AlgoSimBench_mcq.json

[
  {
    "reference": {
      "prob_id": "816b",
      "tags": [
        "Prefix Sums and Difference Arrays|Range Queries|Basics",
        "Prefix Sum|Range Queries Without Updates|Basics"
      ]
    },
    "positive": {
      "prob_id": "276c",
      "tags": [
        "Prefix Sums and Difference Arrays|Range Queries|Basics",
        "Prefix Sum|Range Queries Without Updates|Basics"
      ]
    },
    "negative": [
      {
        "prob_id": "814d",
        "tags": [
          "Trees and Tree Traversals|Graphs|Basics"
        ]
      },
      {
        "prob_id": "814c",
        "tags": [
          "Two Pointers|Leveraging Monotonicity|Basics"
        ]
      },
      {
        "prob_id": "585f",
        "tags": [
          "Aho-Corasick|String Structures|Advanced Data Structures"
        ]
      }
    ]
  },
  {
    "reference": {
      "prob_id": "1349a",
      "tags": [
        "Divisibility||",
        "Prime Factorization|Number Theory|Basics"
      ]
    },
    "positive": {
      "prob_id": "div game169",
      "tags": [
        "Divisibility||",
        "Prime Factorization|Number Theory|Basics"
      ]
    },
    "negative": [
      {
        "prob_id": "1361c",
        "tags": [
          "Euler Tour||"
        ]
      },
      {
        "prob_id": "1349d",
        "tags": [
          "Martingale|Probabilities and Expected Values|Math",
          "Probabilities and Expected Values|Probabilities and Expected Values|Math",
          "The Slime Trick|Probabilities and Expected Values|Math",
          "Thomas Algorithm|Linear Algebra|Math"
        ]
      },
      {
        "prob_id": "1436e",
        "tags": [
          "[Problem] MEX of all Subarrays|Miscellaneous|Miscellaneous"
        ]
      }
    ]
  },
  {
    "reference": {
      "prob_id": "1140g",
      "tags": [
        "Binary Lifting||Advanced Data Structures",
        "Lowest Common Ancestor|Lowest common ancestor|Graphs"
      ]
    },
    "positive": {
      "prob_id": "588e",
      "tags": [
        "Binary Lifting||Advanced Data Structures",
        "Lowest Common Ancestor|Lowest common ancestor|Graphs"
      ]
    },
    "negative": [
      {
        "prob_id": "1108e2",
        "tags": [
          "Sorted Set||"
        ]
      },
      {
        "prob_id": "1190a",
        "tags": [
          "Global Deltaing||Advanced Data Structures"
        ]
      },
      {
        "prob_id": "1327d",
        "tags": [
          "Permutations and Permutation Cycles|Permutations|Miscellaneous"
        ]
      }
    ]
  },
  {
    "reference": {
      "prob_id": "588e",
      "tags": [
        "Binary Lifting||Advanced Data Structures",
        "Lowest Common Ancestor|Lowest common ancestor|Graphs"
      ]
    },
    "positive": {
      "prob_id": "609e",
      "tags": [
        "Binary Lifting||Advanced Data Structures",
        "Lowest Common Ancestor|Lowest common ancestor|Graphs"
      ]
    },
    "negative": [
      {
        "prob_id": "585d",
        "tags": [
          "Meet in the Middle|Techniques|Miscellaneous"
        ]
      },
      {
        "prob_id": "585f",
        "tags": [
          "Aho-Corasick|String Structures|Advanced Data Structures"
        ]
      },
      {
        "prob_id": "793c",
        "tags": [
          "Vectors and Applications|Geometry|Basics"
        ]
      }
    ]
  },
  {
    "reference": {
      "prob_id": "1217f",
      "tags": [
        "Dynamic Connectivity Problem (Offline)|Dynamic Connectivity|Data Structures (DS)",
        "Offline Deletion||"
      ]
    },
    "positive": {
      "prob_id": "601e",
      "tags": [
        "Dynamic Connectivity Problem (Offline)|Dynamic Connectivity|Data Structures (DS)",
        "Offline Deletion||"
      ]
    },
    "negative": [
      {
        "prob_id": "1051f",
        "tags": [
          "Euler Tour||",
          "Tree Euler||"
        ]
      },
      {
        "prob_id": "842d",
        "tags": [
          "MEX task (Minimal Excluded element in an array)|Sequences|Miscellaneous"
        ]
      },
      {
        "prob_id": "1416c",
        "tags": [
          "Trie on Strings and Bits|Trie|Data Structures (DS)"
        ]
      }
    ]
  },
  {
    "reference": {
      "prob_id": "235c",
      "tags": [
        "String Suffix||",
        "Suffix Automaton|Suffix Structures|Strings"
      ]
    },
    "positive": {
      "prob_id": "1063f",
      "tags": [
        "String Suffix||",
        "Suffix Automaton|Suffix Structures|Strings"
      ]
    },
    "negative": [
      {
        "prob_id": "1081c",
        "tags": [
          "Counting with DP||Intermediate"
        ]
      },
      {
        "prob_id": "704b",
        "tags": [
          "Connected Component DP|Miscellaneous|Dynamic Programming (DP)"
        ]
      },
      {
        "prob_id": "911g",
        "tags": [
          "Pragmas|Miscellaneous|Miscellaneous",
          "Splay Trees and Treaps|Augmenting Data Structures|Advanced Data Structures"
        ]
      }
    ]
  },
  {
    "reference": {
      "prob_id": "755114",
      "tags": [
        "Backtracking|Miscellaneous|Miscellaneous"
      ]
    },
    "positive": {
      "prob_id": "292c",
      "tags": [
        "Backtracking|Miscellaneous|Miscellaneous"
      ]
    },
    "negative": [
      {
        "prob_id": "754e",
        "tags": [
          "2D FFT / NTT|Convolution|Math",
          "String Matching using Bitsets|String Matching|Strings"
        ]
      },
      {
        "prob_id": "modularness156",
        "tags": [
          "Divisibility, Modular Arithmetic|Number Theory|Basics"
        ]
      },
      {
        "prob_id": "snuke line068",
        "tags": [
          "Harmonic Lemma||Intermediate"
        ]
      }
    ]
  },
  {
    "reference": {
      "prob_id": "1478c",
      "tags": [
        "Prefix Sum|Range Queries Without Updates|Basics"
      ]
    },
    "positive": {
      "prob_id": "17e",
      "tags": [
        "Prefix Sum|Range Queries Without Updates|Basics"
      ]
    },
    "negative": [
      {
        "prob_id": "1580c",
        "tags": [
          "Splitting Objects into Light and Heavy / Case Processing|Square Root Decomposition|Data Structures (DS)"
        ]
      },
      {
        "prob_id": "161d",
        "tags": [
          "DP on Trees and DAGs|Dp on Different Structures|Dynamic Programming (DP)"
        ]
      },
      {
        "prob_id": "1476a",
        "tags": [
          "Basic Constructive Algorithms|Greedy and Constructive|Basics"
        ]
      }
    ]
  },
  {
    "reference": {
      "prob_id": "363b",
      "tags": [
        "Prefix Sum|Range Queries Without Updates|Basics"
      ]
    },
    "positive": {
      "prob_id": "multiple of 2019164",
      "tags": [
        "Prefix Sum|Range Queries Without Updates|Basics"
      ]
    },
    "negative": [
      {
        "prob_id": "22e",
        "tags": [
          "Strongly Connected Components (SCC)|Graph Connectivity|Graph Theory"
        ]
      },
      {
        "prob_id": "471d",
        "tags": [
          "Z algorithm|String Matching|Strings"
        ]
      },
      {
        "prob_id": "241c",
        "tags": [
          "Reflections on a Rectangle|Miscellaneous|Geometry"
        ]
      }
    ]
  },
  {
    "reference": {
      "prob_id": "847b",
      "tags": [
        "Binary Search|Leveraging Monotonicity|Basics"
      ]
    },
    "positive": {
      "prob_id": "590b",
      "tags": [
        "Binary Search|Leveraging Monotonicity|Basics"
      ]
    },
    "negative": [
      {
        "prob_id": "809d",
        "tags": [
          "Splay Trees and Treaps|Augmenting Data Structures|Advanced Data Structures"
        ]
      },
      {
        "prob_id": "873f",
        "tags": [
          "Suffix Array|Suffix Structures|Strings"
        ]
      },
      {
        "prob_id": "893f",
        "tags": [
          "Tree Flattening||Advanced Data Structures"
        ]
      }
    ]
  },
  {
    "reference": {
      "prob_id": "818f",
      "tags": [
        "Binary Search|Leveraging Monotonicity|Basics"
      ]
    },
    "positive": {
      "prob_id": "590b",
      "tags": [
        "Binary Search|Leveraging Monotonicity|Basics"
      ]
    },
    "negative": [
      {
        "prob_id": "825f",
        "tags": [
          "Z algorithm|String Matching|Strings"
        ]
      },
      {
        "prob_id": "1100b",
        "tags": [
          "Amortized Analysis||Intermediate"
        ]
      },
      {
        "prob_id": "839d",
        "tags": [
          "Principle of Inclusion-Exclusion|Counting|Intermediate"
        ]
      }
    ]
  },
  {
    "reference": {
      "prob_id": "1468d",
      "tags": [
        "Binary Search|Leveraging Monotonicity|Basics"
      ]
    },
    "positive": {
      "prob_id": "water bottle144",
      "tags": [
        "Binary Search|Leveraging Monotonicity|Basics"
      ]
    },
    "negative": [
      {
        "prob_id": "1368d",
        "tags": [
          "Basic Greedy Algorithms|Greedy and Constructive|Basics",
          "Bit Manipulation|Bit Manipulation|Basics"
        ]
      },
      {
        "prob_id": "576a",
        "tags": [
          "Prime Factorization|Number Theory|Basics"
        ]
      },
      {
        "prob_id": "1387b2",
        "tags": [
          "Centroid Decomposition|Centroid Decomposition|Data Structures (DS)",
          "Small to Large||"
        ]
      }
    ]
  },
  {
    "reference": {
      "prob_id": "triangles143",
      "tags": [
        "Binary Search|Leveraging Monotonicity|Basics"
      ]
    },
    "positive": {
      "prob_id": "590b",
      "tags": [
        "Binary Search|Leveraging Monotonicity|Basics"
      ]
    },
    "negative": [
      {
        "prob_id": "digit sum044",
        "tags": [
          "Harmonic Lemma||Intermediate",
          "Square-Root Discrimination||Intermediate"
        ]
      },
      {
        "prob_id": "meaningful mean075",
        "tags": [
          "Order Statistics||Intermediate"
        ]
      },
      {
        "prob_id": "one-stroke path054",
        "tags": [
          "All-Permutations|Complete Search|Basics"
        ]
      }
    ]
  },
  {
    "reference": {
      "prob_id": "580b",
      "tags": [
        "Binary Search|Leveraging Monotonicity|Basics"
      ]
    },
    "positive": {
      "prob_id": "distinct numbers143",
      "tags": [
        "Binary Search|Leveraging Monotonicity|Basics"
      ]
    },
    "negative": [
      {
        "prob_id": "568b",
        "tags": [
          "Partitions of a Set / Bell Number|Partitions|Combinatorics"
        ]
      },
      {
        "prob_id": "567e",
        "tags": [
          "Dijkstra's Algorithm|Shortest Paths|Graph Theory"
        ]
      },
      {
        "prob_id": "392c",
        "tags": [
          "Berlekamp Messey|Linear Recurrence|Math",
          "Generating Functions|Polynomial|Math",
          "Matrix Exponentiation|Matrices|Math"
        ]
      }
    ]
  },
  {
    "reference": {
      "prob_id": "165b",
      "tags": [
        "Binary Search|Leveraging Monotonicity|Basics"
      ]
    },
    "positive": {
      "prob_id": "water bottle144",
      "tags": [
        "Binary Search|Leveraging Monotonicity|Basics"
      ]
    },
    "negative": [
      {
        "prob_id": "868f",
        "tags": [
          "Divide and Conquer DP|DP optimizations|Dynamic Programming"
        ]
      },
      {
        "prob_id": "632d",
        "tags": [
          "Divisor Sieve|Number Theory|Intermediate",
          "Harmonic Lemma||Intermediate"
        ]
      },
      {
        "prob_id": "148d",
        "tags": [
          "Markov Chains|Probability and Expectation|Intermediate"
        ]
      }
    ]
  },
  {
    "reference": {
      "prob_id": "6e",
      "tags": [
        "Two Pointers|Leveraging Monotonicity|Basics"
      ]
    },
    "positive": {
      "prob_id": "1073c",
      "tags": [
        "Two Pointers|Leveraging Monotonicity|Basics"
      ]
    },
    "negative": [
      {
        "prob_id": "117c",
        "tags": [
          "Tournament Graph|Special Graphs|Graph Theory"
        ]
      },
      {
        "prob_id": "498e",
        "tags": [
          "Binary Exponentiation|Fundamentals|Algebra"
        ]
      },
      {
        "prob_id": "873d",
        "tags": [
          "Divide and Conquer DP|DP optimizations|Dynamic Programming"
        ]
      }
    ]
  },
  {
    "reference": {
      "prob_id": "689b",
      "tags": [
        "Breadth First Search|Graph traversal|Graphs"
      ]
    },
    "positive": {
      "prob_id": "1205b",
      "tags": [
        "Breadth First Search|Graph traversal|Graphs"
      ]
    },
    "negative": [
      {
        "prob_id": "10d",
        "tags": [
          "Longest Increasing Subsequence (LIS)|Classic DP|Dynamic Programming (DP)"
        ]
      },
      {
        "prob_id": "1375d",
        "tags": [
          "MEX task (Minimal Excluded element in an array)|Sequences|Miscellaneous"
        ]
      },
      {
        "prob_id": "449d",
        "tags": [
          "Bit Manipulation|Bit Manipulation|Basics",
          "Sum Over Subsets DP (SOS DP)|Bit Related DP|Dynamic Programming (DP)"
        ]
      }
    ]
  },
  {
    "reference": {
      "prob_id": "769c",
      "tags": [
        "Breadth First Search|Graph traversal|Graphs"
      ]
    },
    "positive": {
      "prob_id": "1407e",
      "tags": [
        "Breadth First Search|Graph traversal|Graphs"
      ]
    },
    "negative": [
      {
        "prob_id": "1061f",
        "tags": [
          "Interactive Problems|Interactive Problems|Miscellaneous",
          "Probabilities and Expected Values|Probabilities and Expected Values|Math"
        ]
      },
      {
        "prob_id": "588e",
        "tags": [
          "Binary Lifting||Advanced Data Structures",
          "Lowest Common Ancestor|Lowest common ancestor|Graphs"
        ]
      },
      {
        "prob_id": "1328a",
        "tags": [
          "Big O Notation and Time and Space Complexity Analysis|Complexity Analysis|Basics"
        ]
      }
    ]
  },
  {
    "reference": {
      "prob_id": "579a",
      "tags": [
        "Bit Manipulation|Bit Manipulation|Basics"
      ]
    },
    "positive": {
      "prob_id": "291c",
      "tags": [
        "Bit Manipulation|Bit Manipulation|Basics"
      ]
    },
    "negative": [
      {
        "prob_id": "431c",
        "tags": [
          "Counting with DP||Intermediate"
        ]
      },
      {
        "prob_id": "803d",
        "tags": [
          "Basic Greedy Algorithms|Greedy and Constructive|Basics",
          "Binary Search|Leveraging Monotonicity|Basics",
          "Two Pointers|Leveraging Monotonicity|Basics"
        ]
      },
      {
        "prob_id": "741d",
        "tags": [
          "Small to Large||"
        ]
      }
    ]
  },
  {
    "reference": {
      "prob_id": "1420b",
      "tags": [
        "Bit Manipulation|Bit Manipulation|Basics"
      ]
    },
    "positive": {
      "prob_id": "291c",
      "tags": [
        "Bit Manipulation|Bit Manipulation|Basics"
      ]
    },
    "negative": [
      {
        "prob_id": "14d",
        "tags": [
          "Tree Centers and Diameter||Advanced Data Structures",
          "Trees and Tree Traversals|Graphs|Basics"
        ]
      },
      {
        "prob_id": "1555e",
        "tags": [
          "Segment Tree with Lazy Propagation|Segment Tree|Data Structures (DS)"
        ]
      },
      {
        "prob_id": "1365d",
        "tags": [
          "Flood Fill||"
        ]
      }
    ]
  },
  {
    "reference": {
      "prob_id": "1225d",
      "tags": [
        "Divisibility||"
      ]
    },
    "positive": {
      "prob_id": "906d",
      "tags": [
        "Divisibility||"
      ]
    },
    "negative": [
      {
        "prob_id": "161d",
        "tags": [
          "DP on Trees and DAGs|Dp on Different Structures|Dynamic Programming (DP)"
        ]
      },
      {
        "prob_id": "1454e",
        "tags": [
          "Cycle Detection|Cycles|Graph Theory"
        ]
      },
      {
        "prob_id": "1475f",
        "tags": [
          "2-SAT|DFS Lowlink|Advanced Algorithms",
          "Strongly Connected Components (SCC)|Graph Connectivity|Graph Theory"
        ]
      }
    ]
  },
  {
    "reference": {
      "prob_id": "1536c",
      "tags": [
        "Divisibility||"
      ]
    },
    "positive": {
      "prob_id": "906d",
      "tags": [
        "Divisibility||"
      ]
    },
    "negative": [
      {
        "prob_id": "1085e",
        "tags": [
          "Tree Centers and Diameter||Advanced Data Structures"
        ]
      },
      {
        "prob_id": "630h",
        "tags": [
          "Binomial Coefficients|Fundamentals|Combinatorics"
        ]
      },
      {
        "prob_id": "1375d",
        "tags": [
          "MEX task (Minimal Excluded element in an array)|Sequences|Miscellaneous"
        ]
      }
    ]
  },
  {
    "reference": {
      "prob_id": "675e",
      "tags": [
        "Knapsack DP||"
      ]
    },
    "positive": {
      "prob_id": "common subsequence130",
      "tags": [
        "Knapsack DP||"
      ]
    },
    "negative": [
      {
        "prob_id": "785e",
        "tags": [
          "2D BIT|Binary Indexed Tree (BIT)|Data Structures (DS)"
        ]
      },
      {
        "prob_id": "877e",
        "tags": [
          "Tree Flattening||Advanced Data Structures"
        ]
      },
      {
        "prob_id": "653f",
        "tags": [
          "String Suffix||",
          "Suffix Automaton|Suffix Structures|Strings"
        ]
      }
    ]
  },
  {
    "reference": {
      "prob_id": "797f",
      "tags": [
        "Knapsack DP||"
      ]
    },
    "positive": {
      "prob_id": "909c",
      "tags": [
        "Knapsack DP||"
      ]
    },
    "negative": [
      {
        "prob_id": "510b",
        "tags": [
          "Grid Graphs|Graphs|Basics"
        ]
      },
      {
        "prob_id": "702e",
        "tags": [
          "Binary Lifting||Advanced Data Structures"
        ]
      },
      {
        "prob_id": "662a",
        "tags": [
          "Vector Space and XOR Basis|Linear Algebra|Math"
        ]
      }
    ]
  },
  {
    "reference": {
      "prob_id": "735e",
      "tags": [
        "DP on Trees and DAGs|Dp on Different Structures|Dynamic Programming (DP)"
      ]
    },
    "positive": {
      "prob_id": "1402c",
      "tags": [
        "DP on Trees and DAGs|Dp on Different Structures|Dynamic Programming (DP)"
      ]
    },
    "negative": [
      {
        "prob_id": "1240f",
        "tags": [
          "Edge Coloring of Bipartite Graph|Graph Coloring|Graph Theory"
        ]
      },
      {
        "prob_id": "708d",
        "tags": [
          "Flow with Lower Bounds||"
        ]
      },
      {
        "prob_id": "1594c",
        "tags": [
          "Harmonic Number|Very Basic Math|Basics"
        ]
      }
    ]
  },
  {
    "reference": {
      "prob_id": "766e",
      "tags": [
        "DP on Trees and DAGs|Dp on Different Structures|Dynamic Programming (DP)"
      ]
    },
    "positive": {
      "prob_id": "ki138",
      "tags": [
        "DP on Trees and DAGs|Dp on Different Structures|Dynamic Programming (DP)"
      ]
    },
    "negative": [
      {
        "prob_id": "833b",
        "tags": [
          "Divide and Conquer DP|DP optimizations|Dynamic Programming",
          "Knapsack DP||"
        ]
      },
      {
        "prob_id": "786b",
        "tags": [
          "[Trick] Dijkstra on Segment Tree|Shortest Paths|Graph Theory"
        ]
      },
      {
        "prob_id": "323b",
        "tags": [
          "Tournament Graph|Special Graphs|Graph Theory"
        ]
      }
    ]
  },
  {
    "reference": {
      "prob_id": "decayed bridges120",
      "tags": [
        "Disjoint Set Union||"
      ]
    },
    "positive": {
      "prob_id": "150b",
      "tags": [
        "Disjoint Set Union||"
      ]
    },
    "negative": [
      {
        "prob_id": "honestorunkind2147",
        "tags": [
          "All-Subsets / Binary Choice|Complete Search|Basics"
        ]
      },
      {
        "prob_id": "fork in the road144",
        "tags": [
          "Markov Chains|Probability and Expectation|Intermediate"
        ]
      },
      {
        "prob_id": "tak and hotels060",
        "tags": [
          "Binary Lifting||Advanced Data Structures"
        ]
      }
    ]
  },
  {
    "reference": {
      "prob_id": "585d",
      "tags": [
        "Meet in the Middle|Techniques|Miscellaneous"
      ]
    },
    "positive": {
      "prob_id": "1257f",
      "tags": [
        "Meet in the Middle|Techniques|Miscellaneous"
      ]
    },
    "negative": [
      {
        "prob_id": "702f",
        "tags": [
          "Splay Trees and Treaps|Augmenting Data Structures|Advanced Data Structures"
        ]
      },
      {
        "prob_id": "1203f1",
        "tags": [
          "Exchange Argument|Greedy|Basics"
        ]
      },
      {
        "prob_id": "1000e",
        "tags": [
          "Articulation Bridges and Bridge Tree|Graph Connectivity|Graph Theory"
        ]
      }
    ]
  },
  {
    "reference": {
      "prob_id": "1105e",
      "tags": [
        "Meet in the Middle|Techniques|Miscellaneous"
      ]
    },
    "positive": {
      "prob_id": "1257f",
      "tags": [
        "Meet in the Middle|Techniques|Miscellaneous"
      ]
    },
    "negative": [
      {
        "prob_id": "633c",
        "tags": [
          "Trie||"
        ]
      },
      {
        "prob_id": "414b",
        "tags": [
          "Counting with DP||Intermediate"
        ]
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        "tags": [
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      {
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        "tags": [
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        "tags": [
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      },
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        ]
      },
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        "tags": [
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        ]
      },
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          "Graph Traversal||"
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  },
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    },
    "negative": [
      {
        "prob_id": "727e",
        "tags": [
          "String Hashing|Hashing|Strings"
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      {
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        "Lowest Common Ancestor|Lowest common ancestor|Graphs"
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    },
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        "Binary Lifting||Advanced Data Structures"
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    },
    "negative": [
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        "tags": [
          "Chinese Remainder Theorem|Modular arithmetic|Algebra"
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      },
      {
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          "Probabilities and Expected Values|Probabilities and Expected Values|Math"
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      },
      {
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      }
    ]
  },
  {
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    },
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    },
    "negative": [
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        "tags": [
          "Basic Constructive Algorithms|Greedy and Constructive|Basics"
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      },
      {
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          "Graph Traversal||"
        ]
      },
      {
        "prob_id": "1272e",
        "tags": [
          "Breadth First Search|Graph traversal|Graphs"
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      }
    ]
  },
  {
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      "tags": [
        "Sieve of Eratosthenes|Prime numbers|Algebra"
      ]
    },
    "positive": {
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      "tags": [
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    },
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        "tags": [
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      },
      {
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        "tags": [
          "Interactive Problems|Interactive Problems|Miscellaneous"
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      },
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        "prob_id": "176e",
        "tags": [
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      }
    ]
  },
  {
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    },
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    },
    "negative": [
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        "tags": [
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      },
      {
        "prob_id": "616c",
        "tags": [
          "Grid Graphs|Graphs|Basics"
        ]
      },
      {
        "prob_id": "540c",
        "tags": [
          "Grid Graphs|Graphs|Basics"
        ]
      }
    ]
  },
  {
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    },
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      "tags": [
        "Sieve of Eratosthenes|Prime numbers|Algebra"
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    },
    "negative": [
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        "tags": [
          "Prefix Sums and Difference Arrays|Range Queries|Basics"
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      },
      {
        "prob_id": "286e",
        "tags": [
          "Fast Fourier Transform (FFT)|Convolution|Math"
        ]
      },
      {
        "prob_id": "715c",
        "tags": [
          "Centroid Decomposition|Centroid Decomposition|Data Structures (DS)"
        ]
      }
    ]
  },
  {
    "reference": {
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      "tags": [
        "Prime and Composite, Primality Testing, Prime Sieves, Finding Divisors|Number Theory|Preliminaries"
      ]
    },
    "positive": {
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        "Sieve of Eratosthenes|Prime numbers|Algebra"
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    },
    "negative": [
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        "tags": [
          "DP on Trees and DAGs|Dp on Different Structures|Dynamic Programming (DP)"
        ]
      },
      {
        "prob_id": "242e",
        "tags": [
          "Segment Tree with Lazy Propagation|Segment Tree|Data Structures (DS)"
        ]
      },
      {
        "prob_id": "243a",
        "tags": [
          "Bit Manipulation|Bit Manipulation|Basics"
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      }
    ]
  },
  {
    "reference": {
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        "Sum Over Subsets DP (SOS DP)|Bit Related DP|Dynamic Programming (DP)"
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    },
    "positive": {
      "prob_id": "165e",
      "tags": [
        "Bit Manipulation|Bit Manipulation|Basics",
        "Sum Over Subsets DP (SOS DP)|Bit Related DP|Dynamic Programming (DP)"
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    },
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      },
      {
        "prob_id": "1552i",
        "tags": [
          "PQ Tree|More Tree Data Structures|Data Structures (DS)"
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      },
      {
        "prob_id": "1240f",
        "tags": [
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      }
    ]
  },
  {
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        "Sum Over Subsets DP (SOS DP)|Bit Related DP|Dynamic Programming (DP)"
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    },
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    },
    "negative": [
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        "prob_id": "280d",
        "tags": [
          "Min Cost Max Flow (MCMF)|Flow and Min Cut|Graph Theory"
        ]
      },
      {
        "prob_id": "691e",
        "tags": [
          "Matrix Exponentiation|Matrices|Math"
        ]
      },
      {
        "prob_id": "272d",
        "tags": [
          "Permutations with Repetition, Stars and Bars|Counting|Intermediate"
        ]
      }
    ]
  },
  {
    "reference": {
      "prob_id": "1240f",
      "tags": [
        "Edge Coloring of Bipartite Graph|Graph Coloring|Graph Theory"
      ]
    },
    "positive": {
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      "tags": [
        "Bipartite Matching|Network Flows|Advanced Algorithms"
      ]
    },
    "negative": [
      {
        "prob_id": "1394d",
        "tags": [
          "DP on Trees and DAGs|Dp on Different Structures|Dynamic Programming (DP)"
        ]
      },
      {
        "prob_id": "1572f",
        "tags": [
          "Segment Tree Beats|Segment Tree|Data Structures (DS)",
          "Segment Tree Beats|Segment Tree|Data Structures (DS)"
        ]
      },
      {
        "prob_id": "1158a",
        "tags": [
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          "Basic Greedy Algorithms|Greedy and Constructive|Basics"
        ]
      }
    ]
  },
  {
    "reference": {
      "prob_id": "212a",
      "tags": [
        "Edge Coloring of Bipartite Graph|Graph Coloring|Graph Theory"
      ]
    },
    "positive": {
      "prob_id": "600f",
      "tags": [
        "Bipartite Matching|Network Flows|Advanced Algorithms"
      ]
    },
    "negative": [
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        "prob_id": "25c",
        "tags": [
          "All-Pairs|Shortest Paths|Intermediate"
        ]
      },
      {
        "prob_id": "440d",
        "tags": [
          "DP on Trees and DAGs|Dp on Different Structures|Dynamic Programming (DP)"
        ]
      },
      {
        "prob_id": "190e",
        "tags": [
          "Inverse Graph|Graph Traversal|Graph Theory"
        ]
      }
    ]
  },
  {
    "reference": {
      "prob_id": "1288c",
      "tags": [
        "Stars and Bars|Basic Counting|Basics"
      ]
    },
    "positive": {
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      "tags": [
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    },
    "negative": [
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        "prob_id": "1476a",
        "tags": [
          "Basic Constructive Algorithms|Greedy and Constructive|Basics"
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      },
      {
        "prob_id": "1364c",
        "tags": [
          "Basic Constructive Algorithms|Greedy and Constructive|Basics"
        ]
      },
      {
        "prob_id": "1093d",
        "tags": [
          "Depth First Search (DFS)|Graph Traversal|Graph Theory"
        ]
      }
    ]
  },
  {
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    },
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    },
    "negative": [
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        "prob_id": "850f",
        "tags": [
          "Martingale|Probabilities and Expected Values|Math"
        ]
      },
      {
        "prob_id": "506d",
        "tags": [
          "Splitting Objects into Light and Heavy / Case Processing|Square Root Decomposition|Data Structures (DS)"
        ]
      },
      {
        "prob_id": "650d",
        "tags": [
          "O(n lg n) Longest Monotonic Subsequence|DP Optimization|Advanced Algorithms"
        ]
      }
    ]
  },
  {
    "reference": {
      "prob_id": "510b",
      "tags": [
        "Grid Graphs|Graphs|Basics"
      ]
    },
    "positive": {
      "prob_id": "129c",
      "tags": [
        "Direction Array|Techniques|Miscellaneous"
      ]
    },
    "negative": [
      {
        "prob_id": "778a",
        "tags": [
          "Binary Search|Leveraging Monotonicity|Basics"
        ]
      },
      {
        "prob_id": "506d",
        "tags": [
          "Splitting Objects into Light and Heavy / Case Processing|Square Root Decomposition|Data Structures (DS)"
        ]
      },
      {
        "prob_id": "786c",
        "tags": [
          "Sqrt Decomposition|Trees|Data Structures"
        ]
      }
    ]
  },
  {
    "reference": {
      "prob_id": "305e",
      "tags": [
        "Nim, Sprague-Grundy Theorem|Combinatorial Games|Advanced Mathematics"
      ]
    },
    "positive": {
      "prob_id": "1312f",
      "tags": [
        "Nim, Sprague-Grundy Theorem|Combinatorial Games|Advanced Mathematics"
      ]
    },
    "negative": [
      {
        "prob_id": "1303e",
        "tags": [
          "Interval DP / Range DP / Matrix Chain Multiplication (MCM)|Classic DP|Dynamic Programming (DP)"
        ]
      },
      {
        "prob_id": "237e",
        "tags": [
          "Min Cost Max Flow (MCMF)|Flow and Min Cut|Graph Theory",
          "Min Cost Max Flow (MCMF)|Flow and Min Cut|Graph Theory"
        ]
      },
      {
        "prob_id": "245h",
        "tags": [
          "Interval DP / Range DP / Matrix Chain Multiplication (MCM)|Classic DP|Dynamic Programming (DP)"
        ]
      }
    ]
  },
  {
    "reference": {
      "prob_id": "1328a",
      "tags": [
        "Big O Notation and Time and Space Complexity Analysis|Complexity Analysis|Basics"
      ]
    },
    "positive": {
      "prob_id": "486a",
      "tags": [
        "Sums and Asymptotics||Preliminaries"
      ]
    },
    "negative": [
      {
        "prob_id": "1311d",
        "tags": [
          "Harmonic Lemma||Intermediate"
        ]
      },
      {
        "prob_id": "1499d",
        "tags": [
          "Divisibility||"
        ]
      },
      {
        "prob_id": "1439b",
        "tags": [
          "Maximum Clique|Cliques and Independent Sets|Graph Theory"
        ]
      }
    ]
  },
  {
    "reference": {
      "prob_id": "508d",
      "tags": [
        "Euler Tour||"
      ]
    },
    "positive": {
      "prob_id": "528c",
      "tags": [
        "Euler Tour||"
      ]
    },
    "negative": [
      {
        "prob_id": "152c",
        "tags": [
          "Rule of Sum, Rule of Product|Counting|Basics"
        ]
      },
      {
        "prob_id": "825f",
        "tags": [
          "Z algorithm|String Matching|Strings"
        ]
      },
      {
        "prob_id": "727e",
        "tags": [
          "String Hashing|Hashing|Strings"
        ]
      }
    ]
  },
  {
    "reference": {
      "prob_id": "528c",
      "tags": [
        "Euler Tour||"
      ]
    },
    "positive": {
      "prob_id": "508d",
      "tags": [
        "Euler Tour||"
      ]
    },
    "negative": [
      {
        "prob_id": "292b",
        "tags": [
          "Basic Graph Theory|Graphs|Basics"
        ]
      },
      {
        "prob_id": "434d",
        "tags": [
          "Network Flows||Advanced Algorithms"
        ]
      },
      {
        "prob_id": "631d",
        "tags": [
          "Knuth Morris Pratt (KMP) and Prefix Automaton|String Matching|Strings"
        ]
      }
    ]
  },
  {
    "reference": {
      "prob_id": "653d",
      "tags": [
        "Network Flows||Advanced Algorithms"
      ]
    },
    "positive": {
      "prob_id": "808f",
      "tags": [
        "Maximum Flow / Minimum Cut|Flow and Min Cut|Graph Theory"
      ]
    },
    "negative": [
      {
        "prob_id": "629e",
        "tags": [
          "Binary Lifting||Advanced Data Structures"
        ]
      },
      {
        "prob_id": "1083a",
        "tags": [
          "DP on Trees and DAGs|Dp on Different Structures|Dynamic Programming (DP)"
        ]
      },
      {
        "prob_id": "319c",
        "tags": [
          "Convex Hull Trick|DP Optimization|Advanced Algorithms"
        ]
      }
    ]
  },
  {
    "reference": {
      "prob_id": "704d",
      "tags": [
        "Flow with Lower Bounds||"
      ]
    },
    "positive": {
      "prob_id": "708d",
      "tags": [
        "Flow with Lower Bounds||"
      ]
    },
    "negative": [
      {
        "prob_id": "506d",
        "tags": [
          "Splitting Objects into Light and Heavy / Case Processing|Square Root Decomposition|Data Structures (DS)"
        ]
      },
      {
        "prob_id": "704e",
        "tags": [
          "Heavy Light Decomposition (HLD) ft Subtrees and Path Query|Heavy Light Decomposition (HLD)|Data Structures (DS)"
        ]
      },
      {
        "prob_id": "484e",
        "tags": [
          "Persistent Segment Tree|Segment Tree|Data Structures (DS)"
        ]
      }
    ]
  },
  {
    "reference": {
      "prob_id": "1510b",
      "tags": [
        "Min Cost Max Flow (MCMF)|Flow and Min Cut|Graph Theory"
      ]
    },
    "positive": {
      "prob_id": "818g",
      "tags": [
        "Min Cost Max Flow (MCMF)|Flow and Min Cut|Graph Theory"
      ]
    },
    "negative": [
      {
        "prob_id": "479e",
        "tags": [
          "Knapsack DP||"
        ]
      },
      {
        "prob_id": "1381a2",
        "tags": [
          "Basic Constructive Algorithms|Greedy and Constructive|Basics"
        ]
      },
      {
        "prob_id": "1446a",
        "tags": [
          "Knapsack DP||"
        ]
      }
    ]
  },
  {
    "reference": {
      "prob_id": "280d",
      "tags": [
        "Min Cost Max Flow (MCMF)|Flow and Min Cut|Graph Theory"
      ]
    },
    "positive": {
      "prob_id": "818g",
      "tags": [
        "Min Cost Max Flow (MCMF)|Flow and Min Cut|Graph Theory"
      ]
    },
    "negative": [
      {
        "prob_id": "868f",
        "tags": [
          "Divide and Conquer DP|DP optimizations|Dynamic Programming",
          "Divide and Conquer DP|DP optimizations|Dynamic Programming"
        ]
      },
      {
        "prob_id": "863d",
        "tags": [
          "Splay Trees and Treaps|Augmenting Data Structures|Advanced Data Structures"
        ]
      },
      {
        "prob_id": "1176e",
        "tags": [
          "Bipartite Graph||",
          "Graph Traversal||"
        ]
      }
    ]
  },
  {
    "reference": {
      "prob_id": "802c",
      "tags": [
        "Min Cost Max Flow (MCMF)|Flow and Min Cut|Graph Theory"
      ]
    },
    "positive": {
      "prob_id": "818g",
      "tags": [
        "Min Cost Max Flow (MCMF)|Flow and Min Cut|Graph Theory"
      ]
    },
    "negative": [
      {
        "prob_id": "802j",
        "tags": [
          "Depth First Search (DFS)|Graph Traversal|Graph Theory"
        ]
      },
      {
        "prob_id": "587e",
        "tags": [
          "Vector Space and XOR Basis|Linear Algebra|Math"
        ]
      },
      {
        "prob_id": "599e",
        "tags": [
          "Sum Over Subsets DP (SOS DP)|Bit Related DP|Dynamic Programming (DP)"
        ]
      }
    ]
  },
  {
    "reference": {
      "prob_id": "818g",
      "tags": [
        "Min Cost Max Flow (MCMF)|Flow and Min Cut|Graph Theory"
      ]
    },
    "positive": {
      "prob_id": "1510b",
      "tags": [
        "Min Cost Max Flow (MCMF)|Flow and Min Cut|Graph Theory"
      ]
    },
    "negative": [
      {
        "prob_id": "486e",
        "tags": [
          "O(n lg n) Longest Monotonic Subsequence|DP Optimization|Advanced Algorithms"
        ]
      },
      {
        "prob_id": "868f",
        "tags": [
          "Divide and Conquer DP|DP optimizations|Dynamic Programming",
          "Divide and Conquer DP|DP optimizations|Dynamic Programming"
        ]
      },
      {
        "prob_id": "577b",
        "tags": [
          "Counting with DP||Intermediate"
        ]
      }
    ]
  },
  {
    "reference": {
      "prob_id": "884f",
      "tags": [
        "Min Cost Max Flow (MCMF)|Flow and Min Cut|Graph Theory"
      ]
    },
    "positive": {
      "prob_id": "818g",
      "tags": [
        "Min Cost Max Flow (MCMF)|Flow and Min Cut|Graph Theory"
      ]
    },
    "negative": [
      {
        "prob_id": "825f",
        "tags": [
          "Z algorithm|String Matching|Strings"
        ]
      },
      {
        "prob_id": "900e",
        "tags": [
          "Knapsack DP||"
        ]
      },
      {
        "prob_id": "1303e",
        "tags": [
          "Interval DP / Range DP / Matrix Chain Multiplication (MCM)|Classic DP|Dynamic Programming (DP)"
        ]
      }
    ]
  },
  {
    "reference": {
      "prob_id": "547c",
      "tags": [
        "M\u00f6bius Inversion||Advanced Mathematics",
        "Principle of Inclusion-Exclusion|Counting|Intermediate"
      ]
    },
    "positive": {
      "prob_id": "920g",
      "tags": [
        "Mobius Function and Mobius Inversion|Multiplicative Functions|Number Theory"
      ]
    },
    "negative": [
      {
        "prob_id": "343d",
        "tags": [
          "Small to Large||",
          "Tree Flattening||Advanced Data Structures"
        ]
      },
      {
        "prob_id": "587e",
        "tags": [
          "Vector Space and XOR Basis|Linear Algebra|Math"
        ]
      },
      {
        "prob_id": "689b",
        "tags": [
          "Breadth First Search|Graph traversal|Graphs"
        ]
      }
    ]
  },
  {
    "reference": {
      "prob_id": "1527e",
      "tags": [
        "Divide and Conquer DP|DP optimizations|Dynamic Programming"
      ]
    },
    "positive": {
      "prob_id": "321e",
      "tags": [
        "Divide and Conquer DP|DP optimizations|Dynamic Programming"
      ]
    },
    "negative": [
      {
        "prob_id": "1175d",
        "tags": [
          "Prefix Sums and Difference Arrays|Range Queries|Basics"
        ]
      },
      {
        "prob_id": "1006c",
        "tags": [
          "Two Pointers|Leveraging Monotonicity|Basics"
        ]
      },
      {
        "prob_id": "1101g",
        "tags": [
          "XOR Basis||"
        ]
      }
    ]
  },
  {
    "reference": {
      "prob_id": "321e",
      "tags": [
        "Divide and Conquer DP|DP optimizations|Dynamic Programming"
      ]
    },
    "positive": {
      "prob_id": "1527e",
      "tags": [
        "Divide and Conquer DP|DP optimizations|Dynamic Programming"
      ]
    },
    "negative": [
      {
        "prob_id": "295b",
        "tags": [
          "All-Pairs|Shortest Paths|Intermediate"
        ]
      },
      {
        "prob_id": "689b",
        "tags": [
          "Breadth First Search|Graph traversal|Graphs"
        ]
      },
      {
        "prob_id": "177c2",
        "tags": [
          "Depth-First Traversal, Connectivity, Cycle Detection|Graphs|Basics"
        ]
      }
    ]
  },
  {
    "reference": {
      "prob_id": "833b",
      "tags": [
        "Divide and Conquer DP|DP optimizations|Dynamic Programming",
        "Knapsack DP||"
      ]
    },
    "positive": {
      "prob_id": "1527e",
      "tags": [
        "Divide and Conquer DP|DP optimizations|Dynamic Programming"
      ]
    },
    "negative": [
      {
        "prob_id": "833e",
        "tags": [
          "Sweep Line||"
        ]
      },
      {
        "prob_id": "707b",
        "tags": [
          "Single Source|Shortest Paths|Intermediate"
        ]
      },
      {
        "prob_id": "793a",
        "tags": [
          "Divisibility, Modular Arithmetic|Number Theory|Basics"
        ]
      }
    ]
  },
  {
    "reference": {
      "prob_id": "868f",
      "tags": [
        "Divide and Conquer DP|DP optimizations|Dynamic Programming"
      ]
    },
    "positive": {
      "prob_id": "321e",
      "tags": [
        "Divide and Conquer DP|DP optimizations|Dynamic Programming"
      ]
    },
    "negative": [
      {
        "prob_id": "280d",
        "tags": [
          "Min Cost Max Flow (MCMF)|Flow and Min Cut|Graph Theory"
        ]
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          "DFS||"
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        "DP on Subsets||Intermediate"
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      },
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      },
      {
        "prob_id": "816b",
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      {
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      },
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      },
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      }
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  },
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    },
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      },
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        "tags": [
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    },
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      },
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  },
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    },
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    },
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      {
        "prob_id": "580c",
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      },
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      }
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        "tags": [
          "Backtracking|Miscellaneous|Miscellaneous"
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      {
        "prob_id": "1143f",
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    ]
  },
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    },
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    },
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        "prob_id": "1328a",
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      },
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        "prob_id": "1439b",
        "tags": [
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      }
    ]
  },
  {
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      ]
    },
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      "tags": [
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        "tags": [
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        ]
      },
      {
        "prob_id": "739e",
        "tags": [
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      },
      {
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      }
    ]
  },
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      "tags": [
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    },
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      "tags": [
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    },
    "negative": [
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        "prob_id": "639a",
        "tags": [
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      },
      {
        "prob_id": "1552i",
        "tags": [
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    ]
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      }
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  },
  {
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        "Principle of Inclusion-Exclusion|Counting|Intermediate"
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    },
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    },
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      },
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      },
      {
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  },
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      },
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      },
      {
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      }
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  },
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      "tags": [
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      ]
    },
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      "tags": [
        "Counting with DP||Intermediate"
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    },
    "negative": [
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        "tags": [
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          "String Suffix||"
        ]
      },
      {
        "prob_id": "223a",
        "tags": [
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        ]
      },
      {
        "prob_id": "380c",
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      "tags": [
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        "DP on Digits||Intermediate"
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    },
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      },
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      },
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      }
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  },
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        "DP on Digits||Intermediate"
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    },
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      "tags": [
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    },
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        "tags": [
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      },
      {
        "prob_id": "1085e",
        "tags": [
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        "prob_id": "898f",
        "tags": [
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    ]
  },
  {
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      "tags": [
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    },
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      "tags": [
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    },
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        "tags": [
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      },
      {
        "prob_id": "827f",
        "tags": [
          "Dijkstra's Algorithm|Shortest Paths|Graph Theory"
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      },
      {
        "prob_id": "114b",
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      }
    ]
  },
  {
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      "tags": [
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      ]
    },
    "positive": {
      "prob_id": "609e",
      "tags": [
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        "Lowest Common Ancestor|Lowest common ancestor|Graphs"
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    },
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        "prob_id": "727e",
        "tags": [
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      },
      {
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      },
      {
        "prob_id": "1572f",
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          "Segment Tree Beats|Segment Tree|Data Structures (DS)"
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      }
    ]
  },
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      ]
    },
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      "tags": [
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    },
    "negative": [
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        "tags": [
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      },
      {
        "prob_id": "414b",
        "tags": [
          "Counting with DP||Intermediate"
        ]
      },
      {
        "prob_id": "1081d",
        "tags": [
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          "Centroid Decomposition|Centroid Decomposition|Data Structures (DS)"
        ]
      }
    ]
  },
  {
    "reference": {
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      "tags": [
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      ]
    },
    "positive": {
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      "tags": [
        "Z algorithm|String Matching|Strings"
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    },
    "negative": [
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        "prob_id": "202a",
        "tags": [
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      },
      {
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        "tags": [
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      },
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        "tags": [
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      }
    ]
  },
  {
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    },
    "positive": {
      "prob_id": "432d",
      "tags": [
        "Z algorithm|String Matching|Strings"
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    },
    "negative": [
      {
        "prob_id": "547c",
        "tags": [
          "Principle of Inclusion-Exclusion|Counting|Intermediate",
          "M\u00f6bius Inversion||Advanced Mathematics"
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      },
      {
        "prob_id": "650d",
        "tags": [
          "O(n lg n) Longest Monotonic Subsequence|DP Optimization|Advanced Algorithms"
        ]
      },
      {
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        "tags": [
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        ]
      }
    ]
  },
  {
    "reference": {
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      "tags": [
        "Tree Centers and Diameter||Advanced Data Structures"
      ]
    },
    "positive": {
      "prob_id": "835f",
      "tags": [
        "Tree Centers and Diameter||Advanced Data Structures"
      ]
    },
    "negative": [
      {
        "prob_id": "832d",
        "tags": [
          "Lowest Common Ancestor|Lowest common ancestor|Graphs"
        ]
      },
      {
        "prob_id": "515c",
        "tags": [
          "Structural Argument|Greedy|Basics"
        ]
      },
      {
        "prob_id": "232d",
        "tags": [
          "Suffix Array|Suffix Structures|Strings"
        ]
      }
    ]
  },
  {
    "reference": {
      "prob_id": "1252l",
      "tags": [
        "Network Flows||Advanced Algorithms"
      ]
    },
    "positive": {
      "prob_id": "269c",
      "tags": [
        "Network Flows||Advanced Algorithms"
      ]
    },
    "negative": [
      {
        "prob_id": "1394d",
        "tags": [
          "DP on Trees and DAGs|Dp on Different Structures|Dynamic Programming (DP)"
        ]
      },
      {
        "prob_id": "330b",
        "tags": [
          "Basic Graph Theory|Graphs|Basics"
        ]
      },
      {
        "prob_id": "456e",
        "tags": [
          "Tree Diameter|Graph Traversal|Graph Theory"
        ]
      }
    ]
  },
  {
    "reference": {
      "prob_id": "1082g",
      "tags": [
        "Network Flows||Advanced Algorithms"
      ]
    },
    "positive": {
      "prob_id": "1252l",
      "tags": [
        "Network Flows||Advanced Algorithms"
      ]
    },
    "negative": [
      {
        "prob_id": "1249f",
        "tags": [
          "Combining Subtrees Technique|More Tricks|Dynamic Programming (DP)"
        ]
      },
      {
        "prob_id": "1446a",
        "tags": [
          "Knapsack DP||"
        ]
      },
      {
        "prob_id": "1000g",
        "tags": [
          "Lowest Common Ancestor|Lowest common ancestor|Graphs",
          "Binary Lifting||Advanced Data Structures"
        ]
      }
    ]
  },
  {
    "reference": {
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      "tags": [
        "Bipartite Matching|Network Flows|Advanced Algorithms"
      ]
    },
    "positive": {
      "prob_id": "808f",
      "tags": [
        "Maximum Flow / Minimum Cut|Flow and Min Cut|Graph Theory"
      ]
    },
    "negative": [
      {
        "prob_id": "845e",
        "tags": [
          "Sweep Line||"
        ]
      },
      {
        "prob_id": "780g",
        "tags": [
          "Sweep Line||"
        ]
      },
      {
        "prob_id": "60b",
        "tags": [
          "Grid Graphs|Graphs|Basics"
        ]
      }
    ]
  },
  {
    "reference": {
      "prob_id": "808f",
      "tags": [
        "Maximum Flow / Minimum Cut|Flow and Min Cut|Graph Theory"
      ]
    },
    "positive": {
      "prob_id": "78e",
      "tags": [
        "Bipartite Matching|Network Flows|Advanced Algorithms"
      ]
    },
    "negative": [
      {
        "prob_id": "817e",
        "tags": [
          "Trie on Strings and Bits|Trie|Data Structures (DS)"
        ]
      },
      {
        "prob_id": "706c",
        "tags": [
          "Common Patterns|DP|Intermediate"
        ]
      },
      {
        "prob_id": "4d",
        "tags": [
          "DP with Solution Construction||Intermediate"
        ]
      }
    ]
  },
  {
    "reference": {
      "prob_id": "362e",
      "tags": [
        "Bipartite Matching|Network Flows|Advanced Algorithms"
      ]
    },
    "positive": {
      "prob_id": "818g",
      "tags": [
        "Min Cost Max Flow (MCMF)|Flow and Min Cut|Graph Theory"
      ]
    },
    "negative": [
      {
        "prob_id": "13e",
        "tags": [
          "Square Root Decomposition||"
        ]
      },
      {
        "prob_id": "505d",
        "tags": [
          "Strongly Connected Components (SCC)|Graph Connectivity|Graph Theory"
        ]
      },
      {
        "prob_id": "101b",
        "tags": [
          "Sweep Line||",
          "Knapsack DP||"
        ]
      }
    ]
  },
  {
    "reference": {
      "prob_id": "543a",
      "tags": [
        "Flatten the Memo|DP Optimization|Advanced Algorithms"
      ]
    },
    "positive": {
      "prob_id": "191a",
      "tags": [
        "Flatten the Memo|DP Optimization|Advanced Algorithms"
      ]
    },
    "negative": [
      {
        "prob_id": "280d",
        "tags": [
          "Min Cost Max Flow (MCMF)|Flow and Min Cut|Graph Theory"
        ]
      },
      {
        "prob_id": "177d2",
        "tags": [
          "Prefix Sums and Difference Arrays|Range Queries|Basics"
        ]
      },
      {
        "prob_id": "915g",
        "tags": [
          "Mobius Function and Mobius Inversion|Multiplicative Functions|Number Theory"
        ]
      }
    ]
  },
  {
    "reference": {
      "prob_id": "478d",
      "tags": [
        "Eliminate Dependent Parameters|DP Optimization|Advanced Algorithms",
        "Flatten the Memo|DP Optimization|Advanced Algorithms"
      ]
    },
    "positive": {
      "prob_id": "837d",
      "tags": [
        "Knapsack DP||"
      ]
    },
    "negative": [
      {
        "prob_id": "702c",
        "tags": [
          "Binary Search|Leveraging Monotonicity|Basics",
          "Binary Search|Leveraging Monotonicity|Basics",
          "Two Pointers|Leveraging Monotonicity|Basics"
        ]
      },
      {
        "prob_id": "704d",
        "tags": [
          "Flow with Lower Bounds||"
        ]
      },
      {
        "prob_id": "15d",
        "tags": [
          "Level Ancestors and Lowest Common Ancestors||Advanced Data Structures"
        ]
      }
    ]
  },
  {
    "reference": {
      "prob_id": "837d",
      "tags": [
        "Knapsack DP||"
      ]
    },
    "positive": {
      "prob_id": "478d",
      "tags": [
        "Eliminate Dependent Parameters|DP Optimization|Advanced Algorithms",
        "Flatten the Memo|DP Optimization|Advanced Algorithms"
      ]
    },
    "negative": [
      {
        "prob_id": "617e",
        "tags": [
          "Mo's Algorithm|Square Root Decomposition|Data Structures (DS)"
        ]
      },
      {
        "prob_id": "372d",
        "tags": [
          "Binary Lifting||Advanced Data Structures"
        ]
      },
      {
        "prob_id": "280d",
        "tags": [
          "Min Cost Max Flow (MCMF)|Flow and Min Cut|Graph Theory"
        ]
      }
    ]
  },
  {
    "reference": {
      "prob_id": "191a",
      "tags": [
        "Flatten the Memo|DP Optimization|Advanced Algorithms"
      ]
    },
    "positive": {
      "prob_id": "543a",
      "tags": [
        "Flatten the Memo|DP Optimization|Advanced Algorithms"
      ]
    },
    "negative": [
      {
        "prob_id": "191c",
        "tags": [
          "Level Ancestors and Lowest Common Ancestors||Advanced Data Structures"
        ]
      },
      {
        "prob_id": "226e",
        "tags": [
          "Persistent Segment Tree|Segment Tree|Data Structures (DS)"
        ]
      },
      {
        "prob_id": "878c",
        "tags": [
          "Sorted Set||",
          "Sorted Set||"
        ]
      }
    ]
  },
  {
    "reference": {
      "prob_id": "213c",
      "tags": [
        "Eliminate Dependent Parameters|DP Optimization|Advanced Algorithms",
        "Flatten the Memo|DP Optimization|Advanced Algorithms"
      ]
    },
    "positive": {
      "prob_id": "837d",
      "tags": [
        "Knapsack DP||"
      ]
    },
    "negative": [
      {
        "prob_id": "214a",
        "tags": [
          "Iterative|Complete Search|Basics"
        ]
      },
      {
        "prob_id": "610d",
        "tags": [
          "Sweep Line||"
        ]
      },
      {
        "prob_id": "715c",
        "tags": [
          "Centroid Decomposition|Centroid Decomposition|Data Structures (DS)"
        ]
      }
    ]
  },
  {
    "reference": {
      "prob_id": "213c",
      "tags": [
        "Eliminate Dependent Parameters|DP Optimization|Advanced Algorithms",
        "Flatten the Memo|DP Optimization|Advanced Algorithms"
      ]
    },
    "positive": {
      "prob_id": "478d",
      "tags": [
        "Eliminate Dependent Parameters|DP Optimization|Advanced Algorithms",
        "Flatten the Memo|DP Optimization|Advanced Algorithms"
      ]
    },
    "negative": [
      {
        "prob_id": "214a",
        "tags": [
          "Iterative|Complete Search|Basics"
        ]
      },
      {
        "prob_id": "610d",
        "tags": [
          "Sweep Line||"
        ]
      },
      {
        "prob_id": "715c",
        "tags": [
          "Centroid Decomposition|Centroid Decomposition|Data Structures (DS)"
        ]
      }
    ]
  },
  {
    "reference": {
      "prob_id": "17c",
      "tags": [
        "Eliminate Dependent Parameters|DP Optimization|Advanced Algorithms",
        "Skip Useless States|DP Optimization|Advanced Algorithms"
      ]
    },
    "positive": {
      "prob_id": "268d",
      "tags": [
        "Eliminate Dependent Parameters|DP Optimization|Advanced Algorithms"
      ]
    },
    "negative": [
      {
        "prob_id": "204e",
        "tags": [
          "Suffix Array|Suffix Structures|Strings"
        ]
      },
      {
        "prob_id": "706c",
        "tags": [
          "Common Patterns|DP|Intermediate"
        ]
      },
      {
        "prob_id": "1012d",
        "tags": [
          "Tricky Implementation||Intermediate"
        ]
      }
    ]
  },
  {
    "reference": {
      "prob_id": "268d",
      "tags": [
        "Eliminate Dependent Parameters|DP Optimization|Advanced Algorithms"
      ]
    },
    "positive": {
      "prob_id": "17c",
      "tags": [
        "Eliminate Dependent Parameters|DP Optimization|Advanced Algorithms",
        "Skip Useless States|DP Optimization|Advanced Algorithms"
      ]
    },
    "negative": [
      {
        "prob_id": "15d",
        "tags": [
          "Level Ancestors and Lowest Common Ancestors||Advanced Data Structures"
        ]
      },
      {
        "prob_id": "363b",
        "tags": [
          "Prefix Sum|Range Queries Without Updates|Basics"
        ]
      },
      {
        "prob_id": "849b",
        "tags": [
          "Basics|Geometry|Basics"
        ]
      }
    ]
  },
  {
    "reference": {
      "prob_id": "17c",
      "tags": [
        "Eliminate Dependent Parameters|DP Optimization|Advanced Algorithms",
        "Skip Useless States|DP Optimization|Advanced Algorithms"
      ]
    },
    "positive": {
      "prob_id": "729f",
      "tags": [
        "Skip Useless States|DP Optimization|Advanced Algorithms"
      ]
    },
    "negative": [
      {
        "prob_id": "204e",
        "tags": [
          "Suffix Array|Suffix Structures|Strings"
        ]
      },
      {
        "prob_id": "706c",
        "tags": [
          "Common Patterns|DP|Intermediate"
        ]
      },
      {
        "prob_id": "1012d",
        "tags": [
          "Tricky Implementation||Intermediate"
        ]
      }
    ]
  },
  {
    "reference": {
      "prob_id": "629d",
      "tags": [
        "O(n lg n) Longest Monotonic Subsequence|DP Optimization|Advanced Algorithms"
      ]
    },
    "positive": {
      "prob_id": "314c",
      "tags": [
        "O(n lg n) Longest Monotonic Subsequence|DP Optimization|Advanced Algorithms"
      ]
    },
    "negative": [
      {
        "prob_id": "621c",
        "tags": [
          "Basics|Probability and Expectation|Basics"
        ]
      },
      {
        "prob_id": "794b",
        "tags": [
          "Basics|Geometry|Basics"
        ]
      },
      {
        "prob_id": "623d",
        "tags": [
          "Probabilities and Expected Values|Probabilities and Expected Values|Math"
        ]
      }
    ]
  },
  {
    "reference": {
      "prob_id": "474e",
      "tags": [
        "O(n lg n) Longest Monotonic Subsequence|DP Optimization|Advanced Algorithms"
      ]
    },
    "positive": {
      "prob_id": "76f",
      "tags": [
        "O(n lg n) Longest Monotonic Subsequence|DP Optimization|Advanced Algorithms"
      ]
    },
    "negative": [
      {
        "prob_id": "363b",
        "tags": [
          "Prefix Sum|Range Queries Without Updates|Basics"
        ]
      },
      {
        "prob_id": "587e",
        "tags": [
          "Vector Space and XOR Basis|Linear Algebra|Math"
        ]
      },
      {
        "prob_id": "338d",
        "tags": [
          "Chinese Remainder Theorem|Modular arithmetic|Algebra"
        ]
      }
    ]
  },
  {
    "reference": {
      "prob_id": "650d",
      "tags": [
        "O(n lg n) Longest Monotonic Subsequence|DP Optimization|Advanced Algorithms"
      ]
    },
    "positive": {
      "prob_id": "946g",
      "tags": [
        "O(n lg n) Longest Monotonic Subsequence|DP Optimization|Advanced Algorithms"
      ]
    },
    "negative": [
      {
        "prob_id": "576a",
        "tags": [
          "Prime Factorization|Number Theory|Basics"
        ]
      },
      {
        "prob_id": "59e",
        "tags": [
          "Breadth First Search|Graph traversal|Graphs"
        ]
      },
      {
        "prob_id": "653e",
        "tags": [
          "Breadth First Search|Graph traversal|Graphs"
        ]
      }
    ]
  },
  {
    "reference": {
      "prob_id": "946g",
      "tags": [
        "O(n lg n) Longest Monotonic Subsequence|DP Optimization|Advanced Algorithms"
      ]
    },
    "positive": {
      "prob_id": "76f",
      "tags": [
        "O(n lg n) Longest Monotonic Subsequence|DP Optimization|Advanced Algorithms"
      ]
    },
    "negative": [
      {
        "prob_id": "722c",
        "tags": [
          "Disjoint Set Union||"
        ]
      },
      {
        "prob_id": "446a",
        "tags": [
          "Classic Problems with Twists|DP|Intermediate"
        ]
      },
      {
        "prob_id": "1476a",
        "tags": [
          "Basic Constructive Algorithms|Greedy and Constructive|Basics"
        ]
      }
    ]
  },
  {
    "reference": {
      "prob_id": "76f",
      "tags": [
        "O(n lg n) Longest Monotonic Subsequence|DP Optimization|Advanced Algorithms"
      ]
    },
    "positive": {
      "prob_id": "946g",
      "tags": [
        "O(n lg n) Longest Monotonic Subsequence|DP Optimization|Advanced Algorithms"
      ]
    },
    "negative": [
      {
        "prob_id": "340c",
        "tags": [
          "Binomial Coefficients|Fundamentals|Combinatorics"
        ]
      },
      {
        "prob_id": "1209h",
        "tags": [
          "Slope Trick|More Tricks|Dynamic Programming (DP)"
        ]
      },
      {
        "prob_id": "704e",
        "tags": [
          "Heavy Light Decomposition (HLD) ft Subtrees and Path Query|Heavy Light Decomposition (HLD)|Data Structures (DS)"
        ]
      }
    ]
  },
  {
    "reference": {
      "prob_id": "833b",
      "tags": [
        "Divide and Conquer DP|DP optimizations|Dynamic Programming",
        "Knapsack DP||"
      ]
    },
    "positive": {
      "prob_id": "487b",
      "tags": [
        "Knapsack DP||"
      ]
    },
    "negative": [
      {
        "prob_id": "833e",
        "tags": [
          "Sweep Line||"
        ]
      },
      {
        "prob_id": "707b",
        "tags": [
          "Single Source|Shortest Paths|Intermediate"
        ]
      },
      {
        "prob_id": "793a",
        "tags": [
          "Divisibility, Modular Arithmetic|Number Theory|Basics"
        ]
      }
    ]
  },
  {
    "reference": {
      "prob_id": "909c",
      "tags": [
        "Knapsack DP||"
      ]
    },
    "positive": {
      "prob_id": "18e",
      "tags": [
        "Knapsack DP||"
      ]
    },
    "negative": [
      {
        "prob_id": "1391e",
        "tags": [
          "DFS Tree|Graph Traversal|Graph Theory"
        ]
      },
      {
        "prob_id": "1175b",
        "tags": [
          "Lists, Stacks, Queues|Basic Data Structures|Basics"
        ]
      },
      {
        "prob_id": "1454e",
        "tags": [
          "Cycle Detection|Cycles|Graph Theory"
        ]
      }
    ]
  },
  {
    "reference": {
      "prob_id": "372c",
      "tags": [
        "Knapsack DP||"
      ]
    },
    "positive": {
      "prob_id": "900e",
      "tags": [
        "Knapsack DP||"
      ]
    },
    "negative": [
      {
        "prob_id": "704e",
        "tags": [
          "Heavy Light Decomposition (HLD) ft Subtrees and Path Query|Heavy Light Decomposition (HLD)|Data Structures (DS)"
        ]
      },
      {
        "prob_id": "150e",
        "tags": [
          "Centroid Decomposition|Centroid Decomposition|Data Structures (DS)"
        ]
      },
      {
        "prob_id": "269c",
        "tags": [
          "Network Flows||Advanced Algorithms"
        ]
      }
    ]
  },
  {
    "reference": {
      "prob_id": "487b",
      "tags": [
        "Knapsack DP||"
      ]
    },
    "positive": {
      "prob_id": "833b",
      "tags": [
        "Divide and Conquer DP|DP optimizations|Dynamic Programming",
        "Knapsack DP||"
      ]
    },
    "negative": [
      {
        "prob_id": "506d",
        "tags": [
          "Splitting Objects into Light and Heavy / Case Processing|Square Root Decomposition|Data Structures (DS)"
        ]
      },
      {
        "prob_id": "468b",
        "tags": [
          "Disjoint Set Union||"
        ]
      },
      {
        "prob_id": "243a",
        "tags": [
          "Bit Manipulation|Bit Manipulation|Basics"
        ]
      }
    ]
  },
  {
    "reference": {
      "prob_id": "264c",
      "tags": [
        "Knapsack DP||"
      ]
    },
    "positive": {
      "prob_id": "909c",
      "tags": [
        "Knapsack DP||"
      ]
    },
    "negative": [
      {
        "prob_id": "348c",
        "tags": [
          "Square-Root Discrimination||Intermediate"
        ]
      },
      {
        "prob_id": "242e",
        "tags": [
          "Segment Tree with Lazy Propagation|Segment Tree|Data Structures (DS)"
        ]
      },
      {
        "prob_id": "193b",
        "tags": [
          "Backtracking|Miscellaneous|Miscellaneous"
        ]
      }
    ]
  },
  {
    "reference": {
      "prob_id": "101b",
      "tags": [
        "Knapsack DP||",
        "Sweep Line||"
      ]
    },
    "positive": {
      "prob_id": "487b",
      "tags": [
        "Knapsack DP||"
      ]
    },
    "negative": [
      {
        "prob_id": "340c",
        "tags": [
          "Binomial Coefficients|Fundamentals|Combinatorics"
        ]
      },
      {
        "prob_id": "362e",
        "tags": [
          "Bipartite Matching|Network Flows|Advanced Algorithms"
        ]
      },
      {
        "prob_id": "567e",
        "tags": [
          "Dijkstra's Algorithm|Shortest Paths|Graph Theory"
        ]
      }
    ]
  },
  {
    "reference": {
      "prob_id": "38e",
      "tags": [
        "Knapsack DP||"
      ]
    },
    "positive": {
      "prob_id": "909c",
      "tags": [
        "Knapsack DP||"
      ]
    },
    "negative": [
      {
        "prob_id": "843d",
        "tags": [
          "Dijkstra's Algorithm|Shortest Paths|Graph Theory"
        ]
      },
      {
        "prob_id": "420d",
        "tags": [
          "Splay Trees and Treaps|Augmenting Data Structures|Advanced Data Structures"
        ]
      },
      {
        "prob_id": "241c",
        "tags": [
          "Reflections on a Rectangle|Miscellaneous|Geometry"
        ]
      }
    ]
  },
  {
    "reference": {
      "prob_id": "115e",
      "tags": [
        "Knapsack DP||"
      ]
    },
    "positive": {
      "prob_id": "900e",
      "tags": [
        "Knapsack DP||"
      ]
    },
    "negative": [
      {
        "prob_id": "638c",
        "tags": [
          "Depth First Search|Graph traversal|Graphs"
        ]
      },
      {
        "prob_id": "700c",
        "tags": [
          "Finding Bridges in O(N+M)|Connected components, bridges, articulations points|Graphs"
        ]
      },
      {
        "prob_id": "427e",
        "tags": [
          "Ternary Search|Search Techniques|Miscellaneous"
        ]
      }
    ]
  },
  {
    "reference": {
      "prob_id": "900e",
      "tags": [
        "Knapsack DP||"
      ]
    },
    "positive": {
      "prob_id": "115e",
      "tags": [
        "Knapsack DP||"
      ]
    },
    "negative": [
      {
        "prob_id": "152c",
        "tags": [
          "Rule of Sum, Rule of Product|Counting|Basics"
        ]
      },
      {
        "prob_id": "528d",
        "tags": [
          "Fast Fourier Transform (FFT)|Convolution|Math"
        ]
      },
      {
        "prob_id": "305e",
        "tags": [
          "Nim, Sprague-Grundy Theorem|Combinatorial Games|Advanced Mathematics"
        ]
      }
    ]
  },
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