $\\mathit {\\Sigma }$ is obvious.","Next, the fact that the partition, denoted by $\\zeta _f$ , is measurable follows by taking $\\mathfrak {D}_f$ to be the collection of pre-images under $f^\\ast $ of open intervals with rational endpoints in $\\mathbb {R}$ .","Because $f^\\ast $ is measurable, each pre-image $(f^\\ast )^{-1}([a,b])$ , where $a$ and $b$ are rational numbers, is measurable.","Every set of this type is clearly separated into sets of the form $(f^\\ast )^{-1}(\\lbrace c\\rbrace )$ , $c\\in \\mathbb {R}$ .","This implies that every element of $\\mathfrak {D}_f$ is a union of elements of $\\zeta _f$ .","Furthermore, because the set of all rational numbers is dense in $\\mathbb {R}$ , for any pair $\\alpha ,\\beta \\in \\mathbb {R}$ satisfying $\\alpha <\\beta $ , there exist two rational numbers $\\underline{a},\\overline{a}$ such that $\\alpha <\\underline{a}<\\beta <\\overline{a}$ .","The pre-image $(f^\\ast )^{-1}([\\underline{a},\\overline{a}])$ is an element of $\\mathfrak {D}_f$ which we denote by $D$ .","Obviously, we see $A_\\alpha \\subset {D}^{\\rm c}$ and $A_\\beta \\subset {D}$ .","Thus, it follows that $\\mathfrak {D}_f$ is a basis for $\\zeta _f$ , and we conclude that $\\zeta _f$ is measurable."],["Proof of Theorem ","Let $A$ be an element of $\\zeta {e}$ .","For a.e.","point $x\\in A$ , the time-average $f^\\ast (x)$ exists for all $f\\in \\mathcal {C}(X)$ .","Thus, the following linear functional $L_{A}$ on $\\mathcal {C}(X)$ is well-defined: $L_{A}(f):=\\lim _{T\\rightarrow \\infty }\\frac{1}{T}\\int ^T_0 f({S}^t (x)){d} t \\qquad x\\in A.$ Then, because $L_{A}$ is a positive linear functional and $L_{A}(1)=1$ , by Riesz's Representation Theorem (I.8.4 in [23]) there exists a unique probability measure $\\mu _{A}$ on $X$ such that $\\int _{X} f{d}\\mu _A=L_{A}(f),$ for all $f\\in \\mathcal {C}(X)$ .","Note that $\\mu _{A}$ is invariant for ${S}^t$ .","To prove this, for all $t\\in \\mathbb {R}$ we have $\\int _{X} f\\circ {S}^t\\,{d}\\mu _{A}=L_{A}(f\\circ {S}^t)=L_{A}(f)=\\int _{X} f{d}\\mu _{A}.$ The second equality is a consequence of (REF ).","For the above operation, the continuity of ${S}^t$ is required.","Because $\\mathcal {C}(X)$ is dense in $\\mathcal {L}^1_{\\mu _A}(X)$ , $\\mu _{A}$ is invariant.","Now, we prove that $\\mu _{A}$ is a probability measure on $A$ .","There is a sequence of compact sets $A^{\\rm c}_n$ , subsets of $A^{\\rm c}$ , such that ${A}^{\\rm c}_1\\subset \\cdots \\subset {A}^{\\rm c}_{n}\\subset {A}^{\\rm c}_{n+1}\\subset \\cdots , \\qquad \\mu _{A}\\left(A^{\\rm c}\\setminus \\bigcup _{n\\ge 1}A^{\\rm c}_n\\right)=0.","\\qquad $ Here, we can show $\\mu _{A}(A^{\\rm c}_n)=0$ for every $A^{\\rm c}_n$ .","To do this, note that by Urysohn's Lemma, for every $A^{\\rm c}_n$ , there is a continuous, positive function $f_n$ on $X$ that is equal (i) to one on $A^{\\rm c}_n$ and (ii) to zero outside of $A^{\\rm c}_{n+1}$ .","Clearly, we see $f_n=0$ on $A$ .","Therefore, because of $\\int _{X}f_n{d}\\mu _{A}=\\int _{X\\setminus A^{\\rm c}_{n+1}}f_n{d}\\mu _{A}+\\int _{A^{\\rm c}_{n+1}\\setminus A^{\\rm c}_n}f_n{d}\\mu _{A}+\\int _{A^{\\rm c}_n}f_n{d}\\mu _{A}$ and the positiveness of $f_n$ , we have $0\\le \\mu _{A}(A^{\\rm c}_n)\\le \\int _{X} f_n{d}\\mu _{A}=L_{A}(f_n)=0.$ The measure of a union of the countable number of sets with measure zero is zero: $\\mu _{A}\\left(\\bigcup _{n\\ge 1}A^{\\rm c}_n\\right)=0.$ Therefore, by (REF ) and (REF ), we have $\\mu _{A}(A^{\\rm c})=0$ .","It follows from $\\mu _{A}(X)=1$ that $\\mu _{A}$ is a probability measure on $A$ .","Next, let us prove that $\\mu _{A}$ is an ergodic measure on $A$ .","First, observe that the set of all restrictions of functions in $\\mathcal {C}(X)$ to $A$ , denoted by $\\mathcal {C}(X)|_{A}$ , is dense in the set of all $\\mu _{A}$ -integrable functions on $A$ , denoted by $\\mathcal {L}^1_{\\mu _{A}}(A)$ .","To show this, note that $\\mathcal {C}(X)$ is dense in $\\mathcal {L}^1_{\\mu _{A}}(X)$ .","Let $f$ be an element of $\\mathcal {L}^1_{\\mu _{A}}(A)$ .","Consider the extension of $f$ to $X$ , $\\bar{f}$ , such that $\\bar{f}=f$ on $A$ and $\\bar{f}=0$ elsewhere.","Then, we have $\\bar{f}\\in \\mathcal {L}^1_{\\mu _{A}}(X)$ because the following integral exists: $\\int _{X}\\bar{f}{d}\\mu _{A}=\\int _{A}f{d}\\mu _{A}.$ Here, since $\\mathcal {C}(X)$ is dense in $\\mathcal {L}^1_{\\mu _{A}}(X)$ , there is a sequence of functions in $\\mathcal {C}(X)$ , $\\lbrace f_n\\rbrace $ , converging to $\\bar{f}$ .","Thus, the corresponding sequence of restrictions, $\\lbrace f_n|_{A}\\rbrace $ , converges to $f$ .","Therefore, we observe that $\\mathcal {C}(X)|_{A}$ is dense in $\\mathcal {L}^1_{\\mu _{A}}(A)$ .","Now, by the same argument as (REF ), for all $f\\in \\mathcal {C}(X)|_{A}$ we have $\\int _{A}f{d}\\mu _{A}&= L_A(f) \\nonumber \\\\&= f^\\ast (x) \\qquad x\\in A.$ Since (REF ) holds for the dense set $\\mathcal {C}(X)|_A$ in $\\mathcal {L}^1_{\\mu _{A}}(A)$ , ${S}^t|_{A}$ is ergodic: see Proposition 2.2 in Chapter II of [23] for discrete-time systems.","This proposition can be naturally extended to continuous-time systems.","Hence, we complete the proof that there indeed exists an ergodic measure $\\mu _{A}$ for any element $A$ of the partition $\\zeta {e}$ .","Finally, we consider (REF ) and that $A$ is invariant.","The equality (REF ) is obtained with the proof of Theorem 6.4 in Chapter II of [23].","The proof is obtained for discrete-time systems and is extended to continuous-time systems.","By construction, the fact that $A$ is invariant is obvious.","This completes the proof of Theorem REF ."],["Symplectic Integration of Time-Dependent Hamiltonian Systems","In Section REF , it is required to numerically simulate the Hamiltonian system (REF ) with the time-dependent Hamiltonian function ${H}(\\delta ,\\omega ,t)$ .","Symplectic integrator [30] is normally formulated in the case of time-independent Hamiltonian functions.","However, one can exploit the integrator in the case of time-dependent Hamiltonian functions by augmenting the original Hamiltonian system.","Consider the $N$ degree-of-freedom Hamiltonian system with the Hamiltonian function ${H}(q,p,t)$ : for $i=1,2,\\ldots ,N$ , $\\frac{dq_i}{dt}=\\frac{}{p_i}{H}(q,p,t), \\qquad \\frac{dp_i}{dt}=-\\frac{}{q_i}{H}(q,p,t)$ where $q=(q_1,q_2,\\ldots ,q_N)^\\top $ , $p=(p_1,p_2,\\ldots ,p_N)^\\top $ , and $t\\in \\mathbb {R}$ .","Now, by replacing the time variable $t$ with one new variable $q_0$ and defining the other new variable $dp_0/dt:=-H/t$ , we have the augmented Hamiltonian function $\\bar{H}(q_0,p_0,q,p)$ as follows: $\\bar{H}(q_0,p_0,q,p):=p_0+{H}(q,p,q_0).$ Thus, the augmented Hamiltonian system of the time-independent Hamiltonian function $\\bar{H}$ is derived as $\\frac{dq_i}{dt}=\\frac{}{p_i}\\bar{H}(q_0,p_0,q,p), \\qquad \\frac{dp_i}{dt}=-\\frac{}{q_i}\\bar{H}(q_0,p_0,q,p)$ where $i=0,1,\\ldots ,N$ .","The flow induced by trajectories of the augmented system (REF ) is divergence-free and conserves the value of the Hamiltonian function $\\bar{H}$ .","Thus, by using the integrator for the augmented system, numerical simulations of the original system (REF ) are indirectly performed.","Note that the accuracy of numerical integration of (REF ) is checked by estimating the value of $\\bar{H}$ .","This idea is applicable to the case of non-periodic time-dependent Hamiltonian functions."]],"string":"[\n [\n \"Invariant Sets in Quasiperiodically Forced Dynamical Systems\"\n ],\n [\n \"Abstract This paper addresses structures of state space in quasiperiodically forced dynamical systems.\",\n \"We develop a theory of ergodic partition of state space in a class of measure-preserving and dissipative flows, which is a natural extension of the existing theory for measure-preserving maps.\",\n \"The ergodic partition result is based on eigenspace at eigenvalue 0 of the associated Koopman operator, which is realized via time-averages of observables, and provides a constructive way to visualize a low-dimensional slice through a high-dimensional invariant set.\",\n \"We apply the result to the systems with a finite number of attractors and show that the time-average of a continuous observable is well-defined and reveals the invariant sets, namely, a finite number of basins of attraction.\",\n \"We provide a characterization of invariant sets in the quasiperiodically forced systems.\",\n \"A theoretical result on uniform boundedness of the invariant sets is presented.\",\n \"The series of theoretical results enables numerical analysis of invariant sets in the quasiperiodically forced systems based on the ergodic partition and time-averages.\",\n \"Using this, we analyze a nonlinear model of complex power grids that represents the short-term swing instability, named the coherent swing instability.\",\n \"We show that our theoretical results can be used to understand stability regions in such complex systems.\"\n ],\n [\n \"Introduction\",\n \"Methods of ergodic theory [1], [2] have been applied for solving a number of problems in science and technology such as control of mixing of fluid flows [3], analysis of traffic jams [4] and of quantum many-body models [5], performance evaluation of ecosystem models [6].\",\n \"A visualization method for invariant sets of discrete-time dynamical systems (or maps) based on ergodic theory was developed in [7].\",\n \"This method is based on the ergodic partition or ergodic decomposition [8] of state space in a given dynamical system, which is associated with the eigenspace at eigenvalue 1 of the Koopman operator [9].\",\n \"This method partitions a (compact and metric) state space $X$ using joint level sets of time-averages of a basis of functions defined on $X$ .\",\n \"This leads to an approximation of invariant sets on which the dynamics of the system are ergodic.\",\n \"In an ergodic invariant set, almost all points are accessible in the sense that the initial conditions in this set thoroughly sample the set.\",\n \"The method enables the visualization of low-dimensional (in practice usually two-dimensional) slices through high-dimensional invariant structures and offers a computationally tractable method for analyzing global structures of state space in discrete-time nonlinear models.\",\n \"The numerical methods associated with the ergodic partition theory have been developed: see [9] and references therein.\",\n \"The purpose of this paper is to extend the ergodic partition theory and exploit it to provide global analysis of state space in quasiperiodically forced dynamical systems.\",\n \"Quasiperiodicity is one of the three types of commonly observed dynamics in both deterministic models and experiments: see, e.g., [10], [11].\",\n \"Quasiperiodically forced systems are important objects in nonlinear dynamics and have been studied by many groups of researchers: phenomenology of chaotic attractors [12], analysis of dynamics described by the Schrödinger equations [13] and of quantum chaos [14], and dynamical systems analysis with applications to fluid mixing [15], [16].\",\n \"In the present paper, we study the global structure of state space in quasiperiodically forced systems from the viewpoint of ergodic partition.\",\n \"First, we develop a theory of ergodic partition of state space in measure-preserving and dissipative, continuous-time dynamical systems or flows.\",\n \"This is a natural extension of the existing theory for measure-preserving maps in [7].\",\n \"The ergodic partition is again based on eigenspace at eigenvalue 0 of the associated Koopman group.\",\n \"Related analyses of time-periodic flows via spectral properties of linear operators are reported in [17], [18].\",\n \"Since an arbitrary quasiperiodically forced system with a smooth invariant measure is transformed into an autonomous system determining a measure-preserving flow (see Section ), the ergodic partition theory is used for visualization of invariant sets for the quasiperiodically forced system.\",\n \"Next, we provide a characterization of invariant sets in quasiperiodically forced systems.\",\n \"In general, understanding invariant structures in the quasiperiodically forced systems is not easy because the associated portraits of state space change with time in an aperiodic manner.\",\n \"Here, we introduce a notion of uniform boundedness of invariant sets.\",\n \"A theoretical result on uniform boundedness of invariant sets is presented in this paper: see Theorem REF , and Corollaries REF and REF .\",\n \"This clarifies a dynamical feature of invariant sets in the quasiperiodically forced systems that is directly determined by the ergodic partition and its associated visualization technique.\",\n \"Also, we apply the theory developed above to analysis of a nonlinear model of complex power grids that represents the short-term swing instability, which we name the Coherent Swing Instability (CSI) [19].\",\n \"Preliminary results in this paper were published in the conference proceedings [20], [21].\",\n \"This paper contains detailed discussion of the theory with a new proof and a set of new numerical simulations.\",\n \"The rest of this paper is organized as follows.\",\n \"In Section we introduce set-up and mathematical preliminaries from dynamical systems theory.\",\n \"A theory of ergodic partition in measure-preserving flows is developed in Section .\",\n \"Based on the result, the basin of attraction in the dissipative case is explored in ().\",\n \"In Section we provide a new characterization of invariant sets in the quasiperiodically forced dynamical systems.\",\n \"The developed theory is applied in Section to two simple examples of the quasiperiodically forced system and to a nonlinear model of the CSI phenomenon of a power grid.\",\n \"Conclusions of this paper are presented in Section .\"\n ],\n [\n \"Quasiperiodically Forced Dynamical Systems\",\n \"Throughout this paper, we address the following quasiperiodically forced dynamical system that evolves on a finite-dimensional metric space $M$ : for ${m}\\\\in {M}$ and $t\\\\in \\\\mathbb {R}$ , $\\\\frac{{d}{m}}{{d} t}={g}({m},t).$ The function $g$ is assumed to be quasiperiodic on $t$ in the sense of Moser [22], that is, ${g}({m},t)={G}({m},\\\\mathit {\\\\Omega }_1t,\\\\mathit {\\\\Omega }_2t,\\\\ldots ,\\\\mathit {\\\\Omega }_Nt),$ where ${G}({m},\\\\theta _1,\\\\theta _2,\\\\ldots ,\\\\theta _N)$ is assumed to be a smooth vector-valued function and of period $2\\\\pi $ in $\\\\theta _1,\\\\theta _2,\\\\ldots ,\\\\theta _N$ .\",\n \"The real numbers $\\\\mathit {\\\\Omega }_1,\\\\mathit {\\\\Omega }_2,\\\\ldots ,\\\\mathit {\\\\Omega }_N$ are the $N$ basic angular frequencies and assumed to be rationally independent: there exists no $N$ -dimensional integer vector $(k_1,k_2,\\\\ldots ,k_N)^\\\\top $ ($\\\\top $ stands for the transpose operation of real-valued vectors) in which the entries do not all vanish, satisfying the resonance relation: $\\\\sum ^{N}_{i=1}k_i\\\\mathit {\\\\Omega }_i=0.$ The system (REF ) is non-autonomous and transformed into an autonomous system defined on the augmented state space $X:=M\\\\times \\\\mathbb {T}^N$ , in the same manner as in [16].\",\n \"By introducing the $N$ variables $\\\\theta _i=\\\\mathit {\\\\Omega }_it\\\\in \\\\mathbb {T}$ ($i=1,2,\\\\ldots ,N$ ), we have $\\\\frac{d{m}}{dt}={G}({m},\\\\theta _1,\\\\theta _2,\\\\ldots ,\\\\theta _N), \\\\qquad \\\\frac{d\\\\theta _i}{dt}=\\\\mathit {\\\\Omega }_i \\\\qquad i=1,2,\\\\ldots ,N.$ In this way, by taking trajectories of the augmented system (REF ) on the augmented state variable ${x}:=({m},\\\\theta _1,\\\\theta _2,\\\\ldots ,\\\\theta _N)\\\\in X$ , a flow, that is, one-parameter group of diffeomorphisms is defined for the quasiperiodically forced system (REF ).\"\n ],\n [\n \"Measure-Preserving Flows and Partition of State Space\",\n \"In Section , as one type of flows induced by (REF ), we will investigate a measure-preserving flow defined on a class of probability spaces, where ergodic theory has been developed [8], [23].\",\n \"The probability space considered here corresponds to the tuple $({X},\\\\mathfrak {B}_{X},\\\\mu )$ , where ${X}$ is a compact metric space, $\\\\mathfrak {B}_{X}$ is the Borel $\\\\sigma $ -algebra of ${X}$ , and $\\\\mu $ is a probability measure.\",\n \"A measure-preserving flow of the probability space is a one-parameter group of measure-preserving diffeomorphisms ${S}^t: {X}\\\\rightarrow {X}$ , $t\\\\in \\\\mathbb {R}$ such that ${S}^t$ is measure-preserving (for all $A\\\\in \\\\mathfrak {B}_{X}$ we have $\\\\mu ({S}^{-t}(A))=\\\\mu (A)$ ), ${S}^0$ the identity, and ${S}^{t_1+t_2}={S}^{t_1}\\\\circ {S}^{t_2}$ for $t_1,t_2\\\\in \\\\mathbb {R}$ .\",\n \"The important notion which we utilize throughout this paper is the partition of state space in dynamical systems.\",\n \"One motivation behind the following definitions is that we aim to locate such a partition that plays a role in the data-driven estimation of certain statistical properties of non-ergodic measure-preserving dynamical systems; we will show this later as the ergodic partition.\",\n \"The definitions of partition and measurable partition from [7] are the following.\",\n \"Definition 1 A partition of ${X}$ is a family $\\\\zeta $ of sets satisfying $A,B\\\\in \\\\zeta \\\\quad \\\\Rightarrow \\\\quad \\\\mu (A\\\\cap B)=0, \\\\quad \\\\mu \\\\left({X}\\\\setminus \\\\bigcup _{A\\\\in \\\\zeta }A\\\\right)=0.$ The element of $\\\\zeta $ containing a point ${x}\\\\in {X}$ is denoted by $\\\\zeta ({x})$ .\",\n \"A partition $\\\\zeta $ of ${X}$ is said to be measurable if there exists a countable family $\\\\mathfrak {D}$ of measurable sets $\\\\lbrace D_i\\\\rbrace $ such that every $D_i$ is a union of elements of $\\\\zeta $ , and for any pair $A_1,A_2$ of elements of $\\\\zeta $ there exists $D_j\\\\in \\\\mathfrak {D}$ such that $A_1\\\\subset D_j$ and $A_2\\\\subset D_j^{\\\\rm c}$ , where $D_j^{\\\\rm c}$ stands for the complement of $D_j$ in ${X}$ .\",\n \"The family $\\\\mathfrak {D}$ is called a basis for $\\\\zeta $ .\",\n \"In Section we use a product operation on the set of partitions of ${X}$ .\",\n \"The product operation is defined in [23] as follows.\",\n \"Definition 2 If $\\\\zeta _n$ , $n=1,\\\\ldots ,N$ are partitions, we then define their product $\\\\vee ^{N}_{n=1}\\\\zeta _n$ as the partition whose elements are the sets of the form $\\\\cap ^{N}_{i=1}{A_i}$ , for $A_i\\\\in \\\\zeta _i$ , $i=1,\\\\ldots ,N$ , satisfying $\\\\mu \\\\left(\\\\cap ^{N}_{i=1}A_i\\\\right)\\\\ne 0$ .\",\n \"For a countable sequence of partitions, the notation $\\\\vee ^{\\\\infty }_{n=1}\\\\zeta _n$ will be used for the $\\\\sigma $ -algebra generated by $\\\\cup ^\\\\infty _{n=1}\\\\zeta _n$ .\"\n ],\n [\n \"Koopman Group\",\n \"Consider a flow ${S}^t: {X}\\\\rightarrow {X}$ ($t\\\\in \\\\mathbb {R}$ ) on a finite-dimensional space ${X}$ and denote by $\\\\mathcal {F}$ a space of scalar-valued functions defined on ${X}$ : $\\\\mathcal {F}\\\\ni f: {X}\\\\rightarrow \\\\mathbb {C}$ , which we call the space of observables.\",\n \"In the following, we suppose that the existence and uniqueness of solutions associated with the flow hold for all $t$ .\",\n \"Then, we define the one-parameter group of linear operators $U^t$ ($t\\\\in \\\\mathbb {R}$ ) as a group of composition operators with ${S}^t$ : for $f\\\\in \\\\mathcal {F}$ , $(U^tf)({x}):=f({S}^t({x}))=(f\\\\circ {S}^t)({x}).$ where it is known as the Koopman group [24], [9].\",\n \"For spaces as $\\\\mathcal {F}=\\\\mathcal {C}^0(X)$ and $\\\\mathcal {L}^p(X)$ , the Koopman group is strongly continuous.\",\n \"Although the original flow ${S}^t$ is possibly described by a nonlinear differential equation and evolves on the finite-dimensional space ${X}$ , the operators $U^t$ are linear but evolve on the infinite-dimensional space $\\\\mathcal {F}$ .\",\n \"The eigenvalue $\\\\lambda \\\\in \\\\mathbb {C}$ and eigenfunction $\\\\phi _\\\\lambda \\\\in \\\\mathcal {F}\\\\setminus \\\\lbrace 0\\\\rbrace $ of the Koopman group are defined as follows: $(U^t\\\\phi _\\\\lambda )({x})=\\\\exp (\\\\lambda t)\\\\phi _\\\\lambda ({x}).$ The notion of the Koopman group and its spectral characterization will be used in the following sections.\"\n ],\n [\n \"Ergodic Partition in Measure-Preserving Flows\",\n \"For completeness, in this section we extend the theory of ergodic partition in [7] to the measure-preserving flow ${S}^t$ on the probability space $(X,\\\\mathfrak {B}_{X},\\\\mu )$ , which is introduced in Section REF .\",\n \"This section consists of a few lemmas and a theorem.\",\n \"Their proofs are almost identical to the proofs for maps [7] and appear in Appendices and .\",\n \"We denote by $\\\\mathcal {L}^1_\\\\mu ({X})$ the space of all $\\\\mu $ -integrable functions on ${X}$ and by $\\\\mathcal {C}({X})$ the space of all real-valued continuous functions on ${X}$ endowed with the sup norm.\",\n \"Roughly speaking, an ergodic partition is a partition of the state space ${X}$ into invariant sets on which the dynamics are ergodic.\",\n \"The precise definition of ergodic partition is the following.\",\n \"Definition 3 A measurable partition $\\\\zeta $ of ${X}$ is said to be ergodic under the flow ${S}^t$ if for any element $A$ of $\\\\zeta $ , (i) $A$ is invariant under ${S}^t$ , and (ii) there exists an invariant probability measure $\\\\mu _A$ on $A$ such that the restriction of ${S}^t$ to $A$ , denoted by ${S}^t|_A$ , is an ergodic diffeomorphism on $A$ , and for all $f\\\\in \\\\mathcal {L}^1_\\\\mu ({X})$ , $\\\\int _{X} f{d}\\\\mu =\\\\int _{X}\\\\left[\\\\int _{A=:\\\\zeta ({x})} f|_A{d}\\\\mu _A\\\\right]{d}\\\\mu ({x}),$ where $f|_A$ stands for the restriction of $f$ to the ergodic element $A$ , and $\\\\mu ({x})$ is again a probability measure on $X$ .\",\n \"It is remarked that the ergodic partition is defined for the entire state space $X$ with $\\\\sigma $ -algebra, but it can be relaxed in context of the $\\\\sigma $ -algebra of invariant sets which can be parts of the state space.\",\n \"Here, we denote by $f^\\\\ast $ the time-average of $f$ under the flow ${S}^t$ if the right-hand side of $f^\\\\ast ({x}):=\\\\lim _{T\\\\rightarrow \\\\infty }\\\\frac{1}{T}\\\\int ^T_0 f({S}^t({x})){d} t,$ exists for almost every (a.e.)\",\n \"point ${x}\\\\in {X}$ with respect to the measure $\\\\mu $ .\",\n \"Birkhoff's Ergodic Theorem [25], [1] shows that for all $f\\\\in \\\\mathcal {L}^1_\\\\mu ({X})$ , (i) $f^\\\\ast ({x})$ exists for a.e.\",\n \"point ${x}\\\\in {X}$ (with respect to $\\\\mu $ ); (ii) $f^\\\\ast ({S}^t({x}))=f^\\\\ast ({x})$ for a.e.\",\n \"point ${x}\\\\in {X}$ ; and (iii) $\\\\int _{X}f{d}\\\\mu =\\\\int _{X}f^\\\\ast {d}\\\\mu $ .\",\n \"Here, we let $\\\\mathit {\\\\Sigma }$ be the set of points ${x}\\\\in {X}$ such that $f^\\\\ast ({x})$ exists for all $f\\\\in \\\\mathcal {C}({X})$ , and $\\\\mathit {\\\\Sigma }(f)$ the set of points ${x}\\\\in {X}$ such that $f^\\\\ast ({x})$ exists for a particular $f\\\\in \\\\mathcal {C}({X})$ .\",\n \"The following lemma is standard (see page 129 of [23] for discrete-time dynamical systems (maps): it is obvious for flows).\",\n \"Lemma 4 For a countable and dense set $S$ in $\\\\mathcal {C}({X})$ , $\\\\mathit {\\\\Sigma }=\\\\bigcap _{f\\\\in S}\\\\mathit {\\\\Sigma }(f).$ Now, we have a set $\\\\mathit {\\\\Sigma }$ such that the time-averages of all continuous functions on ${X}$ exist.\",\n \"Then, the complimentary set $\\\\mathit {\\\\Sigma }^{\\\\rm c}$ is of measure zero.\",\n \"This is because according to Birkhoff's Ergodic Theorem (i), each $\\\\lbrace \\\\mathit {\\\\Sigma }(f)\\\\rbrace ^{\\\\rm c}$ is of measure zero, and thus $\\\\mathit {\\\\Sigma }^{\\\\rm c}:=\\\\cup _{f\\\\in S}\\\\lbrace \\\\mathit {\\\\Sigma }(f)\\\\rbrace ^{\\\\rm c}$ is a countable union of sets with zero measure, which is again of measure zero.\",\n \"The next lemma shows that the time-average of a continuous function induces a measurable partition on ${X}$ .\",\n \"Lemma 5 Let $f$ be a continuous function on ${X}$ .\",\n \"The family of level sets of $f^\\\\ast $ , $A_\\\\alpha :=\\\\lbrace {x} : {x}\\\\in \\\\mathit {\\\\Sigma }, f^\\\\ast ({x})=\\\\alpha \\\\rbrace ,~~~\\\\alpha \\\\in \\\\mathbb {R},$ is a measurable partition of ${X}$ .\",\n \"We denote by $\\\\zeta _f$ this partition and call it the partition induced by the function $f$ .\",\n \"See Appedix .\",\n \"Now, we can prove the first theorem stating that $\\\\zeta _f$ induces the ergodic partition of ${X}$ , which holds for general measure-preserving flows including the augmented system (REF ) for the quasiperiodically forced system (REF ).\",\n \"Theorem 6 Consider the measure-preserving flow ${S}^t: X\\\\rightarrow X$ ($t\\\\in \\\\mathbb {R}$ ) associated with $\\\\left.\\\\frac{d{S}^t({x})}{dt}\\\\right|_{t=0} ={F}({x}) \\\\quad \\\\textrm {for each}~{x}\\\\in X$ where ${F}: X\\\\rightarrow \\\\mathrm {T}X$ (tangent bundle of $X$ ) is a nonlinear vector field.\",\n \"Let $\\\\zeta {e}$ be the product of measurable partitions of ${X}$ induced by every $f\\\\in S$ : $\\\\zeta {e}=\\\\bigvee _{f\\\\in S}\\\\zeta _f.$ Then, $\\\\zeta {e}$ is the ergodic partition of ${X}$ .\",\n \"See Appendix .\",\n \"By combining the above theorem and lemma 20 in [26], we have the following corollary based on a finite number of basis functions.\",\n \"Corollary 7 Assume there exists a complete system of functions $\\\\lbrace f_i\\\\rbrace $ , $f_i\\\\in \\\\mathcal {C}(X)$ , $i\\\\in \\\\mathbb {N}_{>0}$ (set of all natural numbers except for 0) i.e.\",\n \"finite linear combinations of $f_i$ are dense in $\\\\mathcal {C}(X)$ .\",\n \"The ergodic partition $\\\\zeta {e}$ is $\\\\zeta {e}=\\\\bigvee _{i\\\\in \\\\mathbb {N}_{>0}}\\\\zeta _{f_i}.$ For a given function $f$ , it is obvious that each level set $A_\\\\alpha =\\\\lbrace {x} : {x}\\\\in \\\\mathit {\\\\Sigma }, f^\\\\ast ({x})=\\\\alpha \\\\rbrace $ ($\\\\alpha \\\\in \\\\mathbb {R}$ ) as an element of $\\\\zeta _f$ is invariant under ${S}^t$ .\",\n \"In [7] the authors proposed to use the level sets $A_\\\\alpha $ , which are directly computed with the time-averaging of $f$ under ${S}^t$ , for identifying and visualizing invariant sets in discrete-time dynamical systems possessing a smooth invariant measure.\",\n \"Theorem REF implies that the same computation can be used for visualization of invariant sets for the measure-preserving flow, namely, continuous-time dynamical systems with a smooth invariant measure.\",\n \"Since, as shown in Section , the quasiperiodically forced system (REF ) defined on the state space $M$ is transformed to the flow defined on the augmented state space $X=M\\\\times \\\\mathbb {T}^N$ , the ergodic partition theory is applicable to visualization of invariant sets in $X$ for the original quasiperiodically forced system (REF ) possessing a smooth invariant measure.\",\n \"However, in the quasiperiodically forced system (REF ) we are interested in dynamics on $M$ not on $X=M\\\\times \\\\mathbb {T}^N$ .\",\n \"While for $N=1$ it is clear that every intersection of the invariant set in $M\\\\times \\\\mathbb {T}^1$ with $M$ is an invariant set of the associated Poincaré map.\",\n \"The relationship in the quasiperiodically forced system (namely, $N\\\\ge 2$ ) is much less clear.\",\n \"We make it precise in Section .\",\n \"Here, let us introduce the connection of time-average $f^\\\\ast $ and the Koopman group.\",\n \"It is obvious that for any $f\\\\in \\\\mathcal {L}^1_\\\\mu (X)$ its time-average $f^\\\\ast $ is an eigenfuction of the operators $U^t$ at eigenvalue 0: for a.e.\",\n \"point ${x}\\\\in {X}$ with respect to $\\\\mu $ , $(U^tf^\\\\ast )({x})=f^\\\\ast ({S}^t({x}))=f^\\\\ast ({x})=\\\\exp (0t)f^\\\\ast ({x}).$ Since the eigenfunctions at 0 are invariant under the flow, we have a complete characterization of (possibly non-smooth) invariant sets.\"\n ],\n [\n \"Basin of Attraction in Dissipative Flows\",\n \"In the last of the previous section, we indicate that the time-average of an observable enables the visualization of low-dimensional slices through high-dimensional invariant sets of the quasiperiodically forced system with a smooth invariant measure.\",\n \"Here, we consider a more general class of the quasiperiodically forced systems with dissipation and will point out that the use of time-average works for characterizing an important invariant set—basin of attraction.\",\n \"In Section we considered the measure-preserving flow on a compact metric space.\",\n \"Every continuous flow on a compact metric space preserves a measure.\",\n \"This is the content of the so-called Krylov-Bogolyubov theorem [27].\",\n \"However, this might be a useless statement if our intent is to study the behavior of trajectories from their time-averages, by the method used in the measure-preserving cases in Section .\",\n \"For example, let us take a simple dissipative flow, $\\\\frac{{d} x}{{d} t}=-\\\\lambda x \\\\qquad x\\\\in \\\\mathbb {R}, \\\\qquad \\\\lambda >0.$ All of the initial conditions converge to $x=0$ as $t\\\\rightarrow \\\\infty $ along the flow $S^t(x)=\\\\exp (-\\\\lambda t)x$ .\",\n \"The invariant measure clearly exists and is the Dirac measure supported at $x=0$ (to which any “initial\\\" measure defined on $\\\\mathbb {R}$ evolves).\",\n \"Thus, the following statement holds: for a function $f:\\\\mathbb {R}\\\\rightarrow \\\\mathbb {C}$ , $f^\\\\ast (x)=\\\\int _{\\\\mathbb {R}}f{d}\\\\mu ,$ for a.e.\",\n \"point $x\\\\in \\\\mathbb {R}$ with respect to the Dirac measure.\",\n \"However, that excludes the whole real line except for the origin itself.\",\n \"The situation feels better though.\",\n \"For example, it is easy to see that the time-average of any continuous function on $\\\\mathbb {R}$ is just its value at 0: $f^*(x)=f(0)$ .\",\n \"Note that for the measure-preserving case in Section , we basically considered integrable functions for which time-averages exist.\",\n \"We can not do that in the dissipative case.\",\n \"A function that is 1 everywhere on a finite interval $(\\\\underline{a},\\\\overline{a})$ containing 0, except at 0 where its value is 0, is clearly integrable (being a union of two simple functions).\",\n \"Then, its time-average under the dissipative flow is identically 1, and its integral with respect to the invariant Dirac measure is 0.\",\n \"It turns out that in dissipative systems, it is best for the time-average to work with continuous functions.\",\n \"Adopting that idea, the whole method of ergodic partitioning can be extended to capture basins of attraction for continuous flows that are not necessarily measure-preserving.\",\n \"Consider a continuous flow ${S}^t: X\\\\rightarrow X$ , where $X$ is not necessary compact, and label by $F$ the set on which time averages of continuous functions do not exist.\",\n \"On the set $X\\\\setminus F$ , consider a set $C$ on which time-averages of continuous functions (or, better, a countable, separating set of continuous functions) are constant.There can be uncountably many such sets inside $X\\\\setminus F$ : for example, let ${S}^t$ be the identity map, mapping every point into itself.\",\n \"Then, by the same construction described in Section and Appendix , there is an invariant measure $\\\\mu _C$ such that $f^*({x})=\\\\int _C f d\\\\mu _C$ for any continuous function $f$ , ${x}\\\\in C$ .\",\n \"Such a measure is called a physical measure [28], provided $M$ is equipped with a measure $\\\\mu $ that ${S}^t$ does not preserve (but we are interested in)—say Lebesgue measure—and $C$ contains an open (and thus positive measure) set in $M$ .In light of this discussion, one might say that the notion of “ergodic measure\\\" should be preserved for such constructs even in dissipative systems.\",\n \"Namely, many ergodic measures we obtain in measure-preserving system are certainly physically important, although their support does not contain an open set.\",\n \"The tricky part is that due to Birkhoff's Ergodic Theorem, ergodic measures are associated with time averages of integrable functions.\",\n \"In dissipative systems, this does not work (see the above example).\",\n \"We could keep the requirement of equality of space and time averages, like in (REF ), but for continuous functions only.\",\n \"Physical measures would then be ergodic measures with support that contains an open set.\",\n \"Suppose that the state space of a continuous flow admits a finite number of attractors, and that the union of their basins of attraction is of full measure.\",\n \"We state the result for flows (continuous-time systems) to emphasize that considerations here work for both discrete and continuous time.\",\n \"Theorem 8 Let the system $\\\\left.\\\\frac{d{S}^t({x})}{dt}\\\\right|_{t=0}={F}({x}), \\\\quad {x}\\\\in X\\\\subset \\\\mathbb {R}^n$ with a flow ${S}^t$ have a finite number $N$ of attractors with basins of attraction $A_j, j=1,...,N$ , such that $\\\\mu (\\\\cup _j A_j)=\\\\mu (X)$ , where $\\\\mu $ is the Lebesgue measure.\",\n \"Also, let the time averages of continuous functions exist everywhere on a set $X\\\\setminus F$ , where $\\\\mu (F)=0$ .\",\n \"Then, the time-average $h^*({x})$ of a continuous function $h\\\\in \\\\mathcal {C}(X)$ is a piecewise constant eigenfunction of the Koopman operators $U^t$ at eigenvalue 0 defined a.e.\",\n \"point with respect to $\\\\mu $ .\",\n \"For any point ${x}\\\\in A_j,$ there exists a set $C$ such that $(U^t h^*)({x})&=U^t \\\\lim _{T\\\\rightarrow \\\\infty }\\\\frac{1}{T}\\\\int _0^{T} h({S}^\\\\tau ({x}))d\\\\tau \\\\nonumber \\\\\\\\&= \\\\int _C (U^t h) d\\\\mu _C\\\\nonumber \\\\\\\\&=\\\\int _C (h\\\\circ {S}^t) d\\\\mu _C\\\\nonumber \\\\\\\\&=\\\\int _C h d\\\\mu _C \\\\nonumber \\\\\\\\&=h^*({x}), \\\\nonumber $ where in the second line we utilized (REF ), and transitioning from line 3 to 4 we used the fact that $\\\\mu _C$ is invariant under ${S}^t$ .\",\n \"Since $(U^t h^*)({x})=h^*({x})$ is exactly the equation which $h^*({x})$ has to satisfy in order to be an eigenfunction of $U^t$ , and the basin of attraction $A_j$ is arbitrary, by taking into account that $\\\\mu (\\\\cup _j A_j)=\\\\mu (X)$ the proposition is proven.\",\n \"The concept of the ergodic partition in dissipative systems is applied to a more general class of systems than those treated in Theorem REF .\",\n \"It provides ergodic measures on invariant sets that are not necessarily attractors—because they do not have an open set of initial conditions converging to them.\",\n \"Yet, these sets are dynamically important and can not be refined without losing the equivalence of space and time averages.\",\n \"More precisely, the ergodic partition $\\\\zeta {e}$ of $M$ under ${S}^t$ (not necessarily measure-$\\\\mu $ preserving) is a partition into sets $C_{\\\\alpha }$ , where $\\\\alpha $ is a member of an indexing set, such that on each set $C_{\\\\alpha }$ (ergodic set) there exists an ergodic measure $\\\\mu _{C_\\\\alpha }$ such that $\\\\mu _{C_{\\\\alpha }}(C_{\\\\alpha })=1,$ For every $f\\\\in \\\\mathcal {C}(M),f^{\\\\ast }({x}\\\\in C_{\\\\alpha })=\\\\int _{C_{\\\\alpha }}fd\\\\mu _{C_{\\\\alpha }},$ for a.e ${x}$ point with respect to $\\\\mu $ .\",\n \"The last condition emphasizes the role of the measure $\\\\mu $ —the time averages are equal to space averages with respect to $\\\\mu _{C_\\\\alpha }$ , but their equality is almost everywhere with respect to measure $\\\\mu $ that might be of our interest.\",\n \"In this way, we have escaped the realm of the Krylov-Bogolyubov Theorem, which, in the case when dynamics are not measure-preserving, neglects dynamics on possibly large swaths of the state space.\"\n ],\n [\n \"Sample-Based Characterization of Invariant Sets\",\n \"In this section, we provide a characterization of invariant sets in the quasiperiodically forced system (REF ).\",\n \"Generally speaking, understanding invariant structures of (REF ) is not easy because the associated portrait of $M$ changes with time in an aperiodic manner.\",\n \"As shown in Section , the ergodic partition theory enables one to visualize an invariant set as its low-dimensional slice in $M$ by the time-averaging technique.\",\n \"Also, in () it was shown that the time-averaging technique works for dissipative case to visualize the basin of attraction.\",\n \"The obtained slice here is just a sample of the invariant set at a particular initial time (in other words, an initial condition in $\\\\mathbb {T}^N$ ).\",\n \"Because of the aperiodic nature, such a sample does not seem to provide complete information on the entire structure of invariant set in the augmented state space $M\\\\times \\\\mathbb {T}^N$ .\",\n \"However, we will prove that a boundedness property of the invariant set in $M\\\\times \\\\mathbb {T}^N$ is captured by means of one sample of it in $M$ (see Theorem REF ).\",\n \"In the following, in order to encompass both the cases in Section and (), we consider the original state space $M$ that is metric and not necessarily compact.\",\n \"A continuous, linear skew-product flow ${S}^t$ (diffeomorphism) on the augmented state space $X:=M\\\\times \\\\mathbb {T}^N$ derived from the quasiperiodically forced system (REF ) is described below: for ${x}=({m},{\\\\theta })\\\\in M\\\\times \\\\mathbb {T}^N$ , ${S}^t({x}):=\\\\left({S}^t_{M}({m},{\\\\theta }),{S}^t_\\\\mathit {\\\\Omega }({\\\\theta })\\\\right),$ where ${S}^t_{M}: M\\\\times \\\\mathbb {T}^N\\\\rightarrow M$ is the continuous map defined by trajectories of the original system (REF ), and ${S}^t_\\\\mathit {\\\\Omega }: \\\\mathbb {T}^N\\\\rightarrow \\\\mathbb {T}^N$ the linear (continuous) flow on the torus $\\\\mathbb {T}^N$ .\",\n \"For a fixed ${\\\\theta }_0\\\\in \\\\mathbb {T}^N$ , we will write the orbit on $\\\\mathbb {T}^N$ through ${\\\\theta }_0$ as $\\\\mathcal {O}_{\\\\mathbb {T}^N}({\\\\theta }_0):=\\\\lbrace {\\\\theta } : {\\\\theta }={S}^t_\\\\mathit {\\\\Omega }({\\\\theta }_0)\\\\in \\\\mathbb {T}^N, t\\\\in \\\\mathbb {R}\\\\rbrace $ .\",\n \"Since the $N$ basic frequencies in the original system (REF ) are rationally independent, the following lemma is obvious.\",\n \"Lemma 9 For any ${\\\\theta }_0\\\\in \\\\mathbb {T}^N$ , $\\\\mathcal {O}_{\\\\mathbb {T}^N}({\\\\theta }_0)$ is dense in $\\\\mathbb {T}^N$ .\",\n \"Now, we investigate an invariant set in the augmented system (REF ).\",\n \"Let $I\\\\subseteq X$ be a positively invariant set of (REF ) that is supposed to be closedIf $I$ is invariant and open, then its closure $\\\\mathrm {cl}(I)$ is still invariant..\",\n \"It follows from Lemma REF and by closure that the following decomposition of $I$ holds: $I=\\\\bigcup _{\\\\theta \\\\in \\\\mathbb {T}^N} A_\\\\theta \\\\times \\\\lbrace {\\\\theta }\\\\rbrace ,$ where $A_{\\\\theta }$ is a closed subset of $M$ .\",\n \"Let us denote by $A_{\\\\theta _0}$ an intersection or sample of the invariant set $I$ at ${\\\\theta }={\\\\theta }_0$ .\",\n \"The next lemma provides a topological property of the intersections $A_\\\\theta $ .\",\n \"Lemma 10 For each $\\\\epsilon >0$ there exists a positive constant $\\\\delta $ so that $|{\\\\theta }-{\\\\theta }_0|_{\\\\mathbb {T}^N}<\\\\delta $ implies $d(A_\\\\theta ,A_{\\\\theta _0})<\\\\epsilon $ , where $d(A_{\\\\theta },A_{\\\\theta _0})$ is the Hausdorff distance that induces a topology on the family of all closed subsets of $M$ .\",\n \"Assume not.\",\n \"Then, for each $\\\\delta >0$ there exist a sequence of times $\\\\lbrace t_j\\\\rbrace $ , where $t_j\\\\rightarrow \\\\infty $ as $j\\\\rightarrow \\\\infty $ , and a positive constant $\\\\epsilon _1$ such that ${\\\\theta }_j={S}^{t_j}_\\\\mathit {\\\\Omega }({\\\\theta }_0)$ satisfies $|{\\\\theta }_j-{\\\\theta }_0|_{\\\\mathbb {T}^N}<\\\\delta $ for every $j>J$ ($J$ is an integer), while $d(A_{\\\\theta _k},A_{\\\\theta _0})>\\\\epsilon _1$ holds for some $k>J$ .\",\n \"Here, from the continuity of ${S}^t_M$ in $t$ , for each ${m}\\\\in A_{\\\\theta _0}$ and $\\\\epsilon >0$ , there exists a positive constant $\\\\delta $ such that $|{\\\\theta }_k-{\\\\theta }_0|_{\\\\mathbb {T}^N}<\\\\delta $ implies $|{m}_k-{m}|_{M}<\\\\epsilon $ , where ${m}_k:={S}^{t_k}_{M}({m},{\\\\theta }_0)\\\\in A_{\\\\theta _k}$ .\",\n \"This gives us a contradiction of $d(A_{\\\\theta _k},A_{\\\\theta _0})>\\\\epsilon _1$ by taking $\\\\epsilon =\\\\epsilon _1$ and from the definition of Hausdorff distance.\",\n \"This lemma suggests that the invariant set $I$ is topologically connected with respect to ${\\\\theta }$ .\",\n \"One trivial example is provided by taking as $I$ the closure of a single trajectory starting at a point $({m},{\\\\theta }_0)$ , for which $A_{\\\\theta _0}$ consists of a single point.\",\n \"In this case, $I$ is topologically regarded as a product set of a single point in $M$ and the torus $\\\\mathbb {T}^N$ .\",\n \"This is rigorously stated in the next theorem.\",\n \"Theorem 11 Suppose that $A_{\\\\theta _0}$ is bounded in $M$ .\",\n \"Then, $A_{\\\\theta _0}\\\\times \\\\mathbb {T}^N$ is homeomorphic to $I$ .\",\n \"Now, let us construct the mapping $h_{\\\\theta _0}: A_{\\\\theta _0}\\\\times \\\\mathcal {O}_{\\\\mathbb {T}^N}({\\\\theta }_0)\\\\rightarrow I$ in the following manner.\",\n \"For each $({m},{\\\\theta })\\\\in A_{\\\\theta _0}\\\\times \\\\mathcal {O}_{\\\\mathbb {T}^N}({\\\\theta }_0)$ , by choosing time $\\\\tau _\\\\theta $ that satisfies ${\\\\theta }={S}^{\\\\tau _\\\\theta }_{\\\\it \\\\Omega }({\\\\theta }_0)$ , we define $h_{\\\\theta _0}({m},{\\\\theta })$ as $({S}^{\\\\tau _\\\\theta }_M({m},{\\\\theta }_0),{\\\\theta })$ .\",\n \"We here prove the continuity of $h_{\\\\theta _0}$ .\",\n \"For each $\\\\epsilon >0$ and ${\\\\theta }\\\\in \\\\mathcal {O}_{\\\\mathbb {T}^N}({\\\\theta }_0)$ , there exists a sequence of $\\\\lbrace t_j\\\\rbrace $ (where $t_j\\\\rightarrow \\\\infty $ as $j\\\\rightarrow \\\\infty $ ) such that ${\\\\theta }_j={S}^{t_j}_\\\\mathit {\\\\Omega }({\\\\theta }_0)$ satisfies $\\\\left|{\\\\theta }_j-{\\\\theta }\\\\right|_{\\\\mathbb {T}^N}<\\\\epsilon /2$ for every $j>J$ ($J$ is an integer).\",\n \"Furthermore, because of the continuity of ${S}^t_{M}$ in $t$ , for each ${m}_j\\\\in A_{\\\\theta _0}$ there exists a positive constant $\\\\delta _j$ such that $\\\\left|{m}-{m}_j\\\\right|_M<\\\\delta _j$ implies $\\\\left|{S}_M^{\\\\tau _\\\\theta }({m},{\\\\theta }_0)-{S}_M^{\\\\tau _{\\\\theta _j}}({m}_j,{\\\\theta }_0)\\\\right|_M<\\\\epsilon /2$ .\",\n \"Here, because for every $j>J$ $\\\\left|({m},{\\\\theta })-({m}_j,{\\\\theta }_j)\\\\right|_{M\\\\times \\\\mathbb {T}^N}\\\\le \\\\left|{m}-{m}_j\\\\right|_{M}+\\\\left|{\\\\theta }-{\\\\theta }_j\\\\right|_{\\\\mathbb {T}^N}< \\\\delta _j+\\\\frac{\\\\epsilon }{2},$ we set $\\\\delta :=\\\\delta _j+\\\\epsilon /2$ .\",\n \"Thus, if $\\\\left|({m},{\\\\theta })-({m}_j,{\\\\theta }_j)\\\\right|_{M\\\\times \\\\mathbb {T}^N}<\\\\delta $ , then $\\\\left|h_{\\\\theta _0}({m},{\\\\theta })-h_{\\\\theta _0}({m}_j,{\\\\theta }_j)\\\\right|_{M\\\\times \\\\mathbb {T}^N}&\\\\le \\\\left|{S}_M^{\\\\tau _\\\\theta }({m},{\\\\theta }_0)-{S}_M^{\\\\tau _{\\\\theta _j}}({m}_j,{\\\\theta }_0)\\\\right|_M+\\\\left|{\\\\theta }-{\\\\theta }_j\\\\right|_{\\\\mathbb {T}^N}\\\\nonumber \\\\\\\\& <\\\\frac{\\\\epsilon }{2}+\\\\frac{\\\\epsilon }{2}=\\\\epsilon .\\\\nonumber $ This shows that $h_{\\\\theta _0}$ is continuous.\",\n \"We next prove that $h_{\\\\theta _0}$ is uniformly continuous.\",\n \"Assume not.\",\n \"Then, for each $\\\\delta >0$ there exists a positive constant $\\\\epsilon _1$ such that $\\\\left|({m}^{\\\\prime }_j,{\\\\theta }^{\\\\prime }_j)-({m}_j,{\\\\theta }_j)\\\\right|_{M\\\\times \\\\mathbb {T}^N}<\\\\delta $ implies $\\\\left|h_{\\\\theta _0}({m}^{\\\\prime }_j,{\\\\theta }^{\\\\prime }_j)-h_{\\\\theta _0}({m}_j,{\\\\theta }_j)\\\\right|_{M\\\\times \\\\mathbb {T}^N}\\\\ge \\\\epsilon _1$ for some $({m}_j,{\\\\theta }_j) $ and $({m}^{\\\\prime }_j,{\\\\theta }^{\\\\prime }_j)\\\\in A_{\\\\theta _0}\\\\times \\\\mathcal {O}_{\\\\mathbb {T}^N}({\\\\theta }_0)$ .\",\n \"We here set $\\\\delta =1/j$ .\",\n \"Since it is supposed that $A_{\\\\theta _0}$ is bounded and closed (and $\\\\mathcal {O}_{\\\\mathbb {T}^N}({\\\\theta }_0)\\\\subset \\\\mathbb {T}^N$ ), the two sequences of points $\\\\lbrace ({m}_j,{\\\\theta }_j)\\\\rbrace $ and $\\\\lbrace ({m}^{\\\\prime }_j,{\\\\theta }^{\\\\prime }_j)\\\\rbrace $ have convergent subsequences, denoted as $\\\\lbrace ({m}_{j_k},{\\\\theta }_{j_k})\\\\rbrace $ and $\\\\lbrace ({m}^{\\\\prime }_{j_k},{\\\\theta }^{\\\\prime }_{j_k})\\\\rbrace $ .\",\n \"Because of $\\\\left|({m}^{\\\\prime }_j,{\\\\theta }^{\\\\prime }_j)-({m}_j,{\\\\theta }_j)\\\\right|_{M\\\\times \\\\mathbb {T}^N}<1/j$ , both the subsequences have the same convergent point, denoted as $({m}^\\\\ast ,{\\\\theta }^\\\\ast )$ .\",\n \"That is, for each $\\\\delta _1>0$ , there exists a positive integer $J_k$ such that both the inequalities $\\\\left|({m}^{\\\\prime }_{j_k},{\\\\theta }^{\\\\prime }_{j_k})-({m}^\\\\ast ,{\\\\theta }^\\\\ast )\\\\right|_{M\\\\times \\\\mathbb {T}^N}<\\\\delta _1$ and $\\\\left|({m}_{j_k},{\\\\theta }_{j_k})-({m}^\\\\ast ,{\\\\theta }^\\\\ast )\\\\right|_{M\\\\times \\\\mathbb {T}^N}<\\\\delta _1$ hold for every $j_k\\\\ge J_k$ .\",\n \"Here, since $h_{\\\\theta _0}$ is proven to be continuous at $({m}^\\\\ast ,{\\\\theta }^\\\\ast )$ , for each $\\\\epsilon >0$ , there exists a positive constant $\\\\delta _2$ such that $\\\\left|({m},{\\\\theta })-({m}^\\\\ast ,{\\\\theta }^\\\\ast )\\\\right|_{M\\\\times \\\\mathbb {T}^N}<\\\\delta _2$ implies $\\\\left|h_{\\\\theta _0}({m},{\\\\theta })-h_{\\\\theta _0}({m}^\\\\ast ,{\\\\theta }^\\\\ast )\\\\right|_{M\\\\times \\\\mathbb {T}^N}<\\\\epsilon /2$ .\",\n \"By taking $\\\\epsilon =\\\\epsilon _1$ and $\\\\delta _1=\\\\delta _2$ thus choosing $J_k$ appropriately, we see that both the inequalities $\\\\left|h_{\\\\theta _0}({m}^{\\\\prime }_{j_k},{\\\\theta }^{\\\\prime }_{j_k})-h_{\\\\theta _0}({m}^\\\\ast ,{\\\\theta }^\\\\ast )\\\\right|_{M\\\\times \\\\mathbb {T}^M}<\\\\epsilon _1/2$ and $\\\\left|h_{\\\\theta _0}({m}_{j_k},{\\\\theta }_{j_k})-h_{\\\\theta _0}({m}^\\\\ast ,{\\\\theta }^\\\\ast )\\\\right|_{M\\\\times \\\\mathbb {T}^N}<\\\\epsilon _1/2$ hold for every $j_k\\\\ge J_k$ , and $\\\\begin{aligned}\\\\left|h_{\\\\theta _0}({m}^{\\\\prime }_{j_k},{\\\\theta }^{\\\\prime }_{j_k})-h_{\\\\theta _0}({m}_{j_k},{\\\\theta }_{j_k})\\\\right|_{M\\\\times \\\\mathbb {T}^M}\\\\le &\\\\left|h_{\\\\theta _0}({m}^{\\\\prime }_{j_k},{\\\\theta }^{\\\\prime }_{j_k})-h_{\\\\theta _0}({m}^\\\\ast ,{\\\\theta }^\\\\ast )\\\\right|_{M\\\\times \\\\mathbb {T}^N} \\\\\\\\& + \\\\left|h_{\\\\theta _0}({m}^\\\\ast ,{\\\\theta }^\\\\ast )-h_{\\\\theta _0}({m}_{j_k},{\\\\theta }_{j_k})\\\\right|_{M\\\\times \\\\mathbb {T}^N} \\\\\\\\< & \\\\frac{\\\\epsilon _1}{2}+\\\\frac{\\\\epsilon _1}{2} = \\\\epsilon _1,\\\\end{aligned}$ implying the contradiction for $\\\\left|h_{\\\\theta _0}({m}^{\\\\prime }_{j_k},{\\\\theta }^{\\\\prime }_{j_k})-h_{\\\\theta _0}({m}_{j_k},{\\\\theta }_{j_k})\\\\right|_{M\\\\times \\\\mathbb {T}^N}\\\\ge \\\\epsilon _1$ .\",\n \"Thus, $h_{\\\\theta _0}$ is uniformly continuous.\",\n \"By virtue of Theorem 3.45 in page 136 of [29], the uniformly-continuous $h_{\\\\theta _0}: A_{\\\\theta _0}\\\\times \\\\mathcal {O}_{\\\\mathbb {T}^N}({\\\\theta }_0)\\\\rightarrow I$ is extended to the mapping $\\\\hat{h}_{\\\\theta _0}: A_{\\\\theta _0}\\\\times \\\\mathbb {T}^N\\\\rightarrow I$ that is also (uniformly) continuous.\",\n \"We now prove that $\\\\hat{h}_{\\\\theta _0}$ is a bijection.\",\n \"For each ${\\\\theta }\\\\in \\\\mathcal {O}_{\\\\mathbb {T}^N}({\\\\theta }_0)$ , the statement is obvious from the construction of $h_{\\\\theta _0}$ (see its dependence on ${\\\\theta }$ ).\",\n \"It is thus enough to check the case ${\\\\theta }\\\\in \\\\mathbb {T}^N\\\\setminus \\\\mathcal {O}_{\\\\mathbb {T}^N}({\\\\theta }_0)$ .\",\n \"For each ${\\\\theta }\\\\in \\\\mathbb {T}^N\\\\setminus \\\\mathcal {O}_{\\\\mathbb {T}^N}({\\\\theta }_0)$ , there exists a sequence of times $\\\\lbrace t_j\\\\rbrace $ (where $t_j\\\\rightarrow \\\\infty $ as $j\\\\rightarrow \\\\infty $ ) such that ${\\\\theta }_j={S}^{t_j}_\\\\mathit {\\\\Omega }({\\\\theta }_0)$ converges to ${\\\\theta }$ .\",\n \"Because $\\\\hat{h}_{\\\\theta _0}$ is continuous, the sequence $\\\\lbrace \\\\hat{h}_{\\\\theta _0}({m},{\\\\theta }_j)=h_{\\\\theta _0}({m},{\\\\theta }_j)\\\\rbrace $ converges to $\\\\hat{h}_{\\\\theta _0}({m},{\\\\theta })$ , which is represented as $\\\\displaystyle \\\\left(\\\\lim _{t_j\\\\rightarrow \\\\infty }{S}_M^{t_j}({m},{\\\\theta }_0),{\\\\theta }\\\\right)$ .\",\n \"Namely, the limit exists for every ${m}\\\\in A_{\\\\theta _0}$ .\",\n \"Assume that $\\\\hat{h}_{\\\\theta _0}$ is not an injection.\",\n \"Then, there exist ${m},{m}^{\\\\prime }\\\\in A_{\\\\theta _0}$ satisfying ${m}\\\\ne {m}^{\\\\prime }$ such that for each $\\\\epsilon >0$ , $\\\\left|{S}_M^{t_j}({m},{\\\\theta }_0)-{S}_M^{t_j}({m}^{\\\\prime },{\\\\theta }_0)\\\\right|_M<\\\\epsilon $ holds for every $j>J$ ($J$ is an integer and depends on $\\\\epsilon $ ).\",\n \"Here, the fundamental property of uniqueness of trajectories in (REF ) (or ${S}^t$ diffeomorphsim) implies that for each $t_j$ , $\\\\left|{S}_M^{t_j}({m},{\\\\theta }_0)-{S}_M^{t_j}({m}^{\\\\prime },{\\\\theta }_0)\\\\right|_M\\\\ge \\\\epsilon _j$ holds ($\\\\epsilon _j$ is a positive constant and depends on $t_j$ ), showing the contradiction by taking $\\\\epsilon =\\\\epsilon _j$ .\",\n \"Hence, $\\\\hat{h}_{\\\\theta _0}$ is an injection.\",\n \"Regarding $\\\\hat{h}_{\\\\theta _0}$ surjective, it is obvious that for each ${\\\\theta }\\\\in \\\\mathcal {O}_{\\\\mathbb {T}^N}({\\\\theta }_0)$ , $\\\\hat{h}_{\\\\theta _0}(A_{\\\\theta _0},{\\\\theta })=h_{\\\\theta _0}(A_{\\\\theta _0},{\\\\theta })=\\\\left({S}_M^{\\\\tau _\\\\theta }(A_{\\\\theta _0},{\\\\theta }_0),{\\\\theta }\\\\right)=(A_\\\\theta ,{\\\\theta })$ .\",\n \"For each ${\\\\theta }\\\\in \\\\mathbb {T}^N\\\\setminus \\\\mathcal {O}_{\\\\mathbb {T}^N}({\\\\theta }_0)$ we see $\\\\displaystyle \\\\hat{h}_{\\\\theta _0}(A_{\\\\theta _0},{\\\\theta })=\\\\left(\\\\lim _{t_j\\\\rightarrow \\\\infty }{S}_M^{t_j}(A_{\\\\theta _0},{\\\\theta }_0),{\\\\theta }\\\\right)$ , and the limit converges to $A_\\\\theta $ from Lemma REF .\",\n \"Hence, from (REF ), $\\\\hat{h}_{\\\\theta _0}$ is a surjection.\",\n \"Finally, the inverse of $\\\\hat{h}_{\\\\theta _0}$ exists according to the above construction and is continuous.\",\n \"Therefore, $\\\\hat{h}_{\\\\theta _0}: A_{\\\\theta _0}\\\\times \\\\mathbb {T}^N\\\\rightarrow I$ is a homeomorphism, namely, $A_{\\\\theta _0}\\\\times \\\\mathbb {T}^N$ is homeomorphic to $I$ .\",\n \"The following two corollaries provide a way of characterization of boundedness of $I$ by means of one sample of it.\",\n \"Corollary 12 Suppose that $A_{\\\\theta _0}$ is bounded in $M$ .\",\n \"Then, $I$ is bounded in $X$ .\",\n \"Corollary 13 Suppose that a closed subset $B_{\\\\theta _0}$ of $A_{\\\\theta _0}$ is bounded in $M$ .\",\n \"Then, $\\\\hat{h}_{\\\\theta _0}(B_{\\\\theta _0}\\\\times \\\\mathbb {T}^N)$ is a subset of $I$ and bounded in $X$ .\",\n \"Corollaries REF and REF imply that by computing a slice of the invariant set in $M$ at one sample onset, it is possible to determine whether the invariant set or its subset is bounded in the augmented state space $X$ .\",\n \"For a bounded invariant set $I$ in $X$ , any cross-section $I_{\\\\theta _0}=A_{\\\\theta _0}\\\\times \\\\lbrace {\\\\theta }_0\\\\rbrace $ (${\\\\theta }_0\\\\in \\\\mathbb {T}^N$ ) is bounded in terms of $M$ , in other words, the boundedness property of $A_{\\\\theta _0}$ does not depend on the choice of ${\\\\theta }_0$ .\",\n \"In this way, we call the bounded invariant set $I$ in $X$ uniformly bounded invariant set.\"\n ],\n [\n \"Example Studies\",\n \"In this section, we provide three examples of the application of the above theoretical results.\",\n \"The last example is related to practical problems on stability of power grids.\"\n ],\n [\n \"Forced Linear Harmonic Oscillator\",\n \"First, we will consider the following one-degree-of-freedom linear harmonic oscillator with quasi-periodic forcing of (multiple) $N$ frequencies: $\\\\frac{dm_1}{dt}=m_2, \\\\qquad \\\\frac{dm_2}{dt}=-m_1+\\\\sum ^{N}_{i=1}F_i\\\\sin (\\\\mathit {\\\\Omega }_it+\\\\theta _{i0}),$ or $\\\\frac{dm_1}{dt}=m_2, \\\\qquad \\\\frac{dm_2}{dt}=-m_1+\\\\sum ^{N}_{i=1}F_i\\\\sin \\\\theta _i, \\\\qquad \\\\frac{d\\\\theta _i}{dt}=\\\\mathit {\\\\Omega }_i \\\\qquad i=1,2,\\\\ldots ,N$ where $F_i$ ($>0$ ) ($i=1,2,\\\\ldots ,N$ ) are the amplitudes of periodic forces, $\\\\mathit {\\\\Omega }_i$ ($>0$ ) the angular frequencies of the forces, and $\\\\theta _{i0}\\\\in \\\\mathbb {T}$ the initial phases.\",\n \"We assume there is no resonance: the $N$ angular frequencies $\\\\mathit {\\\\Omega }_i$ are rationally independent; and $\\\\mathit {\\\\Omega }_i\\\\ne 1$ ($i=1,2,\\\\ldots ,N$ ).\",\n \"The solution of (REF ) with initial condition $(m_{10},m_{20},\\\\theta _{10},\\\\theta _{20},\\\\ldots ,\\\\theta _{N0})$ is analytically represented as follows: $\\\\left.\\\\begin{aligned}m_1(t) &=C_1\\\\cos t+C_2\\\\sin t+\\\\sum ^{N}_{i=1}\\\\frac{F_i}{1-\\\\mathit {\\\\Omega }^2_i}\\\\cos (\\\\mathit {\\\\Omega }_it+\\\\theta _{i0}), \\\\\\\\m_2(t) &=\\\\displaystyle \\\\frac{{d}}{{d} t}m_1(t), \\\\\\\\{\\\\vspace{14.22636pt}}\\\\theta _i(t) &= \\\\mathit {\\\\Omega }_it+\\\\theta _{i0} \\\\qquad i=1,2,\\\\ldots ,N,\\\\end{aligned}\\\\right\\\\rbrace $ with $C_1:=m_{10}-\\\\sum ^{N}_{i=1}\\\\frac{F_i}{1-\\\\mathit {\\\\Omega }^2_i}\\\\cos \\\\theta _{i0}, \\\\qquad C_2:=m_{20}+\\\\sum ^{N}_{i=1}\\\\frac{F_i\\\\mathit {\\\\Omega }_i}{1-\\\\mathit {\\\\Omega }^2_i}\\\\sin \\\\theta _{i0}.$ Now, we estimate the time-averages of an observable under the dynamics described by the linear system (REF ).\",\n \"We define the following quadratic observable $f: \\\\mathbb {R}^2\\\\times \\\\mathbb {T}^N\\\\rightarrow \\\\mathbb {R}$ , related to the potential energy $m^2_1/2$ : $f(m_1,m_2,\\\\theta _1,\\\\theta _2,\\\\ldots ,\\\\theta _N)=m^2_1.$ Then, we calculate its time-average under the solution as $\\\\lim _{T\\\\rightarrow \\\\infty }\\\\frac{1}{T}\\\\int ^T_0(U^tf)(m_{10},m_{20},\\\\theta _{10},\\\\theta _{20},\\\\ldots ,\\\\theta _{N0}){d} t=\\\\frac{1}{2}\\\\left\\\\lbrace C^2_1+C^2_2+\\\\sum ^{N}_{i=1}\\\\frac{F^2_i}{(1-\\\\mathit {\\\\Omega }^2_i)^2}\\\\right\\\\rbrace .$ Here, as shown in (REF ), the constants $C_1$ and $C_2$ are determined by the initial condition $(m_{10},m_{20},\\\\theta _{10},\\\\theta _{20},\\\\ldots ,\\\\theta _{N0})$ .\",\n \"The level set, parameterized by a real-valued constant $c\\\\in \\\\mathbb {R}$ , of the time-average in the two-dimensional initial plane $(m_{10},m_{20})$ at fixed $(\\\\theta _{10},\\\\theta _{20},\\\\ldots ,\\\\theta _{N0})$ becomes a circle with center of $\\\\left(\\\\sum ^{N}_{i=1}\\\\frac{F_i}{1-\\\\mathit {\\\\Omega }^2_i}\\\\cos \\\\theta _{i0},-\\\\sum ^{N}_{i=1}\\\\frac{F_i\\\\mathit {\\\\Omega }_i}{1-\\\\mathit {\\\\Omega }^2_i}\\\\sin \\\\theta _{i0}\\\\right),$ if the following inequality holds: $r^2:=2c-\\\\sum ^{N}_{i=1}\\\\frac{F^2_i}{(1-\\\\mathit {\\\\Omega }^2_i)^2}>0$ Thus, the constant $r$ corresponds to the radius of the circle of the level set for $c$ .\",\n \"The center of the level set seems to stay close to the origin and is shifted with the choice of initial phases $(\\\\theta _{10},\\\\theta _{20},\\\\ldots ,\\\\theta _{N0})$ .\",\n \"Numerical results of the level sets for $N=2$ , $\\\\mathit {\\\\Omega }_1=\\\\pi /3$ , $\\\\mathit {\\\\Omega }_2=11/10$ , and $F_1=F_2=0.2$ are presented in Figure REF .\",\n \"Figure: Level sets of the quasiperiodically-forced one-degree-of-freedom harmonic oscillator () for different (θ 10 ,θ 20 )(\\\\theta _{10},\\\\theta _{20}):(a) (0,0)(0,0), (b) (π/2,π/2)(\\\\pi /2,\\\\pi /2), (c) (π,π)(\\\\pi ,\\\\pi ), and (d) (3π/2,3π/2)(3\\\\pi /2,3\\\\pi /2).The horizontal (or vertical) axis for each figure is x 1 ∈[-10,10]x_1\\\\in [-10,10] (or x 2 ∈[-10,10]x_2\\\\in [-10,10]).\"\n ],\n [\n \"One-Dimensional Dissipative System\",\n \"Next, we consider the following linear dissipative system with quasi-periodic forcing: $\\\\frac{{d} m}{{d} t}=-\\\\lambda m+\\\\sum ^{N}_{i=1}F_i\\\\sin (\\\\mathit {\\\\Omega }_it+\\\\theta _{i0}).$ or $\\\\frac{dm}{dt}=-\\\\lambda m+\\\\sum ^{N}_{i=1}F_i\\\\sin \\\\theta _i, \\\\qquad \\\\frac{d\\\\theta _i}{dt}=\\\\mathit {\\\\Omega }_i \\\\qquad i=1,2,\\\\ldots ,N$ where $\\\\lambda >0$ .\",\n \"In the same manner as above, we suppose the $N$ angular frequencies $\\\\mathit {\\\\Omega }_i$ are rationally independent.\",\n \"The solution of (REF ) with initial condition $(m_{0},\\\\theta _{10},\\\\theta _{20},\\\\ldots ,\\\\theta _{N0})$ is analytically represented as follows: $\\\\left.\\\\begin{array}{ccl}m(t) &=& \\\\displaystyle C^{-\\\\lambda t}+\\\\sum ^{N}_{i=1}\\\\frac{F_i}{\\\\lambda ^2+\\\\mathit {\\\\Omega }^2_i}\\\\lbrace \\\\lambda \\\\sin (\\\\mathit {\\\\Omega }_it+\\\\theta _{i0})-\\\\mathit {\\\\Omega }_i\\\\cos (\\\\mathit {\\\\Omega }_it+\\\\theta _{i0})\\\\rbrace ,\\\\\\\\{\\\\vspace{11.38109pt}}\\\\theta _i(t) &=& \\\\mathit {\\\\Omega }_it+\\\\theta _{i0}~~~(i=1,2,\\\\ldots ,N),\\\\end{array}\\\\right\\\\rbrace $ with $C:=m_0-\\\\sum ^{N}_{i=1}\\\\frac{F_i}{\\\\lambda ^2+\\\\mathit {\\\\Omega }^2_i}\\\\lbrace \\\\lambda \\\\sin \\\\theta _{i0}-\\\\mathit {\\\\Omega }_i\\\\cos \\\\theta _{i0}\\\\rbrace .$ Here, we estimate the time-averages of an observable under the dynamics described by the linear system (REF ).\",\n \"We consider the quadratic observable as $f(m,\\\\theta _1,\\\\theta _2,\\\\ldots ,\\\\theta _N)=m^2.$ Then, its time-average under the solution is directly calculated as follows: $\\\\lim _{T\\\\rightarrow \\\\infty }\\\\frac{1}{T}\\\\int ^T_0(U^tf)(m_0,\\\\theta _{10},\\\\theta _{20},\\\\ldots ,\\\\theta _{N0}){d} t=\\\\frac{1}{2}\\\\sum ^{N}_{i=1}\\\\frac{F^2_i}{\\\\lambda ^2+\\\\mathit {\\\\Omega }^2_i}.$ Although this is trivial, since the transient term in the solution is filtered out, the value of the time-average is constant in the augmented state space $\\\\mathbb {R}\\\\times \\\\mathbb {T}^N$ , in other words, does not depend on the initial condition $(m_0,\\\\theta _{10},\\\\theta _{20},\\\\ldots ,\\\\theta _{N0})$ .\",\n \"This implies that for any choice of sample at $(\\\\theta _{10},\\\\theta _{20},\\\\ldots ,\\\\theta _{N0})^\\\\top \\\\in \\\\mathbb {T}^N$ , the partition of $M=\\\\mathbb {R}$ based on the level set of the time-averages is just one and corresponds to $\\\\mathbb {R}$ itself.\",\n \"Also, the above observation on the partition holds when we pick up a polynomial observable like $f(m,\\\\theta _1,\\\\theta _2,\\\\ldots ,\\\\theta _N)=\\\\sum ^{n}_{j=0}a_jm^j,$ with coefficients $a_j\\\\in \\\\mathbb {R}$ and finite integer $n$ .\",\n \"For every continuous function defined on a closed interval in $\\\\mathbb {R}$ , from the Weierstrass Approximation Theorem, it can be uniformly approximated on that interval by polynomials to any degree of accuracy.\",\n \"This implies that the above observation of the partition holds for every continuous function, which is an example of Theorem REF .\",\n \"Thus, the augmented state space $\\\\mathbb {R}\\\\times \\\\mathbb {T}^N$ is the basin of attraction for the system (REF ), that is to say, the torus $\\\\mathbb {T}^N$ is the unique attractor for the system.\"\n ],\n [\n \"Two-Dimensional Nonlinear Model of a Power Grid\",\n \"To show the effectiveness of the theory beyond the basic examples, we apply the developed theory for analyzing the CSI phenomenon in a loop power grid.\",\n \"CSI is a undesirable and emergent phenomenon of synchronous machines in a power grid, in which most of the machines coherently lose synchronism with the rest of the grid after being subjected to a finite disturbance [19].\",\n \"In the case of small dissipation, this phenomenon generally does not happen upon an infinitesimally small perturbation around a steady operating state (stable equilibrium).If we have no dissipation, nonlinear stability properties are not understood in the high-dimensional cases.\",\n \"However, it encompasses the situation when the grid's operating state escapes a predefined, positive-measure set around the equilibrium.\",\n \"In this way, the notion of instability that we address here is non-local.\",\n \"In [19], we derived a reduced-order dynamical system that described averaged dynamics of machines in a simple loop power grid and explained the non-local instability.\",\n \"The reduced-order system has an quasiperiodic forcing (so it is non-autonomous), and its solutions define a measure-preserving flow.\",\n \"The goal of this section is to characterize the non-local instability by analyzing invariant sets of the quasiperiodically forced system, which is introduced in (REF ) below.\"\n ],\n [\n \"Mathematical Model\",\n \"Consider short-term (zero to ten seconds) swing dynamics of a rudimentary power grid with the loop topology shown in Figure REF , where the small gray circles denote synchronous generators.\",\n \"The loop part of the grid consists of $N{G}$ small, identical generators, encompassed by the dotted box, which operate in the grid and are connected to the infinite busAn ideal voltage source of constant voltage and constant frequency.\",\n \"The loss-less transmission lines joining the infinite bus and a generator are much longer than those joining two generators in the loop part.\",\n \"Thus, the magnitude of interaction between the infinite bus and a generator is much smaller than that between two neighboring generators on the loop part.\",\n \"We call the model of Figure REF the loop power grid in the following.\",\n \"Here, we assume that the lengths of transmission lines between two neighboring generators are identical.\",\n \"In [19], we showed that the CSI phenomenon in the loop power grid was accurately captured by the following dynamical system defined on the cylindrical state space $\\\\mathbb {T}^1\\\\times \\\\mathbb {R}$ : $\\\\frac{d\\\\delta }{dt}=\\\\omega , \\\\qquad \\\\frac{d\\\\omega }{dt}=p{m}-\\\\frac{b}{N{G}}\\\\sum _{i=1}^{N{G}}\\\\sin \\\\left(\\\\sum _{j\\\\in \\\\mathcal {J}}e_{ij}c_j\\\\cos \\\\mathit {\\\\Omega }_jt+\\\\delta \\\\right)$ where $e_{ij}=\\\\sqrt{\\\\frac{2}{N{G}}}\\\\cos \\\\left(\\\\frac{2\\\\pi ij}{N{G}}+\\\\frac{\\\\pi }{4}\\\\right),\\\\qquad \\\\mathit {\\\\Omega }_j=2\\\\sqrt{|b{int}|}\\\\left|\\\\sin \\\\frac{\\\\pi j}{N{G}}\\\\right|.$ The system (REF ) represents spatially-averaged dynamics of the $N{G}$ generators in the loop power grid.\",\n \"The variable $\\\\delta \\\\in \\\\mathbb {T}^1$ is the average of angular positions of rotors (with respect to the infinite bus) of the $N{G}$ generators, and $\\\\omega \\\\in \\\\mathbb {R}$ is the average of deviations of rotor speeds in the $N{G}$ generators relative to the system angular frequency.\",\n \"The parameter $p{m}$ stands for the mechanical input power to a generator, $b$ for the maximum transmission power between the infinite bus and a generator, and $b{int}$ for the maximum transmission power between two neighboring generators in the loop power grid.\",\n \"The constants $e_{ij}$ are the eigenfunctions of linear modal oscillations between coupled generators in the loop part, $\\\\mathit {\\\\Omega }_j$ their eigen-(angular) frequency, and $c_j$ the strengths of modal oscillations.\",\n \"The finite index set $\\\\mathcal {J}$ determines which modes are excited in the loop part.\",\n \"The system (REF ) is derived under the observation that the linear modal oscillations in the loop part act as perturbations on the spatially-averaged dynamics of the $N{G}$ generators: see [19] and references therein.\",\n \"Note that (REF ) is the Hamiltonian system: $\\\\frac{d\\\\delta }{dt}=\\\\frac{}{\\\\omega }{H}(\\\\delta ,\\\\omega ,t), \\\\qquad \\\\frac{d\\\\omega }{dt}=-\\\\frac{}{\\\\delta }{H}(\\\\delta ,\\\\omega ,t),$ with the time-dependent Hamiltonian function ${H}(\\\\delta ,\\\\omega ,t)$ , given by ${H}(\\\\delta ,\\\\omega ,t):=\\\\frac{1}{2}\\\\omega ^2-p{m}\\\\delta -\\\\frac{b}{N{G}}\\\\sum ^{N{G}}_{i=1}\\\\cos \\\\left(\\\\sum _{j\\\\in \\\\mathcal {J}}e_{ij}c_j\\\\cos \\\\mathit {\\\\Omega }_jt+\\\\delta \\\\right).$ Because the flow defined here is divergence-free, i.e.\",\n \"$({H}/\\\\delta )({d}\\\\delta /{d} t)+({H}/\\\\omega )({d}\\\\omega /{d} t)=0$ , the system (REF ) preserves the Liouville measure ${d}\\\\delta {d}\\\\omega $ .\",\n \"Note that it does not conserve the Hamiltonian function ${H}$ , because of ${d}{H}/{d} t\\\\ne 0$ if $c_j\\\\ne 0$ .\",\n \"In this way, the augmented system for (REF ), namely $\\\\frac{d\\\\delta }{dt}=\\\\omega , \\\\quad \\\\frac{d\\\\omega }{dt}=p{m}-\\\\frac{b}{N{G}}\\\\sum _{i=1}^{N{G}}\\\\sin \\\\left(\\\\sum _{j\\\\in \\\\mathcal {J}}e_{ij}c_j\\\\cos \\\\theta _j+\\\\delta \\\\right), \\\\quad \\\\frac{d\\\\theta _j}{dt}=\\\\mathit {\\\\Omega }_j \\\\qquad j\\\\in \\\\mathcal {J},$ defines a measure-preserving flow on $M\\\\times \\\\mathbb {T}^{|{\\\\cal J}|}$ with $M=\\\\mathbb {T}^1\\\\times \\\\mathbb {R}$ , where $|{\\\\cal J}|$ stands for the cardinality of ${\\\\cal J}$ .\"\n ],\n [\n \"Results and Implications\",\n \"It was shown in [19] that the unbounded motion in the quasiperiodically forced system (REF ) corresponds to the CSI phenomenon.\",\n \"Because of $\\\\delta \\\\in \\\\mathbb {T}^1$ , the unbounded motion involves the unbounded trajectory in the $\\\\omega $ -direction, that is, the average of deviation of rotor speeds.\",\n \"We use Corollaries REF and REF to analyze invariant sets of the flow (defined by the system (REF )), in which all the generators show bounded deviation of rotor speeds in time.\",\n \"The analysis is crucial to understanding the so-called stability region of the loop power grid, i.e.\",\n \"how the grid's behavior depends on initial conditions representing failures in the grid.\",\n \"Numerical simulations are performed for analysis of invariant sets.\",\n \"To do so, we need to fix (i) a function $f$ , (ii) the subset of state space on which we identify invariant sets, and (iii) the exit time $T{ex}$ to obtain an approximation of each time-average $f^\\\\ast $ .\",\n \"We use the function $f(\\\\delta )=\\\\sin 2\\\\delta $ and the grid of $401\\\\times 401$ of initial conditions $(\\\\delta ,\\\\omega )$ on $[1,2]\\\\times [-0.15,0.15]$ .\",\n \"The averaging operation of a single function can be used for the identification of invariant sets.\",\n \"Numerical integration of the system (REF ) is performed with the 4th-order symplectic integrator [30] with time step $h$ : see Appendix for details.\",\n \"The parameter settings are the following: $p{m}=0.95, \\\\quad b=1, \\\\quad N{G}=20, \\\\quad b{int}=100, \\\\quad h=\\\\frac{2\\\\pi }{\\\\mathit {\\\\Omega }_1}\\\\frac{1}{N},\\\\quad T{ex}=\\\\frac{2\\\\pi }{\\\\mathit {\\\\Omega }_1}\\\\times 2000,$ where $N=8$ or 16 depending on the setting of $\\\\mathcal {J}$ .\",\n \"The values of $p{m}$ , $b$ , $N{G}$ , and $ b{int}$ are the same as in [19].\",\n \"Figure: Analysis of invariant sets of the measure-preserving flow defined by ()–I: Initial phase θ 0 =0{\\\\theta }_0={0} and multiple setting of 𝒥\\\\mathcal {J} i.e.\",\n \"(a) 𝒥={1}\\\\mathcal {J}=\\\\lbrace 1\\\\rbrace , (b) 𝒥={1,2}\\\\mathcal {J}=\\\\lbrace 1,2\\\\rbrace , (c) 𝒥={1,2,3}\\\\mathcal {J}=\\\\lbrace 1,2,3\\\\rbrace , (d) 𝒥={1,2,3,4}\\\\mathcal {J}=\\\\lbrace 1,2,3,4\\\\rbrace , (e) 𝒥={1,2,3,4,5}\\\\mathcal {J}=\\\\lbrace 1,2,3,4,5\\\\rbrace , and (f) 𝒥={1,2,3,4,5,6}\\\\mathcal {J}=\\\\lbrace 1,2,3,4,5,6\\\\rbrace .The horizontal axis for each figure is δ∈[1,2]\\\\delta \\\\in [1,2], and the vertical axis is ω∈[-0.15,0.15]\\\\omega \\\\in [-0.15,0.15].For the outer dark-green regions, every trajectory starting from them is unbounded in time.Figure: Analysis of invariant sets of the measure-preserving flow defined by ()–II: 𝒥={1,2}\\\\mathcal {J}=\\\\lbrace 1,2\\\\rbrace and multiple setting of initial phase θ 0 =(θ 10 ,θ 20 ) ⊤ =(2πkΩ 1 /Ω 2 ,0) ⊤ {\\\\theta }_0=(\\\\theta _{10},\\\\theta _{20})^\\\\top =(2\\\\pi k\\\\mathit {\\\\Omega }_1/\\\\mathit {\\\\Omega }_2,0)^\\\\top for (a) k=0k=0 (same as Figure (b)), (b) k=1k=1, (c) k=2k=2, (d) k=3k=3, (e) k=4k=4, and (f) k=5k=5.The horizontal axis for each figure is δ∈[0,π]\\\\delta \\\\in [0,\\\\pi ], and the vertical axis is ω∈[-0.15,0.15]\\\\omega \\\\in [-0.15,0.15].For the outer dark-green regions, every trajectory starting from them is unbounded in time.Figure REF shows numerical results on analysis of invariant sets of the flow defined by the quasiperiodically forced system (REF ).\",\n \"In this figure we change the number of excitation modes, i.e.\",\n \"$\\\\mathcal {J}$ ; (a) $\\\\mathcal {J}=\\\\lbrace 1\\\\rbrace $ , (b) $\\\\mathcal {J}=\\\\lbrace 1,2\\\\rbrace $ , (c) $\\\\mathcal {J}=\\\\lbrace 1,2,3\\\\rbrace $ , (d) $\\\\mathcal {J}=\\\\lbrace 1,2,3,4\\\\rbrace $ , (e) $\\\\mathcal {J}=\\\\lbrace 1,2,3,4,5\\\\rbrace $ , and (f) $\\\\mathcal {J}=\\\\lbrace 1,2,3,4,5,6\\\\rbrace $ .\",\n \"The amplitude $c_j$ in the figure is common and satisfies $\\\\displaystyle \\\\sqrt{\\\\sum _{j\\\\in \\\\mathcal {J}}c_j^2}=1.5$ , implying that the root-means-square of the forcing term does not change for any setting of $\\\\mathcal {J}$ .\",\n \"All the initial phases ${\\\\theta }_0$ are set to zero.\",\n \"For the outer dark-green region in each figure, every trajectory starting from it is unbounded in time.\",\n \"Except for the dark-green on the bottom, the color bar attached to each figure denotes the value of time-average $f^\\\\ast (\\\\delta )$ .\",\n \"The level sets of $f^\\\\ast (\\\\delta )$ are colored by the same color.\",\n \"That is, the set of the same color belongs to one invariant set.\",\n \"By Corollary REF , the fact that the level set is bounded in these figures implies that the associated subset of invariant set is bounded in the augmented state space $M\\\\times \\\\mathbb {T}^{|\\\\mathcal {J}|}$ .\",\n \"In the figures (a,b,c), we see that the color plot of the level sets forms concentric rings.\",\n \"However, in the figures (d,e,f), we see that the color plot does does not necessarily exhibit concentric rings and does become scattered in the bands close to the outer dark-green regions.\",\n \"This implies that the structure of invariant sets is complicated in the bands.\",\n \"We anticipate this results from the so-called resonance phenomenon (see, e.g., [31], [32]) as an interaction between a family of bounded oscillations in the unforced system and quasiperiodic forcing.\",\n \"Figure REF shows other numerical results on analysis of invariant sets.\",\n \"In this figure, we consider the forcing term with two frequencies, $\\\\mathcal {J}=\\\\lbrace 1,2\\\\rbrace $ , and we change the initial phases ${\\\\theta }_0=(\\\\theta _{10},\\\\theta _{20})^\\\\top =(2\\\\pi k\\\\mathit {\\\\Omega }_1/\\\\mathit {\\\\Omega }_2,0)$ where $k=0,1,\\\\ldots ,5$ .\",\n \"The color plots here are conducted in the same manner as in Figure REF , and for the outer dark-green regions every trajectory starting from them is unbounded in time.\",\n \"By Corollary REF , the fact that the level set with same color is bounded in these figures implies that the associated subset of invariant set is bounded in the augmented state space.\",\n \"Here, under the current setting of parameters, it is conjectured that outside the outer dark-green regions (i.e., outside the computational domain of the analysis), there exists no state from which trajectory is bounded in time.\",\n \"This is true in the unperturbed case because there exists one homoclinic orbit separating the bounded and unbounded trajectories inside the computational domain.\",\n \"Thus, it can be inferred that the level sets discussed above are bounded in $M=\\\\mathbb {T}^1\\\\times \\\\mathbb {R}$ , implying by Corollary REF that the whole of the corresponding invariant sets are uniformly bounded in the augmented state space.\",\n \"The intersection of uniformly bounded invariant sets for all ${\\\\theta }$ corresponds to the stability region of the loop power grid, in which all the generators show bounded deviation of rotor speeds in time.\"\n ],\n [\n \"Conclusions\",\n \"In this paper, we studied the ergodic partition and invariant sets of the quasiperiodically forced dynamical system (REF ).\",\n \"The main theoretical contributions of this paper are twofold.\",\n \"One is to provide a theory of ergodic partition of state space for smooth flows.\",\n \"The theory is a natural extension of that in [7] and is applicable to measure-preserving and dissipative flows arising in various physical and engineering systems.\",\n \"Examples of them include dynamical systems induced by time-dependent Hamiltonians and incompressible fluid flows with time-dependent velocity profiles.\",\n \"The other is to provide a new characterization of invariant sets in the the quasiperiodically forced system (REF ), in which we introduced a concept of uniformly bounded invariant sets.\",\n \"The developed theory was applied to characterize the CSI phenomenon of a rudimentary power grid.\",\n \"We have speculated that the phenomenon can be characterized, in particular, the stability region corresponds to the intersection of uniformly bounded sets for all initial phases; or a sufficient condition for the phenomenon is that the operating state of the grid is placed outside of the bounded sets at a particular initial phase or time like $t=0$ .\"\n ],\n [\n \"Acknowledgments\",\n \"Y.S.\",\n \"thanks Dr. Marko $\\\\rm Budi\\\\check{s}i\\\\acute{c}$ for his introduction to theory and computation of ergodic partition and fruitful discussions.\",\n \"The authors also appreciate the reviewers for their valuable suggestion of the manuscript.\",\n \"During part of the work on this paper, Y.S.\",\n \"was at the Department of Mechanical Engineering, University of California, Santa Barbara, United States, and at the Department of Electrical Engineering, Kyoto University, Japan.\"\n ],\n [\n \"Proof of Lemma \",\n \"A continuous function $f$ on $X$ is measurable and, from the assumption that $X$ is compact for the theoretical analysis in Section , $f$ is bounded on $X$ .\",\n \"Here, we note that the time-average $f^\\\\ast $ of the measurable function $f$ is measurable as a limit of measurable functions $f_T$ on $X$ , defined as $f_T(x):=\\\\frac{1}{T}\\\\int ^T_0 f({S}^t(x)){d}t \\\\qquad T>0.$ Since we consider the sets $A_\\\\alpha $ on $\\\\mathit {\\\\Sigma }\\\\subset X$ , the fact that the family of $A_\\\\alpha $ is a partition of $\\\\mathit {\\\\Sigma }$ is obvious.\",\n \"Next, the fact that the partition, denoted by $\\\\zeta _f$ , is measurable follows by taking $\\\\mathfrak {D}_f$ to be the collection of pre-images under $f^\\\\ast $ of open intervals with rational endpoints in $\\\\mathbb {R}$ .\",\n \"Because $f^\\\\ast $ is measurable, each pre-image $(f^\\\\ast )^{-1}([a,b])$ , where $a$ and $b$ are rational numbers, is measurable.\",\n \"Every set of this type is clearly separated into sets of the form $(f^\\\\ast )^{-1}(\\\\lbrace c\\\\rbrace )$ , $c\\\\in \\\\mathbb {R}$ .\",\n \"This implies that every element of $\\\\mathfrak {D}_f$ is a union of elements of $\\\\zeta _f$ .\",\n \"Furthermore, because the set of all rational numbers is dense in $\\\\mathbb {R}$ , for any pair $\\\\alpha ,\\\\beta \\\\in \\\\mathbb {R}$ satisfying $\\\\alpha <\\\\beta $ , there exist two rational numbers $\\\\underline{a},\\\\overline{a}$ such that $\\\\alpha <\\\\underline{a}<\\\\beta <\\\\overline{a}$ .\",\n \"The pre-image $(f^\\\\ast )^{-1}([\\\\underline{a},\\\\overline{a}])$ is an element of $\\\\mathfrak {D}_f$ which we denote by $D$ .\",\n \"Obviously, we see $A_\\\\alpha \\\\subset {D}^{\\\\rm c}$ and $A_\\\\beta \\\\subset {D}$ .\",\n \"Thus, it follows that $\\\\mathfrak {D}_f$ is a basis for $\\\\zeta _f$ , and we conclude that $\\\\zeta _f$ is measurable.\"\n ],\n [\n \"Proof of Theorem \",\n \"Let $A$ be an element of $\\\\zeta {e}$ .\",\n \"For a.e.\",\n \"point $x\\\\in A$ , the time-average $f^\\\\ast (x)$ exists for all $f\\\\in \\\\mathcal {C}(X)$ .\",\n \"Thus, the following linear functional $L_{A}$ on $\\\\mathcal {C}(X)$ is well-defined: $L_{A}(f):=\\\\lim _{T\\\\rightarrow \\\\infty }\\\\frac{1}{T}\\\\int ^T_0 f({S}^t (x)){d} t \\\\qquad x\\\\in A.$ Then, because $L_{A}$ is a positive linear functional and $L_{A}(1)=1$ , by Riesz's Representation Theorem (I.8.4 in [23]) there exists a unique probability measure $\\\\mu _{A}$ on $X$ such that $\\\\int _{X} f{d}\\\\mu _A=L_{A}(f),$ for all $f\\\\in \\\\mathcal {C}(X)$ .\",\n \"Note that $\\\\mu _{A}$ is invariant for ${S}^t$ .\",\n \"To prove this, for all $t\\\\in \\\\mathbb {R}$ we have $\\\\int _{X} f\\\\circ {S}^t\\\\,{d}\\\\mu _{A}=L_{A}(f\\\\circ {S}^t)=L_{A}(f)=\\\\int _{X} f{d}\\\\mu _{A}.$ The second equality is a consequence of (REF ).\",\n \"For the above operation, the continuity of ${S}^t$ is required.\",\n \"Because $\\\\mathcal {C}(X)$ is dense in $\\\\mathcal {L}^1_{\\\\mu _A}(X)$ , $\\\\mu _{A}$ is invariant.\",\n \"Now, we prove that $\\\\mu _{A}$ is a probability measure on $A$ .\",\n \"There is a sequence of compact sets $A^{\\\\rm c}_n$ , subsets of $A^{\\\\rm c}$ , such that ${A}^{\\\\rm c}_1\\\\subset \\\\cdots \\\\subset {A}^{\\\\rm c}_{n}\\\\subset {A}^{\\\\rm c}_{n+1}\\\\subset \\\\cdots , \\\\qquad \\\\mu _{A}\\\\left(A^{\\\\rm c}\\\\setminus \\\\bigcup _{n\\\\ge 1}A^{\\\\rm c}_n\\\\right)=0.\",\n \"\\\\qquad $ Here, we can show $\\\\mu _{A}(A^{\\\\rm c}_n)=0$ for every $A^{\\\\rm c}_n$ .\",\n \"To do this, note that by Urysohn's Lemma, for every $A^{\\\\rm c}_n$ , there is a continuous, positive function $f_n$ on $X$ that is equal (i) to one on $A^{\\\\rm c}_n$ and (ii) to zero outside of $A^{\\\\rm c}_{n+1}$ .\",\n \"Clearly, we see $f_n=0$ on $A$ .\",\n \"Therefore, because of $\\\\int _{X}f_n{d}\\\\mu _{A}=\\\\int _{X\\\\setminus A^{\\\\rm c}_{n+1}}f_n{d}\\\\mu _{A}+\\\\int _{A^{\\\\rm c}_{n+1}\\\\setminus A^{\\\\rm c}_n}f_n{d}\\\\mu _{A}+\\\\int _{A^{\\\\rm c}_n}f_n{d}\\\\mu _{A}$ and the positiveness of $f_n$ , we have $0\\\\le \\\\mu _{A}(A^{\\\\rm c}_n)\\\\le \\\\int _{X} f_n{d}\\\\mu _{A}=L_{A}(f_n)=0.$ The measure of a union of the countable number of sets with measure zero is zero: $\\\\mu _{A}\\\\left(\\\\bigcup _{n\\\\ge 1}A^{\\\\rm c}_n\\\\right)=0.$ Therefore, by (REF ) and (REF ), we have $\\\\mu _{A}(A^{\\\\rm c})=0$ .\",\n \"It follows from $\\\\mu _{A}(X)=1$ that $\\\\mu _{A}$ is a probability measure on $A$ .\",\n \"Next, let us prove that $\\\\mu _{A}$ is an ergodic measure on $A$ .\",\n \"First, observe that the set of all restrictions of functions in $\\\\mathcal {C}(X)$ to $A$ , denoted by $\\\\mathcal {C}(X)|_{A}$ , is dense in the set of all $\\\\mu _{A}$ -integrable functions on $A$ , denoted by $\\\\mathcal {L}^1_{\\\\mu _{A}}(A)$ .\",\n \"To show this, note that $\\\\mathcal {C}(X)$ is dense in $\\\\mathcal {L}^1_{\\\\mu _{A}}(X)$ .\",\n \"Let $f$ be an element of $\\\\mathcal {L}^1_{\\\\mu _{A}}(A)$ .\",\n \"Consider the extension of $f$ to $X$ , $\\\\bar{f}$ , such that $\\\\bar{f}=f$ on $A$ and $\\\\bar{f}=0$ elsewhere.\",\n \"Then, we have $\\\\bar{f}\\\\in \\\\mathcal {L}^1_{\\\\mu _{A}}(X)$ because the following integral exists: $\\\\int _{X}\\\\bar{f}{d}\\\\mu _{A}=\\\\int _{A}f{d}\\\\mu _{A}.$ Here, since $\\\\mathcal {C}(X)$ is dense in $\\\\mathcal {L}^1_{\\\\mu _{A}}(X)$ , there is a sequence of functions in $\\\\mathcal {C}(X)$ , $\\\\lbrace f_n\\\\rbrace $ , converging to $\\\\bar{f}$ .\",\n \"Thus, the corresponding sequence of restrictions, $\\\\lbrace f_n|_{A}\\\\rbrace $ , converges to $f$ .\",\n \"Therefore, we observe that $\\\\mathcal {C}(X)|_{A}$ is dense in $\\\\mathcal {L}^1_{\\\\mu _{A}}(A)$ .\",\n \"Now, by the same argument as (REF ), for all $f\\\\in \\\\mathcal {C}(X)|_{A}$ we have $\\\\int _{A}f{d}\\\\mu _{A}&= L_A(f) \\\\nonumber \\\\\\\\&= f^\\\\ast (x) \\\\qquad x\\\\in A.$ Since (REF ) holds for the dense set $\\\\mathcal {C}(X)|_A$ in $\\\\mathcal {L}^1_{\\\\mu _{A}}(A)$ , ${S}^t|_{A}$ is ergodic: see Proposition 2.2 in Chapter II of [23] for discrete-time systems.\",\n \"This proposition can be naturally extended to continuous-time systems.\",\n \"Hence, we complete the proof that there indeed exists an ergodic measure $\\\\mu _{A}$ for any element $A$ of the partition $\\\\zeta {e}$ .\",\n \"Finally, we consider (REF ) and that $A$ is invariant.\",\n \"The equality (REF ) is obtained with the proof of Theorem 6.4 in Chapter II of [23].\",\n \"The proof is obtained for discrete-time systems and is extended to continuous-time systems.\",\n \"By construction, the fact that $A$ is invariant is obvious.\",\n \"This completes the proof of Theorem REF .\"\n ],\n [\n \"Symplectic Integration of Time-Dependent Hamiltonian Systems\",\n \"In Section REF , it is required to numerically simulate the Hamiltonian system (REF ) with the time-dependent Hamiltonian function ${H}(\\\\delta ,\\\\omega ,t)$ .\",\n \"Symplectic integrator [30] is normally formulated in the case of time-independent Hamiltonian functions.\",\n \"However, one can exploit the integrator in the case of time-dependent Hamiltonian functions by augmenting the original Hamiltonian system.\",\n \"Consider the $N$ degree-of-freedom Hamiltonian system with the Hamiltonian function ${H}(q,p,t)$ : for $i=1,2,\\\\ldots ,N$ , $\\\\frac{dq_i}{dt}=\\\\frac{}{p_i}{H}(q,p,t), \\\\qquad \\\\frac{dp_i}{dt}=-\\\\frac{}{q_i}{H}(q,p,t)$ where $q=(q_1,q_2,\\\\ldots ,q_N)^\\\\top $ , $p=(p_1,p_2,\\\\ldots ,p_N)^\\\\top $ , and $t\\\\in \\\\mathbb {R}$ .\",\n \"Now, by replacing the time variable $t$ with one new variable $q_0$ and defining the other new variable $dp_0/dt:=-H/t$ , we have the augmented Hamiltonian function $\\\\bar{H}(q_0,p_0,q,p)$ as follows: $\\\\bar{H}(q_0,p_0,q,p):=p_0+{H}(q,p,q_0).$ Thus, the augmented Hamiltonian system of the time-independent Hamiltonian function $\\\\bar{H}$ is derived as $\\\\frac{dq_i}{dt}=\\\\frac{}{p_i}\\\\bar{H}(q_0,p_0,q,p), \\\\qquad \\\\frac{dp_i}{dt}=-\\\\frac{}{q_i}\\\\bar{H}(q_0,p_0,q,p)$ where $i=0,1,\\\\ldots ,N$ .\",\n \"The flow induced by trajectories of the augmented system (REF ) is divergence-free and conserves the value of the Hamiltonian function $\\\\bar{H}$ .\",\n \"Thus, by using the integrator for the augmented system, numerical simulations of the original system (REF ) are indirectly performed.\",\n \"Note that the accuracy of numerical integration of (REF ) is checked by estimating the value of $\\\\bar{H}$ .\",\n \"This idea is applicable to the case of non-periodic time-dependent Hamiltonian functions.\"\n ]\n]","isConversational":false},"id":{"kind":"string","value":"1808.08340"}}},{"rowIdx":197,"cells":{"text":{"kind":"list like","value":[["How do Convolutional Neural Networks Learn Design?"],["Abstract In this paper, we aim to understand the design principles in book cover images which are carefully crafted by experts.","Book covers are designed in a unique way, specific to genres which convey important information to their readers.","By using Convolutional Neural Networks (CNN) to predict book genres from cover images, visual cues which distinguish genres can be highlighted and analyzed.","In order to understand these visual clues contributing towards the decision of a genre, we present the application of Layer-wise Relevance Propagation (LRP) on the book cover image classification results.","We use LRP to explain the pixel-wise contributions of book cover design and highlight the design elements contributing towards particular genres.","In addition, with the use of state-of-the-art object and text detection methods, insights about genre-specific book cover designs are discovered."],["Introduction","Visual design renders specific impressions to transmit information which enriches the product's value.","However, these visual designs despite of being important are not analyzed objectively or statistically.","Analyzing these visual designs enables us to understand the contained information carried by them.","An interesting target of visual design analysis is book cover image design where the design of a book cover can infer the genre.","Each book cover is carefully designed by typographers and their designs represent the book contents in an intuitive way for better sales.","This association of books to specific genres is based on the differences in their underlying book cover designs [1].","The slight change in book cover design can reflect changes in book genre which makes design learning a challenging task for book covers.","In order to understand the design elements used for machine aided book cover classification, we employ Convolutional Neural Networks (CNN) [2].","In recent years, CNNs have achieved state-of-the-art results in isolated character recognition [3], [4] and large-scale image recognition [5], [6].","Notably, Iwana et al.","[1] demonstrated that CNNs can be used for genre classification based on book cover image, although with a high level of difficulty.","However, that study was subjective and not enough explanation is given as to why the CNN performed as it did.","To interpret the reasoning behind a CNN's prediction we used a method called Layer-wise Relevance Propagation (LRP) [7].","LRP decomposes output function on its input variables and highlights input pixels contributing towards the network decision.","It produces a layer-wise relevance heatmap by recursively multiplying the relevance of higher layers by the normalized feature maps of the target layer.","The heatmaps can help us to discover the input image elements which have an effect on the classification result.","The main contributions of this paper are threefold.","Firstly, we classified the book cover images using one-vs-others classification with CNNs.","Secondly, the models built by the CNNs are analyzed using LRP.","With LRP, we demonstrate design elements specifically relevant to classification of the book cover images.","We show that certain objects have a strong relevance to particular genres.","Finally, we use state-of-the-art object detection and text detection methods, namely Single Shot Multibox Detector (SSD) [8] and Efficient and Accurate Scene Text Detector [9], to quantitatively enforce the results found by LRP.","This reveals the specific elements in which CNNs classify book cover images for genre classification.","The organization is as follows.","Section provides related works in design understanding and genre classification as well as feature visualization of CNNs.","Section reviews the data and tools used for understanding book cover design.","Section presents analysis of CNN's understanding of book cover design.","In Section , we demonstrate the use of LRP combined with SSD and EAST for quantitative analysis.","Finally, Section draws a conclusion.","Artistic style understanding and subjective genre classification is a budding field in machine learning.","For example, recent attempts have been done to identify artistic styles and quality of paintings and photographs [10], [11] with neural network models.","In addition, there have been trials to classify music by genre [12], [13], book covers by genre [1], movie posters by genre [14], paintings by genre [15], and text by genre [16], [17].","Also, in a general sense, document classification can be considered genre classification and deep CNNs are the state-of-the-art in the document classification domain [18], [19], [20]."],["Visualization inside of CNNs","There is a desire to visualize features and determine pixel-wise attention and relevance within the hidden layers of CNNs.","However, this is a not a straightforward task [21].","Erhan et al.","[21] proposed using gradient decent to maximize a node's activation to visualize the employed features.","Similar work has been done for large-scale image classification [22].","Zeiler and Fergus [23] used deconvolutional neural networks to visualize features learned by CNNs.","In addition, they created heatmaps by monitoring class changes systematic cover up of portions of the images.","Class Activation Maps (CAM) [24], GradCAM [25], and GradCAM++ [26] reveal the parts of images which are most important to a class using global average pooling (GAP).","Recently, LRP has been used in the fields of text [27] where classification scores were projected back to input features for extracting relevant words for a specific prediction.","The method has also shown successes in model understanding in fields of sentiment analysis [28], action recognition [29], and age and gender classification [30].","As far as the authors are aware, this is the first time LRP has been used for the understanding of genre or design classification."],["Amazon Book Cover Dataset","We used the Book Cover Image to Genre datasethttps://github.com/uchidalab/book-dataset Task 1.A.","The dataset consists of 57,000 book cover images divided into 30 classes of equal sizes.","In the experiments, we used the predefined training set and test set modified for one-vs-others classification.","In this way, genre-wise training sets were prepared with an equal distribution of positive and negative data samples."],["Convolution Neural Networks","CNNs are able to tackle image recognition by implementing convolutions of learned filter-like shared weights which maintain the structural qualities of images while acting as feature extractors [2].","For the experiment, we implement CNNs to tackle book genre classification.","To use the book cover images with a CNN, they were preprocessed by scaling them to 112$\\times $ 112 pixels by 3 color channel images and by normalizing the values to be between -1 and 1.","The CNN used for the experiments has six convolutional layers with Rectified Linear Units (ReLU) activations and a softmax output layer.","The convolutional layers consisted of three layers of 10 nodes with $5\\times 5$ convolutional filters, one layer of 25 nodes with a $4\\times 4$ filter, one layer of 50 nodes with a $3\\times 3$ filter, and one layer of 100 nodes with a $1\\times 1$ filter.","A $2\\times 2$ maxpooling layer with stride 2 was used between each convolution layer.","Finally, the CNNs were trained using gradient decent with a batch size of 25 at a learning rate of 0.001 for 50,000 iterations.","The accuracy results for each genre is summarized in Fig.","REF .","In particular, the CNNs had difficulties with the reference classes, such as \"Engineering & Transportation,\" \"Health, Fitness and Dieting,\" \"History,\" \"Medical Books,\" and \"Reference.\"","Conversely, \"Children Books,\" \"Romance,\" and \"Test Preparation\" had high accuracies.","However, more than just classification accuracy, the purpose of this paper is to understand why the CNN's performed as such and reveal the relevant parts of the images.","Figure: CNN accuracy by genre."],["Layer-wise Relevance Propagation","The LRP algorithm and the LRP toolbox [31] aims to explain the reasoning behind the decision made by a network model which allows its users to validate classifier results.","LRP is mainly derived from Deep Taylor Decomposition [32], a method of decomposing network's output predictions onto its input variable.","The results after such a decomposition is visualized in the form of a heatmap highlighting each pixel's importance for the prediction.","LRP explains output function, i.e.","classifier's decision, which helps us to derive all of the crucial pixels for a particular prediction.","In Fig.","REF , the technique is shown in which the output value given by the network is decomposed backwards layer by layer until it reaches the input.","This backward decomposition of network's prediction uses local redistribution rules for assigning relevance values $R_i$ to each neuron contributing towards the output, namely $\\sum _{i} R_i = \\sum _{j} R_j =\\dots = \\sum _{k}R_k = f(x),$ where $f(x)$ is the prediction function, $R_i$ is the relevance of node $i$ in the target layer, $R_j$ is the relevance of node $j$ of the previous layer, and $R_k$ is the relevance of node $k$ of the highest layer.","The total amount of relevance is conserved in this equation.","Figure: Feed forward neural network with the (left) forward pass and the (right) backward relevance calculation.","The function f(x)f(x) is the prediction outcome given input xx.","The variables a i a_i and a j a_j are the inputs for node ii and jj, respectively.","R i R_i is the relevance of node ii and R j R_j is the relevance of node jj.For the experiment, we used the $\\alpha -\\beta $ decomposition formula defined by $R_i = \\sum _{j} \\left(\\alpha \\frac{(a_i w_{ij})^+}{\\sum _{i} ( a_i w_{ij}) ^ +} + \\beta \\frac{(a_i w_{ij})^-}{\\sum _{i} ( a_i w_{ij}) ^ -} \\right) R_j,$ where $\\alpha $ and $\\beta $ are hyperparameters to weight the positive values of $\\frac{(a_i w_{ij})}{\\sum _{i} ( a_i w_{ij})}$ and the negative values of $\\frac{(a_i w_{ij})}{\\sum _{i} ( a_i w_{ij})}$ , respectively.","Furthermore, $w_{ij}$ is the weight between nodes $i$ and $j$ and $a_i$ is the input to node $i$ .","This decomposition allows for the separation of the positive connections and the negative connections.","Values inside positive bracket indicates propagation of activating input messages while negative weight connections indicate deactivating input values."],["Single-Shot Multibox Detector","To develop a better understanding of the objects within book cover images, we employed SSD [8], a state-of-the-art deep neural network based object detection method.","SSD is a feed forward CNN which produces a multi-scale collection of fixed size bounding boxes and scores for object detection within the boxes.","A final non-maximal suppression step determines the final detections.","The result of SSD is bounding box regions with object classification labels.","Using SSD, it is possible to accurately detect multiple objects of different classes within images."],["Efficient and Accurate Scene Text Detector","For humans, text is an important component of book covers; it is where the title, authors, and additional information is conveyed.","However, a CNN may place a different importance on text than humans.","Thus, to analyze the relevance of text in book covers, we use EAST [9] as a text detector.","EAST uses a multi-channel Fully Convolutional Network (FCN) and non-maximal suppression on predicted geometric shapes to detect multi-orient text-line and word boxes."],["How CNNs Understand Book Cover Design: Qualitative Analysis","In this section, we have presented LRP results from main genres.","The analysis helped us to deduce book cover design elements contributing towards a prediction by CNN.","We used $\\alpha -\\beta $ decomposition formula with values of $\\alpha =2$ and $\\beta =-1$ which is suggested for networks using ReLU activation functions because it emphasizes the positive elements and de-emphasizes the negative ones [7].","This is important due to the ReLU activation function setting negative values to zero.","In the heatmaps generated by LRP under this decomposition, pixels adding positive contribution are represented in red color and the ones adding negative contribution are represented by blue color."],["Sports & Outdoors","Under this genre, many book covers with pictures of players playing indoor and outdoor games were seen.","Figure REF (a) shows LRP results on these covers, which presents significance of player's picture on the cover.","The first image in Fig.","REF (a) supports this fact with LRP being centered on players who are either playing a sport or showing some player like gesture, with car in background adding no contribution.","The second image in Fig.","REF (a) emphasizes the animal's importance for this genre's prediction.","Figure: Correctly recognized book covers.","Object classes by SSD and text by EAST are highlighted."],["Engineering & Transportation","For this genre, almost all the covers with vehicle pictures on their covers were classified correctly by the network.","With LRP in Fig.","REF (b), part of image containing cars or motorbikes seem to add more relevance than others.","The last image in the Fig.","REF (b) presents the cases when contribution of person image was dominated by vehicle in the image."],["Romance","Its obvious from the genre name that pictures of couples on the cover are going to have more relevance and LRP results showed this fact to be true.","However, among pictures presented in Fig.","REF (c), LRP depicted girls to add more relevance than men or other things.","The reason could reside in their physical appearance, hairs, and choice of dresses.","The same was demonstrated in last picture of Fig.","REF (c) in which girl's hair are seen to add more relevance with zero relevance coming from animal part on book cover."],["Children's Books","Almost all the children book covers contain pictures of cartoon characters.","LRP on covers from this genre showed these cartoon characters to have higher relevance.","An interesting result is shown in first picture of Fig.","REF (d), where person is depicted as an adversarial identity and importance of cartoons in cover is highlighted.","Some covers showed more relevance for one object in the set of objects.","Like, in Fig.","REF (d) some cartoons in last picture have higher relevance.","It can be because of the object placement and their orientations.","With the help of this information, one can make smart choices for different characters, cartoons and color patterns."],["Cookbooks, Food & Wine Books","Book covers in this genre most commonly contained pictures of different kinds of food.","The results in Fig.","REF (e) showed these food pictures as containment of higher relevance for this genre.","However, carefully analyzing LRP results we discovered shapes of dishes like bowls or spoons adding significant relevance for the genre's prediction.","So, this marks significance of dish shape designs on covers from this genre."],["Test Preparation","The genre contained covers with both text and pictorial information as shown in Fig.","REF (f).","With most contribution coming from big text content on covers.","Images of Fig.","REF (f) presents big texts to add more relevance than images of people.","In first image of Fig.","REF (f), despite of big girl face, relevance is concentrated on text area of book cover.","Such analysis helped us to find design elements specific to the presented genres.","To get more familiar with design, we also presented some cases where the network was not able to correctly classify the genre.","Figure REF shows some of these misclassifications, mainly from the presented genres.","The correct genre names are written below the image.","From the analysis presented above, one can easily decode the reason behind their misclassification because the designs on these book covers are not aligned with their genres which makes it obvious for network to mis-classify.","Here, cover from \"Sports & Outdoors\" contains birds, \"Romance\" cover contains text, \"Cookbooks, Food & Wine Books\" contain no food picture and \"Test Preparation\" cover is also without any significant feature.","LRP justifies all these covers misclassification by highlighting these mentioned objects contributing towards the \"other\" class in one-vs-others."],["Experiment Setup","In order to quantitatively analyze LRP, we propose using SSD to detect objects and EAST to detect text within the book cover images.","We then use LRP to compare the relevance of objects bound by the detection methods.","The SSD was trained on the 2012 PASCAL Visual Object Classes (VOC) Challenge dataset [33].","The VOC dataset contains 20 classes, including \"person,\" six animal classes, eight vehicle classes, and seven indoor object classes.","While SSD trained with VOC is intended for natural scene images, it can be used with book cover images because book covers often contain many of the shared classes, such as \"person\" and \"car.\"","Similarly, EAST was trained on the 2015 ICDAR Robust Reading Competition dataset [34] meant for scene text detection.","Despite being trained for scene text, shown in Fig.","REF , EAST performs remarkably well on book covers for detecting text.","To extract object and text bounding box information, the book covers were prepared by scaling the images to $512\\times 512$ pixels by 3 color channels.","It is important to note that the images used for SSD and EAST were larger than the images used by the CNN used for genre classification.","This is due to the detection methods being much more effective at the higher resolution.","To accommodate this, the bounding boxes were scaled post detection and projected onto the LRP heatmaps.","The relevance of an object $R_{\\mathrm {obj}}$ is calculated using the sum of the relevance within the bounding box, or $R_{\\mathrm {obj}} = \\sum _{(n,m)\\in \\mathcal {B}}{R_{(n,m)}},$ where $R_{(n,m)}$ is the relevance at pixel coordinates $(n,m)$ within bounding box $\\mathcal {B}$ .","Figure: “Misclassified” book covers with correct genre names written below each book cover."],["LRP with Object Detection","A macro view of the genres can be seen by viewing the average relevance of object classes.","Figure REF illustrates the average object-wise relevance of each object class as detected by SSD and EAST for each book genre using the test set book cover images.","It should be noted that detected objects such as \"bottle\" and \"tvmonitor\" were overfit to certain book cover images because many books have plain covers which resemble bottle labels or televisions.","However, this does not mean that the information is useless.","For example, from Fig.","REF , \"bottle\" is more relevant for reference and nonfiction genres where plain covers are common.","Figure: Average object-wise relevance for text detected by EAST and each object class detected by SSD for each book genre.","Only object-genre combinations with five or more data points are shown.In addition, by examining the distribution of the $R_{\\mathrm {obj}}$ of specific object classes, such as \"person,\" it is possible to create associations between genres and detected objects.","For example, the relevance of \"person\" $R_{\\mathrm {person}}$ for each genre is shown in Fig.","REF .","The figure demonstrates that detected \"person\"s within certain genres are more relevant than other genres.","For instance, the genres of \"Romance\" and \"Mystery, Thriller & Suspense\" put a high average relevance in \"person.\"","This indicates that \"person\" is important for the CNNs of those categories.","In addition, mentioned in Section REF and shown in Fig.","REF (f), people are common in \"Test Preparation\" but are not necessarily relevant.","This is supported by Fig.","REF which indicates that on average, \"person\" has very little relevance.","Distributions for the other object classes are provided in the supplemental material.","Figure: Box plot of relevance of \"person\" R person R_{\\mathrm {person}} for each genre.","The boxes represent the first through third quartile and the mean is in red.","The whiskers mark the minimum and maximum datum."],["LRP with Text Detection","Figure REF also reveals that the average relevance of text is low.","The reasoning behind this phenomenon can be explained by Fig.","REF .","The figure shows that the majority of the detected text boxes have a very small relevance $R_{\\mathrm {text}}$ , but there are some text boxes have a higher relevance.","For most genres, the title text contains a significant amount of relevance determined by LRP, but the small descriptive text carries very little relevance.","Figure REF (f) in particular demonstrates this with the large title text having a high relevance and much of the smaller descriptive text having near zero relevance.","Figure: Box plot of relevance of \"text\" R text R_{\\mathrm {text}} for each genre.","The boxes represent the first through third quartile and the mean is in red.","The whiskers mark the minimum and maximum datum."],["Conclusion","In this paper, we presented importance of design in book covers belonging to a specific genre.","The application of LRP on the book cover dataset showed genre specific book cover features.","The method described most relevant parts of input book cover contributing towards a genre prediction by CNN.","We also presented quantitative analysis of LRP using an object detection method, SSD, and a text detection method, EAST.","The analysis further demonstrates that genre classification heavily relies on specific objects for each genres."],["ACKNOWLEDGEMENT","This research was partially supported by MEXT-Japan (Grant No.J17H06100)."]],"string":"[\n [\n \"How do Convolutional Neural Networks Learn Design?\"\n ],\n [\n \"Abstract In this paper, we aim to understand the design principles in book cover images which are carefully crafted by experts.\",\n \"Book covers are designed in a unique way, specific to genres which convey important information to their readers.\",\n \"By using Convolutional Neural Networks (CNN) to predict book genres from cover images, visual cues which distinguish genres can be highlighted and analyzed.\",\n \"In order to understand these visual clues contributing towards the decision of a genre, we present the application of Layer-wise Relevance Propagation (LRP) on the book cover image classification results.\",\n \"We use LRP to explain the pixel-wise contributions of book cover design and highlight the design elements contributing towards particular genres.\",\n \"In addition, with the use of state-of-the-art object and text detection methods, insights about genre-specific book cover designs are discovered.\"\n ],\n [\n \"Introduction\",\n \"Visual design renders specific impressions to transmit information which enriches the product's value.\",\n \"However, these visual designs despite of being important are not analyzed objectively or statistically.\",\n \"Analyzing these visual designs enables us to understand the contained information carried by them.\",\n \"An interesting target of visual design analysis is book cover image design where the design of a book cover can infer the genre.\",\n \"Each book cover is carefully designed by typographers and their designs represent the book contents in an intuitive way for better sales.\",\n \"This association of books to specific genres is based on the differences in their underlying book cover designs [1].\",\n \"The slight change in book cover design can reflect changes in book genre which makes design learning a challenging task for book covers.\",\n \"In order to understand the design elements used for machine aided book cover classification, we employ Convolutional Neural Networks (CNN) [2].\",\n \"In recent years, CNNs have achieved state-of-the-art results in isolated character recognition [3], [4] and large-scale image recognition [5], [6].\",\n \"Notably, Iwana et al.\",\n \"[1] demonstrated that CNNs can be used for genre classification based on book cover image, although with a high level of difficulty.\",\n \"However, that study was subjective and not enough explanation is given as to why the CNN performed as it did.\",\n \"To interpret the reasoning behind a CNN's prediction we used a method called Layer-wise Relevance Propagation (LRP) [7].\",\n \"LRP decomposes output function on its input variables and highlights input pixels contributing towards the network decision.\",\n \"It produces a layer-wise relevance heatmap by recursively multiplying the relevance of higher layers by the normalized feature maps of the target layer.\",\n \"The heatmaps can help us to discover the input image elements which have an effect on the classification result.\",\n \"The main contributions of this paper are threefold.\",\n \"Firstly, we classified the book cover images using one-vs-others classification with CNNs.\",\n \"Secondly, the models built by the CNNs are analyzed using LRP.\",\n \"With LRP, we demonstrate design elements specifically relevant to classification of the book cover images.\",\n \"We show that certain objects have a strong relevance to particular genres.\",\n \"Finally, we use state-of-the-art object detection and text detection methods, namely Single Shot Multibox Detector (SSD) [8] and Efficient and Accurate Scene Text Detector [9], to quantitatively enforce the results found by LRP.\",\n \"This reveals the specific elements in which CNNs classify book cover images for genre classification.\",\n \"The organization is as follows.\",\n \"Section provides related works in design understanding and genre classification as well as feature visualization of CNNs.\",\n \"Section reviews the data and tools used for understanding book cover design.\",\n \"Section presents analysis of CNN's understanding of book cover design.\",\n \"In Section , we demonstrate the use of LRP combined with SSD and EAST for quantitative analysis.\",\n \"Finally, Section draws a conclusion.\",\n \"Artistic style understanding and subjective genre classification is a budding field in machine learning.\",\n \"For example, recent attempts have been done to identify artistic styles and quality of paintings and photographs [10], [11] with neural network models.\",\n \"In addition, there have been trials to classify music by genre [12], [13], book covers by genre [1], movie posters by genre [14], paintings by genre [15], and text by genre [16], [17].\",\n \"Also, in a general sense, document classification can be considered genre classification and deep CNNs are the state-of-the-art in the document classification domain [18], [19], [20].\"\n ],\n [\n \"Visualization inside of CNNs\",\n \"There is a desire to visualize features and determine pixel-wise attention and relevance within the hidden layers of CNNs.\",\n \"However, this is a not a straightforward task [21].\",\n \"Erhan et al.\",\n \"[21] proposed using gradient decent to maximize a node's activation to visualize the employed features.\",\n \"Similar work has been done for large-scale image classification [22].\",\n \"Zeiler and Fergus [23] used deconvolutional neural networks to visualize features learned by CNNs.\",\n \"In addition, they created heatmaps by monitoring class changes systematic cover up of portions of the images.\",\n \"Class Activation Maps (CAM) [24], GradCAM [25], and GradCAM++ [26] reveal the parts of images which are most important to a class using global average pooling (GAP).\",\n \"Recently, LRP has been used in the fields of text [27] where classification scores were projected back to input features for extracting relevant words for a specific prediction.\",\n \"The method has also shown successes in model understanding in fields of sentiment analysis [28], action recognition [29], and age and gender classification [30].\",\n \"As far as the authors are aware, this is the first time LRP has been used for the understanding of genre or design classification.\"\n ],\n [\n \"Amazon Book Cover Dataset\",\n \"We used the Book Cover Image to Genre datasethttps://github.com/uchidalab/book-dataset Task 1.A.\",\n \"The dataset consists of 57,000 book cover images divided into 30 classes of equal sizes.\",\n \"In the experiments, we used the predefined training set and test set modified for one-vs-others classification.\",\n \"In this way, genre-wise training sets were prepared with an equal distribution of positive and negative data samples.\"\n ],\n [\n \"Convolution Neural Networks\",\n \"CNNs are able to tackle image recognition by implementing convolutions of learned filter-like shared weights which maintain the structural qualities of images while acting as feature extractors [2].\",\n \"For the experiment, we implement CNNs to tackle book genre classification.\",\n \"To use the book cover images with a CNN, they were preprocessed by scaling them to 112$\\\\times $ 112 pixels by 3 color channel images and by normalizing the values to be between -1 and 1.\",\n \"The CNN used for the experiments has six convolutional layers with Rectified Linear Units (ReLU) activations and a softmax output layer.\",\n \"The convolutional layers consisted of three layers of 10 nodes with $5\\\\times 5$ convolutional filters, one layer of 25 nodes with a $4\\\\times 4$ filter, one layer of 50 nodes with a $3\\\\times 3$ filter, and one layer of 100 nodes with a $1\\\\times 1$ filter.\",\n \"A $2\\\\times 2$ maxpooling layer with stride 2 was used between each convolution layer.\",\n \"Finally, the CNNs were trained using gradient decent with a batch size of 25 at a learning rate of 0.001 for 50,000 iterations.\",\n \"The accuracy results for each genre is summarized in Fig.\",\n \"REF .\",\n \"In particular, the CNNs had difficulties with the reference classes, such as \\\"Engineering & Transportation,\\\" \\\"Health, Fitness and Dieting,\\\" \\\"History,\\\" \\\"Medical Books,\\\" and \\\"Reference.\\\"\",\n \"Conversely, \\\"Children Books,\\\" \\\"Romance,\\\" and \\\"Test Preparation\\\" had high accuracies.\",\n \"However, more than just classification accuracy, the purpose of this paper is to understand why the CNN's performed as such and reveal the relevant parts of the images.\",\n \"Figure: CNN accuracy by genre.\"\n ],\n [\n \"Layer-wise Relevance Propagation\",\n \"The LRP algorithm and the LRP toolbox [31] aims to explain the reasoning behind the decision made by a network model which allows its users to validate classifier results.\",\n \"LRP is mainly derived from Deep Taylor Decomposition [32], a method of decomposing network's output predictions onto its input variable.\",\n \"The results after such a decomposition is visualized in the form of a heatmap highlighting each pixel's importance for the prediction.\",\n \"LRP explains output function, i.e.\",\n \"classifier's decision, which helps us to derive all of the crucial pixels for a particular prediction.\",\n \"In Fig.\",\n \"REF , the technique is shown in which the output value given by the network is decomposed backwards layer by layer until it reaches the input.\",\n \"This backward decomposition of network's prediction uses local redistribution rules for assigning relevance values $R_i$ to each neuron contributing towards the output, namely $\\\\sum _{i} R_i = \\\\sum _{j} R_j =\\\\dots = \\\\sum _{k}R_k = f(x),$ where $f(x)$ is the prediction function, $R_i$ is the relevance of node $i$ in the target layer, $R_j$ is the relevance of node $j$ of the previous layer, and $R_k$ is the relevance of node $k$ of the highest layer.\",\n \"The total amount of relevance is conserved in this equation.\",\n \"Figure: Feed forward neural network with the (left) forward pass and the (right) backward relevance calculation.\",\n \"The function f(x)f(x) is the prediction outcome given input xx.\",\n \"The variables a i a_i and a j a_j are the inputs for node ii and jj, respectively.\",\n \"R i R_i is the relevance of node ii and R j R_j is the relevance of node jj.For the experiment, we used the $\\\\alpha -\\\\beta $ decomposition formula defined by $R_i = \\\\sum _{j} \\\\left(\\\\alpha \\\\frac{(a_i w_{ij})^+}{\\\\sum _{i} ( a_i w_{ij}) ^ +} + \\\\beta \\\\frac{(a_i w_{ij})^-}{\\\\sum _{i} ( a_i w_{ij}) ^ -} \\\\right) R_j,$ where $\\\\alpha $ and $\\\\beta $ are hyperparameters to weight the positive values of $\\\\frac{(a_i w_{ij})}{\\\\sum _{i} ( a_i w_{ij})}$ and the negative values of $\\\\frac{(a_i w_{ij})}{\\\\sum _{i} ( a_i w_{ij})}$ , respectively.\",\n \"Furthermore, $w_{ij}$ is the weight between nodes $i$ and $j$ and $a_i$ is the input to node $i$ .\",\n \"This decomposition allows for the separation of the positive connections and the negative connections.\",\n \"Values inside positive bracket indicates propagation of activating input messages while negative weight connections indicate deactivating input values.\"\n ],\n [\n \"Single-Shot Multibox Detector\",\n \"To develop a better understanding of the objects within book cover images, we employed SSD [8], a state-of-the-art deep neural network based object detection method.\",\n \"SSD is a feed forward CNN which produces a multi-scale collection of fixed size bounding boxes and scores for object detection within the boxes.\",\n \"A final non-maximal suppression step determines the final detections.\",\n \"The result of SSD is bounding box regions with object classification labels.\",\n \"Using SSD, it is possible to accurately detect multiple objects of different classes within images.\"\n ],\n [\n \"Efficient and Accurate Scene Text Detector\",\n \"For humans, text is an important component of book covers; it is where the title, authors, and additional information is conveyed.\",\n \"However, a CNN may place a different importance on text than humans.\",\n \"Thus, to analyze the relevance of text in book covers, we use EAST [9] as a text detector.\",\n \"EAST uses a multi-channel Fully Convolutional Network (FCN) and non-maximal suppression on predicted geometric shapes to detect multi-orient text-line and word boxes.\"\n ],\n [\n \"How CNNs Understand Book Cover Design: Qualitative Analysis\",\n \"In this section, we have presented LRP results from main genres.\",\n \"The analysis helped us to deduce book cover design elements contributing towards a prediction by CNN.\",\n \"We used $\\\\alpha -\\\\beta $ decomposition formula with values of $\\\\alpha =2$ and $\\\\beta =-1$ which is suggested for networks using ReLU activation functions because it emphasizes the positive elements and de-emphasizes the negative ones [7].\",\n \"This is important due to the ReLU activation function setting negative values to zero.\",\n \"In the heatmaps generated by LRP under this decomposition, pixels adding positive contribution are represented in red color and the ones adding negative contribution are represented by blue color.\"\n ],\n [\n \"Sports & Outdoors\",\n \"Under this genre, many book covers with pictures of players playing indoor and outdoor games were seen.\",\n \"Figure REF (a) shows LRP results on these covers, which presents significance of player's picture on the cover.\",\n \"The first image in Fig.\",\n \"REF (a) supports this fact with LRP being centered on players who are either playing a sport or showing some player like gesture, with car in background adding no contribution.\",\n \"The second image in Fig.\",\n \"REF (a) emphasizes the animal's importance for this genre's prediction.\",\n \"Figure: Correctly recognized book covers.\",\n \"Object classes by SSD and text by EAST are highlighted.\"\n ],\n [\n \"Engineering & Transportation\",\n \"For this genre, almost all the covers with vehicle pictures on their covers were classified correctly by the network.\",\n \"With LRP in Fig.\",\n \"REF (b), part of image containing cars or motorbikes seem to add more relevance than others.\",\n \"The last image in the Fig.\",\n \"REF (b) presents the cases when contribution of person image was dominated by vehicle in the image.\"\n ],\n [\n \"Romance\",\n \"Its obvious from the genre name that pictures of couples on the cover are going to have more relevance and LRP results showed this fact to be true.\",\n \"However, among pictures presented in Fig.\",\n \"REF (c), LRP depicted girls to add more relevance than men or other things.\",\n \"The reason could reside in their physical appearance, hairs, and choice of dresses.\",\n \"The same was demonstrated in last picture of Fig.\",\n \"REF (c) in which girl's hair are seen to add more relevance with zero relevance coming from animal part on book cover.\"\n ],\n [\n \"Children's Books\",\n \"Almost all the children book covers contain pictures of cartoon characters.\",\n \"LRP on covers from this genre showed these cartoon characters to have higher relevance.\",\n \"An interesting result is shown in first picture of Fig.\",\n \"REF (d), where person is depicted as an adversarial identity and importance of cartoons in cover is highlighted.\",\n \"Some covers showed more relevance for one object in the set of objects.\",\n \"Like, in Fig.\",\n \"REF (d) some cartoons in last picture have higher relevance.\",\n \"It can be because of the object placement and their orientations.\",\n \"With the help of this information, one can make smart choices for different characters, cartoons and color patterns.\"\n ],\n [\n \"Cookbooks, Food & Wine Books\",\n \"Book covers in this genre most commonly contained pictures of different kinds of food.\",\n \"The results in Fig.\",\n \"REF (e) showed these food pictures as containment of higher relevance for this genre.\",\n \"However, carefully analyzing LRP results we discovered shapes of dishes like bowls or spoons adding significant relevance for the genre's prediction.\",\n \"So, this marks significance of dish shape designs on covers from this genre.\"\n ],\n [\n \"Test Preparation\",\n \"The genre contained covers with both text and pictorial information as shown in Fig.\",\n \"REF (f).\",\n \"With most contribution coming from big text content on covers.\",\n \"Images of Fig.\",\n \"REF (f) presents big texts to add more relevance than images of people.\",\n \"In first image of Fig.\",\n \"REF (f), despite of big girl face, relevance is concentrated on text area of book cover.\",\n \"Such analysis helped us to find design elements specific to the presented genres.\",\n \"To get more familiar with design, we also presented some cases where the network was not able to correctly classify the genre.\",\n \"Figure REF shows some of these misclassifications, mainly from the presented genres.\",\n \"The correct genre names are written below the image.\",\n \"From the analysis presented above, one can easily decode the reason behind their misclassification because the designs on these book covers are not aligned with their genres which makes it obvious for network to mis-classify.\",\n \"Here, cover from \\\"Sports & Outdoors\\\" contains birds, \\\"Romance\\\" cover contains text, \\\"Cookbooks, Food & Wine Books\\\" contain no food picture and \\\"Test Preparation\\\" cover is also without any significant feature.\",\n \"LRP justifies all these covers misclassification by highlighting these mentioned objects contributing towards the \\\"other\\\" class in one-vs-others.\"\n ],\n [\n \"Experiment Setup\",\n \"In order to quantitatively analyze LRP, we propose using SSD to detect objects and EAST to detect text within the book cover images.\",\n \"We then use LRP to compare the relevance of objects bound by the detection methods.\",\n \"The SSD was trained on the 2012 PASCAL Visual Object Classes (VOC) Challenge dataset [33].\",\n \"The VOC dataset contains 20 classes, including \\\"person,\\\" six animal classes, eight vehicle classes, and seven indoor object classes.\",\n \"While SSD trained with VOC is intended for natural scene images, it can be used with book cover images because book covers often contain many of the shared classes, such as \\\"person\\\" and \\\"car.\\\"\",\n \"Similarly, EAST was trained on the 2015 ICDAR Robust Reading Competition dataset [34] meant for scene text detection.\",\n \"Despite being trained for scene text, shown in Fig.\",\n \"REF , EAST performs remarkably well on book covers for detecting text.\",\n \"To extract object and text bounding box information, the book covers were prepared by scaling the images to $512\\\\times 512$ pixels by 3 color channels.\",\n \"It is important to note that the images used for SSD and EAST were larger than the images used by the CNN used for genre classification.\",\n \"This is due to the detection methods being much more effective at the higher resolution.\",\n \"To accommodate this, the bounding boxes were scaled post detection and projected onto the LRP heatmaps.\",\n \"The relevance of an object $R_{\\\\mathrm {obj}}$ is calculated using the sum of the relevance within the bounding box, or $R_{\\\\mathrm {obj}} = \\\\sum _{(n,m)\\\\in \\\\mathcal {B}}{R_{(n,m)}},$ where $R_{(n,m)}$ is the relevance at pixel coordinates $(n,m)$ within bounding box $\\\\mathcal {B}$ .\",\n \"Figure: “Misclassified” book covers with correct genre names written below each book cover.\"\n ],\n [\n \"LRP with Object Detection\",\n \"A macro view of the genres can be seen by viewing the average relevance of object classes.\",\n \"Figure REF illustrates the average object-wise relevance of each object class as detected by SSD and EAST for each book genre using the test set book cover images.\",\n \"It should be noted that detected objects such as \\\"bottle\\\" and \\\"tvmonitor\\\" were overfit to certain book cover images because many books have plain covers which resemble bottle labels or televisions.\",\n \"However, this does not mean that the information is useless.\",\n \"For example, from Fig.\",\n \"REF , \\\"bottle\\\" is more relevant for reference and nonfiction genres where plain covers are common.\",\n \"Figure: Average object-wise relevance for text detected by EAST and each object class detected by SSD for each book genre.\",\n \"Only object-genre combinations with five or more data points are shown.In addition, by examining the distribution of the $R_{\\\\mathrm {obj}}$ of specific object classes, such as \\\"person,\\\" it is possible to create associations between genres and detected objects.\",\n \"For example, the relevance of \\\"person\\\" $R_{\\\\mathrm {person}}$ for each genre is shown in Fig.\",\n \"REF .\",\n \"The figure demonstrates that detected \\\"person\\\"s within certain genres are more relevant than other genres.\",\n \"For instance, the genres of \\\"Romance\\\" and \\\"Mystery, Thriller & Suspense\\\" put a high average relevance in \\\"person.\\\"\",\n \"This indicates that \\\"person\\\" is important for the CNNs of those categories.\",\n \"In addition, mentioned in Section REF and shown in Fig.\",\n \"REF (f), people are common in \\\"Test Preparation\\\" but are not necessarily relevant.\",\n \"This is supported by Fig.\",\n \"REF which indicates that on average, \\\"person\\\" has very little relevance.\",\n \"Distributions for the other object classes are provided in the supplemental material.\",\n \"Figure: Box plot of relevance of \\\"person\\\" R person R_{\\\\mathrm {person}} for each genre.\",\n \"The boxes represent the first through third quartile and the mean is in red.\",\n \"The whiskers mark the minimum and maximum datum.\"\n ],\n [\n \"LRP with Text Detection\",\n \"Figure REF also reveals that the average relevance of text is low.\",\n \"The reasoning behind this phenomenon can be explained by Fig.\",\n \"REF .\",\n \"The figure shows that the majority of the detected text boxes have a very small relevance $R_{\\\\mathrm {text}}$ , but there are some text boxes have a higher relevance.\",\n \"For most genres, the title text contains a significant amount of relevance determined by LRP, but the small descriptive text carries very little relevance.\",\n \"Figure REF (f) in particular demonstrates this with the large title text having a high relevance and much of the smaller descriptive text having near zero relevance.\",\n \"Figure: Box plot of relevance of \\\"text\\\" R text R_{\\\\mathrm {text}} for each genre.\",\n \"The boxes represent the first through third quartile and the mean is in red.\",\n \"The whiskers mark the minimum and maximum datum.\"\n ],\n [\n \"Conclusion\",\n \"In this paper, we presented importance of design in book covers belonging to a specific genre.\",\n \"The application of LRP on the book cover dataset showed genre specific book cover features.\",\n \"The method described most relevant parts of input book cover contributing towards a genre prediction by CNN.\",\n \"We also presented quantitative analysis of LRP using an object detection method, SSD, and a text detection method, EAST.\",\n \"The analysis further demonstrates that genre classification heavily relies on specific objects for each genres.\"\n ],\n [\n \"ACKNOWLEDGEMENT\",\n \"This research was partially supported by MEXT-Japan (Grant No.J17H06100).\"\n ]\n]","isConversational":false},"id":{"kind":"string","value":"1808.08402"}}},{"rowIdx":198,"cells":{"text":{"kind":"list like","value":[["Towards Tight Approximation Bounds for Graph Diameter and Eccentricities"],["Abstract Among the most important graph parameters is the Diameter, the largest distance between any two vertices.","There are no known very efficient algorithms for computing the Diameter exactly.","Thus, much research has been devoted to how fast this parameter can be approximated.","Chechik et al.","showed that the diameter can be approximated within a multiplicative factor of $3/2$ in $\\tilde{O}(m^{3/2})$ time.","Furthermore, Roditty and Vassilevska W. showed that unless the Strong Exponential Time Hypothesis (SETH) fails, no $O(n^{2-\\epsilon})$ time algorithm can achieve an approximation factor better than $3/2$ in sparse graphs.","Thus the above algorithm is essentially optimal for sparse graphs for approximation factors less than $3/2$.","It was, however, completely plausible that a $3/2$-approximation is possible in linear time.","In this work we conditionally rule out such a possibility by showing that unless SETH fails no $O(m^{3/2-\\epsilon})$ time algorithm can achieve an approximation factor better than $5/3$.","Another fundamental set of graph parameters are the Eccentricities.","The Eccentricity of a vertex $v$ is the distance between $v$ and the farthest vertex from $v$.","Chechik et al.","showed that the Eccentricities of all vertices can be approximated within a factor of $5/3$ in $\\tilde{O}(m^{3/2})$ time and Abboud et al.","showed that no $O(n^{2-\\epsilon})$ algorithm can achieve better than $5/3$ approximation in sparse graphs.","We show that the runtime of the $5/3$ approximation algorithm is also optimal under SETH.","We also show that no near-linear time algorithm can achieve a better than $2$ approximation for the Eccentricities and that this is essentially tight: we give an algorithm that approximates Eccentricities within a $2+\\delta$ factor in $\\tilde{O}(m/\\delta)$ time for any $0<\\delta<1$.","This beats all Eccentricity algorithms in Cairo et al."],["Introduction","Among the most important graph parameters are the graph's Diameter and the Eccentricities of its vertices.","The Eccentricity of a vertex $v$ is the (shortest path) distance to the furthest vertex from $v$ , and the Diameter is the largest Eccentricity over all vertices in the graph.","The Eccentricities and Diameter measure how fast information can spread in networks.","Efficient algorithms for their computation are highly desired (see e.g.","[40], [9], [33]).","Unfortunately, the fastest known algorithms for these parameters are very slow on large graphs.","For unweighted graphs on $n$ vertices and $m$ edges, the fastest Diameter algorithm runs in $\\tilde{O}(\\min \\lbrace mn,n^\\omega \\rbrace )$ $\\tilde{O}$ notation hides polylogarithmic factors time [16] where $\\omega <2.373$ is the exponent of square matrix multiplication [52], [32], [45].","For weighted graphs, the fastest Eccentricity and Diameter algorithms actually compute all distances in the graph, i.e.","they solve the All-Pairs Shortest Paths (APSP) problem.","The fastest known algorithms for APSP in weighted graphs run in $\\min \\lbrace \\tilde{O}(mn),n^3/\\exp (\\sqrt{\\log n})\\rbrace $ [53], [35], [37].","Whether one can solve Diameter faster than APSP is a well-known open problem (e.g.","see Problem 6.1 in [20] and [2], [17]).","Whether one can solve Eccentricities faster than APSP was addressed by [50] (for dense graphs) and by [34] (for sparse graphs).","Vassilevska W. and Williams [50] showed that Eccentricities and APSP are equivalent under subcubic reductions, so that either both of them admit $O(n^{3-\\varepsilon })$ time algorithms for $\\varepsilon >0$ , or neither of them do.","Lincoln et al.","[34] proved that under a popular conjecture about the complexity of weighted Clique, the $O(mn)$ runtime for Eccentricities cannot be beaten by any polynomial factor for any sparsity of the form $m=\\Theta (n^{1+1/k})$ for integer $k$ .","Due to the hardness of exact computation, efficient approximation algorithms are sought.","A folklore $\\tilde{O}(m+n)$ time algorithm achieves a 2-approximation for Diameter in directed weighted graphs and a 3-approximation for Eccentricities in undirected weighted graphs.","Aingworth et al.","[2] presented an almost-$3/2$ approximation An almost-$c$ approximation of $X$ is an estimate $X^{\\prime }$ so that $X\\le X^{\\prime }\\le cX+O(1)$ .","algorithm for Diameter running in $\\tilde{O}(n^2+m\\sqrt{n})$ time.","Roditty and Vassilevska W. [41] improved the result of [2] with an $\\tilde{O}(m\\sqrt{n})$ expected time almost-$3/2$ approximation algorithm.","Chechik et al.","[21] obtained a (genuine) $3/2$ approximation algorithm for Diameter (in directed graphs) and a (genuine) $5/3$ -approximation algorithm for Eccentricities (in undirected graphs), running in $\\tilde{O}(\\min \\lbrace m^{3/2},mn^{2/3}\\rbrace )$ time.","These are the only known non-trivial approximation algorithms for Diameter in directed graphs.","So far, there are no known faster than $mn$ algorithms for approximating Eccentrities in directed graphs within any constant factor.","Cairo et al.","[15] generalized the above results for undirected graphs and obtained a time-approximation tradeoff: for every $k\\ge 1$ they obtained an $\\tilde{O}(mn^{1/(k+1)})$ time algorithm that achieves an almost-$2-1/2^k$ approximation for Diameter and an almost $3-4/(2^k+1)$ -approximation for Eccentricities."],["Our contributions.","We address the following natural question: Main Question: Are the known approximation algorithms for Diameter and Eccentricities optimal?","A partial answer is known.","Under the Strong Exponential Time Hypothesis (SETH), every $3/2-\\varepsilon $ approximation algorithm (for $\\varepsilon >0$ ) for Diameter in undirected unweighted graphs with $O(n)$ nodes and edges must use $n^{2-o(1)}$ time [41].","Similarly, every $5/3-\\varepsilon $ approximation algorithm for the Eccentricities of undirected unweighted graphs with $O(n)$ nodes and edges must use use $n^{2-o(1)}$ time [7].","This however does not answer the question of whether the runtimes of the known $3/2$ and $5/3$ approximation algorithms can be improved.","It is completely plausible that there is a $3/2$ -approximation algorithm for Diameter or a $5/3$ -approximation for Eccentricities running in linear time.","We address our Main Question for both sparse and dense graphs.","Our results are shown in Table REF .","Table: Our results.","All of the lower bounds hold even for sparse graphs.","SS-TT Diameter is a variant of Diameter introduced later in this section."],["Sparse graphs.","Our first result (restated as Theorem REF ) regards approximating Diameter in undirected unweighted sparse graphs.","Theorem 1 ($3/2$ -Diameter Approx.","is Tight) Under SETH, no $O(n^{3/2-\\delta })$ time algorithm for $\\delta >0$ can output a $8/5-\\varepsilon $ approximation for $\\varepsilon >0$ for the Diameter of an undirected unweighted sparse graph.","In particular, any $3/2$ -approximation algorithm in sparse graphs must take $n^{3/2-o(1)}$ time.","Hence the $\\tilde{O}(m^{3/2})$ time $3/2$ -approximation algorithm of [41], [21] is optimal in two ways: improving the approximation ratio to $3/2-\\varepsilon $ causes a runtime blow-up to $n^{2-o(1)}$ ([41]) and improving the runtime to $O(m^{3/2-\\delta })$ causes an approximation ratio blow-up to $8/5$ .","Our lower bound instance says that in $O(m^{3/2-\\delta })$ time one cannot return 6 when the Diameter is 8.","One may be tempted to extend the above lower bound, by showing that, say, in $O(m^{4/3-\\delta })$ time one cannot even return 5 when the Diameter is 8.","This approach, however fails: in Theorem REF we give an $O(m^2/n)$ time algorithm that does return 5 in this case, and in general when the Diameter is $2h$ , it returns at least $h+1$ .","Notice that when the Diameter is $2h$ , the folklore linear time algorithm returns an estimate of only $h$ .","Hence for sparse graphs, our algorithm runs in linear time and outperforms the folklore algorithm.","Also, for constant even Diameter, it gives a better than 2 approximation.","We obtain stronger Diameter hardness results for weighted graphs and for directed unweighted graphs.","In particular, assuming SETH: For weighted sparse graphs, no $O(n^{3/2-\\delta })$ time algorithm for $\\delta >0$ can output a $5/3-\\varepsilon $ Diameter approximation (for $\\varepsilon >0$ ) (Theorem REF ).","For directed unweighted sparse graphs, using a general time-accuracy tradeoff lower bound (Theorem REF ), we show that no near-linear time algorithm can achieve an approximation factor better than $5/3$ .","We summarize our Diameter lower bounds and compare them to the known upper bounds in Figure REF .","Figure: Our hardness results for Diameter.","The xx-axis is the approximation factor and the yy-axis is the runtime exponent.","Black lines represent lower bounds.","Black dots represent existing algorithms.","Blue dots represent existing algorithms whose approximation is potentially off by an additive term (the algorithms of ).","Transparent dots represent algorithms that might exist and would be tight with our lower bounds.We address our Main Question for Eccentricities as well.","Our main result for Eccentricities is Theorem REF .","Its first consequence is as follows: Theorem 2 ($5/3$ -Eccentricities Alg.","is Tight) Under SETH, no $O(n^{3/2-\\delta })$ time algorithm for $\\delta >0$ can output a $9/5-\\varepsilon $ approximation for $\\varepsilon >0$ for the Eccentricities of an undirected unweighted sparse graph.","In other words, the $\\tilde{O}(m^{3/2})$ time $5/3$ -approximation algorithm of [41], [21] is tight in two ways.","Improving the approximation ratio to $5/3-\\varepsilon $ causes a runtime blow-up to $n^{2-o(1)}$ ([7]) and improving the runtime to $O(m^{3/2-\\delta })$ causes an approximation ratio blow-up to $9/5$ .","More generally, we prove (in Theorem REF ): for every $k\\ge 2$ , under SETH, distinguishing between Eccentricities $2k-1$ and $4k-3$ in unweighted undirected sparse graphs requires $n^{1+1/(k-1)-o(1)}$ time.","Thus, no near-linear time algorithm can achieve a $2-\\varepsilon $ -approximation for Eccentricities for $\\varepsilon >0$ .","The best (folklore) near-linear time approximation algorithm for Eccentricities currently only achieves a 3-approximation, and only in undirected graphs.","There is no known constant factor approximation algorithm for directed graphs!","Is our limitation result for linear time Eccentricity algorithms far from the truth?","We show that our lower bound result is essentially tight, for both directed and undirected graphs by producing the first non-trivial near-linear time approximation algorithm for the Eccentricities in weighted directed graphs (Theorem REF ).","Theorem 3 (2-Approx.","for Eccentricities in near-linear time.)","Under SETH, no $n^{1+o(1)}$ time algorithm can output a $2-\\varepsilon $ approximation for $\\varepsilon >0$ for the Eccentricities of an undirected unweighted sparse graph.","For every $\\delta >0$ , there is an $\\tilde{O}(m/\\delta )$ time algorithm that produces a $(2+\\delta )$ -approximation for the Eccentricities of any directed weighted graph.","The approximation hardness result is the first result within fine-grained complexity that gives tight hardness for near linear time algorithms.","The $2+\\delta $ approximation ratio that our algorithm produces beats all approximation ratios for Eccentricities given by Cairo et al. [15].","It also constitutes the first known constant factor approximation algorithm for Eccentricities in directed graphs.","Our approximation algorithm also implies as a corollary an approximation algorithm for the Source Radius problemThe Source Radius problem is a natural extension of the undirected Radius definition.","The goal is to return $\\min _x\\max _vd(x,v)$ .","studied in [7] with the same runtime and approximation factor ($2+\\delta $ ).","Abboud et al.","[7] showed that, under the Hitting Set Conjecture, any $(2-\\varepsilon )$ -approximation algorithm (for $\\varepsilon >0$ ) for Source Radius requires $n^{2-o(1)}$ time, and hence our Source Radius algorithm is also essentially tight.","Our lower bound in Theorem REF holds already for undirected unweighted graphs, and the upper bound works even for directed weighted graphs.","The algorithm produces a $(2+\\delta )$ -approximation, which while close, is not quite a 2-approximation.","We design (in Theorem REF ) a genuine 2-approximation algorithm running in $\\tilde{O}(m\\sqrt{n})$ time that also works for directed weighted graphs.","We then complement it (in Theorem REF ) with a tight lower bound under SETH: in sparse directed graphs, if you go below factor 2 in the accuracy, the runtime blows up to quadratic.","Theorem 4 (Tight 2-Approx.","for Eccentricities) Under SETH, no $n^{2-\\delta }$ time algorithm for $\\delta >0$ can output a $2-\\varepsilon $ approximation for the Eccentricities of a directed unweighted sparse graph.","There is an $\\tilde{O}(m\\sqrt{n})$ time algorithm that produces a 2-approximation for the Eccentricities of any directed weighted graph.","We thus give an essentially complete answer to our Main Question for Eccentricities.","Our results are summarized in Figures REF and REF .","Figure: Our algorithms and hardness results for Eccentricities.","The lower bounds are for unweighted graphs and the upper bounds are for weighted graphs.","The xx-axis is the approximation factor and the yy-axis is the runtime exponent.","Black lines represent lower bounds.","Black dots represent existing algorithms (including our algorithm at (2,3/2)(2,3/2) in figure b).","Blue dots represent existing algorithms whose position may not be exactly as it appears in the figure.","Here, the blue dots represent our (2+δ)(2+\\delta )-approximation algorithm running in O ˜(m/δ)\\tilde{O}(m/\\delta ) time.","Transparent dots represent algorithms that might exist and would be tight with our lower bounds.Our conditional lower bounds for both Diameter and Eccentricities are all based on a common construction: a conditional lower bound for a problem called $S$ -$T$ Diameter.","In $S$ -$T$ Diameter, the input is a graph $G=(V,E)$ and two subsets $S,T\\subseteq V$ , not necessarily disjoint, and the output is $D_{S,T}:=\\max _{s\\in S,t\\in T} d(s,t)$ .","$S$ -$T$ Diameter is a problem of independent interest.","It is related to the bichromatic furthest pair problem studied in geometry (e.g.","as in [29]), but for graphs (if we set $T=V\\setminus S$ ).","It is easy to see that if one can compute the $S$ -$T$ Diameter, then one can also compute the Diameter in the same time: just set $S=T=V$ .","We show that actually, when it comes to exact computation, the $S$ -$T$ Diameter and Diameter in weighted graphs are computationally equivalent (Theorem REF ).","We show that $S$ -$T$ Diameter also has similar approximation algorithms to Diameter.","We give a 3-approximation running in linear time (Claim REF based on the folklore Diameter 2-approximation algorithm), and a 2-approximation running in $\\tilde{O}(m^{3/2})$ time (Theorem REF based on the $3/2$ -approximation algorithm of [41], [21]).","We prove the following lower bound for $S$ -$T$ Diameter (restated as Theorem REF ), the proof of which is the starting point for all of our conditional lower bounds.","Theorem 5 Under SETH, for every $k\\ge 2$ , every algorithm that can distinguish between $S$ -$T$ Diameter $k$ and $3k-2$ in undirected unweighted graphs requires $n^{1+1/(k-1)-o(1)}$ time.","Theorem REF implies that under SETH, our aforementioned 2 and 3-approximation algorithms are optimal.","For all of our lower bounds, we also address the question of whether they can be extended to higher values of Diameter and Eccentricities.","All of our lower bounds, with the exception of directed Eccentricities, are of the form “any algorithm that can distinguish between Diameter (or Eccentricity) $a$ and $b$ requires a certain amount of time\" for small values of $a$ and $b$ .","This doesn't exclude the possibility of an algorithm that distinguishes between higher Diameters (or Eccentricities) of the same ratio i.e.","between $a\\ell $ and $b\\ell $ for some $\\ell $ .","For weighted Diameter, our lower bound easily extends to higher values of Diameter by simply scaling up the edge weights.","For $S-T$ Diameter and undirected Eccentricities, our lower bounds easily extend to higher values of Diameter and Eccentricities by simply subdividing the edges.","For unweighted directed Diameter, our lower bound extends to higher values of Diameter with a slight loss in approximation factor by subdividing some of the edges.","For unweighted undirected Diameter, our lower bound does not seem to easily extend to higher values of Diameter."],["Dense graphs.","Can we address our Main Question for dense graphs as well?","In particular, can we extend our runtime lower bounds of the form $n^{1+1/\\ell -o(1)}$ to $mn^{1/\\ell -o(1)}$ , thus matching the known algorithms for larger values of $m$ ?","We show that the answer is “no”.","For undirected unweighted graphs, we obtain $\\tilde{O}(n^2)$ time algorithms for Diameter achieving an almost $3/2$ -approximation (Theorem REF ), and for all Eccentricities achieving an almost $5/3$ -approximation algorithm (Theorem REF ).","These algorithms run in near-linear time in dense graphs, improving the previous best runtime of $\\tilde{O}(m\\sqrt{n})$ by Roditty and Vassilevska W. [41], and subsuming (for dense unweighted graphs) the results of Cairo et al. [15].","Theorem 6 There is an expected $O(n^2\\log n)$ time algorithm that for any undirected unweighted graph with Diameter $D=3h+z$ for $h\\ge 0,z\\in \\lbrace 0,1,2\\rbrace $ , returns an extimate $D^{\\prime }$ such that $2h-1\\le D^{\\prime }\\le D$ if $z=0,1$ and $2h\\le D^{\\prime }\\le D$ if $z=2$ .","There is an expected $O(n^2\\log n)$ time algorithm that for any undirected unweighted graph returns estimates $\\varepsilon ^{\\prime }(v)$ of the Eccentricities $\\varepsilon (v)$ of all vertices such that $3\\varepsilon (v)/5-1\\le \\varepsilon ^{\\prime }(v)\\le \\varepsilon (v)$ for all $v$ .","We also show (in Theorem REF ) that one can improve the estimates slightly with an $O(n^{2.05})$ time algorithm."],["Related work","The fastest known algorithm for APSP in dense weighted graphs is by R. Williams [53] and runs in $O(n^3/2^{\\Theta (\\sqrt{\\log n})})$ time.","For sparse undirected graphs, the fastest known APSP algorithm is by Pettie [35] running in $O(mn+n^2\\log \\log n)$ time.","The fastest APSP algorithm for sparse undirected weighted graphs is by Pettie and Ramachandran [37] and runs in $O(mn\\log \\alpha (m,n))$ time.","For APSP on undirected unweighted graphs with $m>n\\log \\log n$ , Chan [17] presented an $O(mn \\log \\log n/\\log n)$ time algorithm.","In graphs with small integer edge weights bounded in absolute value by $M$ , APSP can be computed in $\\tilde{O}(Mn^\\omega )$ time (by Shoshan and Zwick [46] building upon Seidel [43] and Alon, Galil and Margalit [6]) in undirected graphs and in $\\tilde{O}(M^{0.681}n^{2.5302})$ time (by Zwick [55]) in directed graphs.","Zwick [55] also showed that APSP in directed weighted graphs admits an $(1+\\varepsilon )$ -approximation algorithm for any $\\varepsilon >0$ , running in time $\\tilde{O}(n^\\omega /\\varepsilon \\log (M/\\varepsilon ))$ .","For Diameter in graphs with integer edge weights bounded by $M$ , Cygan et al.","[16] obtained an algorithm running in time $\\tilde{O}(Mn^\\omega )$ .","The pioneering work of Aingworth et al.","[2] on Diameter and shortest paths approximation was the root to many subsequent works.","Building upon Aingworth et al.","[2], Dor, Halperin and Zwick [24] presented additive approximation algorithms for APSP in undirected unweighted graphs, achieving among other things, an additive 2-approximation in $\\tilde{O}(n^{7/3})$ time (notably, the best known bound on $\\omega $ is $>7/3$ ).","They also presented an $\\tilde{O}(n^2)$ time additive $O(\\log n)$ -approximation algorithm.","These algorithms were generalized by Cohen and Zwick [23] who showed that in undirected weighted graphs APSP has a (multiplicative) 3-approximation in $\\tilde{O}(n^2)$ time, a $7/3$ -approximation in $\\tilde{O}(n^{7/3})$ time, and a 2-approximation in $\\tilde{O}(n \\sqrt{mn})$ time.","Baswana and Kavitha [11] presented an $\\tilde{O}(m\\sqrt{n}+n^2)$ time multiplicative 2-approximation algorithm and an $\\tilde{O}(m^{2/3} n+n^2)$ time $7/3$ -approximation algorithm for APSP in weighted undirected graphs.","Spanners are closely related to shortest paths approximation.","A subgraph $H$ is an $(\\alpha , \\beta )$ -spanner of $G=(V,E)$ if for every $u,v\\in V$ , $d_H(u,v) \\le \\alpha \\cdot d_G(u,v) + \\beta $ , where $d_{G^{\\prime }}(u,v)$ is the distance between $u$ and $v$ in $G^{\\prime }$ .","Any weighted undirected graph has a $(2k-1,0)$ -spanner with $O(n^{1+1/k})$ edges [3].","Baswana and Sen [14] presented a randomized linear time algorithm for constructing a $(2k-1,0)$ -spanner with $O(kn^{1+1/k})$ edges.","Dor, Halperin and Zwick [24] showed that a $(1,2)$ -spanner with $O(n^{1.5})$ edges can be constructed in $\\tilde{O}(n^2)$ time.","Elkin and Peleg [25] showed that for every integer $k\\ge 1$ and $\\varepsilon >0$ there is a $(1+\\varepsilon , \\beta )$ -spanner with $O(\\beta n^{1+1/k})$ edges, where $\\beta $ depends on $k$ and $\\varepsilon $ but is independent of $n$ .","Baswana et.","al [12] presented a $(1,6)$ -spanner with $O(n^{4/3})$ edges.","Woodruff [54] presented an $\\tilde{O}(n^2)$ time algorithm that computes a $(1,6)$ -spanner with $O(n^{4/3})$ edges.","Chechik [18] presented a $(1,4)$ -spanner with $O(n^{7/5})$ edges.","Recently, Abboud and Bodwin[1] showed that there is no additive spanner with constant error and $O(n^{4/3-\\varepsilon })$ edges.","Thorup and Zwick [48] introduced the notion of distance oracles, a data structure that stores approximate distances for a weighted undirected graph.","Thorup and Zwick designed a distance oracle that for any $k$ takes $O(mn^{1/k})$ time to construct and, is of size is $O(kn^{1+1/k})$ , and given a pair of vertices $u,v \\in V$ it returns in $O(k)$ time a $(2k-1)$ -approximation for $d(u,v)$ .","Baswana and Sen [13] improved the construction time to $O(n^2)$ for unweighted graphs.","Baswana and Kavitha [11] extended the $O(n^2)$ construction time to weighted graphs.","Subsequently, Baswana, Gaur, Sen, and Upadhyay [10] obtained subquadratic construction time in unweighted graphs, at the price of having additive constant error in addition to the $2k-1$ multiplicative error.","Chechik [19] gave an oracle with space $O(n^{1+1/k})$ and $O(1)$ query time, which like previous work, returns a $(2k-1)$ -approximation.","Pǎtraşcu and Roditty [38] obtained a distance oracle that uses $\\tilde{O}(n^{5/3})$ space, has $O(1)$ query time, and returns a $(2k+1)$ -approximation.","Sommer [44] presented an $\\tilde{O}(n^2)$ time algorithm that constructs such a distance oracle.","The construction time was recently improved to $O(n^2)$ by Knudsen [30].","Pǎtraşcu et.","al [39] presented infinity many distance oracles with fractional approximation factors that for graphs with $m=\\tilde{O}(n)$ converge exactly to the integral stretch factors and the corresponding space bound of Thorup and Zwick.","Thorup and Zwick [47] also extended their techniques from [48] to compact routing schemes.","The lower bounds presented in this paper were inspired by a lower bound by Pǎtraşcu and Roditty [38] who showed conditional hardness based on a conjecture on the hardness of a set intersection problem for the space usage of any distance oracle that can distinguish between distances 3 and 7."],["Organization","In Section we prove our lower bounds for $S$ -$T$ Diameter which serve as a basis for the rest of our lower bounds.","We also show equivalence between Diameter and $S$ -$T$ Diameter.","In Section we prove our lower bounds for Eccentricities: one for directed graphs and one for undirected graphs.","In Section we prove our lower bounds for Diameter.","This section is divided into four subsections, one for each of the results in Table REF .","In Section we describe our algorithms for sparse graphs: our 2-approximation and $(2+\\delta )$ -approximation for Eccentricities, our 2-approximation and 3-approximation for $S$ -$T$ Diameter, and our less than 2-approximation for Diameter.","In Section we describe our algorithms for dense graphs: our nearly $3/2$ -approximations for Diameter and our nearly $5/3$ -approximations for Eccentricities."],["Preliminaries","Let $G=(V,E)$ be a graph, where $|V|=n$ and $|E|=m$ .","For every $u,v\\in V$ let $d_G(u,v)$ be the length of the shortest path from $u$ to $v$ .","When the graph $G$ is clear from the context we omit the subscript $G$ .","The eccentricity $\\varepsilon (v)$ of a vertex $v$ is defined as $\\max _{u\\in V} d(v,u)$ .","The diameter $D$ of a graph is $\\max _{v\\in V}\\varepsilon (v)$ .","In a directed graph we have $\\varepsilon ^{out}(v)=\\max _{u\\in V} d(v,u)$ (resp., $\\varepsilon ^{in}(v)=\\max _{u\\in V} d(u,v)$ ).","Let $deg(v)$ be the degree of $v$ and let $N_s(u)$ be the set of the $s$ closest vertices of $v$ , where ties are broken by taking the vertex with the smaller ID.","In a directed graph let $deg^{out}(v)$ (resp., $deg^{in}(v)$ ) be the outgoing (incoming) degree of $v$ .","Let $N^{\\text{out}}_s(v)$ (resp., $N^{\\text{in}}_s(v)$ ) be the set of the $s$ closest outgoing (incoming) vertices of $v$ , where ties are broken by taking the vertex with the smaller ID.","For a subset $S\\subseteq V$ of vertices and a vertex $v \\in V$ we write $d(S,v):=\\min _{s \\in S}d(s,v)$ to denote the distance from the set $S$ to the vertex $v$ .","Let $k\\ge 2$ .","The $k$ -Orthogonal Vectors Problem ($k$ -OV) is as follows: given $k$ sets $S_1,\\ldots ,S_k$ , where each $S_i$ contains $n$ vectors in $\\lbrace 0,1\\rbrace ^d$ , determine whether there exist $v_1\\in S_1,\\ldots ,v_k\\in S_k$ so that their generalized inner product is 0, i.e.","$\\sum _{i=1}^d \\prod _{j=1}^k v_j[i]=0$ .","R. Williams [51] (see also [49]) showed that if for some $\\varepsilon >0$ there is an $n^{k-\\varepsilon } \\text{\\rm poly~} (d)$ time algorithm for $k$ -OV, then CNF-SAT on formulas with $N$ variables and $m$ clauses can be solved in $2^{N(1-\\varepsilon /k)} \\text{\\rm poly~} (m)$ time.","In particular, such an algorithm would contradict the Strong Exponential Time Hypothesis (SETH) of Impagliazzo, Paturi and Zane [28] which states that for every $\\varepsilon >0$ there is a $K$ such that $K$ -SAT on $N$ variables cannot be solved in $2^{(1-\\varepsilon )N} \\text{\\rm poly~} N$ time (say, on a word-RAM with $O(\\log N)$ bit words).","This also motivates the following $k$ -OV Conjectures (implied by SETH) for all constants $k\\ge 2$ : $k$ -OV requires $n^{k-o(1)}$ time on a word-RAM with $O(\\log n)$ bit words.","Most of our conditional lower bounds are based on the $k$ -OV Conjecture for a particular constant $k$ , and thus they also hold under SETH.","A main motivation behind SETH is that despite decades of research, the best upper bounds for $K$ -SAT on $N$ variables and $M$ clauses remain of the form $2^{N(1-c/K)} \\text{\\rm poly~} {(M)}$ for constant $c$ (see e.g.","[27], [36], [42]).","The best algorithms for the $k$ -OV problem for any constant $k\\ge 2$ on $n$ vectors and dimension $c\\log n$ run in time $n^{2-1/O(\\log c)}$ (Abboud, Williams and Yu [8] and Chan and Williams [22])."],["$S$ -{{formula:678000a5-c252-477a-976e-b4f5fbe676e0}} Diameter hardness","In the following we will prove that under SETH, our $S$ -$T$ Diameter algorithms are essentially optimal.","We prove the following theorem: Theorem 7 Let $k\\ge 2$ be an integer.","There is an $O(k n^{k-1} d^{k-1})$ time reduction that transforms any instance of $k$ -OV on sets of $n$ $d$ -dimensional vectors into a graph on $O(n^{k-1}+k n^{k-2} d^{k-1})$ nodes and $O(k n^{k-1} d^{k-1})$ edges and two disjoint sets $S$ and $T$ on $n^{k-1}$ nodes each, so that if the $k$ -OV instance has a solution, then $D_{S,T}\\ge 3k-2$ , and if it does not, $D_{S,T}\\le k$ .","From Theorem REF we get that if there is some $k\\ge 2$ , $\\varepsilon >0$ and $\\delta >0$ so that there is an $O(M^{1+1/(k-1)-\\varepsilon })$ time $(3-2/k-\\delta )$ -approximation algorithm for $S$ -$T$ Diameter in $M$ -edge graphs, then $k$ -OV has an $n^{k-\\gamma } \\text{\\rm poly~} (d)$ -time algorithm for some $\\gamma >0$ and SETH is false.","We obtain an immediate corollary.","Corollary 8 For $S$ -$T$ Diameter, under SETH, there is no $(2-\\varepsilon )$ -approximation algorithm running in $O(m^{2-\\delta })$ time for any $\\varepsilon >0,\\delta >0$ , no $O(m^{3/2-\\delta })$ time $7/3-\\varepsilon $ -approximation algorithm for any $\\varepsilon >0,\\delta >0$ , no $(m+n)^{1+o(1)}$ time, $3-\\varepsilon $ -approximation algorithm for any $\\varepsilon >0$ .","We will prove the following more detailed theorem, which will be useful for our Diameter lower bounds.","Theorem 9 Let $k\\ge 2$ .","Given a $k$ -OV instance consisting of sets $W_0,W_1,\\dots ,W_{k-1} \\subseteq \\lbrace 0,1\\rbrace ^d$ , each of size $n$ , we can in $O(kn^{k-1}d^{k-1})$ time construct an unweighted, undirected graph with $O(n^{k-1}+k n^{k-2} d^{k-1})$ vertices and $O(k n^{k-1} d^{k-1})$ edges that satisfies the following properties.","The graph consists of $k+1$ layers of vertices $S=L_0,L_1,L_2,\\dots ,L_k=T$ .","The number of nodes in the sets is $|S|=|T|=n^{k-1}$ and $|L_1|,|L_2|,\\dots ,|L_{k-1}|\\le n^{k-2}d^{k-1}$ .","$S$ consists of all tuples $(a_0,a_1,\\ldots , a_{k-2})$ where for each $i$ , $a_i\\in W_i$ .","Similarly, $T$ consists of all tuples $(b_1,b_2,\\ldots , b_{k-1})$ where for each $i$ , $b_i\\in W_i$ .","If the $k$ -OV instance has no solution, then $d(u,v)=k$ for all $u \\in S$ and $v \\in T$ .","If the $k$ -OV instance has a solution $a_0, a_1,\\dots ,a_{k-1}$ where for each $i$ , $a_i\\in W_i$ then if $\\alpha =(a_0,\\dots a_{k-2})\\in S$ and $\\beta = (a_1,\\dots ,a_{k-1}) \\in T$ , then $d(\\alpha ,\\beta )\\ge 3k-2$ .","Suppose the $k$ -OV instance has a solution $a_0, a_1,\\dots ,a_{k-1}$ where for each $i$ , $a_i\\in W_i$ .","Let $t=k-2$ .","Let $s$ be such that $0\\le s\\le t$ .","Let $b_{t-s+j}\\in W_{t-s+j}$ for all $j\\in [1,\\ldots ,s]$ be some other vectors, potentially different from $a_{t-s+j}$ .","Consider $\\alpha =(a_0,a_1,\\ldots ,a_{t-s},b_{t-s+1},\\ldots ,b_{t})\\in L_0$ and $\\beta =(a_{1},\\ldots ,a_{t+1})\\in L_{t+2}$ .","Then the distance between $\\alpha $ and $\\beta $ is at least $3t-2s+4$ .","Symmetrically, let $c_{j}\\in W_{j}$ for all $j\\in [1,\\ldots ,s]$ be some other vectors, potentially different from $a_{j}$ .","Consider $\\alpha =(a_0,a_1,\\ldots ,a_{t})\\in L_0$ and $\\beta =(c_{1},\\ldots ,c_s,a_{s+1},\\dots , a_{t+1})\\in L_{t+2}$ .","Then the distance between $\\alpha $ and $\\beta $ is at least $3t-2s+4$ .","For all $i$ from 1 to $k-1$ , for all $v \\in L_i$ there exists a vertex in $L_{i-1}$ adjacent to $v$ and a vertex in $L_{i+1}$ adjacent to $v$ ."],["Proof of Theorem ","We will prove the theorem for $k=t+2$ for any $t\\ge 0$ .","We will create a layered graph $G$ on $t+3$ layers, $L_0,\\ldots ,L_{t+2}$ , where the edges go only between adjacent layers $L_i,L_{i+1}$ .","We will set $S=L_0$ and $T=L_{t+2}$ for the $S$ -$T$ Diameter instance.","In particular, $D_{S,T}\\ge t+2$ because of the layering.","Let us describe the vertices of $G$ .","$L_0$ consists of $n^{t+1}$ vertices, each corresponding to a $t+1$ -tuple $(a_0,a_1,\\ldots , a_t)$ where for each $i$ , $a_i\\in W_i$ .","Similarly, $L_{t+2}$ consists of $n^{t+1}$ vertices, each corresponding to a $t+1$ -tuple $(b_1,b_2,\\ldots , b_{t+1})$ where for each $i$ , $b_i\\in W_i$ .","Layer $L_1$ consists of $n^t d^{t+1}$ vertices, each corresponding to a tuple $(a_0,\\ldots ,a_{t-1}, \\bar{x})$ where for each $i$ , $a_i\\in W_i$ and $\\bar{x}=(x_0,\\ldots ,x_t)$ is a $(t+1)$ -tuple of coordinates in $[d]$ .","Similarly, $L_{t+1}$ consists of $n^t d^{t+1}$ vertices, each corresponding to a tuple $(b_2,\\ldots ,b_{t+1}, \\bar{x})$ where for each $i$ , $b_i\\in W_i$ and $\\bar{x}$ is a $(t+1)$ -tuple of coordinates.","For every $j\\in \\lbrace 2,\\ldots ,t\\rbrace $ , $L_j$ consists of $n^t d^{t+1}$ vertices $(a_0, \\ldots , a_{t-j},b_{t+3-j},\\ldots , b_{t+1},\\bar{x})$ , where for each $i$ , $a_i\\in W_i$ , $b_i\\in W_i$ and $\\bar{x}=(x_0,\\ldots ,x_t)$ is a $(t+1)$ -tuple of coordinates in $[d]$ .","In other words, there is a vector from $W_i$ for every $i\\notin \\lbrace t-j+1,t-j+2\\rbrace $ .","Now let us define the edges.","Consider a node $(a_0,\\ldots ,a_t)\\in L_0$ .","For every $\\bar{x}=(x_0,\\ldots ,x_t)$ , connect $(a_0,\\ldots ,a_t)$ to $(a_0,\\ldots ,a_{t-1},\\bar{x})\\in L_1$ if and only if for every $j\\in \\lbrace 0,\\ldots ,t\\rbrace $ , $a_j$ is 1 in coordinates $x_0,\\ldots ,x_{t-j}$ .","For any $i\\in \\lbrace 1,\\ldots ,t\\rbrace $ let's define the edges between $L_i$ and $L_{i+1}$ .","For $(a_0,\\ldots ,a_{t-i},b_{t+3-i},\\ldots ,b_{t+1}, \\bar{x})\\in L_i$ Here if $i=1$ , there are no $b$ 's in the tuple.","and for any $c_{t+2-i}\\in W_{t+2-i}$ , add an edge to $(a_0,\\ldots ,a_{t-i-1},c_{t+2-i},b_{t+3-i},\\ldots ,b_{t+1}, \\bar{x})\\in L_{i+1}$ .","Here we “forget” vector $a_{t-i}$ and replace it with $c_{t+2-i}$ , leaving everything else the same.","Finally, the edges between $L_{t+1}$ and $L_{t+2}$ are as follows.","Consider some $(b_1,\\ldots ,b_{t+1})\\in L_{t+2}$ .","For every $\\bar{x}=(x_0,\\ldots ,x_t)$ , connect $(b_1,\\ldots ,b_{t+1})$ to $(b_2,\\ldots ,b_{t+1},\\bar{x})\\in L_{t+1}$ if and only if for every $j\\in \\lbrace 1,\\ldots ,{t+1}\\rbrace $ , $b_j$ is 1 in coordinates $x_{t+1-j},\\ldots ,x_t$ .","Figure REF shows the construction of the graph for $t=2$ .","Figure: The reduction graph from (t+2)(t+2)-OV for t=2t=2.","The figure depicts when a path of length t+2t+2 exists between arbitrary a 0 a 1 a 2 ∈L 0 a_0a_1a_2\\in L_0 and b 1 b 2 b 3 ∈L t+2 b_1b_2b_3\\in L_{t+2}.","It also shows that when there is a path of length t+2t+2 between a 0 a 1 a 2 ∈L 0 a_0a_1a_2\\in L_0 and a 1 a 2 a 3 ∈L t+2 a_1a_2a_3\\in L_{t+2}, a 0 ,a 1 ,a 2 ,a 3 a_0,a_1,a_2,a_3 cannot be an orthogonal 4-tuple.An important claim is as follows: Claim 10 For every $\\bar{x}$ , each $(a_0,\\ldots ,a_{t-1},\\bar{x})\\in L_1$ is at distance $t$ to every $(b_2,\\ldots ,b_t,\\bar{x})\\in L_{t+1}$ .","Consider the path starting from $(a_0,\\ldots ,a_{t-1},\\bar{x})$ , and then for each $i\\ge 1$ following the edges $(a_0,\\ldots ,a_{t-i},b_{t+3-i},\\ldots ,b_{t+1},\\bar{x})\\in L_i$ to $(a_0,\\ldots ,a_{t-1-i},b_{t+2-i},\\ldots ,b_{t+1},\\bar{x})\\in L_{i+1}$ , until we reach $(b_2,\\ldots ,b_{t+1},\\bar{x})\\in L_{t+1}$ .","This path exists by construction and has length $t$ .","Now we proceed to prove the bounds on the $S$ -$T$ Diameter.","Lemma 11 (Property 3 of Theorem REF ) If the $(t+2)$ -OV instance has no solution, then $D_{S,T}=t+2$ .","If the $(t+2)$ -OV instance has no solution, then for every $c_0\\in W_0,c_1\\in W_1,\\ldots ,c_{t+1}\\in W_{t+1}$ , there is some coordinate $x$ such that $c_0[x]=c_1[x]=\\ldots =c_{t+1}[x]=1$ .","Now consider the graph and any $(a_0,\\ldots ,a_t)\\in L_0$ , $(b_1,\\ldots ,b_{t+1})\\in L_{t+2}$ .","For every $j\\in \\lbrace 0,\\ldots ,t\\rbrace $ , let $x_j$ be a coordinate so that $a_0,\\ldots ,a_{t-j},b_{t-j+1},\\ldots ,b_{t+1}$ are all 1 in $x_j$ .","Let $\\bar{x}=(x_0,\\ldots ,x_{t})$ .","By construction, $(a_0,\\ldots ,a_t)$ has an edge to $(a_0,\\ldots ,a_{t-1},\\bar{x})$ and $(b_2,\\ldots ,b_{t+1},\\bar{x})$ has an edge to $(b_1,\\ldots ,b_{t+1})$ .","Also, by Claim REF , $(a_0,\\ldots ,a_{t-1},\\bar{x})$ has a path of length $t$ to $(b_2,\\ldots ,b_{t+1},\\bar{x})$ .","This shows that $D_{S,T}\\le t+2$ ; equality follows because the graph is layered.","Now we prove the guarantee for the case when an orthogonal tuple exists.","Lemma 12 (Property 4 of Theorem REF ) If there exist $a_0\\in W_0,\\ldots ,a_{t+1}\\in W_{t+1}$ that are orthogonal, then $D_{S,T}\\ge 3t+4$ .","To prove the lemma, we will actually prove a more general claim: Property 5 of Theorem REF .","Claim 13 (Property 5 of Theorem REF ) Suppose that $a_0\\in W_0,\\ldots ,a_{t+1}\\in W_{t+1}$ are orthogonal.","Let $s$ be such that $0\\le s\\le t$ .","Let $b_{t-s+j}\\in W_{t-s+j}$ for all $j\\in [1,\\ldots ,s]$ be some other vectors, potentially different from $a_{t-s+j}$ .","Consider $\\alpha =(a_0,a_1,\\ldots ,a_{t-s},b_{t-s+1},\\ldots ,b_{t})\\in L_0$ and $\\beta =(a_{1},\\ldots ,a_{t+1})\\in L_{t+2}$ .","Then the distance between $\\alpha $ and $\\beta $ is at least $3t-2s+4$ .","Symmetrically, let $c_{j}\\in W_{j}$ for all $j\\in [1,\\ldots ,s]$ be some other vectors, potentially different from $a_{j}$ .","Consider $\\alpha =(a_0,a_1,\\ldots ,a_{t})\\in L_0$ and $\\beta =(c_{1},\\ldots ,c_s,a_{s+1},\\dots , a_{t+1})\\in L_{t+2}$ .","Then the distance between $\\alpha $ and $\\beta $ is at least $3t-2s+4$ .","If the claim is true, then using $s=0$ we get that the Diameter is at least $3t+4$ so Lemma REF is true.","The claim for $s>0$ is useful for the rest of our constructions.","We will show that the distance between $\\alpha =(a_0,a_1,\\ldots ,a_{t-s},b_{t-s+1},\\ldots ,b_{t})\\in L_0$ and $\\beta =(a_{1},\\ldots ,a_{t+1})\\in L_{t+2}$ is strictly more than $3t+2-2s$ .","Because the graph is layered and hence bipartite and $t+2\\equiv 3t+2\\mod {2}$ , the distance must be at least $3t-2s+4$ .","Let's assume for contradiction that the shortest path $P$ between $\\alpha $ and $\\beta $ is of length $\\le 3t+2-2s$ .","First let's look at any subpath $P^{\\prime }$ of $P$ strictly within $M=L_1\\cup \\ldots \\cup L_{t+1}$ .","All nodes on $P^{\\prime }$ must share the same $\\bar{x}$ .","Furthermore, if $P^{\\prime }$ starts with a node of $L_1$ and ends with a node of $L_{t+1}$ , as $P^{\\prime }$ needs to be a shortest path and by Claim REF , $P^{\\prime }$ must be of length exactly $t$ .","Next, notice that $P$ cannot go from $L_0$ to $L_{t+2}$ and then back to $L_0$ .","This is because it needs to end up in $L_{t+2}$ and any time it crosses over $M$ , it would need to pay a distance of $t+2$ , so $P$ would have to have length at least $3t+6>3t+2-2s$ .","Hence, $P$ must be of the following form: a path from $\\alpha $ through $L_0\\cup M$ back to $L_0$ (possibly containing only $\\alpha $ ), followed by a path crossing $M$ to reach $L_{t+2}$ , followed by a path through $L_{t+2}\\cup M$ to $L_{t+2}$ (possibly empty).","We will show that if $P$ has length $\\le 3t+2-2s$ then $P$ must contain a length $t+2$ subpath $Q$ between a node $(a_0,\\ldots ,a_{q},w_{q+1},\\ldots , w_t)\\in L_0$ , for some choices of the $w$ 's and some $q\\le t-s$ , and a node $(v_1,\\ldots ,v_q,a_{q+1},\\ldots ,a_{t+1})\\in L_{t+2}$ , for some choices of $v$ 's.","That is, this path traverses $M$ without weaving, by following $(a_0,\\ldots ,a_{q},w_{q+1},\\ldots , w_{t-1},\\bar{x})\\in L_1$ , $(a_0, \\ldots , a_{q},w_{q+1},\\ldots , w_{t-2},a_{t+1},\\bar{x})\\in L_2$ , $\\ldots $ , $(v_2, \\ldots , v_q, a_{q+1}, \\ldots , a_{t-s}, \\ldots , a_{t+1}, \\bar{x})\\in L_{t+1}$ .","Suppose we show that such a subpath exists.","Then by the construction of our graph we have that for every $i\\in \\lbrace 0,\\ldots ,q\\rbrace $ , $a_i[x_j]=1$ for all $j\\in \\lbrace 0,\\ldots ,t-i\\rbrace $ , and that for all $i\\in \\lbrace q+1,\\ldots ,t+1\\rbrace $ , $a_i[x_j]=1$ for all $j\\in \\lbrace t+1-i,\\ldots ,t\\rbrace $ .","That is, for all $i$ , $a_i[x_{t-q}]=1$ , and we get a contradiction since the $a_i$ were supposed to be orthogonal.","Now let $\\alpha ^*$ be the last node from $L_0$ on $P$ and let $\\beta ^*$ be the first node of $L_{t+1}$ of $P$ .","Let $a^*\\in L_1$ be the node right after $\\alpha ^*$ and let $b^*\\in L_{t+1}$ be the node right before $\\beta ^*$ .","Since the subpath of $P$ between $a^*$ and $b^*$ is within $M$ , it must share the same $\\bar{x}$ , and it must have length exactly $t$ by Claim REF .","We will show that the subpath $Q$ that we are looking for is the subpath of $P$ between $\\alpha ^*$ and $\\beta ^*$ .","Its length is exactly what we want: $t+2$ .","It remains to show that for some $q\\le t-s$ and some choices of $w$ 's and $v$ 's, $\\alpha ^*=(a_0,\\ldots ,a_{q},w_{q+1},\\ldots , w_t)$ and $\\beta ^*=(v_1,\\ldots ,v_q,a_{q+1},\\ldots ,a_{t+1})$ .","Consider the path $P_1$ between $\\alpha =(a_0,a_1,\\ldots ,a_{t-s},b_{t-s+1},\\ldots ,b_{t})$ and $\\alpha ^*$ and the path $P_2$ between $\\beta =(a_1,\\ldots ,a_{t+1})$ and $\\beta ^*$ .","Let $L_i$ be the layer in $M$ with largest $i$ that $P_1$ touches and let $L_j$ be the layer in $M$ with smallest $j$ that $P_2$ touches.","For convenience, let us define $j^{\\prime }=t+2-j$ .","The length of $P_1$ is then at least $2i$ and the length of $P_2$ is at least $2j^{\\prime }$ .","The length $|P|$ of $P$ equals $t+2+|P_1|+|P_2|\\ge t+2+2i+2j^{\\prime }=t+2+2(i+j^{\\prime })$ .","Since we have assumed that $|P|\\le 3t+2-2s$ , we must have that $t+2+2(i+j^{\\prime })\\le 3t+2-2s$ and hence $i+j^{\\prime }\\le t-s$ .","Now, since $P_1$ goes at most to $L_i$ , then from getting from $\\alpha $ to $\\alpha ^*$ , at most the last $i$ elements of $(a_0,a_1,\\ldots ,a_{t-s},b_{t-s+1},\\ldots ,b_{t})$ can have been “forgotten”.","Hence, $\\alpha ^*=(a_0,\\ldots ,a_{t-\\max \\lbrace s,i\\rbrace },b_{t-\\max \\lbrace s,i\\rbrace +1},\\ldots ,b_{t-i},w_{t-i+1},\\ldots ,w_{t})$ for some $w$ 's.","(If $i\\ge s$ , the $b$ 's do not appear.)","Similarly, between $\\beta $ and $\\beta ^*$ , at most the first $j^{\\prime }$ elements of $\\beta $ can have been forgotten.","Thus, we have that $\\beta ^*=(v_1,\\ldots ,v_{j^{\\prime }},a_{j^{\\prime }+1},\\ldots ,a_{t+1})$ for some $v$ 's.","Now, since $i+j^{\\prime }\\le t-s$ , we must have that $j^{\\prime }\\le t-s-i\\le t-\\max \\lbrace s,i\\rbrace $ , and hence the path between $\\alpha ^*$ and $\\beta ^*$ is the path $Q$ we are searching for.","See Figure REF for an illustration of $L_i$ , $L_j$ etc.","in the case when $s=0$ .","Figure: Here PP contains at least 2 nodes in L 0 L_0 and at least 2 in L t+2 L_{t+2}, and s=0s=0."],["Equivalence between Diameter and $S$ -{{formula:bdb52e68-e562-4944-b84c-7c1359b488ce}} Diameter","Here we will prove that when it comes to exact computation, $S$ -$T$ -Diameter and Diameter in weighted graphs are equivalent.","The proof for directed graphs is much simpler, so we focus on the equivalence for undirected graphs.","Also, it is clear that if one can solve $S$ -$T$ Diameter, one can also solve Diameter in the same running time since one can simply set $S=T=V$ .","We prove: Theorem 14 Suppose that there is a $T(n,m)$ time algorithm that can compute the Diameter of an $n$ node, $m$ edge graph with nonnegative integer edge weights.","Then, the $S$ -$T$ Diameter of any $n$ node $m$ edge graph with nonnegative integer edge weights can be computed in $T(O(n),O(m))$ time.","Let $G=(V,E),S,T$ be the $S$ -$T$ Diameter instance; let $w:E\\rightarrow \\lbrace 0,\\ldots ,M\\rbrace $ be the edge weights.","First, we can always assume that $M$ is even: if it is not, multiply all edge weights by 2; all distances (and hence also the $S$ -$T$ Diameter) double.","Let $S=\\lbrace s_1,\\ldots ,s_k\\rbrace $ and $T=\\lbrace t_1,\\ldots ,t_{\\ell }\\rbrace $ .","Now, let $W=Mn$ .","First add $|S|=k$ new nodes $S^{\\prime }=\\lbrace v_1,\\ldots ,v_k\\rbrace $ .","For each $i\\in \\lbrace 1,\\ldots ,k\\rbrace $ add a new edge $(v_i,s_i)$ of weight $W$ .","Let $G_S$ be this new graph.","Let's consider the Diameter of $G_S$ .","For every pair of nodes $u,v\\notin S^{\\prime }$ , the distance is the same as in $G$ .","For $v_i\\in S^{\\prime }$ and $x\\notin S^{\\prime }$ , the distance is $W+d_G(s_i,x)\\le W+M(n-1)<2W$ .","For $v_i,v_j\\in S^{\\prime }$ , the distance is $2W+d(s_i,s_j)$ .","Hence the Diameter of $G_S$ is $2W+\\max _{s_i,s_j\\in S} d(s_i,s_j)$ .","Hence by computing the Diameter of $G_S$ , we can compute $D_S=\\max _{s_i,s_j\\in S} d(s_i,s_j)$ .","We can create a similar graph $G_T$ whose Diameter will allow us to compute $D_T=\\max _{t_i,t_j\\in S} d(t_i,t_j)$ .","After this, let's create a graph $G_{S,T}$ as follows.","Add new nodes $S^{\\prime }=\\lbrace v_1,\\ldots ,v_k\\rbrace $ and $T^{\\prime }=\\lbrace u_1,\\ldots ,u_\\ell \\rbrace $ .","For each $i\\in \\lbrace 1,\\ldots ,k\\rbrace $ add a new edge $(v_i,s_i)$ of weight $W$ .","For each $j\\in \\lbrace 1,\\ldots ,\\ell \\rbrace $ add a new edge $(u_j,t_j)$ of weight $W$ .","With a similar argument as above, the Diameter $D^{\\prime }$ of $G_{S,T}$ is $D^{\\prime }=2W+\\max _{u,v\\in S\\cup T} d_G(u,v)$ .","Let's assume without loss of generality that $D_S\\ge D_T$ .","If $D^{\\prime }> D_S$ , then $D^{\\prime }=2W+\\max _{u\\in S,v\\in T} d_G(u,v)$ , and we can compute the $S$ -$T$ Diameter of $G$ by subtracting $2W$ .","Now suppose that we get $D^{\\prime }\\le D_S$ ; we must have then actually gotten $D^{\\prime }=D_S$ .","The $S$ -$T$ Diameter of $G$ might be strictly smaller than $D_S$ .","We add two new nodes $x$ and $y$ to $G_{S,T}$ .","We add an edge $(x,y)$ of weight $2W$ , edges $(x,v_i)$ for every $v_i\\in S^{\\prime }$ of weight $D_S/2$ and (symmetrically) edges $(y,u_j)$ for every $u_j\\in T^{\\prime }$ of weight $D_S/2$ .","The construction of $G^{\\prime }$ is depicted in Figure REF .","Figure: A depiction of the construction of G ' G^{\\prime }.Let us consider the distances in this new graph $G^{\\prime }$ .","For every $a,b\\notin S^{\\prime }\\cup T^{\\prime }\\cup \\lbrace x,y\\rbrace $ , $d(a,b)=d_G(a,b)<W$ as any path not in $G$ would have to use an edge of weight $W>d(a,b)$ .","For every $b\\notin S^{\\prime }\\cup T^{\\prime }\\cup \\lbrace x,y\\rbrace $ , $d(x,b)=W+D_S/2 +\\min _{a\\in S} d_G(a,b)\\le 2W+D_S/2$ .","Similarly, $d(y,b)=W+D_S/2 +\\min _{a\\in T} d_G(a,b)\\le 2W+D_S/2$ .","For every $v_i\\in S^{\\prime }$ , $d(x,v_i)=D_S/2$ , and $d(y,v_i)=2W+D_S/2$ .","For every $u_i\\in T^{\\prime }$ , $d(y,u_i)=D_S/2$ , and $d(x,u_i)=2W+D_S/2$ .","For every $v_i,v_j\\in S^{\\prime }$ , $d(v_i,v_j)=D_S$ .","For every $u_i,u_j\\in T^{\\prime }$ , $d(u_i,u_j)=D_S$ .","For every $v_i\\in S^{\\prime }$ and $b\\notin S^{\\prime }\\cup T^{\\prime }\\cup \\lbrace x,y\\rbrace $ , $d(v_i,b)\\le W+d(s_i,b)\\le 2W.$ For every $u_i\\in T^{\\prime }$ and $b\\notin S^{\\prime }\\cup T^{\\prime }\\cup \\lbrace x,y\\rbrace $ , $d(u_i,b)\\le W+d(t_i,b)\\le 2W.$ For every $v_i\\in S^{\\prime }$ and $u_j\\in T^{\\prime }$ , $d(v_i,u_j)$ is the minimum of $D_S+2W, D_S+2W+\\min _{s\\in S} d_G(s,t_j), D_S+2W+\\min _{t\\in t} d_G(t,s_i)$ and $2W+d_G(s_i,t_j)$ .","The middle two terms are $\\ge D_S+2W$ , and hence $d(v_i,u_j)=2W+\\min \\lbrace D_S,d_G(s_i,t_j)\\rbrace $ .","Consider $s_i\\in S, t_j\\in T$ that are the end points of the $S$ -$T$ Diameter $D$ in $G$ .","Then $D=d_G(s_i,t_j)$ .","Now, we have from before that $D\\le D_S$ , as otherwise we have computed $D$ already.","Hence in $G^{\\prime }$ , the distance $d(u_i,v_j)$ equals $2W+\\min \\lbrace D_S,D\\rbrace =2W+D$ We note that for any $s_i,s_j\\in S$ , and any $t\\in T$ , $d_G(s_i,s_j)\\le d_G(s_i,t)+d_G(t,s_j)\\le 2\\max _{s\\in S,t\\in T} d_G(s,t) = 2D$ .","Thus, $D\\ge D_S/2$ .","The distances in cases (1) to (5) are all $\\le 2W+D_S/2\\le 2W+D$ .","Hence the Diameter of $G^{\\prime }$ is actually exactly $2W+D$ ."],["Undirected graphs","Theorem 15 Let $k\\ge 2$ .","Under the $k$ -OV conjecture, every algorithm that can distinguish between eccentricity at most $2k-1$ and eccentricity at least $4k-3$ for every vertex in an $O(n)$ edge and node undirected graph, requires at least $n^{1+1/(k-1)-o(1)}$ time on a $O(\\log n)$ -bit word-RAM.","Figure: The undirected Eccentricities lower bound for k=5k=5.Let's start with the $S$ -$T$ -diameter construction for $k$ obtained from a given $k$ -OV instance.","We remove any internal nodes if they don't have edges to one of their adjacent layers - they don't hurt the instance.","We have a graph on $O(n^{k-1}d^{k-2})$ vertices and edges with the following properties: (1) Suppose that the $k$ -OV instance has no $k$ -OV solution.","Then for every $s\\in S,t\\in T$ , $d(s,t)=k$ .","Also, for every $s\\in S$ and $u\\notin S\\cup T$ , $d(s,u)\\le (k-1)+k=2k-1$ since we can take a $\\le (k-1)$ length path from $u$ to some node $t\\in T$ and since $d(s,t)=k$ .","(2) If there is a $k$ -OV solution, there are two nodes $s\\in S,t\\in T$ with $d(s,t)\\ge 3k-2$ .","We modify the construction as follows.","For every $s\\in S$ , we create an undirected path on $k-2$ new vertices $s_1\\rightarrow s_2\\rightarrow \\ldots \\rightarrow s_{k-2}$ and add an edge $(s,s_1)$ ; let's call $s$ by $s_0$ .","Now, the distance between $s_0$ and $s_i$ is $i$ .","Add a new node $y$ and create edges $(s_{k-2},y)$ for every $s\\in S$ .","Now, $d(y,s_0)=k-1$ for every $s\\in S$ , and also for every $s,s^{\\prime }\\in S$ and all $i,j\\in \\lbrace 0,\\ldots ,k-2\\rbrace $ , we have that $d(s_i,s^{\\prime }_j)\\le 2k-2$ .","For every $s\\in S$ , $t\\in T$ , there is now potentially a new path between them, from $s$ to $y$ in $k-1$ steps, then to some other $s^{\\prime }$ in $k-1$ steps and then to $t$ using $\\ge k$ steps.","The length is $\\ge 2(k-1)+k=3k-2$ , so when there is a $k$ -OV solution, there is still a pair $s,t$ at distance at least $3k-2$ .","Now, we also attach paths to the nodes in $T$ .","In particular, for each $t$ , add an undirected path $t\\rightarrow t_1\\rightarrow \\ldots \\rightarrow t_{k-1}$ .","The distance between any $s\\in S$ and any $t_i$ is $i+d(s,t)$ .","Hence when there is no $k$ -OV solution, the Eccentricities of all $s_0$ for $s\\in S$ are $\\le k+(k-1)=2k-1$ , and when there is a $k$ -OV solution, there is $s\\in S, t\\in T$ so that $d(s_0,t_{k-1})\\ge (3k-2)+(k-1)=4k-3$ ."],["Directed graphs","Theorem 16 Under the 2-OV conjecture, for any $\\delta >0$ , any $(2-\\delta )$ -approximation algorithm for all Eccentricities in an $n$ node, $O(n)$ -edge directed unweighted graph, requires $n^{2-o(1)}$ time on a $O(\\log n)$ -bit word-RAM.","Figure: The directed Eccentricities lower bound.Suppose we are given an instance of 2-OV: two sets of vectors $U,V$ over $\\lbrace 0,1\\rbrace ^d$ and we want to know whether there are $u\\in U,v\\in V$ with $u\\cdot v=0$ .","Let $L\\ge 1$ be any integer.","Let us create a directed unweighted graph $G$ ; an illustration can be found in Figure REF .","$G$ will have a vertex $u$ for every $u\\in U$ and a vertex $c$ for every $c\\in [d]$ .","Every $v\\in V$ will be represented by a directed path $v_0\\rightarrow v_1\\rightarrow \\ldots \\rightarrow v_L$ .","In addition, there is a directed path $P_x$ on $L$ extra nodes, $x_1\\rightarrow \\ldots \\rightarrow x_L$ so that every $u\\in U$ has directed edges $(u,x_1)$ and $(x_L,u)$ .","For every $u\\in U$ and every $c$ for which $u[c]=1$ , we add a directed edge $(u,c)$ .","For every $v$ and every $c$ for which $v[c]=1$ , we add a directed edge $(c,v_0)$ .","Remove any $c$ that does not have at least one edge coming from $U$ .","Let us consider the eccentricity of any node $u\\in U$ .","First, for all $u^{\\prime }\\in U$ , $d(u,u^{\\prime })\\le L+1$ since one can go through the path $P_x$ .","For every $c\\in [d]$ , there is at least one edge coming from some $u^{\\prime }\\in U$ , and so one can reach $c$ from $u$ by first taking $P_x$ to $u^{\\prime }$ and then using the edge $(u^{\\prime },c)$ .","Hence, $d(u,c)\\le L+2$ for all $c\\in [d]$ .","For any $v\\in V$ and $i\\in \\lbrace 1,\\ldots , L\\rbrace $ , the distance $d(u,v_i)=i+d(u,v_0)$ , and so we consider $d(u,v_0)$ .","If there is a $c$ for which $u[c]=v[c]=1$ , then $d(u,v_0)=2$ , and hence for all $i$ , $d(u,v_i)\\le L+2$ .","If no such $c$ exists and so if $u$ and $v$ are orthogonal, the only way to reach $v_0$ is potentially via $P_x$ to some other $u^{\\prime }\\in U$ which is at distance 2 to $v_0$ .","Hence if $u$ and $v$ are orthogonal, $d(u,v_0)=L+3$ , and hence $d(u,v_L)=2L+3$ .","In other words, the eccentricity of $u$ is $L+2$ if it is not orthogonal to any vectors in $V$ and it is $\\ge 2L+3$ if there is some $v$ that is orthogonal to $u$ .","The number of vertices in the graph is $O(nL+d)$ and the number of edges is $O(nL+nd)$ .","Suppose that there is a $(2-\\varepsilon )$ -approximation algorithm for all Eccentricities in graphs with $O(m)$ nodes and edges running in $O(m^{2-\\delta })$ time for some $\\varepsilon ,\\delta >0$ .","Then, we construct the above instance for $L=\\lceil 1/\\varepsilon \\rceil $ and run the algorithm on it.","The approximation returned is at least as good as a $(2-1/L)$ -approximation.","Hence if the diameter is at least $2L+3$ , the algorithm will return an estimate that is at least $L(2L+3)/(2L-1) > L+2$ .","Thus the algorithm can solve 2-OV in time $O((nL+nd)^{2-\\delta }) = O(n^{2-\\delta }d^{2-\\delta })$ , contradicting the 2-OV conjecture."],["Diameter lower bounds","For all of our constructions we begin with the $S$ -$T$ diameter lower bound construction from Theorem REF .","Here, if the $k$ -OV instance has no solution, $D_{S,T}\\le k$ and if the instance has a solution $D_{S,T}\\ge 3k-2$ .","To adapt this construction to Diameter, we need to ensure that if the OV instance has no solution then all pairs of vertices have small enough distance.","We begin by augmenting the $S$ -$T$ Diameter construction by adding a matching between $S$ and a new set $S^{\\prime }$ as well as a matching between $T$ and a new set $T^{\\prime }$ .","Without any further modifications, pairs of vertices $u,v\\in S\\cup S^{\\prime }$ (or $u,v\\in T\\cup T^{\\prime }$ ) could be far from one another.","The challenge is to add extra gadgetry to make these pairs close for “no\" instances while maintaining that in “yes\" instances the distance between the diameter endpoints $s^{\\prime }\\in S^{\\prime },t^{\\prime }\\in T^{\\prime }$ is large.","That is, for “yes\" instances, we want a shortest path between the diameter endpoints $s^{\\prime }$ and $t^{\\prime }$ to contain the vertex $s\\in S$ matched to $s^{\\prime }$ and the vertex $t\\in T$ matched to $t^{\\prime }$ so that we can use use the fact that $d(s,t)\\ge 3k-2$ .","In other words, we do not want there to be a shortcut from $s^{\\prime }$ to some vertex in $S$ that allows us to use a path of length $k$ from $S$ to $T$ .","For example, we cannot simply create a vertex $x$ and connect it to all vertices in $S\\cup S^{\\prime }$ because this would introduce shortcuts from $S^{\\prime }$ to $S$ .","We will describe some intuition for the augmentations to the graph regarding 3-OV for simplicity.","Recall that $s^{\\prime }\\in S^{\\prime }, t^{\\prime }\\in T^{\\prime }$ are the endpoints of the diameter and let $t$ be the vertex matched to $t^{\\prime }$ .","To solve the problem outlined in the above paragraph, we observe that in the “yes\" case there are three types of vertices $s\\in S$ .","(1) close: $d(s,t)=3$ , (2) far: $d(s,t)\\ge 7$ (property 4 of Theorem REF ), and (3) intermediate: $d(s,t)\\ge 5$ (property 5 of Theorem REF ).","For close $s$ , we need $d(s^{\\prime },s)$ to be large so that there is no shortcut from $s^{\\prime }$ to $t^{\\prime }$ through $s$ .","For far $s$ , it is ok if $d(s^{\\prime },s)$ is small because $d(s,t)$ is large enough to ensure that paths from $s^{\\prime }$ to $t^{\\prime }$ through $s$ are still long enough.","For intermediate $s$ , $d(s^{\\prime },s)$ cannot be small, but it also need not be large.","To fulfill these specifications, we add a small clique (the graph is still sparse) and connect each of its vertices to only some of the vertices in $S$ and/or $S^{\\prime }$ according to the implications of property 5 of Theorem REF .","When $s$ is close, we ensure that $d(s^{\\prime },s)$ is large by requiring that a shortest path from $s^{\\prime }$ to $s$ goes from $s^{\\prime }$ to the clique, uses an edge inside of the clique, and then goes from the clique to $s$ .","When $s$ is intermediate, we ensure that $d(s^{\\prime },s)$ is not too small by requiring that a shortest path from $s^{\\prime }$ to $s$ goes from $s^{\\prime }$ to the clique and then from the clique to $s$ (without using an edge inside of the clique).","These intermediate $s$ are important as they allow every vertex in the clique to have an edge to some vertex in $S$ and thus be close enough to the $T$ side of the graph in the “no\" case."],["5 vs 8 unweighted undirected construction","In this section we show that under the 3-OV Hypothesis, any algorithm that can distinguish between Diameter 5 and 8 in sparse undirected unweighted graphs, requires $\\Omega (n^{3/2-o(1)})$ time.","Theorem REF gives us the following theorem.","Theorem 17 Given a 3-OV instance consisting of three sets $A,B,C \\subseteq \\lbrace 0,1\\rbrace ^d$ , $|A|=|B|=|C|=n$ , we can in $O(n^2d^2)$ time construct an unweighted, undirected graph with $O(n^2+nd^2)$ vertices and $O(n^2d^2)$ edges that satisfies the following properties.","The graph consists of 4 layers of vertices $S,L_1,L_2,T$ .","The number of nodes in the sets is $|S|=|T|=n^2$ and $|L_1|,|L_2|\\le nd^2$ .","$S$ consists of all tuples $(a,b)$ of vertices $a \\in A$ and $b \\in B$ .","Similarly, $T$ consists of all tuples $(b,c)$ of vertices $b \\in B$ and $c \\in C$ .","If the 3-OV instance has no solution, then $d(u,v)=3$ for all $u \\in S$ and $v \\in T$ .","If the 3-OV instance has a solution $a \\in A, b \\in B, c \\in C$ with $a,b,c$ orthogonal, then $d((a,b)\\in S,(b,c) \\in T)\\ge 7$ .","Setting $k=3,s=1$ in Property 5 of Theorem REF : for any $b^{\\prime } \\in B$ we have $d((a,b) \\in S,(b^{\\prime },c) \\in T)\\ge 5$ and $d((a,b^{\\prime }) \\in S,(b,c) \\in T)\\ge 5$ .","For any vertex $u \\in L_1$ there exists a vertex $s \\in S$ that is adjacent to $u$ .","Similarly, for any vertex $v \\in L_2$ there exists a vertex $t \\in T$ that is adjacent to $v$ .","We can assume that this property holds because we can remove all vertices that do not satisfy this property from the graph and the resulting graph will still satisfy the other properties.","In the rest of the section we use Theorem REF to prove the following result.","Theorem 18 Given a 3-OV instance, we can in $O(n^2d^2)$ time construct an unweighted, undirected graph with $O(n^2+nd^2)$ vertices and $O(n^2d^2)$ edges that satisfies the following two properties.","If the 3-OV instance has no solution, then for all pairs of vertices $u$ and $v$ we have $d(u,v)\\le 5$ .","If the 3-OV instance has a solution, then there exists a pair of vertices $u$ and $v$ such that $d(u,v)\\ge 8$ ."],["Construction of the graph","We construct a graph with the required properties by starting with the graph from Thereom REF and adding more vertices and edges.","Figure REF illustrates the construction of the graph.","Figure: The illustration for the 5 vs 8 construction.","The edges between sets S,L 1 ,L 2 S,L_1,L_2 and TT are not depicted.","The edges between vertices in S ' S^{\\prime } and SS (TT and T ' T^{\\prime }) form a matching.","Vertices in S '' S^{\\prime \\prime } (T '' T^{\\prime \\prime }) form a clique.We start by adding a set $S^{\\prime }$ of $n^2$ vertices.","$S^{\\prime }$ consists of all tuples $(a,b)$ of vertices $a \\in A$ and $b \\in B$ .","We connect every $(a,b) \\in S^{\\prime }$ to its counterpart $(a,b) \\in S$ .","Thus, there is a matching between the sets of vertices $S$ and $S^{\\prime }$ .","We also add another set $S^{\\prime \\prime }$ of $n$ vertices.","$S^{\\prime \\prime }$ contains one vertex $a$ for every $a \\in A$ .","For every pair of vertices from $S^{\\prime \\prime }$ we add an edge between the vertices.","Thus, the $n$ vertices form a clique.","Furthermore, for every vertex $a \\in S^{\\prime \\prime }$ we add an edge to $(a,b) \\in S$ for all $b \\in B$ .","In total we added $n^2+n=O(n^2)$ vertices and $\\binom{n}{2}+2n^2=O(n^2)$ edges.","We do a similar construction for the set $T$ of vertices.","We add a set $T^{\\prime }$ of $n^2$ vertices - one vertex for every tuple $(b,c)$ of vertices $b \\in B$ and $c \\in C$ .","We connect every $(b,c)\\in T^{\\prime }$ to $(b,c) \\in T$ .","Finally, we add a set $T^{\\prime \\prime }$ of $n$ vertices.","$T^{\\prime \\prime }$ contains one vertex for every vector $c \\in C$ .","For every pair of vertices from $T^{\\prime \\prime }$ we add an edge between the vertices.","We connect every $c \\in T^{\\prime \\prime }$ to $(b,c) \\in T$ for all $b \\in B$ .","This finishes the construction of the graph.","In the rest of the section we show that the construction satisfies the promised two properties."],["Correctness of the construction","We need to consider two cases."],["Case 1: the 3-OV instance has no solution","In this case we want to show that for all pairs of vertices $u$ and $v$ we have $d(u,v)\\le 5$ .","We consider three subcases."],["Case 1.1: $u \\in S \\cup S^{\\prime } \\cup S^{\\prime \\prime } \\cup L_1$ and {{formula:e49ba0d1-2dce-4e1e-bc34-0f08bcbafce8}}","We observe that there exists $s \\in S$ that has $d(u,s)\\le 1$ .","Indeed, if $u \\in S$ , then $s=u$ works.","If $u \\in S^{\\prime } \\cup S^{\\prime \\prime }$ , then we are done by the construction.","On the other hand, if $u \\in L_1$ , then there exists such an $s \\in S$ by property 6 from Theorem REF .","Similarly we can show that there exists $t \\in T$ such that $d(v,t)\\le 1$ .","Finally, by property 3 we have that $d(s,t)=3$ .","Thus, we can upper bound the distance between $u$ and $v$ by $d(u,v)\\le d(u,s)+d(s,t)+d(t,v)\\le 1+3+1=5$ as required."],["Case 1.2: $u,v \\in S \\cup S^{\\prime } \\cup S^{\\prime \\prime } \\cup L_1$","From the previous case we know that there are two vertices $s_1,s_2 \\in S$ such that $d(u,s_1)\\le 1$ and $d(s_2,v)\\le 1$ .","To show that $d(u,v)\\le 5$ it is sufficient to show that $d(s_1,s_2)\\le 3$ .","This is indeed true since both vertices $s_1$ and $s_2$ are connected to some two vertices in $S^{\\prime \\prime }$ and every two vertices in $S^{\\prime \\prime }$ are at distance at most 1 from each other."],["Case 1.3: $u,v \\in T \\cup T^{\\prime } \\cup T^{\\prime \\prime } \\cup L_2$","The case is analogous to the previous case."],["Case 2: the 3-OV instance has a solution","In this case we want to show that there is a pair of vertices $u,v$ with $d(u,v)\\ge 8$ .","Let $a \\in A, b \\in B, c \\in C$ be a solution to the 3-OV instance.","We claim that $d((a,b) \\in S^{\\prime },(b,c) \\in T^{\\prime })\\ge 8$ .","Let $P$ be an optimal path between $u=((a,b) \\in S^{\\prime })$ and $v=((b,c) \\in T^{\\prime })$ that achieves the smallest distance.","We want to show that $P$ uses at least 8 edges.","Let $t \\in T$ be the first vertex from the set $T$ that is on path $P$ .","Let $s \\in S$ be the last vertex on path $P$ that belongs to $S$ and precedes $t$ in $P$ .","We can easily check that, if $s \\ne ((a,b) \\in S)$ , then $d(u,s)\\ge 3$ and, similarly, if $t \\ne ((b,c) \\in T)$ , then $d(t,v)\\ge 3$ .","We consider three subcases."],["Case 2.1: $s \\ne ((a,b) \\in S)$ and {{formula:bf5aaac7-5b1b-4304-83ad-0a1fd19bda3a}}","Since $s$ and $t$ are separated by two layers of vertices, we must have $d(s,t)\\ge 3$ .","Thus we get lower bound $d(u,v)\\ge d(u,s)+d(s,t)+d(t,v)\\ge 3+3+3=9>8$ as required."],["Case 2.2: $s = ((a,b) \\in S)$ and {{formula:a28d868f-f38c-4534-b9c1-df1ca7ddd683}}","In this case we use property 4 and conclude $d(u,v)\\ge d(u,s)+d(s,t)+d(t,v)=1+d((a,b) \\in S,(b,c) \\in T)+1\\ge 1+7+1=9>8$ as required."],["Case 2.3: either $s = ((a,b) \\in S)$ or {{formula:1d8b5c54-e4d4-4cf8-8777-ab4e928d855f}} holds but not both","W.l.o.g.","$s \\ne ((a,b) \\in S)$ and $t = ((b,c) \\in T)$ .","If the path uses an edge in the clique on $S^{\\prime \\prime }$ before arriving at $s$ , then $d(u,s)\\ge 4$ and we get that $d(u,v)\\ge d(u,s)+d(s,t)+d(t,v)\\ge 4+3+1=8$ .","On the other hand, if the path does not use any edge of the clique, then $s=((a,b^{\\prime }) \\in S)$ for some $b^{\\prime } \\in B$ .","By property 5 we have $d(s,t)=d((a,b^{\\prime })\\in S,(b,c)\\in T)\\ge 5$ .","We conclude that $d(u,v)\\ge d(u,s)+d(s,t)+d(t,v)\\ge 3+5+1=9>8$ as required."],["6 vs 10 weighted undirected construction","In this section we change the construction from Theorem REF to show that under the 3-OV Hypothesis, any algorithm that can distinguish between diameter 6 and 10 in sparse undirected weighted graphs requires $\\Omega (n^{3/2-o(1)})$ time.","We get the following theorem.","Theorem 19 Given a 3-OV instance, we can in $O(n^2d^2)$ time construct an weighted, undirected graph with $O(n^2+nd^2)$ vertices and $O(n^2d^2)$ edges that satisfies the following two properties.","If the 3-OV instance has no solution, then for all pairs of vertices $u$ and $v$ we have $d(u,v)\\le 6$ .","If the 3-OV instance has a solution, then there exists a pair of vertices $u$ and $v$ such that $d(u,v)\\ge 10$ .","Each edge of the graph has weight either 1 or 2."],["Construction of the graph","The construction of the graph is the same as in Theorem REF except all edges connecting vertices between sets $L_1$ and $L_2$ have weight 2 and all edges inside the cliques on nodes $S^{\\prime \\prime }$ and $T^{\\prime \\prime }$ have weight 2.","All the remaining edges have weight 1."],["Correctness of the construction","The correctness proof is essentially the same as for Theorem REF .","As before we consider two cases."],["Case 1: the 3-OV instance has no solution","In this case we want to show that for all pairs of vertices $u$ and $v$ we have $d(u,v)\\le 6$ .","In the analysis of Case 1 in Theorem REF we show a path between $u$ and $v$ such that the path involves at most one edge from the cliques or between sets $L_1$ and $L_2$ .","Since we added weight 2 to the latter edges, the length of the path increased by at most 1 as a result.","So we have upper bound $d(u,v)\\le 6$ for all pairs $u$ and $v$ of vertices."],["Case 2: the 3-OV instance has a solution","In this case we want to show that there is a pair of vertices $u,v$ with $d(u,v)\\ge 10$ .","Similarly to Theorem REF we will show that $d((a,b) \\in S^{\\prime },(b,c) \\in T^{\\prime })\\ge 10$ , where $a \\in A, b \\in B, c \\in C$ is a solution to the 3-OV instance.","The analysis of the subcases is essentially the same as in Theorem REF .","For cases 2.1 and 2.2 in the proof of Theorem REF we had $d((a,b) \\in S^{\\prime },(b,c) \\in T^{\\prime })\\ge 9$ .","Since we increased edge weights between $L_1$ and $L_2$ to 2 and every path from $(a,b) \\in S^{\\prime }$ to $(b,c) \\in T^{\\prime }$ must cross the layer between $L_1$ and $L_2$ , we also increased the lower bound of the length of the path from 9 to 10 for cases 2.1 and 2.2.","It remains to consider Case 2.3.","As in the proof of Theorem REF , w.l.o.g.","$s \\ne ((a,b) \\in S)$ and $t = ((b,c) \\in T)$ .","If the path uses an edge in the clique on $S^{\\prime \\prime }$ before arriving at $s$ , then $d(u,s)\\ge 5$ and we get lower bound $d(u,v)\\ge d(u,s)+d(s,t)+d(t,v)\\ge 5+4+1=10$ .","On the other hand, if the path does not use any edge of the clique, then $s=((a,b^{\\prime }) \\in S)$ for some $b^{\\prime } \\in B$ .","By property 5 and because we increased edge weights between $L_1$ and $L_2$ to 2, we have $d(s,t)=d((a,b^{\\prime })\\in S,(b,c)\\in T)\\ge 6$ .","We conclude that $d(u,v)\\ge d(u,s)+d(s,t)+d(t,v)\\ge 3+6+1=10$ as required."],["$3k-4$ vs {{formula:cef9204d-0a6e-4579-9476-41520f256b04}} unweighted directed construction","In this section, we show that under SETH, for every $k\\ge 3$ , every algorithm that can distinguish between Diameter $3k-4$ and $5k-7$ in directed unweighted graphs requires $\\Omega (n^{1+1/(k-1)-o(1)})$ time.","Theorem REF gives us the following theorem.","Theorem 20 Given a $k$ -OV instance consisting of $k\\ge 2$ sets $W_0,W_1,\\dots ,W_{k-1} \\subseteq \\lbrace 0,1\\rbrace ^d$ , each of size $n$ , we can in $O(kn^{k-1}d^{k-1})$ time construct an unweighted, undirected graph with $O(n^{k-1}+k n^{k-2} d^{k-1})$ vertices and $O(k n^{k-1} d^{k-1})$ edges that satisfies the following properties.","The graph consists of $k+1$ layers of vertices $S=L_0,L_1,L_2,\\dots ,L_k=T$ .","The number of nodes in the sets is $|S|=|T|=n^{k-1}$ and $|L_1|,|L_2|,\\dots ,|L_{k-1}|\\le n^{k-2}d^{k-1}$ .","$S$ consists of all tuples $(a_0,a_1,\\ldots , a_{k-2})$ where for each $i$ , $a_i\\in W_i$ .","Similarly, $T$ consists of all tuples $(b_1,b_2,\\ldots , b_{k-1})$ where for each $i$ , $b_i\\in W_i$ .","If the $k$ -OV instance has no solution, then $d(u,v)=k$ for all $u \\in S$ and $v \\in T$ .","If the $k$ -OV instance has a solution $a_0, a_1,\\dots ,a_{k-1}$ where for each $i$ , $a_i\\in W_i$ then if $\\alpha =(a_0,\\dots a_{k-2})\\in S$ and $\\beta = (a_1,\\dots ,a_{k-1}) \\in T$ , then $d(\\alpha ,\\beta )\\ge 3k-2$ .","Setting $s=k-2$ in Property 5 of Theorem REF : If the $k$ -OV instance has a solution $a_0, a_1,\\dots ,a_{k-1}$ where for each $i$ , $a_i\\in W_i$ then for any tuple $(b_1,\\dots ,b_{k-2})$ , if $\\alpha =(a_0,b_1,\\dots ,b_{k-2})\\in S$ and $\\beta =(a_1,\\dots ,a_{k-1})\\in T$ , then $d(\\alpha ,\\beta )\\ge k+2$ .","Symmetrically, if $\\alpha =(a_0,a_1,\\dots ,a_{k-2})\\in S$ and $\\beta =(b_1,\\dots ,b_{k-2},a_{k-1})\\in T$ , then $d(\\alpha ,\\beta )\\ge k+2$ .","For all $i$ from 1 to $k-1$ , for all $v \\in L_i$ there exists a vertex in $L_{i-1}$ adjacent to $v$ and a vertex in $L_{i+1}$ adjacent to $v$ .","We can assume that this property holds because we can remove all vertices that do not satisfy this property from the graph and the resulting graph will still satisfy the previous three properties.","In the rest of the section we use Theorem REF to prove the following result.","Theorem 21 Given a $k$ -OV instance, we can in $O(kn^{k-1}d^{k-1})$ time construct an unweighted, directed graph with $O(k n^{k-1}+k n^{k-2} d^{k-1})$ vertices and $O(kn^{k-1}d^{k-1})$ edges that satisfies the following two properties.","If the $k$ -OV instance has no solution, then for all pairs of vertices $u$ and $v$ we have $d(u,v)\\le 3k-4$ .","If the $k$ -OV instance has a solution, then there exists a pair of vertices $u$ and $v$ such that $d(u,v)\\ge 5k-7$ ."],["Construction of the graph","We construct a graph with the required properties by starting with the graph from Thereom REF and adding more vertices and edges.","First we will construct a weighted graph and then we will make it unweighted.","Figure REF illustrates the construction of the graph for the special case $k=4$ .","Figure: The 3k-43k-4 vs 5k-75k-7 construction for the special case k=4k=4.","The edges between sets S,L 1 ,L 2 ,L 3 S,L_1,L_2,L_3 and TT are not depicted.","The matching between sets SS and S ' S^{\\prime } consists of unweighted paths of length k-2=2k-2=2.","The edges between sets SS and S '' S^{\\prime \\prime } consists of unweighted paths of length k-2=2k-2=2.","Similarly for the right side.We start by adding a set $S^{\\prime }$ of $n^{k-1}$ vertices.","$S^{\\prime }$ consists of all tuples $(a_0,a_1,\\ldots , a_{k-2})$ where for each $i$ , $a_i\\in W_i$ .","We connect every $(a_0,a_1,\\ldots , a_{k-2}) \\in S^{\\prime }$ to its counterpart $(a_0,a_1,\\ldots , a_{k-2}) \\in S$ with an undirected edge of weight $k-2$ to form a matching.","We also add another set $S^{\\prime \\prime }$ of $n$ vertices.","$S^{\\prime \\prime }$ contains one vertex $a_0$ for every $a_0 \\in W_0$ .","For every pair of vertices in $S^{\\prime \\prime }$ we add an undirected edge of weight 1 between the vertices.","Thus, the $n$ vertices form a clique.","Furthermore, for every vertex $a_0 \\in S^{\\prime \\prime }$ we add an undirected edge of weight $k-2$ to $(a_0,b_1,\\ldots , b_{k-2}) \\in S$ for all $b_1,\\dots , b_{k-2}$ .","Finally for every vertex $a_0 \\in S^{\\prime \\prime }$ we add a directed edge of weight 1 towards $(a_0,b_1,\\ldots , b_{k-2}) \\in S$ for all $b_1,\\dots , b_{k-2}$ .","Some of the edges that we added have weight $k-2$ .","We make those unweighted by subdividing them into edges of weight 1.","Let $S^{\\prime \\prime \\prime }$ be the set of newly added vertices.","In total we added $O(kn^{k-1})$ vertices and $O(kn^{k-1})$ edges.","We do a similar construction for the set $T$ of vertices.","We add a set $T^{\\prime }$ of $n^{k-1}$ vertices — one vertex for every tuple $(a_1,\\dots ,a_{k-1})$ where for each $i$ , $a_i\\in W_i$ .","We connect every $(a_1,\\dots ,a_{k-1})\\in T^{\\prime }$ to $(a_1,\\dots ,a_{k-1}) \\in T$ by an undirected edge of weight $k-2$ .","Finally, we add a set $T^{\\prime \\prime }$ of $n$ vertices.","$T^{\\prime \\prime }$ contains one vertex for every vector $a_{k-1} \\in W_{k-1}$ .","We connect every pair of vertices in $T^{\\prime \\prime }$ by an undirected edge of weight 1.","We connect every vertex $a_{k-1} \\in T^{\\prime \\prime }$ to $(b_1,\\dots ,b_{k-2},a_{k-1}) \\in T$ by an undirected edge of weight $k-2$ for all $b_1,\\dots ,b_{k-2}$ .","Also, for every vertex $a_{k-1} \\in T^{\\prime \\prime }$ we add a directed edge of weight 1 from $(b_1,\\dots ,b_{k-2},a_{k-1}) \\in T^{\\prime }$ to $a_{k-1}$ for all $b_1,\\dots ,b_{k-2}$ .","Some of the edges that we just added have weight $k-2$ .","We make those unweighted by subdividing them into edges of weight 1.","Let $T^{\\prime \\prime \\prime }$ be the set of newly added vertices.","This finishes the construction of the graph.","In the rest of the section we show that the construction satisfies the promised two properties stated in Theorem REF ."],["Correctness of the construction","We need to consider two cases."],["Case 1: the $k$ -OV instance has no solution","In this case we want to show that for all pairs of vertices $u$ and $v$ we have $d(u,v)\\le 3k-4$ .","We consider subcases."],["Case 1.1: $u \\in S \\cup S^{\\prime } \\cup S^{\\prime \\prime } \\cup S^{\\prime \\prime \\prime } \\cup L_i$ for {{formula:341c2d42-7167-4db5-bd6a-4b4996ef5fb1}} and {{formula:b26c11ee-44d0-4a3f-8d06-2943a8aaf115}} for {{formula:b2ebe27d-5f57-42a2-b20b-50932506480e}}","We observe that there exists $s \\in S$ that has $d(u,s)\\le k-2$ .","Similarly, there exists $t \\in T$ with $d(t,v)\\le k-2$ .","By property 3 from Theorem REF we have that $d(s,t)\\le k$ .","This gives us upper bound $d(u,v)\\le d(u,s)+d(s,t)+d(t,v)\\le (k-2)+k+(k-2)=3k-4$ as required.","The proof when the sets for $u$ and $v$ are swapped is identical since we only use paths on unweighted edges."],["Case 1.2: $u,v \\in S \\cup S^{\\prime } \\cup S^{\\prime \\prime } \\cup S^{\\prime \\prime \\prime } \\cup L_1$","We note that there is some vertex $s\\in S^{\\prime \\prime }$ with $d(u,s)\\le 2(k-2)$ (via undirected edges).","Also, there is some vertex $s^{\\prime }\\in S^{\\prime \\prime }$ with $d(s^{\\prime },v)\\le k-1$ (possibly using directed edges).","$S^{\\prime \\prime }$ is a clique so $d(s,s^{\\prime })\\le 1$ .","Thus, $d(u,v)\\le d(u,s)+d(s,s^{\\prime })+d(s^{\\prime },v)\\le 2(k-2)+1+(k-1)=3k-4$ ."],["Case 1.3: $u,v \\in T \\cup T^{\\prime } \\cup T^{\\prime \\prime } \\cup T^{\\prime \\prime \\prime } \\cup L_{k-1}$","This case is similar to the previous case.","We note that there is some vertex $t\\in T^{\\prime \\prime }$ with $d(t,v)\\le 2(k-2)$ (via undirected edges).","Also, there is some vertex $t^{\\prime }\\in T^{\\prime \\prime }$ with $d(u,t^{\\prime })\\le k-1$ (possibly using directed edges).","$S^{\\prime \\prime }$ is a clique so $d(t^{\\prime },t)\\le 1$ .","Thus, $d(u,v)\\le d(u,t^{\\prime })+d(t^{\\prime },t)+d(t,v)\\le (k-1)+1+2(k-2)=3k-4$ ."],["Case 2: the $k$ -OV instance has a solution","In this case we want to show that there is a pair of vertices $u,v$ with $d(u,v)\\ge 5k-7$ .","Let $(a_0,a_1,\\ldots , a_{k-1})$ be a solution to the $k$ -OV instance where for each $i$ , $a_i\\in W_i$ .","We claim that $d((a_0,\\dots ,a_{k-2}) \\in S^{\\prime },(a_1,\\dots ,a_{k-1}) \\in T^{\\prime })\\ge 5k-7$ .","Let $P$ be an shortest path between $u=((a_0,\\dots ,a_{k-2}) \\in S^{\\prime })$ and $v=((a_1,\\dots ,a_{k-1}) \\in T^{\\prime })$ .","We want to show that $P$ uses at least $5k-7$ edges.","Let $s \\in S$ be the first vertex on path $P$ that belongs to $S$ and let $t \\in T$ be the last vertex from the set $T$ that is on path $P$ .","We observe that due to the directionality of the edges, $s$ and $t$ must be the counterparts of $u$ and $v$ respectively; that is, $s=((a_0,\\dots ,a_{k-2}) \\in S)$ and $t=((a_1,\\dots ,a_{k-1}) \\in T)$ .","Note that these definitions of $s$ and $t$ differ from the definitions of $s$ and $t$ in previous proofs.","We consider three subcases."],["Case 2.1: A vertex in $S^{\\prime }\\cup S^{\\prime \\prime }\\cup S^{\\prime \\prime \\prime }$ appears after {{formula:6406b760-7ede-4dc4-b0f6-a59f46a09387}} on the path {{formula:a6c998cf-9396-4ea3-832a-1c6dca88cdb2}}","We observe that if $s_1,s_2\\in S$ is a pair of vertices on the path $P$ such that no vertex in $S$ appears between them on $P$ , then the portion of $P$ between $s_1$ and $s_2$ either contains only vertices in $S^{\\prime }\\cup S^{\\prime \\prime }\\cup S^{\\prime \\prime \\prime }$ or contains no vertices in $S^{\\prime }\\cup S^{\\prime \\prime }\\cup S^{\\prime \\prime \\prime }$ .","Let $s_1,s_2\\in S$ be such that the portion of $P$ between them contains only vertices in $S^{\\prime }\\cup S^{\\prime \\prime }\\cup S^{\\prime \\prime \\prime }$ .","Such $s_1,s_2$ exist by the specification of this case.","If $s_1=s_2$ then $P$ is not a shortest path.","Otherwise, the portion of $P$ between $s_1$ and $s_2$ must include a vertex in $S^{\\prime \\prime }$ .","Thus, $d(s_1,s_2)\\ge 2(k-2)$ .","We consider three subcases.","$s_1\\ne s$ .","The distance between any pair of vertices in $S$ is at least 2 so $d(s,s_1)\\ge 2$ .","Then, $d(u,v)\\ge d(u,s)+d(s,s_1)+d(s_1,s_2)+d(s_2,t)+d(t,v)\\ge (k-2)+2+2(k-2)+k+(k-2)=5k-6$ .","$s_1=s$ and $s_2=((a_0,b_1,\\dots ,b_{k-2}) \\in S)$ for some $b_1,\\dots ,b_{k-2}$ .","In this case, by property 5 we have $d(s_2,t)\\ge k+2$ .","Thus, $d(u,v)\\ge d(u,s_1)+d(s_1,s_2)+d(s_2,t)+d(t,v)\\ge (k-2)+2(k-2)+(k+2)+(k-2)=5k-6$ .","$s_1=s$ and $s_2=((b_0,\\dots ,b_{k-2} \\in S)$ for some with $b_0\\ne a_0$ .","In this case, the path from $s_1$ to $s_2$ must include an edge in the clique $S^{\\prime \\prime }$ since these are the only edges among vertices in $S^{\\prime }\\cup S^{\\prime \\prime }\\cup S^{\\prime \\prime \\prime }$ for which adjacent tuples can differ with respect to their first element.","Thus, $d(s_1,s_2)\\ge 2(k-2)+1\\ge 2k-3$ .","Therefore, $d(u,v)\\ge d(u,s_1)+d(s_1,s_2)+d(s_2,t)+d(t,v)\\ge (k-2)+(2k-3)+k+(k-2)=5k-7$ ."],["Case 2.2: A vertex in $T^{\\prime }\\cup T^{\\prime \\prime }\\cup T^{\\prime \\prime \\prime }$ appears before {{formula:2e52092d-2023-4c53-a764-a41805329e43}} on the path {{formula:d86d9c20-8168-4be0-8d10-99eb7834ec04}}","This case is analogous to the previous case."],["Case 2.3: The portion of the path $P$ between {{formula:558fd34a-9236-4d24-8a2c-f5022f0a499a}} and {{formula:c0c71b02-54cb-42b0-8842-f210ddc962a7}} contains no vertices in {{formula:7c23b5b2-7499-4d20-8946-fe3d3dacb80d}}","By property 4, $d(s,t)\\ge 3k-2$ .","Thus, $d(u,v)\\ge d(u,s)+d(s,t)+d(t,v)\\ge (k-2)+(3k-2)+(k-2)=5k-6$ .","We note that a slight modification of this construction gives a lower bound for higher values of Diameter.","For any L, we can get an $L(3k-4)$ vs $L(5k-8)+1$ construction by subdividing all of the edges in the construction (into paths of length $L$ ) except for the directed edges and the edges within the cliques."],["2-Approximation for Eccentricities in $\\tilde{O}(m\\sqrt{n})$ time","Theorem 22 Given a weighted, directed $m$ edge $n$ node graph, in $\\tilde{O}(m\\sqrt{n})$ time we can output quantities $\\epsilon ^{\\prime }(v)$ such that for all $v \\in V$ we have $\\epsilon (v)/2\\le \\epsilon ^{\\prime }(v)\\le \\epsilon (v)$ .","The algorithm is inspired by the 2-approximation algorithm for directed radius of Abboud, Vassilevska W., and Wang [7].","We claim that the following algorithm achieves the above guarantees.","Sample a random subset $S \\subset V$ of size $|S|=\\Theta (\\sqrt{n} \\log n)$ .","With high probability for every $u \\in V$ we have $N^{\\text{in}}_{\\sqrt{n}}(u)\\cap S \\ne \\emptyset $ .","Let $w$ be a vertex that maximizes $d(S,w)$ , which we find using Dijkstra's algorithm.","Let $S^{\\prime }:=N^{\\text{in}}_{\\sqrt{n}}(w)$ .","For every vertex $v \\in S^{\\prime }$ we output $\\epsilon ^{\\prime }(v)=\\epsilon (v)$ by running Dijkstra's algorithm and following the outgoing edges.","For every vertex $v \\notin S^{\\prime }$ we output the estimate $\\epsilon ^{\\prime }(v)=\\max _{s \\in S\\cup \\lbrace w\\rbrace }d(v,s)$ .","We can determine all these quantities by running Dijkstra's algorithm out of all vertices in $S \\cup \\lbrace w\\rbrace $ and following the incoming edges."],["Correctness","Consider an arbitrary vertex $v\\notin S^{\\prime }$ (if $v \\in S^{\\prime }$ , then we are done by the third step).","If there exists $s \\in S$ such that $d(v,s)\\ge \\epsilon (v)/2$ , then we are done since $\\epsilon ^{\\prime }(v)\\ge d(v,s)\\ge \\epsilon (v)/2$ .","Otherwise, we have $d(v,s)<\\epsilon (v)/2$ for all $s \\in S$ .","Let $v^{\\prime }$ be a vertex that achieves $d(v,v^{\\prime })=\\epsilon (v)$ .","By the triangle inequality we have $d(s,v^{\\prime })>\\epsilon (v)/2$ for all $s \\in S$ .","Equivalently, $d(S,v^{\\prime })>\\epsilon (v)/2$ .","This implies that $d(S,w)>\\epsilon (v)/2$ by our choice of $w$ .","Since $d(S,w)>\\epsilon (v)/2$ and $S^{\\prime }=N^{\\text{in}}_{\\sqrt{n}}(w)$ intersects $S$ , we must have that $S^{\\prime }$ contains all vertices $u$ with $d(u,w)\\le \\epsilon (v)/2$ .","Since $v \\notin S^{\\prime }$ , we must have $d(v,w)>\\epsilon (v)/2$ and we are done since $\\epsilon ^{\\prime }(v)\\ge d(v,w)>\\epsilon (v)/2$ ."],["Almost 2-Approximation for Eccentricities in almost linear time","In contrast to our $\\tilde{O}(m\\sqrt{n})$ time algorithm from the previous section, our near-linear time $(2+\\delta )$ -approximation algorithm is very different from all previously known algorithms.","Our algorithm proceeds in iterations and maintains a set $S$ of nodes for which we still do not have a good eccentricity estimate.","In each iteration either we get a good estimate for many new vertices and hence remove them from $S$ , or we remove all vertices from $S$ that have large eccentricities, and for the remaining nodes in $S$ we have a better upper bound on their eccentricities.","After a small number of iterations we have a good estimate for all vertices of the graph.","Theorem 23 Suppose that we are given a weighted, directed $m$ edge $n$ node graph.","The weights of all edge are non-negative integers bounded by $n^{O(1)}$ .","For any $1>\\tau >0$ we can in $\\tilde{O}(m/\\tau )$ time output quantities $\\varepsilon ^{\\prime }(v)$ such that for all $v \\in V$ we have ${1-\\tau \\over 2}\\varepsilon (v)\\le \\varepsilon ^{\\prime }(v)\\le \\varepsilon (v)$ .","We maintain a subset $S\\subseteq V$ of vertices $v$ for which we still do not have an estimate $\\varepsilon ^{\\prime }(v)$ .","Initially $S=V$ and we will end with $|S|\\le O(1)$ .","When $|S|\\le O(1)$ we can evaluate $\\varepsilon (v)$ for all $v \\in S$ in the total time of $O(m)$ .","Also we maintain a value $D$ that upper bounds the largest eccentricity of a vertex in $S$ .","That is, $\\varepsilon (v)\\le D$ for all $v \\in S$ .","Initially we set $D=n^C$ for some large enough constant $C>0$ (we assume that the input graph is strongly connected).","The algorithm proceeds in phases.","Each phase takes $\\tilde{O}(m)$ time and either $|S|$ decreases by a factor of at least 2 or $D$ decreases by a factor of at least $1/(1-\\tau )$ .","After $O(\\log (n)/\\tau )$ phases either $|S|\\le O(1)$ or $D<1$ .","For a subset $S \\subseteq V$ of vertices and a vertex $x \\in V$ we define a set $S_x\\subseteq S$ to contain those $|S_x|=|S|/2$ vertices from $S$ that are closest to $x$ (according to distance $d(\\cdot ,x)$ ).","The ties are broken by taking the vertex with the smaller id.","Given a subset $S \\subseteq V$ of vertices and a threshold $D$ , a phase proceeds as follows.","[itemsep=0mm] We sample a set $A \\subseteq S$ of $O(\\log n)$ random vertices from the set $S$ .","With high probability for all $x \\in V$ we have $A \\cap S_x \\ne \\emptyset $ .","Let $w \\in V$ be a vertex that maximizes $d(A,w)$ .","We can find it using Dijkstra's algorithm.","We consider two cases.","Case $d(S\\setminus S_w,w)\\ge {1-\\tau \\over 2}D$ .","For all $x \\in S\\setminus S_w$ we have ${1-\\tau \\over 2} D \\le \\varepsilon (x)\\le D$ and we assign the estimate $\\varepsilon ^{\\prime }(x)={1-\\tau \\over 2}D$ .","This gives us that ${1-\\tau \\over 2} \\varepsilon (x)\\le {1-\\tau \\over 2}D=\\varepsilon ^{\\prime }(x)\\le \\varepsilon (x)$ for all $x\\in S\\setminus S_w$ .","We update $S$ to be $S_w$ .","This decreases the size of $S$ by a factor of 2 as required.","Case $d(S\\setminus S_w,w)<{1-\\tau \\over 2}D$ .","Set $S^{\\prime }=S$ .","For every vertex $v \\in S$ evaluate $r_v:=\\max _{x \\in A}d(v,x)$ .","We can evaluate these quantities by running Dijkstra's algorithm from every vertex in $A$ and following the incoming edges.","If $r_v\\ge {1-\\tau \\over 2}D$ , then assign the estimate $\\varepsilon ^{\\prime }(v)={1-\\tau \\over 2}D$ and remove $v$ from $S^{\\prime }$ .","Similarly as in the previous case we have ${1-\\tau \\over 2} \\varepsilon (v)\\le \\varepsilon ^{\\prime }(v)\\le \\varepsilon (v)$ for all $v \\in S\\setminus S^{\\prime }$ .","Below we will show that for every $v \\in S^{\\prime }$ we have $\\varepsilon (v)\\le (1-\\tau )D$ .","Thus we can update $S=S^{\\prime }$ and decrease the threshold $D$ to $(1-\\tau )D$ as required.","Correctness We have to show that, if there exists $v \\in S^{\\prime }$ such that $\\varepsilon (v)>(1-\\tau )D$ , then we will end up in the first case (this is the contrapositive of the claim in the second case).","Since $v \\in S^{\\prime }$ we must have that $d(v,x)\\le {1-\\tau \\over 2}D$ for all $x \\in A$ .","Since $\\varepsilon (v)>(1-\\tau )D$ , we must have that there exists $v^{\\prime }$ such that $d(v,v^{\\prime })>(1-\\tau )D$ .","By the triangle inequality we get that $d(x,v^{\\prime })>{1 - \\tau \\over 2}D$ for every $x \\in A$ .","By choice of $w$ we have $d(A,w)>{1-\\tau \\over 2}D$ .","Since $A \\cap S_w \\ne \\emptyset $ , we have $d(S\\setminus S_w,w)\\ge {1-\\tau \\over 2}D$ and we will end up in the first case.","The guarantee on the approximation factor follows from the description.","As a corollary, we get an algorithm for Source Radius with the same runtime and approximation ratio as Theorem REF .","First, run the Eccentricities algorithm and let $v$ be the vertex with the smallest estimated eccentricity $\\epsilon ^{\\prime }(v)$ .","Then run Dijkstra's algorithm from $v$ and report $\\epsilon (v)$ as the Radius estimate $R^{\\prime }$ .","Let $R$ be the true radius of the graph and let $x$ be the vertex with minimum Eccentricity i.e.","$\\epsilon (x)=R$ .","If $\\alpha $ is the approximation ratio for the Eccentricities algorithm then $\\epsilon (v)\\le \\alpha \\epsilon ^{\\prime }(v)\\le \\alpha \\epsilon (v)$ and $\\epsilon (x)\\le \\alpha \\epsilon ^{\\prime }(x)\\le \\alpha \\epsilon (x)$ .","By choice of $v$ , $\\epsilon ^{\\prime }(v)\\le \\epsilon ^{\\prime }(x)$ .","Thus, $\\alpha R=\\alpha \\epsilon (x)\\ge \\alpha \\epsilon ^{\\prime }(x)\\ge \\alpha \\epsilon ^{\\prime }(v)\\ge \\epsilon (v)=R^{\\prime }$ .","Clearly $R^{\\prime }\\ge R$ , so $R\\le R^{\\prime }\\le \\alpha R$ .","$S$ -$T$ Diameter algorithms Recall that the $S$ -$T$ diameter problem is as follows: Given an undirected graph $G=(V,E)$ and two sets $S\\subseteq V, T\\subseteq V$ , determine $D_{S,T}=\\max _{s\\in S, t\\in T} d(s,t)$ .","Here we will outline two algorithms for the problem.","Let us first consider a fast 3-approximation algorithm.","Claim 24 There is an $O(m+n)$ time algorithm that for any $n$ node $m$ edge graph $G=(V,E)$ and $S\\subseteq V, T\\subseteq V$ , computes an estimate $D^{\\prime }$ such that $D_{S,T}/3\\le D^{\\prime }\\le D_{S,T}$ and two nodes $s\\in S$ , $t\\in T$ such that $d(s,t)=D^{\\prime }$ .","In graphs with nonnegative weights, the same estimate can be achieved in $O(m+n\\log n)$ time.","The algorithm is extremely simple: pick arbitrary nodes $s\\in S$ and $t\\in T$ , compute BFS($s$ ) and BFS($t$ ) and return $\\max \\lbrace \\max _{t^{\\prime }\\in T} d(s,t^{\\prime }),\\max _{s^{\\prime }\\in S} d(s^{\\prime },t)\\rbrace $ (also returning the two nodes achieving the maximum).","For weighted graphs, run Dijkstra's algorithm instead of BFS.","Let's see why this algorithm provides the promised guarantee.","Suppose that for every $t^{\\prime }\\in T$ , $d(s,t^{\\prime })<D_{S,T}/3$ (otherwise we are done).","Then for every $t^{\\prime },t^{\\prime \\prime }\\in T$ , $d(t^{\\prime },t^{\\prime \\prime })\\le d(t^{\\prime },s)+d(s,t^{\\prime \\prime })<2D_{S,T}/3$ .","In particular, for all $t^{\\prime }\\in T$ , $d(t,t^{\\prime })<2D_{S,T}/3$ .","If we also had that for every $s^{\\prime }\\in S$ , $d(t,s^{\\prime })<D_{S,T}/3$ , then we'd get that for all $s^{\\prime }\\in S, t^{\\prime }\\in T$ , $d(s^{\\prime },t^{\\prime })\\le d(s^{\\prime },t)+d(t,t^{\\prime })<D_{S,T}$ , contradicting the definition of $D_{S,T}$ .","Thus, $\\max \\lbrace \\max _{t^{\\prime }\\in T} d(s,t^{\\prime }),\\max _{s^{\\prime }\\in S} d(s^{\\prime },t)\\rbrace \\ge D_{S,T}/3$ .","We will now show an analogue to the $\\tilde{O}(m\\sqrt{n})$ time almost-$3/2$ -approximation diameter algorithm of Roditty and Vassilevska W. [41] for $S$ -$T$ Diameter giving a 2-approximation.","Using a trick from Chechik et al.","[21] we also obtain a true 2 approximation algorithm running in $\\tilde{O}(m^{3/2})$ .","2-Approximation for $S$ -$T$ Diameter [1] 2-Approx $X$ - random sample of nodes, $|X|=\\Theta (\\sqrt{n} \\log n)$ $D_1:=0$ every $x\\in X$ Run BFS($x$ ) Let $t_x$ be the closest node to $x$ in $T$ Run BFS($t_x$ ) $D_1 = \\max \\lbrace D_1,\\max _{s\\in S} d(s,t_x)\\rbrace $ [t]@X@ Let $\\bar{t}$ be the furthest node of $T$ from $X$ (computed above) Run BFS($\\bar{t}$ ) $D_2=\\max _{s\\in S} d(s,\\bar{t})$ .","Let $Y$ be the closest $\\sqrt{n}$ nodes to $\\bar{t}$ .","every $y\\in Y$ Run BFS($y$ ) Let $s_y$ be the closest node to $y$ in $S$ Run BFS($s_y$ ) $D_2 = \\max \\lbrace D_2,\\max _{t\\in T} d(s_y,t)\\rbrace $ $\\max \\lbrace D_1,D_2\\rbrace $ We use Algorithm 2-Approx to prove: Theorem 25 In $\\tilde{O}(m\\sqrt{n})$ time one can obtain an estimate $D^{\\prime }$ to the $S$ -$T$ diameter $D$ of an $m$ edge $n$ node unweighted undirected graph such that $2\\lfloor D/4\\rfloor \\le D^{\\prime }\\le D$ .","In $\\tilde{O}(m^{3/2})$ time one can obtain an estimate $D^{\\prime \\prime }$ such that $D/2\\le D^{\\prime \\prime }\\le D$ .","First we analyze Algorithm 2-Approx.","Let $s^*\\in S$ and $t^*\\in T$ be the end points of the $S$ -$T$ Diameter path so that $d(s^*,t^*)=D$ .","Let $d=\\lfloor D/4\\rfloor $ .","Suppose first that for some $x\\in X$ , $d(x,t^*)\\le d$ .","Then, $d(x,t_x)\\le d(x,t^*)\\le d$ and hence $d(t_x,t^*)\\le d(t_x,x)+d(x,t^*)\\le 2d$ .","However, then $d(t_x,s^*)\\ge d(t^*,s^*)-d(t^*,t_x)\\ge D-2d\\ge D/2$ .","Thus, if $D_1<D/2$ , it must be that for every $x\\in X$ , $d(x,t^*)\\ge d+1$ .","Hence, for every $x \\in X$ , $d(x,\\bar{t})\\ge d(x,t^*)\\ge d+1$ by the definition of $\\bar{t}$ .","If $d(\\bar{t},s^*)\\ge D/2$ , then $D_2\\ge D/2$ and we are done, so let us assume that $d(\\bar{t},s^*)\\le D/2$ .","Now, as $X$ is random of size $c\\sqrt{n}\\log n$ for large enough $c$ , with high probability, $X$ hits the $\\sqrt{n}$ -neighborhoods of all nodes.","In particular, $X\\cap Y\\ne \\emptyset $ .","However, since $d(x,\\bar{t})\\ge d+1$ for every $x\\in X$ , it must be that $Y$ contains all nodes at distance $d$ from $\\bar{t}$ as it contains all nodes closer to $\\bar{t}$ than $x\\in Y\\cap X$ .","If $s^*\\in Y$ , then we would have run BFS from $s^*$ and returned $D$ .","Hence $d(\\bar{t},s^*)>d$ .","Let $a$ be the node on the shortest path between $\\bar{t}$ and $s^*$ with $d(\\bar{t},a)=d$ .","We thus have that $a\\in Y$ .","Also, since $d(\\bar{t},s^*)\\le D/2$ , $d(a,s^*)\\le D/2-d$ and hence $d(a,s_a)\\le D/2-d$ , so that $d(s_a,t^*)\\ge D-2(D/2-d)\\ge 2d$ .","This finishes the argument that 2-Approx returns an estimate $D^{\\prime }$ with $2\\lfloor D/4\\rfloor \\le D^{\\prime }\\le D$ .","It is not too hard to see that the only time that we might get an estimate that is less than $D/2$ is in the last part of the argument and only if the diameter is of the form $4d+3$ .","(We will prove the algorithm guarantees formally soon.)","The analysis fails to work in that case because $Y$ is guaranteed to contain only the nodes at distance $d$ from $\\bar{t}$ .","In particular, if $Y$ contains all nodes at distance $d+1$ from $\\bar{t}$ instead of just those at distance at most $d$ , we could consider $a$ to be the node on the shortest path between $\\bar{t}$ and $s^*$ with $d(\\bar{t},a)=d+1$ , and $a\\in Y$ .","Now since $d(\\bar{t},s^*)\\le 2d+1$ (as otherwise we'd be done), $d(a,s_a)\\le d(a,s^*)\\le 2d+1-d-1 =d$ , so that $d(s_a,t^*)\\ge 2d+3$ .","Hence everything would work out.","We handle this issue with a trick from Chechik et al. [21].","First, we make graph have constant degree by blowing up the number of nodes and adding 0 weight edges as follows.","Let $v$ be an original node and suppose it has degree $d(v)$ .","Replace $v$ with a $d(v)$ -cycle of 0 weight edges so that each of the cycle nodes is connected to a one of the neighbors of $v$ , where each neighbor has a cycle node corresponding to it.","This makes every node have degree 3 and increases the number of vertices to $O(m)$ .","Now, we run algorithm 2-Approx with two changes.","The first is that instead of BFS we use Dijkstra's algorithm because the edges now have weights.","The second change is that we redefine $Y$ as follows.","Let $Z$ be the closest $\\sqrt{m}$ nodes $Z$ to $\\bar{t}$ .","Define $Y$ to be $Z$ , together with all nodes that have an edge of weight 1 to some node of $Z$ .","Now, consider the shortest path $P$ between $\\bar{t}$ and $s^*$ .","Consider the last node $a$ of $P$ (in the direction from $\\bar{t}$ to $s^*$ ) that has distance $\\le d$ from $\\bar{t}$ .","The node $a^{\\prime }$ after $a$ on $P$ must be connected to $a$ by an edge $(a,a^{\\prime })$ of weight 1, as otherwise $a^{\\prime }$ would be at distance at most $d$ from $\\bar{t}$ .","Since $a\\in Z$ , we must have $a^{\\prime }\\in Y$ and we know also that $d(\\bar{t},a^{\\prime })=d+1$ .","In the last case when $d(\\bar{t},s^*)\\le 2d+1$ , we get that $Y$ contains a node $a^{\\prime }$ of distance $\\le (2d+1)-(d+1) = d$ from $s^*$ and hence $d(a^{\\prime },s_{a^{\\prime }})\\le d$ and hence $d(s_{a^{\\prime }},t^*)\\ge 2d+3$ .","Since every node has degree 3, the number of nodes in $Y$ is $\\le 4|Z|\\le O(\\sqrt{m})$ and hence we can afford to run Dijkstra from each of them and complete the algorithm in $\\tilde{O}(m^{3/2})$ time.","Let us now formally analyze the guarantees of the algorithm.","Suppose that $D=4d+z$ where $z\\in \\lbrace 0,1,2,3\\rbrace $ ; we will show that the estimate that the algorithm gives is always at least $\\lceil D/2\\rceil =2d+\\lceil z/2\\rceil $ .","If some node $x\\in X$ has $d(x,t^*)\\le d$ , we get that $d(t_x,s^*)\\ge D-2d=2d+z\\ge \\lceil D/2\\rceil $ .","If we are not done, all nodes of $X$ have $d(x,\\bar{t})\\ge d(x,t^*)\\ge d+1$ and $Z$ contains all nodes at distance $\\le d$ from $\\bar{t}$ .","If $s^*\\in Z$ , we are done so we must have $d(s^*,\\bar{t})\\ge d+1$ .","Consider the last node $a^{\\prime }$ on the $\\bar{t}$ to $s^*$ shortest path (in the direction towards $s^*$ ) for which $d(\\bar{t},a^{\\prime })\\le d$ .","We have that $a^{\\prime }\\in Z$ .","Also, the node $a$ after $a^{\\prime }$ on the $\\bar{t}$ to $s^*$ shortest path must be in $Y$ since by the choice of $a^{\\prime }$ , $d(\\bar{t},a)=t+1$ and $(a,a^{\\prime })$ is an edge of weight 1.","If $d(\\bar{t},s^*)\\ge 2d+\\lceil z/2\\rceil $ , we are done, so we get that $d(a,s^*)\\le 2d+\\lceil z/2\\rceil -1 - (d+1) =d+\\lceil z/2\\rceil -2$ .","Hence, $d(s_a,t^*)\\ge D-2(d+\\lceil z/2\\rceil -2)=2d + (z+4-2\\lceil z/2\\rceil )\\ge 2d+z$ .","It is not hard to extend the $S$ -$T$ Diameter algorithms to work for weighted undirected graphs as well.","The basic idea is to use Dijkstra's algorithm instead of BFS in Algorithm 1.","This gives an almost 2-approximation.","In particular, let $a^{\\prime }$ be the last node on the $\\bar{t}$ to $s^*$ shortest path that is at distance $\\le d$ from $\\bar{t}$ and let $a$ be the node after $a^{\\prime }$ .","The weighted version of Algorithm 1 achieves a guarantee $D^{\\prime }$ of the $S$ -$T$ Diameter, such that $D/2-2w(a,a^{\\prime })\\le D^{\\prime }\\le D$ .","We can obtain an $\\tilde{O}(m^{3/2})$ true 2-approximation algorithm similar to the proof above.","Let $Z$ be the closest $\\sqrt{m}$ nodes to $\\bar{t}$ and extend $Z$ to $Y$ by adding all nodes that have a non-zero edge to a node of $Z$ .","With this modification, the node $a$ is guaranteed to be in $Y$ and hence the estimate for the diameter is at least $D/2$ .","Corollary 26 In $\\tilde{O}(m\\sqrt{n})$ time one can obtain an estimate $D^{\\prime }$ to the $S$ -$T$ diameter $D$ of an $m$ edge $n$ node undirected graph with nonnegative edge weights such that $D/2 - 2w(a,a^{\\prime }) \\le D^{\\prime }\\le D$ for some edge $(a,a^{\\prime })$ .","In $\\tilde{O}(m^{3/2})$ time one can obtain an estimate $D^{\\prime \\prime }$ such that $D/2\\le D^{\\prime \\prime }\\le D$ .","It is an easy exercise to see that when $D=2h+1$ then the value $\\max \\lbrace \\epsilon ^{in}(v), \\epsilon ^{out}(v) \\rbrace $ of an arbitrary vertex $v \\in V$ is an estimation to the diameter which is at least $h+1$ and at most $D$ .","In this section we present a deterministic algorithm that gets a directed unweighted graph $G$ with $D=2h$ and computes in $O(m^2/n)$ time an estimation $\\hat{D}$ such that $h+1\\le \\hat{D} \\le D$ .","The algorithm works as follows.","A variable $\\hat{D}$ is set to zero.","The algorithm searches for a vertex $v$ that its sum of in and out degree is minimal.","Then the algorithm computes the in and out eccentricity of $v$ and every vertex that has an edge with $v$ (incoming or outgoing).","The algorithm outputs the maximum of all the in and out eccentricities that were computed.","Theorem 27 Let $G=(V,E)$ an unweighted directed graph and let $D=2h$ .","Algorithm REF returns in $O(m^2/n)$ time an estimate $\\hat{D}$ such that $h+1\\le \\hat{D} \\le D$ .","We start with the running time analysis.","Consider the graph $G$ and ignore the edge directions.","For every $v\\in V$ let $deg(v) = deg^{in}(v)+deg^{out}(v)$ .","Since $m = \\frac{1}{2}\\sum _{v\\in V} deg(v)$ a vertex $v$ of minimal degree satisfies $deg(v)\\le 2m/n$ .","Therefore, the cost of computing in and out eccentricities for all vertices in the set $N(v) \\cup \\lbrace v \\rbrace $ is $O(\\frac{m}{n} \\times m)$ .","We now turn to bound $\\hat{D}$ .","Let $a, b\\in V$ and let $d(a,b)=2h$ .","Let $P(a,b)$ be a shortest path from $a$ to $b$ and let $v\\in P(a,b)$ .","If $d(a,v)\\le h-1$ then $\\epsilon ^{out}(v)\\ge h+1$ .","Similarly, if $d(v,b)\\le h-1$ then $\\epsilon ^{in}(v)\\ge h+1$ .","Consider now the case that $d(a,v)=h$ and $d(v,b)=h$ .","Let $u\\in P(a,b)$ be the vertex that precedes $v$ on the shortest path from $a$ to $b$ .","Since $u$ has an incoming edge to $v$ it follows that $u\\in N(v)$ and $\\epsilon ^{out}(u)$ and $\\epsilon ^{in}(u)$ are computed.","Since $d(a,v)=h$ it follows that $d(a,u)=h-1$ , $\\epsilon ^{out}(u)\\ge h+1$ and $\\hat{D}$ is at least $h+1$ .","Fast approximation of the diameter [1] Diam-Approx$G$ $\\hat{D} = 0$ $v = \\arg \\min _{x\\in V} deg^{in}(x)+deg^{out}(x)$ every $w \\in N^{\\text{in}}(v) \\cup N^{\\text{out}}(v) \\cup \\lbrace v \\rbrace $ compute $\\epsilon ^{in}(w)$ and $\\epsilon ^{out}(w)$ $\\hat{D}= \\max \\lbrace \\hat{D},\\epsilon ^{in}(w),\\epsilon ^{out}(w)\\rbrace $ $\\hat{D}$ Algorithms for dense graphs Algorithm overview The almost-$3/2$ Diameter approximation algorithm of Aingworth et al.","[2] runs in $\\tilde{O}(n^2+m\\sqrt{n})$ time.","Roditty and Vassilevska W. [41] removed the $\\tilde{O}(n^2)$ term to obtain an $\\tilde{O}(m\\sqrt{n})$ expected time almost-$3/2$ approximation algorithm.","For every graph with $\\Omega (n^{1.5})$ edges the running time of the latter algorithm is not better than the running time of the former algorithm.","Therefore, it is interesting to consider the opposite question to the one considered by [41].","Can the $\\tilde{O}(m\\sqrt{n})$ term be removed?","We show that this can be done for undirected unweighted graphs and present an $O(n^2 \\log n)$ expected time algorithm.","For a graph of Diameter $D=3h+z$ , where $z\\in [0,1,2]$ our algorithm returns an estimation $\\hat{D}$ such that $2h-1\\le \\hat{D} \\le D$ , when $z \\in [0,1]$ and $2h\\le \\hat{D} \\le D$ , when $z=2$ .","Interestingly, our algorithm is obtained by using ideas developed originally for distance oracles and compact routing schemes.","As we are allowed to use quadratic time, we try to estimate the distance between every pair of vertices.","To enable this approach we can no longer sample $A$ naively.","Instead, we adapt a recursive sampling algorithm to compute $A$ , that was introduced by Thorup and Zwick [47] in the context of compact routing schemes.","The expected running time of their algorithm is $\\tilde{O}(mn/|A|)$ .","We provide a new implementation of their algorithm that runs in expected $\\tilde{O}(n(n/|A|)^2)$ time.","The set $A$ has the following important property, for every vertex $w\\in V$ , its cluster (see [48]) $\\lbrace u \\mid d(u,w)<d(u,A) \\rbrace $ is of size $O(n/ |A|)$ .","Consider now a pair of vertices $u$ and $v$ that are in the cluster of $w$ .","For any such pair we can efficiently compute their exact distance.","Moreover, we show that for all pairs $u,v$ that are not in the same cluster of any vertex, we can bound $d(u,v)$ from below with $d(u,A)+d(v,A)-1$ .","This, combined with some other ideas, gives our approximation guarantees.","We extend our approach to also provide an almost $5/3$ -approximation for all Eccentricities.","The idea of using the bounded clusters of Thorup and Zwick [47] has been used in prior work to obtain improved distance oracles [38], [5], approximate shortest paths [11] and compact routing schemes [4].","A simple approach with additive error Let's first consider a simple approach obtaining an $\\tilde{O}(n^2)$ time approximation algorithm for Diameter, Eccentricities or $S$ -$T$ Diameter.","Suppose that we have an algorithm ALG that can compute in $\\tilde{O}(m\\sqrt{n})$ time, for any graph $G^{\\prime }$ , an estimate $D^{\\prime }$ of its Diameter $D$ such that $p\\cdot D - q\\le D^{\\prime }\\le D$ , estimates $e(v)$ of $\\epsilon (v)$ for all $v$ so that $r\\epsilon (v)-s\\le e(v)\\le \\epsilon (v)$ , and an estimate $D^{\\prime \\prime }$ of the $S$ -$T$ Diameter $D_{S,T}$ so that $t\\cdot D_{S,T}-u\\le D^{\\prime \\prime }\\le D_{S,T}$ .","Now, Dor, Halperin and Zwick [24] showed that in $\\tilde{O}(n^2)$ time one can compute for any $n$ node $G$ , an additive 2 spanner $H$ on $\\tilde{O}(n^{1.5})$ edges.","In fact Knudsen [30] recently showed that in $O(n^2)$ time one can get $H$ on $O(n^{1.5})$ edges (i.e.","he removed all logs!).","Let's compute $H$ for our given graph and run ALG on $H$ .","The runtime is $\\tilde{O}(n^{1.5}\\cdot \\sqrt{n})\\le \\tilde{O}(n^2)$ since $H$ has $\\le O(n^{1.5})$ edges.","Let $D^{\\prime }_H,e_H(\\cdot ),D^{\\prime \\prime }_H$ be the estimates that we obtain respectively for the Diameter $D_H$ of $H$ , the Eccentricities $\\epsilon _H(\\cdot )$ of $H$ and the $S,T$ Diameter $D^H_{S,T}$ .","Let's return $D^{\\prime }_H-2,e_H(\\cdot )-2,D^{\\prime \\prime }_H-2$ for our estimates for the Diameter, Eccentricities and $S$ -$T$ Diameter of $G$ .","Notice that $pD-q\\le p\\cdot D_H - q\\le D^{\\prime }_H\\le D_H\\le D+2$ and so $pD-2-q \\le D^{\\prime }_H-2\\le D$ .","Similarly since $r\\epsilon (v)-s\\le r\\epsilon _H(v)-s\\le e_H(v)\\le \\epsilon _H(v)\\le \\epsilon (v)+2$ , we get $r\\epsilon (v)-s-2\\le e_H(v)-2\\le \\epsilon (v)$ .","Finally since $t\\cdot D_{S,T}-u\\le t\\cdot D^H_{S,T}-u\\le D^{\\prime \\prime }_H\\le D^H_{S,T}\\le D_{S,T}+2$ , we get $t\\cdot D_{S,T}-u-2\\le D^{\\prime \\prime }_H-2\\le D_{S,T}$ .","Thus, in $\\tilde{O}(n^2)$ time we get almost the same guarantees as in the $\\tilde{O}(m\\sqrt{n})$ time algorithms, except for an extra additive loss of 2 in the quality.","Below we show how to make the additive loss in quality smaller for Diameter and Eccentricities.","This is especially important when these parameters are constant, which is the hard case of the problems anyway.","Near linear almost 3/2-approximation for Diameter Thorup and Zwick [48] introduced distance oracles, a succinct data structure for answering approximate distance queries efficiently.","Among the tools they use are clusters and bunches.","Let $A\\subseteq V$ , let $p_A(u)$ be the closest vertex to $u$ from $A$ , where ties are broken in favor of the vertex with a smaller identifier and let $d(u,A)=d(u,p_A(u))$ .","For every $v\\in V$ , let $B_A(u)=\\lbrace v\\in V\\mid d(u,v) <d(u,A \\rbrace $ be the bunch of $u$ .","For every $w\\in V\\setminus A$ , let $C_A(w)= \\lbrace v \\mid w\\in B_A(v)\\rbrace $ be the cluster of $w$ .","Thorup and Zwick [48] showed that if a set $A$ is formed by adding every vertex of $V$ to $A$ with probability $p$ then the expected size of $B_A(v)$ is $O(1/p)$ , for every $v\\in V$ .","They also showed, in the context of compact routing schemes [47], that if the set $A$ is constructed by a recursive sampling algorithm then it is possible to bound the maximum size of a cluster as well.","They also showed, in the context of compact routing schemes [47], that if the set $A$ is constructed by a recursive sampling algorithm then it is possible to bound the maximum size of a cluster as well.","Their algorithm works as follows.","It sets $A$ to the empty set and $W$ to $V$ .","Next, as long as the set $W$ is not empty the algorithm samples from $W$ vertices with probability $p$ and adds the sampled vertices to $A$ .","The algorithm computes $C_A(w)$ for every $w\\in W$ and removes from $W$ all the vertices whose cluster has at most $4/p$ vertices with respect to the updated $A$ .","The pseudo-code is given in Algorithm REF .","Thorup and Zwick center algorithm [1] center$G,p$ $A= \\emptyset $ $W =V$ $W\\ne \\emptyset $ $X$ - random sample of nodes from $W$ , $|X|=|W|p$ $A= A \\cup X$ $W = \\lbrace w\\in V \\mid |C_A(w)|> 4/p \\rbrace $ $A$ Thorup and Zwick proved the following Theorem: Theorem 28 (Theorem 3.1 from [47]) The expected size of the set $A$ returned by Algorithm REF is at most $2np \\log n$ .","For every $w \\in V$ we then have $|C_A(w)|\\le 4/p$ .","Thorup and Zwick claimed that the expected running time of Algorithm REF is $O(mnp \\log n)$ .","They did not provide the details and refer the reader to [48].","However, an educated guess is that they compute clusters for the vertices currently in $W$ in each iteration of the while loop, which results in the claimed running time.","The starting point of the Diameter and Eccentricities algorithms presented in this section is an $O(n/p^2 \\log n)$ expected time implementation of Algorithm REF .","The first idea behind our implementation is that, as opposed to what Thorup and Zwick did, we will compute the bunches and use them to compute the clusters and the set $W$ .","This can be done as follows.","Once we have computed $B_A(v)$ for every $v\\in V$ , we can scan $B_A(v)$ , and for every $w\\in B_A(v)$ we can add $v$ to $C_A(w)$ .","The cost of this process is $O(|\\cup _{v\\in V} B_A(v)|)$ and since the clusters are by definition the inverse of the bunches, at the end of this process we have $C_A(w)$ and $|C_A(w)|$ , for every $w\\in V$ and we can compute $W$ (as needed in Algorithm REF ).","However, in the current implementation only the expected size of a bunch is bounded, and since the Thorup-Zwick bound on the number of iterations is $O(\\log n)$ in expectation as well, we cannot apply this idea directly to deduce a good expected running time.","To this end, more ideas are needed.","The following simple observation helps us to achieve our goal.","Observation 29 Let $A_i$ be the set $A$ after updating it in the beginning of the $i$ -th iteration of the while loop in Algorithm REF .","Let $A^*$ be a set such that $A^*\\subseteq A_i$ , for every $i\\ge 1$ .","For every $v\\in V$ it holds that $B_{A_i}(v) \\subseteq B_{A^*}(v)$ .","It follows from this observation that we only need to pick the first set $A_1$ such that $|B_{A_1}(v)|\\le O(1/p)$ for every $v\\in V$ .","It is folklore that the $s$ closest vertices $N_{s}(v)$ to a vertex $v$ can be computed in $O(s^2)$ time [24].","This implies that we can compute $N_{1/p}(v)$ for every $v\\in V$ in $O(n/p^2)$ time.","It is not hard to see that, given the sets $N_{1/p}(v)$ of all $v\\in V$ , one can (deterministically) compute a “hitting” set $A$ of size $O(np \\log n)$ in $O(n+n/p)$ worst case time, so that $N_{1/p}(v)\\cap A\\ne \\emptyset $ for every $v\\in V$ (a greedy algorithm works; e.g.","see [48]).","The second idea behind our implementation is that we first compute the sets $N_{1/p}(v)$ for every $v\\in V$ and the hitting set $A$ , as described above.","Then, using these sets, we initialize Algorithm REF with a set $A$ such that $|B_A(v)|=O(1/p)$ , for every $v\\in V$ .","In more detail, our algorithm works as follows.","For every $v\\in V$ it computes the set $N_{1/p}(v)$ in $O(n/p^2)$ time.","Then it finds a set $A$ such that $N_{1/p}(v)\\cap A\\ne \\emptyset $ for every $v\\in V$ .","Given the hitting set $A$ , it computes $d(v,A)$ and $p_A(v)$ for every $v\\in V$ .","Using $d(v,p_A(v))$ and $N_{1/p}(v)$ it computes for every $v\\in V$ the bunch $B_A(v)$ .","Finally, it computes the clusters and $W$ using the bunches as we described above.","The rest of the algorithm is almost identical to Algorithm REF .","The only difference is that we compute the bunches and use them to compute the clusters and the set $W$ .","The pseudo-code is given in Algorithm REF .","New implementation of Thorup and Zwick center algorithm [1] center$G,p$ compute $N_{1/p}(v)$ for every $v\\in V$ .","$A=$ hitting set of the sets $N_{1/p}(v)$ , where $v\\in V$ .","compute $d(v,A)$ and $p_A(v)$ for every $v\\in V$ .","compute $B_A(v)$ using $N_{1/p}(v)$ and $d(v,p_A(v))$ .","every $u\\in V$ compute $C_A(u)$ using $B_A(\\cdot )$ $W = \\lbrace w\\in V \\mid |C_A(w)|> 4/p \\rbrace $ $W\\ne \\emptyset $ $X$ - random sample of nodes from $W$ , $|X|=np$ $A= A \\cup X$ every $v\\in V$ compute $B_A(v)$ every $u\\in V$ compute $C_A(u)$ using $B_A(\\cdot )$ $W = \\lbrace w\\in V \\mid |C_A(w)|> 4/p \\rbrace $ $A$ We show: Lemma 30 Algorithm REF computes in $O(n/p^2 \\log n)$ expected time a set $A$ of expected size $O((n/p)\\cdot \\log n)$ that guarantees for every vertex $w\\in V \\setminus A$ that $|C_A(w)|=O(1/p)$ , and for every $v\\in V$ that $|B_A(v)|=O(1/p)$ .","The cost of computing $N_{1/p}(v)$ for every $v\\in V$ is $O(n (1/p)^2)$ [24].","The cost of computing $A$ is $O(n/p)$ time [48].","Computing $d(v,A)$ and $p_A(v)$ for every $v\\in V$ in $O(m)$ time is straightforward by running shortest paths tree computation from a dummy vertex that is connected to the set $A$ .","To compute $B_A(v)$ using $N_{1/p}(v)$ we only scan $N_{1/p}(v)$ , thus, the total cost is $O(n(1/p))$ .","As we explained earlier the cost of computing clusters using bunches is $O(|\\cup _{v\\in V} B_A(v)|)$ .","Since for every $v\\in V$ we have $B_A(v)\\subseteq N_{1/p}(v)$ the total cost is $O(n(1/p))$ .","This completes the analysis of the part that precedes the while loop.","Next, we analyze the cost of the while loop.","Let $A^*$ be the set $A$ that was computed before the while loop and let $A_i$ be the set $A$ after updating it in the beginning of the $i$ -th iteration of the while loop.","From Observation REF it follows that $B_{A_i}(v) \\subseteq B_{A^*}(v)$ and therefore in every iteration the cost of computing bunches from scratch is at most $O(n (1/p)^2)$ as $|B_{A^*}(v)|=O(1/p)$ , for every $v\\in V$ .","One can also compute $B_{A_{i+1}}(v)$ from $B_{A_i}(v)$ by first computing $d(v,A_{i+1})$ and if $d(v,A_{i+1})<d(v,A_i)$ to prune $B_{A_{i+1}}(v)$ accordingly at a smaller cost of $O(n(1/p) + m)$ , however this does not affect the overall complexity.","Thorup and Zwick [47] proved that the expected number of iterations is $O(\\log n)$ .","The fact that the set $A$ from which we start is different does not affect their proofThey prove that in each iteration with probability $1/2$ the size of $W$ decreases by a factor of 2.","For this argument they only require that the set $A$ in each iteration will be chosen from $W$ uniformly at random with probability $p$ as we do., therefore there are only $O(\\log n)$ iterations in expectation.","This implies that a set $A$ of expected size $(np \\log n)$ is returned in $O(n/p^2 \\log n)$ expected time.","The algorithm stops only when there are no large clusters, thus the bound on the cluster size follows.","As we mentioned above the algorithm starts with bunches that satisfy the required bound and their size can only decrease afterwards, thus the bound on the bunches follows.","almost $3/2$ -Approximation for Diameter [1] $3/2$ -Approx-DiamG $M$ - $n\\times n$ matrix whose entries are set to $n$ $A =$ CENTER$(G,1/\\sqrt{n})$ every $w\\in V$ Step 1 every $\\langle u, v\\rangle \\in C_A(w) \\times C_A(w)$ , s.t.","$u\\ne v\\;$ $M(u,v) = \\min (M(u,v), d(u,w)+d(v,w))$ every $\\langle u, v\\rangle \\in V \\times V$ , s.t.","$M(u,v)=n$ Step 2 $M(u,v) = d(u,A)+d(v,A)-1$ $H$ - an additive 2 spanner of $G$ Step 3 every $u \\in A$ compute shortest paths tree for $u$ in $H$ and set $\\epsilon _H(u)$ , the Eccentricity of $u$ in $H$ $D_1 = \\max _{\\langle u, v\\rangle \\in V \\times V} M(u,v)$ $D_2 = \\max _{u\\in A} \\epsilon _H(u)$ $\\hat{D} = \\max (D_1,D_2-2)$ $\\hat{D}$ We can now turn to describe the new Diameter algorithm.","The algorithm works as follows.","All entries of an $n \\times n$ matrix $M$ are set to $n$ .","A set $A$ of centers is computed using the algorithm of Thorup and Zwick [47].","For every vertex $w\\in V$ and every pair $\\langle u, v\\rangle \\in C_A(w)\\times C_A(w)$ the algorithm sets $M(u,v)$ to $\\min (M(u,v), d(u,w)+d(v,w))$ (Step 1).","Next, the algorithm searches the matrix $M$ for entries whose value is still $n$ .","Given a pair $\\langle u, v\\rangle \\in V \\times V$ for which $M(u,v)=n$ the algorithm sets $M(u,v)$ to $d(u,A)+d(v,A)-1$ (Step 2).","Finally, the algorithm computes an additive 2 spanner $H$ of the input graph $G$ and for every $u\\in A$ it computes $\\epsilon _H(u)$ , the Eccentricity of $u$ in $H$ (Step 3).","The algorithm outputs the maximum between $D_1$ and $D_2-2$ , where $D_1$ is $\\max _{\\langle u, v\\rangle \\in V \\times V} M(u,v)$ and $D_2$ is $\\max _{u\\in A} \\epsilon _H(u)$ .","Next, we bound the value returned by Algorithm REF .","Theorem 31 Let $D=3h+z$ , where $z\\in [0,1,2]$ .","The value $\\hat{D}$ returned by Algorithm REF satisfies: $\\begin{array}{ll}2h-1 & \\mbox{if } z \\in [0,1] \\\\2h & \\mbox{if } z=2\\end{array}\\le \\hat{D} \\le D$ We start with the following Lemma: Lemma 32 Let $u,v\\in V$ and let $P(u,v)$ be a shortest path between $u$ and $v$ .","If $B_A(u) \\cap B_A(v)\\ne \\emptyset $ then $(B_A(u) \\cap B_A(v)) \\cap P(u,v) \\ne \\emptyset $ .","If $v\\in B_A(u)$ then the claim trivially holds so we can assume that $v\\notin B_A(u)$ .","Let $w$ be the vertex farthest from $u$ that is in $B_A(u) \\cap P(u,v)$ .","From the definition of $w$ it follows that $d(u,w)=d(u,A)-1$ .","Assume, towards a contradiction, that $(B_A(u) \\cap B_A(v)) \\cap P(u,v) = \\emptyset $ .","This implies that $w\\notin B_A(v)$ and $d(v,A)-1<d(v,w)=d(u,v)-d(u,w)$ .","However, since $B_A(u) \\cap B_A(v)\\ne \\emptyset $ there is a vertex $w^{\\prime }$ such that $d(u,w^{\\prime })\\le d(u,w)$ and $d(v,w^{\\prime })\\le d(v,A)-1 < d(u,v)-d(u,w)$ .","This implies that $d(u,w^{\\prime })+d(v,w^{\\prime })< d(u,v)$ , a contradiction to the triangle inequality.","Lemma 33 Let $u,v\\in V$ .","If $B_A(u) \\cap B_A(v)= \\emptyset $ then $d(u,A)+d(v,A)-1 \\le d(u,v)$ .","Notice first that $B_A(u)$ (resp., $B_A(v)$ ) contains all the vertices at distance $d(u,A)-1$ (resp., $d(v,A)-1$ ).","Let $P(u,v)$ be a shortest path between $u$ and $v$ .","Let $w$ be the vertex farthest from $u$ on $P(u,v)$ that is also in $B_A(u)$ .","Similarly, let $w^{\\prime }$ be the vertex farthest from $v$ on $P(u,v)$ that is also in $B_A(v)$ .","Since $B_A(u) \\cap B_A(v)= \\emptyset $ it holds that $w\\ne w^{\\prime }$ .","Therefore: $d(u,v) = d(u,A)-1+d(v,A)-1+d(w,w^{\\prime })\\ge d(u,A)+d(v,A)-1.$ Let $a$ and $b$ be the Diameter endpoints, that is $d(a,b)=D=3h+z$ , where $z\\in [0,1,2]$ .","Let $P(a,b)$ be a shortest path between $a$ and $b$ .","Assume first that $B_A(a) \\cap B_A(b)\\ne \\emptyset $ .","It follows from Lemma REF that there is a vertex $w\\in P(a,b)$ such that $\\langle a, b\\rangle \\in C(w) \\times C(w)$ .","Therefore, $M(a,b)=D$ after Step 1.","After the update in Step 2 it follows from Lemma REF that $M(u,v)\\le d(u,v)$ for every $u ,v\\in V$ .","Therefore, the maximum value in the matrix is $d(a,b)$ and $D_1=D$ .","Let $x = \\operatornamewithlimits{arg\\,max}_{y\\in A} \\epsilon _H(y)$ .","Since $H$ is an additive 2 spanner it holds that $\\epsilon _H(x)\\le D+2$ , hence, we have $D_2 \\le D$ and the algorithm returns the exact value of the Diameter.","Assume now that $B_A(a) \\cap B_A(b) = \\emptyset $ .","From the discussion of the previous case it follows that in this case $\\hat{D}\\le D$ as well.","Thus, it is only left to prove the lower bound.","Assume that $z\\in [0,1]$ .","Consider first the case that $d(a,A)\\ge h$ and $d(b,A)\\ge h$ then from Lemma REF it follows that $M(a,b)\\ge 2h-1$ after Step 2 and $D_1$ is at least $2h-1$ .","If this is not the case then either $d(a,A)< h$ or $d(b,A)< h$ (or both).","Assume, wlog, that $d(a,A)< h$ .","In this case the Eccentricity in $H$ of at least one vertex from $A$ is at least $2h+1$ and hence $D_2-2$ is at least $2h-1$ .","Assume now that $z=2$ .","If either $d(a,A)\\ge h$ and $d(b,A)> h$ or $d(a,A)>h$ and $d(b,A)\\ge h$ then from Lemma REF it follows that $M(a,b)\\ge 2h$ after Step 2 and $D_1$ is at least $2h$ .","If this is not the case then either $d(a,A)\\le h$ or $d(b,A)\\le h$ (or both).","Assume, wlog, that $d(a,A)\\le h$ .","In this case the Eccentricity in $H$ of at least one vertex from $A$ is at least $2h+2$ and hence $D_2-2$ is at least $2h$ .","We now turn to we analyze the running time of Algorithm REF .","Theorem 34 The expected running time of Algorithm REF is $O(n^2 \\log n)$ .","The set $A$ is computed by the center algorithm presented in Algorithm REF with $p=1/\\sqrt{n}$ .","From Lemma REF it follows that the size of the set $A$ is $O(\\sqrt{n} \\log n)$ and its construction time is $O(n^2\\log n)$ in expectation.","For every $w\\in V$ the size of $C_A(w)$ is $O(\\sqrt{n})$ .","Therefore, Step 1 takes $O(n \\times |C_A(w)|^2)=O(n^2)$ .","Step 2 takes $O(n^2)$ time as well.","In Step 3 we first compute an additive 2 spanner $H$ on $O(n^{1.5})$ edges.","Knudsen [30], following Dor, Halperin and Zwick [24] showed how to do this in $O(n^2)$ time.","We also compute $|A|$ shortest paths trees in $H$ .","As $H$ has $O(n^{1.5})$ edges, this step takes $O(n^2 \\log n)$ time.","Near linear almost 5/3-approximation for Eccentricities almost $5/3$ -Approximation for all Eccentricities [1] $5/3$ -Approx-EccG Run lines 2-11 of Algorithm REF , with $H$ augmented with shortest paths trees for $B_A(u) \\cup \\lbrace p(u) \\rbrace $ for every $u\\in V$ every $u \\in V$ $\\epsilon _1(u) = \\max _{v\\in V} M(u,v)$ $\\epsilon _2(u) = \\epsilon _H(p(u))-d(u,p(u))-2$ $\\epsilon _3(u) = d_H(u,y)-2$ , where $y=\\operatornamewithlimits{arg\\,max}_{x\\in A} d_H(u,x)$ $\\epsilon ^{\\prime }(u) = \\max (\\epsilon _1(u),\\epsilon _2(u),\\epsilon _3(u))$ Next, we show how to update Algorithm REF to obtain an almost $5/3$ approximation for all Eccentricities.","We run lines 2-11 of Algorithm REF .","The only difference is that $H$ is augmented with the edges of the shortest paths tree that span the set $B_A(u) \\cup \\lbrace p(u) \\rbrace $ for every $u\\in V$ .","Then, for every $u\\in V$ we compute $\\epsilon _1(u)$ , $\\epsilon _2(u)$ and $\\epsilon _3(u)$ , which are defined as follows: $\\epsilon _1(u) = \\max _{v\\in V} M(u,v)$ , $\\epsilon _2(u) = \\epsilon _H(p(u))-d(u,p(u))-2$ and $\\epsilon _3(u) = d_H(u,y)-2$ , where $y=\\operatornamewithlimits{arg\\,max}_{x\\in A} d_H(u,x)$ .","The algorithm sets $\\epsilon ^{\\prime }(u)$ to $\\max \\lbrace \\epsilon _1(u), \\epsilon _2(u), \\epsilon _3(u) \\rbrace $ for every $u\\in V$ as an estimation to $\\epsilon (u)$ .","The pseudo-code is given in Algorithm REF .","We now prove: Theorem 35 For every $u\\in V$ , Algorithm REF computes in $O(n^2 \\log n)$ expected time a value $\\epsilon ^{\\prime }(u)$ that satisfies: $\\frac{3\\epsilon (u)}{5}-1 \\le \\epsilon ^{\\prime }(u)\\le \\epsilon (u).$ We start by analyzing the running time.","Lines 2-11 of the algorithm are the same as Algorithm REF , with one difference, the spanner $H$ is augmented with the edges of the shortest paths tree that span the set $B_A(u) \\cup \\lbrace p(u) \\rbrace $ for every $u\\in V$ .","This adds at most $O(n^{1.5})$ edges to $H$ and hence the cost of these lines remain $O(n^2 \\log n)$ time in expectation.","The computation of $\\epsilon _1(u)$ , $\\epsilon _2(u)$ and $\\epsilon _3(u)$ for every $u\\in V$ costs $O(n^2)$ time in total.","Let $u\\in V$ be an arbitrary vertex and let $\\epsilon (u)=d(u,t)$ .","We now turn to bound $\\epsilon ^{\\prime }(u)$ .","In our analysis we will use the following simple observation: Observation 36 In an undirected graph it holds for every $u,v \\in V$ that $\\epsilon (u)\\ge \\epsilon (v)-d(u,v)$ .","It is straightforward to see that both $\\epsilon _2(u)$ and $\\epsilon _3(u)$ are at most $\\epsilon (u)$ .","Recall that $\\epsilon _2(u) = \\epsilon _H(p(u))-d(u,p(u))-2\\le \\epsilon (p(u))-d(u,p(u)\\le \\epsilon (u)$ and $\\epsilon _3(u) = d_H(u,y)-2\\le d(u,y)\\le \\epsilon (u)$ .","We distinguish between two cases.","Case 1: $B_A(u) \\cap B_A(t)\\ne \\emptyset $ .","It follows from Lemma REF that $P(u,t)\\cap (B_A(u) \\cap B_A(t)) \\ne \\emptyset $ and $M(u,t)= \\epsilon (u)$ .","From Lemma REF it follows that $M(u,w)\\le d(u,w)$ for every $w\\in V$ after Step 2.","Therefore, $\\epsilon _1(u)=\\epsilon (u)$ .","Since $\\epsilon _2(u)\\le \\epsilon (u)$ and $\\epsilon _3(u)\\le \\epsilon (u)$ we get that $\\epsilon ^{\\prime }(u)=\\epsilon (u)$ .","Case 2: $B_A(u) \\cap B_A(t) = \\emptyset $ .","Consider first the case that $d(u,p(u))\\le \\frac{\\epsilon (u)}{5}-1$ .","From Observation REF we get that $\\epsilon _H(p(u))\\ge \\epsilon _H(u)-d_H(u,p(u))$ .","As we augmented $H$ with a shortest paths tree that spans $B_A(u) \\cup \\lbrace p(u) \\rbrace $ we have $d(u,p(u))=d_H(u,p(u))$ and we get $\\epsilon _H(p(u))\\ge \\epsilon _H(u)-d(u,p(u))$ .","Hence, we get that $\\epsilon _2(u) = \\epsilon _H(p(u))-d(u,p(u))-2 \\ge \\epsilon _H(u)-2d(u,p(u))-2\\ge \\epsilon (u)-2d(u,p(u))-2$ .","As before we have $\\epsilon _2(u) \\le \\epsilon (u)$ .","Using $d(u,p(u))\\le \\frac{\\epsilon (u)}{5}-1$ we get that: $\\epsilon _2(u) \\ge \\epsilon (u)-\\frac{2\\epsilon (u)}{5}\\ge \\frac{3\\epsilon (u)}{5}.$ Assume now that $d(u,p(u))\\ge \\frac{\\epsilon (u)}{5}$ .","This means that $d(u,A)-1\\ge \\frac{\\epsilon (u)}{5}-1$ .","Let $S$ be the set of all vertices $v\\in V$ such that $B_A(u) \\cap B_A(v)= \\emptyset $ , that is, $S=V \\setminus \\cup _{w\\in B_A(u)} C_A(w)$ .","Let $t^{\\prime } = \\operatornamewithlimits{arg\\,max}_{x\\in S} d(x,A)-1$ .","If $d(t^{\\prime },A)-1 \\ge \\frac{2\\epsilon (u)}{5}-1$ we get from Lemma REF that $M(u,t^{\\prime })\\ge \\frac{3\\epsilon (u)}{5}-1$ .","Assume now that $d(t^{\\prime },A) < \\frac{2\\epsilon (u)}{5}$ .","As $t^{\\prime }$ is the farthest vertex from $A$ we get that $d(t,p(t))<\\frac{2\\epsilon (u)}{5}$ and $d(u,p(t))>\\frac{3\\epsilon (u)}{5}$ .","Therefore, $\\epsilon _3(u) = d_H(u,y)-2\\ge d(u,p(t))-2\\ge \\frac{3\\epsilon (u)}{5}-1$ .","From Lemma REF and Lemma REF it follows that $\\epsilon _1(u)\\le \\epsilon (u)$ and the bound follows.","Algorithms for dense graphs using matrix multiplication Here we will give $O(n^{2.05})$ time approximation algorithms for Diameter and Eccentricities in dense unweighted undirected graphs.","The approximation guarantees of these algorithms are slightly better than those in our $O(n^2\\log n)$ time algorithm.","In fact, the guarantees are exactly the same as in the $\\tilde{O}(m\\sqrt{n})$ time algorithms for Diameter and Eccentricities of Roditty and V. Williams [41] and Cairo et al. [15].","To achieve this, we give an efficient implementation using fast matrix multiplication of the $\\tilde{O}(m\\sqrt{n})$ time algorithms of [15] and [41].","The main overhead of the $\\tilde{O}(m\\sqrt{n})$ time algorithms [15], [41] is in computing the distances from a set $S=W\\cup \\lbrace w\\rbrace \\cup T$ of $O(\\sqrt{n}\\log n)$ nodes: the set $S$ itself can be computed in linear time using random sampling to form a set $W$ , BFS from a dummy node to find the node $w$ farthest from $W$ and then BFS from $w$ to find the set $T$ of closest $\\sqrt{n}$ nodes to $w$ .","After one knows all distances from every $s\\in S$ to every $v\\in V$ , it takes linear time to output the Diameter and Eccentricity estimates.","The main idea of our algorithms is as follows.","If the Diameter is of size $\\le O(\\log n)$ , then one does not need all distances between $S$ and $V$ , but only those that are $O(\\log n)$ .","Small distances are easy to compute with matrix multiplication.","Let $A$ be the adjacency matrix and $A_S$ be its submatrix formed by just the rows in $S$ .","Then we can find the distances for all pairs in $S\\times V$ at distance $\\le t$ by computing $A_S\\times A^{t-1}$ , which can be computed by performing $t-1$ matrix products of dimension $|S|\\times n$ by $n\\times n$ , and this can be accomplished in $O(tn^{2.05})$ time [26], [31].","If on the other hand the Diameter is $D\\ge 100\\log n$ , then one can use an $\\tilde{O}(n^2)$ time algorithm by Dor et al.","[24] to compute estimates of all pairwise distances with an additive error at most $4\\log n$ .","The maximum distance estimate computed, minus $4\\log n$ , will be between $0.96 D$ and $D$ , giving a really good approximation already.","A similar argument works for Eccentricities, and also for $S$ -$T$ Diameter.","Below we recap the guarentees of the $\\tilde{O}(m\\sqrt{n})$ time approximation algorithms of [15], [41].","Theorem 37 ([15], [41]) The following can be computed in $\\tilde{O}(m\\sqrt{n})$ time: an estimate $\\hat{D}$ of the graph Diameter $D$ , such that $\\frac{2}{3} D-\\frac{1}{3}\\le \\hat{D}\\le D$ , for every node $v$ , an estimate $e(v)$ of its Eccentricity $\\epsilon (v)$ , such that $\\frac{3}{5} \\epsilon (v)-\\frac{1}{5}\\le e(v)\\le \\epsilon (v)$ .","Using Seidel's algorithm [43] we can compute all the distances exactly, and hence the above parameters as well, all in $O(n^\\omega )$ time for $\\omega < 2.373$ .","We will show that for dense graphs, we can obtain the same approximation guarantees as in Theorem REF , in time $O(n^{2.05})$ .","Let us compare to our $O(n^2\\log n)$ time algorithms.","For Diameter $D=3h+z$ , the $O(n^2\\log n)$ time algorithm returns an estimate $2h-1$ when $z=0,1$ and $2h$ when $z=2$ .","The estimate $\\hat{D}$ here is $\\ge (2D-1)/3= 2h+ (2z-1)/3$ , which is $\\ge 2h$ when $z=0$ and $\\ge 2h+1$ when $z=1,2$ .","For Eccentricities, the $O(n^2\\log n)$ time algorithm returns estimates $e(v)\\ge 3\\epsilon (v)/5 -1$ , and here we return a better estimate $e(v)\\ge (3\\epsilon -1)/5$ .","We will rely on two known algorithms.","The first is from a paper by Dor, Halperin and Zwick [24] on additive approximations of All-Pairs Shortest Paths (APSP).","Among many other results, [24] show that in $\\tilde{O}(n^2)$ time, one can compute for all pairs of vertices $u,v$ , an estimate $d^{\\prime }(u,v)$ of their distance $d(u,v)$ so that $d(u,v)\\le d^{\\prime }(u,v)\\le d(u,v)+a\\log n$ for an explicit constant $a\\le 4$ .","The second is an algorithm for the following truncated multi-source shortest paths problem: given an integer $Q$ , a graph $G=(V,E)$ and a set $S$ , compute the distances $d(s,v)$ for every $s\\in S$ and $v\\in V$ for which $d(s,v)\\le Q$ .","The algorithm uses fast matrix multiplication and is quite straightforward.","Let $A$ be the $n\\times n$ Boolean matrix with rows and columns indexed by $V$ , so that $A[u,v]=1$ if there is an edge between $u$ and $v$ or $u=v$ , and $A[u,v]=0$ otherwise; i.e.","$A$ is the adjacency matrix added to the identity matrix.","Let $A_S$ be the $|S|\\times n$ submatrix of $A$ consisting of the rows indexed by nodes of $S$ .","For an integer $i\\ge 1$ , let $A^i$ be the $i$ -th power of $A$ under the Boolean matrix product.","Here, $A^i[u,v]=1$ if and only if the distance between $u$ and $v$ is at most $i$ .","Define $A^0$ as the identity matrix.","Consider $A_S \\cdot A^i$ for any choice of $i\\ge 0$ (under the Boolean matrix product).","Here, $(A_S \\cdot A^i)[s,v]=1$ if and only if the distance between $s$ and $v$ is at most $i+1$ .","Thus, if we compute $D_i:=A_S\\cdot A^i$ for every $1\\le i< Q$ , we would know the distance from every $s\\in S$ to every $v\\in V$ , whenever this distance is at most $Q$ .","Computing these matrix products can easily be done by performing the following $Q-1$ Boolean products of an $|S|\\times n$ matrix by an $n\\times n$ matrix: let $D_0=A_S$ ; then for each $i$ from 1 to $t-1$ , compute $D_i:=D_{i-1}\\cdot A$ .","Thus, the running time is $O(Q\\cdot M(|S|,n,n))$ where $M(|S|,n,n)$ is the runtime of multiplying an $|S|\\times n$ matrix by an $n\\times n$ matrix.","Armed with these two algorithms, let us recap Roditty et al.","'s (and Cairo et al.","'s) approximation algorithm and see how to modify it.","The algorithm proceeds as follows: Let $D$ , $R$ and $\\epsilon (v)$ denote the Diameter and Radius of $G$ and the Eccentricity of node $v$ , respectively.","RV/CGR Algorithm [1] Using BFS (see [41] and [15]), in $O(m+n)$ time compute $W,w,T$ , where $W\\subseteq V$ is a uniformly chosen subset of size $O(\\sqrt{n} \\log n)$ , $w$ is the furthest node from $W$ and $T$ are the closest $\\sqrt{n}$ nodes to $w$ .","Let $S=\\lbrace w\\rbrace \\cup W\\cup T$ .","For every $s\\in S$ and every $v\\in V$ , compute the distance $d(s,v)$ between $s$ and $v$ ; set $\\epsilon (s)=\\max _v d(s,v)$ .","Set $\\tilde{D}=\\max _{x\\in S} \\epsilon (x)$ .","Set for every $v\\in V$ , $\\tilde{\\epsilon }(v)=\\max \\lbrace d(w,v),\\max _{x\\in W} d(x,v), \\max _{x\\in T} (\\epsilon (x)-d(x,v))\\rbrace $ .","The runtime bottleneck in the above algorithm is step (2) which runs in $\\tilde{O}(mn^{1/2})$ time if one uses BFS through each node of $S$ .","Let us describe how to modify the algorithm.","We will replace (2) with a truncated distance computation and also use the algorithm of Dor, Halperin and Zwick to handle large distances that we might have ignored in the truncated computation.","Our Modified Approximation.","[1] FasterApproximation First part: Handle Large Distances: [t]@X@ Use Dor, Halperin and Zwick's algorithm to compute distance estimates $d^{\\prime }(\\cdot ,\\cdot )$ so that for every $u,v\\in V$ , $d(u,v)\\le d^{\\prime }(u,v)\\le d(u,v)+a\\log n$ .","Let $X=3a\\log n$ .","Set $\\tilde{D}_1 = \\max _{u,v\\in V} d^{\\prime }(u,v) - a\\log n$ .","For every $v\\in V$ , set $\\tilde{\\epsilon }_1(v) = \\max _{u} d^{\\prime }(u,v) - a\\log n$ .","Second Part: Handle Small Distances: [t]@X@ Using BFS (see [41] and [15]), in $O(m+n)$ time compute $W,w,T$ , where $W\\subseteq V$ is a uniformly chosen subset of size $O(\\sqrt{n} \\log n)$ , $w$ is the furthest node from $W$ and $T$ are the closest $\\sqrt{n}$ nodes to $w$ .","Let $S=\\lbrace w\\rbrace \\cup W\\cup T$ .","[t]@X@ Let $Q=2(X+a\\log n)=8a\\log n$ .","For every $s\\in S$ and every $v\\in V$ whose distance $d(s,v)$ is at most $Q$ , compute $d(s,v)$ .","Let $d_{\\le }(s,v)$ denote $d(s,v)$ if we have computed it, and $\\infty $ otherwise.","Set $\\epsilon _{\\le }(s)=\\max _v d_{\\le }(s,v)$ .","Set $\\tilde{D}_2=\\max _{x\\in S} \\epsilon _{\\le }(x)$ .","[t]@X@ $\\forall v\\in V$ , set $\\tilde{\\epsilon }_2(v)=\\max \\lbrace d_\\le (w,v), \\max _{x\\in W} d_\\le (x,v), \\max _{y\\in T} (\\epsilon _\\le (y)-d_\\le (y,v))\\rbrace $ .","Third Part: Set $\\tilde{D},\\tilde{\\epsilon }(\\cdot )$: If $\\tilde{D}_1\\ge X$ , set $\\tilde{D}=\\tilde{D}_1$ , and otherwise set $\\tilde{D}=\\tilde{D}_2$ .","[t]@X@ For every $v\\in V$ , if there exists some $x\\in S$ such that $d^{\\prime }(x,v)\\ge X+a\\log n$ , then set $\\tilde{\\epsilon }(v)=\\tilde{\\epsilon }_1(v)$ , and otherwise $\\tilde{\\epsilon }(v)=\\tilde{\\epsilon }_2(v)$ .","Consider our modified algorithm, FasterApproximation.","Now we will prove several claims.","Claim 38 The running time of algorithm FasterApproximation is $\\tilde{O}(M(\\sqrt{n},n,n))$ .","The Dor, Halperin, Zwick part of the algorithm (Step 3) runs in $\\tilde{O}(n^2)$ time.","Step 8 runs in $O((X+a\\log n)\\cdot M(|S|,n,n))$ time where $S=\\lbrace w\\rbrace \\cup W\\cup T$ , using the iterated rectangular matrix product algorithm.","Recall that $X+a\\log n = O(\\log n)$ .","Thus Step 8 runs in $\\tilde{O}(M(|S|,n,n))$ time.","Since $|S|=\\tilde{O}(\\sqrt{n})$ and we can partition an $|S|\\times n\\times n$ matrix product into $ \\text{\\rm polylog~} n$ , $n^{1/2}\\times n\\times n$ matrix products, the runtime of the step is $\\tilde{O}(M(n^{1/(2)},n,n))$ .","Steps 10 and 13 run in $O(n|S|)<\\tilde{O}(n^2)$ time.","The rest of the steps run in linear time.","Since $M(n^{1/2},n,n)\\ge n^2$ (one must at least read the input), the total running time is $\\tilde{O}(M(n^{1/2},n,n))$ .","Claim 39 $\\frac{2D-1}{3} \\le \\tilde{D}\\le D$ .","Suppose that $\\tilde{D}_1\\ge X$ .","The algorithm returns $\\tilde{D}=\\tilde{D}_1=\\max _{u,v} d^{\\prime }(u,v)-a\\log n$ .","By the guarantee on $d^{\\prime }$ , we have $D-a\\log n \\le \\tilde{D}_1\\le D$ .","Hence $\\tilde{D}\\ge D(1-(a\\log n)/D)\\ge D(1-(a\\log n)/X) =2D/3 \\ge (2D-1)/3$ .","Suppose now that $\\tilde{D}_1< X$ .","This means that $D<X+a\\log n$ and every distance in the graph is $\\le X+a\\log n$ .","In the second part of the algorithm we set $Q=2(X+a\\log n)$ , and hence every distance is computed exactly: for every $s\\in S$ , $v\\in V$ , $d_\\le (s,v)=d(s,v)$ .","Hence the second part of the algorithm will be identical to the RV/CGR algorithm and hence we get the same guarantees: $(2D-1)/3\\le \\tilde{D}\\le D$ .","Claim 40 For every node $v$ , $\\frac{3\\epsilon (v)-1}{5} \\epsilon (v)\\le \\tilde{\\epsilon }(v)\\le \\epsilon (v)$ .","Fix $v$ .","Suppose first that there exists some $x$ such that $d^{\\prime }(x,v)\\ge X+a\\log n$ .","Then $\\epsilon (v)\\ge \\tilde{\\epsilon }_1(v)=\\max _u d^{\\prime }(u,v) - a\\log n \\ge \\epsilon (v)-a\\log n = \\epsilon (v)(1-a\\log n/\\epsilon (v))$ .","Since $\\epsilon (v)\\ge d(x,v)\\ge d^{\\prime }(x,v)-a\\log n \\ge X$ , we get that $\\tilde{\\epsilon }_1(v)\\ge \\epsilon (v)(1-a\\log n / X) = 2\\epsilon (v)/3$ .","Now suppose that for all $x\\in V$ , $d^{\\prime }(x,v)<X+a\\log n$ .","Then, also for all $x\\in V$ , $d(x,v)<X+a\\log n$ and $\\epsilon (v)<X+a\\log n$ .","Consider all the quantities needed in the second part of the algorithm to compute $\\tilde{\\epsilon }_2(v)$ : $d_\\le (w,v)$ : since $\\forall x\\in V$ , $d(x,v)<X+a\\log n$ , $d_\\le (w_i,v)=d(w_i,v)$ for each $w_i$ ; $d_\\le (x,v)$ for every $x\\in W$ : as above, $d_\\le (x,v)=d(x,v)$ ; $\\epsilon _\\le (x)-d_\\le (x,v)$ for all $x\\in T$ : here, $\\epsilon (x)\\le \\epsilon (v)+d(x,v)\\le 2\\epsilon (v)<2(X+a\\log n)$ .","Since we compute all distances from nodes in $S$ up to $2(X+a\\log n)$ and $x\\in S$ , $\\epsilon _\\le (x)=\\epsilon (x)$ .","Also as in the above bullets, $d_\\le (x,v)=d(x,v)$ .","Thus all the quantities needed are the correct ones and $\\tilde{\\epsilon }(v)=\\tilde{\\epsilon }_2(v)$ inherits the same guarantees as in the algorithm by Cairo et al.","From Le Gall and Urrutia [26] (see also, [31]) we obtain that $M(\\sqrt{n},n,n)\\le O(n^{2.044183})$ .","We obtain: Theorem 41 In $O(n^{2.045})$ time, one can obtain an almost $3/2$ -approximation $\\tilde{D}$ to the Diameter $D$ and almost $5/3$ -approximations $e(v)$ to all Eccentricities $\\epsilon (v)$ : $\\frac{2D-1}{3} \\le \\tilde{D}\\le D$ .","For every node $v$ , $\\frac{3\\epsilon (v)-1}{5}\\le \\tilde{\\epsilon }(v)\\le \\epsilon (v)$ .","Finally we note that our approach also works to speed up our almost 2-approximation algorithm for $S$ -$T$ Diameter as well, giving an $O(n^{2.045})$ time almost-2 approximation algorithm.","The main reason is that, like in the Diameter approximation algorithm, if the $S$ -$T$ Diameter is very large (say $D_{S,T}>100a\\log n$ ), then the $+a\\log n$ APSP algorithm with $a\\log n$ subtracted will return an estimate that is at least $D_{S,T}-a\\log n>0.99D_{S,T}$ .","On the other hand, our $S$ -$T$ Diameter approximation algorithm only needs to know the distances up to $D_{S,T}$ to compute an estimate of $D_{S,T}$ , and so if $D_{S,T}\\le 100a\\log n$ , then we only need to compute $O(\\log n)$ matrix products of dimension $O(\\sqrt{n} \\log n)\\times n\\times n$ again."],["Algorithm overview","The almost-$3/2$ Diameter approximation algorithm of Aingworth et al.","[2] runs in $\\tilde{O}(n^2+m\\sqrt{n})$ time.","Roditty and Vassilevska W. [41] removed the $\\tilde{O}(n^2)$ term to obtain an $\\tilde{O}(m\\sqrt{n})$ expected time almost-$3/2$ approximation algorithm.","For every graph with $\\Omega (n^{1.5})$ edges the running time of the latter algorithm is not better than the running time of the former algorithm.","Therefore, it is interesting to consider the opposite question to the one considered by [41].","Can the $\\tilde{O}(m\\sqrt{n})$ term be removed?","We show that this can be done for undirected unweighted graphs and present an $O(n^2 \\log n)$ expected time algorithm.","For a graph of Diameter $D=3h+z$ , where $z\\in [0,1,2]$ our algorithm returns an estimation $\\hat{D}$ such that $2h-1\\le \\hat{D} \\le D$ , when $z \\in [0,1]$ and $2h\\le \\hat{D} \\le D$ , when $z=2$ .","Interestingly, our algorithm is obtained by using ideas developed originally for distance oracles and compact routing schemes.","As we are allowed to use quadratic time, we try to estimate the distance between every pair of vertices.","To enable this approach we can no longer sample $A$ naively.","Instead, we adapt a recursive sampling algorithm to compute $A$ , that was introduced by Thorup and Zwick [47] in the context of compact routing schemes.","The expected running time of their algorithm is $\\tilde{O}(mn/|A|)$ .","We provide a new implementation of their algorithm that runs in expected $\\tilde{O}(n(n/|A|)^2)$ time.","The set $A$ has the following important property, for every vertex $w\\in V$ , its cluster (see [48]) $\\lbrace u \\mid d(u,w)<d(u,A) \\rbrace $ is of size $O(n/ |A|)$ .","Consider now a pair of vertices $u$ and $v$ that are in the cluster of $w$ .","For any such pair we can efficiently compute their exact distance.","Moreover, we show that for all pairs $u,v$ that are not in the same cluster of any vertex, we can bound $d(u,v)$ from below with $d(u,A)+d(v,A)-1$ .","This, combined with some other ideas, gives our approximation guarantees.","We extend our approach to also provide an almost $5/3$ -approximation for all Eccentricities.","The idea of using the bounded clusters of Thorup and Zwick [47] has been used in prior work to obtain improved distance oracles [38], [5], approximate shortest paths [11] and compact routing schemes [4]."],["A simple approach with additive error","Let's first consider a simple approach obtaining an $\\tilde{O}(n^2)$ time approximation algorithm for Diameter, Eccentricities or $S$ -$T$ Diameter.","Suppose that we have an algorithm ALG that can compute in $\\tilde{O}(m\\sqrt{n})$ time, for any graph $G^{\\prime }$ , an estimate $D^{\\prime }$ of its Diameter $D$ such that $p\\cdot D - q\\le D^{\\prime }\\le D$ , estimates $e(v)$ of $\\epsilon (v)$ for all $v$ so that $r\\epsilon (v)-s\\le e(v)\\le \\epsilon (v)$ , and an estimate $D^{\\prime \\prime }$ of the $S$ -$T$ Diameter $D_{S,T}$ so that $t\\cdot D_{S,T}-u\\le D^{\\prime \\prime }\\le D_{S,T}$ .","Now, Dor, Halperin and Zwick [24] showed that in $\\tilde{O}(n^2)$ time one can compute for any $n$ node $G$ , an additive 2 spanner $H$ on $\\tilde{O}(n^{1.5})$ edges.","In fact Knudsen [30] recently showed that in $O(n^2)$ time one can get $H$ on $O(n^{1.5})$ edges (i.e.","he removed all logs!).","Let's compute $H$ for our given graph and run ALG on $H$ .","The runtime is $\\tilde{O}(n^{1.5}\\cdot \\sqrt{n})\\le \\tilde{O}(n^2)$ since $H$ has $\\le O(n^{1.5})$ edges.","Let $D^{\\prime }_H,e_H(\\cdot ),D^{\\prime \\prime }_H$ be the estimates that we obtain respectively for the Diameter $D_H$ of $H$ , the Eccentricities $\\epsilon _H(\\cdot )$ of $H$ and the $S,T$ Diameter $D^H_{S,T}$ .","Let's return $D^{\\prime }_H-2,e_H(\\cdot )-2,D^{\\prime \\prime }_H-2$ for our estimates for the Diameter, Eccentricities and $S$ -$T$ Diameter of $G$ .","Notice that $pD-q\\le p\\cdot D_H - q\\le D^{\\prime }_H\\le D_H\\le D+2$ and so $pD-2-q \\le D^{\\prime }_H-2\\le D$ .","Similarly since $r\\epsilon (v)-s\\le r\\epsilon _H(v)-s\\le e_H(v)\\le \\epsilon _H(v)\\le \\epsilon (v)+2$ , we get $r\\epsilon (v)-s-2\\le e_H(v)-2\\le \\epsilon (v)$ .","Finally since $t\\cdot D_{S,T}-u\\le t\\cdot D^H_{S,T}-u\\le D^{\\prime \\prime }_H\\le D^H_{S,T}\\le D_{S,T}+2$ , we get $t\\cdot D_{S,T}-u-2\\le D^{\\prime \\prime }_H-2\\le D_{S,T}$ .","Thus, in $\\tilde{O}(n^2)$ time we get almost the same guarantees as in the $\\tilde{O}(m\\sqrt{n})$ time algorithms, except for an extra additive loss of 2 in the quality.","Below we show how to make the additive loss in quality smaller for Diameter and Eccentricities.","This is especially important when these parameters are constant, which is the hard case of the problems anyway."],["Near linear almost 3/2-approximation for Diameter","Thorup and Zwick [48] introduced distance oracles, a succinct data structure for answering approximate distance queries efficiently.","Among the tools they use are clusters and bunches.","Let $A\\subseteq V$ , let $p_A(u)$ be the closest vertex to $u$ from $A$ , where ties are broken in favor of the vertex with a smaller identifier and let $d(u,A)=d(u,p_A(u))$ .","For every $v\\in V$ , let $B_A(u)=\\lbrace v\\in V\\mid d(u,v) <d(u,A \\rbrace $ be the bunch of $u$ .","For every $w\\in V\\setminus A$ , let $C_A(w)= \\lbrace v \\mid w\\in B_A(v)\\rbrace $ be the cluster of $w$ .","Thorup and Zwick [48] showed that if a set $A$ is formed by adding every vertex of $V$ to $A$ with probability $p$ then the expected size of $B_A(v)$ is $O(1/p)$ , for every $v\\in V$ .","They also showed, in the context of compact routing schemes [47], that if the set $A$ is constructed by a recursive sampling algorithm then it is possible to bound the maximum size of a cluster as well.","They also showed, in the context of compact routing schemes [47], that if the set $A$ is constructed by a recursive sampling algorithm then it is possible to bound the maximum size of a cluster as well.","Their algorithm works as follows.","It sets $A$ to the empty set and $W$ to $V$ .","Next, as long as the set $W$ is not empty the algorithm samples from $W$ vertices with probability $p$ and adds the sampled vertices to $A$ .","The algorithm computes $C_A(w)$ for every $w\\in W$ and removes from $W$ all the vertices whose cluster has at most $4/p$ vertices with respect to the updated $A$ .","The pseudo-code is given in Algorithm REF .","Thorup and Zwick center algorithm [1] center$G,p$ $A= \\emptyset $ $W =V$ $W\\ne \\emptyset $ $X$ - random sample of nodes from $W$ , $|X|=|W|p$ $A= A \\cup X$ $W = \\lbrace w\\in V \\mid |C_A(w)|> 4/p \\rbrace $ $A$ Thorup and Zwick proved the following Theorem: Theorem 28 (Theorem 3.1 from [47]) The expected size of the set $A$ returned by Algorithm REF is at most $2np \\log n$ .","For every $w \\in V$ we then have $|C_A(w)|\\le 4/p$ .","Thorup and Zwick claimed that the expected running time of Algorithm REF is $O(mnp \\log n)$ .","They did not provide the details and refer the reader to [48].","However, an educated guess is that they compute clusters for the vertices currently in $W$ in each iteration of the while loop, which results in the claimed running time.","The starting point of the Diameter and Eccentricities algorithms presented in this section is an $O(n/p^2 \\log n)$ expected time implementation of Algorithm REF .","The first idea behind our implementation is that, as opposed to what Thorup and Zwick did, we will compute the bunches and use them to compute the clusters and the set $W$ .","This can be done as follows.","Once we have computed $B_A(v)$ for every $v\\in V$ , we can scan $B_A(v)$ , and for every $w\\in B_A(v)$ we can add $v$ to $C_A(w)$ .","The cost of this process is $O(|\\cup _{v\\in V} B_A(v)|)$ and since the clusters are by definition the inverse of the bunches, at the end of this process we have $C_A(w)$ and $|C_A(w)|$ , for every $w\\in V$ and we can compute $W$ (as needed in Algorithm REF ).","However, in the current implementation only the expected size of a bunch is bounded, and since the Thorup-Zwick bound on the number of iterations is $O(\\log n)$ in expectation as well, we cannot apply this idea directly to deduce a good expected running time.","To this end, more ideas are needed.","The following simple observation helps us to achieve our goal.","Observation 29 Let $A_i$ be the set $A$ after updating it in the beginning of the $i$ -th iteration of the while loop in Algorithm REF .","Let $A^*$ be a set such that $A^*\\subseteq A_i$ , for every $i\\ge 1$ .","For every $v\\in V$ it holds that $B_{A_i}(v) \\subseteq B_{A^*}(v)$ .","It follows from this observation that we only need to pick the first set $A_1$ such that $|B_{A_1}(v)|\\le O(1/p)$ for every $v\\in V$ .","It is folklore that the $s$ closest vertices $N_{s}(v)$ to a vertex $v$ can be computed in $O(s^2)$ time [24].","This implies that we can compute $N_{1/p}(v)$ for every $v\\in V$ in $O(n/p^2)$ time.","It is not hard to see that, given the sets $N_{1/p}(v)$ of all $v\\in V$ , one can (deterministically) compute a “hitting” set $A$ of size $O(np \\log n)$ in $O(n+n/p)$ worst case time, so that $N_{1/p}(v)\\cap A\\ne \\emptyset $ for every $v\\in V$ (a greedy algorithm works; e.g.","see [48]).","The second idea behind our implementation is that we first compute the sets $N_{1/p}(v)$ for every $v\\in V$ and the hitting set $A$ , as described above.","Then, using these sets, we initialize Algorithm REF with a set $A$ such that $|B_A(v)|=O(1/p)$ , for every $v\\in V$ .","In more detail, our algorithm works as follows.","For every $v\\in V$ it computes the set $N_{1/p}(v)$ in $O(n/p^2)$ time.","Then it finds a set $A$ such that $N_{1/p}(v)\\cap A\\ne \\emptyset $ for every $v\\in V$ .","Given the hitting set $A$ , it computes $d(v,A)$ and $p_A(v)$ for every $v\\in V$ .","Using $d(v,p_A(v))$ and $N_{1/p}(v)$ it computes for every $v\\in V$ the bunch $B_A(v)$ .","Finally, it computes the clusters and $W$ using the bunches as we described above.","The rest of the algorithm is almost identical to Algorithm REF .","The only difference is that we compute the bunches and use them to compute the clusters and the set $W$ .","The pseudo-code is given in Algorithm REF .","New implementation of Thorup and Zwick center algorithm [1] center$G,p$ compute $N_{1/p}(v)$ for every $v\\in V$ .","$A=$ hitting set of the sets $N_{1/p}(v)$ , where $v\\in V$ .","compute $d(v,A)$ and $p_A(v)$ for every $v\\in V$ .","compute $B_A(v)$ using $N_{1/p}(v)$ and $d(v,p_A(v))$ .","every $u\\in V$ compute $C_A(u)$ using $B_A(\\cdot )$ $W = \\lbrace w\\in V \\mid |C_A(w)|> 4/p \\rbrace $ $W\\ne \\emptyset $ $X$ - random sample of nodes from $W$ , $|X|=np$ $A= A \\cup X$ every $v\\in V$ compute $B_A(v)$ every $u\\in V$ compute $C_A(u)$ using $B_A(\\cdot )$ $W = \\lbrace w\\in V \\mid |C_A(w)|> 4/p \\rbrace $ $A$ We show: Lemma 30 Algorithm REF computes in $O(n/p^2 \\log n)$ expected time a set $A$ of expected size $O((n/p)\\cdot \\log n)$ that guarantees for every vertex $w\\in V \\setminus A$ that $|C_A(w)|=O(1/p)$ , and for every $v\\in V$ that $|B_A(v)|=O(1/p)$ .","The cost of computing $N_{1/p}(v)$ for every $v\\in V$ is $O(n (1/p)^2)$ [24].","The cost of computing $A$ is $O(n/p)$ time [48].","Computing $d(v,A)$ and $p_A(v)$ for every $v\\in V$ in $O(m)$ time is straightforward by running shortest paths tree computation from a dummy vertex that is connected to the set $A$ .","To compute $B_A(v)$ using $N_{1/p}(v)$ we only scan $N_{1/p}(v)$ , thus, the total cost is $O(n(1/p))$ .","As we explained earlier the cost of computing clusters using bunches is $O(|\\cup _{v\\in V} B_A(v)|)$ .","Since for every $v\\in V$ we have $B_A(v)\\subseteq N_{1/p}(v)$ the total cost is $O(n(1/p))$ .","This completes the analysis of the part that precedes the while loop.","Next, we analyze the cost of the while loop.","Let $A^*$ be the set $A$ that was computed before the while loop and let $A_i$ be the set $A$ after updating it in the beginning of the $i$ -th iteration of the while loop.","From Observation REF it follows that $B_{A_i}(v) \\subseteq B_{A^*}(v)$ and therefore in every iteration the cost of computing bunches from scratch is at most $O(n (1/p)^2)$ as $|B_{A^*}(v)|=O(1/p)$ , for every $v\\in V$ .","One can also compute $B_{A_{i+1}}(v)$ from $B_{A_i}(v)$ by first computing $d(v,A_{i+1})$ and if $d(v,A_{i+1})<d(v,A_i)$ to prune $B_{A_{i+1}}(v)$ accordingly at a smaller cost of $O(n(1/p) + m)$ , however this does not affect the overall complexity.","Thorup and Zwick [47] proved that the expected number of iterations is $O(\\log n)$ .","The fact that the set $A$ from which we start is different does not affect their proofThey prove that in each iteration with probability $1/2$ the size of $W$ decreases by a factor of 2.","For this argument they only require that the set $A$ in each iteration will be chosen from $W$ uniformly at random with probability $p$ as we do., therefore there are only $O(\\log n)$ iterations in expectation.","This implies that a set $A$ of expected size $(np \\log n)$ is returned in $O(n/p^2 \\log n)$ expected time.","The algorithm stops only when there are no large clusters, thus the bound on the cluster size follows.","As we mentioned above the algorithm starts with bunches that satisfy the required bound and their size can only decrease afterwards, thus the bound on the bunches follows.","almost $3/2$ -Approximation for Diameter [1] $3/2$ -Approx-DiamG $M$ - $n\\times n$ matrix whose entries are set to $n$ $A =$ CENTER$(G,1/\\sqrt{n})$ every $w\\in V$ Step 1 every $\\langle u, v\\rangle \\in C_A(w) \\times C_A(w)$ , s.t.","$u\\ne v\\;$ $M(u,v) = \\min (M(u,v), d(u,w)+d(v,w))$ every $\\langle u, v\\rangle \\in V \\times V$ , s.t.","$M(u,v)=n$ Step 2 $M(u,v) = d(u,A)+d(v,A)-1$ $H$ - an additive 2 spanner of $G$ Step 3 every $u \\in A$ compute shortest paths tree for $u$ in $H$ and set $\\epsilon _H(u)$ , the Eccentricity of $u$ in $H$ $D_1 = \\max _{\\langle u, v\\rangle \\in V \\times V} M(u,v)$ $D_2 = \\max _{u\\in A} \\epsilon _H(u)$ $\\hat{D} = \\max (D_1,D_2-2)$ $\\hat{D}$ We can now turn to describe the new Diameter algorithm.","The algorithm works as follows.","All entries of an $n \\times n$ matrix $M$ are set to $n$ .","A set $A$ of centers is computed using the algorithm of Thorup and Zwick [47].","For every vertex $w\\in V$ and every pair $\\langle u, v\\rangle \\in C_A(w)\\times C_A(w)$ the algorithm sets $M(u,v)$ to $\\min (M(u,v), d(u,w)+d(v,w))$ (Step 1).","Next, the algorithm searches the matrix $M$ for entries whose value is still $n$ .","Given a pair $\\langle u, v\\rangle \\in V \\times V$ for which $M(u,v)=n$ the algorithm sets $M(u,v)$ to $d(u,A)+d(v,A)-1$ (Step 2).","Finally, the algorithm computes an additive 2 spanner $H$ of the input graph $G$ and for every $u\\in A$ it computes $\\epsilon _H(u)$ , the Eccentricity of $u$ in $H$ (Step 3).","The algorithm outputs the maximum between $D_1$ and $D_2-2$ , where $D_1$ is $\\max _{\\langle u, v\\rangle \\in V \\times V} M(u,v)$ and $D_2$ is $\\max _{u\\in A} \\epsilon _H(u)$ .","Next, we bound the value returned by Algorithm REF .","Theorem 31 Let $D=3h+z$ , where $z\\in [0,1,2]$ .","The value $\\hat{D}$ returned by Algorithm REF satisfies: $\\begin{array}{ll}2h-1 & \\mbox{if } z \\in [0,1] \\\\2h & \\mbox{if } z=2\\end{array}\\le \\hat{D} \\le D$ We start with the following Lemma: Lemma 32 Let $u,v\\in V$ and let $P(u,v)$ be a shortest path between $u$ and $v$ .","If $B_A(u) \\cap B_A(v)\\ne \\emptyset $ then $(B_A(u) \\cap B_A(v)) \\cap P(u,v) \\ne \\emptyset $ .","If $v\\in B_A(u)$ then the claim trivially holds so we can assume that $v\\notin B_A(u)$ .","Let $w$ be the vertex farthest from $u$ that is in $B_A(u) \\cap P(u,v)$ .","From the definition of $w$ it follows that $d(u,w)=d(u,A)-1$ .","Assume, towards a contradiction, that $(B_A(u) \\cap B_A(v)) \\cap P(u,v) = \\emptyset $ .","This implies that $w\\notin B_A(v)$ and $d(v,A)-1<d(v,w)=d(u,v)-d(u,w)$ .","However, since $B_A(u) \\cap B_A(v)\\ne \\emptyset $ there is a vertex $w^{\\prime }$ such that $d(u,w^{\\prime })\\le d(u,w)$ and $d(v,w^{\\prime })\\le d(v,A)-1 < d(u,v)-d(u,w)$ .","This implies that $d(u,w^{\\prime })+d(v,w^{\\prime })< d(u,v)$ , a contradiction to the triangle inequality.","Lemma 33 Let $u,v\\in V$ .","If $B_A(u) \\cap B_A(v)= \\emptyset $ then $d(u,A)+d(v,A)-1 \\le d(u,v)$ .","Notice first that $B_A(u)$ (resp., $B_A(v)$ ) contains all the vertices at distance $d(u,A)-1$ (resp., $d(v,A)-1$ ).","Let $P(u,v)$ be a shortest path between $u$ and $v$ .","Let $w$ be the vertex farthest from $u$ on $P(u,v)$ that is also in $B_A(u)$ .","Similarly, let $w^{\\prime }$ be the vertex farthest from $v$ on $P(u,v)$ that is also in $B_A(v)$ .","Since $B_A(u) \\cap B_A(v)= \\emptyset $ it holds that $w\\ne w^{\\prime }$ .","Therefore: $d(u,v) = d(u,A)-1+d(v,A)-1+d(w,w^{\\prime })\\ge d(u,A)+d(v,A)-1.$ Let $a$ and $b$ be the Diameter endpoints, that is $d(a,b)=D=3h+z$ , where $z\\in [0,1,2]$ .","Let $P(a,b)$ be a shortest path between $a$ and $b$ .","Assume first that $B_A(a) \\cap B_A(b)\\ne \\emptyset $ .","It follows from Lemma REF that there is a vertex $w\\in P(a,b)$ such that $\\langle a, b\\rangle \\in C(w) \\times C(w)$ .","Therefore, $M(a,b)=D$ after Step 1.","After the update in Step 2 it follows from Lemma REF that $M(u,v)\\le d(u,v)$ for every $u ,v\\in V$ .","Therefore, the maximum value in the matrix is $d(a,b)$ and $D_1=D$ .","Let $x = \\operatornamewithlimits{arg\\,max}_{y\\in A} \\epsilon _H(y)$ .","Since $H$ is an additive 2 spanner it holds that $\\epsilon _H(x)\\le D+2$ , hence, we have $D_2 \\le D$ and the algorithm returns the exact value of the Diameter.","Assume now that $B_A(a) \\cap B_A(b) = \\emptyset $ .","From the discussion of the previous case it follows that in this case $\\hat{D}\\le D$ as well.","Thus, it is only left to prove the lower bound.","Assume that $z\\in [0,1]$ .","Consider first the case that $d(a,A)\\ge h$ and $d(b,A)\\ge h$ then from Lemma REF it follows that $M(a,b)\\ge 2h-1$ after Step 2 and $D_1$ is at least $2h-1$ .","If this is not the case then either $d(a,A)< h$ or $d(b,A)< h$ (or both).","Assume, wlog, that $d(a,A)< h$ .","In this case the Eccentricity in $H$ of at least one vertex from $A$ is at least $2h+1$ and hence $D_2-2$ is at least $2h-1$ .","Assume now that $z=2$ .","If either $d(a,A)\\ge h$ and $d(b,A)> h$ or $d(a,A)>h$ and $d(b,A)\\ge h$ then from Lemma REF it follows that $M(a,b)\\ge 2h$ after Step 2 and $D_1$ is at least $2h$ .","If this is not the case then either $d(a,A)\\le h$ or $d(b,A)\\le h$ (or both).","Assume, wlog, that $d(a,A)\\le h$ .","In this case the Eccentricity in $H$ of at least one vertex from $A$ is at least $2h+2$ and hence $D_2-2$ is at least $2h$ .","We now turn to we analyze the running time of Algorithm REF .","Theorem 34 The expected running time of Algorithm REF is $O(n^2 \\log n)$ .","The set $A$ is computed by the center algorithm presented in Algorithm REF with $p=1/\\sqrt{n}$ .","From Lemma REF it follows that the size of the set $A$ is $O(\\sqrt{n} \\log n)$ and its construction time is $O(n^2\\log n)$ in expectation.","For every $w\\in V$ the size of $C_A(w)$ is $O(\\sqrt{n})$ .","Therefore, Step 1 takes $O(n \\times |C_A(w)|^2)=O(n^2)$ .","Step 2 takes $O(n^2)$ time as well.","In Step 3 we first compute an additive 2 spanner $H$ on $O(n^{1.5})$ edges.","Knudsen [30], following Dor, Halperin and Zwick [24] showed how to do this in $O(n^2)$ time.","We also compute $|A|$ shortest paths trees in $H$ .","As $H$ has $O(n^{1.5})$ edges, this step takes $O(n^2 \\log n)$ time."],["Near linear almost 5/3-approximation for Eccentricities","almost $5/3$ -Approximation for all Eccentricities [1] $5/3$ -Approx-EccG Run lines 2-11 of Algorithm REF , with $H$ augmented with shortest paths trees for $B_A(u) \\cup \\lbrace p(u) \\rbrace $ for every $u\\in V$ every $u \\in V$ $\\epsilon _1(u) = \\max _{v\\in V} M(u,v)$ $\\epsilon _2(u) = \\epsilon _H(p(u))-d(u,p(u))-2$ $\\epsilon _3(u) = d_H(u,y)-2$ , where $y=\\operatornamewithlimits{arg\\,max}_{x\\in A} d_H(u,x)$ $\\epsilon ^{\\prime }(u) = \\max (\\epsilon _1(u),\\epsilon _2(u),\\epsilon _3(u))$ Next, we show how to update Algorithm REF to obtain an almost $5/3$ approximation for all Eccentricities.","We run lines 2-11 of Algorithm REF .","The only difference is that $H$ is augmented with the edges of the shortest paths tree that span the set $B_A(u) \\cup \\lbrace p(u) \\rbrace $ for every $u\\in V$ .","Then, for every $u\\in V$ we compute $\\epsilon _1(u)$ , $\\epsilon _2(u)$ and $\\epsilon _3(u)$ , which are defined as follows: $\\epsilon _1(u) = \\max _{v\\in V} M(u,v)$ , $\\epsilon _2(u) = \\epsilon _H(p(u))-d(u,p(u))-2$ and $\\epsilon _3(u) = d_H(u,y)-2$ , where $y=\\operatornamewithlimits{arg\\,max}_{x\\in A} d_H(u,x)$ .","The algorithm sets $\\epsilon ^{\\prime }(u)$ to $\\max \\lbrace \\epsilon _1(u), \\epsilon _2(u), \\epsilon _3(u) \\rbrace $ for every $u\\in V$ as an estimation to $\\epsilon (u)$ .","The pseudo-code is given in Algorithm REF .","We now prove: Theorem 35 For every $u\\in V$ , Algorithm REF computes in $O(n^2 \\log n)$ expected time a value $\\epsilon ^{\\prime }(u)$ that satisfies: $\\frac{3\\epsilon (u)}{5}-1 \\le \\epsilon ^{\\prime }(u)\\le \\epsilon (u).$ We start by analyzing the running time.","Lines 2-11 of the algorithm are the same as Algorithm REF , with one difference, the spanner $H$ is augmented with the edges of the shortest paths tree that span the set $B_A(u) \\cup \\lbrace p(u) \\rbrace $ for every $u\\in V$ .","This adds at most $O(n^{1.5})$ edges to $H$ and hence the cost of these lines remain $O(n^2 \\log n)$ time in expectation.","The computation of $\\epsilon _1(u)$ , $\\epsilon _2(u)$ and $\\epsilon _3(u)$ for every $u\\in V$ costs $O(n^2)$ time in total.","Let $u\\in V$ be an arbitrary vertex and let $\\epsilon (u)=d(u,t)$ .","We now turn to bound $\\epsilon ^{\\prime }(u)$ .","In our analysis we will use the following simple observation: Observation 36 In an undirected graph it holds for every $u,v \\in V$ that $\\epsilon (u)\\ge \\epsilon (v)-d(u,v)$ .","It is straightforward to see that both $\\epsilon _2(u)$ and $\\epsilon _3(u)$ are at most $\\epsilon (u)$ .","Recall that $\\epsilon _2(u) = \\epsilon _H(p(u))-d(u,p(u))-2\\le \\epsilon (p(u))-d(u,p(u)\\le \\epsilon (u)$ and $\\epsilon _3(u) = d_H(u,y)-2\\le d(u,y)\\le \\epsilon (u)$ .","We distinguish between two cases.","Case 1: $B_A(u) \\cap B_A(t)\\ne \\emptyset $ .","It follows from Lemma REF that $P(u,t)\\cap (B_A(u) \\cap B_A(t)) \\ne \\emptyset $ and $M(u,t)= \\epsilon (u)$ .","From Lemma REF it follows that $M(u,w)\\le d(u,w)$ for every $w\\in V$ after Step 2.","Therefore, $\\epsilon _1(u)=\\epsilon (u)$ .","Since $\\epsilon _2(u)\\le \\epsilon (u)$ and $\\epsilon _3(u)\\le \\epsilon (u)$ we get that $\\epsilon ^{\\prime }(u)=\\epsilon (u)$ .","Case 2: $B_A(u) \\cap B_A(t) = \\emptyset $ .","Consider first the case that $d(u,p(u))\\le \\frac{\\epsilon (u)}{5}-1$ .","From Observation REF we get that $\\epsilon _H(p(u))\\ge \\epsilon _H(u)-d_H(u,p(u))$ .","As we augmented $H$ with a shortest paths tree that spans $B_A(u) \\cup \\lbrace p(u) \\rbrace $ we have $d(u,p(u))=d_H(u,p(u))$ and we get $\\epsilon _H(p(u))\\ge \\epsilon _H(u)-d(u,p(u))$ .","Hence, we get that $\\epsilon _2(u) = \\epsilon _H(p(u))-d(u,p(u))-2 \\ge \\epsilon _H(u)-2d(u,p(u))-2\\ge \\epsilon (u)-2d(u,p(u))-2$ .","As before we have $\\epsilon _2(u) \\le \\epsilon (u)$ .","Using $d(u,p(u))\\le \\frac{\\epsilon (u)}{5}-1$ we get that: $\\epsilon _2(u) \\ge \\epsilon (u)-\\frac{2\\epsilon (u)}{5}\\ge \\frac{3\\epsilon (u)}{5}.$ Assume now that $d(u,p(u))\\ge \\frac{\\epsilon (u)}{5}$ .","This means that $d(u,A)-1\\ge \\frac{\\epsilon (u)}{5}-1$ .","Let $S$ be the set of all vertices $v\\in V$ such that $B_A(u) \\cap B_A(v)= \\emptyset $ , that is, $S=V \\setminus \\cup _{w\\in B_A(u)} C_A(w)$ .","Let $t^{\\prime } = \\operatornamewithlimits{arg\\,max}_{x\\in S} d(x,A)-1$ .","If $d(t^{\\prime },A)-1 \\ge \\frac{2\\epsilon (u)}{5}-1$ we get from Lemma REF that $M(u,t^{\\prime })\\ge \\frac{3\\epsilon (u)}{5}-1$ .","Assume now that $d(t^{\\prime },A) < \\frac{2\\epsilon (u)}{5}$ .","As $t^{\\prime }$ is the farthest vertex from $A$ we get that $d(t,p(t))<\\frac{2\\epsilon (u)}{5}$ and $d(u,p(t))>\\frac{3\\epsilon (u)}{5}$ .","Therefore, $\\epsilon _3(u) = d_H(u,y)-2\\ge d(u,p(t))-2\\ge \\frac{3\\epsilon (u)}{5}-1$ .","From Lemma REF and Lemma REF it follows that $\\epsilon _1(u)\\le \\epsilon (u)$ and the bound follows."],["Algorithms for dense graphs using matrix multiplication","Here we will give $O(n^{2.05})$ time approximation algorithms for Diameter and Eccentricities in dense unweighted undirected graphs.","The approximation guarantees of these algorithms are slightly better than those in our $O(n^2\\log n)$ time algorithm.","In fact, the guarantees are exactly the same as in the $\\tilde{O}(m\\sqrt{n})$ time algorithms for Diameter and Eccentricities of Roditty and V. Williams [41] and Cairo et al. [15].","To achieve this, we give an efficient implementation using fast matrix multiplication of the $\\tilde{O}(m\\sqrt{n})$ time algorithms of [15] and [41].","The main overhead of the $\\tilde{O}(m\\sqrt{n})$ time algorithms [15], [41] is in computing the distances from a set $S=W\\cup \\lbrace w\\rbrace \\cup T$ of $O(\\sqrt{n}\\log n)$ nodes: the set $S$ itself can be computed in linear time using random sampling to form a set $W$ , BFS from a dummy node to find the node $w$ farthest from $W$ and then BFS from $w$ to find the set $T$ of closest $\\sqrt{n}$ nodes to $w$ .","After one knows all distances from every $s\\in S$ to every $v\\in V$ , it takes linear time to output the Diameter and Eccentricity estimates.","The main idea of our algorithms is as follows.","If the Diameter is of size $\\le O(\\log n)$ , then one does not need all distances between $S$ and $V$ , but only those that are $O(\\log n)$ .","Small distances are easy to compute with matrix multiplication.","Let $A$ be the adjacency matrix and $A_S$ be its submatrix formed by just the rows in $S$ .","Then we can find the distances for all pairs in $S\\times V$ at distance $\\le t$ by computing $A_S\\times A^{t-1}$ , which can be computed by performing $t-1$ matrix products of dimension $|S|\\times n$ by $n\\times n$ , and this can be accomplished in $O(tn^{2.05})$ time [26], [31].","If on the other hand the Diameter is $D\\ge 100\\log n$ , then one can use an $\\tilde{O}(n^2)$ time algorithm by Dor et al.","[24] to compute estimates of all pairwise distances with an additive error at most $4\\log n$ .","The maximum distance estimate computed, minus $4\\log n$ , will be between $0.96 D$ and $D$ , giving a really good approximation already.","A similar argument works for Eccentricities, and also for $S$ -$T$ Diameter.","Below we recap the guarentees of the $\\tilde{O}(m\\sqrt{n})$ time approximation algorithms of [15], [41].","Theorem 37 ([15], [41]) The following can be computed in $\\tilde{O}(m\\sqrt{n})$ time: an estimate $\\hat{D}$ of the graph Diameter $D$ , such that $\\frac{2}{3} D-\\frac{1}{3}\\le \\hat{D}\\le D$ , for every node $v$ , an estimate $e(v)$ of its Eccentricity $\\epsilon (v)$ , such that $\\frac{3}{5} \\epsilon (v)-\\frac{1}{5}\\le e(v)\\le \\epsilon (v)$ .","Using Seidel's algorithm [43] we can compute all the distances exactly, and hence the above parameters as well, all in $O(n^\\omega )$ time for $\\omega < 2.373$ .","We will show that for dense graphs, we can obtain the same approximation guarantees as in Theorem REF , in time $O(n^{2.05})$ .","Let us compare to our $O(n^2\\log n)$ time algorithms.","For Diameter $D=3h+z$ , the $O(n^2\\log n)$ time algorithm returns an estimate $2h-1$ when $z=0,1$ and $2h$ when $z=2$ .","The estimate $\\hat{D}$ here is $\\ge (2D-1)/3= 2h+ (2z-1)/3$ , which is $\\ge 2h$ when $z=0$ and $\\ge 2h+1$ when $z=1,2$ .","For Eccentricities, the $O(n^2\\log n)$ time algorithm returns estimates $e(v)\\ge 3\\epsilon (v)/5 -1$ , and here we return a better estimate $e(v)\\ge (3\\epsilon -1)/5$ .","We will rely on two known algorithms.","The first is from a paper by Dor, Halperin and Zwick [24] on additive approximations of All-Pairs Shortest Paths (APSP).","Among many other results, [24] show that in $\\tilde{O}(n^2)$ time, one can compute for all pairs of vertices $u,v$ , an estimate $d^{\\prime }(u,v)$ of their distance $d(u,v)$ so that $d(u,v)\\le d^{\\prime }(u,v)\\le d(u,v)+a\\log n$ for an explicit constant $a\\le 4$ .","The second is an algorithm for the following truncated multi-source shortest paths problem: given an integer $Q$ , a graph $G=(V,E)$ and a set $S$ , compute the distances $d(s,v)$ for every $s\\in S$ and $v\\in V$ for which $d(s,v)\\le Q$ .","The algorithm uses fast matrix multiplication and is quite straightforward.","Let $A$ be the $n\\times n$ Boolean matrix with rows and columns indexed by $V$ , so that $A[u,v]=1$ if there is an edge between $u$ and $v$ or $u=v$ , and $A[u,v]=0$ otherwise; i.e.","$A$ is the adjacency matrix added to the identity matrix.","Let $A_S$ be the $|S|\\times n$ submatrix of $A$ consisting of the rows indexed by nodes of $S$ .","For an integer $i\\ge 1$ , let $A^i$ be the $i$ -th power of $A$ under the Boolean matrix product.","Here, $A^i[u,v]=1$ if and only if the distance between $u$ and $v$ is at most $i$ .","Define $A^0$ as the identity matrix.","Consider $A_S \\cdot A^i$ for any choice of $i\\ge 0$ (under the Boolean matrix product).","Here, $(A_S \\cdot A^i)[s,v]=1$ if and only if the distance between $s$ and $v$ is at most $i+1$ .","Thus, if we compute $D_i:=A_S\\cdot A^i$ for every $1\\le i< Q$ , we would know the distance from every $s\\in S$ to every $v\\in V$ , whenever this distance is at most $Q$ .","Computing these matrix products can easily be done by performing the following $Q-1$ Boolean products of an $|S|\\times n$ matrix by an $n\\times n$ matrix: let $D_0=A_S$ ; then for each $i$ from 1 to $t-1$ , compute $D_i:=D_{i-1}\\cdot A$ .","Thus, the running time is $O(Q\\cdot M(|S|,n,n))$ where $M(|S|,n,n)$ is the runtime of multiplying an $|S|\\times n$ matrix by an $n\\times n$ matrix.","Armed with these two algorithms, let us recap Roditty et al.","'s (and Cairo et al.","'s) approximation algorithm and see how to modify it.","The algorithm proceeds as follows: Let $D$ , $R$ and $\\epsilon (v)$ denote the Diameter and Radius of $G$ and the Eccentricity of node $v$ , respectively.","RV/CGR Algorithm [1] Using BFS (see [41] and [15]), in $O(m+n)$ time compute $W,w,T$ , where $W\\subseteq V$ is a uniformly chosen subset of size $O(\\sqrt{n} \\log n)$ , $w$ is the furthest node from $W$ and $T$ are the closest $\\sqrt{n}$ nodes to $w$ .","Let $S=\\lbrace w\\rbrace \\cup W\\cup T$ .","For every $s\\in S$ and every $v\\in V$ , compute the distance $d(s,v)$ between $s$ and $v$ ; set $\\epsilon (s)=\\max _v d(s,v)$ .","Set $\\tilde{D}=\\max _{x\\in S} \\epsilon (x)$ .","Set for every $v\\in V$ , $\\tilde{\\epsilon }(v)=\\max \\lbrace d(w,v),\\max _{x\\in W} d(x,v), \\max _{x\\in T} (\\epsilon (x)-d(x,v))\\rbrace $ .","The runtime bottleneck in the above algorithm is step (2) which runs in $\\tilde{O}(mn^{1/2})$ time if one uses BFS through each node of $S$ .","Let us describe how to modify the algorithm.","We will replace (2) with a truncated distance computation and also use the algorithm of Dor, Halperin and Zwick to handle large distances that we might have ignored in the truncated computation.","Our Modified Approximation.","[1] FasterApproximation First part: Handle Large Distances: [t]@X@ Use Dor, Halperin and Zwick's algorithm to compute distance estimates $d^{\\prime }(\\cdot ,\\cdot )$ so that for every $u,v\\in V$ , $d(u,v)\\le d^{\\prime }(u,v)\\le d(u,v)+a\\log n$ .","Let $X=3a\\log n$ .","Set $\\tilde{D}_1 = \\max _{u,v\\in V} d^{\\prime }(u,v) - a\\log n$ .","For every $v\\in V$ , set $\\tilde{\\epsilon }_1(v) = \\max _{u} d^{\\prime }(u,v) - a\\log n$ .","Second Part: Handle Small Distances: [t]@X@ Using BFS (see [41] and [15]), in $O(m+n)$ time compute $W,w,T$ , where $W\\subseteq V$ is a uniformly chosen subset of size $O(\\sqrt{n} \\log n)$ , $w$ is the furthest node from $W$ and $T$ are the closest $\\sqrt{n}$ nodes to $w$ .","Let $S=\\lbrace w\\rbrace \\cup W\\cup T$ .","[t]@X@ Let $Q=2(X+a\\log n)=8a\\log n$ .","For every $s\\in S$ and every $v\\in V$ whose distance $d(s,v)$ is at most $Q$ , compute $d(s,v)$ .","Let $d_{\\le }(s,v)$ denote $d(s,v)$ if we have computed it, and $\\infty $ otherwise.","Set $\\epsilon _{\\le }(s)=\\max _v d_{\\le }(s,v)$ .","Set $\\tilde{D}_2=\\max _{x\\in S} \\epsilon _{\\le }(x)$ .","[t]@X@ $\\forall v\\in V$ , set $\\tilde{\\epsilon }_2(v)=\\max \\lbrace d_\\le (w,v), \\max _{x\\in W} d_\\le (x,v), \\max _{y\\in T} (\\epsilon _\\le (y)-d_\\le (y,v))\\rbrace $ .","Third Part: Set $\\tilde{D},\\tilde{\\epsilon }(\\cdot )$: If $\\tilde{D}_1\\ge X$ , set $\\tilde{D}=\\tilde{D}_1$ , and otherwise set $\\tilde{D}=\\tilde{D}_2$ .","[t]@X@ For every $v\\in V$ , if there exists some $x\\in S$ such that $d^{\\prime }(x,v)\\ge X+a\\log n$ , then set $\\tilde{\\epsilon }(v)=\\tilde{\\epsilon }_1(v)$ , and otherwise $\\tilde{\\epsilon }(v)=\\tilde{\\epsilon }_2(v)$ .","Consider our modified algorithm, FasterApproximation.","Now we will prove several claims.","Claim 38 The running time of algorithm FasterApproximation is $\\tilde{O}(M(\\sqrt{n},n,n))$ .","The Dor, Halperin, Zwick part of the algorithm (Step 3) runs in $\\tilde{O}(n^2)$ time.","Step 8 runs in $O((X+a\\log n)\\cdot M(|S|,n,n))$ time where $S=\\lbrace w\\rbrace \\cup W\\cup T$ , using the iterated rectangular matrix product algorithm.","Recall that $X+a\\log n = O(\\log n)$ .","Thus Step 8 runs in $\\tilde{O}(M(|S|,n,n))$ time.","Since $|S|=\\tilde{O}(\\sqrt{n})$ and we can partition an $|S|\\times n\\times n$ matrix product into $ \\text{\\rm polylog~} n$ , $n^{1/2}\\times n\\times n$ matrix products, the runtime of the step is $\\tilde{O}(M(n^{1/(2)},n,n))$ .","Steps 10 and 13 run in $O(n|S|)<\\tilde{O}(n^2)$ time.","The rest of the steps run in linear time.","Since $M(n^{1/2},n,n)\\ge n^2$ (one must at least read the input), the total running time is $\\tilde{O}(M(n^{1/2},n,n))$ .","Claim 39 $\\frac{2D-1}{3} \\le \\tilde{D}\\le D$ .","Suppose that $\\tilde{D}_1\\ge X$ .","The algorithm returns $\\tilde{D}=\\tilde{D}_1=\\max _{u,v} d^{\\prime }(u,v)-a\\log n$ .","By the guarantee on $d^{\\prime }$ , we have $D-a\\log n \\le \\tilde{D}_1\\le D$ .","Hence $\\tilde{D}\\ge D(1-(a\\log n)/D)\\ge D(1-(a\\log n)/X) =2D/3 \\ge (2D-1)/3$ .","Suppose now that $\\tilde{D}_1< X$ .","This means that $D<X+a\\log n$ and every distance in the graph is $\\le X+a\\log n$ .","In the second part of the algorithm we set $Q=2(X+a\\log n)$ , and hence every distance is computed exactly: for every $s\\in S$ , $v\\in V$ , $d_\\le (s,v)=d(s,v)$ .","Hence the second part of the algorithm will be identical to the RV/CGR algorithm and hence we get the same guarantees: $(2D-1)/3\\le \\tilde{D}\\le D$ .","Claim 40 For every node $v$ , $\\frac{3\\epsilon (v)-1}{5} \\epsilon (v)\\le \\tilde{\\epsilon }(v)\\le \\epsilon (v)$ .","Fix $v$ .","Suppose first that there exists some $x$ such that $d^{\\prime }(x,v)\\ge X+a\\log n$ .","Then $\\epsilon (v)\\ge \\tilde{\\epsilon }_1(v)=\\max _u d^{\\prime }(u,v) - a\\log n \\ge \\epsilon (v)-a\\log n = \\epsilon (v)(1-a\\log n/\\epsilon (v))$ .","Since $\\epsilon (v)\\ge d(x,v)\\ge d^{\\prime }(x,v)-a\\log n \\ge X$ , we get that $\\tilde{\\epsilon }_1(v)\\ge \\epsilon (v)(1-a\\log n / X) = 2\\epsilon (v)/3$ .","Now suppose that for all $x\\in V$ , $d^{\\prime }(x,v)<X+a\\log n$ .","Then, also for all $x\\in V$ , $d(x,v)<X+a\\log n$ and $\\epsilon (v)<X+a\\log n$ .","Consider all the quantities needed in the second part of the algorithm to compute $\\tilde{\\epsilon }_2(v)$ : $d_\\le (w,v)$ : since $\\forall x\\in V$ , $d(x,v)<X+a\\log n$ , $d_\\le (w_i,v)=d(w_i,v)$ for each $w_i$ ; $d_\\le (x,v)$ for every $x\\in W$ : as above, $d_\\le (x,v)=d(x,v)$ ; $\\epsilon _\\le (x)-d_\\le (x,v)$ for all $x\\in T$ : here, $\\epsilon (x)\\le \\epsilon (v)+d(x,v)\\le 2\\epsilon (v)<2(X+a\\log n)$ .","Since we compute all distances from nodes in $S$ up to $2(X+a\\log n)$ and $x\\in S$ , $\\epsilon _\\le (x)=\\epsilon (x)$ .","Also as in the above bullets, $d_\\le (x,v)=d(x,v)$ .","Thus all the quantities needed are the correct ones and $\\tilde{\\epsilon }(v)=\\tilde{\\epsilon }_2(v)$ inherits the same guarantees as in the algorithm by Cairo et al.","From Le Gall and Urrutia [26] (see also, [31]) we obtain that $M(\\sqrt{n},n,n)\\le O(n^{2.044183})$ .","We obtain: Theorem 41 In $O(n^{2.045})$ time, one can obtain an almost $3/2$ -approximation $\\tilde{D}$ to the Diameter $D$ and almost $5/3$ -approximations $e(v)$ to all Eccentricities $\\epsilon (v)$ : $\\frac{2D-1}{3} \\le \\tilde{D}\\le D$ .","For every node $v$ , $\\frac{3\\epsilon (v)-1}{5}\\le \\tilde{\\epsilon }(v)\\le \\epsilon (v)$ .","Finally we note that our approach also works to speed up our almost 2-approximation algorithm for $S$ -$T$ Diameter as well, giving an $O(n^{2.045})$ time almost-2 approximation algorithm.","The main reason is that, like in the Diameter approximation algorithm, if the $S$ -$T$ Diameter is very large (say $D_{S,T}>100a\\log n$ ), then the $+a\\log n$ APSP algorithm with $a\\log n$ subtracted will return an estimate that is at least $D_{S,T}-a\\log n>0.99D_{S,T}$ .","On the other hand, our $S$ -$T$ Diameter approximation algorithm only needs to know the distances up to $D_{S,T}$ to compute an estimate of $D_{S,T}$ , and so if $D_{S,T}\\le 100a\\log n$ , then we only need to compute $O(\\log n)$ matrix products of dimension $O(\\sqrt{n} \\log n)\\times n\\times n$ again."]],"string":"[\n [\n \"Towards Tight Approximation Bounds for Graph Diameter and Eccentricities\"\n ],\n [\n \"Abstract Among the most important graph parameters is the Diameter, the largest distance between any two vertices.\",\n \"There are no known very efficient algorithms for computing the Diameter exactly.\",\n \"Thus, much research has been devoted to how fast this parameter can be approximated.\",\n \"Chechik et al.\",\n \"showed that the diameter can be approximated within a multiplicative factor of $3/2$ in $\\\\tilde{O}(m^{3/2})$ time.\",\n \"Furthermore, Roditty and Vassilevska W. showed that unless the Strong Exponential Time Hypothesis (SETH) fails, no $O(n^{2-\\\\epsilon})$ time algorithm can achieve an approximation factor better than $3/2$ in sparse graphs.\",\n \"Thus the above algorithm is essentially optimal for sparse graphs for approximation factors less than $3/2$.\",\n \"It was, however, completely plausible that a $3/2$-approximation is possible in linear time.\",\n \"In this work we conditionally rule out such a possibility by showing that unless SETH fails no $O(m^{3/2-\\\\epsilon})$ time algorithm can achieve an approximation factor better than $5/3$.\",\n \"Another fundamental set of graph parameters are the Eccentricities.\",\n \"The Eccentricity of a vertex $v$ is the distance between $v$ and the farthest vertex from $v$.\",\n \"Chechik et al.\",\n \"showed that the Eccentricities of all vertices can be approximated within a factor of $5/3$ in $\\\\tilde{O}(m^{3/2})$ time and Abboud et al.\",\n \"showed that no $O(n^{2-\\\\epsilon})$ algorithm can achieve better than $5/3$ approximation in sparse graphs.\",\n \"We show that the runtime of the $5/3$ approximation algorithm is also optimal under SETH.\",\n \"We also show that no near-linear time algorithm can achieve a better than $2$ approximation for the Eccentricities and that this is essentially tight: we give an algorithm that approximates Eccentricities within a $2+\\\\delta$ factor in $\\\\tilde{O}(m/\\\\delta)$ time for any $0<\\\\delta<1$.\",\n \"This beats all Eccentricity algorithms in Cairo et al.\"\n ],\n [\n \"Introduction\",\n \"Among the most important graph parameters are the graph's Diameter and the Eccentricities of its vertices.\",\n \"The Eccentricity of a vertex $v$ is the (shortest path) distance to the furthest vertex from $v$ , and the Diameter is the largest Eccentricity over all vertices in the graph.\",\n \"The Eccentricities and Diameter measure how fast information can spread in networks.\",\n \"Efficient algorithms for their computation are highly desired (see e.g.\",\n \"[40], [9], [33]).\",\n \"Unfortunately, the fastest known algorithms for these parameters are very slow on large graphs.\",\n \"For unweighted graphs on $n$ vertices and $m$ edges, the fastest Diameter algorithm runs in $\\\\tilde{O}(\\\\min \\\\lbrace mn,n^\\\\omega \\\\rbrace )$ $\\\\tilde{O}$ notation hides polylogarithmic factors time [16] where $\\\\omega <2.373$ is the exponent of square matrix multiplication [52], [32], [45].\",\n \"For weighted graphs, the fastest Eccentricity and Diameter algorithms actually compute all distances in the graph, i.e.\",\n \"they solve the All-Pairs Shortest Paths (APSP) problem.\",\n \"The fastest known algorithms for APSP in weighted graphs run in $\\\\min \\\\lbrace \\\\tilde{O}(mn),n^3/\\\\exp (\\\\sqrt{\\\\log n})\\\\rbrace $ [53], [35], [37].\",\n \"Whether one can solve Diameter faster than APSP is a well-known open problem (e.g.\",\n \"see Problem 6.1 in [20] and [2], [17]).\",\n \"Whether one can solve Eccentricities faster than APSP was addressed by [50] (for dense graphs) and by [34] (for sparse graphs).\",\n \"Vassilevska W. and Williams [50] showed that Eccentricities and APSP are equivalent under subcubic reductions, so that either both of them admit $O(n^{3-\\\\varepsilon })$ time algorithms for $\\\\varepsilon >0$ , or neither of them do.\",\n \"Lincoln et al.\",\n \"[34] proved that under a popular conjecture about the complexity of weighted Clique, the $O(mn)$ runtime for Eccentricities cannot be beaten by any polynomial factor for any sparsity of the form $m=\\\\Theta (n^{1+1/k})$ for integer $k$ .\",\n \"Due to the hardness of exact computation, efficient approximation algorithms are sought.\",\n \"A folklore $\\\\tilde{O}(m+n)$ time algorithm achieves a 2-approximation for Diameter in directed weighted graphs and a 3-approximation for Eccentricities in undirected weighted graphs.\",\n \"Aingworth et al.\",\n \"[2] presented an almost-$3/2$ approximation An almost-$c$ approximation of $X$ is an estimate $X^{\\\\prime }$ so that $X\\\\le X^{\\\\prime }\\\\le cX+O(1)$ .\",\n \"algorithm for Diameter running in $\\\\tilde{O}(n^2+m\\\\sqrt{n})$ time.\",\n \"Roditty and Vassilevska W. [41] improved the result of [2] with an $\\\\tilde{O}(m\\\\sqrt{n})$ expected time almost-$3/2$ approximation algorithm.\",\n \"Chechik et al.\",\n \"[21] obtained a (genuine) $3/2$ approximation algorithm for Diameter (in directed graphs) and a (genuine) $5/3$ -approximation algorithm for Eccentricities (in undirected graphs), running in $\\\\tilde{O}(\\\\min \\\\lbrace m^{3/2},mn^{2/3}\\\\rbrace )$ time.\",\n \"These are the only known non-trivial approximation algorithms for Diameter in directed graphs.\",\n \"So far, there are no known faster than $mn$ algorithms for approximating Eccentrities in directed graphs within any constant factor.\",\n \"Cairo et al.\",\n \"[15] generalized the above results for undirected graphs and obtained a time-approximation tradeoff: for every $k\\\\ge 1$ they obtained an $\\\\tilde{O}(mn^{1/(k+1)})$ time algorithm that achieves an almost-$2-1/2^k$ approximation for Diameter and an almost $3-4/(2^k+1)$ -approximation for Eccentricities.\"\n ],\n [\n \"Our contributions.\",\n \"We address the following natural question: Main Question: Are the known approximation algorithms for Diameter and Eccentricities optimal?\",\n \"A partial answer is known.\",\n \"Under the Strong Exponential Time Hypothesis (SETH), every $3/2-\\\\varepsilon $ approximation algorithm (for $\\\\varepsilon >0$ ) for Diameter in undirected unweighted graphs with $O(n)$ nodes and edges must use $n^{2-o(1)}$ time [41].\",\n \"Similarly, every $5/3-\\\\varepsilon $ approximation algorithm for the Eccentricities of undirected unweighted graphs with $O(n)$ nodes and edges must use use $n^{2-o(1)}$ time [7].\",\n \"This however does not answer the question of whether the runtimes of the known $3/2$ and $5/3$ approximation algorithms can be improved.\",\n \"It is completely plausible that there is a $3/2$ -approximation algorithm for Diameter or a $5/3$ -approximation for Eccentricities running in linear time.\",\n \"We address our Main Question for both sparse and dense graphs.\",\n \"Our results are shown in Table REF .\",\n \"Table: Our results.\",\n \"All of the lower bounds hold even for sparse graphs.\",\n \"SS-TT Diameter is a variant of Diameter introduced later in this section.\"\n ],\n [\n \"Sparse graphs.\",\n \"Our first result (restated as Theorem REF ) regards approximating Diameter in undirected unweighted sparse graphs.\",\n \"Theorem 1 ($3/2$ -Diameter Approx.\",\n \"is Tight) Under SETH, no $O(n^{3/2-\\\\delta })$ time algorithm for $\\\\delta >0$ can output a $8/5-\\\\varepsilon $ approximation for $\\\\varepsilon >0$ for the Diameter of an undirected unweighted sparse graph.\",\n \"In particular, any $3/2$ -approximation algorithm in sparse graphs must take $n^{3/2-o(1)}$ time.\",\n \"Hence the $\\\\tilde{O}(m^{3/2})$ time $3/2$ -approximation algorithm of [41], [21] is optimal in two ways: improving the approximation ratio to $3/2-\\\\varepsilon $ causes a runtime blow-up to $n^{2-o(1)}$ ([41]) and improving the runtime to $O(m^{3/2-\\\\delta })$ causes an approximation ratio blow-up to $8/5$ .\",\n \"Our lower bound instance says that in $O(m^{3/2-\\\\delta })$ time one cannot return 6 when the Diameter is 8.\",\n \"One may be tempted to extend the above lower bound, by showing that, say, in $O(m^{4/3-\\\\delta })$ time one cannot even return 5 when the Diameter is 8.\",\n \"This approach, however fails: in Theorem REF we give an $O(m^2/n)$ time algorithm that does return 5 in this case, and in general when the Diameter is $2h$ , it returns at least $h+1$ .\",\n \"Notice that when the Diameter is $2h$ , the folklore linear time algorithm returns an estimate of only $h$ .\",\n \"Hence for sparse graphs, our algorithm runs in linear time and outperforms the folklore algorithm.\",\n \"Also, for constant even Diameter, it gives a better than 2 approximation.\",\n \"We obtain stronger Diameter hardness results for weighted graphs and for directed unweighted graphs.\",\n \"In particular, assuming SETH: For weighted sparse graphs, no $O(n^{3/2-\\\\delta })$ time algorithm for $\\\\delta >0$ can output a $5/3-\\\\varepsilon $ Diameter approximation (for $\\\\varepsilon >0$ ) (Theorem REF ).\",\n \"For directed unweighted sparse graphs, using a general time-accuracy tradeoff lower bound (Theorem REF ), we show that no near-linear time algorithm can achieve an approximation factor better than $5/3$ .\",\n \"We summarize our Diameter lower bounds and compare them to the known upper bounds in Figure REF .\",\n \"Figure: Our hardness results for Diameter.\",\n \"The xx-axis is the approximation factor and the yy-axis is the runtime exponent.\",\n \"Black lines represent lower bounds.\",\n \"Black dots represent existing algorithms.\",\n \"Blue dots represent existing algorithms whose approximation is potentially off by an additive term (the algorithms of ).\",\n \"Transparent dots represent algorithms that might exist and would be tight with our lower bounds.We address our Main Question for Eccentricities as well.\",\n \"Our main result for Eccentricities is Theorem REF .\",\n \"Its first consequence is as follows: Theorem 2 ($5/3$ -Eccentricities Alg.\",\n \"is Tight) Under SETH, no $O(n^{3/2-\\\\delta })$ time algorithm for $\\\\delta >0$ can output a $9/5-\\\\varepsilon $ approximation for $\\\\varepsilon >0$ for the Eccentricities of an undirected unweighted sparse graph.\",\n \"In other words, the $\\\\tilde{O}(m^{3/2})$ time $5/3$ -approximation algorithm of [41], [21] is tight in two ways.\",\n \"Improving the approximation ratio to $5/3-\\\\varepsilon $ causes a runtime blow-up to $n^{2-o(1)}$ ([7]) and improving the runtime to $O(m^{3/2-\\\\delta })$ causes an approximation ratio blow-up to $9/5$ .\",\n \"More generally, we prove (in Theorem REF ): for every $k\\\\ge 2$ , under SETH, distinguishing between Eccentricities $2k-1$ and $4k-3$ in unweighted undirected sparse graphs requires $n^{1+1/(k-1)-o(1)}$ time.\",\n \"Thus, no near-linear time algorithm can achieve a $2-\\\\varepsilon $ -approximation for Eccentricities for $\\\\varepsilon >0$ .\",\n \"The best (folklore) near-linear time approximation algorithm for Eccentricities currently only achieves a 3-approximation, and only in undirected graphs.\",\n \"There is no known constant factor approximation algorithm for directed graphs!\",\n \"Is our limitation result for linear time Eccentricity algorithms far from the truth?\",\n \"We show that our lower bound result is essentially tight, for both directed and undirected graphs by producing the first non-trivial near-linear time approximation algorithm for the Eccentricities in weighted directed graphs (Theorem REF ).\",\n \"Theorem 3 (2-Approx.\",\n \"for Eccentricities in near-linear time.)\",\n \"Under SETH, no $n^{1+o(1)}$ time algorithm can output a $2-\\\\varepsilon $ approximation for $\\\\varepsilon >0$ for the Eccentricities of an undirected unweighted sparse graph.\",\n \"For every $\\\\delta >0$ , there is an $\\\\tilde{O}(m/\\\\delta )$ time algorithm that produces a $(2+\\\\delta )$ -approximation for the Eccentricities of any directed weighted graph.\",\n \"The approximation hardness result is the first result within fine-grained complexity that gives tight hardness for near linear time algorithms.\",\n \"The $2+\\\\delta $ approximation ratio that our algorithm produces beats all approximation ratios for Eccentricities given by Cairo et al. [15].\",\n \"It also constitutes the first known constant factor approximation algorithm for Eccentricities in directed graphs.\",\n \"Our approximation algorithm also implies as a corollary an approximation algorithm for the Source Radius problemThe Source Radius problem is a natural extension of the undirected Radius definition.\",\n \"The goal is to return $\\\\min _x\\\\max _vd(x,v)$ .\",\n \"studied in [7] with the same runtime and approximation factor ($2+\\\\delta $ ).\",\n \"Abboud et al.\",\n \"[7] showed that, under the Hitting Set Conjecture, any $(2-\\\\varepsilon )$ -approximation algorithm (for $\\\\varepsilon >0$ ) for Source Radius requires $n^{2-o(1)}$ time, and hence our Source Radius algorithm is also essentially tight.\",\n \"Our lower bound in Theorem REF holds already for undirected unweighted graphs, and the upper bound works even for directed weighted graphs.\",\n \"The algorithm produces a $(2+\\\\delta )$ -approximation, which while close, is not quite a 2-approximation.\",\n \"We design (in Theorem REF ) a genuine 2-approximation algorithm running in $\\\\tilde{O}(m\\\\sqrt{n})$ time that also works for directed weighted graphs.\",\n \"We then complement it (in Theorem REF ) with a tight lower bound under SETH: in sparse directed graphs, if you go below factor 2 in the accuracy, the runtime blows up to quadratic.\",\n \"Theorem 4 (Tight 2-Approx.\",\n \"for Eccentricities) Under SETH, no $n^{2-\\\\delta }$ time algorithm for $\\\\delta >0$ can output a $2-\\\\varepsilon $ approximation for the Eccentricities of a directed unweighted sparse graph.\",\n \"There is an $\\\\tilde{O}(m\\\\sqrt{n})$ time algorithm that produces a 2-approximation for the Eccentricities of any directed weighted graph.\",\n \"We thus give an essentially complete answer to our Main Question for Eccentricities.\",\n \"Our results are summarized in Figures REF and REF .\",\n \"Figure: Our algorithms and hardness results for Eccentricities.\",\n \"The lower bounds are for unweighted graphs and the upper bounds are for weighted graphs.\",\n \"The xx-axis is the approximation factor and the yy-axis is the runtime exponent.\",\n \"Black lines represent lower bounds.\",\n \"Black dots represent existing algorithms (including our algorithm at (2,3/2)(2,3/2) in figure b).\",\n \"Blue dots represent existing algorithms whose position may not be exactly as it appears in the figure.\",\n \"Here, the blue dots represent our (2+δ)(2+\\\\delta )-approximation algorithm running in O ˜(m/δ)\\\\tilde{O}(m/\\\\delta ) time.\",\n \"Transparent dots represent algorithms that might exist and would be tight with our lower bounds.Our conditional lower bounds for both Diameter and Eccentricities are all based on a common construction: a conditional lower bound for a problem called $S$ -$T$ Diameter.\",\n \"In $S$ -$T$ Diameter, the input is a graph $G=(V,E)$ and two subsets $S,T\\\\subseteq V$ , not necessarily disjoint, and the output is $D_{S,T}:=\\\\max _{s\\\\in S,t\\\\in T} d(s,t)$ .\",\n \"$S$ -$T$ Diameter is a problem of independent interest.\",\n \"It is related to the bichromatic furthest pair problem studied in geometry (e.g.\",\n \"as in [29]), but for graphs (if we set $T=V\\\\setminus S$ ).\",\n \"It is easy to see that if one can compute the $S$ -$T$ Diameter, then one can also compute the Diameter in the same time: just set $S=T=V$ .\",\n \"We show that actually, when it comes to exact computation, the $S$ -$T$ Diameter and Diameter in weighted graphs are computationally equivalent (Theorem REF ).\",\n \"We show that $S$ -$T$ Diameter also has similar approximation algorithms to Diameter.\",\n \"We give a 3-approximation running in linear time (Claim REF based on the folklore Diameter 2-approximation algorithm), and a 2-approximation running in $\\\\tilde{O}(m^{3/2})$ time (Theorem REF based on the $3/2$ -approximation algorithm of [41], [21]).\",\n \"We prove the following lower bound for $S$ -$T$ Diameter (restated as Theorem REF ), the proof of which is the starting point for all of our conditional lower bounds.\",\n \"Theorem 5 Under SETH, for every $k\\\\ge 2$ , every algorithm that can distinguish between $S$ -$T$ Diameter $k$ and $3k-2$ in undirected unweighted graphs requires $n^{1+1/(k-1)-o(1)}$ time.\",\n \"Theorem REF implies that under SETH, our aforementioned 2 and 3-approximation algorithms are optimal.\",\n \"For all of our lower bounds, we also address the question of whether they can be extended to higher values of Diameter and Eccentricities.\",\n \"All of our lower bounds, with the exception of directed Eccentricities, are of the form “any algorithm that can distinguish between Diameter (or Eccentricity) $a$ and $b$ requires a certain amount of time\\\" for small values of $a$ and $b$ .\",\n \"This doesn't exclude the possibility of an algorithm that distinguishes between higher Diameters (or Eccentricities) of the same ratio i.e.\",\n \"between $a\\\\ell $ and $b\\\\ell $ for some $\\\\ell $ .\",\n \"For weighted Diameter, our lower bound easily extends to higher values of Diameter by simply scaling up the edge weights.\",\n \"For $S-T$ Diameter and undirected Eccentricities, our lower bounds easily extend to higher values of Diameter and Eccentricities by simply subdividing the edges.\",\n \"For unweighted directed Diameter, our lower bound extends to higher values of Diameter with a slight loss in approximation factor by subdividing some of the edges.\",\n \"For unweighted undirected Diameter, our lower bound does not seem to easily extend to higher values of Diameter.\"\n ],\n [\n \"Dense graphs.\",\n \"Can we address our Main Question for dense graphs as well?\",\n \"In particular, can we extend our runtime lower bounds of the form $n^{1+1/\\\\ell -o(1)}$ to $mn^{1/\\\\ell -o(1)}$ , thus matching the known algorithms for larger values of $m$ ?\",\n \"We show that the answer is “no”.\",\n \"For undirected unweighted graphs, we obtain $\\\\tilde{O}(n^2)$ time algorithms for Diameter achieving an almost $3/2$ -approximation (Theorem REF ), and for all Eccentricities achieving an almost $5/3$ -approximation algorithm (Theorem REF ).\",\n \"These algorithms run in near-linear time in dense graphs, improving the previous best runtime of $\\\\tilde{O}(m\\\\sqrt{n})$ by Roditty and Vassilevska W. [41], and subsuming (for dense unweighted graphs) the results of Cairo et al. [15].\",\n \"Theorem 6 There is an expected $O(n^2\\\\log n)$ time algorithm that for any undirected unweighted graph with Diameter $D=3h+z$ for $h\\\\ge 0,z\\\\in \\\\lbrace 0,1,2\\\\rbrace $ , returns an extimate $D^{\\\\prime }$ such that $2h-1\\\\le D^{\\\\prime }\\\\le D$ if $z=0,1$ and $2h\\\\le D^{\\\\prime }\\\\le D$ if $z=2$ .\",\n \"There is an expected $O(n^2\\\\log n)$ time algorithm that for any undirected unweighted graph returns estimates $\\\\varepsilon ^{\\\\prime }(v)$ of the Eccentricities $\\\\varepsilon (v)$ of all vertices such that $3\\\\varepsilon (v)/5-1\\\\le \\\\varepsilon ^{\\\\prime }(v)\\\\le \\\\varepsilon (v)$ for all $v$ .\",\n \"We also show (in Theorem REF ) that one can improve the estimates slightly with an $O(n^{2.05})$ time algorithm.\"\n ],\n [\n \"Related work\",\n \"The fastest known algorithm for APSP in dense weighted graphs is by R. Williams [53] and runs in $O(n^3/2^{\\\\Theta (\\\\sqrt{\\\\log n})})$ time.\",\n \"For sparse undirected graphs, the fastest known APSP algorithm is by Pettie [35] running in $O(mn+n^2\\\\log \\\\log n)$ time.\",\n \"The fastest APSP algorithm for sparse undirected weighted graphs is by Pettie and Ramachandran [37] and runs in $O(mn\\\\log \\\\alpha (m,n))$ time.\",\n \"For APSP on undirected unweighted graphs with $m>n\\\\log \\\\log n$ , Chan [17] presented an $O(mn \\\\log \\\\log n/\\\\log n)$ time algorithm.\",\n \"In graphs with small integer edge weights bounded in absolute value by $M$ , APSP can be computed in $\\\\tilde{O}(Mn^\\\\omega )$ time (by Shoshan and Zwick [46] building upon Seidel [43] and Alon, Galil and Margalit [6]) in undirected graphs and in $\\\\tilde{O}(M^{0.681}n^{2.5302})$ time (by Zwick [55]) in directed graphs.\",\n \"Zwick [55] also showed that APSP in directed weighted graphs admits an $(1+\\\\varepsilon )$ -approximation algorithm for any $\\\\varepsilon >0$ , running in time $\\\\tilde{O}(n^\\\\omega /\\\\varepsilon \\\\log (M/\\\\varepsilon ))$ .\",\n \"For Diameter in graphs with integer edge weights bounded by $M$ , Cygan et al.\",\n \"[16] obtained an algorithm running in time $\\\\tilde{O}(Mn^\\\\omega )$ .\",\n \"The pioneering work of Aingworth et al.\",\n \"[2] on Diameter and shortest paths approximation was the root to many subsequent works.\",\n \"Building upon Aingworth et al.\",\n \"[2], Dor, Halperin and Zwick [24] presented additive approximation algorithms for APSP in undirected unweighted graphs, achieving among other things, an additive 2-approximation in $\\\\tilde{O}(n^{7/3})$ time (notably, the best known bound on $\\\\omega $ is $>7/3$ ).\",\n \"They also presented an $\\\\tilde{O}(n^2)$ time additive $O(\\\\log n)$ -approximation algorithm.\",\n \"These algorithms were generalized by Cohen and Zwick [23] who showed that in undirected weighted graphs APSP has a (multiplicative) 3-approximation in $\\\\tilde{O}(n^2)$ time, a $7/3$ -approximation in $\\\\tilde{O}(n^{7/3})$ time, and a 2-approximation in $\\\\tilde{O}(n \\\\sqrt{mn})$ time.\",\n \"Baswana and Kavitha [11] presented an $\\\\tilde{O}(m\\\\sqrt{n}+n^2)$ time multiplicative 2-approximation algorithm and an $\\\\tilde{O}(m^{2/3} n+n^2)$ time $7/3$ -approximation algorithm for APSP in weighted undirected graphs.\",\n \"Spanners are closely related to shortest paths approximation.\",\n \"A subgraph $H$ is an $(\\\\alpha , \\\\beta )$ -spanner of $G=(V,E)$ if for every $u,v\\\\in V$ , $d_H(u,v) \\\\le \\\\alpha \\\\cdot d_G(u,v) + \\\\beta $ , where $d_{G^{\\\\prime }}(u,v)$ is the distance between $u$ and $v$ in $G^{\\\\prime }$ .\",\n \"Any weighted undirected graph has a $(2k-1,0)$ -spanner with $O(n^{1+1/k})$ edges [3].\",\n \"Baswana and Sen [14] presented a randomized linear time algorithm for constructing a $(2k-1,0)$ -spanner with $O(kn^{1+1/k})$ edges.\",\n \"Dor, Halperin and Zwick [24] showed that a $(1,2)$ -spanner with $O(n^{1.5})$ edges can be constructed in $\\\\tilde{O}(n^2)$ time.\",\n \"Elkin and Peleg [25] showed that for every integer $k\\\\ge 1$ and $\\\\varepsilon >0$ there is a $(1+\\\\varepsilon , \\\\beta )$ -spanner with $O(\\\\beta n^{1+1/k})$ edges, where $\\\\beta $ depends on $k$ and $\\\\varepsilon $ but is independent of $n$ .\",\n \"Baswana et.\",\n \"al [12] presented a $(1,6)$ -spanner with $O(n^{4/3})$ edges.\",\n \"Woodruff [54] presented an $\\\\tilde{O}(n^2)$ time algorithm that computes a $(1,6)$ -spanner with $O(n^{4/3})$ edges.\",\n \"Chechik [18] presented a $(1,4)$ -spanner with $O(n^{7/5})$ edges.\",\n \"Recently, Abboud and Bodwin[1] showed that there is no additive spanner with constant error and $O(n^{4/3-\\\\varepsilon })$ edges.\",\n \"Thorup and Zwick [48] introduced the notion of distance oracles, a data structure that stores approximate distances for a weighted undirected graph.\",\n \"Thorup and Zwick designed a distance oracle that for any $k$ takes $O(mn^{1/k})$ time to construct and, is of size is $O(kn^{1+1/k})$ , and given a pair of vertices $u,v \\\\in V$ it returns in $O(k)$ time a $(2k-1)$ -approximation for $d(u,v)$ .\",\n \"Baswana and Sen [13] improved the construction time to $O(n^2)$ for unweighted graphs.\",\n \"Baswana and Kavitha [11] extended the $O(n^2)$ construction time to weighted graphs.\",\n \"Subsequently, Baswana, Gaur, Sen, and Upadhyay [10] obtained subquadratic construction time in unweighted graphs, at the price of having additive constant error in addition to the $2k-1$ multiplicative error.\",\n \"Chechik [19] gave an oracle with space $O(n^{1+1/k})$ and $O(1)$ query time, which like previous work, returns a $(2k-1)$ -approximation.\",\n \"Pǎtraşcu and Roditty [38] obtained a distance oracle that uses $\\\\tilde{O}(n^{5/3})$ space, has $O(1)$ query time, and returns a $(2k+1)$ -approximation.\",\n \"Sommer [44] presented an $\\\\tilde{O}(n^2)$ time algorithm that constructs such a distance oracle.\",\n \"The construction time was recently improved to $O(n^2)$ by Knudsen [30].\",\n \"Pǎtraşcu et.\",\n \"al [39] presented infinity many distance oracles with fractional approximation factors that for graphs with $m=\\\\tilde{O}(n)$ converge exactly to the integral stretch factors and the corresponding space bound of Thorup and Zwick.\",\n \"Thorup and Zwick [47] also extended their techniques from [48] to compact routing schemes.\",\n \"The lower bounds presented in this paper were inspired by a lower bound by Pǎtraşcu and Roditty [38] who showed conditional hardness based on a conjecture on the hardness of a set intersection problem for the space usage of any distance oracle that can distinguish between distances 3 and 7.\"\n ],\n [\n \"Organization\",\n \"In Section we prove our lower bounds for $S$ -$T$ Diameter which serve as a basis for the rest of our lower bounds.\",\n \"We also show equivalence between Diameter and $S$ -$T$ Diameter.\",\n \"In Section we prove our lower bounds for Eccentricities: one for directed graphs and one for undirected graphs.\",\n \"In Section we prove our lower bounds for Diameter.\",\n \"This section is divided into four subsections, one for each of the results in Table REF .\",\n \"In Section we describe our algorithms for sparse graphs: our 2-approximation and $(2+\\\\delta )$ -approximation for Eccentricities, our 2-approximation and 3-approximation for $S$ -$T$ Diameter, and our less than 2-approximation for Diameter.\",\n \"In Section we describe our algorithms for dense graphs: our nearly $3/2$ -approximations for Diameter and our nearly $5/3$ -approximations for Eccentricities.\"\n ],\n [\n \"Preliminaries\",\n \"Let $G=(V,E)$ be a graph, where $|V|=n$ and $|E|=m$ .\",\n \"For every $u,v\\\\in V$ let $d_G(u,v)$ be the length of the shortest path from $u$ to $v$ .\",\n \"When the graph $G$ is clear from the context we omit the subscript $G$ .\",\n \"The eccentricity $\\\\varepsilon (v)$ of a vertex $v$ is defined as $\\\\max _{u\\\\in V} d(v,u)$ .\",\n \"The diameter $D$ of a graph is $\\\\max _{v\\\\in V}\\\\varepsilon (v)$ .\",\n \"In a directed graph we have $\\\\varepsilon ^{out}(v)=\\\\max _{u\\\\in V} d(v,u)$ (resp., $\\\\varepsilon ^{in}(v)=\\\\max _{u\\\\in V} d(u,v)$ ).\",\n \"Let $deg(v)$ be the degree of $v$ and let $N_s(u)$ be the set of the $s$ closest vertices of $v$ , where ties are broken by taking the vertex with the smaller ID.\",\n \"In a directed graph let $deg^{out}(v)$ (resp., $deg^{in}(v)$ ) be the outgoing (incoming) degree of $v$ .\",\n \"Let $N^{\\\\text{out}}_s(v)$ (resp., $N^{\\\\text{in}}_s(v)$ ) be the set of the $s$ closest outgoing (incoming) vertices of $v$ , where ties are broken by taking the vertex with the smaller ID.\",\n \"For a subset $S\\\\subseteq V$ of vertices and a vertex $v \\\\in V$ we write $d(S,v):=\\\\min _{s \\\\in S}d(s,v)$ to denote the distance from the set $S$ to the vertex $v$ .\",\n \"Let $k\\\\ge 2$ .\",\n \"The $k$ -Orthogonal Vectors Problem ($k$ -OV) is as follows: given $k$ sets $S_1,\\\\ldots ,S_k$ , where each $S_i$ contains $n$ vectors in $\\\\lbrace 0,1\\\\rbrace ^d$ , determine whether there exist $v_1\\\\in S_1,\\\\ldots ,v_k\\\\in S_k$ so that their generalized inner product is 0, i.e.\",\n \"$\\\\sum _{i=1}^d \\\\prod _{j=1}^k v_j[i]=0$ .\",\n \"R. Williams [51] (see also [49]) showed that if for some $\\\\varepsilon >0$ there is an $n^{k-\\\\varepsilon } \\\\text{\\\\rm poly~} (d)$ time algorithm for $k$ -OV, then CNF-SAT on formulas with $N$ variables and $m$ clauses can be solved in $2^{N(1-\\\\varepsilon /k)} \\\\text{\\\\rm poly~} (m)$ time.\",\n \"In particular, such an algorithm would contradict the Strong Exponential Time Hypothesis (SETH) of Impagliazzo, Paturi and Zane [28] which states that for every $\\\\varepsilon >0$ there is a $K$ such that $K$ -SAT on $N$ variables cannot be solved in $2^{(1-\\\\varepsilon )N} \\\\text{\\\\rm poly~} N$ time (say, on a word-RAM with $O(\\\\log N)$ bit words).\",\n \"This also motivates the following $k$ -OV Conjectures (implied by SETH) for all constants $k\\\\ge 2$ : $k$ -OV requires $n^{k-o(1)}$ time on a word-RAM with $O(\\\\log n)$ bit words.\",\n \"Most of our conditional lower bounds are based on the $k$ -OV Conjecture for a particular constant $k$ , and thus they also hold under SETH.\",\n \"A main motivation behind SETH is that despite decades of research, the best upper bounds for $K$ -SAT on $N$ variables and $M$ clauses remain of the form $2^{N(1-c/K)} \\\\text{\\\\rm poly~} {(M)}$ for constant $c$ (see e.g.\",\n \"[27], [36], [42]).\",\n \"The best algorithms for the $k$ -OV problem for any constant $k\\\\ge 2$ on $n$ vectors and dimension $c\\\\log n$ run in time $n^{2-1/O(\\\\log c)}$ (Abboud, Williams and Yu [8] and Chan and Williams [22]).\"\n ],\n [\n \"$S$ -{{formula:678000a5-c252-477a-976e-b4f5fbe676e0}} Diameter hardness\",\n \"In the following we will prove that under SETH, our $S$ -$T$ Diameter algorithms are essentially optimal.\",\n \"We prove the following theorem: Theorem 7 Let $k\\\\ge 2$ be an integer.\",\n \"There is an $O(k n^{k-1} d^{k-1})$ time reduction that transforms any instance of $k$ -OV on sets of $n$ $d$ -dimensional vectors into a graph on $O(n^{k-1}+k n^{k-2} d^{k-1})$ nodes and $O(k n^{k-1} d^{k-1})$ edges and two disjoint sets $S$ and $T$ on $n^{k-1}$ nodes each, so that if the $k$ -OV instance has a solution, then $D_{S,T}\\\\ge 3k-2$ , and if it does not, $D_{S,T}\\\\le k$ .\",\n \"From Theorem REF we get that if there is some $k\\\\ge 2$ , $\\\\varepsilon >0$ and $\\\\delta >0$ so that there is an $O(M^{1+1/(k-1)-\\\\varepsilon })$ time $(3-2/k-\\\\delta )$ -approximation algorithm for $S$ -$T$ Diameter in $M$ -edge graphs, then $k$ -OV has an $n^{k-\\\\gamma } \\\\text{\\\\rm poly~} (d)$ -time algorithm for some $\\\\gamma >0$ and SETH is false.\",\n \"We obtain an immediate corollary.\",\n \"Corollary 8 For $S$ -$T$ Diameter, under SETH, there is no $(2-\\\\varepsilon )$ -approximation algorithm running in $O(m^{2-\\\\delta })$ time for any $\\\\varepsilon >0,\\\\delta >0$ , no $O(m^{3/2-\\\\delta })$ time $7/3-\\\\varepsilon $ -approximation algorithm for any $\\\\varepsilon >0,\\\\delta >0$ , no $(m+n)^{1+o(1)}$ time, $3-\\\\varepsilon $ -approximation algorithm for any $\\\\varepsilon >0$ .\",\n \"We will prove the following more detailed theorem, which will be useful for our Diameter lower bounds.\",\n \"Theorem 9 Let $k\\\\ge 2$ .\",\n \"Given a $k$ -OV instance consisting of sets $W_0,W_1,\\\\dots ,W_{k-1} \\\\subseteq \\\\lbrace 0,1\\\\rbrace ^d$ , each of size $n$ , we can in $O(kn^{k-1}d^{k-1})$ time construct an unweighted, undirected graph with $O(n^{k-1}+k n^{k-2} d^{k-1})$ vertices and $O(k n^{k-1} d^{k-1})$ edges that satisfies the following properties.\",\n \"The graph consists of $k+1$ layers of vertices $S=L_0,L_1,L_2,\\\\dots ,L_k=T$ .\",\n \"The number of nodes in the sets is $|S|=|T|=n^{k-1}$ and $|L_1|,|L_2|,\\\\dots ,|L_{k-1}|\\\\le n^{k-2}d^{k-1}$ .\",\n \"$S$ consists of all tuples $(a_0,a_1,\\\\ldots , a_{k-2})$ where for each $i$ , $a_i\\\\in W_i$ .\",\n \"Similarly, $T$ consists of all tuples $(b_1,b_2,\\\\ldots , b_{k-1})$ where for each $i$ , $b_i\\\\in W_i$ .\",\n \"If the $k$ -OV instance has no solution, then $d(u,v)=k$ for all $u \\\\in S$ and $v \\\\in T$ .\",\n \"If the $k$ -OV instance has a solution $a_0, a_1,\\\\dots ,a_{k-1}$ where for each $i$ , $a_i\\\\in W_i$ then if $\\\\alpha =(a_0,\\\\dots a_{k-2})\\\\in S$ and $\\\\beta = (a_1,\\\\dots ,a_{k-1}) \\\\in T$ , then $d(\\\\alpha ,\\\\beta )\\\\ge 3k-2$ .\",\n \"Suppose the $k$ -OV instance has a solution $a_0, a_1,\\\\dots ,a_{k-1}$ where for each $i$ , $a_i\\\\in W_i$ .\",\n \"Let $t=k-2$ .\",\n \"Let $s$ be such that $0\\\\le s\\\\le t$ .\",\n \"Let $b_{t-s+j}\\\\in W_{t-s+j}$ for all $j\\\\in [1,\\\\ldots ,s]$ be some other vectors, potentially different from $a_{t-s+j}$ .\",\n \"Consider $\\\\alpha =(a_0,a_1,\\\\ldots ,a_{t-s},b_{t-s+1},\\\\ldots ,b_{t})\\\\in L_0$ and $\\\\beta =(a_{1},\\\\ldots ,a_{t+1})\\\\in L_{t+2}$ .\",\n \"Then the distance between $\\\\alpha $ and $\\\\beta $ is at least $3t-2s+4$ .\",\n \"Symmetrically, let $c_{j}\\\\in W_{j}$ for all $j\\\\in [1,\\\\ldots ,s]$ be some other vectors, potentially different from $a_{j}$ .\",\n \"Consider $\\\\alpha =(a_0,a_1,\\\\ldots ,a_{t})\\\\in L_0$ and $\\\\beta =(c_{1},\\\\ldots ,c_s,a_{s+1},\\\\dots , a_{t+1})\\\\in L_{t+2}$ .\",\n \"Then the distance between $\\\\alpha $ and $\\\\beta $ is at least $3t-2s+4$ .\",\n \"For all $i$ from 1 to $k-1$ , for all $v \\\\in L_i$ there exists a vertex in $L_{i-1}$ adjacent to $v$ and a vertex in $L_{i+1}$ adjacent to $v$ .\"\n ],\n [\n \"Proof of Theorem \",\n \"We will prove the theorem for $k=t+2$ for any $t\\\\ge 0$ .\",\n \"We will create a layered graph $G$ on $t+3$ layers, $L_0,\\\\ldots ,L_{t+2}$ , where the edges go only between adjacent layers $L_i,L_{i+1}$ .\",\n \"We will set $S=L_0$ and $T=L_{t+2}$ for the $S$ -$T$ Diameter instance.\",\n \"In particular, $D_{S,T}\\\\ge t+2$ because of the layering.\",\n \"Let us describe the vertices of $G$ .\",\n \"$L_0$ consists of $n^{t+1}$ vertices, each corresponding to a $t+1$ -tuple $(a_0,a_1,\\\\ldots , a_t)$ where for each $i$ , $a_i\\\\in W_i$ .\",\n \"Similarly, $L_{t+2}$ consists of $n^{t+1}$ vertices, each corresponding to a $t+1$ -tuple $(b_1,b_2,\\\\ldots , b_{t+1})$ where for each $i$ , $b_i\\\\in W_i$ .\",\n \"Layer $L_1$ consists of $n^t d^{t+1}$ vertices, each corresponding to a tuple $(a_0,\\\\ldots ,a_{t-1}, \\\\bar{x})$ where for each $i$ , $a_i\\\\in W_i$ and $\\\\bar{x}=(x_0,\\\\ldots ,x_t)$ is a $(t+1)$ -tuple of coordinates in $[d]$ .\",\n \"Similarly, $L_{t+1}$ consists of $n^t d^{t+1}$ vertices, each corresponding to a tuple $(b_2,\\\\ldots ,b_{t+1}, \\\\bar{x})$ where for each $i$ , $b_i\\\\in W_i$ and $\\\\bar{x}$ is a $(t+1)$ -tuple of coordinates.\",\n \"For every $j\\\\in \\\\lbrace 2,\\\\ldots ,t\\\\rbrace $ , $L_j$ consists of $n^t d^{t+1}$ vertices $(a_0, \\\\ldots , a_{t-j},b_{t+3-j},\\\\ldots , b_{t+1},\\\\bar{x})$ , where for each $i$ , $a_i\\\\in W_i$ , $b_i\\\\in W_i$ and $\\\\bar{x}=(x_0,\\\\ldots ,x_t)$ is a $(t+1)$ -tuple of coordinates in $[d]$ .\",\n \"In other words, there is a vector from $W_i$ for every $i\\\\notin \\\\lbrace t-j+1,t-j+2\\\\rbrace $ .\",\n \"Now let us define the edges.\",\n \"Consider a node $(a_0,\\\\ldots ,a_t)\\\\in L_0$ .\",\n \"For every $\\\\bar{x}=(x_0,\\\\ldots ,x_t)$ , connect $(a_0,\\\\ldots ,a_t)$ to $(a_0,\\\\ldots ,a_{t-1},\\\\bar{x})\\\\in L_1$ if and only if for every $j\\\\in \\\\lbrace 0,\\\\ldots ,t\\\\rbrace $ , $a_j$ is 1 in coordinates $x_0,\\\\ldots ,x_{t-j}$ .\",\n \"For any $i\\\\in \\\\lbrace 1,\\\\ldots ,t\\\\rbrace $ let's define the edges between $L_i$ and $L_{i+1}$ .\",\n \"For $(a_0,\\\\ldots ,a_{t-i},b_{t+3-i},\\\\ldots ,b_{t+1}, \\\\bar{x})\\\\in L_i$ Here if $i=1$ , there are no $b$ 's in the tuple.\",\n \"and for any $c_{t+2-i}\\\\in W_{t+2-i}$ , add an edge to $(a_0,\\\\ldots ,a_{t-i-1},c_{t+2-i},b_{t+3-i},\\\\ldots ,b_{t+1}, \\\\bar{x})\\\\in L_{i+1}$ .\",\n \"Here we “forget” vector $a_{t-i}$ and replace it with $c_{t+2-i}$ , leaving everything else the same.\",\n \"Finally, the edges between $L_{t+1}$ and $L_{t+2}$ are as follows.\",\n \"Consider some $(b_1,\\\\ldots ,b_{t+1})\\\\in L_{t+2}$ .\",\n \"For every $\\\\bar{x}=(x_0,\\\\ldots ,x_t)$ , connect $(b_1,\\\\ldots ,b_{t+1})$ to $(b_2,\\\\ldots ,b_{t+1},\\\\bar{x})\\\\in L_{t+1}$ if and only if for every $j\\\\in \\\\lbrace 1,\\\\ldots ,{t+1}\\\\rbrace $ , $b_j$ is 1 in coordinates $x_{t+1-j},\\\\ldots ,x_t$ .\",\n \"Figure REF shows the construction of the graph for $t=2$ .\",\n \"Figure: The reduction graph from (t+2)(t+2)-OV for t=2t=2.\",\n \"The figure depicts when a path of length t+2t+2 exists between arbitrary a 0 a 1 a 2 ∈L 0 a_0a_1a_2\\\\in L_0 and b 1 b 2 b 3 ∈L t+2 b_1b_2b_3\\\\in L_{t+2}.\",\n \"It also shows that when there is a path of length t+2t+2 between a 0 a 1 a 2 ∈L 0 a_0a_1a_2\\\\in L_0 and a 1 a 2 a 3 ∈L t+2 a_1a_2a_3\\\\in L_{t+2}, a 0 ,a 1 ,a 2 ,a 3 a_0,a_1,a_2,a_3 cannot be an orthogonal 4-tuple.An important claim is as follows: Claim 10 For every $\\\\bar{x}$ , each $(a_0,\\\\ldots ,a_{t-1},\\\\bar{x})\\\\in L_1$ is at distance $t$ to every $(b_2,\\\\ldots ,b_t,\\\\bar{x})\\\\in L_{t+1}$ .\",\n \"Consider the path starting from $(a_0,\\\\ldots ,a_{t-1},\\\\bar{x})$ , and then for each $i\\\\ge 1$ following the edges $(a_0,\\\\ldots ,a_{t-i},b_{t+3-i},\\\\ldots ,b_{t+1},\\\\bar{x})\\\\in L_i$ to $(a_0,\\\\ldots ,a_{t-1-i},b_{t+2-i},\\\\ldots ,b_{t+1},\\\\bar{x})\\\\in L_{i+1}$ , until we reach $(b_2,\\\\ldots ,b_{t+1},\\\\bar{x})\\\\in L_{t+1}$ .\",\n \"This path exists by construction and has length $t$ .\",\n \"Now we proceed to prove the bounds on the $S$ -$T$ Diameter.\",\n \"Lemma 11 (Property 3 of Theorem REF ) If the $(t+2)$ -OV instance has no solution, then $D_{S,T}=t+2$ .\",\n \"If the $(t+2)$ -OV instance has no solution, then for every $c_0\\\\in W_0,c_1\\\\in W_1,\\\\ldots ,c_{t+1}\\\\in W_{t+1}$ , there is some coordinate $x$ such that $c_0[x]=c_1[x]=\\\\ldots =c_{t+1}[x]=1$ .\",\n \"Now consider the graph and any $(a_0,\\\\ldots ,a_t)\\\\in L_0$ , $(b_1,\\\\ldots ,b_{t+1})\\\\in L_{t+2}$ .\",\n \"For every $j\\\\in \\\\lbrace 0,\\\\ldots ,t\\\\rbrace $ , let $x_j$ be a coordinate so that $a_0,\\\\ldots ,a_{t-j},b_{t-j+1},\\\\ldots ,b_{t+1}$ are all 1 in $x_j$ .\",\n \"Let $\\\\bar{x}=(x_0,\\\\ldots ,x_{t})$ .\",\n \"By construction, $(a_0,\\\\ldots ,a_t)$ has an edge to $(a_0,\\\\ldots ,a_{t-1},\\\\bar{x})$ and $(b_2,\\\\ldots ,b_{t+1},\\\\bar{x})$ has an edge to $(b_1,\\\\ldots ,b_{t+1})$ .\",\n \"Also, by Claim REF , $(a_0,\\\\ldots ,a_{t-1},\\\\bar{x})$ has a path of length $t$ to $(b_2,\\\\ldots ,b_{t+1},\\\\bar{x})$ .\",\n \"This shows that $D_{S,T}\\\\le t+2$ ; equality follows because the graph is layered.\",\n \"Now we prove the guarantee for the case when an orthogonal tuple exists.\",\n \"Lemma 12 (Property 4 of Theorem REF ) If there exist $a_0\\\\in W_0,\\\\ldots ,a_{t+1}\\\\in W_{t+1}$ that are orthogonal, then $D_{S,T}\\\\ge 3t+4$ .\",\n \"To prove the lemma, we will actually prove a more general claim: Property 5 of Theorem REF .\",\n \"Claim 13 (Property 5 of Theorem REF ) Suppose that $a_0\\\\in W_0,\\\\ldots ,a_{t+1}\\\\in W_{t+1}$ are orthogonal.\",\n \"Let $s$ be such that $0\\\\le s\\\\le t$ .\",\n \"Let $b_{t-s+j}\\\\in W_{t-s+j}$ for all $j\\\\in [1,\\\\ldots ,s]$ be some other vectors, potentially different from $a_{t-s+j}$ .\",\n \"Consider $\\\\alpha =(a_0,a_1,\\\\ldots ,a_{t-s},b_{t-s+1},\\\\ldots ,b_{t})\\\\in L_0$ and $\\\\beta =(a_{1},\\\\ldots ,a_{t+1})\\\\in L_{t+2}$ .\",\n \"Then the distance between $\\\\alpha $ and $\\\\beta $ is at least $3t-2s+4$ .\",\n \"Symmetrically, let $c_{j}\\\\in W_{j}$ for all $j\\\\in [1,\\\\ldots ,s]$ be some other vectors, potentially different from $a_{j}$ .\",\n \"Consider $\\\\alpha =(a_0,a_1,\\\\ldots ,a_{t})\\\\in L_0$ and $\\\\beta =(c_{1},\\\\ldots ,c_s,a_{s+1},\\\\dots , a_{t+1})\\\\in L_{t+2}$ .\",\n \"Then the distance between $\\\\alpha $ and $\\\\beta $ is at least $3t-2s+4$ .\",\n \"If the claim is true, then using $s=0$ we get that the Diameter is at least $3t+4$ so Lemma REF is true.\",\n \"The claim for $s>0$ is useful for the rest of our constructions.\",\n \"We will show that the distance between $\\\\alpha =(a_0,a_1,\\\\ldots ,a_{t-s},b_{t-s+1},\\\\ldots ,b_{t})\\\\in L_0$ and $\\\\beta =(a_{1},\\\\ldots ,a_{t+1})\\\\in L_{t+2}$ is strictly more than $3t+2-2s$ .\",\n \"Because the graph is layered and hence bipartite and $t+2\\\\equiv 3t+2\\\\mod {2}$ , the distance must be at least $3t-2s+4$ .\",\n \"Let's assume for contradiction that the shortest path $P$ between $\\\\alpha $ and $\\\\beta $ is of length $\\\\le 3t+2-2s$ .\",\n \"First let's look at any subpath $P^{\\\\prime }$ of $P$ strictly within $M=L_1\\\\cup \\\\ldots \\\\cup L_{t+1}$ .\",\n \"All nodes on $P^{\\\\prime }$ must share the same $\\\\bar{x}$ .\",\n \"Furthermore, if $P^{\\\\prime }$ starts with a node of $L_1$ and ends with a node of $L_{t+1}$ , as $P^{\\\\prime }$ needs to be a shortest path and by Claim REF , $P^{\\\\prime }$ must be of length exactly $t$ .\",\n \"Next, notice that $P$ cannot go from $L_0$ to $L_{t+2}$ and then back to $L_0$ .\",\n \"This is because it needs to end up in $L_{t+2}$ and any time it crosses over $M$ , it would need to pay a distance of $t+2$ , so $P$ would have to have length at least $3t+6>3t+2-2s$ .\",\n \"Hence, $P$ must be of the following form: a path from $\\\\alpha $ through $L_0\\\\cup M$ back to $L_0$ (possibly containing only $\\\\alpha $ ), followed by a path crossing $M$ to reach $L_{t+2}$ , followed by a path through $L_{t+2}\\\\cup M$ to $L_{t+2}$ (possibly empty).\",\n \"We will show that if $P$ has length $\\\\le 3t+2-2s$ then $P$ must contain a length $t+2$ subpath $Q$ between a node $(a_0,\\\\ldots ,a_{q},w_{q+1},\\\\ldots , w_t)\\\\in L_0$ , for some choices of the $w$ 's and some $q\\\\le t-s$ , and a node $(v_1,\\\\ldots ,v_q,a_{q+1},\\\\ldots ,a_{t+1})\\\\in L_{t+2}$ , for some choices of $v$ 's.\",\n \"That is, this path traverses $M$ without weaving, by following $(a_0,\\\\ldots ,a_{q},w_{q+1},\\\\ldots , w_{t-1},\\\\bar{x})\\\\in L_1$ , $(a_0, \\\\ldots , a_{q},w_{q+1},\\\\ldots , w_{t-2},a_{t+1},\\\\bar{x})\\\\in L_2$ , $\\\\ldots $ , $(v_2, \\\\ldots , v_q, a_{q+1}, \\\\ldots , a_{t-s}, \\\\ldots , a_{t+1}, \\\\bar{x})\\\\in L_{t+1}$ .\",\n \"Suppose we show that such a subpath exists.\",\n \"Then by the construction of our graph we have that for every $i\\\\in \\\\lbrace 0,\\\\ldots ,q\\\\rbrace $ , $a_i[x_j]=1$ for all $j\\\\in \\\\lbrace 0,\\\\ldots ,t-i\\\\rbrace $ , and that for all $i\\\\in \\\\lbrace q+1,\\\\ldots ,t+1\\\\rbrace $ , $a_i[x_j]=1$ for all $j\\\\in \\\\lbrace t+1-i,\\\\ldots ,t\\\\rbrace $ .\",\n \"That is, for all $i$ , $a_i[x_{t-q}]=1$ , and we get a contradiction since the $a_i$ were supposed to be orthogonal.\",\n \"Now let $\\\\alpha ^*$ be the last node from $L_0$ on $P$ and let $\\\\beta ^*$ be the first node of $L_{t+1}$ of $P$ .\",\n \"Let $a^*\\\\in L_1$ be the node right after $\\\\alpha ^*$ and let $b^*\\\\in L_{t+1}$ be the node right before $\\\\beta ^*$ .\",\n \"Since the subpath of $P$ between $a^*$ and $b^*$ is within $M$ , it must share the same $\\\\bar{x}$ , and it must have length exactly $t$ by Claim REF .\",\n \"We will show that the subpath $Q$ that we are looking for is the subpath of $P$ between $\\\\alpha ^*$ and $\\\\beta ^*$ .\",\n \"Its length is exactly what we want: $t+2$ .\",\n \"It remains to show that for some $q\\\\le t-s$ and some choices of $w$ 's and $v$ 's, $\\\\alpha ^*=(a_0,\\\\ldots ,a_{q},w_{q+1},\\\\ldots , w_t)$ and $\\\\beta ^*=(v_1,\\\\ldots ,v_q,a_{q+1},\\\\ldots ,a_{t+1})$ .\",\n \"Consider the path $P_1$ between $\\\\alpha =(a_0,a_1,\\\\ldots ,a_{t-s},b_{t-s+1},\\\\ldots ,b_{t})$ and $\\\\alpha ^*$ and the path $P_2$ between $\\\\beta =(a_1,\\\\ldots ,a_{t+1})$ and $\\\\beta ^*$ .\",\n \"Let $L_i$ be the layer in $M$ with largest $i$ that $P_1$ touches and let $L_j$ be the layer in $M$ with smallest $j$ that $P_2$ touches.\",\n \"For convenience, let us define $j^{\\\\prime }=t+2-j$ .\",\n \"The length of $P_1$ is then at least $2i$ and the length of $P_2$ is at least $2j^{\\\\prime }$ .\",\n \"The length $|P|$ of $P$ equals $t+2+|P_1|+|P_2|\\\\ge t+2+2i+2j^{\\\\prime }=t+2+2(i+j^{\\\\prime })$ .\",\n \"Since we have assumed that $|P|\\\\le 3t+2-2s$ , we must have that $t+2+2(i+j^{\\\\prime })\\\\le 3t+2-2s$ and hence $i+j^{\\\\prime }\\\\le t-s$ .\",\n \"Now, since $P_1$ goes at most to $L_i$ , then from getting from $\\\\alpha $ to $\\\\alpha ^*$ , at most the last $i$ elements of $(a_0,a_1,\\\\ldots ,a_{t-s},b_{t-s+1},\\\\ldots ,b_{t})$ can have been “forgotten”.\",\n \"Hence, $\\\\alpha ^*=(a_0,\\\\ldots ,a_{t-\\\\max \\\\lbrace s,i\\\\rbrace },b_{t-\\\\max \\\\lbrace s,i\\\\rbrace +1},\\\\ldots ,b_{t-i},w_{t-i+1},\\\\ldots ,w_{t})$ for some $w$ 's.\",\n \"(If $i\\\\ge s$ , the $b$ 's do not appear.)\",\n \"Similarly, between $\\\\beta $ and $\\\\beta ^*$ , at most the first $j^{\\\\prime }$ elements of $\\\\beta $ can have been forgotten.\",\n \"Thus, we have that $\\\\beta ^*=(v_1,\\\\ldots ,v_{j^{\\\\prime }},a_{j^{\\\\prime }+1},\\\\ldots ,a_{t+1})$ for some $v$ 's.\",\n \"Now, since $i+j^{\\\\prime }\\\\le t-s$ , we must have that $j^{\\\\prime }\\\\le t-s-i\\\\le t-\\\\max \\\\lbrace s,i\\\\rbrace $ , and hence the path between $\\\\alpha ^*$ and $\\\\beta ^*$ is the path $Q$ we are searching for.\",\n \"See Figure REF for an illustration of $L_i$ , $L_j$ etc.\",\n \"in the case when $s=0$ .\",\n \"Figure: Here PP contains at least 2 nodes in L 0 L_0 and at least 2 in L t+2 L_{t+2}, and s=0s=0.\"\n ],\n [\n \"Equivalence between Diameter and $S$ -{{formula:bdb52e68-e562-4944-b84c-7c1359b488ce}} Diameter\",\n \"Here we will prove that when it comes to exact computation, $S$ -$T$ -Diameter and Diameter in weighted graphs are equivalent.\",\n \"The proof for directed graphs is much simpler, so we focus on the equivalence for undirected graphs.\",\n \"Also, it is clear that if one can solve $S$ -$T$ Diameter, one can also solve Diameter in the same running time since one can simply set $S=T=V$ .\",\n \"We prove: Theorem 14 Suppose that there is a $T(n,m)$ time algorithm that can compute the Diameter of an $n$ node, $m$ edge graph with nonnegative integer edge weights.\",\n \"Then, the $S$ -$T$ Diameter of any $n$ node $m$ edge graph with nonnegative integer edge weights can be computed in $T(O(n),O(m))$ time.\",\n \"Let $G=(V,E),S,T$ be the $S$ -$T$ Diameter instance; let $w:E\\\\rightarrow \\\\lbrace 0,\\\\ldots ,M\\\\rbrace $ be the edge weights.\",\n \"First, we can always assume that $M$ is even: if it is not, multiply all edge weights by 2; all distances (and hence also the $S$ -$T$ Diameter) double.\",\n \"Let $S=\\\\lbrace s_1,\\\\ldots ,s_k\\\\rbrace $ and $T=\\\\lbrace t_1,\\\\ldots ,t_{\\\\ell }\\\\rbrace $ .\",\n \"Now, let $W=Mn$ .\",\n \"First add $|S|=k$ new nodes $S^{\\\\prime }=\\\\lbrace v_1,\\\\ldots ,v_k\\\\rbrace $ .\",\n \"For each $i\\\\in \\\\lbrace 1,\\\\ldots ,k\\\\rbrace $ add a new edge $(v_i,s_i)$ of weight $W$ .\",\n \"Let $G_S$ be this new graph.\",\n \"Let's consider the Diameter of $G_S$ .\",\n \"For every pair of nodes $u,v\\\\notin S^{\\\\prime }$ , the distance is the same as in $G$ .\",\n \"For $v_i\\\\in S^{\\\\prime }$ and $x\\\\notin S^{\\\\prime }$ , the distance is $W+d_G(s_i,x)\\\\le W+M(n-1)<2W$ .\",\n \"For $v_i,v_j\\\\in S^{\\\\prime }$ , the distance is $2W+d(s_i,s_j)$ .\",\n \"Hence the Diameter of $G_S$ is $2W+\\\\max _{s_i,s_j\\\\in S} d(s_i,s_j)$ .\",\n \"Hence by computing the Diameter of $G_S$ , we can compute $D_S=\\\\max _{s_i,s_j\\\\in S} d(s_i,s_j)$ .\",\n \"We can create a similar graph $G_T$ whose Diameter will allow us to compute $D_T=\\\\max _{t_i,t_j\\\\in S} d(t_i,t_j)$ .\",\n \"After this, let's create a graph $G_{S,T}$ as follows.\",\n \"Add new nodes $S^{\\\\prime }=\\\\lbrace v_1,\\\\ldots ,v_k\\\\rbrace $ and $T^{\\\\prime }=\\\\lbrace u_1,\\\\ldots ,u_\\\\ell \\\\rbrace $ .\",\n \"For each $i\\\\in \\\\lbrace 1,\\\\ldots ,k\\\\rbrace $ add a new edge $(v_i,s_i)$ of weight $W$ .\",\n \"For each $j\\\\in \\\\lbrace 1,\\\\ldots ,\\\\ell \\\\rbrace $ add a new edge $(u_j,t_j)$ of weight $W$ .\",\n \"With a similar argument as above, the Diameter $D^{\\\\prime }$ of $G_{S,T}$ is $D^{\\\\prime }=2W+\\\\max _{u,v\\\\in S\\\\cup T} d_G(u,v)$ .\",\n \"Let's assume without loss of generality that $D_S\\\\ge D_T$ .\",\n \"If $D^{\\\\prime }> D_S$ , then $D^{\\\\prime }=2W+\\\\max _{u\\\\in S,v\\\\in T} d_G(u,v)$ , and we can compute the $S$ -$T$ Diameter of $G$ by subtracting $2W$ .\",\n \"Now suppose that we get $D^{\\\\prime }\\\\le D_S$ ; we must have then actually gotten $D^{\\\\prime }=D_S$ .\",\n \"The $S$ -$T$ Diameter of $G$ might be strictly smaller than $D_S$ .\",\n \"We add two new nodes $x$ and $y$ to $G_{S,T}$ .\",\n \"We add an edge $(x,y)$ of weight $2W$ , edges $(x,v_i)$ for every $v_i\\\\in S^{\\\\prime }$ of weight $D_S/2$ and (symmetrically) edges $(y,u_j)$ for every $u_j\\\\in T^{\\\\prime }$ of weight $D_S/2$ .\",\n \"The construction of $G^{\\\\prime }$ is depicted in Figure REF .\",\n \"Figure: A depiction of the construction of G ' G^{\\\\prime }.Let us consider the distances in this new graph $G^{\\\\prime }$ .\",\n \"For every $a,b\\\\notin S^{\\\\prime }\\\\cup T^{\\\\prime }\\\\cup \\\\lbrace x,y\\\\rbrace $ , $d(a,b)=d_G(a,b)<W$ as any path not in $G$ would have to use an edge of weight $W>d(a,b)$ .\",\n \"For every $b\\\\notin S^{\\\\prime }\\\\cup T^{\\\\prime }\\\\cup \\\\lbrace x,y\\\\rbrace $ , $d(x,b)=W+D_S/2 +\\\\min _{a\\\\in S} d_G(a,b)\\\\le 2W+D_S/2$ .\",\n \"Similarly, $d(y,b)=W+D_S/2 +\\\\min _{a\\\\in T} d_G(a,b)\\\\le 2W+D_S/2$ .\",\n \"For every $v_i\\\\in S^{\\\\prime }$ , $d(x,v_i)=D_S/2$ , and $d(y,v_i)=2W+D_S/2$ .\",\n \"For every $u_i\\\\in T^{\\\\prime }$ , $d(y,u_i)=D_S/2$ , and $d(x,u_i)=2W+D_S/2$ .\",\n \"For every $v_i,v_j\\\\in S^{\\\\prime }$ , $d(v_i,v_j)=D_S$ .\",\n \"For every $u_i,u_j\\\\in T^{\\\\prime }$ , $d(u_i,u_j)=D_S$ .\",\n \"For every $v_i\\\\in S^{\\\\prime }$ and $b\\\\notin S^{\\\\prime }\\\\cup T^{\\\\prime }\\\\cup \\\\lbrace x,y\\\\rbrace $ , $d(v_i,b)\\\\le W+d(s_i,b)\\\\le 2W.$ For every $u_i\\\\in T^{\\\\prime }$ and $b\\\\notin S^{\\\\prime }\\\\cup T^{\\\\prime }\\\\cup \\\\lbrace x,y\\\\rbrace $ , $d(u_i,b)\\\\le W+d(t_i,b)\\\\le 2W.$ For every $v_i\\\\in S^{\\\\prime }$ and $u_j\\\\in T^{\\\\prime }$ , $d(v_i,u_j)$ is the minimum of $D_S+2W, D_S+2W+\\\\min _{s\\\\in S} d_G(s,t_j), D_S+2W+\\\\min _{t\\\\in t} d_G(t,s_i)$ and $2W+d_G(s_i,t_j)$ .\",\n \"The middle two terms are $\\\\ge D_S+2W$ , and hence $d(v_i,u_j)=2W+\\\\min \\\\lbrace D_S,d_G(s_i,t_j)\\\\rbrace $ .\",\n \"Consider $s_i\\\\in S, t_j\\\\in T$ that are the end points of the $S$ -$T$ Diameter $D$ in $G$ .\",\n \"Then $D=d_G(s_i,t_j)$ .\",\n \"Now, we have from before that $D\\\\le D_S$ , as otherwise we have computed $D$ already.\",\n \"Hence in $G^{\\\\prime }$ , the distance $d(u_i,v_j)$ equals $2W+\\\\min \\\\lbrace D_S,D\\\\rbrace =2W+D$ We note that for any $s_i,s_j\\\\in S$ , and any $t\\\\in T$ , $d_G(s_i,s_j)\\\\le d_G(s_i,t)+d_G(t,s_j)\\\\le 2\\\\max _{s\\\\in S,t\\\\in T} d_G(s,t) = 2D$ .\",\n \"Thus, $D\\\\ge D_S/2$ .\",\n \"The distances in cases (1) to (5) are all $\\\\le 2W+D_S/2\\\\le 2W+D$ .\",\n \"Hence the Diameter of $G^{\\\\prime }$ is actually exactly $2W+D$ .\"\n ],\n [\n \"Undirected graphs\",\n \"Theorem 15 Let $k\\\\ge 2$ .\",\n \"Under the $k$ -OV conjecture, every algorithm that can distinguish between eccentricity at most $2k-1$ and eccentricity at least $4k-3$ for every vertex in an $O(n)$ edge and node undirected graph, requires at least $n^{1+1/(k-1)-o(1)}$ time on a $O(\\\\log n)$ -bit word-RAM.\",\n \"Figure: The undirected Eccentricities lower bound for k=5k=5.Let's start with the $S$ -$T$ -diameter construction for $k$ obtained from a given $k$ -OV instance.\",\n \"We remove any internal nodes if they don't have edges to one of their adjacent layers - they don't hurt the instance.\",\n \"We have a graph on $O(n^{k-1}d^{k-2})$ vertices and edges with the following properties: (1) Suppose that the $k$ -OV instance has no $k$ -OV solution.\",\n \"Then for every $s\\\\in S,t\\\\in T$ , $d(s,t)=k$ .\",\n \"Also, for every $s\\\\in S$ and $u\\\\notin S\\\\cup T$ , $d(s,u)\\\\le (k-1)+k=2k-1$ since we can take a $\\\\le (k-1)$ length path from $u$ to some node $t\\\\in T$ and since $d(s,t)=k$ .\",\n \"(2) If there is a $k$ -OV solution, there are two nodes $s\\\\in S,t\\\\in T$ with $d(s,t)\\\\ge 3k-2$ .\",\n \"We modify the construction as follows.\",\n \"For every $s\\\\in S$ , we create an undirected path on $k-2$ new vertices $s_1\\\\rightarrow s_2\\\\rightarrow \\\\ldots \\\\rightarrow s_{k-2}$ and add an edge $(s,s_1)$ ; let's call $s$ by $s_0$ .\",\n \"Now, the distance between $s_0$ and $s_i$ is $i$ .\",\n \"Add a new node $y$ and create edges $(s_{k-2},y)$ for every $s\\\\in S$ .\",\n \"Now, $d(y,s_0)=k-1$ for every $s\\\\in S$ , and also for every $s,s^{\\\\prime }\\\\in S$ and all $i,j\\\\in \\\\lbrace 0,\\\\ldots ,k-2\\\\rbrace $ , we have that $d(s_i,s^{\\\\prime }_j)\\\\le 2k-2$ .\",\n \"For every $s\\\\in S$ , $t\\\\in T$ , there is now potentially a new path between them, from $s$ to $y$ in $k-1$ steps, then to some other $s^{\\\\prime }$ in $k-1$ steps and then to $t$ using $\\\\ge k$ steps.\",\n \"The length is $\\\\ge 2(k-1)+k=3k-2$ , so when there is a $k$ -OV solution, there is still a pair $s,t$ at distance at least $3k-2$ .\",\n \"Now, we also attach paths to the nodes in $T$ .\",\n \"In particular, for each $t$ , add an undirected path $t\\\\rightarrow t_1\\\\rightarrow \\\\ldots \\\\rightarrow t_{k-1}$ .\",\n \"The distance between any $s\\\\in S$ and any $t_i$ is $i+d(s,t)$ .\",\n \"Hence when there is no $k$ -OV solution, the Eccentricities of all $s_0$ for $s\\\\in S$ are $\\\\le k+(k-1)=2k-1$ , and when there is a $k$ -OV solution, there is $s\\\\in S, t\\\\in T$ so that $d(s_0,t_{k-1})\\\\ge (3k-2)+(k-1)=4k-3$ .\"\n ],\n [\n \"Directed graphs\",\n \"Theorem 16 Under the 2-OV conjecture, for any $\\\\delta >0$ , any $(2-\\\\delta )$ -approximation algorithm for all Eccentricities in an $n$ node, $O(n)$ -edge directed unweighted graph, requires $n^{2-o(1)}$ time on a $O(\\\\log n)$ -bit word-RAM.\",\n \"Figure: The directed Eccentricities lower bound.Suppose we are given an instance of 2-OV: two sets of vectors $U,V$ over $\\\\lbrace 0,1\\\\rbrace ^d$ and we want to know whether there are $u\\\\in U,v\\\\in V$ with $u\\\\cdot v=0$ .\",\n \"Let $L\\\\ge 1$ be any integer.\",\n \"Let us create a directed unweighted graph $G$ ; an illustration can be found in Figure REF .\",\n \"$G$ will have a vertex $u$ for every $u\\\\in U$ and a vertex $c$ for every $c\\\\in [d]$ .\",\n \"Every $v\\\\in V$ will be represented by a directed path $v_0\\\\rightarrow v_1\\\\rightarrow \\\\ldots \\\\rightarrow v_L$ .\",\n \"In addition, there is a directed path $P_x$ on $L$ extra nodes, $x_1\\\\rightarrow \\\\ldots \\\\rightarrow x_L$ so that every $u\\\\in U$ has directed edges $(u,x_1)$ and $(x_L,u)$ .\",\n \"For every $u\\\\in U$ and every $c$ for which $u[c]=1$ , we add a directed edge $(u,c)$ .\",\n \"For every $v$ and every $c$ for which $v[c]=1$ , we add a directed edge $(c,v_0)$ .\",\n \"Remove any $c$ that does not have at least one edge coming from $U$ .\",\n \"Let us consider the eccentricity of any node $u\\\\in U$ .\",\n \"First, for all $u^{\\\\prime }\\\\in U$ , $d(u,u^{\\\\prime })\\\\le L+1$ since one can go through the path $P_x$ .\",\n \"For every $c\\\\in [d]$ , there is at least one edge coming from some $u^{\\\\prime }\\\\in U$ , and so one can reach $c$ from $u$ by first taking $P_x$ to $u^{\\\\prime }$ and then using the edge $(u^{\\\\prime },c)$ .\",\n \"Hence, $d(u,c)\\\\le L+2$ for all $c\\\\in [d]$ .\",\n \"For any $v\\\\in V$ and $i\\\\in \\\\lbrace 1,\\\\ldots , L\\\\rbrace $ , the distance $d(u,v_i)=i+d(u,v_0)$ , and so we consider $d(u,v_0)$ .\",\n \"If there is a $c$ for which $u[c]=v[c]=1$ , then $d(u,v_0)=2$ , and hence for all $i$ , $d(u,v_i)\\\\le L+2$ .\",\n \"If no such $c$ exists and so if $u$ and $v$ are orthogonal, the only way to reach $v_0$ is potentially via $P_x$ to some other $u^{\\\\prime }\\\\in U$ which is at distance 2 to $v_0$ .\",\n \"Hence if $u$ and $v$ are orthogonal, $d(u,v_0)=L+3$ , and hence $d(u,v_L)=2L+3$ .\",\n \"In other words, the eccentricity of $u$ is $L+2$ if it is not orthogonal to any vectors in $V$ and it is $\\\\ge 2L+3$ if there is some $v$ that is orthogonal to $u$ .\",\n \"The number of vertices in the graph is $O(nL+d)$ and the number of edges is $O(nL+nd)$ .\",\n \"Suppose that there is a $(2-\\\\varepsilon )$ -approximation algorithm for all Eccentricities in graphs with $O(m)$ nodes and edges running in $O(m^{2-\\\\delta })$ time for some $\\\\varepsilon ,\\\\delta >0$ .\",\n \"Then, we construct the above instance for $L=\\\\lceil 1/\\\\varepsilon \\\\rceil $ and run the algorithm on it.\",\n \"The approximation returned is at least as good as a $(2-1/L)$ -approximation.\",\n \"Hence if the diameter is at least $2L+3$ , the algorithm will return an estimate that is at least $L(2L+3)/(2L-1) > L+2$ .\",\n \"Thus the algorithm can solve 2-OV in time $O((nL+nd)^{2-\\\\delta }) = O(n^{2-\\\\delta }d^{2-\\\\delta })$ , contradicting the 2-OV conjecture.\"\n ],\n [\n \"Diameter lower bounds\",\n \"For all of our constructions we begin with the $S$ -$T$ diameter lower bound construction from Theorem REF .\",\n \"Here, if the $k$ -OV instance has no solution, $D_{S,T}\\\\le k$ and if the instance has a solution $D_{S,T}\\\\ge 3k-2$ .\",\n \"To adapt this construction to Diameter, we need to ensure that if the OV instance has no solution then all pairs of vertices have small enough distance.\",\n \"We begin by augmenting the $S$ -$T$ Diameter construction by adding a matching between $S$ and a new set $S^{\\\\prime }$ as well as a matching between $T$ and a new set $T^{\\\\prime }$ .\",\n \"Without any further modifications, pairs of vertices $u,v\\\\in S\\\\cup S^{\\\\prime }$ (or $u,v\\\\in T\\\\cup T^{\\\\prime }$ ) could be far from one another.\",\n \"The challenge is to add extra gadgetry to make these pairs close for “no\\\" instances while maintaining that in “yes\\\" instances the distance between the diameter endpoints $s^{\\\\prime }\\\\in S^{\\\\prime },t^{\\\\prime }\\\\in T^{\\\\prime }$ is large.\",\n \"That is, for “yes\\\" instances, we want a shortest path between the diameter endpoints $s^{\\\\prime }$ and $t^{\\\\prime }$ to contain the vertex $s\\\\in S$ matched to $s^{\\\\prime }$ and the vertex $t\\\\in T$ matched to $t^{\\\\prime }$ so that we can use use the fact that $d(s,t)\\\\ge 3k-2$ .\",\n \"In other words, we do not want there to be a shortcut from $s^{\\\\prime }$ to some vertex in $S$ that allows us to use a path of length $k$ from $S$ to $T$ .\",\n \"For example, we cannot simply create a vertex $x$ and connect it to all vertices in $S\\\\cup S^{\\\\prime }$ because this would introduce shortcuts from $S^{\\\\prime }$ to $S$ .\",\n \"We will describe some intuition for the augmentations to the graph regarding 3-OV for simplicity.\",\n \"Recall that $s^{\\\\prime }\\\\in S^{\\\\prime }, t^{\\\\prime }\\\\in T^{\\\\prime }$ are the endpoints of the diameter and let $t$ be the vertex matched to $t^{\\\\prime }$ .\",\n \"To solve the problem outlined in the above paragraph, we observe that in the “yes\\\" case there are three types of vertices $s\\\\in S$ .\",\n \"(1) close: $d(s,t)=3$ , (2) far: $d(s,t)\\\\ge 7$ (property 4 of Theorem REF ), and (3) intermediate: $d(s,t)\\\\ge 5$ (property 5 of Theorem REF ).\",\n \"For close $s$ , we need $d(s^{\\\\prime },s)$ to be large so that there is no shortcut from $s^{\\\\prime }$ to $t^{\\\\prime }$ through $s$ .\",\n \"For far $s$ , it is ok if $d(s^{\\\\prime },s)$ is small because $d(s,t)$ is large enough to ensure that paths from $s^{\\\\prime }$ to $t^{\\\\prime }$ through $s$ are still long enough.\",\n \"For intermediate $s$ , $d(s^{\\\\prime },s)$ cannot be small, but it also need not be large.\",\n \"To fulfill these specifications, we add a small clique (the graph is still sparse) and connect each of its vertices to only some of the vertices in $S$ and/or $S^{\\\\prime }$ according to the implications of property 5 of Theorem REF .\",\n \"When $s$ is close, we ensure that $d(s^{\\\\prime },s)$ is large by requiring that a shortest path from $s^{\\\\prime }$ to $s$ goes from $s^{\\\\prime }$ to the clique, uses an edge inside of the clique, and then goes from the clique to $s$ .\",\n \"When $s$ is intermediate, we ensure that $d(s^{\\\\prime },s)$ is not too small by requiring that a shortest path from $s^{\\\\prime }$ to $s$ goes from $s^{\\\\prime }$ to the clique and then from the clique to $s$ (without using an edge inside of the clique).\",\n \"These intermediate $s$ are important as they allow every vertex in the clique to have an edge to some vertex in $S$ and thus be close enough to the $T$ side of the graph in the “no\\\" case.\"\n ],\n [\n \"5 vs 8 unweighted undirected construction\",\n \"In this section we show that under the 3-OV Hypothesis, any algorithm that can distinguish between Diameter 5 and 8 in sparse undirected unweighted graphs, requires $\\\\Omega (n^{3/2-o(1)})$ time.\",\n \"Theorem REF gives us the following theorem.\",\n \"Theorem 17 Given a 3-OV instance consisting of three sets $A,B,C \\\\subseteq \\\\lbrace 0,1\\\\rbrace ^d$ , $|A|=|B|=|C|=n$ , we can in $O(n^2d^2)$ time construct an unweighted, undirected graph with $O(n^2+nd^2)$ vertices and $O(n^2d^2)$ edges that satisfies the following properties.\",\n \"The graph consists of 4 layers of vertices $S,L_1,L_2,T$ .\",\n \"The number of nodes in the sets is $|S|=|T|=n^2$ and $|L_1|,|L_2|\\\\le nd^2$ .\",\n \"$S$ consists of all tuples $(a,b)$ of vertices $a \\\\in A$ and $b \\\\in B$ .\",\n \"Similarly, $T$ consists of all tuples $(b,c)$ of vertices $b \\\\in B$ and $c \\\\in C$ .\",\n \"If the 3-OV instance has no solution, then $d(u,v)=3$ for all $u \\\\in S$ and $v \\\\in T$ .\",\n \"If the 3-OV instance has a solution $a \\\\in A, b \\\\in B, c \\\\in C$ with $a,b,c$ orthogonal, then $d((a,b)\\\\in S,(b,c) \\\\in T)\\\\ge 7$ .\",\n \"Setting $k=3,s=1$ in Property 5 of Theorem REF : for any $b^{\\\\prime } \\\\in B$ we have $d((a,b) \\\\in S,(b^{\\\\prime },c) \\\\in T)\\\\ge 5$ and $d((a,b^{\\\\prime }) \\\\in S,(b,c) \\\\in T)\\\\ge 5$ .\",\n \"For any vertex $u \\\\in L_1$ there exists a vertex $s \\\\in S$ that is adjacent to $u$ .\",\n \"Similarly, for any vertex $v \\\\in L_2$ there exists a vertex $t \\\\in T$ that is adjacent to $v$ .\",\n \"We can assume that this property holds because we can remove all vertices that do not satisfy this property from the graph and the resulting graph will still satisfy the other properties.\",\n \"In the rest of the section we use Theorem REF to prove the following result.\",\n \"Theorem 18 Given a 3-OV instance, we can in $O(n^2d^2)$ time construct an unweighted, undirected graph with $O(n^2+nd^2)$ vertices and $O(n^2d^2)$ edges that satisfies the following two properties.\",\n \"If the 3-OV instance has no solution, then for all pairs of vertices $u$ and $v$ we have $d(u,v)\\\\le 5$ .\",\n \"If the 3-OV instance has a solution, then there exists a pair of vertices $u$ and $v$ such that $d(u,v)\\\\ge 8$ .\"\n ],\n [\n \"Construction of the graph\",\n \"We construct a graph with the required properties by starting with the graph from Thereom REF and adding more vertices and edges.\",\n \"Figure REF illustrates the construction of the graph.\",\n \"Figure: The illustration for the 5 vs 8 construction.\",\n \"The edges between sets S,L 1 ,L 2 S,L_1,L_2 and TT are not depicted.\",\n \"The edges between vertices in S ' S^{\\\\prime } and SS (TT and T ' T^{\\\\prime }) form a matching.\",\n \"Vertices in S '' S^{\\\\prime \\\\prime } (T '' T^{\\\\prime \\\\prime }) form a clique.We start by adding a set $S^{\\\\prime }$ of $n^2$ vertices.\",\n \"$S^{\\\\prime }$ consists of all tuples $(a,b)$ of vertices $a \\\\in A$ and $b \\\\in B$ .\",\n \"We connect every $(a,b) \\\\in S^{\\\\prime }$ to its counterpart $(a,b) \\\\in S$ .\",\n \"Thus, there is a matching between the sets of vertices $S$ and $S^{\\\\prime }$ .\",\n \"We also add another set $S^{\\\\prime \\\\prime }$ of $n$ vertices.\",\n \"$S^{\\\\prime \\\\prime }$ contains one vertex $a$ for every $a \\\\in A$ .\",\n \"For every pair of vertices from $S^{\\\\prime \\\\prime }$ we add an edge between the vertices.\",\n \"Thus, the $n$ vertices form a clique.\",\n \"Furthermore, for every vertex $a \\\\in S^{\\\\prime \\\\prime }$ we add an edge to $(a,b) \\\\in S$ for all $b \\\\in B$ .\",\n \"In total we added $n^2+n=O(n^2)$ vertices and $\\\\binom{n}{2}+2n^2=O(n^2)$ edges.\",\n \"We do a similar construction for the set $T$ of vertices.\",\n \"We add a set $T^{\\\\prime }$ of $n^2$ vertices - one vertex for every tuple $(b,c)$ of vertices $b \\\\in B$ and $c \\\\in C$ .\",\n \"We connect every $(b,c)\\\\in T^{\\\\prime }$ to $(b,c) \\\\in T$ .\",\n \"Finally, we add a set $T^{\\\\prime \\\\prime }$ of $n$ vertices.\",\n \"$T^{\\\\prime \\\\prime }$ contains one vertex for every vector $c \\\\in C$ .\",\n \"For every pair of vertices from $T^{\\\\prime \\\\prime }$ we add an edge between the vertices.\",\n \"We connect every $c \\\\in T^{\\\\prime \\\\prime }$ to $(b,c) \\\\in T$ for all $b \\\\in B$ .\",\n \"This finishes the construction of the graph.\",\n \"In the rest of the section we show that the construction satisfies the promised two properties.\"\n ],\n [\n \"Correctness of the construction\",\n \"We need to consider two cases.\"\n ],\n [\n \"Case 1: the 3-OV instance has no solution\",\n \"In this case we want to show that for all pairs of vertices $u$ and $v$ we have $d(u,v)\\\\le 5$ .\",\n \"We consider three subcases.\"\n ],\n [\n \"Case 1.1: $u \\\\in S \\\\cup S^{\\\\prime } \\\\cup S^{\\\\prime \\\\prime } \\\\cup L_1$ and {{formula:e49ba0d1-2dce-4e1e-bc34-0f08bcbafce8}}\",\n \"We observe that there exists $s \\\\in S$ that has $d(u,s)\\\\le 1$ .\",\n \"Indeed, if $u \\\\in S$ , then $s=u$ works.\",\n \"If $u \\\\in S^{\\\\prime } \\\\cup S^{\\\\prime \\\\prime }$ , then we are done by the construction.\",\n \"On the other hand, if $u \\\\in L_1$ , then there exists such an $s \\\\in S$ by property 6 from Theorem REF .\",\n \"Similarly we can show that there exists $t \\\\in T$ such that $d(v,t)\\\\le 1$ .\",\n \"Finally, by property 3 we have that $d(s,t)=3$ .\",\n \"Thus, we can upper bound the distance between $u$ and $v$ by $d(u,v)\\\\le d(u,s)+d(s,t)+d(t,v)\\\\le 1+3+1=5$ as required.\"\n ],\n [\n \"Case 1.2: $u,v \\\\in S \\\\cup S^{\\\\prime } \\\\cup S^{\\\\prime \\\\prime } \\\\cup L_1$\",\n \"From the previous case we know that there are two vertices $s_1,s_2 \\\\in S$ such that $d(u,s_1)\\\\le 1$ and $d(s_2,v)\\\\le 1$ .\",\n \"To show that $d(u,v)\\\\le 5$ it is sufficient to show that $d(s_1,s_2)\\\\le 3$ .\",\n \"This is indeed true since both vertices $s_1$ and $s_2$ are connected to some two vertices in $S^{\\\\prime \\\\prime }$ and every two vertices in $S^{\\\\prime \\\\prime }$ are at distance at most 1 from each other.\"\n ],\n [\n \"Case 1.3: $u,v \\\\in T \\\\cup T^{\\\\prime } \\\\cup T^{\\\\prime \\\\prime } \\\\cup L_2$\",\n \"The case is analogous to the previous case.\"\n ],\n [\n \"Case 2: the 3-OV instance has a solution\",\n \"In this case we want to show that there is a pair of vertices $u,v$ with $d(u,v)\\\\ge 8$ .\",\n \"Let $a \\\\in A, b \\\\in B, c \\\\in C$ be a solution to the 3-OV instance.\",\n \"We claim that $d((a,b) \\\\in S^{\\\\prime },(b,c) \\\\in T^{\\\\prime })\\\\ge 8$ .\",\n \"Let $P$ be an optimal path between $u=((a,b) \\\\in S^{\\\\prime })$ and $v=((b,c) \\\\in T^{\\\\prime })$ that achieves the smallest distance.\",\n \"We want to show that $P$ uses at least 8 edges.\",\n \"Let $t \\\\in T$ be the first vertex from the set $T$ that is on path $P$ .\",\n \"Let $s \\\\in S$ be the last vertex on path $P$ that belongs to $S$ and precedes $t$ in $P$ .\",\n \"We can easily check that, if $s \\\\ne ((a,b) \\\\in S)$ , then $d(u,s)\\\\ge 3$ and, similarly, if $t \\\\ne ((b,c) \\\\in T)$ , then $d(t,v)\\\\ge 3$ .\",\n \"We consider three subcases.\"\n ],\n [\n \"Case 2.1: $s \\\\ne ((a,b) \\\\in S)$ and {{formula:bf5aaac7-5b1b-4304-83ad-0a1fd19bda3a}}\",\n \"Since $s$ and $t$ are separated by two layers of vertices, we must have $d(s,t)\\\\ge 3$ .\",\n \"Thus we get lower bound $d(u,v)\\\\ge d(u,s)+d(s,t)+d(t,v)\\\\ge 3+3+3=9>8$ as required.\"\n ],\n [\n \"Case 2.2: $s = ((a,b) \\\\in S)$ and {{formula:a28d868f-f38c-4534-b9c1-df1ca7ddd683}}\",\n \"In this case we use property 4 and conclude $d(u,v)\\\\ge d(u,s)+d(s,t)+d(t,v)=1+d((a,b) \\\\in S,(b,c) \\\\in T)+1\\\\ge 1+7+1=9>8$ as required.\"\n ],\n [\n \"Case 2.3: either $s = ((a,b) \\\\in S)$ or {{formula:1d8b5c54-e4d4-4cf8-8777-ab4e928d855f}} holds but not both\",\n \"W.l.o.g.\",\n \"$s \\\\ne ((a,b) \\\\in S)$ and $t = ((b,c) \\\\in T)$ .\",\n \"If the path uses an edge in the clique on $S^{\\\\prime \\\\prime }$ before arriving at $s$ , then $d(u,s)\\\\ge 4$ and we get that $d(u,v)\\\\ge d(u,s)+d(s,t)+d(t,v)\\\\ge 4+3+1=8$ .\",\n \"On the other hand, if the path does not use any edge of the clique, then $s=((a,b^{\\\\prime }) \\\\in S)$ for some $b^{\\\\prime } \\\\in B$ .\",\n \"By property 5 we have $d(s,t)=d((a,b^{\\\\prime })\\\\in S,(b,c)\\\\in T)\\\\ge 5$ .\",\n \"We conclude that $d(u,v)\\\\ge d(u,s)+d(s,t)+d(t,v)\\\\ge 3+5+1=9>8$ as required.\"\n ],\n [\n \"6 vs 10 weighted undirected construction\",\n \"In this section we change the construction from Theorem REF to show that under the 3-OV Hypothesis, any algorithm that can distinguish between diameter 6 and 10 in sparse undirected weighted graphs requires $\\\\Omega (n^{3/2-o(1)})$ time.\",\n \"We get the following theorem.\",\n \"Theorem 19 Given a 3-OV instance, we can in $O(n^2d^2)$ time construct an weighted, undirected graph with $O(n^2+nd^2)$ vertices and $O(n^2d^2)$ edges that satisfies the following two properties.\",\n \"If the 3-OV instance has no solution, then for all pairs of vertices $u$ and $v$ we have $d(u,v)\\\\le 6$ .\",\n \"If the 3-OV instance has a solution, then there exists a pair of vertices $u$ and $v$ such that $d(u,v)\\\\ge 10$ .\",\n \"Each edge of the graph has weight either 1 or 2.\"\n ],\n [\n \"Construction of the graph\",\n \"The construction of the graph is the same as in Theorem REF except all edges connecting vertices between sets $L_1$ and $L_2$ have weight 2 and all edges inside the cliques on nodes $S^{\\\\prime \\\\prime }$ and $T^{\\\\prime \\\\prime }$ have weight 2.\",\n \"All the remaining edges have weight 1.\"\n ],\n [\n \"Correctness of the construction\",\n \"The correctness proof is essentially the same as for Theorem REF .\",\n \"As before we consider two cases.\"\n ],\n [\n \"Case 1: the 3-OV instance has no solution\",\n \"In this case we want to show that for all pairs of vertices $u$ and $v$ we have $d(u,v)\\\\le 6$ .\",\n \"In the analysis of Case 1 in Theorem REF we show a path between $u$ and $v$ such that the path involves at most one edge from the cliques or between sets $L_1$ and $L_2$ .\",\n \"Since we added weight 2 to the latter edges, the length of the path increased by at most 1 as a result.\",\n \"So we have upper bound $d(u,v)\\\\le 6$ for all pairs $u$ and $v$ of vertices.\"\n ],\n [\n \"Case 2: the 3-OV instance has a solution\",\n \"In this case we want to show that there is a pair of vertices $u,v$ with $d(u,v)\\\\ge 10$ .\",\n \"Similarly to Theorem REF we will show that $d((a,b) \\\\in S^{\\\\prime },(b,c) \\\\in T^{\\\\prime })\\\\ge 10$ , where $a \\\\in A, b \\\\in B, c \\\\in C$ is a solution to the 3-OV instance.\",\n \"The analysis of the subcases is essentially the same as in Theorem REF .\",\n \"For cases 2.1 and 2.2 in the proof of Theorem REF we had $d((a,b) \\\\in S^{\\\\prime },(b,c) \\\\in T^{\\\\prime })\\\\ge 9$ .\",\n \"Since we increased edge weights between $L_1$ and $L_2$ to 2 and every path from $(a,b) \\\\in S^{\\\\prime }$ to $(b,c) \\\\in T^{\\\\prime }$ must cross the layer between $L_1$ and $L_2$ , we also increased the lower bound of the length of the path from 9 to 10 for cases 2.1 and 2.2.\",\n \"It remains to consider Case 2.3.\",\n \"As in the proof of Theorem REF , w.l.o.g.\",\n \"$s \\\\ne ((a,b) \\\\in S)$ and $t = ((b,c) \\\\in T)$ .\",\n \"If the path uses an edge in the clique on $S^{\\\\prime \\\\prime }$ before arriving at $s$ , then $d(u,s)\\\\ge 5$ and we get lower bound $d(u,v)\\\\ge d(u,s)+d(s,t)+d(t,v)\\\\ge 5+4+1=10$ .\",\n \"On the other hand, if the path does not use any edge of the clique, then $s=((a,b^{\\\\prime }) \\\\in S)$ for some $b^{\\\\prime } \\\\in B$ .\",\n \"By property 5 and because we increased edge weights between $L_1$ and $L_2$ to 2, we have $d(s,t)=d((a,b^{\\\\prime })\\\\in S,(b,c)\\\\in T)\\\\ge 6$ .\",\n \"We conclude that $d(u,v)\\\\ge d(u,s)+d(s,t)+d(t,v)\\\\ge 3+6+1=10$ as required.\"\n ],\n [\n \"$3k-4$ vs {{formula:cef9204d-0a6e-4579-9476-41520f256b04}} unweighted directed construction\",\n \"In this section, we show that under SETH, for every $k\\\\ge 3$ , every algorithm that can distinguish between Diameter $3k-4$ and $5k-7$ in directed unweighted graphs requires $\\\\Omega (n^{1+1/(k-1)-o(1)})$ time.\",\n \"Theorem REF gives us the following theorem.\",\n \"Theorem 20 Given a $k$ -OV instance consisting of $k\\\\ge 2$ sets $W_0,W_1,\\\\dots ,W_{k-1} \\\\subseteq \\\\lbrace 0,1\\\\rbrace ^d$ , each of size $n$ , we can in $O(kn^{k-1}d^{k-1})$ time construct an unweighted, undirected graph with $O(n^{k-1}+k n^{k-2} d^{k-1})$ vertices and $O(k n^{k-1} d^{k-1})$ edges that satisfies the following properties.\",\n \"The graph consists of $k+1$ layers of vertices $S=L_0,L_1,L_2,\\\\dots ,L_k=T$ .\",\n \"The number of nodes in the sets is $|S|=|T|=n^{k-1}$ and $|L_1|,|L_2|,\\\\dots ,|L_{k-1}|\\\\le n^{k-2}d^{k-1}$ .\",\n \"$S$ consists of all tuples $(a_0,a_1,\\\\ldots , a_{k-2})$ where for each $i$ , $a_i\\\\in W_i$ .\",\n \"Similarly, $T$ consists of all tuples $(b_1,b_2,\\\\ldots , b_{k-1})$ where for each $i$ , $b_i\\\\in W_i$ .\",\n \"If the $k$ -OV instance has no solution, then $d(u,v)=k$ for all $u \\\\in S$ and $v \\\\in T$ .\",\n \"If the $k$ -OV instance has a solution $a_0, a_1,\\\\dots ,a_{k-1}$ where for each $i$ , $a_i\\\\in W_i$ then if $\\\\alpha =(a_0,\\\\dots a_{k-2})\\\\in S$ and $\\\\beta = (a_1,\\\\dots ,a_{k-1}) \\\\in T$ , then $d(\\\\alpha ,\\\\beta )\\\\ge 3k-2$ .\",\n \"Setting $s=k-2$ in Property 5 of Theorem REF : If the $k$ -OV instance has a solution $a_0, a_1,\\\\dots ,a_{k-1}$ where for each $i$ , $a_i\\\\in W_i$ then for any tuple $(b_1,\\\\dots ,b_{k-2})$ , if $\\\\alpha =(a_0,b_1,\\\\dots ,b_{k-2})\\\\in S$ and $\\\\beta =(a_1,\\\\dots ,a_{k-1})\\\\in T$ , then $d(\\\\alpha ,\\\\beta )\\\\ge k+2$ .\",\n \"Symmetrically, if $\\\\alpha =(a_0,a_1,\\\\dots ,a_{k-2})\\\\in S$ and $\\\\beta =(b_1,\\\\dots ,b_{k-2},a_{k-1})\\\\in T$ , then $d(\\\\alpha ,\\\\beta )\\\\ge k+2$ .\",\n \"For all $i$ from 1 to $k-1$ , for all $v \\\\in L_i$ there exists a vertex in $L_{i-1}$ adjacent to $v$ and a vertex in $L_{i+1}$ adjacent to $v$ .\",\n \"We can assume that this property holds because we can remove all vertices that do not satisfy this property from the graph and the resulting graph will still satisfy the previous three properties.\",\n \"In the rest of the section we use Theorem REF to prove the following result.\",\n \"Theorem 21 Given a $k$ -OV instance, we can in $O(kn^{k-1}d^{k-1})$ time construct an unweighted, directed graph with $O(k n^{k-1}+k n^{k-2} d^{k-1})$ vertices and $O(kn^{k-1}d^{k-1})$ edges that satisfies the following two properties.\",\n \"If the $k$ -OV instance has no solution, then for all pairs of vertices $u$ and $v$ we have $d(u,v)\\\\le 3k-4$ .\",\n \"If the $k$ -OV instance has a solution, then there exists a pair of vertices $u$ and $v$ such that $d(u,v)\\\\ge 5k-7$ .\"\n ],\n [\n \"Construction of the graph\",\n \"We construct a graph with the required properties by starting with the graph from Thereom REF and adding more vertices and edges.\",\n \"First we will construct a weighted graph and then we will make it unweighted.\",\n \"Figure REF illustrates the construction of the graph for the special case $k=4$ .\",\n \"Figure: The 3k-43k-4 vs 5k-75k-7 construction for the special case k=4k=4.\",\n \"The edges between sets S,L 1 ,L 2 ,L 3 S,L_1,L_2,L_3 and TT are not depicted.\",\n \"The matching between sets SS and S ' S^{\\\\prime } consists of unweighted paths of length k-2=2k-2=2.\",\n \"The edges between sets SS and S '' S^{\\\\prime \\\\prime } consists of unweighted paths of length k-2=2k-2=2.\",\n \"Similarly for the right side.We start by adding a set $S^{\\\\prime }$ of $n^{k-1}$ vertices.\",\n \"$S^{\\\\prime }$ consists of all tuples $(a_0,a_1,\\\\ldots , a_{k-2})$ where for each $i$ , $a_i\\\\in W_i$ .\",\n \"We connect every $(a_0,a_1,\\\\ldots , a_{k-2}) \\\\in S^{\\\\prime }$ to its counterpart $(a_0,a_1,\\\\ldots , a_{k-2}) \\\\in S$ with an undirected edge of weight $k-2$ to form a matching.\",\n \"We also add another set $S^{\\\\prime \\\\prime }$ of $n$ vertices.\",\n \"$S^{\\\\prime \\\\prime }$ contains one vertex $a_0$ for every $a_0 \\\\in W_0$ .\",\n \"For every pair of vertices in $S^{\\\\prime \\\\prime }$ we add an undirected edge of weight 1 between the vertices.\",\n \"Thus, the $n$ vertices form a clique.\",\n \"Furthermore, for every vertex $a_0 \\\\in S^{\\\\prime \\\\prime }$ we add an undirected edge of weight $k-2$ to $(a_0,b_1,\\\\ldots , b_{k-2}) \\\\in S$ for all $b_1,\\\\dots , b_{k-2}$ .\",\n \"Finally for every vertex $a_0 \\\\in S^{\\\\prime \\\\prime }$ we add a directed edge of weight 1 towards $(a_0,b_1,\\\\ldots , b_{k-2}) \\\\in S$ for all $b_1,\\\\dots , b_{k-2}$ .\",\n \"Some of the edges that we added have weight $k-2$ .\",\n \"We make those unweighted by subdividing them into edges of weight 1.\",\n \"Let $S^{\\\\prime \\\\prime \\\\prime }$ be the set of newly added vertices.\",\n \"In total we added $O(kn^{k-1})$ vertices and $O(kn^{k-1})$ edges.\",\n \"We do a similar construction for the set $T$ of vertices.\",\n \"We add a set $T^{\\\\prime }$ of $n^{k-1}$ vertices — one vertex for every tuple $(a_1,\\\\dots ,a_{k-1})$ where for each $i$ , $a_i\\\\in W_i$ .\",\n \"We connect every $(a_1,\\\\dots ,a_{k-1})\\\\in T^{\\\\prime }$ to $(a_1,\\\\dots ,a_{k-1}) \\\\in T$ by an undirected edge of weight $k-2$ .\",\n \"Finally, we add a set $T^{\\\\prime \\\\prime }$ of $n$ vertices.\",\n \"$T^{\\\\prime \\\\prime }$ contains one vertex for every vector $a_{k-1} \\\\in W_{k-1}$ .\",\n \"We connect every pair of vertices in $T^{\\\\prime \\\\prime }$ by an undirected edge of weight 1.\",\n \"We connect every vertex $a_{k-1} \\\\in T^{\\\\prime \\\\prime }$ to $(b_1,\\\\dots ,b_{k-2},a_{k-1}) \\\\in T$ by an undirected edge of weight $k-2$ for all $b_1,\\\\dots ,b_{k-2}$ .\",\n \"Also, for every vertex $a_{k-1} \\\\in T^{\\\\prime \\\\prime }$ we add a directed edge of weight 1 from $(b_1,\\\\dots ,b_{k-2},a_{k-1}) \\\\in T^{\\\\prime }$ to $a_{k-1}$ for all $b_1,\\\\dots ,b_{k-2}$ .\",\n \"Some of the edges that we just added have weight $k-2$ .\",\n \"We make those unweighted by subdividing them into edges of weight 1.\",\n \"Let $T^{\\\\prime \\\\prime \\\\prime }$ be the set of newly added vertices.\",\n \"This finishes the construction of the graph.\",\n \"In the rest of the section we show that the construction satisfies the promised two properties stated in Theorem REF .\"\n ],\n [\n \"Correctness of the construction\",\n \"We need to consider two cases.\"\n ],\n [\n \"Case 1: the $k$ -OV instance has no solution\",\n \"In this case we want to show that for all pairs of vertices $u$ and $v$ we have $d(u,v)\\\\le 3k-4$ .\",\n \"We consider subcases.\"\n ],\n [\n \"Case 1.1: $u \\\\in S \\\\cup S^{\\\\prime } \\\\cup S^{\\\\prime \\\\prime } \\\\cup S^{\\\\prime \\\\prime \\\\prime } \\\\cup L_i$ for {{formula:341c2d42-7167-4db5-bd6a-4b4996ef5fb1}} and {{formula:b26c11ee-44d0-4a3f-8d06-2943a8aaf115}} for {{formula:b2ebe27d-5f57-42a2-b20b-50932506480e}}\",\n \"We observe that there exists $s \\\\in S$ that has $d(u,s)\\\\le k-2$ .\",\n \"Similarly, there exists $t \\\\in T$ with $d(t,v)\\\\le k-2$ .\",\n \"By property 3 from Theorem REF we have that $d(s,t)\\\\le k$ .\",\n \"This gives us upper bound $d(u,v)\\\\le d(u,s)+d(s,t)+d(t,v)\\\\le (k-2)+k+(k-2)=3k-4$ as required.\",\n \"The proof when the sets for $u$ and $v$ are swapped is identical since we only use paths on unweighted edges.\"\n ],\n [\n \"Case 1.2: $u,v \\\\in S \\\\cup S^{\\\\prime } \\\\cup S^{\\\\prime \\\\prime } \\\\cup S^{\\\\prime \\\\prime \\\\prime } \\\\cup L_1$\",\n \"We note that there is some vertex $s\\\\in S^{\\\\prime \\\\prime }$ with $d(u,s)\\\\le 2(k-2)$ (via undirected edges).\",\n \"Also, there is some vertex $s^{\\\\prime }\\\\in S^{\\\\prime \\\\prime }$ with $d(s^{\\\\prime },v)\\\\le k-1$ (possibly using directed edges).\",\n \"$S^{\\\\prime \\\\prime }$ is a clique so $d(s,s^{\\\\prime })\\\\le 1$ .\",\n \"Thus, $d(u,v)\\\\le d(u,s)+d(s,s^{\\\\prime })+d(s^{\\\\prime },v)\\\\le 2(k-2)+1+(k-1)=3k-4$ .\"\n ],\n [\n \"Case 1.3: $u,v \\\\in T \\\\cup T^{\\\\prime } \\\\cup T^{\\\\prime \\\\prime } \\\\cup T^{\\\\prime \\\\prime \\\\prime } \\\\cup L_{k-1}$\",\n \"This case is similar to the previous case.\",\n \"We note that there is some vertex $t\\\\in T^{\\\\prime \\\\prime }$ with $d(t,v)\\\\le 2(k-2)$ (via undirected edges).\",\n \"Also, there is some vertex $t^{\\\\prime }\\\\in T^{\\\\prime \\\\prime }$ with $d(u,t^{\\\\prime })\\\\le k-1$ (possibly using directed edges).\",\n \"$S^{\\\\prime \\\\prime }$ is a clique so $d(t^{\\\\prime },t)\\\\le 1$ .\",\n \"Thus, $d(u,v)\\\\le d(u,t^{\\\\prime })+d(t^{\\\\prime },t)+d(t,v)\\\\le (k-1)+1+2(k-2)=3k-4$ .\"\n ],\n [\n \"Case 2: the $k$ -OV instance has a solution\",\n \"In this case we want to show that there is a pair of vertices $u,v$ with $d(u,v)\\\\ge 5k-7$ .\",\n \"Let $(a_0,a_1,\\\\ldots , a_{k-1})$ be a solution to the $k$ -OV instance where for each $i$ , $a_i\\\\in W_i$ .\",\n \"We claim that $d((a_0,\\\\dots ,a_{k-2}) \\\\in S^{\\\\prime },(a_1,\\\\dots ,a_{k-1}) \\\\in T^{\\\\prime })\\\\ge 5k-7$ .\",\n \"Let $P$ be an shortest path between $u=((a_0,\\\\dots ,a_{k-2}) \\\\in S^{\\\\prime })$ and $v=((a_1,\\\\dots ,a_{k-1}) \\\\in T^{\\\\prime })$ .\",\n \"We want to show that $P$ uses at least $5k-7$ edges.\",\n \"Let $s \\\\in S$ be the first vertex on path $P$ that belongs to $S$ and let $t \\\\in T$ be the last vertex from the set $T$ that is on path $P$ .\",\n \"We observe that due to the directionality of the edges, $s$ and $t$ must be the counterparts of $u$ and $v$ respectively; that is, $s=((a_0,\\\\dots ,a_{k-2}) \\\\in S)$ and $t=((a_1,\\\\dots ,a_{k-1}) \\\\in T)$ .\",\n \"Note that these definitions of $s$ and $t$ differ from the definitions of $s$ and $t$ in previous proofs.\",\n \"We consider three subcases.\"\n ],\n [\n \"Case 2.1: A vertex in $S^{\\\\prime }\\\\cup S^{\\\\prime \\\\prime }\\\\cup S^{\\\\prime \\\\prime \\\\prime }$ appears after {{formula:6406b760-7ede-4dc4-b0f6-a59f46a09387}} on the path {{formula:a6c998cf-9396-4ea3-832a-1c6dca88cdb2}}\",\n \"We observe that if $s_1,s_2\\\\in S$ is a pair of vertices on the path $P$ such that no vertex in $S$ appears between them on $P$ , then the portion of $P$ between $s_1$ and $s_2$ either contains only vertices in $S^{\\\\prime }\\\\cup S^{\\\\prime \\\\prime }\\\\cup S^{\\\\prime \\\\prime \\\\prime }$ or contains no vertices in $S^{\\\\prime }\\\\cup S^{\\\\prime \\\\prime }\\\\cup S^{\\\\prime \\\\prime \\\\prime }$ .\",\n \"Let $s_1,s_2\\\\in S$ be such that the portion of $P$ between them contains only vertices in $S^{\\\\prime }\\\\cup S^{\\\\prime \\\\prime }\\\\cup S^{\\\\prime \\\\prime \\\\prime }$ .\",\n \"Such $s_1,s_2$ exist by the specification of this case.\",\n \"If $s_1=s_2$ then $P$ is not a shortest path.\",\n \"Otherwise, the portion of $P$ between $s_1$ and $s_2$ must include a vertex in $S^{\\\\prime \\\\prime }$ .\",\n \"Thus, $d(s_1,s_2)\\\\ge 2(k-2)$ .\",\n \"We consider three subcases.\",\n \"$s_1\\\\ne s$ .\",\n \"The distance between any pair of vertices in $S$ is at least 2 so $d(s,s_1)\\\\ge 2$ .\",\n \"Then, $d(u,v)\\\\ge d(u,s)+d(s,s_1)+d(s_1,s_2)+d(s_2,t)+d(t,v)\\\\ge (k-2)+2+2(k-2)+k+(k-2)=5k-6$ .\",\n \"$s_1=s$ and $s_2=((a_0,b_1,\\\\dots ,b_{k-2}) \\\\in S)$ for some $b_1,\\\\dots ,b_{k-2}$ .\",\n \"In this case, by property 5 we have $d(s_2,t)\\\\ge k+2$ .\",\n \"Thus, $d(u,v)\\\\ge d(u,s_1)+d(s_1,s_2)+d(s_2,t)+d(t,v)\\\\ge (k-2)+2(k-2)+(k+2)+(k-2)=5k-6$ .\",\n \"$s_1=s$ and $s_2=((b_0,\\\\dots ,b_{k-2} \\\\in S)$ for some with $b_0\\\\ne a_0$ .\",\n \"In this case, the path from $s_1$ to $s_2$ must include an edge in the clique $S^{\\\\prime \\\\prime }$ since these are the only edges among vertices in $S^{\\\\prime }\\\\cup S^{\\\\prime \\\\prime }\\\\cup S^{\\\\prime \\\\prime \\\\prime }$ for which adjacent tuples can differ with respect to their first element.\",\n \"Thus, $d(s_1,s_2)\\\\ge 2(k-2)+1\\\\ge 2k-3$ .\",\n \"Therefore, $d(u,v)\\\\ge d(u,s_1)+d(s_1,s_2)+d(s_2,t)+d(t,v)\\\\ge (k-2)+(2k-3)+k+(k-2)=5k-7$ .\"\n ],\n [\n \"Case 2.2: A vertex in $T^{\\\\prime }\\\\cup T^{\\\\prime \\\\prime }\\\\cup T^{\\\\prime \\\\prime \\\\prime }$ appears before {{formula:2e52092d-2023-4c53-a764-a41805329e43}} on the path {{formula:d86d9c20-8168-4be0-8d10-99eb7834ec04}}\",\n \"This case is analogous to the previous case.\"\n ],\n [\n \"Case 2.3: The portion of the path $P$ between {{formula:558fd34a-9236-4d24-8a2c-f5022f0a499a}} and {{formula:c0c71b02-54cb-42b0-8842-f210ddc962a7}} contains no vertices in {{formula:7c23b5b2-7499-4d20-8946-fe3d3dacb80d}}\",\n \"By property 4, $d(s,t)\\\\ge 3k-2$ .\",\n \"Thus, $d(u,v)\\\\ge d(u,s)+d(s,t)+d(t,v)\\\\ge (k-2)+(3k-2)+(k-2)=5k-6$ .\",\n \"We note that a slight modification of this construction gives a lower bound for higher values of Diameter.\",\n \"For any L, we can get an $L(3k-4)$ vs $L(5k-8)+1$ construction by subdividing all of the edges in the construction (into paths of length $L$ ) except for the directed edges and the edges within the cliques.\"\n ],\n [\n \"2-Approximation for Eccentricities in $\\\\tilde{O}(m\\\\sqrt{n})$ time\",\n \"Theorem 22 Given a weighted, directed $m$ edge $n$ node graph, in $\\\\tilde{O}(m\\\\sqrt{n})$ time we can output quantities $\\\\epsilon ^{\\\\prime }(v)$ such that for all $v \\\\in V$ we have $\\\\epsilon (v)/2\\\\le \\\\epsilon ^{\\\\prime }(v)\\\\le \\\\epsilon (v)$ .\",\n \"The algorithm is inspired by the 2-approximation algorithm for directed radius of Abboud, Vassilevska W., and Wang [7].\",\n \"We claim that the following algorithm achieves the above guarantees.\",\n \"Sample a random subset $S \\\\subset V$ of size $|S|=\\\\Theta (\\\\sqrt{n} \\\\log n)$ .\",\n \"With high probability for every $u \\\\in V$ we have $N^{\\\\text{in}}_{\\\\sqrt{n}}(u)\\\\cap S \\\\ne \\\\emptyset $ .\",\n \"Let $w$ be a vertex that maximizes $d(S,w)$ , which we find using Dijkstra's algorithm.\",\n \"Let $S^{\\\\prime }:=N^{\\\\text{in}}_{\\\\sqrt{n}}(w)$ .\",\n \"For every vertex $v \\\\in S^{\\\\prime }$ we output $\\\\epsilon ^{\\\\prime }(v)=\\\\epsilon (v)$ by running Dijkstra's algorithm and following the outgoing edges.\",\n \"For every vertex $v \\\\notin S^{\\\\prime }$ we output the estimate $\\\\epsilon ^{\\\\prime }(v)=\\\\max _{s \\\\in S\\\\cup \\\\lbrace w\\\\rbrace }d(v,s)$ .\",\n \"We can determine all these quantities by running Dijkstra's algorithm out of all vertices in $S \\\\cup \\\\lbrace w\\\\rbrace $ and following the incoming edges.\"\n ],\n [\n \"Correctness\",\n \"Consider an arbitrary vertex $v\\\\notin S^{\\\\prime }$ (if $v \\\\in S^{\\\\prime }$ , then we are done by the third step).\",\n \"If there exists $s \\\\in S$ such that $d(v,s)\\\\ge \\\\epsilon (v)/2$ , then we are done since $\\\\epsilon ^{\\\\prime }(v)\\\\ge d(v,s)\\\\ge \\\\epsilon (v)/2$ .\",\n \"Otherwise, we have $d(v,s)<\\\\epsilon (v)/2$ for all $s \\\\in S$ .\",\n \"Let $v^{\\\\prime }$ be a vertex that achieves $d(v,v^{\\\\prime })=\\\\epsilon (v)$ .\",\n \"By the triangle inequality we have $d(s,v^{\\\\prime })>\\\\epsilon (v)/2$ for all $s \\\\in S$ .\",\n \"Equivalently, $d(S,v^{\\\\prime })>\\\\epsilon (v)/2$ .\",\n \"This implies that $d(S,w)>\\\\epsilon (v)/2$ by our choice of $w$ .\",\n \"Since $d(S,w)>\\\\epsilon (v)/2$ and $S^{\\\\prime }=N^{\\\\text{in}}_{\\\\sqrt{n}}(w)$ intersects $S$ , we must have that $S^{\\\\prime }$ contains all vertices $u$ with $d(u,w)\\\\le \\\\epsilon (v)/2$ .\",\n \"Since $v \\\\notin S^{\\\\prime }$ , we must have $d(v,w)>\\\\epsilon (v)/2$ and we are done since $\\\\epsilon ^{\\\\prime }(v)\\\\ge d(v,w)>\\\\epsilon (v)/2$ .\"\n ],\n [\n \"Almost 2-Approximation for Eccentricities in almost linear time\",\n \"In contrast to our $\\\\tilde{O}(m\\\\sqrt{n})$ time algorithm from the previous section, our near-linear time $(2+\\\\delta )$ -approximation algorithm is very different from all previously known algorithms.\",\n \"Our algorithm proceeds in iterations and maintains a set $S$ of nodes for which we still do not have a good eccentricity estimate.\",\n \"In each iteration either we get a good estimate for many new vertices and hence remove them from $S$ , or we remove all vertices from $S$ that have large eccentricities, and for the remaining nodes in $S$ we have a better upper bound on their eccentricities.\",\n \"After a small number of iterations we have a good estimate for all vertices of the graph.\",\n \"Theorem 23 Suppose that we are given a weighted, directed $m$ edge $n$ node graph.\",\n \"The weights of all edge are non-negative integers bounded by $n^{O(1)}$ .\",\n \"For any $1>\\\\tau >0$ we can in $\\\\tilde{O}(m/\\\\tau )$ time output quantities $\\\\varepsilon ^{\\\\prime }(v)$ such that for all $v \\\\in V$ we have ${1-\\\\tau \\\\over 2}\\\\varepsilon (v)\\\\le \\\\varepsilon ^{\\\\prime }(v)\\\\le \\\\varepsilon (v)$ .\",\n \"We maintain a subset $S\\\\subseteq V$ of vertices $v$ for which we still do not have an estimate $\\\\varepsilon ^{\\\\prime }(v)$ .\",\n \"Initially $S=V$ and we will end with $|S|\\\\le O(1)$ .\",\n \"When $|S|\\\\le O(1)$ we can evaluate $\\\\varepsilon (v)$ for all $v \\\\in S$ in the total time of $O(m)$ .\",\n \"Also we maintain a value $D$ that upper bounds the largest eccentricity of a vertex in $S$ .\",\n \"That is, $\\\\varepsilon (v)\\\\le D$ for all $v \\\\in S$ .\",\n \"Initially we set $D=n^C$ for some large enough constant $C>0$ (we assume that the input graph is strongly connected).\",\n \"The algorithm proceeds in phases.\",\n \"Each phase takes $\\\\tilde{O}(m)$ time and either $|S|$ decreases by a factor of at least 2 or $D$ decreases by a factor of at least $1/(1-\\\\tau )$ .\",\n \"After $O(\\\\log (n)/\\\\tau )$ phases either $|S|\\\\le O(1)$ or $D<1$ .\",\n \"For a subset $S \\\\subseteq V$ of vertices and a vertex $x \\\\in V$ we define a set $S_x\\\\subseteq S$ to contain those $|S_x|=|S|/2$ vertices from $S$ that are closest to $x$ (according to distance $d(\\\\cdot ,x)$ ).\",\n \"The ties are broken by taking the vertex with the smaller id.\",\n \"Given a subset $S \\\\subseteq V$ of vertices and a threshold $D$ , a phase proceeds as follows.\",\n \"[itemsep=0mm] We sample a set $A \\\\subseteq S$ of $O(\\\\log n)$ random vertices from the set $S$ .\",\n \"With high probability for all $x \\\\in V$ we have $A \\\\cap S_x \\\\ne \\\\emptyset $ .\",\n \"Let $w \\\\in V$ be a vertex that maximizes $d(A,w)$ .\",\n \"We can find it using Dijkstra's algorithm.\",\n \"We consider two cases.\",\n \"Case $d(S\\\\setminus S_w,w)\\\\ge {1-\\\\tau \\\\over 2}D$ .\",\n \"For all $x \\\\in S\\\\setminus S_w$ we have ${1-\\\\tau \\\\over 2} D \\\\le \\\\varepsilon (x)\\\\le D$ and we assign the estimate $\\\\varepsilon ^{\\\\prime }(x)={1-\\\\tau \\\\over 2}D$ .\",\n \"This gives us that ${1-\\\\tau \\\\over 2} \\\\varepsilon (x)\\\\le {1-\\\\tau \\\\over 2}D=\\\\varepsilon ^{\\\\prime }(x)\\\\le \\\\varepsilon (x)$ for all $x\\\\in S\\\\setminus S_w$ .\",\n \"We update $S$ to be $S_w$ .\",\n \"This decreases the size of $S$ by a factor of 2 as required.\",\n \"Case $d(S\\\\setminus S_w,w)<{1-\\\\tau \\\\over 2}D$ .\",\n \"Set $S^{\\\\prime }=S$ .\",\n \"For every vertex $v \\\\in S$ evaluate $r_v:=\\\\max _{x \\\\in A}d(v,x)$ .\",\n \"We can evaluate these quantities by running Dijkstra's algorithm from every vertex in $A$ and following the incoming edges.\",\n \"If $r_v\\\\ge {1-\\\\tau \\\\over 2}D$ , then assign the estimate $\\\\varepsilon ^{\\\\prime }(v)={1-\\\\tau \\\\over 2}D$ and remove $v$ from $S^{\\\\prime }$ .\",\n \"Similarly as in the previous case we have ${1-\\\\tau \\\\over 2} \\\\varepsilon (v)\\\\le \\\\varepsilon ^{\\\\prime }(v)\\\\le \\\\varepsilon (v)$ for all $v \\\\in S\\\\setminus S^{\\\\prime }$ .\",\n \"Below we will show that for every $v \\\\in S^{\\\\prime }$ we have $\\\\varepsilon (v)\\\\le (1-\\\\tau )D$ .\",\n \"Thus we can update $S=S^{\\\\prime }$ and decrease the threshold $D$ to $(1-\\\\tau )D$ as required.\",\n \"Correctness We have to show that, if there exists $v \\\\in S^{\\\\prime }$ such that $\\\\varepsilon (v)>(1-\\\\tau )D$ , then we will end up in the first case (this is the contrapositive of the claim in the second case).\",\n \"Since $v \\\\in S^{\\\\prime }$ we must have that $d(v,x)\\\\le {1-\\\\tau \\\\over 2}D$ for all $x \\\\in A$ .\",\n \"Since $\\\\varepsilon (v)>(1-\\\\tau )D$ , we must have that there exists $v^{\\\\prime }$ such that $d(v,v^{\\\\prime })>(1-\\\\tau )D$ .\",\n \"By the triangle inequality we get that $d(x,v^{\\\\prime })>{1 - \\\\tau \\\\over 2}D$ for every $x \\\\in A$ .\",\n \"By choice of $w$ we have $d(A,w)>{1-\\\\tau \\\\over 2}D$ .\",\n \"Since $A \\\\cap S_w \\\\ne \\\\emptyset $ , we have $d(S\\\\setminus S_w,w)\\\\ge {1-\\\\tau \\\\over 2}D$ and we will end up in the first case.\",\n \"The guarantee on the approximation factor follows from the description.\",\n \"As a corollary, we get an algorithm for Source Radius with the same runtime and approximation ratio as Theorem REF .\",\n \"First, run the Eccentricities algorithm and let $v$ be the vertex with the smallest estimated eccentricity $\\\\epsilon ^{\\\\prime }(v)$ .\",\n \"Then run Dijkstra's algorithm from $v$ and report $\\\\epsilon (v)$ as the Radius estimate $R^{\\\\prime }$ .\",\n \"Let $R$ be the true radius of the graph and let $x$ be the vertex with minimum Eccentricity i.e.\",\n \"$\\\\epsilon (x)=R$ .\",\n \"If $\\\\alpha $ is the approximation ratio for the Eccentricities algorithm then $\\\\epsilon (v)\\\\le \\\\alpha \\\\epsilon ^{\\\\prime }(v)\\\\le \\\\alpha \\\\epsilon (v)$ and $\\\\epsilon (x)\\\\le \\\\alpha \\\\epsilon ^{\\\\prime }(x)\\\\le \\\\alpha \\\\epsilon (x)$ .\",\n \"By choice of $v$ , $\\\\epsilon ^{\\\\prime }(v)\\\\le \\\\epsilon ^{\\\\prime }(x)$ .\",\n \"Thus, $\\\\alpha R=\\\\alpha \\\\epsilon (x)\\\\ge \\\\alpha \\\\epsilon ^{\\\\prime }(x)\\\\ge \\\\alpha \\\\epsilon ^{\\\\prime }(v)\\\\ge \\\\epsilon (v)=R^{\\\\prime }$ .\",\n \"Clearly $R^{\\\\prime }\\\\ge R$ , so $R\\\\le R^{\\\\prime }\\\\le \\\\alpha R$ .\",\n \"$S$ -$T$ Diameter algorithms Recall that the $S$ -$T$ diameter problem is as follows: Given an undirected graph $G=(V,E)$ and two sets $S\\\\subseteq V, T\\\\subseteq V$ , determine $D_{S,T}=\\\\max _{s\\\\in S, t\\\\in T} d(s,t)$ .\",\n \"Here we will outline two algorithms for the problem.\",\n \"Let us first consider a fast 3-approximation algorithm.\",\n \"Claim 24 There is an $O(m+n)$ time algorithm that for any $n$ node $m$ edge graph $G=(V,E)$ and $S\\\\subseteq V, T\\\\subseteq V$ , computes an estimate $D^{\\\\prime }$ such that $D_{S,T}/3\\\\le D^{\\\\prime }\\\\le D_{S,T}$ and two nodes $s\\\\in S$ , $t\\\\in T$ such that $d(s,t)=D^{\\\\prime }$ .\",\n \"In graphs with nonnegative weights, the same estimate can be achieved in $O(m+n\\\\log n)$ time.\",\n \"The algorithm is extremely simple: pick arbitrary nodes $s\\\\in S$ and $t\\\\in T$ , compute BFS($s$ ) and BFS($t$ ) and return $\\\\max \\\\lbrace \\\\max _{t^{\\\\prime }\\\\in T} d(s,t^{\\\\prime }),\\\\max _{s^{\\\\prime }\\\\in S} d(s^{\\\\prime },t)\\\\rbrace $ (also returning the two nodes achieving the maximum).\",\n \"For weighted graphs, run Dijkstra's algorithm instead of BFS.\",\n \"Let's see why this algorithm provides the promised guarantee.\",\n \"Suppose that for every $t^{\\\\prime }\\\\in T$ , $d(s,t^{\\\\prime })<D_{S,T}/3$ (otherwise we are done).\",\n \"Then for every $t^{\\\\prime },t^{\\\\prime \\\\prime }\\\\in T$ , $d(t^{\\\\prime },t^{\\\\prime \\\\prime })\\\\le d(t^{\\\\prime },s)+d(s,t^{\\\\prime \\\\prime })<2D_{S,T}/3$ .\",\n \"In particular, for all $t^{\\\\prime }\\\\in T$ , $d(t,t^{\\\\prime })<2D_{S,T}/3$ .\",\n \"If we also had that for every $s^{\\\\prime }\\\\in S$ , $d(t,s^{\\\\prime })<D_{S,T}/3$ , then we'd get that for all $s^{\\\\prime }\\\\in S, t^{\\\\prime }\\\\in T$ , $d(s^{\\\\prime },t^{\\\\prime })\\\\le d(s^{\\\\prime },t)+d(t,t^{\\\\prime })<D_{S,T}$ , contradicting the definition of $D_{S,T}$ .\",\n \"Thus, $\\\\max \\\\lbrace \\\\max _{t^{\\\\prime }\\\\in T} d(s,t^{\\\\prime }),\\\\max _{s^{\\\\prime }\\\\in S} d(s^{\\\\prime },t)\\\\rbrace \\\\ge D_{S,T}/3$ .\",\n \"We will now show an analogue to the $\\\\tilde{O}(m\\\\sqrt{n})$ time almost-$3/2$ -approximation diameter algorithm of Roditty and Vassilevska W. [41] for $S$ -$T$ Diameter giving a 2-approximation.\",\n \"Using a trick from Chechik et al.\",\n \"[21] we also obtain a true 2 approximation algorithm running in $\\\\tilde{O}(m^{3/2})$ .\",\n \"2-Approximation for $S$ -$T$ Diameter [1] 2-Approx $X$ - random sample of nodes, $|X|=\\\\Theta (\\\\sqrt{n} \\\\log n)$ $D_1:=0$ every $x\\\\in X$ Run BFS($x$ ) Let $t_x$ be the closest node to $x$ in $T$ Run BFS($t_x$ ) $D_1 = \\\\max \\\\lbrace D_1,\\\\max _{s\\\\in S} d(s,t_x)\\\\rbrace $ [t]@X@ Let $\\\\bar{t}$ be the furthest node of $T$ from $X$ (computed above) Run BFS($\\\\bar{t}$ ) $D_2=\\\\max _{s\\\\in S} d(s,\\\\bar{t})$ .\",\n \"Let $Y$ be the closest $\\\\sqrt{n}$ nodes to $\\\\bar{t}$ .\",\n \"every $y\\\\in Y$ Run BFS($y$ ) Let $s_y$ be the closest node to $y$ in $S$ Run BFS($s_y$ ) $D_2 = \\\\max \\\\lbrace D_2,\\\\max _{t\\\\in T} d(s_y,t)\\\\rbrace $ $\\\\max \\\\lbrace D_1,D_2\\\\rbrace $ We use Algorithm 2-Approx to prove: Theorem 25 In $\\\\tilde{O}(m\\\\sqrt{n})$ time one can obtain an estimate $D^{\\\\prime }$ to the $S$ -$T$ diameter $D$ of an $m$ edge $n$ node unweighted undirected graph such that $2\\\\lfloor D/4\\\\rfloor \\\\le D^{\\\\prime }\\\\le D$ .\",\n \"In $\\\\tilde{O}(m^{3/2})$ time one can obtain an estimate $D^{\\\\prime \\\\prime }$ such that $D/2\\\\le D^{\\\\prime \\\\prime }\\\\le D$ .\",\n \"First we analyze Algorithm 2-Approx.\",\n \"Let $s^*\\\\in S$ and $t^*\\\\in T$ be the end points of the $S$ -$T$ Diameter path so that $d(s^*,t^*)=D$ .\",\n \"Let $d=\\\\lfloor D/4\\\\rfloor $ .\",\n \"Suppose first that for some $x\\\\in X$ , $d(x,t^*)\\\\le d$ .\",\n \"Then, $d(x,t_x)\\\\le d(x,t^*)\\\\le d$ and hence $d(t_x,t^*)\\\\le d(t_x,x)+d(x,t^*)\\\\le 2d$ .\",\n \"However, then $d(t_x,s^*)\\\\ge d(t^*,s^*)-d(t^*,t_x)\\\\ge D-2d\\\\ge D/2$ .\",\n \"Thus, if $D_1<D/2$ , it must be that for every $x\\\\in X$ , $d(x,t^*)\\\\ge d+1$ .\",\n \"Hence, for every $x \\\\in X$ , $d(x,\\\\bar{t})\\\\ge d(x,t^*)\\\\ge d+1$ by the definition of $\\\\bar{t}$ .\",\n \"If $d(\\\\bar{t},s^*)\\\\ge D/2$ , then $D_2\\\\ge D/2$ and we are done, so let us assume that $d(\\\\bar{t},s^*)\\\\le D/2$ .\",\n \"Now, as $X$ is random of size $c\\\\sqrt{n}\\\\log n$ for large enough $c$ , with high probability, $X$ hits the $\\\\sqrt{n}$ -neighborhoods of all nodes.\",\n \"In particular, $X\\\\cap Y\\\\ne \\\\emptyset $ .\",\n \"However, since $d(x,\\\\bar{t})\\\\ge d+1$ for every $x\\\\in X$ , it must be that $Y$ contains all nodes at distance $d$ from $\\\\bar{t}$ as it contains all nodes closer to $\\\\bar{t}$ than $x\\\\in Y\\\\cap X$ .\",\n \"If $s^*\\\\in Y$ , then we would have run BFS from $s^*$ and returned $D$ .\",\n \"Hence $d(\\\\bar{t},s^*)>d$ .\",\n \"Let $a$ be the node on the shortest path between $\\\\bar{t}$ and $s^*$ with $d(\\\\bar{t},a)=d$ .\",\n \"We thus have that $a\\\\in Y$ .\",\n \"Also, since $d(\\\\bar{t},s^*)\\\\le D/2$ , $d(a,s^*)\\\\le D/2-d$ and hence $d(a,s_a)\\\\le D/2-d$ , so that $d(s_a,t^*)\\\\ge D-2(D/2-d)\\\\ge 2d$ .\",\n \"This finishes the argument that 2-Approx returns an estimate $D^{\\\\prime }$ with $2\\\\lfloor D/4\\\\rfloor \\\\le D^{\\\\prime }\\\\le D$ .\",\n \"It is not too hard to see that the only time that we might get an estimate that is less than $D/2$ is in the last part of the argument and only if the diameter is of the form $4d+3$ .\",\n \"(We will prove the algorithm guarantees formally soon.)\",\n \"The analysis fails to work in that case because $Y$ is guaranteed to contain only the nodes at distance $d$ from $\\\\bar{t}$ .\",\n \"In particular, if $Y$ contains all nodes at distance $d+1$ from $\\\\bar{t}$ instead of just those at distance at most $d$ , we could consider $a$ to be the node on the shortest path between $\\\\bar{t}$ and $s^*$ with $d(\\\\bar{t},a)=d+1$ , and $a\\\\in Y$ .\",\n \"Now since $d(\\\\bar{t},s^*)\\\\le 2d+1$ (as otherwise we'd be done), $d(a,s_a)\\\\le d(a,s^*)\\\\le 2d+1-d-1 =d$ , so that $d(s_a,t^*)\\\\ge 2d+3$ .\",\n \"Hence everything would work out.\",\n \"We handle this issue with a trick from Chechik et al. [21].\",\n \"First, we make graph have constant degree by blowing up the number of nodes and adding 0 weight edges as follows.\",\n \"Let $v$ be an original node and suppose it has degree $d(v)$ .\",\n \"Replace $v$ with a $d(v)$ -cycle of 0 weight edges so that each of the cycle nodes is connected to a one of the neighbors of $v$ , where each neighbor has a cycle node corresponding to it.\",\n \"This makes every node have degree 3 and increases the number of vertices to $O(m)$ .\",\n \"Now, we run algorithm 2-Approx with two changes.\",\n \"The first is that instead of BFS we use Dijkstra's algorithm because the edges now have weights.\",\n \"The second change is that we redefine $Y$ as follows.\",\n \"Let $Z$ be the closest $\\\\sqrt{m}$ nodes $Z$ to $\\\\bar{t}$ .\",\n \"Define $Y$ to be $Z$ , together with all nodes that have an edge of weight 1 to some node of $Z$ .\",\n \"Now, consider the shortest path $P$ between $\\\\bar{t}$ and $s^*$ .\",\n \"Consider the last node $a$ of $P$ (in the direction from $\\\\bar{t}$ to $s^*$ ) that has distance $\\\\le d$ from $\\\\bar{t}$ .\",\n \"The node $a^{\\\\prime }$ after $a$ on $P$ must be connected to $a$ by an edge $(a,a^{\\\\prime })$ of weight 1, as otherwise $a^{\\\\prime }$ would be at distance at most $d$ from $\\\\bar{t}$ .\",\n \"Since $a\\\\in Z$ , we must have $a^{\\\\prime }\\\\in Y$ and we know also that $d(\\\\bar{t},a^{\\\\prime })=d+1$ .\",\n \"In the last case when $d(\\\\bar{t},s^*)\\\\le 2d+1$ , we get that $Y$ contains a node $a^{\\\\prime }$ of distance $\\\\le (2d+1)-(d+1) = d$ from $s^*$ and hence $d(a^{\\\\prime },s_{a^{\\\\prime }})\\\\le d$ and hence $d(s_{a^{\\\\prime }},t^*)\\\\ge 2d+3$ .\",\n \"Since every node has degree 3, the number of nodes in $Y$ is $\\\\le 4|Z|\\\\le O(\\\\sqrt{m})$ and hence we can afford to run Dijkstra from each of them and complete the algorithm in $\\\\tilde{O}(m^{3/2})$ time.\",\n \"Let us now formally analyze the guarantees of the algorithm.\",\n \"Suppose that $D=4d+z$ where $z\\\\in \\\\lbrace 0,1,2,3\\\\rbrace $ ; we will show that the estimate that the algorithm gives is always at least $\\\\lceil D/2\\\\rceil =2d+\\\\lceil z/2\\\\rceil $ .\",\n \"If some node $x\\\\in X$ has $d(x,t^*)\\\\le d$ , we get that $d(t_x,s^*)\\\\ge D-2d=2d+z\\\\ge \\\\lceil D/2\\\\rceil $ .\",\n \"If we are not done, all nodes of $X$ have $d(x,\\\\bar{t})\\\\ge d(x,t^*)\\\\ge d+1$ and $Z$ contains all nodes at distance $\\\\le d$ from $\\\\bar{t}$ .\",\n \"If $s^*\\\\in Z$ , we are done so we must have $d(s^*,\\\\bar{t})\\\\ge d+1$ .\",\n \"Consider the last node $a^{\\\\prime }$ on the $\\\\bar{t}$ to $s^*$ shortest path (in the direction towards $s^*$ ) for which $d(\\\\bar{t},a^{\\\\prime })\\\\le d$ .\",\n \"We have that $a^{\\\\prime }\\\\in Z$ .\",\n \"Also, the node $a$ after $a^{\\\\prime }$ on the $\\\\bar{t}$ to $s^*$ shortest path must be in $Y$ since by the choice of $a^{\\\\prime }$ , $d(\\\\bar{t},a)=t+1$ and $(a,a^{\\\\prime })$ is an edge of weight 1.\",\n \"If $d(\\\\bar{t},s^*)\\\\ge 2d+\\\\lceil z/2\\\\rceil $ , we are done, so we get that $d(a,s^*)\\\\le 2d+\\\\lceil z/2\\\\rceil -1 - (d+1) =d+\\\\lceil z/2\\\\rceil -2$ .\",\n \"Hence, $d(s_a,t^*)\\\\ge D-2(d+\\\\lceil z/2\\\\rceil -2)=2d + (z+4-2\\\\lceil z/2\\\\rceil )\\\\ge 2d+z$ .\",\n \"It is not hard to extend the $S$ -$T$ Diameter algorithms to work for weighted undirected graphs as well.\",\n \"The basic idea is to use Dijkstra's algorithm instead of BFS in Algorithm 1.\",\n \"This gives an almost 2-approximation.\",\n \"In particular, let $a^{\\\\prime }$ be the last node on the $\\\\bar{t}$ to $s^*$ shortest path that is at distance $\\\\le d$ from $\\\\bar{t}$ and let $a$ be the node after $a^{\\\\prime }$ .\",\n \"The weighted version of Algorithm 1 achieves a guarantee $D^{\\\\prime }$ of the $S$ -$T$ Diameter, such that $D/2-2w(a,a^{\\\\prime })\\\\le D^{\\\\prime }\\\\le D$ .\",\n \"We can obtain an $\\\\tilde{O}(m^{3/2})$ true 2-approximation algorithm similar to the proof above.\",\n \"Let $Z$ be the closest $\\\\sqrt{m}$ nodes to $\\\\bar{t}$ and extend $Z$ to $Y$ by adding all nodes that have a non-zero edge to a node of $Z$ .\",\n \"With this modification, the node $a$ is guaranteed to be in $Y$ and hence the estimate for the diameter is at least $D/2$ .\",\n \"Corollary 26 In $\\\\tilde{O}(m\\\\sqrt{n})$ time one can obtain an estimate $D^{\\\\prime }$ to the $S$ -$T$ diameter $D$ of an $m$ edge $n$ node undirected graph with nonnegative edge weights such that $D/2 - 2w(a,a^{\\\\prime }) \\\\le D^{\\\\prime }\\\\le D$ for some edge $(a,a^{\\\\prime })$ .\",\n \"In $\\\\tilde{O}(m^{3/2})$ time one can obtain an estimate $D^{\\\\prime \\\\prime }$ such that $D/2\\\\le D^{\\\\prime \\\\prime }\\\\le D$ .\",\n \"It is an easy exercise to see that when $D=2h+1$ then the value $\\\\max \\\\lbrace \\\\epsilon ^{in}(v), \\\\epsilon ^{out}(v) \\\\rbrace $ of an arbitrary vertex $v \\\\in V$ is an estimation to the diameter which is at least $h+1$ and at most $D$ .\",\n \"In this section we present a deterministic algorithm that gets a directed unweighted graph $G$ with $D=2h$ and computes in $O(m^2/n)$ time an estimation $\\\\hat{D}$ such that $h+1\\\\le \\\\hat{D} \\\\le D$ .\",\n \"The algorithm works as follows.\",\n \"A variable $\\\\hat{D}$ is set to zero.\",\n \"The algorithm searches for a vertex $v$ that its sum of in and out degree is minimal.\",\n \"Then the algorithm computes the in and out eccentricity of $v$ and every vertex that has an edge with $v$ (incoming or outgoing).\",\n \"The algorithm outputs the maximum of all the in and out eccentricities that were computed.\",\n \"Theorem 27 Let $G=(V,E)$ an unweighted directed graph and let $D=2h$ .\",\n \"Algorithm REF returns in $O(m^2/n)$ time an estimate $\\\\hat{D}$ such that $h+1\\\\le \\\\hat{D} \\\\le D$ .\",\n \"We start with the running time analysis.\",\n \"Consider the graph $G$ and ignore the edge directions.\",\n \"For every $v\\\\in V$ let $deg(v) = deg^{in}(v)+deg^{out}(v)$ .\",\n \"Since $m = \\\\frac{1}{2}\\\\sum _{v\\\\in V} deg(v)$ a vertex $v$ of minimal degree satisfies $deg(v)\\\\le 2m/n$ .\",\n \"Therefore, the cost of computing in and out eccentricities for all vertices in the set $N(v) \\\\cup \\\\lbrace v \\\\rbrace $ is $O(\\\\frac{m}{n} \\\\times m)$ .\",\n \"We now turn to bound $\\\\hat{D}$ .\",\n \"Let $a, b\\\\in V$ and let $d(a,b)=2h$ .\",\n \"Let $P(a,b)$ be a shortest path from $a$ to $b$ and let $v\\\\in P(a,b)$ .\",\n \"If $d(a,v)\\\\le h-1$ then $\\\\epsilon ^{out}(v)\\\\ge h+1$ .\",\n \"Similarly, if $d(v,b)\\\\le h-1$ then $\\\\epsilon ^{in}(v)\\\\ge h+1$ .\",\n \"Consider now the case that $d(a,v)=h$ and $d(v,b)=h$ .\",\n \"Let $u\\\\in P(a,b)$ be the vertex that precedes $v$ on the shortest path from $a$ to $b$ .\",\n \"Since $u$ has an incoming edge to $v$ it follows that $u\\\\in N(v)$ and $\\\\epsilon ^{out}(u)$ and $\\\\epsilon ^{in}(u)$ are computed.\",\n \"Since $d(a,v)=h$ it follows that $d(a,u)=h-1$ , $\\\\epsilon ^{out}(u)\\\\ge h+1$ and $\\\\hat{D}$ is at least $h+1$ .\",\n \"Fast approximation of the diameter [1] Diam-Approx$G$ $\\\\hat{D} = 0$ $v = \\\\arg \\\\min _{x\\\\in V} deg^{in}(x)+deg^{out}(x)$ every $w \\\\in N^{\\\\text{in}}(v) \\\\cup N^{\\\\text{out}}(v) \\\\cup \\\\lbrace v \\\\rbrace $ compute $\\\\epsilon ^{in}(w)$ and $\\\\epsilon ^{out}(w)$ $\\\\hat{D}= \\\\max \\\\lbrace \\\\hat{D},\\\\epsilon ^{in}(w),\\\\epsilon ^{out}(w)\\\\rbrace $ $\\\\hat{D}$ Algorithms for dense graphs Algorithm overview The almost-$3/2$ Diameter approximation algorithm of Aingworth et al.\",\n \"[2] runs in $\\\\tilde{O}(n^2+m\\\\sqrt{n})$ time.\",\n \"Roditty and Vassilevska W. [41] removed the $\\\\tilde{O}(n^2)$ term to obtain an $\\\\tilde{O}(m\\\\sqrt{n})$ expected time almost-$3/2$ approximation algorithm.\",\n \"For every graph with $\\\\Omega (n^{1.5})$ edges the running time of the latter algorithm is not better than the running time of the former algorithm.\",\n \"Therefore, it is interesting to consider the opposite question to the one considered by [41].\",\n \"Can the $\\\\tilde{O}(m\\\\sqrt{n})$ term be removed?\",\n \"We show that this can be done for undirected unweighted graphs and present an $O(n^2 \\\\log n)$ expected time algorithm.\",\n \"For a graph of Diameter $D=3h+z$ , where $z\\\\in [0,1,2]$ our algorithm returns an estimation $\\\\hat{D}$ such that $2h-1\\\\le \\\\hat{D} \\\\le D$ , when $z \\\\in [0,1]$ and $2h\\\\le \\\\hat{D} \\\\le D$ , when $z=2$ .\",\n \"Interestingly, our algorithm is obtained by using ideas developed originally for distance oracles and compact routing schemes.\",\n \"As we are allowed to use quadratic time, we try to estimate the distance between every pair of vertices.\",\n \"To enable this approach we can no longer sample $A$ naively.\",\n \"Instead, we adapt a recursive sampling algorithm to compute $A$ , that was introduced by Thorup and Zwick [47] in the context of compact routing schemes.\",\n \"The expected running time of their algorithm is $\\\\tilde{O}(mn/|A|)$ .\",\n \"We provide a new implementation of their algorithm that runs in expected $\\\\tilde{O}(n(n/|A|)^2)$ time.\",\n \"The set $A$ has the following important property, for every vertex $w\\\\in V$ , its cluster (see [48]) $\\\\lbrace u \\\\mid d(u,w)<d(u,A) \\\\rbrace $ is of size $O(n/ |A|)$ .\",\n \"Consider now a pair of vertices $u$ and $v$ that are in the cluster of $w$ .\",\n \"For any such pair we can efficiently compute their exact distance.\",\n \"Moreover, we show that for all pairs $u,v$ that are not in the same cluster of any vertex, we can bound $d(u,v)$ from below with $d(u,A)+d(v,A)-1$ .\",\n \"This, combined with some other ideas, gives our approximation guarantees.\",\n \"We extend our approach to also provide an almost $5/3$ -approximation for all Eccentricities.\",\n \"The idea of using the bounded clusters of Thorup and Zwick [47] has been used in prior work to obtain improved distance oracles [38], [5], approximate shortest paths [11] and compact routing schemes [4].\",\n \"A simple approach with additive error Let's first consider a simple approach obtaining an $\\\\tilde{O}(n^2)$ time approximation algorithm for Diameter, Eccentricities or $S$ -$T$ Diameter.\",\n \"Suppose that we have an algorithm ALG that can compute in $\\\\tilde{O}(m\\\\sqrt{n})$ time, for any graph $G^{\\\\prime }$ , an estimate $D^{\\\\prime }$ of its Diameter $D$ such that $p\\\\cdot D - q\\\\le D^{\\\\prime }\\\\le D$ , estimates $e(v)$ of $\\\\epsilon (v)$ for all $v$ so that $r\\\\epsilon (v)-s\\\\le e(v)\\\\le \\\\epsilon (v)$ , and an estimate $D^{\\\\prime \\\\prime }$ of the $S$ -$T$ Diameter $D_{S,T}$ so that $t\\\\cdot D_{S,T}-u\\\\le D^{\\\\prime \\\\prime }\\\\le D_{S,T}$ .\",\n \"Now, Dor, Halperin and Zwick [24] showed that in $\\\\tilde{O}(n^2)$ time one can compute for any $n$ node $G$ , an additive 2 spanner $H$ on $\\\\tilde{O}(n^{1.5})$ edges.\",\n \"In fact Knudsen [30] recently showed that in $O(n^2)$ time one can get $H$ on $O(n^{1.5})$ edges (i.e.\",\n \"he removed all logs!).\",\n \"Let's compute $H$ for our given graph and run ALG on $H$ .\",\n \"The runtime is $\\\\tilde{O}(n^{1.5}\\\\cdot \\\\sqrt{n})\\\\le \\\\tilde{O}(n^2)$ since $H$ has $\\\\le O(n^{1.5})$ edges.\",\n \"Let $D^{\\\\prime }_H,e_H(\\\\cdot ),D^{\\\\prime \\\\prime }_H$ be the estimates that we obtain respectively for the Diameter $D_H$ of $H$ , the Eccentricities $\\\\epsilon _H(\\\\cdot )$ of $H$ and the $S,T$ Diameter $D^H_{S,T}$ .\",\n \"Let's return $D^{\\\\prime }_H-2,e_H(\\\\cdot )-2,D^{\\\\prime \\\\prime }_H-2$ for our estimates for the Diameter, Eccentricities and $S$ -$T$ Diameter of $G$ .\",\n \"Notice that $pD-q\\\\le p\\\\cdot D_H - q\\\\le D^{\\\\prime }_H\\\\le D_H\\\\le D+2$ and so $pD-2-q \\\\le D^{\\\\prime }_H-2\\\\le D$ .\",\n \"Similarly since $r\\\\epsilon (v)-s\\\\le r\\\\epsilon _H(v)-s\\\\le e_H(v)\\\\le \\\\epsilon _H(v)\\\\le \\\\epsilon (v)+2$ , we get $r\\\\epsilon (v)-s-2\\\\le e_H(v)-2\\\\le \\\\epsilon (v)$ .\",\n \"Finally since $t\\\\cdot D_{S,T}-u\\\\le t\\\\cdot D^H_{S,T}-u\\\\le D^{\\\\prime \\\\prime }_H\\\\le D^H_{S,T}\\\\le D_{S,T}+2$ , we get $t\\\\cdot D_{S,T}-u-2\\\\le D^{\\\\prime \\\\prime }_H-2\\\\le D_{S,T}$ .\",\n \"Thus, in $\\\\tilde{O}(n^2)$ time we get almost the same guarantees as in the $\\\\tilde{O}(m\\\\sqrt{n})$ time algorithms, except for an extra additive loss of 2 in the quality.\",\n \"Below we show how to make the additive loss in quality smaller for Diameter and Eccentricities.\",\n \"This is especially important when these parameters are constant, which is the hard case of the problems anyway.\",\n \"Near linear almost 3/2-approximation for Diameter Thorup and Zwick [48] introduced distance oracles, a succinct data structure for answering approximate distance queries efficiently.\",\n \"Among the tools they use are clusters and bunches.\",\n \"Let $A\\\\subseteq V$ , let $p_A(u)$ be the closest vertex to $u$ from $A$ , where ties are broken in favor of the vertex with a smaller identifier and let $d(u,A)=d(u,p_A(u))$ .\",\n \"For every $v\\\\in V$ , let $B_A(u)=\\\\lbrace v\\\\in V\\\\mid d(u,v) <d(u,A \\\\rbrace $ be the bunch of $u$ .\",\n \"For every $w\\\\in V\\\\setminus A$ , let $C_A(w)= \\\\lbrace v \\\\mid w\\\\in B_A(v)\\\\rbrace $ be the cluster of $w$ .\",\n \"Thorup and Zwick [48] showed that if a set $A$ is formed by adding every vertex of $V$ to $A$ with probability $p$ then the expected size of $B_A(v)$ is $O(1/p)$ , for every $v\\\\in V$ .\",\n \"They also showed, in the context of compact routing schemes [47], that if the set $A$ is constructed by a recursive sampling algorithm then it is possible to bound the maximum size of a cluster as well.\",\n \"They also showed, in the context of compact routing schemes [47], that if the set $A$ is constructed by a recursive sampling algorithm then it is possible to bound the maximum size of a cluster as well.\",\n \"Their algorithm works as follows.\",\n \"It sets $A$ to the empty set and $W$ to $V$ .\",\n \"Next, as long as the set $W$ is not empty the algorithm samples from $W$ vertices with probability $p$ and adds the sampled vertices to $A$ .\",\n \"The algorithm computes $C_A(w)$ for every $w\\\\in W$ and removes from $W$ all the vertices whose cluster has at most $4/p$ vertices with respect to the updated $A$ .\",\n \"The pseudo-code is given in Algorithm REF .\",\n \"Thorup and Zwick center algorithm [1] center$G,p$ $A= \\\\emptyset $ $W =V$ $W\\\\ne \\\\emptyset $ $X$ - random sample of nodes from $W$ , $|X|=|W|p$ $A= A \\\\cup X$ $W = \\\\lbrace w\\\\in V \\\\mid |C_A(w)|> 4/p \\\\rbrace $ $A$ Thorup and Zwick proved the following Theorem: Theorem 28 (Theorem 3.1 from [47]) The expected size of the set $A$ returned by Algorithm REF is at most $2np \\\\log n$ .\",\n \"For every $w \\\\in V$ we then have $|C_A(w)|\\\\le 4/p$ .\",\n \"Thorup and Zwick claimed that the expected running time of Algorithm REF is $O(mnp \\\\log n)$ .\",\n \"They did not provide the details and refer the reader to [48].\",\n \"However, an educated guess is that they compute clusters for the vertices currently in $W$ in each iteration of the while loop, which results in the claimed running time.\",\n \"The starting point of the Diameter and Eccentricities algorithms presented in this section is an $O(n/p^2 \\\\log n)$ expected time implementation of Algorithm REF .\",\n \"The first idea behind our implementation is that, as opposed to what Thorup and Zwick did, we will compute the bunches and use them to compute the clusters and the set $W$ .\",\n \"This can be done as follows.\",\n \"Once we have computed $B_A(v)$ for every $v\\\\in V$ , we can scan $B_A(v)$ , and for every $w\\\\in B_A(v)$ we can add $v$ to $C_A(w)$ .\",\n \"The cost of this process is $O(|\\\\cup _{v\\\\in V} B_A(v)|)$ and since the clusters are by definition the inverse of the bunches, at the end of this process we have $C_A(w)$ and $|C_A(w)|$ , for every $w\\\\in V$ and we can compute $W$ (as needed in Algorithm REF ).\",\n \"However, in the current implementation only the expected size of a bunch is bounded, and since the Thorup-Zwick bound on the number of iterations is $O(\\\\log n)$ in expectation as well, we cannot apply this idea directly to deduce a good expected running time.\",\n \"To this end, more ideas are needed.\",\n \"The following simple observation helps us to achieve our goal.\",\n \"Observation 29 Let $A_i$ be the set $A$ after updating it in the beginning of the $i$ -th iteration of the while loop in Algorithm REF .\",\n \"Let $A^*$ be a set such that $A^*\\\\subseteq A_i$ , for every $i\\\\ge 1$ .\",\n \"For every $v\\\\in V$ it holds that $B_{A_i}(v) \\\\subseteq B_{A^*}(v)$ .\",\n \"It follows from this observation that we only need to pick the first set $A_1$ such that $|B_{A_1}(v)|\\\\le O(1/p)$ for every $v\\\\in V$ .\",\n \"It is folklore that the $s$ closest vertices $N_{s}(v)$ to a vertex $v$ can be computed in $O(s^2)$ time [24].\",\n \"This implies that we can compute $N_{1/p}(v)$ for every $v\\\\in V$ in $O(n/p^2)$ time.\",\n \"It is not hard to see that, given the sets $N_{1/p}(v)$ of all $v\\\\in V$ , one can (deterministically) compute a “hitting” set $A$ of size $O(np \\\\log n)$ in $O(n+n/p)$ worst case time, so that $N_{1/p}(v)\\\\cap A\\\\ne \\\\emptyset $ for every $v\\\\in V$ (a greedy algorithm works; e.g.\",\n \"see [48]).\",\n \"The second idea behind our implementation is that we first compute the sets $N_{1/p}(v)$ for every $v\\\\in V$ and the hitting set $A$ , as described above.\",\n \"Then, using these sets, we initialize Algorithm REF with a set $A$ such that $|B_A(v)|=O(1/p)$ , for every $v\\\\in V$ .\",\n \"In more detail, our algorithm works as follows.\",\n \"For every $v\\\\in V$ it computes the set $N_{1/p}(v)$ in $O(n/p^2)$ time.\",\n \"Then it finds a set $A$ such that $N_{1/p}(v)\\\\cap A\\\\ne \\\\emptyset $ for every $v\\\\in V$ .\",\n \"Given the hitting set $A$ , it computes $d(v,A)$ and $p_A(v)$ for every $v\\\\in V$ .\",\n \"Using $d(v,p_A(v))$ and $N_{1/p}(v)$ it computes for every $v\\\\in V$ the bunch $B_A(v)$ .\",\n \"Finally, it computes the clusters and $W$ using the bunches as we described above.\",\n \"The rest of the algorithm is almost identical to Algorithm REF .\",\n \"The only difference is that we compute the bunches and use them to compute the clusters and the set $W$ .\",\n \"The pseudo-code is given in Algorithm REF .\",\n \"New implementation of Thorup and Zwick center algorithm [1] center$G,p$ compute $N_{1/p}(v)$ for every $v\\\\in V$ .\",\n \"$A=$ hitting set of the sets $N_{1/p}(v)$ , where $v\\\\in V$ .\",\n \"compute $d(v,A)$ and $p_A(v)$ for every $v\\\\in V$ .\",\n \"compute $B_A(v)$ using $N_{1/p}(v)$ and $d(v,p_A(v))$ .\",\n \"every $u\\\\in V$ compute $C_A(u)$ using $B_A(\\\\cdot )$ $W = \\\\lbrace w\\\\in V \\\\mid |C_A(w)|> 4/p \\\\rbrace $ $W\\\\ne \\\\emptyset $ $X$ - random sample of nodes from $W$ , $|X|=np$ $A= A \\\\cup X$ every $v\\\\in V$ compute $B_A(v)$ every $u\\\\in V$ compute $C_A(u)$ using $B_A(\\\\cdot )$ $W = \\\\lbrace w\\\\in V \\\\mid |C_A(w)|> 4/p \\\\rbrace $ $A$ We show: Lemma 30 Algorithm REF computes in $O(n/p^2 \\\\log n)$ expected time a set $A$ of expected size $O((n/p)\\\\cdot \\\\log n)$ that guarantees for every vertex $w\\\\in V \\\\setminus A$ that $|C_A(w)|=O(1/p)$ , and for every $v\\\\in V$ that $|B_A(v)|=O(1/p)$ .\",\n \"The cost of computing $N_{1/p}(v)$ for every $v\\\\in V$ is $O(n (1/p)^2)$ [24].\",\n \"The cost of computing $A$ is $O(n/p)$ time [48].\",\n \"Computing $d(v,A)$ and $p_A(v)$ for every $v\\\\in V$ in $O(m)$ time is straightforward by running shortest paths tree computation from a dummy vertex that is connected to the set $A$ .\",\n \"To compute $B_A(v)$ using $N_{1/p}(v)$ we only scan $N_{1/p}(v)$ , thus, the total cost is $O(n(1/p))$ .\",\n \"As we explained earlier the cost of computing clusters using bunches is $O(|\\\\cup _{v\\\\in V} B_A(v)|)$ .\",\n \"Since for every $v\\\\in V$ we have $B_A(v)\\\\subseteq N_{1/p}(v)$ the total cost is $O(n(1/p))$ .\",\n \"This completes the analysis of the part that precedes the while loop.\",\n \"Next, we analyze the cost of the while loop.\",\n \"Let $A^*$ be the set $A$ that was computed before the while loop and let $A_i$ be the set $A$ after updating it in the beginning of the $i$ -th iteration of the while loop.\",\n \"From Observation REF it follows that $B_{A_i}(v) \\\\subseteq B_{A^*}(v)$ and therefore in every iteration the cost of computing bunches from scratch is at most $O(n (1/p)^2)$ as $|B_{A^*}(v)|=O(1/p)$ , for every $v\\\\in V$ .\",\n \"One can also compute $B_{A_{i+1}}(v)$ from $B_{A_i}(v)$ by first computing $d(v,A_{i+1})$ and if $d(v,A_{i+1})<d(v,A_i)$ to prune $B_{A_{i+1}}(v)$ accordingly at a smaller cost of $O(n(1/p) + m)$ , however this does not affect the overall complexity.\",\n \"Thorup and Zwick [47] proved that the expected number of iterations is $O(\\\\log n)$ .\",\n \"The fact that the set $A$ from which we start is different does not affect their proofThey prove that in each iteration with probability $1/2$ the size of $W$ decreases by a factor of 2.\",\n \"For this argument they only require that the set $A$ in each iteration will be chosen from $W$ uniformly at random with probability $p$ as we do., therefore there are only $O(\\\\log n)$ iterations in expectation.\",\n \"This implies that a set $A$ of expected size $(np \\\\log n)$ is returned in $O(n/p^2 \\\\log n)$ expected time.\",\n \"The algorithm stops only when there are no large clusters, thus the bound on the cluster size follows.\",\n \"As we mentioned above the algorithm starts with bunches that satisfy the required bound and their size can only decrease afterwards, thus the bound on the bunches follows.\",\n \"almost $3/2$ -Approximation for Diameter [1] $3/2$ -Approx-DiamG $M$ - $n\\\\times n$ matrix whose entries are set to $n$ $A =$ CENTER$(G,1/\\\\sqrt{n})$ every $w\\\\in V$ Step 1 every $\\\\langle u, v\\\\rangle \\\\in C_A(w) \\\\times C_A(w)$ , s.t.\",\n \"$u\\\\ne v\\\\;$ $M(u,v) = \\\\min (M(u,v), d(u,w)+d(v,w))$ every $\\\\langle u, v\\\\rangle \\\\in V \\\\times V$ , s.t.\",\n \"$M(u,v)=n$ Step 2 $M(u,v) = d(u,A)+d(v,A)-1$ $H$ - an additive 2 spanner of $G$ Step 3 every $u \\\\in A$ compute shortest paths tree for $u$ in $H$ and set $\\\\epsilon _H(u)$ , the Eccentricity of $u$ in $H$ $D_1 = \\\\max _{\\\\langle u, v\\\\rangle \\\\in V \\\\times V} M(u,v)$ $D_2 = \\\\max _{u\\\\in A} \\\\epsilon _H(u)$ $\\\\hat{D} = \\\\max (D_1,D_2-2)$ $\\\\hat{D}$ We can now turn to describe the new Diameter algorithm.\",\n \"The algorithm works as follows.\",\n \"All entries of an $n \\\\times n$ matrix $M$ are set to $n$ .\",\n \"A set $A$ of centers is computed using the algorithm of Thorup and Zwick [47].\",\n \"For every vertex $w\\\\in V$ and every pair $\\\\langle u, v\\\\rangle \\\\in C_A(w)\\\\times C_A(w)$ the algorithm sets $M(u,v)$ to $\\\\min (M(u,v), d(u,w)+d(v,w))$ (Step 1).\",\n \"Next, the algorithm searches the matrix $M$ for entries whose value is still $n$ .\",\n \"Given a pair $\\\\langle u, v\\\\rangle \\\\in V \\\\times V$ for which $M(u,v)=n$ the algorithm sets $M(u,v)$ to $d(u,A)+d(v,A)-1$ (Step 2).\",\n \"Finally, the algorithm computes an additive 2 spanner $H$ of the input graph $G$ and for every $u\\\\in A$ it computes $\\\\epsilon _H(u)$ , the Eccentricity of $u$ in $H$ (Step 3).\",\n \"The algorithm outputs the maximum between $D_1$ and $D_2-2$ , where $D_1$ is $\\\\max _{\\\\langle u, v\\\\rangle \\\\in V \\\\times V} M(u,v)$ and $D_2$ is $\\\\max _{u\\\\in A} \\\\epsilon _H(u)$ .\",\n \"Next, we bound the value returned by Algorithm REF .\",\n \"Theorem 31 Let $D=3h+z$ , where $z\\\\in [0,1,2]$ .\",\n \"The value $\\\\hat{D}$ returned by Algorithm REF satisfies: $\\\\begin{array}{ll}2h-1 & \\\\mbox{if } z \\\\in [0,1] \\\\\\\\2h & \\\\mbox{if } z=2\\\\end{array}\\\\le \\\\hat{D} \\\\le D$ We start with the following Lemma: Lemma 32 Let $u,v\\\\in V$ and let $P(u,v)$ be a shortest path between $u$ and $v$ .\",\n \"If $B_A(u) \\\\cap B_A(v)\\\\ne \\\\emptyset $ then $(B_A(u) \\\\cap B_A(v)) \\\\cap P(u,v) \\\\ne \\\\emptyset $ .\",\n \"If $v\\\\in B_A(u)$ then the claim trivially holds so we can assume that $v\\\\notin B_A(u)$ .\",\n \"Let $w$ be the vertex farthest from $u$ that is in $B_A(u) \\\\cap P(u,v)$ .\",\n \"From the definition of $w$ it follows that $d(u,w)=d(u,A)-1$ .\",\n \"Assume, towards a contradiction, that $(B_A(u) \\\\cap B_A(v)) \\\\cap P(u,v) = \\\\emptyset $ .\",\n \"This implies that $w\\\\notin B_A(v)$ and $d(v,A)-1<d(v,w)=d(u,v)-d(u,w)$ .\",\n \"However, since $B_A(u) \\\\cap B_A(v)\\\\ne \\\\emptyset $ there is a vertex $w^{\\\\prime }$ such that $d(u,w^{\\\\prime })\\\\le d(u,w)$ and $d(v,w^{\\\\prime })\\\\le d(v,A)-1 < d(u,v)-d(u,w)$ .\",\n \"This implies that $d(u,w^{\\\\prime })+d(v,w^{\\\\prime })< d(u,v)$ , a contradiction to the triangle inequality.\",\n \"Lemma 33 Let $u,v\\\\in V$ .\",\n \"If $B_A(u) \\\\cap B_A(v)= \\\\emptyset $ then $d(u,A)+d(v,A)-1 \\\\le d(u,v)$ .\",\n \"Notice first that $B_A(u)$ (resp., $B_A(v)$ ) contains all the vertices at distance $d(u,A)-1$ (resp., $d(v,A)-1$ ).\",\n \"Let $P(u,v)$ be a shortest path between $u$ and $v$ .\",\n \"Let $w$ be the vertex farthest from $u$ on $P(u,v)$ that is also in $B_A(u)$ .\",\n \"Similarly, let $w^{\\\\prime }$ be the vertex farthest from $v$ on $P(u,v)$ that is also in $B_A(v)$ .\",\n \"Since $B_A(u) \\\\cap B_A(v)= \\\\emptyset $ it holds that $w\\\\ne w^{\\\\prime }$ .\",\n \"Therefore: $d(u,v) = d(u,A)-1+d(v,A)-1+d(w,w^{\\\\prime })\\\\ge d(u,A)+d(v,A)-1.$ Let $a$ and $b$ be the Diameter endpoints, that is $d(a,b)=D=3h+z$ , where $z\\\\in [0,1,2]$ .\",\n \"Let $P(a,b)$ be a shortest path between $a$ and $b$ .\",\n \"Assume first that $B_A(a) \\\\cap B_A(b)\\\\ne \\\\emptyset $ .\",\n \"It follows from Lemma REF that there is a vertex $w\\\\in P(a,b)$ such that $\\\\langle a, b\\\\rangle \\\\in C(w) \\\\times C(w)$ .\",\n \"Therefore, $M(a,b)=D$ after Step 1.\",\n \"After the update in Step 2 it follows from Lemma REF that $M(u,v)\\\\le d(u,v)$ for every $u ,v\\\\in V$ .\",\n \"Therefore, the maximum value in the matrix is $d(a,b)$ and $D_1=D$ .\",\n \"Let $x = \\\\operatornamewithlimits{arg\\\\,max}_{y\\\\in A} \\\\epsilon _H(y)$ .\",\n \"Since $H$ is an additive 2 spanner it holds that $\\\\epsilon _H(x)\\\\le D+2$ , hence, we have $D_2 \\\\le D$ and the algorithm returns the exact value of the Diameter.\",\n \"Assume now that $B_A(a) \\\\cap B_A(b) = \\\\emptyset $ .\",\n \"From the discussion of the previous case it follows that in this case $\\\\hat{D}\\\\le D$ as well.\",\n \"Thus, it is only left to prove the lower bound.\",\n \"Assume that $z\\\\in [0,1]$ .\",\n \"Consider first the case that $d(a,A)\\\\ge h$ and $d(b,A)\\\\ge h$ then from Lemma REF it follows that $M(a,b)\\\\ge 2h-1$ after Step 2 and $D_1$ is at least $2h-1$ .\",\n \"If this is not the case then either $d(a,A)< h$ or $d(b,A)< h$ (or both).\",\n \"Assume, wlog, that $d(a,A)< h$ .\",\n \"In this case the Eccentricity in $H$ of at least one vertex from $A$ is at least $2h+1$ and hence $D_2-2$ is at least $2h-1$ .\",\n \"Assume now that $z=2$ .\",\n \"If either $d(a,A)\\\\ge h$ and $d(b,A)> h$ or $d(a,A)>h$ and $d(b,A)\\\\ge h$ then from Lemma REF it follows that $M(a,b)\\\\ge 2h$ after Step 2 and $D_1$ is at least $2h$ .\",\n \"If this is not the case then either $d(a,A)\\\\le h$ or $d(b,A)\\\\le h$ (or both).\",\n \"Assume, wlog, that $d(a,A)\\\\le h$ .\",\n \"In this case the Eccentricity in $H$ of at least one vertex from $A$ is at least $2h+2$ and hence $D_2-2$ is at least $2h$ .\",\n \"We now turn to we analyze the running time of Algorithm REF .\",\n \"Theorem 34 The expected running time of Algorithm REF is $O(n^2 \\\\log n)$ .\",\n \"The set $A$ is computed by the center algorithm presented in Algorithm REF with $p=1/\\\\sqrt{n}$ .\",\n \"From Lemma REF it follows that the size of the set $A$ is $O(\\\\sqrt{n} \\\\log n)$ and its construction time is $O(n^2\\\\log n)$ in expectation.\",\n \"For every $w\\\\in V$ the size of $C_A(w)$ is $O(\\\\sqrt{n})$ .\",\n \"Therefore, Step 1 takes $O(n \\\\times |C_A(w)|^2)=O(n^2)$ .\",\n \"Step 2 takes $O(n^2)$ time as well.\",\n \"In Step 3 we first compute an additive 2 spanner $H$ on $O(n^{1.5})$ edges.\",\n \"Knudsen [30], following Dor, Halperin and Zwick [24] showed how to do this in $O(n^2)$ time.\",\n \"We also compute $|A|$ shortest paths trees in $H$ .\",\n \"As $H$ has $O(n^{1.5})$ edges, this step takes $O(n^2 \\\\log n)$ time.\",\n \"Near linear almost 5/3-approximation for Eccentricities almost $5/3$ -Approximation for all Eccentricities [1] $5/3$ -Approx-EccG Run lines 2-11 of Algorithm REF , with $H$ augmented with shortest paths trees for $B_A(u) \\\\cup \\\\lbrace p(u) \\\\rbrace $ for every $u\\\\in V$ every $u \\\\in V$ $\\\\epsilon _1(u) = \\\\max _{v\\\\in V} M(u,v)$ $\\\\epsilon _2(u) = \\\\epsilon _H(p(u))-d(u,p(u))-2$ $\\\\epsilon _3(u) = d_H(u,y)-2$ , where $y=\\\\operatornamewithlimits{arg\\\\,max}_{x\\\\in A} d_H(u,x)$ $\\\\epsilon ^{\\\\prime }(u) = \\\\max (\\\\epsilon _1(u),\\\\epsilon _2(u),\\\\epsilon _3(u))$ Next, we show how to update Algorithm REF to obtain an almost $5/3$ approximation for all Eccentricities.\",\n \"We run lines 2-11 of Algorithm REF .\",\n \"The only difference is that $H$ is augmented with the edges of the shortest paths tree that span the set $B_A(u) \\\\cup \\\\lbrace p(u) \\\\rbrace $ for every $u\\\\in V$ .\",\n \"Then, for every $u\\\\in V$ we compute $\\\\epsilon _1(u)$ , $\\\\epsilon _2(u)$ and $\\\\epsilon _3(u)$ , which are defined as follows: $\\\\epsilon _1(u) = \\\\max _{v\\\\in V} M(u,v)$ , $\\\\epsilon _2(u) = \\\\epsilon _H(p(u))-d(u,p(u))-2$ and $\\\\epsilon _3(u) = d_H(u,y)-2$ , where $y=\\\\operatornamewithlimits{arg\\\\,max}_{x\\\\in A} d_H(u,x)$ .\",\n \"The algorithm sets $\\\\epsilon ^{\\\\prime }(u)$ to $\\\\max \\\\lbrace \\\\epsilon _1(u), \\\\epsilon _2(u), \\\\epsilon _3(u) \\\\rbrace $ for every $u\\\\in V$ as an estimation to $\\\\epsilon (u)$ .\",\n \"The pseudo-code is given in Algorithm REF .\",\n \"We now prove: Theorem 35 For every $u\\\\in V$ , Algorithm REF computes in $O(n^2 \\\\log n)$ expected time a value $\\\\epsilon ^{\\\\prime }(u)$ that satisfies: $\\\\frac{3\\\\epsilon (u)}{5}-1 \\\\le \\\\epsilon ^{\\\\prime }(u)\\\\le \\\\epsilon (u).$ We start by analyzing the running time.\",\n \"Lines 2-11 of the algorithm are the same as Algorithm REF , with one difference, the spanner $H$ is augmented with the edges of the shortest paths tree that span the set $B_A(u) \\\\cup \\\\lbrace p(u) \\\\rbrace $ for every $u\\\\in V$ .\",\n \"This adds at most $O(n^{1.5})$ edges to $H$ and hence the cost of these lines remain $O(n^2 \\\\log n)$ time in expectation.\",\n \"The computation of $\\\\epsilon _1(u)$ , $\\\\epsilon _2(u)$ and $\\\\epsilon _3(u)$ for every $u\\\\in V$ costs $O(n^2)$ time in total.\",\n \"Let $u\\\\in V$ be an arbitrary vertex and let $\\\\epsilon (u)=d(u,t)$ .\",\n \"We now turn to bound $\\\\epsilon ^{\\\\prime }(u)$ .\",\n \"In our analysis we will use the following simple observation: Observation 36 In an undirected graph it holds for every $u,v \\\\in V$ that $\\\\epsilon (u)\\\\ge \\\\epsilon (v)-d(u,v)$ .\",\n \"It is straightforward to see that both $\\\\epsilon _2(u)$ and $\\\\epsilon _3(u)$ are at most $\\\\epsilon (u)$ .\",\n \"Recall that $\\\\epsilon _2(u) = \\\\epsilon _H(p(u))-d(u,p(u))-2\\\\le \\\\epsilon (p(u))-d(u,p(u)\\\\le \\\\epsilon (u)$ and $\\\\epsilon _3(u) = d_H(u,y)-2\\\\le d(u,y)\\\\le \\\\epsilon (u)$ .\",\n \"We distinguish between two cases.\",\n \"Case 1: $B_A(u) \\\\cap B_A(t)\\\\ne \\\\emptyset $ .\",\n \"It follows from Lemma REF that $P(u,t)\\\\cap (B_A(u) \\\\cap B_A(t)) \\\\ne \\\\emptyset $ and $M(u,t)= \\\\epsilon (u)$ .\",\n \"From Lemma REF it follows that $M(u,w)\\\\le d(u,w)$ for every $w\\\\in V$ after Step 2.\",\n \"Therefore, $\\\\epsilon _1(u)=\\\\epsilon (u)$ .\",\n \"Since $\\\\epsilon _2(u)\\\\le \\\\epsilon (u)$ and $\\\\epsilon _3(u)\\\\le \\\\epsilon (u)$ we get that $\\\\epsilon ^{\\\\prime }(u)=\\\\epsilon (u)$ .\",\n \"Case 2: $B_A(u) \\\\cap B_A(t) = \\\\emptyset $ .\",\n \"Consider first the case that $d(u,p(u))\\\\le \\\\frac{\\\\epsilon (u)}{5}-1$ .\",\n \"From Observation REF we get that $\\\\epsilon _H(p(u))\\\\ge \\\\epsilon _H(u)-d_H(u,p(u))$ .\",\n \"As we augmented $H$ with a shortest paths tree that spans $B_A(u) \\\\cup \\\\lbrace p(u) \\\\rbrace $ we have $d(u,p(u))=d_H(u,p(u))$ and we get $\\\\epsilon _H(p(u))\\\\ge \\\\epsilon _H(u)-d(u,p(u))$ .\",\n \"Hence, we get that $\\\\epsilon _2(u) = \\\\epsilon _H(p(u))-d(u,p(u))-2 \\\\ge \\\\epsilon _H(u)-2d(u,p(u))-2\\\\ge \\\\epsilon (u)-2d(u,p(u))-2$ .\",\n \"As before we have $\\\\epsilon _2(u) \\\\le \\\\epsilon (u)$ .\",\n \"Using $d(u,p(u))\\\\le \\\\frac{\\\\epsilon (u)}{5}-1$ we get that: $\\\\epsilon _2(u) \\\\ge \\\\epsilon (u)-\\\\frac{2\\\\epsilon (u)}{5}\\\\ge \\\\frac{3\\\\epsilon (u)}{5}.$ Assume now that $d(u,p(u))\\\\ge \\\\frac{\\\\epsilon (u)}{5}$ .\",\n \"This means that $d(u,A)-1\\\\ge \\\\frac{\\\\epsilon (u)}{5}-1$ .\",\n \"Let $S$ be the set of all vertices $v\\\\in V$ such that $B_A(u) \\\\cap B_A(v)= \\\\emptyset $ , that is, $S=V \\\\setminus \\\\cup _{w\\\\in B_A(u)} C_A(w)$ .\",\n \"Let $t^{\\\\prime } = \\\\operatornamewithlimits{arg\\\\,max}_{x\\\\in S} d(x,A)-1$ .\",\n \"If $d(t^{\\\\prime },A)-1 \\\\ge \\\\frac{2\\\\epsilon (u)}{5}-1$ we get from Lemma REF that $M(u,t^{\\\\prime })\\\\ge \\\\frac{3\\\\epsilon (u)}{5}-1$ .\",\n \"Assume now that $d(t^{\\\\prime },A) < \\\\frac{2\\\\epsilon (u)}{5}$ .\",\n \"As $t^{\\\\prime }$ is the farthest vertex from $A$ we get that $d(t,p(t))<\\\\frac{2\\\\epsilon (u)}{5}$ and $d(u,p(t))>\\\\frac{3\\\\epsilon (u)}{5}$ .\",\n \"Therefore, $\\\\epsilon _3(u) = d_H(u,y)-2\\\\ge d(u,p(t))-2\\\\ge \\\\frac{3\\\\epsilon (u)}{5}-1$ .\",\n \"From Lemma REF and Lemma REF it follows that $\\\\epsilon _1(u)\\\\le \\\\epsilon (u)$ and the bound follows.\",\n \"Algorithms for dense graphs using matrix multiplication Here we will give $O(n^{2.05})$ time approximation algorithms for Diameter and Eccentricities in dense unweighted undirected graphs.\",\n \"The approximation guarantees of these algorithms are slightly better than those in our $O(n^2\\\\log n)$ time algorithm.\",\n \"In fact, the guarantees are exactly the same as in the $\\\\tilde{O}(m\\\\sqrt{n})$ time algorithms for Diameter and Eccentricities of Roditty and V. Williams [41] and Cairo et al. [15].\",\n \"To achieve this, we give an efficient implementation using fast matrix multiplication of the $\\\\tilde{O}(m\\\\sqrt{n})$ time algorithms of [15] and [41].\",\n \"The main overhead of the $\\\\tilde{O}(m\\\\sqrt{n})$ time algorithms [15], [41] is in computing the distances from a set $S=W\\\\cup \\\\lbrace w\\\\rbrace \\\\cup T$ of $O(\\\\sqrt{n}\\\\log n)$ nodes: the set $S$ itself can be computed in linear time using random sampling to form a set $W$ , BFS from a dummy node to find the node $w$ farthest from $W$ and then BFS from $w$ to find the set $T$ of closest $\\\\sqrt{n}$ nodes to $w$ .\",\n \"After one knows all distances from every $s\\\\in S$ to every $v\\\\in V$ , it takes linear time to output the Diameter and Eccentricity estimates.\",\n \"The main idea of our algorithms is as follows.\",\n \"If the Diameter is of size $\\\\le O(\\\\log n)$ , then one does not need all distances between $S$ and $V$ , but only those that are $O(\\\\log n)$ .\",\n \"Small distances are easy to compute with matrix multiplication.\",\n \"Let $A$ be the adjacency matrix and $A_S$ be its submatrix formed by just the rows in $S$ .\",\n \"Then we can find the distances for all pairs in $S\\\\times V$ at distance $\\\\le t$ by computing $A_S\\\\times A^{t-1}$ , which can be computed by performing $t-1$ matrix products of dimension $|S|\\\\times n$ by $n\\\\times n$ , and this can be accomplished in $O(tn^{2.05})$ time [26], [31].\",\n \"If on the other hand the Diameter is $D\\\\ge 100\\\\log n$ , then one can use an $\\\\tilde{O}(n^2)$ time algorithm by Dor et al.\",\n \"[24] to compute estimates of all pairwise distances with an additive error at most $4\\\\log n$ .\",\n \"The maximum distance estimate computed, minus $4\\\\log n$ , will be between $0.96 D$ and $D$ , giving a really good approximation already.\",\n \"A similar argument works for Eccentricities, and also for $S$ -$T$ Diameter.\",\n \"Below we recap the guarentees of the $\\\\tilde{O}(m\\\\sqrt{n})$ time approximation algorithms of [15], [41].\",\n \"Theorem 37 ([15], [41]) The following can be computed in $\\\\tilde{O}(m\\\\sqrt{n})$ time: an estimate $\\\\hat{D}$ of the graph Diameter $D$ , such that $\\\\frac{2}{3} D-\\\\frac{1}{3}\\\\le \\\\hat{D}\\\\le D$ , for every node $v$ , an estimate $e(v)$ of its Eccentricity $\\\\epsilon (v)$ , such that $\\\\frac{3}{5} \\\\epsilon (v)-\\\\frac{1}{5}\\\\le e(v)\\\\le \\\\epsilon (v)$ .\",\n \"Using Seidel's algorithm [43] we can compute all the distances exactly, and hence the above parameters as well, all in $O(n^\\\\omega )$ time for $\\\\omega < 2.373$ .\",\n \"We will show that for dense graphs, we can obtain the same approximation guarantees as in Theorem REF , in time $O(n^{2.05})$ .\",\n \"Let us compare to our $O(n^2\\\\log n)$ time algorithms.\",\n \"For Diameter $D=3h+z$ , the $O(n^2\\\\log n)$ time algorithm returns an estimate $2h-1$ when $z=0,1$ and $2h$ when $z=2$ .\",\n \"The estimate $\\\\hat{D}$ here is $\\\\ge (2D-1)/3= 2h+ (2z-1)/3$ , which is $\\\\ge 2h$ when $z=0$ and $\\\\ge 2h+1$ when $z=1,2$ .\",\n \"For Eccentricities, the $O(n^2\\\\log n)$ time algorithm returns estimates $e(v)\\\\ge 3\\\\epsilon (v)/5 -1$ , and here we return a better estimate $e(v)\\\\ge (3\\\\epsilon -1)/5$ .\",\n \"We will rely on two known algorithms.\",\n \"The first is from a paper by Dor, Halperin and Zwick [24] on additive approximations of All-Pairs Shortest Paths (APSP).\",\n \"Among many other results, [24] show that in $\\\\tilde{O}(n^2)$ time, one can compute for all pairs of vertices $u,v$ , an estimate $d^{\\\\prime }(u,v)$ of their distance $d(u,v)$ so that $d(u,v)\\\\le d^{\\\\prime }(u,v)\\\\le d(u,v)+a\\\\log n$ for an explicit constant $a\\\\le 4$ .\",\n \"The second is an algorithm for the following truncated multi-source shortest paths problem: given an integer $Q$ , a graph $G=(V,E)$ and a set $S$ , compute the distances $d(s,v)$ for every $s\\\\in S$ and $v\\\\in V$ for which $d(s,v)\\\\le Q$ .\",\n \"The algorithm uses fast matrix multiplication and is quite straightforward.\",\n \"Let $A$ be the $n\\\\times n$ Boolean matrix with rows and columns indexed by $V$ , so that $A[u,v]=1$ if there is an edge between $u$ and $v$ or $u=v$ , and $A[u,v]=0$ otherwise; i.e.\",\n \"$A$ is the adjacency matrix added to the identity matrix.\",\n \"Let $A_S$ be the $|S|\\\\times n$ submatrix of $A$ consisting of the rows indexed by nodes of $S$ .\",\n \"For an integer $i\\\\ge 1$ , let $A^i$ be the $i$ -th power of $A$ under the Boolean matrix product.\",\n \"Here, $A^i[u,v]=1$ if and only if the distance between $u$ and $v$ is at most $i$ .\",\n \"Define $A^0$ as the identity matrix.\",\n \"Consider $A_S \\\\cdot A^i$ for any choice of $i\\\\ge 0$ (under the Boolean matrix product).\",\n \"Here, $(A_S \\\\cdot A^i)[s,v]=1$ if and only if the distance between $s$ and $v$ is at most $i+1$ .\",\n \"Thus, if we compute $D_i:=A_S\\\\cdot A^i$ for every $1\\\\le i< Q$ , we would know the distance from every $s\\\\in S$ to every $v\\\\in V$ , whenever this distance is at most $Q$ .\",\n \"Computing these matrix products can easily be done by performing the following $Q-1$ Boolean products of an $|S|\\\\times n$ matrix by an $n\\\\times n$ matrix: let $D_0=A_S$ ; then for each $i$ from 1 to $t-1$ , compute $D_i:=D_{i-1}\\\\cdot A$ .\",\n \"Thus, the running time is $O(Q\\\\cdot M(|S|,n,n))$ where $M(|S|,n,n)$ is the runtime of multiplying an $|S|\\\\times n$ matrix by an $n\\\\times n$ matrix.\",\n \"Armed with these two algorithms, let us recap Roditty et al.\",\n \"'s (and Cairo et al.\",\n \"'s) approximation algorithm and see how to modify it.\",\n \"The algorithm proceeds as follows: Let $D$ , $R$ and $\\\\epsilon (v)$ denote the Diameter and Radius of $G$ and the Eccentricity of node $v$ , respectively.\",\n \"RV/CGR Algorithm [1] Using BFS (see [41] and [15]), in $O(m+n)$ time compute $W,w,T$ , where $W\\\\subseteq V$ is a uniformly chosen subset of size $O(\\\\sqrt{n} \\\\log n)$ , $w$ is the furthest node from $W$ and $T$ are the closest $\\\\sqrt{n}$ nodes to $w$ .\",\n \"Let $S=\\\\lbrace w\\\\rbrace \\\\cup W\\\\cup T$ .\",\n \"For every $s\\\\in S$ and every $v\\\\in V$ , compute the distance $d(s,v)$ between $s$ and $v$ ; set $\\\\epsilon (s)=\\\\max _v d(s,v)$ .\",\n \"Set $\\\\tilde{D}=\\\\max _{x\\\\in S} \\\\epsilon (x)$ .\",\n \"Set for every $v\\\\in V$ , $\\\\tilde{\\\\epsilon }(v)=\\\\max \\\\lbrace d(w,v),\\\\max _{x\\\\in W} d(x,v), \\\\max _{x\\\\in T} (\\\\epsilon (x)-d(x,v))\\\\rbrace $ .\",\n \"The runtime bottleneck in the above algorithm is step (2) which runs in $\\\\tilde{O}(mn^{1/2})$ time if one uses BFS through each node of $S$ .\",\n \"Let us describe how to modify the algorithm.\",\n \"We will replace (2) with a truncated distance computation and also use the algorithm of Dor, Halperin and Zwick to handle large distances that we might have ignored in the truncated computation.\",\n \"Our Modified Approximation.\",\n \"[1] FasterApproximation First part: Handle Large Distances: [t]@X@ Use Dor, Halperin and Zwick's algorithm to compute distance estimates $d^{\\\\prime }(\\\\cdot ,\\\\cdot )$ so that for every $u,v\\\\in V$ , $d(u,v)\\\\le d^{\\\\prime }(u,v)\\\\le d(u,v)+a\\\\log n$ .\",\n \"Let $X=3a\\\\log n$ .\",\n \"Set $\\\\tilde{D}_1 = \\\\max _{u,v\\\\in V} d^{\\\\prime }(u,v) - a\\\\log n$ .\",\n \"For every $v\\\\in V$ , set $\\\\tilde{\\\\epsilon }_1(v) = \\\\max _{u} d^{\\\\prime }(u,v) - a\\\\log n$ .\",\n \"Second Part: Handle Small Distances: [t]@X@ Using BFS (see [41] and [15]), in $O(m+n)$ time compute $W,w,T$ , where $W\\\\subseteq V$ is a uniformly chosen subset of size $O(\\\\sqrt{n} \\\\log n)$ , $w$ is the furthest node from $W$ and $T$ are the closest $\\\\sqrt{n}$ nodes to $w$ .\",\n \"Let $S=\\\\lbrace w\\\\rbrace \\\\cup W\\\\cup T$ .\",\n \"[t]@X@ Let $Q=2(X+a\\\\log n)=8a\\\\log n$ .\",\n \"For every $s\\\\in S$ and every $v\\\\in V$ whose distance $d(s,v)$ is at most $Q$ , compute $d(s,v)$ .\",\n \"Let $d_{\\\\le }(s,v)$ denote $d(s,v)$ if we have computed it, and $\\\\infty $ otherwise.\",\n \"Set $\\\\epsilon _{\\\\le }(s)=\\\\max _v d_{\\\\le }(s,v)$ .\",\n \"Set $\\\\tilde{D}_2=\\\\max _{x\\\\in S} \\\\epsilon _{\\\\le }(x)$ .\",\n \"[t]@X@ $\\\\forall v\\\\in V$ , set $\\\\tilde{\\\\epsilon }_2(v)=\\\\max \\\\lbrace d_\\\\le (w,v), \\\\max _{x\\\\in W} d_\\\\le (x,v), \\\\max _{y\\\\in T} (\\\\epsilon _\\\\le (y)-d_\\\\le (y,v))\\\\rbrace $ .\",\n \"Third Part: Set $\\\\tilde{D},\\\\tilde{\\\\epsilon }(\\\\cdot )$: If $\\\\tilde{D}_1\\\\ge X$ , set $\\\\tilde{D}=\\\\tilde{D}_1$ , and otherwise set $\\\\tilde{D}=\\\\tilde{D}_2$ .\",\n \"[t]@X@ For every $v\\\\in V$ , if there exists some $x\\\\in S$ such that $d^{\\\\prime }(x,v)\\\\ge X+a\\\\log n$ , then set $\\\\tilde{\\\\epsilon }(v)=\\\\tilde{\\\\epsilon }_1(v)$ , and otherwise $\\\\tilde{\\\\epsilon }(v)=\\\\tilde{\\\\epsilon }_2(v)$ .\",\n \"Consider our modified algorithm, FasterApproximation.\",\n \"Now we will prove several claims.\",\n \"Claim 38 The running time of algorithm FasterApproximation is $\\\\tilde{O}(M(\\\\sqrt{n},n,n))$ .\",\n \"The Dor, Halperin, Zwick part of the algorithm (Step 3) runs in $\\\\tilde{O}(n^2)$ time.\",\n \"Step 8 runs in $O((X+a\\\\log n)\\\\cdot M(|S|,n,n))$ time where $S=\\\\lbrace w\\\\rbrace \\\\cup W\\\\cup T$ , using the iterated rectangular matrix product algorithm.\",\n \"Recall that $X+a\\\\log n = O(\\\\log n)$ .\",\n \"Thus Step 8 runs in $\\\\tilde{O}(M(|S|,n,n))$ time.\",\n \"Since $|S|=\\\\tilde{O}(\\\\sqrt{n})$ and we can partition an $|S|\\\\times n\\\\times n$ matrix product into $ \\\\text{\\\\rm polylog~} n$ , $n^{1/2}\\\\times n\\\\times n$ matrix products, the runtime of the step is $\\\\tilde{O}(M(n^{1/(2)},n,n))$ .\",\n \"Steps 10 and 13 run in $O(n|S|)<\\\\tilde{O}(n^2)$ time.\",\n \"The rest of the steps run in linear time.\",\n \"Since $M(n^{1/2},n,n)\\\\ge n^2$ (one must at least read the input), the total running time is $\\\\tilde{O}(M(n^{1/2},n,n))$ .\",\n \"Claim 39 $\\\\frac{2D-1}{3} \\\\le \\\\tilde{D}\\\\le D$ .\",\n \"Suppose that $\\\\tilde{D}_1\\\\ge X$ .\",\n \"The algorithm returns $\\\\tilde{D}=\\\\tilde{D}_1=\\\\max _{u,v} d^{\\\\prime }(u,v)-a\\\\log n$ .\",\n \"By the guarantee on $d^{\\\\prime }$ , we have $D-a\\\\log n \\\\le \\\\tilde{D}_1\\\\le D$ .\",\n \"Hence $\\\\tilde{D}\\\\ge D(1-(a\\\\log n)/D)\\\\ge D(1-(a\\\\log n)/X) =2D/3 \\\\ge (2D-1)/3$ .\",\n \"Suppose now that $\\\\tilde{D}_1< X$ .\",\n \"This means that $D<X+a\\\\log n$ and every distance in the graph is $\\\\le X+a\\\\log n$ .\",\n \"In the second part of the algorithm we set $Q=2(X+a\\\\log n)$ , and hence every distance is computed exactly: for every $s\\\\in S$ , $v\\\\in V$ , $d_\\\\le (s,v)=d(s,v)$ .\",\n \"Hence the second part of the algorithm will be identical to the RV/CGR algorithm and hence we get the same guarantees: $(2D-1)/3\\\\le \\\\tilde{D}\\\\le D$ .\",\n \"Claim 40 For every node $v$ , $\\\\frac{3\\\\epsilon (v)-1}{5} \\\\epsilon (v)\\\\le \\\\tilde{\\\\epsilon }(v)\\\\le \\\\epsilon (v)$ .\",\n \"Fix $v$ .\",\n \"Suppose first that there exists some $x$ such that $d^{\\\\prime }(x,v)\\\\ge X+a\\\\log n$ .\",\n \"Then $\\\\epsilon (v)\\\\ge \\\\tilde{\\\\epsilon }_1(v)=\\\\max _u d^{\\\\prime }(u,v) - a\\\\log n \\\\ge \\\\epsilon (v)-a\\\\log n = \\\\epsilon (v)(1-a\\\\log n/\\\\epsilon (v))$ .\",\n \"Since $\\\\epsilon (v)\\\\ge d(x,v)\\\\ge d^{\\\\prime }(x,v)-a\\\\log n \\\\ge X$ , we get that $\\\\tilde{\\\\epsilon }_1(v)\\\\ge \\\\epsilon (v)(1-a\\\\log n / X) = 2\\\\epsilon (v)/3$ .\",\n \"Now suppose that for all $x\\\\in V$ , $d^{\\\\prime }(x,v)<X+a\\\\log n$ .\",\n \"Then, also for all $x\\\\in V$ , $d(x,v)<X+a\\\\log n$ and $\\\\epsilon (v)<X+a\\\\log n$ .\",\n \"Consider all the quantities needed in the second part of the algorithm to compute $\\\\tilde{\\\\epsilon }_2(v)$ : $d_\\\\le (w,v)$ : since $\\\\forall x\\\\in V$ , $d(x,v)<X+a\\\\log n$ , $d_\\\\le (w_i,v)=d(w_i,v)$ for each $w_i$ ; $d_\\\\le (x,v)$ for every $x\\\\in W$ : as above, $d_\\\\le (x,v)=d(x,v)$ ; $\\\\epsilon _\\\\le (x)-d_\\\\le (x,v)$ for all $x\\\\in T$ : here, $\\\\epsilon (x)\\\\le \\\\epsilon (v)+d(x,v)\\\\le 2\\\\epsilon (v)<2(X+a\\\\log n)$ .\",\n \"Since we compute all distances from nodes in $S$ up to $2(X+a\\\\log n)$ and $x\\\\in S$ , $\\\\epsilon _\\\\le (x)=\\\\epsilon (x)$ .\",\n \"Also as in the above bullets, $d_\\\\le (x,v)=d(x,v)$ .\",\n \"Thus all the quantities needed are the correct ones and $\\\\tilde{\\\\epsilon }(v)=\\\\tilde{\\\\epsilon }_2(v)$ inherits the same guarantees as in the algorithm by Cairo et al.\",\n \"From Le Gall and Urrutia [26] (see also, [31]) we obtain that $M(\\\\sqrt{n},n,n)\\\\le O(n^{2.044183})$ .\",\n \"We obtain: Theorem 41 In $O(n^{2.045})$ time, one can obtain an almost $3/2$ -approximation $\\\\tilde{D}$ to the Diameter $D$ and almost $5/3$ -approximations $e(v)$ to all Eccentricities $\\\\epsilon (v)$ : $\\\\frac{2D-1}{3} \\\\le \\\\tilde{D}\\\\le D$ .\",\n \"For every node $v$ , $\\\\frac{3\\\\epsilon (v)-1}{5}\\\\le \\\\tilde{\\\\epsilon }(v)\\\\le \\\\epsilon (v)$ .\",\n \"Finally we note that our approach also works to speed up our almost 2-approximation algorithm for $S$ -$T$ Diameter as well, giving an $O(n^{2.045})$ time almost-2 approximation algorithm.\",\n \"The main reason is that, like in the Diameter approximation algorithm, if the $S$ -$T$ Diameter is very large (say $D_{S,T}>100a\\\\log n$ ), then the $+a\\\\log n$ APSP algorithm with $a\\\\log n$ subtracted will return an estimate that is at least $D_{S,T}-a\\\\log n>0.99D_{S,T}$ .\",\n \"On the other hand, our $S$ -$T$ Diameter approximation algorithm only needs to know the distances up to $D_{S,T}$ to compute an estimate of $D_{S,T}$ , and so if $D_{S,T}\\\\le 100a\\\\log n$ , then we only need to compute $O(\\\\log n)$ matrix products of dimension $O(\\\\sqrt{n} \\\\log n)\\\\times n\\\\times n$ again.\"\n ],\n [\n \"Algorithm overview\",\n \"The almost-$3/2$ Diameter approximation algorithm of Aingworth et al.\",\n \"[2] runs in $\\\\tilde{O}(n^2+m\\\\sqrt{n})$ time.\",\n \"Roditty and Vassilevska W. [41] removed the $\\\\tilde{O}(n^2)$ term to obtain an $\\\\tilde{O}(m\\\\sqrt{n})$ expected time almost-$3/2$ approximation algorithm.\",\n \"For every graph with $\\\\Omega (n^{1.5})$ edges the running time of the latter algorithm is not better than the running time of the former algorithm.\",\n \"Therefore, it is interesting to consider the opposite question to the one considered by [41].\",\n \"Can the $\\\\tilde{O}(m\\\\sqrt{n})$ term be removed?\",\n \"We show that this can be done for undirected unweighted graphs and present an $O(n^2 \\\\log n)$ expected time algorithm.\",\n \"For a graph of Diameter $D=3h+z$ , where $z\\\\in [0,1,2]$ our algorithm returns an estimation $\\\\hat{D}$ such that $2h-1\\\\le \\\\hat{D} \\\\le D$ , when $z \\\\in [0,1]$ and $2h\\\\le \\\\hat{D} \\\\le D$ , when $z=2$ .\",\n \"Interestingly, our algorithm is obtained by using ideas developed originally for distance oracles and compact routing schemes.\",\n \"As we are allowed to use quadratic time, we try to estimate the distance between every pair of vertices.\",\n \"To enable this approach we can no longer sample $A$ naively.\",\n \"Instead, we adapt a recursive sampling algorithm to compute $A$ , that was introduced by Thorup and Zwick [47] in the context of compact routing schemes.\",\n \"The expected running time of their algorithm is $\\\\tilde{O}(mn/|A|)$ .\",\n \"We provide a new implementation of their algorithm that runs in expected $\\\\tilde{O}(n(n/|A|)^2)$ time.\",\n \"The set $A$ has the following important property, for every vertex $w\\\\in V$ , its cluster (see [48]) $\\\\lbrace u \\\\mid d(u,w)<d(u,A) \\\\rbrace $ is of size $O(n/ |A|)$ .\",\n \"Consider now a pair of vertices $u$ and $v$ that are in the cluster of $w$ .\",\n \"For any such pair we can efficiently compute their exact distance.\",\n \"Moreover, we show that for all pairs $u,v$ that are not in the same cluster of any vertex, we can bound $d(u,v)$ from below with $d(u,A)+d(v,A)-1$ .\",\n \"This, combined with some other ideas, gives our approximation guarantees.\",\n \"We extend our approach to also provide an almost $5/3$ -approximation for all Eccentricities.\",\n \"The idea of using the bounded clusters of Thorup and Zwick [47] has been used in prior work to obtain improved distance oracles [38], [5], approximate shortest paths [11] and compact routing schemes [4].\"\n ],\n [\n \"A simple approach with additive error\",\n \"Let's first consider a simple approach obtaining an $\\\\tilde{O}(n^2)$ time approximation algorithm for Diameter, Eccentricities or $S$ -$T$ Diameter.\",\n \"Suppose that we have an algorithm ALG that can compute in $\\\\tilde{O}(m\\\\sqrt{n})$ time, for any graph $G^{\\\\prime }$ , an estimate $D^{\\\\prime }$ of its Diameter $D$ such that $p\\\\cdot D - q\\\\le D^{\\\\prime }\\\\le D$ , estimates $e(v)$ of $\\\\epsilon (v)$ for all $v$ so that $r\\\\epsilon (v)-s\\\\le e(v)\\\\le \\\\epsilon (v)$ , and an estimate $D^{\\\\prime \\\\prime }$ of the $S$ -$T$ Diameter $D_{S,T}$ so that $t\\\\cdot D_{S,T}-u\\\\le D^{\\\\prime \\\\prime }\\\\le D_{S,T}$ .\",\n \"Now, Dor, Halperin and Zwick [24] showed that in $\\\\tilde{O}(n^2)$ time one can compute for any $n$ node $G$ , an additive 2 spanner $H$ on $\\\\tilde{O}(n^{1.5})$ edges.\",\n \"In fact Knudsen [30] recently showed that in $O(n^2)$ time one can get $H$ on $O(n^{1.5})$ edges (i.e.\",\n \"he removed all logs!).\",\n \"Let's compute $H$ for our given graph and run ALG on $H$ .\",\n \"The runtime is $\\\\tilde{O}(n^{1.5}\\\\cdot \\\\sqrt{n})\\\\le \\\\tilde{O}(n^2)$ since $H$ has $\\\\le O(n^{1.5})$ edges.\",\n \"Let $D^{\\\\prime }_H,e_H(\\\\cdot ),D^{\\\\prime \\\\prime }_H$ be the estimates that we obtain respectively for the Diameter $D_H$ of $H$ , the Eccentricities $\\\\epsilon _H(\\\\cdot )$ of $H$ and the $S,T$ Diameter $D^H_{S,T}$ .\",\n \"Let's return $D^{\\\\prime }_H-2,e_H(\\\\cdot )-2,D^{\\\\prime \\\\prime }_H-2$ for our estimates for the Diameter, Eccentricities and $S$ -$T$ Diameter of $G$ .\",\n \"Notice that $pD-q\\\\le p\\\\cdot D_H - q\\\\le D^{\\\\prime }_H\\\\le D_H\\\\le D+2$ and so $pD-2-q \\\\le D^{\\\\prime }_H-2\\\\le D$ .\",\n \"Similarly since $r\\\\epsilon (v)-s\\\\le r\\\\epsilon _H(v)-s\\\\le e_H(v)\\\\le \\\\epsilon _H(v)\\\\le \\\\epsilon (v)+2$ , we get $r\\\\epsilon (v)-s-2\\\\le e_H(v)-2\\\\le \\\\epsilon (v)$ .\",\n \"Finally since $t\\\\cdot D_{S,T}-u\\\\le t\\\\cdot D^H_{S,T}-u\\\\le D^{\\\\prime \\\\prime }_H\\\\le D^H_{S,T}\\\\le D_{S,T}+2$ , we get $t\\\\cdot D_{S,T}-u-2\\\\le D^{\\\\prime \\\\prime }_H-2\\\\le D_{S,T}$ .\",\n \"Thus, in $\\\\tilde{O}(n^2)$ time we get almost the same guarantees as in the $\\\\tilde{O}(m\\\\sqrt{n})$ time algorithms, except for an extra additive loss of 2 in the quality.\",\n \"Below we show how to make the additive loss in quality smaller for Diameter and Eccentricities.\",\n \"This is especially important when these parameters are constant, which is the hard case of the problems anyway.\"\n ],\n [\n \"Near linear almost 3/2-approximation for Diameter\",\n \"Thorup and Zwick [48] introduced distance oracles, a succinct data structure for answering approximate distance queries efficiently.\",\n \"Among the tools they use are clusters and bunches.\",\n \"Let $A\\\\subseteq V$ , let $p_A(u)$ be the closest vertex to $u$ from $A$ , where ties are broken in favor of the vertex with a smaller identifier and let $d(u,A)=d(u,p_A(u))$ .\",\n \"For every $v\\\\in V$ , let $B_A(u)=\\\\lbrace v\\\\in V\\\\mid d(u,v) <d(u,A \\\\rbrace $ be the bunch of $u$ .\",\n \"For every $w\\\\in V\\\\setminus A$ , let $C_A(w)= \\\\lbrace v \\\\mid w\\\\in B_A(v)\\\\rbrace $ be the cluster of $w$ .\",\n \"Thorup and Zwick [48] showed that if a set $A$ is formed by adding every vertex of $V$ to $A$ with probability $p$ then the expected size of $B_A(v)$ is $O(1/p)$ , for every $v\\\\in V$ .\",\n \"They also showed, in the context of compact routing schemes [47], that if the set $A$ is constructed by a recursive sampling algorithm then it is possible to bound the maximum size of a cluster as well.\",\n \"They also showed, in the context of compact routing schemes [47], that if the set $A$ is constructed by a recursive sampling algorithm then it is possible to bound the maximum size of a cluster as well.\",\n \"Their algorithm works as follows.\",\n \"It sets $A$ to the empty set and $W$ to $V$ .\",\n \"Next, as long as the set $W$ is not empty the algorithm samples from $W$ vertices with probability $p$ and adds the sampled vertices to $A$ .\",\n \"The algorithm computes $C_A(w)$ for every $w\\\\in W$ and removes from $W$ all the vertices whose cluster has at most $4/p$ vertices with respect to the updated $A$ .\",\n \"The pseudo-code is given in Algorithm REF .\",\n \"Thorup and Zwick center algorithm [1] center$G,p$ $A= \\\\emptyset $ $W =V$ $W\\\\ne \\\\emptyset $ $X$ - random sample of nodes from $W$ , $|X|=|W|p$ $A= A \\\\cup X$ $W = \\\\lbrace w\\\\in V \\\\mid |C_A(w)|> 4/p \\\\rbrace $ $A$ Thorup and Zwick proved the following Theorem: Theorem 28 (Theorem 3.1 from [47]) The expected size of the set $A$ returned by Algorithm REF is at most $2np \\\\log n$ .\",\n \"For every $w \\\\in V$ we then have $|C_A(w)|\\\\le 4/p$ .\",\n \"Thorup and Zwick claimed that the expected running time of Algorithm REF is $O(mnp \\\\log n)$ .\",\n \"They did not provide the details and refer the reader to [48].\",\n \"However, an educated guess is that they compute clusters for the vertices currently in $W$ in each iteration of the while loop, which results in the claimed running time.\",\n \"The starting point of the Diameter and Eccentricities algorithms presented in this section is an $O(n/p^2 \\\\log n)$ expected time implementation of Algorithm REF .\",\n \"The first idea behind our implementation is that, as opposed to what Thorup and Zwick did, we will compute the bunches and use them to compute the clusters and the set $W$ .\",\n \"This can be done as follows.\",\n \"Once we have computed $B_A(v)$ for every $v\\\\in V$ , we can scan $B_A(v)$ , and for every $w\\\\in B_A(v)$ we can add $v$ to $C_A(w)$ .\",\n \"The cost of this process is $O(|\\\\cup _{v\\\\in V} B_A(v)|)$ and since the clusters are by definition the inverse of the bunches, at the end of this process we have $C_A(w)$ and $|C_A(w)|$ , for every $w\\\\in V$ and we can compute $W$ (as needed in Algorithm REF ).\",\n \"However, in the current implementation only the expected size of a bunch is bounded, and since the Thorup-Zwick bound on the number of iterations is $O(\\\\log n)$ in expectation as well, we cannot apply this idea directly to deduce a good expected running time.\",\n \"To this end, more ideas are needed.\",\n \"The following simple observation helps us to achieve our goal.\",\n \"Observation 29 Let $A_i$ be the set $A$ after updating it in the beginning of the $i$ -th iteration of the while loop in Algorithm REF .\",\n \"Let $A^*$ be a set such that $A^*\\\\subseteq A_i$ , for every $i\\\\ge 1$ .\",\n \"For every $v\\\\in V$ it holds that $B_{A_i}(v) \\\\subseteq B_{A^*}(v)$ .\",\n \"It follows from this observation that we only need to pick the first set $A_1$ such that $|B_{A_1}(v)|\\\\le O(1/p)$ for every $v\\\\in V$ .\",\n \"It is folklore that the $s$ closest vertices $N_{s}(v)$ to a vertex $v$ can be computed in $O(s^2)$ time [24].\",\n \"This implies that we can compute $N_{1/p}(v)$ for every $v\\\\in V$ in $O(n/p^2)$ time.\",\n \"It is not hard to see that, given the sets $N_{1/p}(v)$ of all $v\\\\in V$ , one can (deterministically) compute a “hitting” set $A$ of size $O(np \\\\log n)$ in $O(n+n/p)$ worst case time, so that $N_{1/p}(v)\\\\cap A\\\\ne \\\\emptyset $ for every $v\\\\in V$ (a greedy algorithm works; e.g.\",\n \"see [48]).\",\n \"The second idea behind our implementation is that we first compute the sets $N_{1/p}(v)$ for every $v\\\\in V$ and the hitting set $A$ , as described above.\",\n \"Then, using these sets, we initialize Algorithm REF with a set $A$ such that $|B_A(v)|=O(1/p)$ , for every $v\\\\in V$ .\",\n \"In more detail, our algorithm works as follows.\",\n \"For every $v\\\\in V$ it computes the set $N_{1/p}(v)$ in $O(n/p^2)$ time.\",\n \"Then it finds a set $A$ such that $N_{1/p}(v)\\\\cap A\\\\ne \\\\emptyset $ for every $v\\\\in V$ .\",\n \"Given the hitting set $A$ , it computes $d(v,A)$ and $p_A(v)$ for every $v\\\\in V$ .\",\n \"Using $d(v,p_A(v))$ and $N_{1/p}(v)$ it computes for every $v\\\\in V$ the bunch $B_A(v)$ .\",\n \"Finally, it computes the clusters and $W$ using the bunches as we described above.\",\n \"The rest of the algorithm is almost identical to Algorithm REF .\",\n \"The only difference is that we compute the bunches and use them to compute the clusters and the set $W$ .\",\n \"The pseudo-code is given in Algorithm REF .\",\n \"New implementation of Thorup and Zwick center algorithm [1] center$G,p$ compute $N_{1/p}(v)$ for every $v\\\\in V$ .\",\n \"$A=$ hitting set of the sets $N_{1/p}(v)$ , where $v\\\\in V$ .\",\n \"compute $d(v,A)$ and $p_A(v)$ for every $v\\\\in V$ .\",\n \"compute $B_A(v)$ using $N_{1/p}(v)$ and $d(v,p_A(v))$ .\",\n \"every $u\\\\in V$ compute $C_A(u)$ using $B_A(\\\\cdot )$ $W = \\\\lbrace w\\\\in V \\\\mid |C_A(w)|> 4/p \\\\rbrace $ $W\\\\ne \\\\emptyset $ $X$ - random sample of nodes from $W$ , $|X|=np$ $A= A \\\\cup X$ every $v\\\\in V$ compute $B_A(v)$ every $u\\\\in V$ compute $C_A(u)$ using $B_A(\\\\cdot )$ $W = \\\\lbrace w\\\\in V \\\\mid |C_A(w)|> 4/p \\\\rbrace $ $A$ We show: Lemma 30 Algorithm REF computes in $O(n/p^2 \\\\log n)$ expected time a set $A$ of expected size $O((n/p)\\\\cdot \\\\log n)$ that guarantees for every vertex $w\\\\in V \\\\setminus A$ that $|C_A(w)|=O(1/p)$ , and for every $v\\\\in V$ that $|B_A(v)|=O(1/p)$ .\",\n \"The cost of computing $N_{1/p}(v)$ for every $v\\\\in V$ is $O(n (1/p)^2)$ [24].\",\n \"The cost of computing $A$ is $O(n/p)$ time [48].\",\n \"Computing $d(v,A)$ and $p_A(v)$ for every $v\\\\in V$ in $O(m)$ time is straightforward by running shortest paths tree computation from a dummy vertex that is connected to the set $A$ .\",\n \"To compute $B_A(v)$ using $N_{1/p}(v)$ we only scan $N_{1/p}(v)$ , thus, the total cost is $O(n(1/p))$ .\",\n \"As we explained earlier the cost of computing clusters using bunches is $O(|\\\\cup _{v\\\\in V} B_A(v)|)$ .\",\n \"Since for every $v\\\\in V$ we have $B_A(v)\\\\subseteq N_{1/p}(v)$ the total cost is $O(n(1/p))$ .\",\n \"This completes the analysis of the part that precedes the while loop.\",\n \"Next, we analyze the cost of the while loop.\",\n \"Let $A^*$ be the set $A$ that was computed before the while loop and let $A_i$ be the set $A$ after updating it in the beginning of the $i$ -th iteration of the while loop.\",\n \"From Observation REF it follows that $B_{A_i}(v) \\\\subseteq B_{A^*}(v)$ and therefore in every iteration the cost of computing bunches from scratch is at most $O(n (1/p)^2)$ as $|B_{A^*}(v)|=O(1/p)$ , for every $v\\\\in V$ .\",\n \"One can also compute $B_{A_{i+1}}(v)$ from $B_{A_i}(v)$ by first computing $d(v,A_{i+1})$ and if $d(v,A_{i+1})<d(v,A_i)$ to prune $B_{A_{i+1}}(v)$ accordingly at a smaller cost of $O(n(1/p) + m)$ , however this does not affect the overall complexity.\",\n \"Thorup and Zwick [47] proved that the expected number of iterations is $O(\\\\log n)$ .\",\n \"The fact that the set $A$ from which we start is different does not affect their proofThey prove that in each iteration with probability $1/2$ the size of $W$ decreases by a factor of 2.\",\n \"For this argument they only require that the set $A$ in each iteration will be chosen from $W$ uniformly at random with probability $p$ as we do., therefore there are only $O(\\\\log n)$ iterations in expectation.\",\n \"This implies that a set $A$ of expected size $(np \\\\log n)$ is returned in $O(n/p^2 \\\\log n)$ expected time.\",\n \"The algorithm stops only when there are no large clusters, thus the bound on the cluster size follows.\",\n \"As we mentioned above the algorithm starts with bunches that satisfy the required bound and their size can only decrease afterwards, thus the bound on the bunches follows.\",\n \"almost $3/2$ -Approximation for Diameter [1] $3/2$ -Approx-DiamG $M$ - $n\\\\times n$ matrix whose entries are set to $n$ $A =$ CENTER$(G,1/\\\\sqrt{n})$ every $w\\\\in V$ Step 1 every $\\\\langle u, v\\\\rangle \\\\in C_A(w) \\\\times C_A(w)$ , s.t.\",\n \"$u\\\\ne v\\\\;$ $M(u,v) = \\\\min (M(u,v), d(u,w)+d(v,w))$ every $\\\\langle u, v\\\\rangle \\\\in V \\\\times V$ , s.t.\",\n \"$M(u,v)=n$ Step 2 $M(u,v) = d(u,A)+d(v,A)-1$ $H$ - an additive 2 spanner of $G$ Step 3 every $u \\\\in A$ compute shortest paths tree for $u$ in $H$ and set $\\\\epsilon _H(u)$ , the Eccentricity of $u$ in $H$ $D_1 = \\\\max _{\\\\langle u, v\\\\rangle \\\\in V \\\\times V} M(u,v)$ $D_2 = \\\\max _{u\\\\in A} \\\\epsilon _H(u)$ $\\\\hat{D} = \\\\max (D_1,D_2-2)$ $\\\\hat{D}$ We can now turn to describe the new Diameter algorithm.\",\n \"The algorithm works as follows.\",\n \"All entries of an $n \\\\times n$ matrix $M$ are set to $n$ .\",\n \"A set $A$ of centers is computed using the algorithm of Thorup and Zwick [47].\",\n \"For every vertex $w\\\\in V$ and every pair $\\\\langle u, v\\\\rangle \\\\in C_A(w)\\\\times C_A(w)$ the algorithm sets $M(u,v)$ to $\\\\min (M(u,v), d(u,w)+d(v,w))$ (Step 1).\",\n \"Next, the algorithm searches the matrix $M$ for entries whose value is still $n$ .\",\n \"Given a pair $\\\\langle u, v\\\\rangle \\\\in V \\\\times V$ for which $M(u,v)=n$ the algorithm sets $M(u,v)$ to $d(u,A)+d(v,A)-1$ (Step 2).\",\n \"Finally, the algorithm computes an additive 2 spanner $H$ of the input graph $G$ and for every $u\\\\in A$ it computes $\\\\epsilon _H(u)$ , the Eccentricity of $u$ in $H$ (Step 3).\",\n \"The algorithm outputs the maximum between $D_1$ and $D_2-2$ , where $D_1$ is $\\\\max _{\\\\langle u, v\\\\rangle \\\\in V \\\\times V} M(u,v)$ and $D_2$ is $\\\\max _{u\\\\in A} \\\\epsilon _H(u)$ .\",\n \"Next, we bound the value returned by Algorithm REF .\",\n \"Theorem 31 Let $D=3h+z$ , where $z\\\\in [0,1,2]$ .\",\n \"The value $\\\\hat{D}$ returned by Algorithm REF satisfies: $\\\\begin{array}{ll}2h-1 & \\\\mbox{if } z \\\\in [0,1] \\\\\\\\2h & \\\\mbox{if } z=2\\\\end{array}\\\\le \\\\hat{D} \\\\le D$ We start with the following Lemma: Lemma 32 Let $u,v\\\\in V$ and let $P(u,v)$ be a shortest path between $u$ and $v$ .\",\n \"If $B_A(u) \\\\cap B_A(v)\\\\ne \\\\emptyset $ then $(B_A(u) \\\\cap B_A(v)) \\\\cap P(u,v) \\\\ne \\\\emptyset $ .\",\n \"If $v\\\\in B_A(u)$ then the claim trivially holds so we can assume that $v\\\\notin B_A(u)$ .\",\n \"Let $w$ be the vertex farthest from $u$ that is in $B_A(u) \\\\cap P(u,v)$ .\",\n \"From the definition of $w$ it follows that $d(u,w)=d(u,A)-1$ .\",\n \"Assume, towards a contradiction, that $(B_A(u) \\\\cap B_A(v)) \\\\cap P(u,v) = \\\\emptyset $ .\",\n \"This implies that $w\\\\notin B_A(v)$ and $d(v,A)-1<d(v,w)=d(u,v)-d(u,w)$ .\",\n \"However, since $B_A(u) \\\\cap B_A(v)\\\\ne \\\\emptyset $ there is a vertex $w^{\\\\prime }$ such that $d(u,w^{\\\\prime })\\\\le d(u,w)$ and $d(v,w^{\\\\prime })\\\\le d(v,A)-1 < d(u,v)-d(u,w)$ .\",\n \"This implies that $d(u,w^{\\\\prime })+d(v,w^{\\\\prime })< d(u,v)$ , a contradiction to the triangle inequality.\",\n \"Lemma 33 Let $u,v\\\\in V$ .\",\n \"If $B_A(u) \\\\cap B_A(v)= \\\\emptyset $ then $d(u,A)+d(v,A)-1 \\\\le d(u,v)$ .\",\n \"Notice first that $B_A(u)$ (resp., $B_A(v)$ ) contains all the vertices at distance $d(u,A)-1$ (resp., $d(v,A)-1$ ).\",\n \"Let $P(u,v)$ be a shortest path between $u$ and $v$ .\",\n \"Let $w$ be the vertex farthest from $u$ on $P(u,v)$ that is also in $B_A(u)$ .\",\n \"Similarly, let $w^{\\\\prime }$ be the vertex farthest from $v$ on $P(u,v)$ that is also in $B_A(v)$ .\",\n \"Since $B_A(u) \\\\cap B_A(v)= \\\\emptyset $ it holds that $w\\\\ne w^{\\\\prime }$ .\",\n \"Therefore: $d(u,v) = d(u,A)-1+d(v,A)-1+d(w,w^{\\\\prime })\\\\ge d(u,A)+d(v,A)-1.$ Let $a$ and $b$ be the Diameter endpoints, that is $d(a,b)=D=3h+z$ , where $z\\\\in [0,1,2]$ .\",\n \"Let $P(a,b)$ be a shortest path between $a$ and $b$ .\",\n \"Assume first that $B_A(a) \\\\cap B_A(b)\\\\ne \\\\emptyset $ .\",\n \"It follows from Lemma REF that there is a vertex $w\\\\in P(a,b)$ such that $\\\\langle a, b\\\\rangle \\\\in C(w) \\\\times C(w)$ .\",\n \"Therefore, $M(a,b)=D$ after Step 1.\",\n \"After the update in Step 2 it follows from Lemma REF that $M(u,v)\\\\le d(u,v)$ for every $u ,v\\\\in V$ .\",\n \"Therefore, the maximum value in the matrix is $d(a,b)$ and $D_1=D$ .\",\n \"Let $x = \\\\operatornamewithlimits{arg\\\\,max}_{y\\\\in A} \\\\epsilon _H(y)$ .\",\n \"Since $H$ is an additive 2 spanner it holds that $\\\\epsilon _H(x)\\\\le D+2$ , hence, we have $D_2 \\\\le D$ and the algorithm returns the exact value of the Diameter.\",\n \"Assume now that $B_A(a) \\\\cap B_A(b) = \\\\emptyset $ .\",\n \"From the discussion of the previous case it follows that in this case $\\\\hat{D}\\\\le D$ as well.\",\n \"Thus, it is only left to prove the lower bound.\",\n \"Assume that $z\\\\in [0,1]$ .\",\n \"Consider first the case that $d(a,A)\\\\ge h$ and $d(b,A)\\\\ge h$ then from Lemma REF it follows that $M(a,b)\\\\ge 2h-1$ after Step 2 and $D_1$ is at least $2h-1$ .\",\n \"If this is not the case then either $d(a,A)< h$ or $d(b,A)< h$ (or both).\",\n \"Assume, wlog, that $d(a,A)< h$ .\",\n \"In this case the Eccentricity in $H$ of at least one vertex from $A$ is at least $2h+1$ and hence $D_2-2$ is at least $2h-1$ .\",\n \"Assume now that $z=2$ .\",\n \"If either $d(a,A)\\\\ge h$ and $d(b,A)> h$ or $d(a,A)>h$ and $d(b,A)\\\\ge h$ then from Lemma REF it follows that $M(a,b)\\\\ge 2h$ after Step 2 and $D_1$ is at least $2h$ .\",\n \"If this is not the case then either $d(a,A)\\\\le h$ or $d(b,A)\\\\le h$ (or both).\",\n \"Assume, wlog, that $d(a,A)\\\\le h$ .\",\n \"In this case the Eccentricity in $H$ of at least one vertex from $A$ is at least $2h+2$ and hence $D_2-2$ is at least $2h$ .\",\n \"We now turn to we analyze the running time of Algorithm REF .\",\n \"Theorem 34 The expected running time of Algorithm REF is $O(n^2 \\\\log n)$ .\",\n \"The set $A$ is computed by the center algorithm presented in Algorithm REF with $p=1/\\\\sqrt{n}$ .\",\n \"From Lemma REF it follows that the size of the set $A$ is $O(\\\\sqrt{n} \\\\log n)$ and its construction time is $O(n^2\\\\log n)$ in expectation.\",\n \"For every $w\\\\in V$ the size of $C_A(w)$ is $O(\\\\sqrt{n})$ .\",\n \"Therefore, Step 1 takes $O(n \\\\times |C_A(w)|^2)=O(n^2)$ .\",\n \"Step 2 takes $O(n^2)$ time as well.\",\n \"In Step 3 we first compute an additive 2 spanner $H$ on $O(n^{1.5})$ edges.\",\n \"Knudsen [30], following Dor, Halperin and Zwick [24] showed how to do this in $O(n^2)$ time.\",\n \"We also compute $|A|$ shortest paths trees in $H$ .\",\n \"As $H$ has $O(n^{1.5})$ edges, this step takes $O(n^2 \\\\log n)$ time.\"\n ],\n [\n \"Near linear almost 5/3-approximation for Eccentricities\",\n \"almost $5/3$ -Approximation for all Eccentricities [1] $5/3$ -Approx-EccG Run lines 2-11 of Algorithm REF , with $H$ augmented with shortest paths trees for $B_A(u) \\\\cup \\\\lbrace p(u) \\\\rbrace $ for every $u\\\\in V$ every $u \\\\in V$ $\\\\epsilon _1(u) = \\\\max _{v\\\\in V} M(u,v)$ $\\\\epsilon _2(u) = \\\\epsilon _H(p(u))-d(u,p(u))-2$ $\\\\epsilon _3(u) = d_H(u,y)-2$ , where $y=\\\\operatornamewithlimits{arg\\\\,max}_{x\\\\in A} d_H(u,x)$ $\\\\epsilon ^{\\\\prime }(u) = \\\\max (\\\\epsilon _1(u),\\\\epsilon _2(u),\\\\epsilon _3(u))$ Next, we show how to update Algorithm REF to obtain an almost $5/3$ approximation for all Eccentricities.\",\n \"We run lines 2-11 of Algorithm REF .\",\n \"The only difference is that $H$ is augmented with the edges of the shortest paths tree that span the set $B_A(u) \\\\cup \\\\lbrace p(u) \\\\rbrace $ for every $u\\\\in V$ .\",\n \"Then, for every $u\\\\in V$ we compute $\\\\epsilon _1(u)$ , $\\\\epsilon _2(u)$ and $\\\\epsilon _3(u)$ , which are defined as follows: $\\\\epsilon _1(u) = \\\\max _{v\\\\in V} M(u,v)$ , $\\\\epsilon _2(u) = \\\\epsilon _H(p(u))-d(u,p(u))-2$ and $\\\\epsilon _3(u) = d_H(u,y)-2$ , where $y=\\\\operatornamewithlimits{arg\\\\,max}_{x\\\\in A} d_H(u,x)$ .\",\n \"The algorithm sets $\\\\epsilon ^{\\\\prime }(u)$ to $\\\\max \\\\lbrace \\\\epsilon _1(u), \\\\epsilon _2(u), \\\\epsilon _3(u) \\\\rbrace $ for every $u\\\\in V$ as an estimation to $\\\\epsilon (u)$ .\",\n \"The pseudo-code is given in Algorithm REF .\",\n \"We now prove: Theorem 35 For every $u\\\\in V$ , Algorithm REF computes in $O(n^2 \\\\log n)$ expected time a value $\\\\epsilon ^{\\\\prime }(u)$ that satisfies: $\\\\frac{3\\\\epsilon (u)}{5}-1 \\\\le \\\\epsilon ^{\\\\prime }(u)\\\\le \\\\epsilon (u).$ We start by analyzing the running time.\",\n \"Lines 2-11 of the algorithm are the same as Algorithm REF , with one difference, the spanner $H$ is augmented with the edges of the shortest paths tree that span the set $B_A(u) \\\\cup \\\\lbrace p(u) \\\\rbrace $ for every $u\\\\in V$ .\",\n \"This adds at most $O(n^{1.5})$ edges to $H$ and hence the cost of these lines remain $O(n^2 \\\\log n)$ time in expectation.\",\n \"The computation of $\\\\epsilon _1(u)$ , $\\\\epsilon _2(u)$ and $\\\\epsilon _3(u)$ for every $u\\\\in V$ costs $O(n^2)$ time in total.\",\n \"Let $u\\\\in V$ be an arbitrary vertex and let $\\\\epsilon (u)=d(u,t)$ .\",\n \"We now turn to bound $\\\\epsilon ^{\\\\prime }(u)$ .\",\n \"In our analysis we will use the following simple observation: Observation 36 In an undirected graph it holds for every $u,v \\\\in V$ that $\\\\epsilon (u)\\\\ge \\\\epsilon (v)-d(u,v)$ .\",\n \"It is straightforward to see that both $\\\\epsilon _2(u)$ and $\\\\epsilon _3(u)$ are at most $\\\\epsilon (u)$ .\",\n \"Recall that $\\\\epsilon _2(u) = \\\\epsilon _H(p(u))-d(u,p(u))-2\\\\le \\\\epsilon (p(u))-d(u,p(u)\\\\le \\\\epsilon (u)$ and $\\\\epsilon _3(u) = d_H(u,y)-2\\\\le d(u,y)\\\\le \\\\epsilon (u)$ .\",\n \"We distinguish between two cases.\",\n \"Case 1: $B_A(u) \\\\cap B_A(t)\\\\ne \\\\emptyset $ .\",\n \"It follows from Lemma REF that $P(u,t)\\\\cap (B_A(u) \\\\cap B_A(t)) \\\\ne \\\\emptyset $ and $M(u,t)= \\\\epsilon (u)$ .\",\n \"From Lemma REF it follows that $M(u,w)\\\\le d(u,w)$ for every $w\\\\in V$ after Step 2.\",\n \"Therefore, $\\\\epsilon _1(u)=\\\\epsilon (u)$ .\",\n \"Since $\\\\epsilon _2(u)\\\\le \\\\epsilon (u)$ and $\\\\epsilon _3(u)\\\\le \\\\epsilon (u)$ we get that $\\\\epsilon ^{\\\\prime }(u)=\\\\epsilon (u)$ .\",\n \"Case 2: $B_A(u) \\\\cap B_A(t) = \\\\emptyset $ .\",\n \"Consider first the case that $d(u,p(u))\\\\le \\\\frac{\\\\epsilon (u)}{5}-1$ .\",\n \"From Observation REF we get that $\\\\epsilon _H(p(u))\\\\ge \\\\epsilon _H(u)-d_H(u,p(u))$ .\",\n \"As we augmented $H$ with a shortest paths tree that spans $B_A(u) \\\\cup \\\\lbrace p(u) \\\\rbrace $ we have $d(u,p(u))=d_H(u,p(u))$ and we get $\\\\epsilon _H(p(u))\\\\ge \\\\epsilon _H(u)-d(u,p(u))$ .\",\n \"Hence, we get that $\\\\epsilon _2(u) = \\\\epsilon _H(p(u))-d(u,p(u))-2 \\\\ge \\\\epsilon _H(u)-2d(u,p(u))-2\\\\ge \\\\epsilon (u)-2d(u,p(u))-2$ .\",\n \"As before we have $\\\\epsilon _2(u) \\\\le \\\\epsilon (u)$ .\",\n \"Using $d(u,p(u))\\\\le \\\\frac{\\\\epsilon (u)}{5}-1$ we get that: $\\\\epsilon _2(u) \\\\ge \\\\epsilon (u)-\\\\frac{2\\\\epsilon (u)}{5}\\\\ge \\\\frac{3\\\\epsilon (u)}{5}.$ Assume now that $d(u,p(u))\\\\ge \\\\frac{\\\\epsilon (u)}{5}$ .\",\n \"This means that $d(u,A)-1\\\\ge \\\\frac{\\\\epsilon (u)}{5}-1$ .\",\n \"Let $S$ be the set of all vertices $v\\\\in V$ such that $B_A(u) \\\\cap B_A(v)= \\\\emptyset $ , that is, $S=V \\\\setminus \\\\cup _{w\\\\in B_A(u)} C_A(w)$ .\",\n \"Let $t^{\\\\prime } = \\\\operatornamewithlimits{arg\\\\,max}_{x\\\\in S} d(x,A)-1$ .\",\n \"If $d(t^{\\\\prime },A)-1 \\\\ge \\\\frac{2\\\\epsilon (u)}{5}-1$ we get from Lemma REF that $M(u,t^{\\\\prime })\\\\ge \\\\frac{3\\\\epsilon (u)}{5}-1$ .\",\n \"Assume now that $d(t^{\\\\prime },A) < \\\\frac{2\\\\epsilon (u)}{5}$ .\",\n \"As $t^{\\\\prime }$ is the farthest vertex from $A$ we get that $d(t,p(t))<\\\\frac{2\\\\epsilon (u)}{5}$ and $d(u,p(t))>\\\\frac{3\\\\epsilon (u)}{5}$ .\",\n \"Therefore, $\\\\epsilon _3(u) = d_H(u,y)-2\\\\ge d(u,p(t))-2\\\\ge \\\\frac{3\\\\epsilon (u)}{5}-1$ .\",\n \"From Lemma REF and Lemma REF it follows that $\\\\epsilon _1(u)\\\\le \\\\epsilon (u)$ and the bound follows.\"\n ],\n [\n \"Algorithms for dense graphs using matrix multiplication\",\n \"Here we will give $O(n^{2.05})$ time approximation algorithms for Diameter and Eccentricities in dense unweighted undirected graphs.\",\n \"The approximation guarantees of these algorithms are slightly better than those in our $O(n^2\\\\log n)$ time algorithm.\",\n \"In fact, the guarantees are exactly the same as in the $\\\\tilde{O}(m\\\\sqrt{n})$ time algorithms for Diameter and Eccentricities of Roditty and V. Williams [41] and Cairo et al. [15].\",\n \"To achieve this, we give an efficient implementation using fast matrix multiplication of the $\\\\tilde{O}(m\\\\sqrt{n})$ time algorithms of [15] and [41].\",\n \"The main overhead of the $\\\\tilde{O}(m\\\\sqrt{n})$ time algorithms [15], [41] is in computing the distances from a set $S=W\\\\cup \\\\lbrace w\\\\rbrace \\\\cup T$ of $O(\\\\sqrt{n}\\\\log n)$ nodes: the set $S$ itself can be computed in linear time using random sampling to form a set $W$ , BFS from a dummy node to find the node $w$ farthest from $W$ and then BFS from $w$ to find the set $T$ of closest $\\\\sqrt{n}$ nodes to $w$ .\",\n \"After one knows all distances from every $s\\\\in S$ to every $v\\\\in V$ , it takes linear time to output the Diameter and Eccentricity estimates.\",\n \"The main idea of our algorithms is as follows.\",\n \"If the Diameter is of size $\\\\le O(\\\\log n)$ , then one does not need all distances between $S$ and $V$ , but only those that are $O(\\\\log n)$ .\",\n \"Small distances are easy to compute with matrix multiplication.\",\n \"Let $A$ be the adjacency matrix and $A_S$ be its submatrix formed by just the rows in $S$ .\",\n \"Then we can find the distances for all pairs in $S\\\\times V$ at distance $\\\\le t$ by computing $A_S\\\\times A^{t-1}$ , which can be computed by performing $t-1$ matrix products of dimension $|S|\\\\times n$ by $n\\\\times n$ , and this can be accomplished in $O(tn^{2.05})$ time [26], [31].\",\n \"If on the other hand the Diameter is $D\\\\ge 100\\\\log n$ , then one can use an $\\\\tilde{O}(n^2)$ time algorithm by Dor et al.\",\n \"[24] to compute estimates of all pairwise distances with an additive error at most $4\\\\log n$ .\",\n \"The maximum distance estimate computed, minus $4\\\\log n$ , will be between $0.96 D$ and $D$ , giving a really good approximation already.\",\n \"A similar argument works for Eccentricities, and also for $S$ -$T$ Diameter.\",\n \"Below we recap the guarentees of the $\\\\tilde{O}(m\\\\sqrt{n})$ time approximation algorithms of [15], [41].\",\n \"Theorem 37 ([15], [41]) The following can be computed in $\\\\tilde{O}(m\\\\sqrt{n})$ time: an estimate $\\\\hat{D}$ of the graph Diameter $D$ , such that $\\\\frac{2}{3} D-\\\\frac{1}{3}\\\\le \\\\hat{D}\\\\le D$ , for every node $v$ , an estimate $e(v)$ of its Eccentricity $\\\\epsilon (v)$ , such that $\\\\frac{3}{5} \\\\epsilon (v)-\\\\frac{1}{5}\\\\le e(v)\\\\le \\\\epsilon (v)$ .\",\n \"Using Seidel's algorithm [43] we can compute all the distances exactly, and hence the above parameters as well, all in $O(n^\\\\omega )$ time for $\\\\omega < 2.373$ .\",\n \"We will show that for dense graphs, we can obtain the same approximation guarantees as in Theorem REF , in time $O(n^{2.05})$ .\",\n \"Let us compare to our $O(n^2\\\\log n)$ time algorithms.\",\n \"For Diameter $D=3h+z$ , the $O(n^2\\\\log n)$ time algorithm returns an estimate $2h-1$ when $z=0,1$ and $2h$ when $z=2$ .\",\n \"The estimate $\\\\hat{D}$ here is $\\\\ge (2D-1)/3= 2h+ (2z-1)/3$ , which is $\\\\ge 2h$ when $z=0$ and $\\\\ge 2h+1$ when $z=1,2$ .\",\n \"For Eccentricities, the $O(n^2\\\\log n)$ time algorithm returns estimates $e(v)\\\\ge 3\\\\epsilon (v)/5 -1$ , and here we return a better estimate $e(v)\\\\ge (3\\\\epsilon -1)/5$ .\",\n \"We will rely on two known algorithms.\",\n \"The first is from a paper by Dor, Halperin and Zwick [24] on additive approximations of All-Pairs Shortest Paths (APSP).\",\n \"Among many other results, [24] show that in $\\\\tilde{O}(n^2)$ time, one can compute for all pairs of vertices $u,v$ , an estimate $d^{\\\\prime }(u,v)$ of their distance $d(u,v)$ so that $d(u,v)\\\\le d^{\\\\prime }(u,v)\\\\le d(u,v)+a\\\\log n$ for an explicit constant $a\\\\le 4$ .\",\n \"The second is an algorithm for the following truncated multi-source shortest paths problem: given an integer $Q$ , a graph $G=(V,E)$ and a set $S$ , compute the distances $d(s,v)$ for every $s\\\\in S$ and $v\\\\in V$ for which $d(s,v)\\\\le Q$ .\",\n \"The algorithm uses fast matrix multiplication and is quite straightforward.\",\n \"Let $A$ be the $n\\\\times n$ Boolean matrix with rows and columns indexed by $V$ , so that $A[u,v]=1$ if there is an edge between $u$ and $v$ or $u=v$ , and $A[u,v]=0$ otherwise; i.e.\",\n \"$A$ is the adjacency matrix added to the identity matrix.\",\n \"Let $A_S$ be the $|S|\\\\times n$ submatrix of $A$ consisting of the rows indexed by nodes of $S$ .\",\n \"For an integer $i\\\\ge 1$ , let $A^i$ be the $i$ -th power of $A$ under the Boolean matrix product.\",\n \"Here, $A^i[u,v]=1$ if and only if the distance between $u$ and $v$ is at most $i$ .\",\n \"Define $A^0$ as the identity matrix.\",\n \"Consider $A_S \\\\cdot A^i$ for any choice of $i\\\\ge 0$ (under the Boolean matrix product).\",\n \"Here, $(A_S \\\\cdot A^i)[s,v]=1$ if and only if the distance between $s$ and $v$ is at most $i+1$ .\",\n \"Thus, if we compute $D_i:=A_S\\\\cdot A^i$ for every $1\\\\le i< Q$ , we would know the distance from every $s\\\\in S$ to every $v\\\\in V$ , whenever this distance is at most $Q$ .\",\n \"Computing these matrix products can easily be done by performing the following $Q-1$ Boolean products of an $|S|\\\\times n$ matrix by an $n\\\\times n$ matrix: let $D_0=A_S$ ; then for each $i$ from 1 to $t-1$ , compute $D_i:=D_{i-1}\\\\cdot A$ .\",\n \"Thus, the running time is $O(Q\\\\cdot M(|S|,n,n))$ where $M(|S|,n,n)$ is the runtime of multiplying an $|S|\\\\times n$ matrix by an $n\\\\times n$ matrix.\",\n \"Armed with these two algorithms, let us recap Roditty et al.\",\n \"'s (and Cairo et al.\",\n \"'s) approximation algorithm and see how to modify it.\",\n \"The algorithm proceeds as follows: Let $D$ , $R$ and $\\\\epsilon (v)$ denote the Diameter and Radius of $G$ and the Eccentricity of node $v$ , respectively.\",\n \"RV/CGR Algorithm [1] Using BFS (see [41] and [15]), in $O(m+n)$ time compute $W,w,T$ , where $W\\\\subseteq V$ is a uniformly chosen subset of size $O(\\\\sqrt{n} \\\\log n)$ , $w$ is the furthest node from $W$ and $T$ are the closest $\\\\sqrt{n}$ nodes to $w$ .\",\n \"Let $S=\\\\lbrace w\\\\rbrace \\\\cup W\\\\cup T$ .\",\n \"For every $s\\\\in S$ and every $v\\\\in V$ , compute the distance $d(s,v)$ between $s$ and $v$ ; set $\\\\epsilon (s)=\\\\max _v d(s,v)$ .\",\n \"Set $\\\\tilde{D}=\\\\max _{x\\\\in S} \\\\epsilon (x)$ .\",\n \"Set for every $v\\\\in V$ , $\\\\tilde{\\\\epsilon }(v)=\\\\max \\\\lbrace d(w,v),\\\\max _{x\\\\in W} d(x,v), \\\\max _{x\\\\in T} (\\\\epsilon (x)-d(x,v))\\\\rbrace $ .\",\n \"The runtime bottleneck in the above algorithm is step (2) which runs in $\\\\tilde{O}(mn^{1/2})$ time if one uses BFS through each node of $S$ .\",\n \"Let us describe how to modify the algorithm.\",\n \"We will replace (2) with a truncated distance computation and also use the algorithm of Dor, Halperin and Zwick to handle large distances that we might have ignored in the truncated computation.\",\n \"Our Modified Approximation.\",\n \"[1] FasterApproximation First part: Handle Large Distances: [t]@X@ Use Dor, Halperin and Zwick's algorithm to compute distance estimates $d^{\\\\prime }(\\\\cdot ,\\\\cdot )$ so that for every $u,v\\\\in V$ , $d(u,v)\\\\le d^{\\\\prime }(u,v)\\\\le d(u,v)+a\\\\log n$ .\",\n \"Let $X=3a\\\\log n$ .\",\n \"Set $\\\\tilde{D}_1 = \\\\max _{u,v\\\\in V} d^{\\\\prime }(u,v) - a\\\\log n$ .\",\n \"For every $v\\\\in V$ , set $\\\\tilde{\\\\epsilon }_1(v) = \\\\max _{u} d^{\\\\prime }(u,v) - a\\\\log n$ .\",\n \"Second Part: Handle Small Distances: [t]@X@ Using BFS (see [41] and [15]), in $O(m+n)$ time compute $W,w,T$ , where $W\\\\subseteq V$ is a uniformly chosen subset of size $O(\\\\sqrt{n} \\\\log n)$ , $w$ is the furthest node from $W$ and $T$ are the closest $\\\\sqrt{n}$ nodes to $w$ .\",\n \"Let $S=\\\\lbrace w\\\\rbrace \\\\cup W\\\\cup T$ .\",\n \"[t]@X@ Let $Q=2(X+a\\\\log n)=8a\\\\log n$ .\",\n \"For every $s\\\\in S$ and every $v\\\\in V$ whose distance $d(s,v)$ is at most $Q$ , compute $d(s,v)$ .\",\n \"Let $d_{\\\\le }(s,v)$ denote $d(s,v)$ if we have computed it, and $\\\\infty $ otherwise.\",\n \"Set $\\\\epsilon _{\\\\le }(s)=\\\\max _v d_{\\\\le }(s,v)$ .\",\n \"Set $\\\\tilde{D}_2=\\\\max _{x\\\\in S} \\\\epsilon _{\\\\le }(x)$ .\",\n \"[t]@X@ $\\\\forall v\\\\in V$ , set $\\\\tilde{\\\\epsilon }_2(v)=\\\\max \\\\lbrace d_\\\\le (w,v), \\\\max _{x\\\\in W} d_\\\\le (x,v), \\\\max _{y\\\\in T} (\\\\epsilon _\\\\le (y)-d_\\\\le (y,v))\\\\rbrace $ .\",\n \"Third Part: Set $\\\\tilde{D},\\\\tilde{\\\\epsilon }(\\\\cdot )$: If $\\\\tilde{D}_1\\\\ge X$ , set $\\\\tilde{D}=\\\\tilde{D}_1$ , and otherwise set $\\\\tilde{D}=\\\\tilde{D}_2$ .\",\n \"[t]@X@ For every $v\\\\in V$ , if there exists some $x\\\\in S$ such that $d^{\\\\prime }(x,v)\\\\ge X+a\\\\log n$ , then set $\\\\tilde{\\\\epsilon }(v)=\\\\tilde{\\\\epsilon }_1(v)$ , and otherwise $\\\\tilde{\\\\epsilon }(v)=\\\\tilde{\\\\epsilon }_2(v)$ .\",\n \"Consider our modified algorithm, FasterApproximation.\",\n \"Now we will prove several claims.\",\n \"Claim 38 The running time of algorithm FasterApproximation is $\\\\tilde{O}(M(\\\\sqrt{n},n,n))$ .\",\n \"The Dor, Halperin, Zwick part of the algorithm (Step 3) runs in $\\\\tilde{O}(n^2)$ time.\",\n \"Step 8 runs in $O((X+a\\\\log n)\\\\cdot M(|S|,n,n))$ time where $S=\\\\lbrace w\\\\rbrace \\\\cup W\\\\cup T$ , using the iterated rectangular matrix product algorithm.\",\n \"Recall that $X+a\\\\log n = O(\\\\log n)$ .\",\n \"Thus Step 8 runs in $\\\\tilde{O}(M(|S|,n,n))$ time.\",\n \"Since $|S|=\\\\tilde{O}(\\\\sqrt{n})$ and we can partition an $|S|\\\\times n\\\\times n$ matrix product into $ \\\\text{\\\\rm polylog~} n$ , $n^{1/2}\\\\times n\\\\times n$ matrix products, the runtime of the step is $\\\\tilde{O}(M(n^{1/(2)},n,n))$ .\",\n \"Steps 10 and 13 run in $O(n|S|)<\\\\tilde{O}(n^2)$ time.\",\n \"The rest of the steps run in linear time.\",\n \"Since $M(n^{1/2},n,n)\\\\ge n^2$ (one must at least read the input), the total running time is $\\\\tilde{O}(M(n^{1/2},n,n))$ .\",\n \"Claim 39 $\\\\frac{2D-1}{3} \\\\le \\\\tilde{D}\\\\le D$ .\",\n \"Suppose that $\\\\tilde{D}_1\\\\ge X$ .\",\n \"The algorithm returns $\\\\tilde{D}=\\\\tilde{D}_1=\\\\max _{u,v} d^{\\\\prime }(u,v)-a\\\\log n$ .\",\n \"By the guarantee on $d^{\\\\prime }$ , we have $D-a\\\\log n \\\\le \\\\tilde{D}_1\\\\le D$ .\",\n \"Hence $\\\\tilde{D}\\\\ge D(1-(a\\\\log n)/D)\\\\ge D(1-(a\\\\log n)/X) =2D/3 \\\\ge (2D-1)/3$ .\",\n \"Suppose now that $\\\\tilde{D}_1< X$ .\",\n \"This means that $D<X+a\\\\log n$ and every distance in the graph is $\\\\le X+a\\\\log n$ .\",\n \"In the second part of the algorithm we set $Q=2(X+a\\\\log n)$ , and hence every distance is computed exactly: for every $s\\\\in S$ , $v\\\\in V$ , $d_\\\\le (s,v)=d(s,v)$ .\",\n \"Hence the second part of the algorithm will be identical to the RV/CGR algorithm and hence we get the same guarantees: $(2D-1)/3\\\\le \\\\tilde{D}\\\\le D$ .\",\n \"Claim 40 For every node $v$ , $\\\\frac{3\\\\epsilon (v)-1}{5} \\\\epsilon (v)\\\\le \\\\tilde{\\\\epsilon }(v)\\\\le \\\\epsilon (v)$ .\",\n \"Fix $v$ .\",\n \"Suppose first that there exists some $x$ such that $d^{\\\\prime }(x,v)\\\\ge X+a\\\\log n$ .\",\n \"Then $\\\\epsilon (v)\\\\ge \\\\tilde{\\\\epsilon }_1(v)=\\\\max _u d^{\\\\prime }(u,v) - a\\\\log n \\\\ge \\\\epsilon (v)-a\\\\log n = \\\\epsilon (v)(1-a\\\\log n/\\\\epsilon (v))$ .\",\n \"Since $\\\\epsilon (v)\\\\ge d(x,v)\\\\ge d^{\\\\prime }(x,v)-a\\\\log n \\\\ge X$ , we get that $\\\\tilde{\\\\epsilon }_1(v)\\\\ge \\\\epsilon (v)(1-a\\\\log n / X) = 2\\\\epsilon (v)/3$ .\",\n \"Now suppose that for all $x\\\\in V$ , $d^{\\\\prime }(x,v)<X+a\\\\log n$ .\",\n \"Then, also for all $x\\\\in V$ , $d(x,v)<X+a\\\\log n$ and $\\\\epsilon (v)<X+a\\\\log n$ .\",\n \"Consider all the quantities needed in the second part of the algorithm to compute $\\\\tilde{\\\\epsilon }_2(v)$ : $d_\\\\le (w,v)$ : since $\\\\forall x\\\\in V$ , $d(x,v)<X+a\\\\log n$ , $d_\\\\le (w_i,v)=d(w_i,v)$ for each $w_i$ ; $d_\\\\le (x,v)$ for every $x\\\\in W$ : as above, $d_\\\\le (x,v)=d(x,v)$ ; $\\\\epsilon _\\\\le (x)-d_\\\\le (x,v)$ for all $x\\\\in T$ : here, $\\\\epsilon (x)\\\\le \\\\epsilon (v)+d(x,v)\\\\le 2\\\\epsilon (v)<2(X+a\\\\log n)$ .\",\n \"Since we compute all distances from nodes in $S$ up to $2(X+a\\\\log n)$ and $x\\\\in S$ , $\\\\epsilon _\\\\le (x)=\\\\epsilon (x)$ .\",\n \"Also as in the above bullets, $d_\\\\le (x,v)=d(x,v)$ .\",\n \"Thus all the quantities needed are the correct ones and $\\\\tilde{\\\\epsilon }(v)=\\\\tilde{\\\\epsilon }_2(v)$ inherits the same guarantees as in the algorithm by Cairo et al.\",\n \"From Le Gall and Urrutia [26] (see also, [31]) we obtain that $M(\\\\sqrt{n},n,n)\\\\le O(n^{2.044183})$ .\",\n \"We obtain: Theorem 41 In $O(n^{2.045})$ time, one can obtain an almost $3/2$ -approximation $\\\\tilde{D}$ to the Diameter $D$ and almost $5/3$ -approximations $e(v)$ to all Eccentricities $\\\\epsilon (v)$ : $\\\\frac{2D-1}{3} \\\\le \\\\tilde{D}\\\\le D$ .\",\n \"For every node $v$ , $\\\\frac{3\\\\epsilon (v)-1}{5}\\\\le \\\\tilde{\\\\epsilon }(v)\\\\le \\\\epsilon (v)$ .\",\n \"Finally we note that our approach also works to speed up our almost 2-approximation algorithm for $S$ -$T$ Diameter as well, giving an $O(n^{2.045})$ time almost-2 approximation algorithm.\",\n \"The main reason is that, like in the Diameter approximation algorithm, if the $S$ -$T$ Diameter is very large (say $D_{S,T}>100a\\\\log n$ ), then the $+a\\\\log n$ APSP algorithm with $a\\\\log n$ subtracted will return an estimate that is at least $D_{S,T}-a\\\\log n>0.99D_{S,T}$ .\",\n \"On the other hand, our $S$ -$T$ Diameter approximation algorithm only needs to know the distances up to $D_{S,T}$ to compute an estimate of $D_{S,T}$ , and so if $D_{S,T}\\\\le 100a\\\\log n$ , then we only need to compute $O(\\\\log n)$ matrix products of dimension $O(\\\\sqrt{n} \\\\log n)\\\\times n\\\\times n$ again.\"\n ]\n]","isConversational":false},"id":{"kind":"string","value":"1808.08494"}}},{"rowIdx":199,"cells":{"text":{"kind":"list like","value":[["NavigationNet: A Large-scale Interactive Indoor Navigation Dataset"],["Abstract Indoor navigation aims at performing navigation within buildings.","In scenes like home and factory, most intelligent mobile devices require an functionality of routing to guide itself precisely through indoor scenes to complete various tasks in order to serve human.","In most scenarios, we expected an intelligent device capable of navigating itself in unseen environment.","Although several solutions have been proposed to deal with this issue, they usually require pre-installed beacons or a map pre-built with SLAM, which means that they are not capable of working in novel environments.","To address this, we proposed NavigationNet, a computer vision dataset and benchmark to allow the utilization of deep reinforcement learning on scene-understanding-based indoor navigation.","We also proposed and formalized several typical indoor routing problems that are suitable for deep reinforcement learning."],["Introduction","Indoor navigation aims at preforming navigation within buildings, such as train stations, airports, shopping centers, offices and museums.","Most intelligent devices require an indoor routing functionality to guide itself precisely through rooms to complete various tasks.","In most of the scenarios, we expect an intelligent device capable of navigating itself in new environments to serve human.","In the past years, several solutions have been proposed to deal with this problem [1].","Although they can be applied in some special cases, none of them can work smartly in novel environments or are based on scene understanding.","Those solutions require either dense pre-installed beacons (e.g.","WiFi/Bluetooth chips) over the scene or a map pre-built with SLAM in which human workers are required to carry cameras to scan the scene in advance.","Such requirements significantly limit the applications of these solutions.","New scenes have to be well prepared before the robots can work properly in it, which makes it impossible to be put into large-scale application in various indoor scenarios.","Additionally, neither beacons nor maps are helpful in semantically understanding indoor scenes and thus will lead to the failure of the robot to perform many practical tasks.","For example, to complete a regular command like 'to take a cup', a housekeeping android should understand what the scene is like, what a cup is and where a cup usually be, to navigate properly.","We expected a more general solution of indoor navigation problems that requires no preparation of the scene in advance.","Such a high demand requires a more human-like robot that is much more intelligent to implement self-exploration in novel environments.","Recent years, we have witnessed the rapid development of deep reinforcement learning (DRL) [2], [3] and computer vision [4], [5].","Therefore, smart indoor navigation enabled by DRL with visual inputs has been introduced [6].","In this setting, pre-installed beacons or pre-built map are no longer required.","A smart robot with vision will be enough to do the navigation job in different scenes.","To apply reinforcement learning methods on navigation problems, massive trial-and-error is necessary in training the model.","A straight-forward way is to construct a real robot and train Reinforcement Learning (RL) models on real-world scenes.","However, such process is extremely slow and costly.","Additionally, it is often accompanied by robot collision and damage, which means that more time and resources would be wasted.","Even worse, due to the bottleneck of physical movements, to execute any action in the real world requires at least several seconds, which is unacceptable for any machine learning methods since that they usually require millions of times of trial-and-error.","Aside from those drawbacks, such a process raises the problem of reproducibility.","Even if all the background settings of a research are published in detail, it is still very hard for third parties to reproduce the experiment result since it is costly and time-consuming to find or construct a new environment that is identical to the one the original author used.","That makes it impossible for third parties to examine and evaluate a new research result.","We believe, an open-source, low-cost, large-scale dataset and corresponding benchmarks would be a key to largely advance this research field.","Some previous work has tried to train the robot in computer graphics (CG) scenes like AI2-THOR[6], SUNCG[7].","However, we do have noticed that training models on CG environments raised the problem of model transferring as the gap between hand-crafted scenes and the real world is significant.","The visual appearance of images rendered by CG system always look non-realistic.","Existing 3D models are limited in representing real-world scenes and objects.","To deal with this problem, we proposed NavigationNet, a large-scale, real-world, interactive dataset.","In each scene, we use human-size robot with 8 cameras to capture photos towards 8 different directions at all walk-able positions.","Given all the collected images, we can easily tour the room like a CG system by moving among those walk-able positions.","However, different from CG systems, what we observed from this dataset are all real images captured from the real world.","Therefore, we can virtually command a robot to walk in the scene and get the first-perspective view of the robot.","Our dataset is interactive.","That is to say, as we send a command of movement (e.g.","moving forward, turning right), the robot will perform as required and return a new view it sees from the new position.","To our best knowledge, our NavigationNet is the first real-world indoor scene dataset that is interactive in computer vision field.","Our dataset covers about 1500 $m^2$ indoor area with various bedrooms, studies, meeting room and etc.","This makes the bias less in experiments on our dataset.","Based on NavigationNet, we proposed four applications that require indoor navigation.","We will explained them in detail in chapter .","To summarize, we presented a large-scale dataset consisting of visual data collected from real scenarios to allow a low-cost, reproducible training of indoor navigation robots based on reinforcement learning methods.","We also proposed and formalized several practical indoor navigation tasks in the framework of reinforcement learning."],["Related Work","In this chapter, we will discuss some of the previous works that are related to our work or act as a tool in this project.","First we will talk about the technology of Visual Semantic Planning.","A brief introduction to the popular datasets in the field of computer vision, after which we will explain three techniques used in our work in detail: Reinforcement Learning, Convolutional Neural Network(CNN) and Simultaneous Localization and Mapping."],["Visual Semantic Planning","Fundamental tasks in mobile robot controlling include navigation, mapping, grasping, path planning and so on.","Such task of interacting with a visual world and planning a sequence of actions to achieve a certain goal is addressed as Visual Semantic Planning [8].","A series of work [9], [10], [11], [12] has addressed the task-level planning and learning from dynamics.","Classical approaches [13], [14] of motion planning require building map purely geometrically using Light Detection and Ranging (LiDAR) or Structure from Motion (SfM) [15], [16], [17].","Srivastava et al.","[18] provides an interface between task and motion planning, such that the task planner can effectively operate in an abstracted state space that ignores geometry.","Recently proposed end-to-end learning-based approaches [19], [10], [20] can go directly from pixels to actions.","For instance Zhu et al.","[19] adopt feed-forward neural network to target-driven visual navigation in indoor scenes.","Gupta et al.","[10] use online Cognitive Mapping and Planning (CMP) approach for goal direction visual navigation without requiring a pre-constructed map.","Jang et al.","[9] train a deep neural network to perform the semantic grasping task inspired by the \"two-stream hypothesis\" of human vision.","In terms of Complete Path Planning, Kollar et al.","[21] use reinforcement learning to find trajectories and the learnt policy transfers successfully to a real environment."],["Datasets for Computer Vision","Datasets and corresponding benchmarks have played a significant role in many areas such as computer vision and speech recognition.","The evolution from WordNet[22] to ImageNet[23] was one of the many successful examples that proved the power of an effective dataset to accelerate the development of one area.","On the contrary, the lack of appropriate data has become one of the most crucial challenges for deep reinforcement learning, especially when it is dealing with real world problems like visual semantic learning.","Since training and quantitatively evaluating DRL algorithms in real environments is either impractical or costly, this problem has become more urgent.","For RL algorithms dealing with virtual-world problems, the Arcade Learning Environment (ALE) [24] exposed Atari 2600 games as reinforcement learning problems.","Works such as Duan et al.","[25] and Brockman et al.","[26] propose toolkits to qualify progress in reinforcement learning.","Other benchmarks [27], [28], [29], [30], [31], simulators [32] or physics engines [33] are also designed for the development of deep reinforcement learning.","Also, scientists are trying to model the real world for DRL or other visual tasks.","Chang et al.","[34] provides 90 building-scale reconstructed scenes for supervised and self-supervised computer vision tasks such as keypoint matching, view overlap prediction semantic segmentation, scene classification and so on.","AI2-THOR[19], SUNCG[7] and House3D are all CG scenes designed especially for deep reinforcement learning.","The advanced version of AI2-THOR[35] even provides the functionality to let the robot directly interact with the objects in the scenarios.","Different from our NavigationNet, these datasets are either synthetic or reconstructed using computer graphics techniques, which introduce inconsistency distribution between the real-world scenarios and the produced ones."],["Reinforcement Learning","Reinforcement learning (RL) [36] method was proposed in the late 90s.","It provides a technique to allow the robot or any other intelligent systems to build a value function to evaluate policies in given situations from an interactive way of trail-and-error.","Pioneering works from Mnih et al.","[37], Lillicrap et al.","[38], Schulman et al.","[39] and Silver et al [40] introduced and ignited the idea of join Reinforcement Learning and Deep Learning into Deep Reinforcement Learning.","In learning complex behavior skills and solving challenging control tasks in high-dimensional raw sensory state-space, DRL methods have shown tremendous success [40], [41], [3], [42].","Trust Region Policy Optimization (TRPO) [41] makes a series of approximations to the theoretically-justified procedure and has robust performance on a wide range of tasks.","Asynchronous Advantage Actor-Critic (A3C) Algorithm [3], which uses asynchronous gradient descent for optimization, succeeds on task of navigating random 3D mazes as well as a wide variety of continuous motor control problems.","UNsupervised REinforcement and Auxiliary Learning (UNREAL) [43] brings together the A3C framework with auxiliary control tasks and auxiliary reward tasks.","Actor Critic using Kronecker-factored Trust Region (ACKTR) [42] Algorithm uses a Kronecker-factored approximation to natural policy gradient that allows the covariance of the gradient to be inverted efficiently.","Proximal Policy Optimization (PPO) [44] Algorithms use multiple epochs of stochastic gradient ascent to perform each policy update.","Reinforcement Learning (RL) provides a powerful and flexible framework to several applications.","Andrew Y. Ng et al.","[45] describe a successful application of autonomous helicopter.","Finn et al.","[12] present an approach that automates state-space construction by learning a state representation directly from camera images with deep spatial autoencoder.","Heess et al.","[46] explore a rich environment can help to promote the learning of complex behavior.","They normalize observations, scale the reward and use per-batch normalization of the advantages.","Gu et al.","[47] demonstrate a DRL algorithm based on off-policy training of deep Q-function can scale to complex 3D manipulation tasks and can learn deep neural network policies efficiently enough to train on real physical robots.","Denil et al.","[48] find DRL methods can learn to perform the experiments necessary to discover properties such as mass and cohesion of objects."],["Convolutional Neural Network","Deep convolutional neural networks [49], [50] have led to a series of breakthrough for visual tasks.","He et al.","[5] present a Residual Network (ResNet) to ease the training of deep neural network.","We extract feature map with ResNet 101.","Huang et al.","[51] presents Dense Convolutional Network (DenseNet) which connects each layer to every other layer in a feed-forward fashion.","Chen et al.","[52] propose Dual Path Net (DPN) by revealing ResNet and DenseNet within the higher order recurrent neural network framework.","Deep Learning models have been successful in learning powerful representations and understanding stereo context [53], [54].","Shaked et al.","[55] present a three-step pipeline for the stereo matching problem and a network architecture (ResMatch) for computing the matching cost at each possible disparity.","Kendall et al.","[56] propose an end-to-end model (GC-Net) for regressing disparity from a rectified pair of stereo images."],["Simultaneous Localization and Mapping","Visual SLAM plays an important role in autonomous navigation of mobile robots [57].","The emergence of MonoSLAM [58] makes the idea of utilizing one camera become popular.","Parallel Tracking and Mapping (PTaM) [59] uses an approach based on keyframes with two parallel processing threads.","ORB [60] becomes a computationally-efficient replacement to SIFT keypoint detector and descriptor that has similar matching performance.","Keyframe bundle adjustment outperforms filtering, since it gives the most accuracy per unit of computing time [61].","S-PTAM [62] allows accuracy improvement of the mapping process with respect to monocular SLAM and avoiding the well-known bootstrapping problem.","ORB-SLAM2 [63] is a complete SLAM system for monocular, stereo and RGB-D cameras.","RGB-D SLAM [64] can producing high quality globally consistent surface reconstruction with only a low-cost commodity RGB-D sensor.","Mur-Artal et al.","[65] propose a visual-inertial monocular SLAM which is able to close loops and reuse its map to achieve zero-drift localization in already mapped areas."],["NavigationNet","NavigationNet is specifically designed for applying reinforcement learning methods to indoor navigation tasks.","We collected data from the real world and organized it properly to allow the robot to roam in the dataset as if they are in the corresponding real-world scenario.","In this chapter, we will explain its organization and applications in detail to show you the world of NavigationNet."],["Data Organization","Images in NavigationNet are organized hierarchically.","The hightest node is the root of NavigationNet.","The second level nodes, also the primary elements, are scenes.","A scene is a collection of images collected from the same indoor space.","In the next level, a scene is divided into rooms, space connected with doors.","In the current version of NavigationNet, we have 15 scenes, each with 1-3 rooms.","The origin room for each scene is at least 50 $m^2$ in area.","The fourth level is called postion.","A position is from where images are collected.","A room contains hundreds of postions.","When being trained, the robot could move among the adjacent positions to 'see' from that perspective.","When constructing the dataset, our mobile robot iterated over all the walkable points in the room with the granularity of 20cm.","Hence, in the dataset, two adjacent positions are 20cm away from each other.","With such a granularity, one room usually contains thounsands of positions.","From the perspective of information, a position can be seen as a bundle of information that the robot will receive when it is placed at that position.","Usually, we offer 8 images in this bundle.","These images are divided into four directions: front, back, left and right.","Towards each direction, two parallel images are taken to complete a binocular vision system.","Such a setting enables the possibilities like stereo matching and panorama reconstruction.","In addition to the images, we also provide a ground-truth map attached to the scene.","The map can be considered as a binary map indicating where is walkable or non-walkable.","This is essential especially to the Auto-SLAM problem."],["Robot Moving Control","To satisfy the different requirements of potential tasks, we tried our best to provide a more generalizable moving control SDK for NavigationNet.","As is stated in section REF , the fundamental element in NavigationNet is scene, virtual robots are allowed to roam in scenes.","To serve well as an action in a reinforcement learning task, movements should be standard, discrete, limited and complete.","An action should be standard, so that the robot should move the same distance towards the same direction or turning the same angle when a specific instruction of a movement is given.","An action should be discrete, so that the action space can be finite and discrete.","Actions should be limited, so that the algorithm do not need to choose actions from an infinite action space.","Actions should be complete, so that all the actions joined together could describe the whole real-world action space.","In practice, it is impossible to use a finite action space to cover an infinite one.","We need some sort of compromise.","The one we take in this case is to discrete the movement length and the turning angle.","So that we defined six types of movements.","MOVE FORWARD Ask the robot to move forward one position.","For example, go from (9, 12) to (9, 13) when facing north but go from (11, 8) to (10, 8) when facing west.","Since two adjacent nodes are 20cm away, this is to move forward 20cm in real-world scenes.","MOVE BACKWARD Ask the robot to move backward one position.","This is the opposite movement of MOVE FORWARD and can revert the effect of MOVE FORWARD.","MOVE LEFT Ask the robot to move left one position.","For example, go from (5, 4) to (4, 4) when facing north but go from (7, 4) to (7, 3) when facing west.","The real world distance would be the same as MOVE FORWARD and MOVE BACKWARD.","It should be paid attention to that MOVE LEFT is very different from TURNING LEFT as it will move the position of the robot but will not change the facing direction.","MOVE RIGHT Ask the robot to move right one position.","This is the opposite movement of MOVE LEFT and can revert the effect of MOVE LEFT.","TURN LEFT Ask the robot to turn left at 90-degree.","For example, when the robot receives this request when at (5, 7) facing north, it should then facing west still at (5, 7).","It should be paid extra attention that this movement is different from MOVE LEFT that the former changes the facing while the latter change the position.","TURN RIGHT Ask the robot to turn right at 90-degree.","This is the opposite movement of TURN LEFT.","Figure: MovementsAt the same time, we should make it clear that, in any given tasks, according to the specific setting, not all the actions (movement) need to be taken into consideration.","For example, when we would like to simulate a two-wheel one-eye robot, TURN LEFT and TURN RIGHT are a must, otherwise seventy-five percent of the images can never be perceived while MOVE LEFT and MOVE RIGHT should be eliminated since a two-wheel robot can not make the exact movement of moving left 20cm.","Meanwhile, when simulating a robot with panorama vision, we could only take MOVE LEFT and MOVE RIGHT but not TURN LEFT and TURN RIGHT as they are not necessary but can only enlarge the action space unnecessarily and make the task more difficult."],["NavigationNet Construction","NavigationNet is a large-scale dataset with hundreds of thousands of images.","The large amount makes the data collection a task of impossible.","What is worst is that, as to each image, it is required that the position, angle and height where it is taken should be exactly where it is expected so long that the simulation of reality will not offset.","These two factors together make the task of collection much more difficult than expected.","In this chapter, we will talk about how we constructed this dataset efficiently and accurately."],["Collector Mobile Robot","Robots are designed to liberate human beings from the repetitive boring work.","NavigationNet is built for intelligent robots, also built with the help of intelligent robots.","Collecting data for NavigationNet requires heavy labor and high accuracy but little flexibility (most of the possible conditions are under control), which is surprisingly suitable for robots.","In order to reduce the labor requirement, we exploited the possibilities of smart hardware.","Our team developed a dedicated data-collecting mobile robot with Arduino Mega2560 and Raspberry Pi 3 Model B codenamed GoodCar.","Arduino[66] is an open-source physical computing platform based on a simple I/O board and a development environment that implements the Processing/Wiring language.","Arduino boards can be programmed in the same name program language Arduino, which is a C-like area-specific language to receive and send signals from tens of low-voltage I/O ports.","It is specifically suitable for robot controlling.","In this project, we use a Arduino Mega2560 to control the movement of the robot.","Raspberry Pi[67] is a cheap single-board computer system running a specificialy made Linux distribution.","It is originally built for computer programming education but we find it suitable for robot controlling.","It consumes little electricity which allows us to strip the 220V power wires.","It contains all kinds of I/O ports such that we could plug it with Arduinos, cameras, other full-sized computers and many other modules to make it the center of the system.","It runs standard linux distribution such that the possibility of software extension is beyond imagine.","Last but not least, it is cheap so that we could us as many as we want on the robot.","In practice, we use two Raspberry Pi 3B for controlling all other modules and communicating with the upper computer.","To build this robot, we used a two-level structure.","The upper-level is as high as 1.4m to simulate the the height of the eyes of an average adult.","We plugged-in eight cameras on this level, two for each direction to make a stereo vision system and to avoid unnecessary turning-around.","The lower-level is for the motion systems.","The Raspberry Pi and Arduino mentioned above are all on this level.","The Arduino is to control the mobile devices and sensors.","It is linked with four motors to control the movement.","In the meantime, it is linked with a sonar and code plate counters to avoid collision and measure the distance of movement.","Also it is linked with a Raspberry Pi and under its control.","We used two Raspberry Pi 3B in each robot, one is master and the other is slave.","The master computer controls many things.","First it communicates with the Arduino to control the movement indirectly.","Second it is responsible for evaluating the signal Arduino sent back to avoid collision.","Third it is master computer's duty to control four of the cameras to take photo at given position and store the images on the SD card.","Fourth the slave computer is also under its control to take photos when required.","Lastly it communicates with the upper computer, which is controlled by human, via Wi-Fi connection.","That is the control core of our robot.","Under the lower-level is the mobile devices.","We have tried on many types of mobile devices and at last we used four track structures to reduce error.","We had constructed six mobile robots before moving on to collect data.","Figure: Robot"],["Data Collection","To make the data collecting process efficient and accurate, also to make the whole process manageable and under control, we designed and followed the following instructions: Step 1 Measure the size of the target room(s).","Find the most southwestern corner as the starting point.","Step 2 Move the robot to the starting point.","Evaluate and record its distance towards the borders.","Step 3 Let the robot face north and start to collect required photos at this point.","Step 4 After photos are taken, let the robot run 20cm towards north and collect data.","Step 5 Keep iterating over Step 3-4 until the robot reaches the end of the line.","Step 6 Move the robot to the starting point of the line and then move 20cm towards east.","Step 7 Iterating over Step3-6 until the robot reaches the east corner.","Step 8 Complete this scenario by taking photos from those points that the robot did not reach.","Data Processing After the collection, the original data should be pre-processed before taken into production.","The data processing steps we taken includes: Data Cleaning When collecting data with robot, there are always inevitable errors occurring especially when dealing with room corners.","For each room we collect, we will scan over all the collected photos and find errors.","If necessary, we would go and retry collecting data to complete the set.","Map Building When collecting data, we also recorded whether a point is accessible.","With this information, we are able to rebuild a walkable-or-not map for reference.","Stereo Vision For some of the task, to simulate a two-eye system or a RGB-D system, it would be of much help if we could produce depth channel in the very beginning.","To serve this purpose, we use MC-CNN[54] to produce some pre-processed depth channel data.","Applications NavigationNet is designed for indoor navigation problems.","In this chapter, we would introduce four applications base on our dataset and formalize them with the idea of reinforcement learning.","Apart from these four applications, we believe our dataset will spawn more applications.","Target-driven Navigation Motivation In the real scenarios, as for robots, the key problem of indoor navigation is not about going to some place but finding some specific item.","For example, as a home android, we do not ask it to get to the kitchen safely but to fetch the milk for the master.","Also, as a robot in a warehouse, the ability to go to some specific location is never more important than finding a given good.","To achieve this goal, a poorly trained robot control system may have to traverse over everywhere it can reach to search target object, which is neither safe nor efficient, since it has no prior knowledge like milk should be in the kitchen, or tables are usually aligned to the wall.","This usually falls in the field of visual semantic learning.","That is to say, to be well prepared for this kind of tasks, the robots should be able to understand the scene, knowing, for example, what the object it perceived is.","Figure: Target-driven Navigation(Upper: Random walking, Lower: DRL-based) Formalization The formalization of this issue is inspired by Yuke Zhu et al.[6].","To complete the task of target-driven navigation, a robot control system is given two images.","One is the target image, the other is the current first-person perspective image perceived from the environment.","The robot can choose to walk forward, backward, turn left or right according to the input images.","The environment will update the states accordingly and give out a reward.","This process loops until the target image is the same with the perceived one.","To formalize that: Goal The goal is straight-forward, to find the given object.","In practice, it can be converted into make the perceived image and the given image identical.","Reward Different from the the goal, the design of the goal is not that straight-forward and requires more discussion.","One thing that is sure is that a large positive reward will be given when the goal is achieved.","To improve time efficiency, each step it taken should be given a small negative reward.","In this paper, we use the setting of one large positive reward at the goal and minor negative in each step.","Action Space To find an object, the understanding of the scene is the most important factor of the task.","Thus we would like to simulate a one-eye robot in this problem and the Action space would be defined as { MOVE FORWARD, MOVE BACKWARD, TURN LEFT, TURN RIGHT } State Space In this problem, we need not only the current perception as input but also the target image.","Hence the State space would be the set of all possible image pairs.","Evaluation To evaluate such a system, many aspects should be taken into consideration.","The most important two are, trajectory should be short and the robot should not run into the obstacles.","Collisions should be limited strictly as only one collision could greatly damage the robot.","To make a trajectory finite, we consider a trajectory longer than 10000 a loop.","To calculate a overall score on all the test results, we utilize the idea of histogram.","A success trajectory with $n$ steps and $m$ is a vote weighting $max(1-m/10, 0)$ to $n$ .","A overall score is calculated by finding the average over this histogram.","Multi-target Navigation Motivation The target-driven navigation task is focused on finding one object efficiently and safely.","However, in practical scenarios, this is often not enough.","In many cases, we need to find more than one items and we do not care about the order.","For example, in the last example about collecting goods from the different rooms and then putting them all into a box, although it is required that the robot should get to the box in the very end, we do not really care about which item it goes to get first.","This has become a rather different problem since it usually requires the knowledge of adjacency.","For example, milk is usually placed near breads while books are usually far away, so although the item order in the list maybe milk, books and breads, robot should take reads instead of books first to advance the efficiency.","Figure: Multi-target Navigation (Left: Random walking, Right: DRL-based) Formalization We also formalized this problem into a reinforcement learning problem.","To find all the given items, the robot is given a list of items to collect, each one is represented by an image.","The rest will be similar with the target-driven task.","A first-person perspective image is given, an action is executed and then state is updated, reward is given.","To formalize that: Goal The goal is also straight-forward, that the agent should find all the required object no matter the order.","Reward The reward design would be similar with the target-driven navigation task though we would like to the finding of each object a separated but relatively smaller reward.","We believe that such a design would be better for deep reinforcement training[43].","Also a minor negative time time punishment is necessary.","Action Space Since the task type is similar with the target-driven problem.","The same action space remains.","{ MOVE FORWARD, MOVE BACKWARD, TURN LEFT, TURN RIGHT } State Space Different from the previous task, this task requires multiple images of objects to be perceived as targets.","However, a set images with unknown-length is hard to be processed.","Thus in this situation, we would limit the number of target inputs to 2 and the space of states would be the set of all possible package of three images Evaluation Although the task is different from the previous task, the two major meters are the same.","Hence, we use the same metric system in this task as is in the target-driven navigation task.","Sweeper Route Planning Motivation Apart from target-driven navigation, there are also other valuable tasks in the field of indoor navigation.","One widely-discussed but never well-addressed problem is the sweeper route planning problem.","Contrary to the previous target-driven tasks, a sweeper robot should traverse over all the possible places in a given room without collision and less repetition.","The current solutions often lead to stuck at some small space.","Besides sweeper robot, it is general problem we will meet in many scenarios.","This problem is usually considered as coverage path planning (CPP) problem, which is usually dealt with using Boustrophedon method.","However, the sweeper routing planning problem using visual semantic learning is actually very different from CPP problem.","The most important factor is that the robot will not know that some movement is blocked until the collision really happens.","It must see and understand to avoid collisions.","Figure: Sweeper Route Planning (Upper: Random walking, Lower: DRL-based) Formalization Again, we formalized this problem into a reinforcement learning task.","At any point, the robot is given an image, the first-person perception of the robot and an action should be given to the environment indicating which way the robot is going the next time slice.","The environment should give out the reward then according to how much area has been covered by the sweeper.","Typically, the robot should maintain a map of the room inside its system.","To formalize that: Goal As a sweeper robot, surely the goal is to cover all the points without obstacles in least time.","However, this goal is straight-forward but practically impossible to achieve as that many corners are hard to get and there is no need to sweep and to achieve a least time path it usually requires the god's perspective.","Reward Reward design is simple in this problem.","Since we do not pursuit a real all-cover path, we should reward every new position it gets and punish every fruitless movement.","That is to say, we should give a large positive reward for a new position and a minor negative reward for an old position.","Action Space In this task, a sweeper does not really care about which direction it is facing.","The only thing it cares is that whether the place it is positioned and it will be positioned have been cleaned (went before).","So we will not give the robot the ability to turn left/right.","As the result, the action space shrinks to { MOVE FORWARD, MOVE BACKWARD, MOVE LEFT, MOVE RIGHT }.","State Space Different form the previous two tasks, sweeper path planning problem does not require a specific target as an input so that the state space would be as simple as a set of all the images that the robot may see.","Evaluation The target for a sweeper is to sweep as much area as possible as fast as possible though these two aspect are usually contradict to each other.","To deal with this, we defined a score similar to mAP score[68] in object detection tasks to evaluate models.","For the $n$ step in a trajectory, we define $m$ that in the first $n$ steps, $m$ points are covered.","If the $n$ step is a new hit, a score of $m/n$ is given to the step, otherwise 0.","An average over scores of all steps in a trajectory is the final score of the trajectory.","Mean Average Precision (mAP) [H] mAPsteps $score \\leftarrow 0$ $n \\leftarrow 0$ $m \\leftarrow 0$ $step$ in $steps$ $n \\leftarrow n + 1$ $step$ is $NEW\\_POSITION$ $m \\leftarrow m + 1$ $score \\leftarrow score + m/n$ $score \\leftarrow score + 0$ $score \\leftarrow score / n$ $score$ Auto-SLAM with Deep Reinforcement Learning Motivation Simultaneous localization and mapping (SLAM) problem has been paid much attention to since it is introduced decades ago.","Many solutions have been proposed to increase the mapping accuracy.","However, due to the lack of smart robots capable of navigating itself in alien environments, robots for SLAM tasks are still human controlled, which limited the application of SLAM technology significantly in dangerous places.","Additionally, applying SLAM for some large-scale scenes (e.g.","Museum) is effort consuming, which requires us to carry camera walked about all corners.","So, we hope that robots can preform SLAM autonomously.","With the help of NavigationNet, we proposed a Auto-SLAM task, to train a robot capable of navigating itself in an alien environment and complete the SLAM task without collision.","Formalization This task is also formalized as a reinforcement learning problem.","On each step, the robot is fed with images perceived at its current location.","The system could use the visual inputs and history information to improve the map.","Reward should be calculated according to the difference between the constructed map and the ground truth.","An action to lead the robot should be given on the basis of the reward and the current map, upon which the perception is updated.","To formally put it: Goal Surely the goal is to build a 3D reconstruction efficiently and accurately.","Though this would become a much more sophisticated problem than the other three due to the difficulties in evaluating the correctness of a new reconstruction.","Reward Due to the difficulty of evaluation in SLAM problems, we would like to consider more about progress.","By talking about progress, we are assuming total trust in legacy SLAM algorithms.","We would like to give a positive reward for any new reconstructed area after each steps.","In the meantime, a fruitless step is also punished with a minor negative reward.","Action Space Due to the problem which will be talked about in State Space part, the function of turning-around is no longer required.","In the meantime, we need to add the function of moving left/right to the robot.","Hence the action space would become { MOVE FORWARD, MOVE BACKWARD, MOVE LEFT, MOVE RIGHT } State Space Unlike the former three problems, SLAM is a problem far more than path planning.","Most of it is about 3D reconstruction.","Thus only a first-person single-eye perception is far from enough.","In this setting, the environment should provide stereo vision from the four direction with depth information.","Evaluation SLAM tasks are divided into two part, mapping and positioning.","Building maps is a way for positioning.","Hence, we use the accuracy of positioning but not the of mapping as the benchmark.","To evaluate a Auto-SLAM model, we ask the robot to run SLAM task in an alien enviroment with limited time.","After that, 10 images are given as query to the positioning system.","The score is calculated by average over the positioning success rate.","Conclusion We believe, to advance a research field, a suitable training environment and corresponding benchmarks will be the best catalyst.","In this article, we proposed NavigationNet, a large-scale, open-source, low-cost, real-world dataset for indoor navigation.","By introducing this dataset, we hope to construct a platform where researchers can train and evaluate their` own robot controlling system without really constructing the robot.","We extract robot controlling system out from robot itself.","We hope to eliminate the long and high-cost preparation process before really training the system.","Also, by constructing this dataset, we hope to make deep learning methods work on real robots.","On NavigationNet, a robot will see as if it is in the physical world but the speed of action will become acceptable for massive trail-and-error.","We also proposed four possible tasks that can be conducted on our dataset.","We are hoping that more tasks can be proposed to exploit its maximum potential.","Our future work includes increasing the number of scenes in the dataset.","Also we will construct more attributes in the dataset (object segmentation for example)."],["Applications","NavigationNet is designed for indoor navigation problems.","In this chapter, we would introduce four applications base on our dataset and formalize them with the idea of reinforcement learning.","Apart from these four applications, we believe our dataset will spawn more applications."],["Motivation","In the real scenarios, as for robots, the key problem of indoor navigation is not about going to some place but finding some specific item.","For example, as a home android, we do not ask it to get to the kitchen safely but to fetch the milk for the master.","Also, as a robot in a warehouse, the ability to go to some specific location is never more important than finding a given good.","To achieve this goal, a poorly trained robot control system may have to traverse over everywhere it can reach to search target object, which is neither safe nor efficient, since it has no prior knowledge like milk should be in the kitchen, or tables are usually aligned to the wall.","This usually falls in the field of visual semantic learning.","That is to say, to be well prepared for this kind of tasks, the robots should be able to understand the scene, knowing, for example, what the object it perceived is.","Figure: Target-driven Navigation(Upper: Random walking, Lower: DRL-based)The formalization of this issue is inspired by Yuke Zhu et al.[6].","To complete the task of target-driven navigation, a robot control system is given two images.","One is the target image, the other is the current first-person perspective image perceived from the environment.","The robot can choose to walk forward, backward, turn left or right according to the input images.","The environment will update the states accordingly and give out a reward.","This process loops until the target image is the same with the perceived one.","To formalize that: Goal The goal is straight-forward, to find the given object.","In practice, it can be converted into make the perceived image and the given image identical.","Reward Different from the the goal, the design of the goal is not that straight-forward and requires more discussion.","One thing that is sure is that a large positive reward will be given when the goal is achieved.","To improve time efficiency, each step it taken should be given a small negative reward.","In this paper, we use the setting of one large positive reward at the goal and minor negative in each step.","Action Space To find an object, the understanding of the scene is the most important factor of the task.","Thus we would like to simulate a one-eye robot in this problem and the Action space would be defined as { MOVE FORWARD, MOVE BACKWARD, TURN LEFT, TURN RIGHT } State Space In this problem, we need not only the current perception as input but also the target image.","Hence the State space would be the set of all possible image pairs.","To evaluate such a system, many aspects should be taken into consideration.","The most important two are, trajectory should be short and the robot should not run into the obstacles.","Collisions should be limited strictly as only one collision could greatly damage the robot.","To make a trajectory finite, we consider a trajectory longer than 10000 a loop.","To calculate a overall score on all the test results, we utilize the idea of histogram.","A success trajectory with $n$ steps and $m$ is a vote weighting $max(1-m/10, 0)$ to $n$ .","A overall score is calculated by finding the average over this histogram.","The target-driven navigation task is focused on finding one object efficiently and safely.","However, in practical scenarios, this is often not enough.","In many cases, we need to find more than one items and we do not care about the order.","For example, in the last example about collecting goods from the different rooms and then putting them all into a box, although it is required that the robot should get to the box in the very end, we do not really care about which item it goes to get first.","This has become a rather different problem since it usually requires the knowledge of adjacency.","For example, milk is usually placed near breads while books are usually far away, so although the item order in the list maybe milk, books and breads, robot should take reads instead of books first to advance the efficiency.","Figure: Multi-target Navigation (Left: Random walking, Right: DRL-based)We also formalized this problem into a reinforcement learning problem.","To find all the given items, the robot is given a list of items to collect, each one is represented by an image.","The rest will be similar with the target-driven task.","A first-person perspective image is given, an action is executed and then state is updated, reward is given.","To formalize that: Goal The goal is also straight-forward, that the agent should find all the required object no matter the order.","Reward The reward design would be similar with the target-driven navigation task though we would like to the finding of each object a separated but relatively smaller reward.","We believe that such a design would be better for deep reinforcement training[43].","Also a minor negative time time punishment is necessary.","Action Space Since the task type is similar with the target-driven problem.","The same action space remains.","{ MOVE FORWARD, MOVE BACKWARD, TURN LEFT, TURN RIGHT } State Space Different from the previous task, this task requires multiple images of objects to be perceived as targets.","However, a set images with unknown-length is hard to be processed.","Thus in this situation, we would limit the number of target inputs to 2 and the space of states would be the set of all possible package of three images Although the task is different from the previous task, the two major meters are the same.","Hence, we use the same metric system in this task as is in the target-driven navigation task.","Apart from target-driven navigation, there are also other valuable tasks in the field of indoor navigation.","One widely-discussed but never well-addressed problem is the sweeper route planning problem.","Contrary to the previous target-driven tasks, a sweeper robot should traverse over all the possible places in a given room without collision and less repetition.","The current solutions often lead to stuck at some small space.","Besides sweeper robot, it is general problem we will meet in many scenarios.","This problem is usually considered as coverage path planning (CPP) problem, which is usually dealt with using Boustrophedon method.","However, the sweeper routing planning problem using visual semantic learning is actually very different from CPP problem.","The most important factor is that the robot will not know that some movement is blocked until the collision really happens.","It must see and understand to avoid collisions.","Figure: Sweeper Route Planning (Upper: Random walking, Lower: DRL-based)Again, we formalized this problem into a reinforcement learning task.","At any point, the robot is given an image, the first-person perception of the robot and an action should be given to the environment indicating which way the robot is going the next time slice.","The environment should give out the reward then according to how much area has been covered by the sweeper.","Typically, the robot should maintain a map of the room inside its system.","To formalize that: Goal As a sweeper robot, surely the goal is to cover all the points without obstacles in least time.","However, this goal is straight-forward but practically impossible to achieve as that many corners are hard to get and there is no need to sweep and to achieve a least time path it usually requires the god's perspective.","Reward Reward design is simple in this problem.","Since we do not pursuit a real all-cover path, we should reward every new position it gets and punish every fruitless movement.","That is to say, we should give a large positive reward for a new position and a minor negative reward for an old position.","Action Space In this task, a sweeper does not really care about which direction it is facing.","The only thing it cares is that whether the place it is positioned and it will be positioned have been cleaned (went before).","So we will not give the robot the ability to turn left/right.","As the result, the action space shrinks to { MOVE FORWARD, MOVE BACKWARD, MOVE LEFT, MOVE RIGHT }.","State Space Different form the previous two tasks, sweeper path planning problem does not require a specific target as an input so that the state space would be as simple as a set of all the images that the robot may see.","The target for a sweeper is to sweep as much area as possible as fast as possible though these two aspect are usually contradict to each other.","To deal with this, we defined a score similar to mAP score[68] in object detection tasks to evaluate models.","For the $n$ step in a trajectory, we define $m$ that in the first $n$ steps, $m$ points are covered.","If the $n$ step is a new hit, a score of $m/n$ is given to the step, otherwise 0.","An average over scores of all steps in a trajectory is the final score of the trajectory.","Mean Average Precision (mAP) [H] mAPsteps $score \\leftarrow 0$ $n \\leftarrow 0$ $m \\leftarrow 0$ $step$ in $steps$ $n \\leftarrow n + 1$ $step$ is $NEW\\_POSITION$ $m \\leftarrow m + 1$ $score \\leftarrow score + m/n$ $score \\leftarrow score + 0$ $score \\leftarrow score / n$ $score$ Simultaneous localization and mapping (SLAM) problem has been paid much attention to since it is introduced decades ago.","Many solutions have been proposed to increase the mapping accuracy.","However, due to the lack of smart robots capable of navigating itself in alien environments, robots for SLAM tasks are still human controlled, which limited the application of SLAM technology significantly in dangerous places.","Additionally, applying SLAM for some large-scale scenes (e.g.","Museum) is effort consuming, which requires us to carry camera walked about all corners.","So, we hope that robots can preform SLAM autonomously.","With the help of NavigationNet, we proposed a Auto-SLAM task, to train a robot capable of navigating itself in an alien environment and complete the SLAM task without collision.","This task is also formalized as a reinforcement learning problem.","On each step, the robot is fed with images perceived at its current location.","The system could use the visual inputs and history information to improve the map.","Reward should be calculated according to the difference between the constructed map and the ground truth.","An action to lead the robot should be given on the basis of the reward and the current map, upon which the perception is updated.","To formally put it: Goal Surely the goal is to build a 3D reconstruction efficiently and accurately.","Though this would become a much more sophisticated problem than the other three due to the difficulties in evaluating the correctness of a new reconstruction.","Reward Due to the difficulty of evaluation in SLAM problems, we would like to consider more about progress.","By talking about progress, we are assuming total trust in legacy SLAM algorithms.","We would like to give a positive reward for any new reconstructed area after each steps.","In the meantime, a fruitless step is also punished with a minor negative reward.","Action Space Due to the problem which will be talked about in State Space part, the function of turning-around is no longer required.","In the meantime, we need to add the function of moving left/right to the robot.","Hence the action space would become { MOVE FORWARD, MOVE BACKWARD, MOVE LEFT, MOVE RIGHT } State Space Unlike the former three problems, SLAM is a problem far more than path planning.","Most of it is about 3D reconstruction.","Thus only a first-person single-eye perception is far from enough.","In this setting, the environment should provide stereo vision from the four direction with depth information.","SLAM tasks are divided into two part, mapping and positioning.","Building maps is a way for positioning.","Hence, we use the accuracy of positioning but not the of mapping as the benchmark.","To evaluate a Auto-SLAM model, we ask the robot to run SLAM task in an alien enviroment with limited time.","After that, 10 images are given as query to the positioning system.","The score is calculated by average over the positioning success rate."],["Conclusion","We believe, to advance a research field, a suitable training environment and corresponding benchmarks will be the best catalyst.","In this article, we proposed NavigationNet, a large-scale, open-source, low-cost, real-world dataset for indoor navigation.","By introducing this dataset, we hope to construct a platform where researchers can train and evaluate their` own robot controlling system without really constructing the robot.","We extract robot controlling system out from robot itself.","We hope to eliminate the long and high-cost preparation process before really training the system.","Also, by constructing this dataset, we hope to make deep learning methods work on real robots.","On NavigationNet, a robot will see as if it is in the physical world but the speed of action will become acceptable for massive trail-and-error.","We also proposed four possible tasks that can be conducted on our dataset.","We are hoping that more tasks can be proposed to exploit its maximum potential.","Our future work includes increasing the number of scenes in the dataset.","Also we will construct more attributes in the dataset (object segmentation for example)."]],"string":"[\n [\n \"NavigationNet: A Large-scale Interactive Indoor Navigation Dataset\"\n ],\n [\n \"Abstract Indoor navigation aims at performing navigation within buildings.\",\n \"In scenes like home and factory, most intelligent mobile devices require an functionality of routing to guide itself precisely through indoor scenes to complete various tasks in order to serve human.\",\n \"In most scenarios, we expected an intelligent device capable of navigating itself in unseen environment.\",\n \"Although several solutions have been proposed to deal with this issue, they usually require pre-installed beacons or a map pre-built with SLAM, which means that they are not capable of working in novel environments.\",\n \"To address this, we proposed NavigationNet, a computer vision dataset and benchmark to allow the utilization of deep reinforcement learning on scene-understanding-based indoor navigation.\",\n \"We also proposed and formalized several typical indoor routing problems that are suitable for deep reinforcement learning.\"\n ],\n [\n \"Introduction\",\n \"Indoor navigation aims at preforming navigation within buildings, such as train stations, airports, shopping centers, offices and museums.\",\n \"Most intelligent devices require an indoor routing functionality to guide itself precisely through rooms to complete various tasks.\",\n \"In most of the scenarios, we expect an intelligent device capable of navigating itself in new environments to serve human.\",\n \"In the past years, several solutions have been proposed to deal with this problem [1].\",\n \"Although they can be applied in some special cases, none of them can work smartly in novel environments or are based on scene understanding.\",\n \"Those solutions require either dense pre-installed beacons (e.g.\",\n \"WiFi/Bluetooth chips) over the scene or a map pre-built with SLAM in which human workers are required to carry cameras to scan the scene in advance.\",\n \"Such requirements significantly limit the applications of these solutions.\",\n \"New scenes have to be well prepared before the robots can work properly in it, which makes it impossible to be put into large-scale application in various indoor scenarios.\",\n \"Additionally, neither beacons nor maps are helpful in semantically understanding indoor scenes and thus will lead to the failure of the robot to perform many practical tasks.\",\n \"For example, to complete a regular command like 'to take a cup', a housekeeping android should understand what the scene is like, what a cup is and where a cup usually be, to navigate properly.\",\n \"We expected a more general solution of indoor navigation problems that requires no preparation of the scene in advance.\",\n \"Such a high demand requires a more human-like robot that is much more intelligent to implement self-exploration in novel environments.\",\n \"Recent years, we have witnessed the rapid development of deep reinforcement learning (DRL) [2], [3] and computer vision [4], [5].\",\n \"Therefore, smart indoor navigation enabled by DRL with visual inputs has been introduced [6].\",\n \"In this setting, pre-installed beacons or pre-built map are no longer required.\",\n \"A smart robot with vision will be enough to do the navigation job in different scenes.\",\n \"To apply reinforcement learning methods on navigation problems, massive trial-and-error is necessary in training the model.\",\n \"A straight-forward way is to construct a real robot and train Reinforcement Learning (RL) models on real-world scenes.\",\n \"However, such process is extremely slow and costly.\",\n \"Additionally, it is often accompanied by robot collision and damage, which means that more time and resources would be wasted.\",\n \"Even worse, due to the bottleneck of physical movements, to execute any action in the real world requires at least several seconds, which is unacceptable for any machine learning methods since that they usually require millions of times of trial-and-error.\",\n \"Aside from those drawbacks, such a process raises the problem of reproducibility.\",\n \"Even if all the background settings of a research are published in detail, it is still very hard for third parties to reproduce the experiment result since it is costly and time-consuming to find or construct a new environment that is identical to the one the original author used.\",\n \"That makes it impossible for third parties to examine and evaluate a new research result.\",\n \"We believe, an open-source, low-cost, large-scale dataset and corresponding benchmarks would be a key to largely advance this research field.\",\n \"Some previous work has tried to train the robot in computer graphics (CG) scenes like AI2-THOR[6], SUNCG[7].\",\n \"However, we do have noticed that training models on CG environments raised the problem of model transferring as the gap between hand-crafted scenes and the real world is significant.\",\n \"The visual appearance of images rendered by CG system always look non-realistic.\",\n \"Existing 3D models are limited in representing real-world scenes and objects.\",\n \"To deal with this problem, we proposed NavigationNet, a large-scale, real-world, interactive dataset.\",\n \"In each scene, we use human-size robot with 8 cameras to capture photos towards 8 different directions at all walk-able positions.\",\n \"Given all the collected images, we can easily tour the room like a CG system by moving among those walk-able positions.\",\n \"However, different from CG systems, what we observed from this dataset are all real images captured from the real world.\",\n \"Therefore, we can virtually command a robot to walk in the scene and get the first-perspective view of the robot.\",\n \"Our dataset is interactive.\",\n \"That is to say, as we send a command of movement (e.g.\",\n \"moving forward, turning right), the robot will perform as required and return a new view it sees from the new position.\",\n \"To our best knowledge, our NavigationNet is the first real-world indoor scene dataset that is interactive in computer vision field.\",\n \"Our dataset covers about 1500 $m^2$ indoor area with various bedrooms, studies, meeting room and etc.\",\n \"This makes the bias less in experiments on our dataset.\",\n \"Based on NavigationNet, we proposed four applications that require indoor navigation.\",\n \"We will explained them in detail in chapter .\",\n \"To summarize, we presented a large-scale dataset consisting of visual data collected from real scenarios to allow a low-cost, reproducible training of indoor navigation robots based on reinforcement learning methods.\",\n \"We also proposed and formalized several practical indoor navigation tasks in the framework of reinforcement learning.\"\n ],\n [\n \"Related Work\",\n \"In this chapter, we will discuss some of the previous works that are related to our work or act as a tool in this project.\",\n \"First we will talk about the technology of Visual Semantic Planning.\",\n \"A brief introduction to the popular datasets in the field of computer vision, after which we will explain three techniques used in our work in detail: Reinforcement Learning, Convolutional Neural Network(CNN) and Simultaneous Localization and Mapping.\"\n ],\n [\n \"Visual Semantic Planning\",\n \"Fundamental tasks in mobile robot controlling include navigation, mapping, grasping, path planning and so on.\",\n \"Such task of interacting with a visual world and planning a sequence of actions to achieve a certain goal is addressed as Visual Semantic Planning [8].\",\n \"A series of work [9], [10], [11], [12] has addressed the task-level planning and learning from dynamics.\",\n \"Classical approaches [13], [14] of motion planning require building map purely geometrically using Light Detection and Ranging (LiDAR) or Structure from Motion (SfM) [15], [16], [17].\",\n \"Srivastava et al.\",\n \"[18] provides an interface between task and motion planning, such that the task planner can effectively operate in an abstracted state space that ignores geometry.\",\n \"Recently proposed end-to-end learning-based approaches [19], [10], [20] can go directly from pixels to actions.\",\n \"For instance Zhu et al.\",\n \"[19] adopt feed-forward neural network to target-driven visual navigation in indoor scenes.\",\n \"Gupta et al.\",\n \"[10] use online Cognitive Mapping and Planning (CMP) approach for goal direction visual navigation without requiring a pre-constructed map.\",\n \"Jang et al.\",\n \"[9] train a deep neural network to perform the semantic grasping task inspired by the \\\"two-stream hypothesis\\\" of human vision.\",\n \"In terms of Complete Path Planning, Kollar et al.\",\n \"[21] use reinforcement learning to find trajectories and the learnt policy transfers successfully to a real environment.\"\n ],\n [\n \"Datasets for Computer Vision\",\n \"Datasets and corresponding benchmarks have played a significant role in many areas such as computer vision and speech recognition.\",\n \"The evolution from WordNet[22] to ImageNet[23] was one of the many successful examples that proved the power of an effective dataset to accelerate the development of one area.\",\n \"On the contrary, the lack of appropriate data has become one of the most crucial challenges for deep reinforcement learning, especially when it is dealing with real world problems like visual semantic learning.\",\n \"Since training and quantitatively evaluating DRL algorithms in real environments is either impractical or costly, this problem has become more urgent.\",\n \"For RL algorithms dealing with virtual-world problems, the Arcade Learning Environment (ALE) [24] exposed Atari 2600 games as reinforcement learning problems.\",\n \"Works such as Duan et al.\",\n \"[25] and Brockman et al.\",\n \"[26] propose toolkits to qualify progress in reinforcement learning.\",\n \"Other benchmarks [27], [28], [29], [30], [31], simulators [32] or physics engines [33] are also designed for the development of deep reinforcement learning.\",\n \"Also, scientists are trying to model the real world for DRL or other visual tasks.\",\n \"Chang et al.\",\n \"[34] provides 90 building-scale reconstructed scenes for supervised and self-supervised computer vision tasks such as keypoint matching, view overlap prediction semantic segmentation, scene classification and so on.\",\n \"AI2-THOR[19], SUNCG[7] and House3D are all CG scenes designed especially for deep reinforcement learning.\",\n \"The advanced version of AI2-THOR[35] even provides the functionality to let the robot directly interact with the objects in the scenarios.\",\n \"Different from our NavigationNet, these datasets are either synthetic or reconstructed using computer graphics techniques, which introduce inconsistency distribution between the real-world scenarios and the produced ones.\"\n ],\n [\n \"Reinforcement Learning\",\n \"Reinforcement learning (RL) [36] method was proposed in the late 90s.\",\n \"It provides a technique to allow the robot or any other intelligent systems to build a value function to evaluate policies in given situations from an interactive way of trail-and-error.\",\n \"Pioneering works from Mnih et al.\",\n \"[37], Lillicrap et al.\",\n \"[38], Schulman et al.\",\n \"[39] and Silver et al [40] introduced and ignited the idea of join Reinforcement Learning and Deep Learning into Deep Reinforcement Learning.\",\n \"In learning complex behavior skills and solving challenging control tasks in high-dimensional raw sensory state-space, DRL methods have shown tremendous success [40], [41], [3], [42].\",\n \"Trust Region Policy Optimization (TRPO) [41] makes a series of approximations to the theoretically-justified procedure and has robust performance on a wide range of tasks.\",\n \"Asynchronous Advantage Actor-Critic (A3C) Algorithm [3], which uses asynchronous gradient descent for optimization, succeeds on task of navigating random 3D mazes as well as a wide variety of continuous motor control problems.\",\n \"UNsupervised REinforcement and Auxiliary Learning (UNREAL) [43] brings together the A3C framework with auxiliary control tasks and auxiliary reward tasks.\",\n \"Actor Critic using Kronecker-factored Trust Region (ACKTR) [42] Algorithm uses a Kronecker-factored approximation to natural policy gradient that allows the covariance of the gradient to be inverted efficiently.\",\n \"Proximal Policy Optimization (PPO) [44] Algorithms use multiple epochs of stochastic gradient ascent to perform each policy update.\",\n \"Reinforcement Learning (RL) provides a powerful and flexible framework to several applications.\",\n \"Andrew Y. Ng et al.\",\n \"[45] describe a successful application of autonomous helicopter.\",\n \"Finn et al.\",\n \"[12] present an approach that automates state-space construction by learning a state representation directly from camera images with deep spatial autoencoder.\",\n \"Heess et al.\",\n \"[46] explore a rich environment can help to promote the learning of complex behavior.\",\n \"They normalize observations, scale the reward and use per-batch normalization of the advantages.\",\n \"Gu et al.\",\n \"[47] demonstrate a DRL algorithm based on off-policy training of deep Q-function can scale to complex 3D manipulation tasks and can learn deep neural network policies efficiently enough to train on real physical robots.\",\n \"Denil et al.\",\n \"[48] find DRL methods can learn to perform the experiments necessary to discover properties such as mass and cohesion of objects.\"\n ],\n [\n \"Convolutional Neural Network\",\n \"Deep convolutional neural networks [49], [50] have led to a series of breakthrough for visual tasks.\",\n \"He et al.\",\n \"[5] present a Residual Network (ResNet) to ease the training of deep neural network.\",\n \"We extract feature map with ResNet 101.\",\n \"Huang et al.\",\n \"[51] presents Dense Convolutional Network (DenseNet) which connects each layer to every other layer in a feed-forward fashion.\",\n \"Chen et al.\",\n \"[52] propose Dual Path Net (DPN) by revealing ResNet and DenseNet within the higher order recurrent neural network framework.\",\n \"Deep Learning models have been successful in learning powerful representations and understanding stereo context [53], [54].\",\n \"Shaked et al.\",\n \"[55] present a three-step pipeline for the stereo matching problem and a network architecture (ResMatch) for computing the matching cost at each possible disparity.\",\n \"Kendall et al.\",\n \"[56] propose an end-to-end model (GC-Net) for regressing disparity from a rectified pair of stereo images.\"\n ],\n [\n \"Simultaneous Localization and Mapping\",\n \"Visual SLAM plays an important role in autonomous navigation of mobile robots [57].\",\n \"The emergence of MonoSLAM [58] makes the idea of utilizing one camera become popular.\",\n \"Parallel Tracking and Mapping (PTaM) [59] uses an approach based on keyframes with two parallel processing threads.\",\n \"ORB [60] becomes a computationally-efficient replacement to SIFT keypoint detector and descriptor that has similar matching performance.\",\n \"Keyframe bundle adjustment outperforms filtering, since it gives the most accuracy per unit of computing time [61].\",\n \"S-PTAM [62] allows accuracy improvement of the mapping process with respect to monocular SLAM and avoiding the well-known bootstrapping problem.\",\n \"ORB-SLAM2 [63] is a complete SLAM system for monocular, stereo and RGB-D cameras.\",\n \"RGB-D SLAM [64] can producing high quality globally consistent surface reconstruction with only a low-cost commodity RGB-D sensor.\",\n \"Mur-Artal et al.\",\n \"[65] propose a visual-inertial monocular SLAM which is able to close loops and reuse its map to achieve zero-drift localization in already mapped areas.\"\n ],\n [\n \"NavigationNet\",\n \"NavigationNet is specifically designed for applying reinforcement learning methods to indoor navigation tasks.\",\n \"We collected data from the real world and organized it properly to allow the robot to roam in the dataset as if they are in the corresponding real-world scenario.\",\n \"In this chapter, we will explain its organization and applications in detail to show you the world of NavigationNet.\"\n ],\n [\n \"Data Organization\",\n \"Images in NavigationNet are organized hierarchically.\",\n \"The hightest node is the root of NavigationNet.\",\n \"The second level nodes, also the primary elements, are scenes.\",\n \"A scene is a collection of images collected from the same indoor space.\",\n \"In the next level, a scene is divided into rooms, space connected with doors.\",\n \"In the current version of NavigationNet, we have 15 scenes, each with 1-3 rooms.\",\n \"The origin room for each scene is at least 50 $m^2$ in area.\",\n \"The fourth level is called postion.\",\n \"A position is from where images are collected.\",\n \"A room contains hundreds of postions.\",\n \"When being trained, the robot could move among the adjacent positions to 'see' from that perspective.\",\n \"When constructing the dataset, our mobile robot iterated over all the walkable points in the room with the granularity of 20cm.\",\n \"Hence, in the dataset, two adjacent positions are 20cm away from each other.\",\n \"With such a granularity, one room usually contains thounsands of positions.\",\n \"From the perspective of information, a position can be seen as a bundle of information that the robot will receive when it is placed at that position.\",\n \"Usually, we offer 8 images in this bundle.\",\n \"These images are divided into four directions: front, back, left and right.\",\n \"Towards each direction, two parallel images are taken to complete a binocular vision system.\",\n \"Such a setting enables the possibilities like stereo matching and panorama reconstruction.\",\n \"In addition to the images, we also provide a ground-truth map attached to the scene.\",\n \"The map can be considered as a binary map indicating where is walkable or non-walkable.\",\n \"This is essential especially to the Auto-SLAM problem.\"\n ],\n [\n \"Robot Moving Control\",\n \"To satisfy the different requirements of potential tasks, we tried our best to provide a more generalizable moving control SDK for NavigationNet.\",\n \"As is stated in section REF , the fundamental element in NavigationNet is scene, virtual robots are allowed to roam in scenes.\",\n \"To serve well as an action in a reinforcement learning task, movements should be standard, discrete, limited and complete.\",\n \"An action should be standard, so that the robot should move the same distance towards the same direction or turning the same angle when a specific instruction of a movement is given.\",\n \"An action should be discrete, so that the action space can be finite and discrete.\",\n \"Actions should be limited, so that the algorithm do not need to choose actions from an infinite action space.\",\n \"Actions should be complete, so that all the actions joined together could describe the whole real-world action space.\",\n \"In practice, it is impossible to use a finite action space to cover an infinite one.\",\n \"We need some sort of compromise.\",\n \"The one we take in this case is to discrete the movement length and the turning angle.\",\n \"So that we defined six types of movements.\",\n \"MOVE FORWARD Ask the robot to move forward one position.\",\n \"For example, go from (9, 12) to (9, 13) when facing north but go from (11, 8) to (10, 8) when facing west.\",\n \"Since two adjacent nodes are 20cm away, this is to move forward 20cm in real-world scenes.\",\n \"MOVE BACKWARD Ask the robot to move backward one position.\",\n \"This is the opposite movement of MOVE FORWARD and can revert the effect of MOVE FORWARD.\",\n \"MOVE LEFT Ask the robot to move left one position.\",\n \"For example, go from (5, 4) to (4, 4) when facing north but go from (7, 4) to (7, 3) when facing west.\",\n \"The real world distance would be the same as MOVE FORWARD and MOVE BACKWARD.\",\n \"It should be paid attention to that MOVE LEFT is very different from TURNING LEFT as it will move the position of the robot but will not change the facing direction.\",\n \"MOVE RIGHT Ask the robot to move right one position.\",\n \"This is the opposite movement of MOVE LEFT and can revert the effect of MOVE LEFT.\",\n \"TURN LEFT Ask the robot to turn left at 90-degree.\",\n \"For example, when the robot receives this request when at (5, 7) facing north, it should then facing west still at (5, 7).\",\n \"It should be paid extra attention that this movement is different from MOVE LEFT that the former changes the facing while the latter change the position.\",\n \"TURN RIGHT Ask the robot to turn right at 90-degree.\",\n \"This is the opposite movement of TURN LEFT.\",\n \"Figure: MovementsAt the same time, we should make it clear that, in any given tasks, according to the specific setting, not all the actions (movement) need to be taken into consideration.\",\n \"For example, when we would like to simulate a two-wheel one-eye robot, TURN LEFT and TURN RIGHT are a must, otherwise seventy-five percent of the images can never be perceived while MOVE LEFT and MOVE RIGHT should be eliminated since a two-wheel robot can not make the exact movement of moving left 20cm.\",\n \"Meanwhile, when simulating a robot with panorama vision, we could only take MOVE LEFT and MOVE RIGHT but not TURN LEFT and TURN RIGHT as they are not necessary but can only enlarge the action space unnecessarily and make the task more difficult.\"\n ],\n [\n \"NavigationNet Construction\",\n \"NavigationNet is a large-scale dataset with hundreds of thousands of images.\",\n \"The large amount makes the data collection a task of impossible.\",\n \"What is worst is that, as to each image, it is required that the position, angle and height where it is taken should be exactly where it is expected so long that the simulation of reality will not offset.\",\n \"These two factors together make the task of collection much more difficult than expected.\",\n \"In this chapter, we will talk about how we constructed this dataset efficiently and accurately.\"\n ],\n [\n \"Collector Mobile Robot\",\n \"Robots are designed to liberate human beings from the repetitive boring work.\",\n \"NavigationNet is built for intelligent robots, also built with the help of intelligent robots.\",\n \"Collecting data for NavigationNet requires heavy labor and high accuracy but little flexibility (most of the possible conditions are under control), which is surprisingly suitable for robots.\",\n \"In order to reduce the labor requirement, we exploited the possibilities of smart hardware.\",\n \"Our team developed a dedicated data-collecting mobile robot with Arduino Mega2560 and Raspberry Pi 3 Model B codenamed GoodCar.\",\n \"Arduino[66] is an open-source physical computing platform based on a simple I/O board and a development environment that implements the Processing/Wiring language.\",\n \"Arduino boards can be programmed in the same name program language Arduino, which is a C-like area-specific language to receive and send signals from tens of low-voltage I/O ports.\",\n \"It is specifically suitable for robot controlling.\",\n \"In this project, we use a Arduino Mega2560 to control the movement of the robot.\",\n \"Raspberry Pi[67] is a cheap single-board computer system running a specificialy made Linux distribution.\",\n \"It is originally built for computer programming education but we find it suitable for robot controlling.\",\n \"It consumes little electricity which allows us to strip the 220V power wires.\",\n \"It contains all kinds of I/O ports such that we could plug it with Arduinos, cameras, other full-sized computers and many other modules to make it the center of the system.\",\n \"It runs standard linux distribution such that the possibility of software extension is beyond imagine.\",\n \"Last but not least, it is cheap so that we could us as many as we want on the robot.\",\n \"In practice, we use two Raspberry Pi 3B for controlling all other modules and communicating with the upper computer.\",\n \"To build this robot, we used a two-level structure.\",\n \"The upper-level is as high as 1.4m to simulate the the height of the eyes of an average adult.\",\n \"We plugged-in eight cameras on this level, two for each direction to make a stereo vision system and to avoid unnecessary turning-around.\",\n \"The lower-level is for the motion systems.\",\n \"The Raspberry Pi and Arduino mentioned above are all on this level.\",\n \"The Arduino is to control the mobile devices and sensors.\",\n \"It is linked with four motors to control the movement.\",\n \"In the meantime, it is linked with a sonar and code plate counters to avoid collision and measure the distance of movement.\",\n \"Also it is linked with a Raspberry Pi and under its control.\",\n \"We used two Raspberry Pi 3B in each robot, one is master and the other is slave.\",\n \"The master computer controls many things.\",\n \"First it communicates with the Arduino to control the movement indirectly.\",\n \"Second it is responsible for evaluating the signal Arduino sent back to avoid collision.\",\n \"Third it is master computer's duty to control four of the cameras to take photo at given position and store the images on the SD card.\",\n \"Fourth the slave computer is also under its control to take photos when required.\",\n \"Lastly it communicates with the upper computer, which is controlled by human, via Wi-Fi connection.\",\n \"That is the control core of our robot.\",\n \"Under the lower-level is the mobile devices.\",\n \"We have tried on many types of mobile devices and at last we used four track structures to reduce error.\",\n \"We had constructed six mobile robots before moving on to collect data.\",\n \"Figure: Robot\"\n ],\n [\n \"Data Collection\",\n \"To make the data collecting process efficient and accurate, also to make the whole process manageable and under control, we designed and followed the following instructions: Step 1 Measure the size of the target room(s).\",\n \"Find the most southwestern corner as the starting point.\",\n \"Step 2 Move the robot to the starting point.\",\n \"Evaluate and record its distance towards the borders.\",\n \"Step 3 Let the robot face north and start to collect required photos at this point.\",\n \"Step 4 After photos are taken, let the robot run 20cm towards north and collect data.\",\n \"Step 5 Keep iterating over Step 3-4 until the robot reaches the end of the line.\",\n \"Step 6 Move the robot to the starting point of the line and then move 20cm towards east.\",\n \"Step 7 Iterating over Step3-6 until the robot reaches the east corner.\",\n \"Step 8 Complete this scenario by taking photos from those points that the robot did not reach.\",\n \"Data Processing After the collection, the original data should be pre-processed before taken into production.\",\n \"The data processing steps we taken includes: Data Cleaning When collecting data with robot, there are always inevitable errors occurring especially when dealing with room corners.\",\n \"For each room we collect, we will scan over all the collected photos and find errors.\",\n \"If necessary, we would go and retry collecting data to complete the set.\",\n \"Map Building When collecting data, we also recorded whether a point is accessible.\",\n \"With this information, we are able to rebuild a walkable-or-not map for reference.\",\n \"Stereo Vision For some of the task, to simulate a two-eye system or a RGB-D system, it would be of much help if we could produce depth channel in the very beginning.\",\n \"To serve this purpose, we use MC-CNN[54] to produce some pre-processed depth channel data.\",\n \"Applications NavigationNet is designed for indoor navigation problems.\",\n \"In this chapter, we would introduce four applications base on our dataset and formalize them with the idea of reinforcement learning.\",\n \"Apart from these four applications, we believe our dataset will spawn more applications.\",\n \"Target-driven Navigation Motivation In the real scenarios, as for robots, the key problem of indoor navigation is not about going to some place but finding some specific item.\",\n \"For example, as a home android, we do not ask it to get to the kitchen safely but to fetch the milk for the master.\",\n \"Also, as a robot in a warehouse, the ability to go to some specific location is never more important than finding a given good.\",\n \"To achieve this goal, a poorly trained robot control system may have to traverse over everywhere it can reach to search target object, which is neither safe nor efficient, since it has no prior knowledge like milk should be in the kitchen, or tables are usually aligned to the wall.\",\n \"This usually falls in the field of visual semantic learning.\",\n \"That is to say, to be well prepared for this kind of tasks, the robots should be able to understand the scene, knowing, for example, what the object it perceived is.\",\n \"Figure: Target-driven Navigation(Upper: Random walking, Lower: DRL-based) Formalization The formalization of this issue is inspired by Yuke Zhu et al.[6].\",\n \"To complete the task of target-driven navigation, a robot control system is given two images.\",\n \"One is the target image, the other is the current first-person perspective image perceived from the environment.\",\n \"The robot can choose to walk forward, backward, turn left or right according to the input images.\",\n \"The environment will update the states accordingly and give out a reward.\",\n \"This process loops until the target image is the same with the perceived one.\",\n \"To formalize that: Goal The goal is straight-forward, to find the given object.\",\n \"In practice, it can be converted into make the perceived image and the given image identical.\",\n \"Reward Different from the the goal, the design of the goal is not that straight-forward and requires more discussion.\",\n \"One thing that is sure is that a large positive reward will be given when the goal is achieved.\",\n \"To improve time efficiency, each step it taken should be given a small negative reward.\",\n \"In this paper, we use the setting of one large positive reward at the goal and minor negative in each step.\",\n \"Action Space To find an object, the understanding of the scene is the most important factor of the task.\",\n \"Thus we would like to simulate a one-eye robot in this problem and the Action space would be defined as { MOVE FORWARD, MOVE BACKWARD, TURN LEFT, TURN RIGHT } State Space In this problem, we need not only the current perception as input but also the target image.\",\n \"Hence the State space would be the set of all possible image pairs.\",\n \"Evaluation To evaluate such a system, many aspects should be taken into consideration.\",\n \"The most important two are, trajectory should be short and the robot should not run into the obstacles.\",\n \"Collisions should be limited strictly as only one collision could greatly damage the robot.\",\n \"To make a trajectory finite, we consider a trajectory longer than 10000 a loop.\",\n \"To calculate a overall score on all the test results, we utilize the idea of histogram.\",\n \"A success trajectory with $n$ steps and $m$ is a vote weighting $max(1-m/10, 0)$ to $n$ .\",\n \"A overall score is calculated by finding the average over this histogram.\",\n \"Multi-target Navigation Motivation The target-driven navigation task is focused on finding one object efficiently and safely.\",\n \"However, in practical scenarios, this is often not enough.\",\n \"In many cases, we need to find more than one items and we do not care about the order.\",\n \"For example, in the last example about collecting goods from the different rooms and then putting them all into a box, although it is required that the robot should get to the box in the very end, we do not really care about which item it goes to get first.\",\n \"This has become a rather different problem since it usually requires the knowledge of adjacency.\",\n \"For example, milk is usually placed near breads while books are usually far away, so although the item order in the list maybe milk, books and breads, robot should take reads instead of books first to advance the efficiency.\",\n \"Figure: Multi-target Navigation (Left: Random walking, Right: DRL-based) Formalization We also formalized this problem into a reinforcement learning problem.\",\n \"To find all the given items, the robot is given a list of items to collect, each one is represented by an image.\",\n \"The rest will be similar with the target-driven task.\",\n \"A first-person perspective image is given, an action is executed and then state is updated, reward is given.\",\n \"To formalize that: Goal The goal is also straight-forward, that the agent should find all the required object no matter the order.\",\n \"Reward The reward design would be similar with the target-driven navigation task though we would like to the finding of each object a separated but relatively smaller reward.\",\n \"We believe that such a design would be better for deep reinforcement training[43].\",\n \"Also a minor negative time time punishment is necessary.\",\n \"Action Space Since the task type is similar with the target-driven problem.\",\n \"The same action space remains.\",\n \"{ MOVE FORWARD, MOVE BACKWARD, TURN LEFT, TURN RIGHT } State Space Different from the previous task, this task requires multiple images of objects to be perceived as targets.\",\n \"However, a set images with unknown-length is hard to be processed.\",\n \"Thus in this situation, we would limit the number of target inputs to 2 and the space of states would be the set of all possible package of three images Evaluation Although the task is different from the previous task, the two major meters are the same.\",\n \"Hence, we use the same metric system in this task as is in the target-driven navigation task.\",\n \"Sweeper Route Planning Motivation Apart from target-driven navigation, there are also other valuable tasks in the field of indoor navigation.\",\n \"One widely-discussed but never well-addressed problem is the sweeper route planning problem.\",\n \"Contrary to the previous target-driven tasks, a sweeper robot should traverse over all the possible places in a given room without collision and less repetition.\",\n \"The current solutions often lead to stuck at some small space.\",\n \"Besides sweeper robot, it is general problem we will meet in many scenarios.\",\n \"This problem is usually considered as coverage path planning (CPP) problem, which is usually dealt with using Boustrophedon method.\",\n \"However, the sweeper routing planning problem using visual semantic learning is actually very different from CPP problem.\",\n \"The most important factor is that the robot will not know that some movement is blocked until the collision really happens.\",\n \"It must see and understand to avoid collisions.\",\n \"Figure: Sweeper Route Planning (Upper: Random walking, Lower: DRL-based) Formalization Again, we formalized this problem into a reinforcement learning task.\",\n \"At any point, the robot is given an image, the first-person perception of the robot and an action should be given to the environment indicating which way the robot is going the next time slice.\",\n \"The environment should give out the reward then according to how much area has been covered by the sweeper.\",\n \"Typically, the robot should maintain a map of the room inside its system.\",\n \"To formalize that: Goal As a sweeper robot, surely the goal is to cover all the points without obstacles in least time.\",\n \"However, this goal is straight-forward but practically impossible to achieve as that many corners are hard to get and there is no need to sweep and to achieve a least time path it usually requires the god's perspective.\",\n \"Reward Reward design is simple in this problem.\",\n \"Since we do not pursuit a real all-cover path, we should reward every new position it gets and punish every fruitless movement.\",\n \"That is to say, we should give a large positive reward for a new position and a minor negative reward for an old position.\",\n \"Action Space In this task, a sweeper does not really care about which direction it is facing.\",\n \"The only thing it cares is that whether the place it is positioned and it will be positioned have been cleaned (went before).\",\n \"So we will not give the robot the ability to turn left/right.\",\n \"As the result, the action space shrinks to { MOVE FORWARD, MOVE BACKWARD, MOVE LEFT, MOVE RIGHT }.\",\n \"State Space Different form the previous two tasks, sweeper path planning problem does not require a specific target as an input so that the state space would be as simple as a set of all the images that the robot may see.\",\n \"Evaluation The target for a sweeper is to sweep as much area as possible as fast as possible though these two aspect are usually contradict to each other.\",\n \"To deal with this, we defined a score similar to mAP score[68] in object detection tasks to evaluate models.\",\n \"For the $n$ step in a trajectory, we define $m$ that in the first $n$ steps, $m$ points are covered.\",\n \"If the $n$ step is a new hit, a score of $m/n$ is given to the step, otherwise 0.\",\n \"An average over scores of all steps in a trajectory is the final score of the trajectory.\",\n \"Mean Average Precision (mAP) [H] mAPsteps $score \\\\leftarrow 0$ $n \\\\leftarrow 0$ $m \\\\leftarrow 0$ $step$ in $steps$ $n \\\\leftarrow n + 1$ $step$ is $NEW\\\\_POSITION$ $m \\\\leftarrow m + 1$ $score \\\\leftarrow score + m/n$ $score \\\\leftarrow score + 0$ $score \\\\leftarrow score / n$ $score$ Auto-SLAM with Deep Reinforcement Learning Motivation Simultaneous localization and mapping (SLAM) problem has been paid much attention to since it is introduced decades ago.\",\n \"Many solutions have been proposed to increase the mapping accuracy.\",\n \"However, due to the lack of smart robots capable of navigating itself in alien environments, robots for SLAM tasks are still human controlled, which limited the application of SLAM technology significantly in dangerous places.\",\n \"Additionally, applying SLAM for some large-scale scenes (e.g.\",\n \"Museum) is effort consuming, which requires us to carry camera walked about all corners.\",\n \"So, we hope that robots can preform SLAM autonomously.\",\n \"With the help of NavigationNet, we proposed a Auto-SLAM task, to train a robot capable of navigating itself in an alien environment and complete the SLAM task without collision.\",\n \"Formalization This task is also formalized as a reinforcement learning problem.\",\n \"On each step, the robot is fed with images perceived at its current location.\",\n \"The system could use the visual inputs and history information to improve the map.\",\n \"Reward should be calculated according to the difference between the constructed map and the ground truth.\",\n \"An action to lead the robot should be given on the basis of the reward and the current map, upon which the perception is updated.\",\n \"To formally put it: Goal Surely the goal is to build a 3D reconstruction efficiently and accurately.\",\n \"Though this would become a much more sophisticated problem than the other three due to the difficulties in evaluating the correctness of a new reconstruction.\",\n \"Reward Due to the difficulty of evaluation in SLAM problems, we would like to consider more about progress.\",\n \"By talking about progress, we are assuming total trust in legacy SLAM algorithms.\",\n \"We would like to give a positive reward for any new reconstructed area after each steps.\",\n \"In the meantime, a fruitless step is also punished with a minor negative reward.\",\n \"Action Space Due to the problem which will be talked about in State Space part, the function of turning-around is no longer required.\",\n \"In the meantime, we need to add the function of moving left/right to the robot.\",\n \"Hence the action space would become { MOVE FORWARD, MOVE BACKWARD, MOVE LEFT, MOVE RIGHT } State Space Unlike the former three problems, SLAM is a problem far more than path planning.\",\n \"Most of it is about 3D reconstruction.\",\n \"Thus only a first-person single-eye perception is far from enough.\",\n \"In this setting, the environment should provide stereo vision from the four direction with depth information.\",\n \"Evaluation SLAM tasks are divided into two part, mapping and positioning.\",\n \"Building maps is a way for positioning.\",\n \"Hence, we use the accuracy of positioning but not the of mapping as the benchmark.\",\n \"To evaluate a Auto-SLAM model, we ask the robot to run SLAM task in an alien enviroment with limited time.\",\n \"After that, 10 images are given as query to the positioning system.\",\n \"The score is calculated by average over the positioning success rate.\",\n \"Conclusion We believe, to advance a research field, a suitable training environment and corresponding benchmarks will be the best catalyst.\",\n \"In this article, we proposed NavigationNet, a large-scale, open-source, low-cost, real-world dataset for indoor navigation.\",\n \"By introducing this dataset, we hope to construct a platform where researchers can train and evaluate their` own robot controlling system without really constructing the robot.\",\n \"We extract robot controlling system out from robot itself.\",\n \"We hope to eliminate the long and high-cost preparation process before really training the system.\",\n \"Also, by constructing this dataset, we hope to make deep learning methods work on real robots.\",\n \"On NavigationNet, a robot will see as if it is in the physical world but the speed of action will become acceptable for massive trail-and-error.\",\n \"We also proposed four possible tasks that can be conducted on our dataset.\",\n \"We are hoping that more tasks can be proposed to exploit its maximum potential.\",\n \"Our future work includes increasing the number of scenes in the dataset.\",\n \"Also we will construct more attributes in the dataset (object segmentation for example).\"\n ],\n [\n \"Applications\",\n \"NavigationNet is designed for indoor navigation problems.\",\n \"In this chapter, we would introduce four applications base on our dataset and formalize them with the idea of reinforcement learning.\",\n \"Apart from these four applications, we believe our dataset will spawn more applications.\"\n ],\n [\n \"Motivation\",\n \"In the real scenarios, as for robots, the key problem of indoor navigation is not about going to some place but finding some specific item.\",\n \"For example, as a home android, we do not ask it to get to the kitchen safely but to fetch the milk for the master.\",\n \"Also, as a robot in a warehouse, the ability to go to some specific location is never more important than finding a given good.\",\n \"To achieve this goal, a poorly trained robot control system may have to traverse over everywhere it can reach to search target object, which is neither safe nor efficient, since it has no prior knowledge like milk should be in the kitchen, or tables are usually aligned to the wall.\",\n \"This usually falls in the field of visual semantic learning.\",\n \"That is to say, to be well prepared for this kind of tasks, the robots should be able to understand the scene, knowing, for example, what the object it perceived is.\",\n \"Figure: Target-driven Navigation(Upper: Random walking, Lower: DRL-based)The formalization of this issue is inspired by Yuke Zhu et al.[6].\",\n \"To complete the task of target-driven navigation, a robot control system is given two images.\",\n \"One is the target image, the other is the current first-person perspective image perceived from the environment.\",\n \"The robot can choose to walk forward, backward, turn left or right according to the input images.\",\n \"The environment will update the states accordingly and give out a reward.\",\n \"This process loops until the target image is the same with the perceived one.\",\n \"To formalize that: Goal The goal is straight-forward, to find the given object.\",\n \"In practice, it can be converted into make the perceived image and the given image identical.\",\n \"Reward Different from the the goal, the design of the goal is not that straight-forward and requires more discussion.\",\n \"One thing that is sure is that a large positive reward will be given when the goal is achieved.\",\n \"To improve time efficiency, each step it taken should be given a small negative reward.\",\n \"In this paper, we use the setting of one large positive reward at the goal and minor negative in each step.\",\n \"Action Space To find an object, the understanding of the scene is the most important factor of the task.\",\n \"Thus we would like to simulate a one-eye robot in this problem and the Action space would be defined as { MOVE FORWARD, MOVE BACKWARD, TURN LEFT, TURN RIGHT } State Space In this problem, we need not only the current perception as input but also the target image.\",\n \"Hence the State space would be the set of all possible image pairs.\",\n \"To evaluate such a system, many aspects should be taken into consideration.\",\n \"The most important two are, trajectory should be short and the robot should not run into the obstacles.\",\n \"Collisions should be limited strictly as only one collision could greatly damage the robot.\",\n \"To make a trajectory finite, we consider a trajectory longer than 10000 a loop.\",\n \"To calculate a overall score on all the test results, we utilize the idea of histogram.\",\n \"A success trajectory with $n$ steps and $m$ is a vote weighting $max(1-m/10, 0)$ to $n$ .\",\n \"A overall score is calculated by finding the average over this histogram.\",\n \"The target-driven navigation task is focused on finding one object efficiently and safely.\",\n \"However, in practical scenarios, this is often not enough.\",\n \"In many cases, we need to find more than one items and we do not care about the order.\",\n \"For example, in the last example about collecting goods from the different rooms and then putting them all into a box, although it is required that the robot should get to the box in the very end, we do not really care about which item it goes to get first.\",\n \"This has become a rather different problem since it usually requires the knowledge of adjacency.\",\n \"For example, milk is usually placed near breads while books are usually far away, so although the item order in the list maybe milk, books and breads, robot should take reads instead of books first to advance the efficiency.\",\n \"Figure: Multi-target Navigation (Left: Random walking, Right: DRL-based)We also formalized this problem into a reinforcement learning problem.\",\n \"To find all the given items, the robot is given a list of items to collect, each one is represented by an image.\",\n \"The rest will be similar with the target-driven task.\",\n \"A first-person perspective image is given, an action is executed and then state is updated, reward is given.\",\n \"To formalize that: Goal The goal is also straight-forward, that the agent should find all the required object no matter the order.\",\n \"Reward The reward design would be similar with the target-driven navigation task though we would like to the finding of each object a separated but relatively smaller reward.\",\n \"We believe that such a design would be better for deep reinforcement training[43].\",\n \"Also a minor negative time time punishment is necessary.\",\n \"Action Space Since the task type is similar with the target-driven problem.\",\n \"The same action space remains.\",\n \"{ MOVE FORWARD, MOVE BACKWARD, TURN LEFT, TURN RIGHT } State Space Different from the previous task, this task requires multiple images of objects to be perceived as targets.\",\n \"However, a set images with unknown-length is hard to be processed.\",\n \"Thus in this situation, we would limit the number of target inputs to 2 and the space of states would be the set of all possible package of three images Although the task is different from the previous task, the two major meters are the same.\",\n \"Hence, we use the same metric system in this task as is in the target-driven navigation task.\",\n \"Apart from target-driven navigation, there are also other valuable tasks in the field of indoor navigation.\",\n \"One widely-discussed but never well-addressed problem is the sweeper route planning problem.\",\n \"Contrary to the previous target-driven tasks, a sweeper robot should traverse over all the possible places in a given room without collision and less repetition.\",\n \"The current solutions often lead to stuck at some small space.\",\n \"Besides sweeper robot, it is general problem we will meet in many scenarios.\",\n \"This problem is usually considered as coverage path planning (CPP) problem, which is usually dealt with using Boustrophedon method.\",\n \"However, the sweeper routing planning problem using visual semantic learning is actually very different from CPP problem.\",\n \"The most important factor is that the robot will not know that some movement is blocked until the collision really happens.\",\n \"It must see and understand to avoid collisions.\",\n \"Figure: Sweeper Route Planning (Upper: Random walking, Lower: DRL-based)Again, we formalized this problem into a reinforcement learning task.\",\n \"At any point, the robot is given an image, the first-person perception of the robot and an action should be given to the environment indicating which way the robot is going the next time slice.\",\n \"The environment should give out the reward then according to how much area has been covered by the sweeper.\",\n \"Typically, the robot should maintain a map of the room inside its system.\",\n \"To formalize that: Goal As a sweeper robot, surely the goal is to cover all the points without obstacles in least time.\",\n \"However, this goal is straight-forward but practically impossible to achieve as that many corners are hard to get and there is no need to sweep and to achieve a least time path it usually requires the god's perspective.\",\n \"Reward Reward design is simple in this problem.\",\n \"Since we do not pursuit a real all-cover path, we should reward every new position it gets and punish every fruitless movement.\",\n \"That is to say, we should give a large positive reward for a new position and a minor negative reward for an old position.\",\n \"Action Space In this task, a sweeper does not really care about which direction it is facing.\",\n \"The only thing it cares is that whether the place it is positioned and it will be positioned have been cleaned (went before).\",\n \"So we will not give the robot the ability to turn left/right.\",\n \"As the result, the action space shrinks to { MOVE FORWARD, MOVE BACKWARD, MOVE LEFT, MOVE RIGHT }.\",\n \"State Space Different form the previous two tasks, sweeper path planning problem does not require a specific target as an input so that the state space would be as simple as a set of all the images that the robot may see.\",\n \"The target for a sweeper is to sweep as much area as possible as fast as possible though these two aspect are usually contradict to each other.\",\n \"To deal with this, we defined a score similar to mAP score[68] in object detection tasks to evaluate models.\",\n \"For the $n$ step in a trajectory, we define $m$ that in the first $n$ steps, $m$ points are covered.\",\n \"If the $n$ step is a new hit, a score of $m/n$ is given to the step, otherwise 0.\",\n \"An average over scores of all steps in a trajectory is the final score of the trajectory.\",\n \"Mean Average Precision (mAP) [H] mAPsteps $score \\\\leftarrow 0$ $n \\\\leftarrow 0$ $m \\\\leftarrow 0$ $step$ in $steps$ $n \\\\leftarrow n + 1$ $step$ is $NEW\\\\_POSITION$ $m \\\\leftarrow m + 1$ $score \\\\leftarrow score + m/n$ $score \\\\leftarrow score + 0$ $score \\\\leftarrow score / n$ $score$ Simultaneous localization and mapping (SLAM) problem has been paid much attention to since it is introduced decades ago.\",\n \"Many solutions have been proposed to increase the mapping accuracy.\",\n \"However, due to the lack of smart robots capable of navigating itself in alien environments, robots for SLAM tasks are still human controlled, which limited the application of SLAM technology significantly in dangerous places.\",\n \"Additionally, applying SLAM for some large-scale scenes (e.g.\",\n \"Museum) is effort consuming, which requires us to carry camera walked about all corners.\",\n \"So, we hope that robots can preform SLAM autonomously.\",\n \"With the help of NavigationNet, we proposed a Auto-SLAM task, to train a robot capable of navigating itself in an alien environment and complete the SLAM task without collision.\",\n \"This task is also formalized as a reinforcement learning problem.\",\n \"On each step, the robot is fed with images perceived at its current location.\",\n \"The system could use the visual inputs and history information to improve the map.\",\n \"Reward should be calculated according to the difference between the constructed map and the ground truth.\",\n \"An action to lead the robot should be given on the basis of the reward and the current map, upon which the perception is updated.\",\n \"To formally put it: Goal Surely the goal is to build a 3D reconstruction efficiently and accurately.\",\n \"Though this would become a much more sophisticated problem than the other three due to the difficulties in evaluating the correctness of a new reconstruction.\",\n \"Reward Due to the difficulty of evaluation in SLAM problems, we would like to consider more about progress.\",\n \"By talking about progress, we are assuming total trust in legacy SLAM algorithms.\",\n \"We would like to give a positive reward for any new reconstructed area after each steps.\",\n \"In the meantime, a fruitless step is also punished with a minor negative reward.\",\n \"Action Space Due to the problem which will be talked about in State Space part, the function of turning-around is no longer required.\",\n \"In the meantime, we need to add the function of moving left/right to the robot.\",\n \"Hence the action space would become { MOVE FORWARD, MOVE BACKWARD, MOVE LEFT, MOVE RIGHT } State Space Unlike the former three problems, SLAM is a problem far more than path planning.\",\n \"Most of it is about 3D reconstruction.\",\n \"Thus only a first-person single-eye perception is far from enough.\",\n \"In this setting, the environment should provide stereo vision from the four direction with depth information.\",\n \"SLAM tasks are divided into two part, mapping and positioning.\",\n \"Building maps is a way for positioning.\",\n \"Hence, we use the accuracy of positioning but not the of mapping as the benchmark.\",\n \"To evaluate a Auto-SLAM model, we ask the robot to run SLAM task in an alien enviroment with limited time.\",\n \"After that, 10 images are given as query to the positioning system.\",\n \"The score is calculated by average over the positioning success rate.\"\n ],\n [\n \"Conclusion\",\n \"We believe, to advance a research field, a suitable training environment and corresponding benchmarks will be the best catalyst.\",\n \"In this article, we proposed NavigationNet, a large-scale, open-source, low-cost, real-world dataset for indoor navigation.\",\n \"By introducing this dataset, we hope to construct a platform where researchers can train and evaluate their` own robot controlling system without really constructing the robot.\",\n \"We extract robot controlling system out from robot itself.\",\n \"We hope to eliminate the long and high-cost preparation process before really training the system.\",\n \"Also, by constructing this dataset, we hope to make deep learning methods work on real robots.\",\n \"On NavigationNet, a robot will see as if it is in the physical world but the speed of action will become acceptable for massive trail-and-error.\",\n \"We also proposed four possible tasks that can be conducted on our dataset.\",\n \"We are hoping that more tasks can be proposed to exploit its maximum potential.\",\n \"Our future work includes increasing the number of scenes in the dataset.\",\n \"Also we will construct more attributes in the dataset (object segmentation for example).\"\n ]\n]","isConversational":false},"id":{"kind":"string","value":"1808.08374"}}}],"truncated":false,"partial":false},"paginationData":{"pageIndex":1,"numItemsPerPage":100,"numTotalItems":1867602,"offset":100,"length":100}},"jwt":"eyJhbGciOiJFZERTQSJ9.eyJyZWFkIjp0cnVlLCJwZXJtaXNzaW9ucyI6eyJyZXBvLmNvbnRlbnQucmVhZCI6dHJ1ZX0sImlhdCI6MTc3Njc5NDYzMCwic3ViIjoiL2RhdGFzZXRzL2hvd2V5L3VuYXJYaXZlIiwiZXhwIjoxNzc2Nzk4MjMwLCJpc3MiOiJodHRwczovL2h1Z2dpbmdmYWNlLmNvIn0.WOSYCFDMTXMWIxdw0L3a3gaxwVj8zqO28yzteD3xqxyJia1bNpVhUxUuH8oGwhc2YpHNHBhUOBQmx2MKx3n5CQ","displayUrls":true,"splitSizeSummaries":[{"config":"default","split":"train","numRows":1867602,"numBytesParquet":44989201432}]},"discussionsStats":{"closed":0,"open":1,"total":1},"fullWidth":true,"hasGatedAccess":true,"hasFullAccess":true,"isEmbedded":false,"savedQueries":{"community":[],"user":[]}}">
text listlengths 2 2.54k | id stringlengths 9 16 |
|---|---|
[
[
"Graph Learning for Anomaly Analytics: Algorithms, Applications, and\n Challenges"
],
[
"Abstract Anomaly analytics is a popular and vital task in various research contexts, which has been studied for several decades.",
"At the same time, deep learning has shown its capacity in solving many grap... | 2212.05532 |
[
[
"Near-Field Enhancement of Optical Second Harmonic Generation in Hybrid\n Gold-Lithium Niobate Nanostructures"
],
[
"Abstract Nanophotonics research has focused recently on the ability of non-linear optical processes to mediate and transform optical signals in a myriad of novel devices, including op... | 2212.05569 |
[
[
"Designing Human-Centered Algorithms for the Public Sector: A Case Study\n of the U.S. Child-Welfare System"
],
[
"Abstract The U.S. Child Welfare System (CWS) is increasingly seeking to emulate business models of the private sector centered in efficiency, cost reduction, and innovation through the ... | 2212.05556 |
[
[
"A global adaptive velocity space for general discrete velocity framework\n in predictions of rarefied and multi-scale flows"
],
[
"Abstract The rarefied flow and multi-scale flow are crucial for the aerodynamic design of spacecraft, ultra-low orbital vehicles and plumes.",
"By introducing a dis... | 2212.05645 |
[
[
"Reaction Nanoscopy of Ion Emission from Sub-wavelength Propanediol\n Droplets"
],
[
"Abstract Droplets provide unique opportunities for the investigation of laser-induced surface chemistry.",
"Chemical reactions on the surface of charged droplets are ubiquitous in nature and can provide critica... | 2212.05587 |
[
[
"Blockchain Network Analysis: A Comparative Study of Decentralized Banks"
],
[
"Abstract Decentralized finance (DeFi) is known for its unique mechanism design, which applies smart contracts to facilitate peer-to-peer transactions.",
"The decentralized bank is a typical DeFi application.",
"Id... | 2212.05632 |
[
[
"Harmonic maps on weighted foliations"
],
[
"Abstract On foliations, there are two kinds of harmonic maps, that is, transversally harmonic map and $(F,F')$-harmonic map which are equivalent when the foliation is minimal.",
"In this paper, we study transversally f-harmonic and $(F,F')_f$-harmonic ... | 2212.05639 |
[
[
"Off-Policy Deep Reinforcement Learning Algorithms for Handling Various\n Robotic Manipulator Tasks"
],
[
"Abstract In order to avoid conventional controlling methods which created obstacles due to the complexity of systems and intense demand on data density, developing modern and more efficient con... | 2212.05572 |
[
[
"Schur-Sergeev duality for Ariki-Koike algebras"
],
[
"Abstract Let $U_q(\\mathfrak{g})$ be the quantized superalgebra of $\\mathfrak{g}=\\mathfrak{gl}(k_1|\\ell_1)\\oplus\\cdots\\oplus\\mathfrak{gl}(k_m|\\ell_m)$ and $H_{m,n}(q,\\mathbf{Q})$ the cyclotomic Hecke algebra of type $G(m,1,n)$.",
"We... | 1808.08484 |
[
[
"Efficiently Processing Workflow Provenance Queries on SPARK"
],
[
"Abstract In this paper, we investigate how we can leverage Spark platform for efficiently processing provenance queries on large volumes of workflow provenance data.",
"We focus on processing provenance queries at attribute-value... | 1808.08424 |
[
[
"Alignment Strength and Correlation for Graphs"
],
[
"Abstract When two graphs have a correlated Bernoulli distribution, we prove that the alignment strength of their natural bijection strongly converges to a novel measure of graph correlation $\\rho_T$ that neatly combines intergraph with intragraph... | 1808.08502 |
[
[
"Trustworthy and Privacy-Aware Sensing for Internet of Things"
],
[
"Abstract The Internet of Things (IoT) is considered as the key enabling technology for smart services.",
"Security and privacy are particularly open challenges for IoT applications due to the widespread use of commodity devices.... | 1808.08549 |
[
[
"Twisted cohomology pairings of knots III; triple cup products"
],
[
"Abstract Given a representation of a link group, we introduce a trilinear form, as a topological invariant.",
"We show that, if the link is either hyperbolic or a knot with malnormality, then the trilinear form equals the pairi... | 1808.08532 |
[
[
"The energy-critical nonlinear wave equation with an inverse-square\n potential"
],
[
"Abstract We study the energy-critical nonlinear wave equation in the presence of an inverse-square potential in dimensions three and four.",
"In the defocusing case, we prove that arbitrary initial data in the... | 1808.08571 |
[
[
"Commensurability growth of branch groups"
],
[
"Abstract Fixing a subgroup $\\Gamma$ in a group $G$, the commensurability growth function assigns to each $n$ the cardinality of the set of subgroups $\\Delta$ of $G$ with $[\\Gamma: \\Gamma \\cap \\Delta][\\Delta : \\Gamma \\cap \\Delta] = n$.",
"... | 1808.08660 |
[
[
"Non-expansive bijections, uniformities and polyhedral faces"
],
[
"Abstract We extend the result of B. Cascales at al.",
"about expand-contract plasticity of the unit ball of strictly convex Banach space to those spaces whose unit ball is the union of all its finite-dimensional polyhedral extrem... | 1808.08534 |
[
[
"Energy Distribution of Radial Solutions to Energy Subcritical Wave\n Equation with an Application on Scattering Theory"
],
[
"Abstract The topic of this paper is a semi-linear, energy sub-critical, defocusing wave equation $\\partial_t^2 u - \\Delta u = - |u|^{p -1} u$ in the 3-dimensional space ($... | 1808.08656 |
[
[
"On a reverse H\\\"older inequality for Schr\\\"odinger operators"
],
[
"Abstract We obtain a reverse H\\\"older inequality for the eigenfuctions of the Schr\\\"odinger operator with slowly decaying potentials.",
"The class of potentials includes singular potentials which decay like $|x|^{-\\alph... | 1808.08516 |
[
[
"Deep Convolutional Neural Network with Mixup for Environmental Sound\n Classification"
],
[
"Abstract Environmental sound classification (ESC) is an important and challenging problem.",
"In contrast to speech, sound events have noise-like nature and may be produced by a wide variety of sources.... | 1808.08405 |
[
[
"Phonon-induced linewidths of graphene electronic states"
],
[
"Abstract The linewidths of the electronic bands originating from the electron-phonon coupling in graphene are analyzed based on model tight-binding calculations and experimental angle-resolved photoemission spectroscopy (ARPES) data.",
... | 1808.08620 |
[
[
"StreamChain: Rethinking Blockchain for Datacenters"
],
[
"Abstract Permissioned blockchains promise secure decentralized data management in business-to-business use-cases.",
"In contrast to Bitcoin and similar public blockchains which rely on Proof-of-Work for consensus and are deployed on thous... | 1808.08406 |
[
[
"Deep Emotion: A Computational Model of Emotion Using Deep Neural\n Networks"
],
[
"Abstract Emotions are very important for human intelligence.",
"For example, emotions are closely related to the appraisal of the internal bodily state and external stimuli.",
"This helps us to respond quickl... | 1808.08447 |
[
[
"Aperiodic Array Synthesis for Multi-User MIMO Applications"
],
[
"Abstract This paper demonstrates the advantages of aperiodic arrays in multi-user multiple-input multiple-output systems for future mobile communication applications.",
"We propose a novel aperiodic array synthesis method which ac... | 1808.08321 |
[
[
"Swift spectra of AT2018cow: A White Dwarf Tidal Disruption Event?"
],
[
"Abstract The bright transient AT2018cow has been unlike any other known type of transient.",
"Its high brightness, rapid rise and decay and initially nearly featureless spectrum are unprecedented and difficult to explain us... | 1808.08492 |
[
[
"Rain Streak Removal for Single Image via Kernel Guided CNN"
],
[
"Abstract Rain streak removal is an important issue and has recently been investigated extensively.",
"Existing methods, especially the newly emerged deep learning methods, could remove the rain streaks well in many cases.",
"H... | 1808.08545 |
[
[
"Intertwined Spin and Orbital Density Waves in MnP Uncovered by Resonant\n Soft X-ray Scattering"
],
[
"Abstract Unconventional superconductors are often characterized by numerous competing and even intertwined orders in their phase diagrams.",
"In particular, the electronic nematic phases, whic... | 1808.08562 |
[
[
"The effect of retardation in the random networks of excitable nodes\n embeddable in the Euclidean space"
],
[
"Abstract Some features of random networks with excitable nodes that are embeddable in the Euclidean space are not describable in terms of the conventional integrate and fire model (IFM) al... | 1808.08495 |
[
[
"A power-law upper bound on the correlations in the 2D random field Ising\n model"
],
[
"Abstract As first asserted by Y. Imry and S-K Ma, the famed discontinuity of the magnetization as function of the magnetic field in the two dimensional Ising model is eliminated, for all temperatures, through th... | 1808.08351 |
[
[
"Stochastic Collocation with Non-Gaussian Correlated Parameters via a New\n Quadrature Rule"
],
[
"Abstract This paper generalizes stochastic collocation methods to handle correlated non-Gaussian random parameters.",
"The key challenge is to perform a multivariate numerical integration in a corr... | 1808.08381 |
[
[
"Churn Intent Detection in Multilingual Chatbot Conversations and Social\n Media"
],
[
"Abstract We propose a new method to detect when users express the intent to leave a service, also known as churn.",
"While previous work focuses solely on social media, we show that this intent can be detecte... | 1808.08432 |
[
[
"Superconductivity near a nematic quantum critical point -- the interplay\n between hot and lukewarm regions"
],
[
"Abstract We present a strong coupling dynamical theory of the superconducting transition in a metal near a QCP towards $Q = 0$ nematic order.",
"We use a fermion-boson model, in wh... | 1808.08635 |
[
[
"The $K\\bar{K}^*$ interaction in the unitary coupled-channel\n approximation"
],
[
"Abstract The $K\\bar{K}^*$ interaction Lagrangian is constructed when the $SU(3)$ hidden gauge symmetry is taken into account, and then the $K\\bar{K}^*$ potential is obtained.",
"In the low energy region, the $... | 1808.08358 |
[
[
"The hyperbolic Ernst equation in a triangular domain"
],
[
"Abstract The collision of two plane gravitational waves in Einstein's theory of relativity can be described mathematically by a Goursat problem for the hyperbolic Ernst equation in a triangular domain.",
"We use the integrable structure... | 1808.08365 |
[
[
"Additive Volume of Sets Contained in Few Arithmetic Progressions"
],
[
"Abstract A conjecture of Freiman gives an exact formula for the largest volume of a finite set $A$ of integers with given cardinality $k = |A|$ and doubling $T = |2A|$.",
"The formula is known to hold when $T \\le 3k-4$, for... | 1808.08455 |
[
[
"Code Generation for Higher Inductive Types"
],
[
"Abstract Higher inductive types are inductive types that include nontrivial higher-dimensional structure, represented as identifications that are not reflexivity.",
"While work proceeds on type theories with a computational interpretation of univ... | 1808.08330 |
[
[
"SYK/AdS duality with Yang-Baxter deformations"
],
[
"Abstract In this paper, based on the notion of SYK/AdS duality we explore the effects of Yang-Baxter (YB) deformations on the SYK spectrum at strong coupling.",
"In the first part of our analysis, we explore the consequences of YB deformations... | 1808.08380 |
[
[
"Nonlinear $n$-term approximation of harmonic functions from shifts of\n the Newtonian Kernel"
],
[
"Abstract A basic building block in Classical Potential Theory is the fundamental solution of the Laplace equation in ${\\mathbb R}^d$ (Newtonian kernel).",
"The main goal of this article is to st... | 1808.08641 |
[
[
"System of interacting harmonic oscillators in rotationally invariant\n noncommutative phase space"
],
[
"Abstract Rotationally invariant space with noncommutativity of coordinates and noncommutativity of momenta of canonical type is considered.",
"A system of $N$ interacting harmonic oscillator... | 1808.08515 |
[
[
"The Disparate Effects of Strategic Manipulation"
],
[
"Abstract When consequential decisions are informed by algorithmic input, individuals may feel compelled to alter their behavior in order to gain a system's approval.",
"Models of agent responsiveness, termed \"strategic manipulation,\" analy... | 1808.08646 |
[
[
"Spectral Pruning: Compressing Deep Neural Networks via Spectral Analysis\n and its Generalization Error"
],
[
"Abstract Compression techniques for deep neural network models are becoming very important for the efficient execution of high-performance deep learning systems on edge-computing devices."... | 1808.08558 |
[
[
"Considerations on Quantum-Based Methods for Communication Security"
],
[
"Abstract In this paper we provide an intuitive-level discussion of the challenges and opportunities offered by quantum-based methods for supporting secure communications, e.g., over a network.",
"The goal is to distill dow... | 1808.08591 |
[
[
"Experimental demonstration of high-rate measurement-device-independent\n quantum key distribution over asymmetric channels"
],
[
"Abstract Measurement-device-independent quantum key distribution (MDI-QKD) can eliminate all detector side channels and it is practical with current technology.",
"P... | 1808.08584 |
[
[
"Fermion spectral function in hot strongly interacting matter from the\n functional renormalization group"
],
[
"Abstract We present the first calculation of fermion spectral function at finite temperature in quark-meson model in the framework of the functional renormalization group (FRG).",
"We... | 1808.08535 |
[
[
"Primeness and dynamics of some classes of entire functions"
],
[
"Abstract In this paper we investigate the primeness of a class of entire functions and discuss the dynamics of a periodic member f of this class with respect to a transcendental entire function g that permutes with f. In particular we... | 1808.08394 |
[
[
"Lepton masses and mixing in a two-Higgs-doublet model"
],
[
"Abstract Within the framework of the two-Higgs Doublet Model (2HDM), we attempt to find some discrete, non-abelian flavour symmetry which could provide an explanation for the masses and mixing matrix elements of leptons.",
"Unlike the ... | 1808.08384 |
[
[
"An Experimental Comparison of SONC and SOS Certificates for\n Unconstrained Optimization"
],
[
"Abstract Finding the minimum of a multivariate real polynomial is a well-known hard problem with various applications.",
"We present a polynomial time algorithm to approximate such lower bounds via s... | 1808.08431 |
[
[
"A fractional notion of length and an associated nonlocal curvature"
],
[
"Abstract Here a new notion of fractional length of a smooth curve, which depends on a parameter $\\sigma$, is introduced that is analogous to the fractional perimeter functional of sets that has been studied in recent years.",... | 1808.08654 |
[
[
"Unique Solutions of Contractions, CCS, and their HOL Formalisation"
],
[
"Abstract The unique solution of contractions is a proof technique for bisimilarity that overcomes certain syntactic constraints of Milner's \"unique solution of equations\" technique.",
"The paper presents an overview of a... | 1808.08652 |
[
[
"Asymptotically good edge correspondence colouring"
],
[
"Abstract We prove that every simple graph with maximum degree $\\Delta$ has an edge correspondence colouring with $\\Delta+o(\\Delta)$ colours."
],
[
"Introduction",
"Graph colouring is one of the richest and most fundamental field... | 1808.08594 |
[
[
"Analysis of adversarial attacks against CNN-based image forgery\n detectors"
],
[
"Abstract With the ubiquitous diffusion of social networks, images are becoming a dominant and powerful communication channel.",
"Not surprisingly, they are also increasingly subject to manipulations aimed at dist... | 1808.08426 |
[
[
"Resonating quantum three-coloring wavefunctions for the kagome quantum\n antiferromagnet"
],
[
"Abstract Motivated by the recent discovery of a macroscopically degenerate exactly solvable point of the spin-$1/2$ $XXZ$ model for $J_z/J=-1/2$ on the kagome lattice [H. J. Changlani et al.",
"Phys.... | 1808.08633 |
[
[
"Contextual Parameter Generation for Universal Neural Machine Translation"
],
[
"Abstract We propose a simple modification to existing neural machine translation (NMT) models that enables using a single universal model to translate between multiple languages while allowing for language specific param... | 1808.08493 |
[
[
"Hypocoercivity of Piecewise Deterministic Markov Process-Monte Carlo"
],
[
"Abstract In this work, we establish $\\mathrm{L}^2$-exponential convergence for a broad class of Piecewise Deterministic Markov Processes recently proposed in the context of Markov Process Monte Carlo methods and covering in... | 1808.08592 |
[
[
"Directed flow in an extended multiphase transport model"
],
[
"Abstract We have studied the rapidity-odd directed flow in $^{197}$Au+$^{197}$Au collisions in the beam energy range from $\\sqrt{s_{NN}}$ = 7.7 to 39 GeV within the framework of an extended multiphase transport model with both partonic ... | 1808.08334 |
[
[
"Larger mutual inclinations for the shortest-period planets"
],
[
"Abstract The {\\it Kepler} mission revealed a population of compact multiple-planet systems with orbital periods shorter than a year, and occasionally even shorter than a day.",
"By analyzing a sample of 102 {\\it Kepler} and {\\i... | 1808.08475 |
[
[
"Symmetry deduction from spectral fluctuations in complex quantum systems"
],
[
"Abstract The spectral fluctuations of complex quantum systems, in appropriate limit, are known to be consistent with that obtained from random matrices.",
"However, this relation between the spectral fluctuations of ... | 1808.08541 |
[
[
"Rolling and no-slip bouncing in cylinders"
],
[
"Abstract The purpose of this paper is to compare a classical non-holonomic system---a sphere rolling against the inner surface of a vertical cylinder under gravity---and a class of discrete dynamical systems known as no-slip billiards in similar confi... | 1808.08448 |
[
[
"Automatic 3D bi-ventricular segmentation of cardiac images by a\n shape-refined multi-task deep learning approach"
],
[
"Abstract Deep learning approaches have achieved state-of-the-art performance in cardiac magnetic resonance (CMR) image segmentation.",
"However, most approaches have focused ... | 1808.08578 |
[
[
"Secrecy Performance Analysis of UAV Transmissions Subject to\n Eavesdropping and Jamming"
],
[
"Abstract Unmanned aerial vehicles (UAVs) have been undergoing fast development for providing broader signal coverage and more extensive surveillance capabilities in military and civilian applications.",
... | 1808.08628 |
[
[
"New supercharacter theory for Sylow subgroups in orthogonal and\n symplectic groups"
],
[
"Abstract Applying the embedding of $A_{n-1}$ in $B_n$, $C_n$ and $D_n$ we construct a new supercharacter theory for the Sylow subgroups in orthogonal and symplectic groups over a finite field.",
"The cons... | 1808.08539 |
[
[
"Fusion++: Volumetric Object-Level SLAM"
],
[
"Abstract We propose an online object-level SLAM system which builds a persistent and accurate 3D graph map of arbitrary reconstructed objects.",
"As an RGB-D camera browses a cluttered indoor scene, Mask-RCNN instance segmentations are used to initia... | 1808.08378 |
[
[
"Protostellar classification using supervised machine learning algorithms"
],
[
"Abstract Classification of young stellar objects (YSOs) into different evolutionary stages helps us to understand the formation process of new stars and planetary systems.",
"Such classification has traditionally bee... | 1808.08371 |
[
[
"Whitney's Theorem, Triangular Sets and Probabilistic Descent on\n Manifolds"
],
[
"Abstract We examine doing probabilistic descent over manifolds implicitly defined by a set of polynomials with rational coefficients.",
"The system of polynomials is assumed to be triangularized.",
"An applic... | 1808.08548 |
[
[
"On the viability of Planck scale cosmology with quartessence"
],
[
"Abstract In this paper, using a qualitative theory of dynamical systems, we study the stability of a unified dark matter-dark energy framework known as quartessence Chaplygin model (QCM) with three different equation-of-states withi... | 1808.08436 |
[
[
"Random Matrix Theory Model for Mean Notch Depth of the Diagonally Loaded\n MVDR Beamformer for a Single Interferer Case"
],
[
"Abstract Adaptive beamformers (ABFs) suppress interferers by placing a notch in the beampattern at the interferer direction.",
"This suppres- sion improves detection of... | 1808.08352 |
[
[
"Gumbel Central Limit Theorem for Max-Min and Min-Max"
],
[
"Abstract The Max-Min and Min-Max of matrices arise prevalently in science and engineering.",
"However, in many real-world situations the computation of the Max-Min and Min-Max is challenging as matrices are large and full information ab... | 1808.08423 |
[
[
"Deep-Learning Ensembles for Skin-Lesion Segmentation, Analysis,\n Classification: RECOD Titans at ISIC Challenge 2018"
],
[
"Abstract This extended abstract describes the participation of RECOD Titans in parts 1 to 3 of the ISIC Challenge 2018 \"Skin Lesion Analysis Towards Melanoma Detection\" (MI... | 1808.08480 |
[
[
"Comparing CNN and LSTM character-level embeddings in BiLSTM-CRF models\n for chemical and disease named entity recognition"
],
[
"Abstract We compare the use of LSTM-based and CNN-based character-level word embeddings in BiLSTM-CRF models to approach chemical and disease named entity recognition (N... | 1808.08450 |
[
[
"Stellar $^{36,38}$Ar$(n,\\gamma)^{37,39}$Ar reactions and their effect on\n light neutron-rich nuclide synthesis"
],
[
"Abstract The $^{36}$Ar$(n,\\gamma)^{37}$Ar ($t_{1/2}$ = 35 d) and $^{38}$Ar$(n,\\gamma)^{39}$Ar (269 y) reactions were studied for the first time with a quasi-Maxwellian ($kT \\si... | 1808.08556 |
[
[
"Ranked Schr\\\"oder Trees"
],
[
"Abstract In biology, a phylogenetic tree is a tool to represent the evolutionary relationship between species.",
"Unfortunately, the classical Schr\\\"oder tree model is not adapted to take into account the chronology between the branching nodes.",
"In partic... | 1808.08376 |
[
[
"Quantum Reversibility Is Relative, Or Do Quantum Measurements Reset\n Initial Conditions?"
],
[
"Abstract I compare the role of the information in the classical and quantum dynamics by examining the relation between information flows in measurements and the ability of observers to reverse evolution... | 1808.08598 |
[
[
"Vector Approximate Message Passing Algorithm for Structured Perturbed\n Sensing Matrix"
],
[
"Abstract In this paper, we consider a general form of noisy compressive sensing (CS) where the sensing matrix is not precisely known.",
"Such cases exist when there are imperfections or unknown calibra... | 1808.08579 |
[
[
"Optimal Nonparametric Inference with Two-Scale Distributional Nearest\n Neighbors"
],
[
"Abstract The weighted nearest neighbors (WNN) estimator has been popularly used as a flexible and easy-to-implement nonparametric tool for mean regression estimation.",
"The bagging technique is an elegant ... | 1808.08469 |
[
[
"Semi-Supervised Event Extraction with Paraphrase Clusters"
],
[
"Abstract Supervised event extraction systems are limited in their accuracy due to the lack of available training data.",
"We present a method for self-training event extraction systems by bootstrapping additional training data.",
... | 1808.08622 |
[
[
"Painting Outside the Box: Image Outpainting with GANs"
],
[
"Abstract The challenging task of image outpainting (extrapolation) has received comparatively little attention in relation to its cousin, image inpainting (completion).",
"Accordingly, we present a deep learning approach based on Iizuk... | 1808.08483 |
[
[
"Scientific Relation Extraction with Selectively Incorporated Concept\n Embeddings"
],
[
"Abstract This paper describes our submission for the SemEval 2018 Task 7 shared task on semantic relation extraction and classification in scientific papers.",
"We extend the end-to-end relation extraction ... | 1808.08643 |
[
[
"On $J$-Colouring of Chithra Graphs"
],
[
"Abstract The family of Chithra graphs is a wide ranging family of graphs which includes any graph of size at least one.",
"Chithra graphs serve as a graph theoretical model for genetic engineering techniques or for modelling natural mutation within vario... | 1808.08661 |
[
[
"Energy Efficient and Fair Resource Allocation for LTE-Unlicensed Uplink\n Networks: A Two-sided Matching Approach with Partial Information"
],
[
"Abstract LTE-Unlicensed (LTE-U) has recently attracted worldwide interest to meet the explosion in cellular traffic data.",
"By using carrier aggrega... | 1808.08508 |
[
[
"Masses and ages for metal-poor stars: a pilot program combining\n asteroseismology and high-resolution spectroscopic follow-up of RAVE halo\n stars"
],
[
"Abstract Very metal-poor halo stars are the best candidates for being among the oldest objects in our Galaxy.",
"Samples of halo stars with... | 1808.08569 |
[
[
"Gauge dependence of the one-loop divergences in $6D$, ${\\cal N} = (1,0)$\n abelian theory"
],
[
"Abstract We study the gauge dependence of the one-loop effective action for the abelian $6D$, ${\\cal N}=(1,0)$ supersymmetric gauge theory formulated in harmonic superspace.",
"We introduce the su... | 1808.08446 |
[
[
"You are what you eat: A social media study of food identity"
],
[
"Abstract Food preferences not only originate from a person's dietary habits, but also reflect personal values and consumer awareness.",
"This study addresses `food identity' or the relationship between food preferences and person... | 1808.08428 |
[
[
"Kapranov's construction of sh Leibniz algebras"
],
[
"Abstract Motivated by Kapranov's discovery of an sh Lie algebra structure on the tangent complex of a K\\\"{a}hler manifold and Chen-Sti\\'{e}non-Xu's construction of sh Leibniz algebras associated with a Lie pair, we find a general method to con... | 1808.08576 |
[
[
"New Perspective On The Unruh Effect"
],
[
"Abstract In this work, based on the worldline path integral representation of the vacuum energy in spacetime with a Lorentzian metric, we provide a new but complementary interpretation of the Unruh effect.",
"We perform the quantization of the massless ... | 1808.08397 |
[
[
"Holographic control of information and dynamical topology change for\n composite open quantum systems"
],
[
"Abstract We investigate how the compositeness of a quantum system influences the characteristic time of equilibration.",
"We study the dynamics of open composite quantum systems strongly... | 1808.08389 |
[
[
"Word Sense Induction with Neural biLM and Symmetric Patterns"
],
[
"Abstract An established method for Word Sense Induction (WSI) uses a language model to predict probable substitutes for target words, and induces senses by clustering these resulting substitute vectors.",
"We replace the ngram-b... | 1808.08518 |
[
[
"A Comparison of the Taguchi Method and Evolutionary Optimization in\n Multivariate Testing"
],
[
"Abstract Multivariate testing has recently emerged as a promising technique in web interface design.",
"In contrast to the standard A/B testing, multivariate approach aims at evaluating a large num... | 1808.08347 |
[
[
"Energy-preserving continuous-stage Runge-Kutta-Nystr\\\"om methods"
],
[
"Abstract Many practical problems can be described by second-order system $\\ddot{q}=-M\\nabla U(q)$, in which people give special emphasis to some invariants with explicit physical meaning, such as energy, momentum, angular mo... | 1808.08451 |
[
[
"A quasi-local characterisation of $L^p$-Roe algebras"
],
[
"Abstract Very recently, \\v{S}pakula and Tikuisis provide a new characterisation of (uniform) Roe algebras via quasi-locality when the underlying metric spaces have straight finite decomposition complexity.",
"In this paper, we improve ... | 1808.08593 |
[
[
"Dynamics of a spatially developing liquid jet with slower coaxial gas\n flow"
],
[
"Abstract A three-dimensional round liquid jet within a low-speed coaxial gas flow is numerically simulated and explained via vortex dynamics ($\\lambda_2$ method).",
"The instabilities on the liquid-gas interfac... | 1808.08322 |
[
[
"Phase factors of periodically driven two-level systems"
],
[
"Abstract Using a perturbative solution for a periodically driven two-level quantum system, we show how to obtain phase factors for both a two-level quantum system and two two-level quantum systems non-interacting and interacting.",
"T... | 1808.08324 |
[
[
"Highly localized kernels on the sphere induced by Newtonian kernels"
],
[
"Abstract The purpose of this article is to construct highly localized summability kernels on the unit sphere in ${\\mathbb R}^d$ that are restrictions to the sphere of linear combinations of a small number of shifts of the fu... | 1808.08637 |
[
[
"Online Human Activity Recognition using Low-Power Wearable Devices"
],
[
"Abstract Human activity recognition~(HAR) has attracted significant research interest due to its applications in health monitoring and patient rehabilitation.",
"Recent research on HAR focuses on using smartphones due to t... | 1808.08615 |
[
[
"Driven tabu search: a quantum inherent optimisation"
],
[
"Abstract Quantum computers are different from binary digital electronic computers based on transistors.",
"Common digital computing encodes the data into binary digits (bits), each of which is always in one of two definite states (0 or 1... | 1808.08429 |
[
[
"A Simplified Weak Galerkin Finite Element Method: Algorithm and Error\n Estimates"
],
[
"Abstract In this article a simplified weak Galerkin finite element method is developed for the Dirichlet boundary value problem of convection-diffusion-reaction equations.",
"The simplified weak Galerkin me... | 1808.08667 |
[
[
"Symmetries and Mass Degeneracies in the Scalar Sector"
],
[
"Abstract We explore some aspects of models with two and three SU(2) scalar doublets that lead to mass degeneracies among some of the physical scalars.",
"In Higgs sectors with two scalar doublets, the exact degeneracy of scalar masses,... | 1808.08629 |
[
[
"Quasistates and quasiprobabilities"
],
[
"Abstract The quasiprobability representation of quantum states addresses two main concerns, the identification of nonclassical features and the decomposition of the density operator.",
"While the former aspect is a main focus of current research, the lat... | 1808.08471 |
[
[
"Invariant Sets in Quasiperiodically Forced Dynamical Systems"
],
[
"Abstract This paper addresses structures of state space in quasiperiodically forced dynamical systems.",
"We develop a theory of ergodic partition of state space in a class of measure-preserving and dissipative flows, which is a... | 1808.08340 |
[
[
"How do Convolutional Neural Networks Learn Design?"
],
[
"Abstract In this paper, we aim to understand the design principles in book cover images which are carefully crafted by experts.",
"Book covers are designed in a unique way, specific to genres which convey important information to their re... | 1808.08402 |
[
[
"Towards Tight Approximation Bounds for Graph Diameter and Eccentricities"
],
[
"Abstract Among the most important graph parameters is the Diameter, the largest distance between any two vertices.",
"There are no known very efficient algorithms for computing the Diameter exactly.",
"Thus, much... | 1808.08494 |
[
[
"NavigationNet: A Large-scale Interactive Indoor Navigation Dataset"
],
[
"Abstract Indoor navigation aims at performing navigation within buildings.",
"In scenes like home and factory, most intelligent mobile devices require an functionality of routing to guide itself precisely through indoor sc... | 1808.08374 |