diff --git "a/AlgoSimBench_mcq.json" "b/AlgoSimBench_mcq.json" new file mode 100644--- /dev/null +++ "b/AlgoSimBench_mcq.json" @@ -0,0 +1,13885 @@ +[ + { + "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" + ] + }, + { + "prob_id": "1091e", + "tags": [ + "Degree Sequences|Miscellaneous|Graph Theory" + ] + } + ] + }, + { + "reference": { + "prob_id": "245h", + "tags": [ + "Interval DP / Range DP / Matrix Chain Multiplication (MCM)|Classic DP|Dynamic Programming (DP)" + ] + }, + "positive": { + "prob_id": "1025d", + "tags": [ + "Interval DP / Range DP / Matrix Chain Multiplication (MCM)|Classic DP|Dynamic Programming (DP)" + ] + }, + "negative": [ + { + "prob_id": "427c", + "tags": [ + "Strongly Connected Components (SCC)|Graph Connectivity|Graph Theory" + ] + }, + { + "prob_id": "1016b", + "tags": [ + "Prefix Sums and Difference Arrays|Range Queries|Basics" + ] + }, + { + "prob_id": "831c", + "tags": [ + "Prefix Sum|Range Queries Without Updates|Basics" + ] + } + ] + }, + { + "reference": { + "prob_id": "1303e", + "tags": [ + "Interval DP / Range DP / Matrix Chain Multiplication (MCM)|Classic DP|Dynamic Programming (DP)" + ] + }, + "positive": { + "prob_id": "1025d", + "tags": [ + "Interval DP / Range DP / Matrix Chain Multiplication (MCM)|Classic DP|Dynamic Programming (DP)" + ] + }, + "negative": [ + { + "prob_id": "1201c", + "tags": [ + "Binary Search|Leveraging Monotonicity|Basics" + ] + }, + { + "prob_id": "427c", + "tags": [ + "Strongly Connected Components (SCC)|Graph Connectivity|Graph Theory" + ] + }, + { + "prob_id": "1613f", + "tags": [ + "Number Theoretic Transform (NTT)|Convolution|Math" + ] + } + ] + }, + { + "reference": { + "prob_id": "1513d", + "tags": [ + "Minimum Spanning Tree||" + ] + }, + "positive": { + "prob_id": "32c", + "tags": [ + "Minimum Spanning Tree||" + ] + }, + "negative": [ + { + "prob_id": "1536b", + "tags": [ + "Pigeonhole Principle|Basic Counting|Basics" + ] + }, + { + "prob_id": "1619h", + "tags": [ + "Splitting Objects into Light and Heavy / Case Processing|Square Root Decomposition|Data Structures (DS)" + ] + }, + { + "prob_id": "150e", + "tags": [ + "Centroid Decomposition|Centroid Decomposition|Data Structures (DS)" + ] + } + ] + }, + { + "reference": { + "prob_id": "1245d", + "tags": [ + "Minimum Spanning Tree||" + ] + }, + "positive": { + "prob_id": "632f", + "tags": [ + "Minimum Spanning Tree||" + ] + }, + "negative": [ + { + "prob_id": "1252k", + "tags": [ + "Segment Tree with Lazy Propagation|Segment Tree|Data Structures (DS)" + ] + }, + { + "prob_id": "4d", + "tags": [ + "DP with Solution Construction||Intermediate" + ] + }, + { + "prob_id": "1477a", + "tags": [ + "Bezout's Identity|Euclidean Algorithm|Number Theory" + ] + } + ] + }, + { + "reference": { + "prob_id": "125e", + "tags": [ + "Minimum Spanning Tree||" + ] + }, + "positive": { + "prob_id": "1513d", + "tags": [ + "Minimum Spanning Tree||" + ] + }, + "negative": [ + { + "prob_id": "244b", + "tags": [ + "All-Subsets / Binary Choice|Complete Search|Basics" + ] + }, + { + "prob_id": "434d", + "tags": [ + "Network Flows||Advanced Algorithms" + ] + }, + { + "prob_id": "245h", + "tags": [ + "Interval DP / Range DP / Matrix Chain Multiplication (MCM)|Classic DP|Dynamic Programming (DP)" + ] + } + ] + }, + { + "reference": { + "prob_id": "1408e", + "tags": [ + "Minimum Spanning Tree||" + ] + }, + "positive": { + "prob_id": "32c", + "tags": [ + "Minimum Spanning Tree||" + ] + }, + "negative": [ + { + "prob_id": "960g", + "tags": [ + "Stirling Number of the First Kind for Fixed N|Stirling Numbers|Combinatorics" + ] + }, + { + "prob_id": "1288c", + "tags": [ + "Stars and Bars|Basic Counting|Basics" + ] + }, + { + "prob_id": "1108e2", + "tags": [ + "Sorted Set||" + ] + } + ] + }, + { + "reference": { + "prob_id": "510c", + "tags": [ + "Topological Sorting|Graph Traversal|Graph Theory" + ] + }, + "positive": { + "prob_id": "909e", + "tags": [ + "Topological Sorting|Graph Traversal|Graph Theory" + ] + }, + "negative": [ + { + "prob_id": "1085e", + "tags": [ + "Tree Centers and Diameter||Advanced Data Structures" + ] + }, + { + "prob_id": "704d", + "tags": [ + "Flow with Lower Bounds||" + ] + }, + { + "prob_id": "128b", + "tags": [ + "Suffix Automaton|Suffix Structures|Strings" + ] + } + ] + }, + { + "reference": { + "prob_id": "501c", + "tags": [ + "Topological Sorting|Graph Traversal|Graph Theory" + ] + }, + "positive": { + "prob_id": "510c", + "tags": [ + "Topological Sorting|Graph Traversal|Graph Theory" + ] + }, + "negative": [ + { + "prob_id": "402e", + "tags": [ + "Matrix Exponentiation|Matrices|Math" + ] + }, + { + "prob_id": "650d", + "tags": [ + "O(n lg n) Longest Monotonic Subsequence|DP Optimization|Advanced Algorithms" + ] + }, + { + "prob_id": "713d", + "tags": [ + "Level Ancestors and Lowest Common Ancestors||Advanced Data Structures" + ] + } + ] + }, + { + "reference": { + "prob_id": "825e", + "tags": [ + "Topological Sorting|Graph Traversal|Graph Theory" + ] + }, + "positive": { + "prob_id": "510c", + "tags": [ + "Topological Sorting|Graph Traversal|Graph Theory" + ] + }, + "negative": [ + { + "prob_id": "117e", + "tags": [ + "Link Cut Tree (LCT)|Link Cut Tree (LCT)|Data Structures (DS)" + ] + }, + { + "prob_id": "464b", + "tags": [ + "All-Permutations|Complete Search|Basics", + "Backtracking|Miscellaneous|Miscellaneous" + ] + }, + { + "prob_id": "1217f", + "tags": [ + "Dynamic Connectivity Problem (Offline)|Dynamic Connectivity|Data Structures (DS)", + "Offline Deletion||" + ] + } + ] + }, + { + "reference": { + "prob_id": "621e", + "tags": [ + "Matrix Exponentiation|Matrices|Math" + ] + }, + "positive": { + "prob_id": "852b", + "tags": [ + "Matrix Exponentiation|Matrices|Math" + ] + }, + "negative": [ + { + "prob_id": "708d", + "tags": [ + "Flow with Lower Bounds||" + ] + }, + { + "prob_id": "884f", + "tags": [ + "Min Cost Max Flow (MCMF)|Flow and Min Cut|Graph Theory" + ] + }, + { + "prob_id": "616c", + "tags": [ + "Grid Graphs|Graphs|Basics" + ] + } + ] + }, + { + "reference": { + "prob_id": "576d", + "tags": [ + "Matrix Exponentiation|Matrices|Math" + ] + }, + "positive": { + "prob_id": "678d", + "tags": [ + "Matrix Exponentiation|Matrices|Math" + ] + }, + "negative": [ + { + "prob_id": "241c", + "tags": [ + "Reflections on a Rectangle|Miscellaneous|Geometry" + ] + }, + { + "prob_id": "32e", + "tags": [ + "Vectors and Applications|Geometry|Basics" + ] + }, + { + "prob_id": "501d", + "tags": [ + "Factorial Number System|Miscellaneous|Number Theory", + "Segment Tree|Trees|Data Structures", + "Sorted Set||" + ] + } + ] + }, + { + "reference": { + "prob_id": "917c", + "tags": [ + "Matrix Exponentiation|Matrices|Math" + ] + }, + "positive": { + "prob_id": "678d", + "tags": [ + "Matrix Exponentiation|Matrices|Math" + ] + }, + "negative": [ + { + "prob_id": "1110e", + "tags": [ + "Invariants and Monovariants|Techniques|Miscellaneous" + ] + }, + { + "prob_id": "679a", + "tags": [ + "Interactive Problems|Interactive Problems|Miscellaneous" + ] + }, + { + "prob_id": "702e", + "tags": [ + "Binary Lifting||Advanced Data Structures" + ] + } + ] + }, + { + "reference": { + "prob_id": "914d", + "tags": [ + "Segment Tree|Trees|Data Structures" + ] + }, + "positive": { + "prob_id": "1234d", + "tags": [ + "Segment Tree|Trees|Data Structures" + ] + }, + "negative": [ + { + "prob_id": "891e", + "tags": [ + "Generating Functions|Polynomial|Math" + ] + }, + { + "prob_id": "1418e", + "tags": [ + "Probabilities and Expected Values|Probabilities and Expected Values|Math" + ] + }, + { + "prob_id": "755114", + "tags": [ + "Backtracking|Miscellaneous|Miscellaneous" + ] + } + ] + }, + { + "reference": { + "prob_id": "834b", + "tags": [ + "Sweep Line||" + ] + }, + "positive": { + "prob_id": "silver fox vs monster153", + "tags": [ + "Sweep Line||" + ] + }, + "negative": [ + { + "prob_id": "82a", + "tags": [ + "Sums and Asymptotics||Preliminaries" + ] + }, + { + "prob_id": "827f", + "tags": [ + "Dijkstra's Algorithm|Shortest Paths|Graph Theory" + ] + }, + { + "prob_id": "840d", + "tags": [ + "Divide & Conquer - Static Range Queries||" + ] + } + ] + }, + { + "reference": { + "prob_id": "333e", + "tags": [ + "Convex Hull Trick|DP Optimization|Advanced Algorithms" + ] + }, + "positive": { + "prob_id": "631e", + "tags": [ + "Convex Hull Trick|DP Optimization|Advanced Algorithms" + ] + }, + "negative": [ + { + "prob_id": "101b", + "tags": [ + "Knapsack DP||", + "Sweep Line||" + ] + }, + { + "prob_id": "609e", + "tags": [ + "Binary Lifting||Advanced Data Structures", + "Lowest Common Ancestor|Lowest common ancestor|Graphs" + ] + }, + { + "prob_id": "594d", + "tags": [ + "BIT / Fenwick Tree|Binary Indexed Tree (BIT)|Data Structures (DS)", + "Euler's totient function|Number-theoretic functions|Algebra", + "Offline Queries|Techniques|Data Structures (DS)", + "Segment Tree|Trees|Data Structures" + ] + } + ] + }, + { + "reference": { + "prob_id": "682e", + "tags": [ + "Convex Hull Trick|DP Optimization|Advanced Algorithms" + ] + }, + "positive": { + "prob_id": "319c", + "tags": [ + "Convex Hull Trick|DP Optimization|Advanced Algorithms" + ] + }, + "negative": [ + { + "prob_id": "545d", + "tags": [ + "Exchange Argument|Greedy|Basics" + ] + }, + { + "prob_id": "755114", + "tags": [ + "Backtracking|Miscellaneous|Miscellaneous" + ] + }, + { + "prob_id": "915g", + "tags": [ + "Mobius Function and Mobius Inversion|Multiplicative Functions|Number Theory" + ] + } + ] + }, + { + "reference": { + "prob_id": "yakiniku restaurants067", + "tags": [ + "Divide and Conquer DP|DP optimizations|Dynamic Programming" + ] + }, + "positive": { + "prob_id": "873d", + "tags": [ + "Divide and Conquer DP|DP optimizations|Dynamic Programming" + ] + }, + "negative": [ + { + "prob_id": "hopscotch addict132", + "tags": [ + "Breadth First Search|Graph traversal|Graphs" + ] + }, + { + "prob_id": "tak and hotels060", + "tags": [ + "Binary Lifting||Advanced Data Structures" + ] + }, + { + "prob_id": "multiple of 2019164", + "tags": [ + "Prefix Sum|Range Queries Without Updates|Basics" + ] + } + ] + }, + { + "reference": { + "prob_id": "868f", + "tags": [ + "Divide and Conquer DP|DP optimizations|Dynamic Programming" + ] + }, + "positive": { + "prob_id": "768b", + "tags": [ + "Divide and Conquer DP|DP optimizations|Dynamic Programming" + ] + }, + "negative": [ + { + "prob_id": "987e", + "tags": [ + "Permutations and Permutation Cycles|Permutations|Miscellaneous" + ] + }, + { + "prob_id": "86d", + "tags": [ + "Mo's Algorithm|Square Root Decomposition|Data Structures (DS)", + "Square Root Decomposition||" + ] + }, + { + "prob_id": "1006c", + "tags": [ + "Two Pointers|Leveraging Monotonicity|Basics" + ] + } + ] + }, + { + "reference": { + "prob_id": "342e", + "tags": [ + "Centroid Decomposition|Centroid Decomposition|Data Structures (DS)" + ] + }, + "positive": { + "prob_id": "BTREE", + "tags": [ + "Centroid Decomposition|Centroid Decomposition|Data Structures (DS)" + ] + }, + "negative": [ + { + "prob_id": "87e", + "tags": [ + "Minkowski Sum|Convex Polygon|Geometry", + "Point in Convex Polygon in log(N)(2D)|Geometry 2D|Geometry" + ] + }, + { + "prob_id": "371d", + "tags": [ + "Disjoint Set Union||" + ] + }, + { + "prob_id": "1176e", + "tags": [ + "Bipartite Graph||", + "Graph Traversal||" + ] + } + ] + }, + { + "reference": { + "prob_id": "150e", + "tags": [ + "Centroid Decomposition|Centroid Decomposition|Data Structures (DS)" + ] + }, + "positive": { + "prob_id": "708c", + "tags": [ + "Centroid Decomposition|Centroid Decomposition|Data Structures (DS)" + ] + }, + "negative": [ + { + "prob_id": "78c", + "tags": [ + "Winning and Losing Positions, Strategy Stealing|Combinatorial Games|Advanced Algorithms" + ] + }, + { + "prob_id": "703d", + "tags": [ + "Offline Processing of Queries||Advanced Data Structures" + ] + }, + { + "prob_id": "786b", + "tags": [ + "[Trick] Dijkstra on Segment Tree|Shortest Paths|Graph Theory" + ] + } + ] + }, + { + "reference": { + "prob_id": "833d", + "tags": [ + "Centroid Decomposition|Centroid Decomposition|Data Structures (DS)" + ] + }, + "positive": { + "prob_id": "708c", + "tags": [ + "Centroid Decomposition|Centroid Decomposition|Data Structures (DS)" + ] + }, + "negative": [ + { + "prob_id": "827e", + "tags": [ + "Fast Fourier Transform (FFT)|Convolution|Math", + "String Matching with FFT ft Wildcard Matching|String Matching|Strings" + ] + }, + { + "prob_id": "825f", + "tags": [ + "Z algorithm|String Matching|Strings" + ] + }, + { + "prob_id": "828f", + "tags": [ + "Lowest Common Ancestor|Lowest common ancestor|Graphs" + ] + } + ] + }, + { + "reference": { + "prob_id": "tak and hotels060", + "tags": [ + "Binary Lifting||Advanced Data Structures" + ] + }, + "positive": { + "prob_id": "702e", + "tags": [ + "Binary Lifting||Advanced Data Structures" + ] + }, + "negative": [ + { + "prob_id": "sum of gcd of tuples (hard)162", + "tags": [ + "Harmonic Lemma||Intermediate", + "M\u00f6bius Inversion||Advanced Mathematics" + ] + }, + { + "prob_id": "exhausted?076", + "tags": [ + "Bipartite Matching|Network Flows|Advanced Algorithms" + ] + }, + { + "prob_id": "or plus max100", + "tags": [ + "Subset Convolution|Bit Related DP|Dynamic Programming (DP)" + ] + } + ] + }, + { + "reference": { + "prob_id": "1328e", + "tags": [ + "Binary Lifting||Advanced Data Structures" + ] + }, + "positive": { + "prob_id": "teleporter167", + "tags": [ + "Binary Lifting||Advanced Data Structures" + ] + }, + "negative": [ + { + "prob_id": "1175f", + "tags": [ + "XOR Hashing|Hashing|Strings" + ] + }, + { + "prob_id": "120j", + "tags": [ + "Closest Pair of Points|Sweep Line|Geometry" + ] + }, + { + "prob_id": "878c", + "tags": [ + "Sorted Set||" + ] + } + ] + }, + { + "reference": { + "prob_id": "372d", + "tags": [ + "Binary Lifting||Advanced Data Structures" + ] + }, + "positive": { + "prob_id": "teleporter167", + "tags": [ + "Binary Lifting||Advanced Data Structures" + ] + }, + "negative": [ + { + "prob_id": "25c", + "tags": [ + "All-Pairs|Shortest Paths|Intermediate" + ] + }, + { + "prob_id": "372b", + "tags": [ + "Prefix Sums and Difference Arrays|Range Queries|Basics" + ] + }, + { + "prob_id": "1434d", + "tags": [ + "Tree Diameter|Graph Traversal|Graph Theory" + ] + } + ] + }, + { + "reference": { + "prob_id": "iroha loves strings058", + "tags": [ + "Z algorithm|String Matching|Strings" + ] + }, + "positive": { + "prob_id": "471d", + "tags": [ + "Z algorithm|String Matching|Strings" + ] + }, + "negative": [ + { + "prob_id": "19e", + "tags": [ + "DFS Tree|Graph Traversal|Graph Theory", + "Queue Undo Trick|Techniques|Data Structures (DS)" + ] + }, + { + "prob_id": "lisdl091", + "tags": [ + "Longest Increasing Subsequence (LIS)|Classic DP|Dynamic Programming (DP)" + ] + }, + { + "prob_id": "lcms038", + "tags": [ + "Inclusion Exclusion on Multiples|Inclusion Exclusion|Combinatorics", + "Mobius Function and Mobius Inversion|Multiplicative Functions|Number Theory" + ] + } + ] + }, + { + "reference": { + "prob_id": "455b", + "tags": [ + "Trie||" + ] + }, + "positive": { + "prob_id": "1055f", + "tags": [ + "Trie||" + ] + }, + "negative": [ + { + "prob_id": "724c", + "tags": [ + "Reflections on a Rectangle|Miscellaneous|Geometry" + ] + }, + { + "prob_id": "305b", + "tags": [ + "Continued Fractions|Miscellaneous|Math" + ] + }, + { + "prob_id": "464e", + "tags": [ + "Dijkstra's Algorithm|Shortest Paths|Graph Theory" + ] + } + ] + }, + { + "reference": { + "prob_id": "997e", + "tags": [ + "Counting Minimums with Segment Tree||" + ] + }, + "positive": { + "prob_id": "1197e", + "tags": [ + "Counting Minimums with Segment Tree||" + ] + }, + "negative": [ + { + "prob_id": "1205b", + "tags": [ + "Breadth First Search|Graph traversal|Graphs" + ] + }, + { + "prob_id": "1093e", + "tags": [ + "2D BIT|Binary Indexed Tree (BIT)|Data Structures (DS)", + "CDQ Divide and Conquer|Techniques|Data Structures (DS)" + ] + }, + { + "prob_id": "909e", + "tags": [ + "Topological Sorting|Graph Traversal|Graph Theory" + ] + } + ] + }, + { + "reference": { + "prob_id": "117e", + "tags": [ + "Link Cut Tree (LCT)|Link Cut Tree (LCT)|Data Structures (DS)" + ] + }, + "positive": { + "prob_id": "1109f", + "tags": [ + "Link Cut Tree (LCT)|Link Cut Tree (LCT)|Data Structures (DS)" + ] + }, + "negative": [ + { + "prob_id": "963e", + "tags": [ + "Gaussian Elimination|Linear Algebra|Math" + ] + }, + { + "prob_id": "1175d", + "tags": [ + "Prefix Sums and Difference Arrays|Range Queries|Basics" + ] + }, + { + "prob_id": "251a", + "tags": [ + "Binary Search|Leveraging Monotonicity|Basics" + ] + } + ] + }, + { + "reference": { + "prob_id": "1172e", + "tags": [ + "Link Cut Tree (LCT)|Link Cut Tree (LCT)|Data Structures (DS)" + ] + }, + "positive": { + "prob_id": "1109f", + "tags": [ + "Link Cut Tree (LCT)|Link Cut Tree (LCT)|Data Structures (DS)" + ] + }, + "negative": [ + { + "prob_id": "954f", + "tags": [ + "Matrix Exponentiation|Matrices|Math" + ] + }, + { + "prob_id": "1389a", + "tags": [ + "Basic Constructive Algorithms|Greedy and Constructive|Basics", + "Greatest Common Divisor and Least Common Multiple|Number Theory|Basics" + ] + }, + { + "prob_id": "597c", + "tags": [ + "BIT / Fenwick Tree|Binary Indexed Tree (BIT)|Data Structures (DS)" + ] + } + ] + }, + { + "reference": { + "prob_id": "1239d", + "tags": [ + "Strongly Connected Components (SCC)|Graph Connectivity|Graph Theory" + ] + }, + "positive": { + "prob_id": "22e", + "tags": [ + "Strongly Connected Components (SCC)|Graph Connectivity|Graph Theory" + ] + }, + "negative": [ + { + "prob_id": "1442d", + "tags": [ + "Queue Undo Trick|Techniques|Data Structures (DS)", + "SMAWK Algorithm and Min Plus Convolution (Convex and Arbitrary)|Min Plus Convolution|Miscellaneous" + ] + }, + { + "prob_id": "1195f", + "tags": [ + "Convex Hull Trick|DP Optimization|Advanced Algorithms", + "Point Update Range Sum||" + ] + }, + { + "prob_id": "1567b", + "tags": [ + "Bit Manipulation|Bit Manipulation|Basics" + ] + } + ] + }, + { + "reference": { + "prob_id": "505d", + "tags": [ + "Strongly Connected Components (SCC)|Graph Connectivity|Graph Theory" + ] + }, + "positive": { + "prob_id": "22e", + "tags": [ + "Strongly Connected Components (SCC)|Graph Connectivity|Graph Theory" + ] + }, + "negative": [ + { + "prob_id": "598c", + "tags": [ + "Vectors and Applications|Geometry|Basics" + ] + }, + { + "prob_id": "793a", + "tags": [ + "Divisibility, Modular Arithmetic|Number Theory|Basics" + ] + }, + { + "prob_id": "1245a", + "tags": [ + "Chicken McNugget Theorem|Euclidean Algorithm|Number Theory" + ] + } + ] + }, + { + "reference": { + "prob_id": "226e", + "tags": [ + "Persistent Segment Tree|Segment Tree|Data Structures (DS)" + ] + }, + "positive": { + "prob_id": "837g", + "tags": [ + "Persistent Segment Tree|Segment Tree|Data Structures (DS)" + ] + }, + "negative": [ + { + "prob_id": "191a", + "tags": [ + "Flatten the Memo|DP Optimization|Advanced Algorithms" + ] + }, + { + "prob_id": "855c", + "tags": [ + "DP on Trees and DAGs|Dp on Different Structures|Dynamic Programming (DP)", + "Subtree Convolution|DP Optimization|Advanced Algorithms" + ] + }, + { + "prob_id": "18e", + "tags": [ + "Knapsack DP||" + ] + } + ] + }, + { + "reference": { + "prob_id": "484e", + "tags": [ + "Persistent Segment Tree|Segment Tree|Data Structures (DS)" + ] + }, + "positive": { + "prob_id": "837g", + "tags": [ + "Persistent Segment Tree|Segment Tree|Data Structures (DS)" + ] + }, + "negative": [ + { + "prob_id": "361b", + "tags": [ + "Relatively Prime, Euler Totient Function|Number Theory|Intermediate" + ] + }, + { + "prob_id": "87e", + "tags": [ + "Minkowski Sum|Convex Polygon|Geometry", + "Point in Convex Polygon in log(N)(2D)|Geometry 2D|Geometry" + ] + }, + { + "prob_id": "340c", + "tags": [ + "Binomial Coefficients|Fundamentals|Combinatorics" + ] + } + ] + }, + { + "reference": { + "prob_id": "204e", + "tags": [ + "Suffix Array|Suffix Structures|Strings" + ] + }, + "positive": { + "prob_id": "EST", + "tags": [ + "Suffix Array|Suffix Structures|Strings" + ] + }, + "negative": [ + { + "prob_id": "219d", + "tags": [ + "Trees and Tree Traversals|Graphs|Basics" + ] + }, + { + "prob_id": "241e", + "tags": [ + "Single Source with Negative Weights|Shortest Paths|Intermediate" + ] + }, + { + "prob_id": "1016b", + "tags": [ + "Prefix Sums and Difference Arrays|Range Queries|Basics" + ] + } + ] + }, + { + "reference": { + "prob_id": "232d", + "tags": [ + "Suffix Array|Suffix Structures|Strings" + ] + }, + "positive": { + "prob_id": "802i", + "tags": [ + "Suffix Array|Suffix Structures|Strings" + ] + }, + "negative": [ + { + "prob_id": "359c", + "tags": [ + "Greatest Common Divisor and Least Common Multiple|Number Theory|Basics" + ] + }, + { + "prob_id": "47d", + "tags": [ + "Backtracking|Miscellaneous|Miscellaneous" + ] + }, + { + "prob_id": "27d", + "tags": [ + "Bipartite Graph||" + ] + } + ] + }, + { + "reference": { + "prob_id": "504e", + "tags": [ + "Suffix Array|Suffix Structures|Strings" + ] + }, + "positive": { + "prob_id": "802i", + "tags": [ + "Suffix Array|Suffix Structures|Strings" + ] + }, + "negative": [ + { + "prob_id": "613d", + "tags": [ + "Virtual Tree / Auxiliary Tree|More Tree Stuff|Graph Theory" + ] + }, + { + "prob_id": "120j", + "tags": [ + "Closest Pair of Points|Sweep Line|Geometry" + ] + }, + { + "prob_id": "617b", + "tags": [ + "Permutations with Repetition, Stars and Bars|Counting|Intermediate" + ] + } + ] + }, + { + "reference": { + "prob_id": "1307g", + "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": "1392h", + "tags": [ + "Martingale|Probabilities and Expected Values|Math", + "Probabilities and Expected Values|Probabilities and Expected Values|Math" + ] + }, + { + "prob_id": "633f", + "tags": [ + "Tree Diameter|Graph Traversal|Graph Theory" + ] + }, + { + "prob_id": "1083a", + "tags": [ + "DP on Trees and DAGs|Dp on Different Structures|Dynamic Programming (DP)" + ] + } + ] + }, + { + "reference": { + "prob_id": "1194d", + "tags": [ + "Nim, Sprague-Grundy Theorem|Combinatorial Games|Advanced Mathematics" + ] + }, + "positive": { + "prob_id": "1310f", + "tags": [ + "Nim, Sprague-Grundy Theorem|Combinatorial Games|Advanced Mathematics" + ] + }, + "negative": [ + { + "prob_id": "722c", + "tags": [ + "Disjoint Set Union||" + ] + }, + { + "prob_id": "453e", + "tags": [ + "Segment Tree Beats|Segment Tree|Data Structures (DS)" + ] + }, + { + "prob_id": "451e", + "tags": [ + "Generating Functions|Polynomial|Math", + "Principle of Inclusion-Exclusion|Counting|Intermediate" + ] + } + ] + }, + { + "reference": { + "prob_id": "FDIVGAME", + "tags": [ + "Nim, Sprague-Grundy Theorem|Combinatorial Games|Advanced Mathematics" + ] + }, + "positive": { + "prob_id": "1310f", + "tags": [ + "Nim, Sprague-Grundy Theorem|Combinatorial Games|Advanced Mathematics" + ] + }, + "negative": [ + { + "prob_id": "1575f", + "tags": [ + "Martingale|Probabilities and Expected Values|Math" + ] + }, + { + "prob_id": "997e", + "tags": [ + "Counting Minimums with Segment Tree||" + ] + }, + { + "prob_id": "1364c", + "tags": [ + "Basic Constructive Algorithms|Greedy and Constructive|Basics" + ] + } + ] + }, + { + "reference": { + "prob_id": "1312f", + "tags": [ + "Nim, Sprague-Grundy Theorem|Combinatorial Games|Advanced Mathematics" + ] + }, + "positive": { + "prob_id": "1310f", + "tags": [ + "Nim, Sprague-Grundy Theorem|Combinatorial Games|Advanced Mathematics" + ] + }, + "negative": [ + { + "prob_id": "222c", + "tags": [ + "Sieve, Prime Factorization and Divisors|Primes and Divisors|Number Theory" + ] + }, + { + "prob_id": "611c", + "tags": [ + "Prefix Sums and Difference Arrays|Range Queries|Basics" + ] + }, + { + "prob_id": "835f", + "tags": [ + "Tree Centers and Diameter||Advanced Data Structures" + ] + } + ] + }, + { + "reference": { + "prob_id": "1451f", + "tags": [ + "Nim, Sprague-Grundy Theorem|Combinatorial Games|Advanced Mathematics" + ] + }, + "positive": { + "prob_id": "1310f", + "tags": [ + "Nim, Sprague-Grundy Theorem|Combinatorial Games|Advanced Mathematics" + ] + }, + "negative": [ + { + "prob_id": "1006f", + "tags": [ + "Binary Search|Leveraging Monotonicity|Basics", + "Meet in the Middle|Techniques|Miscellaneous" + ] + }, + { + "prob_id": "1364c", + "tags": [ + "Basic Constructive Algorithms|Greedy and Constructive|Basics" + ] + }, + { + "prob_id": "1408e", + "tags": [ + "Minimum Spanning Tree||" + ] + } + ] + }, + { + "reference": { + "prob_id": "305e", + "tags": [ + "Nim, Sprague-Grundy Theorem|Combinatorial Games|Advanced Mathematics" + ] + }, + "positive": { + "prob_id": "1310f", + "tags": [ + "Nim, Sprague-Grundy Theorem|Combinatorial Games|Advanced Mathematics" + ] + }, + "negative": [ + { + "prob_id": "12d", + "tags": [ + "BIT / Fenwick Tree|Binary Indexed Tree (BIT)|Data Structures (DS)" + ] + }, + { + "prob_id": "58a", + "tags": [ + "Greedy Stays Ahead|Greedy|Basics" + ] + }, + { + "prob_id": "231e", + "tags": [ + "Biconnectivity|DFS Lowlink|Advanced Algorithms" + ] + } + ] + }, + { + "reference": { + "prob_id": "xor battle045", + "tags": [ + "XOR Basis||" + ] + }, + "positive": { + "prob_id": "1299d", + "tags": [ + "XOR Basis||" + ] + }, + "negative": [ + { + "prob_id": "grouping067", + "tags": [ + "Harmonic Lemma||Intermediate" + ] + }, + { + "prob_id": "fork in the road144", + "tags": [ + "Markov Chains|Probability and Expectation|Intermediate" + ] + }, + { + "prob_id": "red and green apples160", + "tags": [ + "Exchange Argument|Greedy|Basics" + ] + } + ] + }, + { + "reference": { + "prob_id": "438d", + "tags": [ + "Segment Tree Beats|Segment Tree|Data Structures (DS)" + ] + }, + "positive": { + "prob_id": "855f", + "tags": [ + "Segment Tree Beats|Segment Tree|Data Structures (DS)" + ] + }, + "negative": [ + { + "prob_id": "585d", + "tags": [ + "Meet in the Middle|Techniques|Miscellaneous" + ] + }, + { + "prob_id": "193a", + "tags": [ + "Grid Graphs|Graphs|Basics" + ] + }, + { + "prob_id": "434d", + "tags": [ + "Network Flows||Advanced Algorithms" + ] + } + ] + }, + { + "reference": { + "prob_id": "partition112", + "tags": [ + "Greatest Common Divisor and Least Common Multiple|Number Theory|Basics" + ] + }, + "positive": { + "prob_id": "527a", + "tags": [ + "Greatest Common Divisor and Least Common Multiple|Number Theory|Basics" + ] + }, + "negative": [ + { + "prob_id": "engines139", + "tags": [ + "Vectors and Applications|Geometry|Basics" + ] + }, + { + "prob_id": "many easy problems005", + "tags": [ + "Generating Functions|Polynomial|Math", + "[Problem] Sum of nCr(a[i], K) for each K from 1 to N|Factorials and Binomial Coefficients (nCr)|Combinatorics" + ] + }, + { + "prob_id": "lotus leaves074", + "tags": [ + "Network Flows||Advanced Algorithms" + ] + } + ] + }, + { + "reference": { + "prob_id": "284a", + "tags": [ + "Divisibility, Modular Arithmetic|Number Theory|Basics" + ] + }, + "positive": { + "prob_id": "180b", + "tags": [ + "Divisibility, Modular Arithmetic|Number Theory|Basics" + ] + }, + "negative": [ + { + "prob_id": "580c", + "tags": [ + "Trees and Tree Traversals|Graphs|Basics" + ] + }, + { + "prob_id": "678f", + "tags": [ + "Convex Hull Trick|DP Optimization|Advanced Algorithms" + ] + }, + { + "prob_id": "958f3", + "tags": [ + "Fast Fourier Transform (FFT)|Convolution|Math" + ] + } + ] + }, + { + "reference": { + "prob_id": "793a", + "tags": [ + "Divisibility, Modular Arithmetic|Number Theory|Basics" + ] + }, + "positive": { + "prob_id": "modulo summation103", + "tags": [ + "Divisibility, Modular Arithmetic|Number Theory|Basics" + ] + }, + "negative": [ + { + "prob_id": "722c", + "tags": [ + "Disjoint Set Union||" + ] + }, + { + "prob_id": "1151c", + "tags": [ + "Sums and Asymptotics||Preliminaries" + ] + }, + { + "prob_id": "842d", + "tags": [ + "MEX task (Minimal Excluded element in an array)|Sequences|Miscellaneous" + ] + } + ] + }, + { + "reference": { + "prob_id": "682a", + "tags": [ + "Divisibility, Modular Arithmetic|Number Theory|Basics" + ] + }, + "positive": { + "prob_id": "special trains084", + "tags": [ + "Divisibility, Modular Arithmetic|Number Theory|Basics" + ] + }, + "negative": [ + { + "prob_id": "505c", + "tags": [ + "GP Hash Table|Policy Based Data Structures|Data Structures (DS)" + ] + }, + { + "prob_id": "213c", + "tags": [ + "Eliminate Dependent Parameters|DP Optimization|Advanced Algorithms", + "Flatten the Memo|DP Optimization|Advanced Algorithms" + ] + }, + { + "prob_id": "1288c", + "tags": [ + "Stars and Bars|Basic Counting|Basics" + ] + } + ] + }, + { + "reference": { + "prob_id": "remainder minimization 2019133", + "tags": [ + "Divisibility, Modular Arithmetic|Number Theory|Basics" + ] + }, + "positive": { + "prob_id": "180b", + "tags": [ + "Divisibility, Modular Arithmetic|Number Theory|Basics" + ] + }, + "negative": [ + { + "prob_id": "distinct numbers143", + "tags": [ + "Binary Search|Leveraging Monotonicity|Basics" + ] + }, + { + "prob_id": "iroha loves strings058", + "tags": [ + "Z algorithm|String Matching|Strings" + ] + }, + { + "prob_id": "gcd on blackboard125", + "tags": [ + "Prefix Sum|Range Queries Without Updates|Basics" + ] + } + ] + }, + { + "reference": { + "prob_id": "sequence sum179", + "tags": [ + "Divisibility, Modular Arithmetic|Number Theory|Basics" + ] + }, + "positive": { + "prob_id": "180b", + "tags": [ + "Divisibility, Modular Arithmetic|Number Theory|Basics" + ] + }, + "negative": [ + { + "prob_id": "red and green apples160", + "tags": [ + "Exchange Argument|Greedy|Basics" + ] + }, + { + "prob_id": "league139", + "tags": [ + "Directed Acyclic Graphs|Shortest Paths|Intermediate" + ] + }, + { + "prob_id": "digit sum044", + "tags": [ + "Harmonic Lemma||Intermediate", + "Square-Root Discrimination||Intermediate" + ] + } + ] + }, + { + "reference": { + "prob_id": "18c", + "tags": [ + "Prefix Sums and Difference Arrays|Range Queries|Basics" + ] + }, + "positive": { + "prob_id": "983b", + "tags": [ + "Prefix Sums and Difference Arrays|Range Queries|Basics" + ] + }, + "negative": [ + { + "prob_id": "455d", + "tags": [ + "Splay Trees and Treaps|Augmenting Data Structures|Advanced Data Structures" + ] + }, + { + "prob_id": "546e", + "tags": [ + "Network Flows||Advanced Algorithms" + ] + }, + { + "prob_id": "30e", + "tags": [ + "Suffix Array|Suffix Structures|Strings" + ] + } + ] + }, + { + "reference": { + "prob_id": "atcoder express 2106", + "tags": [ + "Prefix Sums and Difference Arrays|Range Queries|Basics" + ] + }, + "positive": { + "prob_id": "877b", + "tags": [ + "Prefix Sums and Difference Arrays|Range Queries|Basics" + ] + }, + "negative": [ + { + "prob_id": "double factorial148", + "tags": [ + "Prime Factorization|Number Theory|Basics" + ] + }, + { + "prob_id": "sum of gcd of tuples (hard)162", + "tags": [ + "Harmonic Lemma||Intermediate", + "M\u00f6bius Inversion||Advanced Mathematics" + ] + }, + { + "prob_id": "must be rectangular!131", + "tags": [ + "Bipartite Graph||" + ] + } + ] + }, + { + "reference": { + "prob_id": "602b", + "tags": [ + "Prefix Sums and Difference Arrays|Range Queries|Basics" + ] + }, + "positive": { + "prob_id": "get ac122", + "tags": [ + "Prefix Sums and Difference Arrays|Range Queries|Basics" + ] + }, + "negative": [ + { + "prob_id": "854a", + "tags": [ + "Greatest Common Divisor and Least Common Multiple|Number Theory|Basics" + ] + }, + { + "prob_id": "19e", + "tags": [ + "DFS Tree|Graph Traversal|Graph Theory", + "Queue Undo Trick|Techniques|Data Structures (DS)" + ] + }, + { + "prob_id": "891e", + "tags": [ + "Generating Functions|Polynomial|Math" + ] + } + ] + }, + { + "reference": { + "prob_id": "many many paths154", + "tags": [ + "Binomial Coefficients|Fundamentals|Combinatorics" + ] + }, + "positive": { + "prob_id": "760f", + "tags": [ + "Binomial Coefficients|Fundamentals|Combinatorics" + ] + }, + "negative": [ + { + "prob_id": "not divisible170", + "tags": [ + "Prime and Composite, Primality Testing, Prime Sieves, Finding Divisors|Number Theory|Preliminaries" + ] + }, + { + "prob_id": "chat in a circle173", + "tags": [ + "Structural Argument|Greedy|Basics" + ] + }, + { + "prob_id": "or plus max100", + "tags": [ + "Subset Convolution|Bit Related DP|Dynamic Programming (DP)" + ] + } + ] + }, + { + "reference": { + "prob_id": "407c", + "tags": [ + "Binomial Coefficients|Fundamentals|Combinatorics" + ] + }, + "positive": { + "prob_id": "785d", + "tags": [ + "Binomial Coefficients|Fundamentals|Combinatorics" + ] + }, + "negative": [ + { + "prob_id": "914e", + "tags": [ + "Centroid Decomposition|Centroid Decomposition|Data Structures (DS)" + ] + }, + { + "prob_id": "241e", + "tags": [ + "Single Source with Negative Weights|Shortest Paths|Intermediate" + ] + }, + { + "prob_id": "616c", + "tags": [ + "Grid Graphs|Graphs|Basics" + ] + } + ] + }, + { + "reference": { + "prob_id": "535b", + "tags": [ + "All-Subsets / Binary Choice|Complete Search|Basics" + ] + }, + "positive": { + "prob_id": "202a", + "tags": [ + "All-Subsets / Binary Choice|Complete Search|Basics" + ] + }, + "negative": [ + { + "prob_id": "528c", + "tags": [ + "Euler Tour||" + ] + }, + { + "prob_id": "545d", + "tags": [ + "Exchange Argument|Greedy|Basics" + ] + }, + { + "prob_id": "946d", + "tags": [ + "Precalculation, Double DP|DP Common Tricks|Intermediate" + ] + } + ] + }, + { + "reference": { + "prob_id": "shopping street080", + "tags": [ + "All-Subsets / Binary Choice|Complete Search|Basics" + ] + }, + "positive": { + "prob_id": "114b", + "tags": [ + "All-Subsets / Binary Choice|Complete Search|Basics" + ] + }, + "negative": [ + { + "prob_id": "walk on multiplication table144", + "tags": [ + "Prime and Composite, Primality Testing, Prime Sieves, Finding Divisors|Number Theory|Preliminaries" + ] + }, + { + "prob_id": "1004e", + "tags": [ + "Tree Diameter|Graph Traversal|Graph Theory" + ] + }, + { + "prob_id": "1383f", + "tags": [ + "Maximum Flow / Minimum Cut|Flow and Min Cut|Graph Theory" + ] + } + ] + }, + { + "reference": { + "prob_id": "431b", + "tags": [ + "All-Permutations|Complete Search|Basics" + ] + }, + "positive": { + "prob_id": "124b", + "tags": [ + "All-Permutations|Complete Search|Basics" + ] + }, + "negative": [ + { + "prob_id": "82d", + "tags": [ + "DP with Solution Construction||Intermediate" + ] + }, + { + "prob_id": "321c", + "tags": [ + "Centroid Decomposition|Centroid Decomposition|Data Structures (DS)" + ] + }, + { + "prob_id": "862b", + "tags": [ + "Bipartite Graph||", + "Graph Traversal||" + ] + } + ] + }, + { + "reference": { + "prob_id": "124b", + "tags": [ + "All-Permutations|Complete Search|Basics" + ] + }, + "positive": { + "prob_id": "five dishes123", + "tags": [ + "All-Permutations|Complete Search|Basics" + ] + }, + "negative": [ + { + "prob_id": "547e", + "tags": [ + "String Suffix||" + ] + }, + { + "prob_id": "1476a", + "tags": [ + "Basic Constructive Algorithms|Greedy and Constructive|Basics" + ] + }, + { + "prob_id": "1244e", + "tags": [ + "Basic Greedy Algorithms|Greedy and Constructive|Basics", + "Binary Search|Leveraging Monotonicity|Basics", + "Prefix Sum|Range Queries Without Updates|Basics" + ] + } + ] + }, + { + "reference": { + "prob_id": "katana thrower085", + "tags": [ + "Exchange Argument|Greedy|Basics" + ] + }, + "positive": { + "prob_id": "545d", + "tags": [ + "Exchange Argument|Greedy|Basics" + ] + }, + "negative": [ + { + "prob_id": "five dishes123", + "tags": [ + "All-Permutations|Complete Search|Basics" + ] + }, + { + "prob_id": "roaming156", + "tags": [ + "Permutations with Repetition, Stars and Bars|Counting|Intermediate" + ] + }, + { + "prob_id": "blue and red balls132", + "tags": [ + "Permutations with Repetition, Stars and Bars|Counting|Intermediate" + ] + } + ] + }, + { + "reference": { + "prob_id": "545d", + "tags": [ + "Exchange Argument|Greedy|Basics" + ] + }, + "positive": { + "prob_id": "atcodeer and rock-paper046", + "tags": [ + "Exchange Argument|Greedy|Basics" + ] + }, + "negative": [ + { + "prob_id": "828f", + "tags": [ + "Lowest Common Ancestor|Lowest common ancestor|Graphs" + ] + }, + { + "prob_id": "346d", + "tags": [ + "Single Source with Negative Weights|Shortest Paths|Intermediate" + ] + }, + { + "prob_id": "516d", + "tags": [ + "Tree Centers and Diameter||Advanced Data Structures" + ] + } + ] + }, + { + "reference": { + "prob_id": "summer vacation137", + "tags": [ + "Exchange Argument|Greedy|Basics" + ] + }, + "positive": { + "prob_id": "922d", + "tags": [ + "Exchange Argument|Greedy|Basics" + ] + }, + "negative": [ + { + "prob_id": "i hate shortest path problem177", + "tags": [ + "Amortized Analysis||Intermediate" + ] + }, + { + "prob_id": "tak and hotels060", + "tags": [ + "Binary Lifting||Advanced Data Structures" + ] + }, + { + "prob_id": "modularness156", + "tags": [ + "Divisibility, Modular Arithmetic|Number Theory|Basics" + ] + } + ] + }, + { + "reference": { + "prob_id": "979b", + "tags": [ + "Structural Argument|Greedy|Basics" + ] + }, + "positive": { + "prob_id": "strange bank099", + "tags": [ + "Structural Argument|Greedy|Basics" + ] + }, + "negative": [ + { + "prob_id": "588e", + "tags": [ + "Binary Lifting||Advanced Data Structures", + "Lowest Common Ancestor|Lowest common ancestor|Graphs" + ] + }, + { + "prob_id": "506d", + "tags": [ + "Splitting Objects into Light and Heavy / Case Processing|Square Root Decomposition|Data Structures (DS)" + ] + }, + { + "prob_id": "9d", + "tags": [ + "Counting with DP||Intermediate" + ] + } + ] + }, + { + "reference": { + "prob_id": "grand garden116", + "tags": [ + "Structural Argument|Greedy|Basics" + ] + }, + "positive": { + "prob_id": "979b", + "tags": [ + "Structural Argument|Greedy|Basics" + ] + }, + "negative": [ + { + "prob_id": "gcd on blackboard125", + "tags": [ + "Prefix Sum|Range Queries Without Updates|Basics" + ] + }, + { + "prob_id": "atcoder express 2106", + "tags": [ + "Prefix Sums and Difference Arrays|Range Queries|Basics" + ] + }, + { + "prob_id": "lotus leaves074", + "tags": [ + "Network Flows||Advanced Algorithms" + ] + } + ] + }, + { + "reference": { + "prob_id": "contrast178", + "tags": [ + "Structural Argument|Greedy|Basics" + ] + }, + "positive": { + "prob_id": "1007a", + "tags": [ + "Structural Argument|Greedy|Basics" + ] + }, + "negative": [ + { + "prob_id": "unfair nim172", + "tags": [ + "Nim, Sprague-Grundy Theorem|Combinatorial Games|Advanced Mathematics" + ] + }, + { + "prob_id": "roadwork128", + "tags": [ + "Sweep Line||" + ] + }, + { + "prob_id": "distributing integers160", + "tags": [ + "Tree Re-rooting|DP Optimization|Advanced Algorithms" + ] + } + ] + }, + { + "reference": { + "prob_id": "429a", + "tags": [ + "Trees and Tree Traversals|Graphs|Basics" + ] + }, + "positive": { + "prob_id": "814d", + "tags": [ + "Trees and Tree Traversals|Graphs|Basics" + ] + }, + "negative": [ + { + "prob_id": "377e", + "tags": [ + "Convex Hull Trick|DP Optimization|Advanced Algorithms" + ] + }, + { + "prob_id": "338d", + "tags": [ + "Chinese Remainder Theorem|Modular arithmetic|Algebra" + ] + }, + { + "prob_id": "724c", + "tags": [ + "Reflections on a Rectangle|Miscellaneous|Geometry" + ] + } + ] + }, + { + "reference": { + "prob_id": "playing tag on tree148", + "tags": [ + "Trees and Tree Traversals|Graphs|Basics" + ] + }, + "positive": { + "prob_id": "814d", + "tags": [ + "Trees and Tree Traversals|Graphs|Basics" + ] + }, + "negative": [ + { + "prob_id": "next prime149", + "tags": [ + "Prime and Composite, Primality Testing, Prime Sieves, Finding Divisors|Number Theory|Preliminaries" + ] + }, + { + "prob_id": "active infants163", + "tags": [ + "Eliminate Dependent Parameters|DP Optimization|Advanced Algorithms", + "Presorting|DP Common Tricks|Intermediate" + ] + }, + { + "prob_id": "enclose all151", + "tags": [ + "Intersections and Containment|Geometry|Intermediate" + ] + } + ] + }, + { + "reference": { + "prob_id": "540c", + "tags": [ + "Grid Graphs|Graphs|Basics" + ] + }, + "positive": { + "prob_id": "grid repainting088", + "tags": [ + "Grid Graphs|Graphs|Basics" + ] + }, + "negative": [ + { + "prob_id": "241e", + "tags": [ + "Single Source with Negative Weights|Shortest Paths|Intermediate" + ] + }, + { + "prob_id": "232d", + "tags": [ + "Suffix Array|Suffix Structures|Strings" + ] + }, + { + "prob_id": "66e", + "tags": [ + "Global Deltaing||Advanced Data Structures" + ] + } + ] + }, + { + "reference": { + "prob_id": "877d", + "tags": [ + "Grid Graphs|Graphs|Basics" + ] + }, + "positive": { + "prob_id": "329b", + "tags": [ + "Grid Graphs|Graphs|Basics" + ] + }, + "negative": [ + { + "prob_id": "232d", + "tags": [ + "Suffix Array|Suffix Structures|Strings" + ] + }, + { + "prob_id": "839e", + "tags": [ + "Meet in the Middle|Techniques|Miscellaneous" + ] + }, + { + "prob_id": "713c", + "tags": [ + "Slope Trick|More Tricks|Dynamic Programming (DP)" + ] + } + ] + }, + { + "reference": { + "prob_id": "585b", + "tags": [ + "Grid Graphs|Graphs|Basics" + ] + }, + "positive": { + "prob_id": "grid repainting088", + "tags": [ + "Grid Graphs|Graphs|Basics" + ] + }, + "negative": [ + { + "prob_id": "1468d", + "tags": [ + "Binary Search|Leveraging Monotonicity|Basics" + ] + }, + { + "prob_id": "305b", + "tags": [ + "Continued Fractions|Miscellaneous|Math" + ] + }, + { + "prob_id": "1392i", + "tags": [ + "Euler's Formula||", + "Fast Fourier Transform (FFT)|Convolution|Math" + ] + } + ] + }, + { + "reference": { + "prob_id": "31d", + "tags": [ + "Grid Graphs|Graphs|Basics" + ] + }, + "positive": { + "prob_id": "329b", + "tags": [ + "Grid Graphs|Graphs|Basics" + ] + }, + "negative": [ + { + "prob_id": "612c", + "tags": [ + "Balanced Brackets|Miscellaneous|Strings" + ] + }, + { + "prob_id": "1081d", + "tags": [ + "Centroid Decomposition|Centroid Decomposition|Data Structures (DS)", + "Minimum Spanning Tree||" + ] + }, + { + "prob_id": "19c", + "tags": [ + "String Hashing|Hashing|Strings" + ] + } + ] + }, + { + "reference": { + "prob_id": "maze master151", + "tags": [ + "Grid Graphs|Graphs|Basics" + ] + }, + "positive": { + "prob_id": "329b", + "tags": [ + "Grid Graphs|Graphs|Basics" + ] + }, + "negative": [ + { + "prob_id": "tree and constraints152", + "tags": [ + "Principle of Inclusion-Exclusion|Counting|Intermediate" + ] + }, + { + "prob_id": "islands war103", + "tags": [ + "Greedy Stays Ahead|Greedy|Basics" + ] + }, + { + "prob_id": "one-stroke path054", + "tags": [ + "All-Permutations|Complete Search|Basics" + ] + } + ] + }, + { + "reference": { + "prob_id": "741c", + "tags": [ + "Bipartite Graph||" + ] + }, + "positive": { + "prob_id": "must be rectangular!131", + "tags": [ + "Bipartite Graph||" + ] + }, + "negative": [ + { + "prob_id": "735e", + "tags": [ + "DP on Trees and DAGs|Dp on Different Structures|Dynamic Programming (DP)" + ] + }, + { + "prob_id": "739e", + "tags": [ + "Aliens Trick / WQS Binary Search|DP Optimizations|Dynamic Programming (DP)" + ] + }, + { + "prob_id": "57c", + "tags": [ + "Permutations with Repetition, Stars and Bars|Counting|Intermediate" + ] + } + ] + }, + { + "reference": { + "prob_id": "598c", + "tags": [ + "Vectors and Applications|Geometry|Basics" + ] + }, + "positive": { + "prob_id": "75e", + "tags": [ + "Vectors and Applications|Geometry|Basics" + ] + }, + "negative": [ + { + "prob_id": "847b", + "tags": [ + "Binary Search|Leveraging Monotonicity|Basics" + ] + }, + { + "prob_id": "577b", + "tags": [ + "Counting with DP||Intermediate" + ] + }, + { + "prob_id": "576a", + "tags": [ + "Prime Factorization|Number Theory|Basics" + ] + } + ] + }, + { + "reference": { + "prob_id": "850a", + "tags": [ + "Vectors and Applications|Geometry|Basics" + ] + }, + "positive": { + "prob_id": ": (colon)168", + "tags": [ + "Vectors and Applications|Geometry|Basics" + ] + }, + "negative": [ + { + "prob_id": "601e", + "tags": [ + "Dynamic Connectivity Problem (Offline)|Dynamic Connectivity|Data Structures (DS)", + "Offline Deletion||" + ] + }, + { + "prob_id": "505d", + "tags": [ + "Strongly Connected Components (SCC)|Graph Connectivity|Graph Theory" + ] + }, + { + "prob_id": "721c", + "tags": [ + "Directed Acyclic Graphs|Shortest Paths|Intermediate", + "Rotate the Recurrence|DP Optimization|Advanced Algorithms" + ] + } + ] + }, + { + "reference": { + "prob_id": "32e", + "tags": [ + "Vectors and Applications|Geometry|Basics" + ] + }, + "positive": { + "prob_id": ": (colon)168", + "tags": [ + "Vectors and Applications|Geometry|Basics" + ] + }, + "negative": [ + { + "prob_id": "721c", + "tags": [ + "Directed Acyclic Graphs|Shortest Paths|Intermediate", + "Rotate the Recurrence|DP Optimization|Advanced Algorithms" + ] + }, + { + "prob_id": "463d", + "tags": [ + "Longest Common Subsequence (LCS)|Longest Common Subsequence (LCS)|Strings" + ] + }, + { + "prob_id": "919d", + "tags": [ + "Topological Sorting|Graph Traversal|Graph Theory" + ] + } + ] + }, + { + "reference": { + "prob_id": "engines139", + "tags": [ + "Vectors and Applications|Geometry|Basics" + ] + }, + "positive": { + "prob_id": "32e", + "tags": [ + "Vectors and Applications|Geometry|Basics" + ] + }, + "negative": [ + { + "prob_id": "megalomania131", + "tags": [ + "Exchange Argument|Greedy|Basics" + ] + }, + { + "prob_id": "even relation126", + "tags": [ + "Trees and Tree Traversals|Graphs|Basics" + ] + }, + { + "prob_id": "11066", + "tags": [ + "Binomial Coefficients|Fundamentals|Combinatorics" + ] + } + ] + }, + { + "reference": { + "prob_id": "30c", + "tags": [ + "Directed Acyclic Graphs|Shortest Paths|Intermediate" + ] + }, + "positive": { + "prob_id": "league139", + "tags": [ + "Directed Acyclic Graphs|Shortest Paths|Intermediate" + ] + }, + "negative": [ + { + "prob_id": "241c", + "tags": [ + "Reflections on a Rectangle|Miscellaneous|Geometry" + ] + }, + { + "prob_id": "76a", + "tags": [ + "Minimum Spanning Tree||" + ] + }, + { + "prob_id": "679e", + "tags": [ + "Segment Tree Beats|Segment Tree|Data Structures (DS)" + ] + } + ] + }, + { + "reference": { + "prob_id": "25c", + "tags": [ + "All-Pairs|Shortest Paths|Intermediate" + ] + }, + "positive": { + "prob_id": "wall079", + "tags": [ + "All-Pairs|Shortest Paths|Intermediate" + ] + }, + "negative": [ + { + "prob_id": "727d", + "tags": [ + "Bipartite Matching|Network Flows|Advanced Algorithms" + ] + }, + { + "prob_id": "191a", + "tags": [ + "Flatten the Memo|DP Optimization|Advanced Algorithms" + ] + }, + { + "prob_id": "559d", + "tags": [ + "Pick's Theorem|Geometry|Advanced Mathematics" + ] + } + ] + }, + { + "reference": { + "prob_id": "travel by car143", + "tags": [ + "All-Pairs|Shortest Paths|Intermediate" + ] + }, + "positive": { + "prob_id": "1204c", + "tags": [ + "All-Pairs|Shortest Paths|Intermediate" + ] + }, + "negative": [ + { + "prob_id": "an ordinary game048", + "tags": [ + "Winning and Losing Positions, Strategy Stealing|Combinatorial Games|Advanced Algorithms" + ] + }, + { + "prob_id": "five dishes123", + "tags": [ + "All-Permutations|Complete Search|Basics" + ] + }, + { + "prob_id": "summer vacation137", + "tags": [ + "Exchange Argument|Greedy|Basics" + ] + } + ] + }, + { + "reference": { + "prob_id": "31e", + "tags": [ + "DP with Solution Construction||Intermediate" + ] + }, + "positive": { + "prob_id": "match matching118", + "tags": [ + "DP with Solution Construction||Intermediate" + ] + }, + "negative": [ + { + "prob_id": "778a", + "tags": [ + "Binary Search|Leveraging Monotonicity|Basics" + ] + }, + { + "prob_id": "343e", + "tags": [ + "Gomory-Hu Tree|Flow and Min Cut|Graph Theory" + ] + }, + { + "prob_id": "727e", + "tags": [ + "String Hashing|Hashing|Strings" + ] + } + ] + }, + { + "reference": { + "prob_id": "990e", + "tags": [ + "Harmonic Lemma||Intermediate" + ] + }, + "positive": { + "prob_id": "grouping067", + "tags": [ + "Harmonic Lemma||Intermediate" + ] + }, + "negative": [ + { + "prob_id": "877d", + "tags": [ + "Grid Graphs|Graphs|Basics" + ] + }, + { + "prob_id": "1312f", + "tags": [ + "Nim, Sprague-Grundy Theorem|Combinatorial Games|Advanced Mathematics" + ] + }, + { + "prob_id": "1385e", + "tags": [ + "Topological Sorting|Graph Traversal|Graph Theory" + ] + } + ] + }, + { + "reference": { + "prob_id": "580d", + "tags": [ + "DP on Subsets||Intermediate" + ] + }, + "positive": { + "prob_id": "mole and abandoned mine078", + "tags": [ + "DP on Subsets||Intermediate" + ] + }, + "negative": [ + { + "prob_id": "577e", + "tags": [ + "Mo's Algorithm|Square Root Decomposition|Data Structures (DS)" + ] + }, + { + "prob_id": "1179c", + "tags": [ + "Segment Tree|Trees|Data Structures" + ] + }, + { + "prob_id": "727e", + "tags": [ + "String Hashing|Hashing|Strings" + ] + } + ] + }, + { + "reference": { + "prob_id": "678e", + "tags": [ + "DP on Subsets||Intermediate" + ] + }, + "positive": { + "prob_id": "11d", + "tags": [ + "DP on Subsets||Intermediate" + ] + }, + "negative": [ + { + "prob_id": "873d", + "tags": [ + "Divide and Conquer DP|DP optimizations|Dynamic Programming" + ] + }, + { + "prob_id": "167e", + "tags": [ + "Determinant of a Matrix|Linear Algebra|Math" + ] + }, + { + "prob_id": "439d", + "tags": [ + "Ternary Search|Search Techniques|Miscellaneous" + ] + } + ] + }, + { + "reference": { + "prob_id": "543c", + "tags": [ + "DP on Subsets||Intermediate" + ] + }, + "positive": { + "prob_id": "11d", + "tags": [ + "DP on Subsets||Intermediate" + ] + }, + "negative": [ + { + "prob_id": "1476a", + "tags": [ + "Basic Constructive Algorithms|Greedy and Constructive|Basics" + ] + }, + { + "prob_id": "1288f", + "tags": [ + "L-R Flow|Flow and Min Cut|Graph Theory", + "Min Cost Max Flow (MCMF)|Flow and Min Cut|Graph Theory" + ] + }, + { + "prob_id": "702f", + "tags": [ + "Splay Trees and Treaps|Augmenting Data Structures|Advanced Data Structures" + ] + } + ] + }, + { + "reference": { + "prob_id": "52c", + "tags": [ + "Segment Tree with Lazy Propagation|Segment Tree|Data Structures (DS)" + ] + }, + "positive": { + "prob_id": "558e", + "tags": [ + "Segment Tree with Lazy Propagation|Segment Tree|Data Structures (DS)" + ] + }, + "negative": [ + { + "prob_id": "235c", + "tags": [ + "String Suffix||", + "Suffix Automaton|Suffix Structures|Strings" + ] + }, + { + "prob_id": "862b", + "tags": [ + "Bipartite Graph||", + "Graph Traversal||" + ] + }, + { + "prob_id": "852f", + "tags": [ + "Multiplicative Order and Carmichael's Lambda Function|Modular Arithmetic|Number Theory" + ] + } + ] + }, + { + "reference": { + "prob_id": "558e", + "tags": [ + "Segment Tree with Lazy Propagation|Segment Tree|Data Structures (DS)" + ] + }, + "positive": { + "prob_id": "1263e", + "tags": [ + "Segment Tree with Lazy Propagation|Segment Tree|Data Structures (DS)" + ] + }, + "negative": [ + { + "prob_id": "906d", + "tags": [ + "Divisibility||" + ] + }, + { + "prob_id": "1073g", + "tags": [ + "String Suffix||", + "Suffix Automaton|Suffix Structures|Strings" + ] + }, + { + "prob_id": "1037h", + "tags": [ + "String Suffix||", + "Suffix Array|Suffix Structures|Strings" + ] + } + ] + }, + { + "reference": { + "prob_id": "446c", + "tags": [ + "Segment Tree with Lazy Propagation|Segment Tree|Data Structures (DS)" + ] + }, + "positive": { + "prob_id": "1263e", + "tags": [ + "Segment Tree with Lazy Propagation|Segment Tree|Data Structures (DS)" + ] + }, + "negative": [ + { + "prob_id": "277a", + "tags": [ + "Bipartite Graph||" + ] + }, + { + "prob_id": "911g", + "tags": [ + "Pragmas|Miscellaneous|Miscellaneous", + "Splay Trees and Treaps|Augmenting Data Structures|Advanced Data Structures" + ] + }, + { + "prob_id": "612c", + "tags": [ + "Balanced Brackets|Miscellaneous|Strings" + ] + } + ] + }, + { + "reference": { + "prob_id": "817f", + "tags": [ + "Segment Tree with Lazy Propagation|Segment Tree|Data Structures (DS)" + ] + }, + "positive": { + "prob_id": "1263e", + "tags": [ + "Segment Tree with Lazy Propagation|Segment Tree|Data Structures (DS)" + ] + }, + "negative": [ + { + "prob_id": "1398g", + "tags": [ + "Fast Fourier Transform (FFT)|Convolution|Math" + ] + }, + { + "prob_id": "911g", + "tags": [ + "Pragmas|Miscellaneous|Miscellaneous", + "Splay Trees and Treaps|Augmenting Data Structures|Advanced Data Structures" + ] + }, + { + "prob_id": "1603f", + "tags": [ + "q Binomial|Linear Algebra|Math" + ] + } + ] + }, + { + "reference": { + "prob_id": "580e", + "tags": [ + "Segment Tree with Lazy Propagation|Segment Tree|Data Structures (DS)" + ] + }, + "positive": { + "prob_id": "1263e", + "tags": [ + "Segment Tree with Lazy Propagation|Segment Tree|Data Structures (DS)" + ] + }, + "negative": [ + { + "prob_id": "576a", + "tags": [ + "Prime Factorization|Number Theory|Basics" + ] + }, + { + "prob_id": "526g", + "tags": [ + "Long Path Decomposition|Long Path Decomposition|Data Structures (DS)" + ] + }, + { + "prob_id": "1381a2", + "tags": [ + "Basic Constructive Algorithms|Greedy and Constructive|Basics" + ] + } + ] + }, + { + "reference": { + "prob_id": "145e", + "tags": [ + "Segment Tree with Lazy Propagation|Segment Tree|Data Structures (DS)" + ] + }, + "positive": { + "prob_id": "1555e", + "tags": [ + "Segment Tree with Lazy Propagation|Segment Tree|Data Structures (DS)" + ] + }, + "negative": [ + { + "prob_id": "121a", + "tags": [ + "All-Subsets / Binary Choice|Complete Search|Basics" + ] + }, + { + "prob_id": "1218i", + "tags": [ + "2-SAT|DFS Lowlink|Advanced Algorithms" + ] + }, + { + "prob_id": "954i", + "tags": [ + "Fast Fourier Transform (FFT)|Convolution|Math", + "String Matching using Bitsets|String Matching|Strings" + ] + } + ] + }, + { + "reference": { + "prob_id": "1100b", + "tags": [ + "Amortized Analysis||Intermediate" + ] + }, + "positive": { + "prob_id": "47e", + "tags": [ + "Amortized Analysis||Intermediate" + ] + }, + "negative": [ + { + "prob_id": "920g", + "tags": [ + "Mobius Function and Mobius Inversion|Multiplicative Functions|Number Theory" + ] + }, + { + "prob_id": "1392i", + "tags": [ + "Euler's Formula||", + "Fast Fourier Transform (FFT)|Convolution|Math" + ] + }, + { + "prob_id": "1153c", + "tags": [ + "Balanced Brackets|Miscellaneous|Strings" + ] + } + ] + }, + { + "reference": { + "prob_id": "609f", + "tags": [ + "Amortized Analysis||Intermediate" + ] + }, + "positive": { + "prob_id": "i hate shortest path problem177", + "tags": [ + "Amortized Analysis||Intermediate" + ] + }, + "negative": [ + { + "prob_id": "295b", + "tags": [ + "All-Pairs|Shortest Paths|Intermediate" + ] + }, + { + "prob_id": "61e", + "tags": [ + "Order Statistics||Intermediate" + ] + }, + { + "prob_id": "643c", + "tags": [ + "Convex Hull Trick|DP Optimization|Advanced Algorithms", + "Divide and Conquer DP|DP optimizations|Dynamic Programming" + ] + } + ] + }, + { + "reference": { + "prob_id": "1263c", + "tags": [ + "Square-Root Discrimination||Intermediate" + ] + }, + "positive": { + "prob_id": "small products132", + "tags": [ + "Square-Root Discrimination||Intermediate" + ] + }, + "negative": [ + { + "prob_id": "1365d", + "tags": [ + "Flood Fill||" + ] + }, + { + "prob_id": "827e", + "tags": [ + "Fast Fourier Transform (FFT)|Convolution|Math", + "String Matching with FFT ft Wildcard Matching|String Matching|Strings" + ] + }, + { + "prob_id": "114b", + "tags": [ + "All-Subsets / Binary Choice|Complete Search|Basics" + ] + } + ] + }, + { + "reference": { + "prob_id": "808b", + "tags": [ + "Sliding Window|Data Structure Techniques|Intermediate" + ] + }, + "positive": { + "prob_id": "679c", + "tags": [ + "Sliding Window|Data Structure Techniques|Intermediate" + ] + }, + "negative": [ + { + "prob_id": "603e", + "tags": [ + "Link Cut Tree (LCT)|Link Cut Tree (LCT)|Data Structures (DS)" + ] + }, + { + "prob_id": "95e", + "tags": [ + "Knapsack DP||" + ] + }, + { + "prob_id": "932g", + "tags": [ + "Palindromic Tree|Palindromes|Strings", + "String Suffix||" + ] + } + ] + }, + { + "reference": { + "prob_id": "898d", + "tags": [ + "Sliding Window|Data Structure Techniques|Intermediate" + ] + }, + "positive": { + "prob_id": "dice in line154", + "tags": [ + "Sliding Window|Data Structure Techniques|Intermediate" + ] + }, + "negative": [ + { + "prob_id": "932g", + "tags": [ + "Palindromic Tree|Palindromes|Strings", + "String Suffix||" + ] + }, + { + "prob_id": "300c", + "tags": [ + "Binomial Coefficients|Fundamentals|Combinatorics" + ] + }, + { + "prob_id": "895e", + "tags": [ + "Segment Tree with Lazy Propagation|Segment Tree|Data Structures (DS)" + ] + } + ] + }, + { + "reference": { + "prob_id": "1009d", + "tags": [ + "Relatively Prime, Euler Totient Function|Number Theory|Intermediate" + ] + }, + "positive": { + "prob_id": "361b", + "tags": [ + "Relatively Prime, Euler Totient Function|Number Theory|Intermediate" + ] + }, + "negative": [ + { + "prob_id": "1143f", + "tags": [ + "Convex Hull Trick|DP Optimization|Advanced Algorithms" + ] + }, + { + "prob_id": "975d", + "tags": [ + "Vectors and Applications|Geometry|Basics" + ] + }, + { + "prob_id": "1383f", + "tags": [ + "Maximum Flow / Minimum Cut|Flow and Min Cut|Graph Theory" + ] + } + ] + }, + { + "reference": { + "prob_id": "617b", + "tags": [ + "Permutations with Repetition, Stars and Bars|Counting|Intermediate" + ] + }, + "positive": { + "prob_id": "factorization110", + "tags": [ + "Permutations with Repetition, Stars and Bars|Counting|Intermediate" + ] + }, + "negative": [ + { + "prob_id": "318a", + "tags": [ + "Sums and Asymptotics||Preliminaries" + ] + }, + { + "prob_id": "616c", + "tags": [ + "Grid Graphs|Graphs|Basics" + ] + }, + { + "prob_id": "18c", + "tags": [ + "Prefix Sums and Difference Arrays|Range Queries|Basics" + ] + } + ] + }, + { + "reference": { + "prob_id": "roaming156", + "tags": [ + "Permutations with Repetition, Stars and Bars|Counting|Intermediate" + ] + }, + "positive": { + "prob_id": "617b", + "tags": [ + "Permutations with Repetition, Stars and Bars|Counting|Intermediate" + ] + }, + "negative": [ + { + "prob_id": "exhausted?076", + "tags": [ + "Bipartite Matching|Network Flows|Advanced Algorithms" + ] + }, + { + "prob_id": "sandglass082", + "tags": [ + "Global Deltaing||Advanced Data Structures" + ] + }, + { + "prob_id": "travel by car143", + "tags": [ + "All-Pairs|Shortest Paths|Intermediate" + ] + } + ] + }, + { + "reference": { + "prob_id": "ubiquity178", + "tags": [ + "Principle of Inclusion-Exclusion|Counting|Intermediate" + ] + }, + "positive": { + "prob_id": "839d", + "tags": [ + "Principle of Inclusion-Exclusion|Counting|Intermediate" + ] + }, + "negative": [ + { + "prob_id": "joisino's travel073", + "tags": [ + "All-Pairs|Shortest Paths|Intermediate", + "All-Permutations|Complete Search|Basics" + ] + }, + { + "prob_id": "enclose all151", + "tags": [ + "Intersections and Containment|Geometry|Intermediate" + ] + }, + { + "prob_id": "mul085", + "tags": [ + "Network Flows||Advanced Algorithms" + ] + } + ] + }, + { + "reference": { + "prob_id": "414b", + "tags": [ + "Counting with DP||Intermediate" + ] + }, + "positive": { + "prob_id": "we like agc122", + "tags": [ + "Counting with DP||Intermediate" + ] + }, + "negative": [ + { + "prob_id": "431b", + "tags": [ + "All-Permutations|Complete Search|Basics" + ] + }, + { + "prob_id": "362e", + "tags": [ + "Bipartite Matching|Network Flows|Advanced Algorithms" + ] + }, + { + "prob_id": "61d", + "tags": [ + "Trees and Tree Traversals|Graphs|Basics" + ] + } + ] + }, + { + "reference": { + "prob_id": "118d", + "tags": [ + "Counting with DP||Intermediate" + ] + }, + "positive": { + "prob_id": "9d", + "tags": [ + "Counting with DP||Intermediate" + ] + }, + "negative": [ + { + "prob_id": "362e", + "tags": [ + "Bipartite Matching|Network Flows|Advanced Algorithms" + ] + }, + { + "prob_id": "633a", + "tags": [ + "Linear Diophantine Equations|Fundamentals|Algebra" + ] + }, + { + "prob_id": "26a", + "tags": [ + "Sieve of Eratosthenes|Prime numbers|Algebra" + ] + } + ] + }, + { + "reference": { + "prob_id": "149d", + "tags": [ + "Counting with DP||Intermediate" + ] + }, + "positive": { + "prob_id": "9d", + "tags": [ + "Counting with DP||Intermediate" + ] + }, + "negative": [ + { + "prob_id": "776b", + "tags": [ + "Prime and Composite, Primality Testing, Prime Sieves, Finding Divisors|Number Theory|Preliminaries" + ] + }, + { + "prob_id": "21d", + "tags": [ + "Floyd Warshall|Shortest Paths|Graph Theory" + ] + }, + { + "prob_id": "55e", + "tags": [ + "Point in Convex Polygon in log(N)(2D)|Geometry 2D|Geometry" + ] + } + ] + }, + { + "reference": { + "prob_id": "518d", + "tags": [ + "Markov Chains|Probability and Expectation|Intermediate" + ] + }, + "positive": { + "prob_id": "1245e", + "tags": [ + "Markov Chains|Probability and Expectation|Intermediate" + ] + }, + "negative": [ + { + "prob_id": "530g", + "tags": [ + "Edit Distance|Classic DP|Dynamic Programming (DP)" + ] + }, + { + "prob_id": "825e", + "tags": [ + "Topological Sorting|Graph Traversal|Graph Theory" + ] + }, + { + "prob_id": "114b", + "tags": [ + "All-Subsets / Binary Choice|Complete Search|Basics" + ] + } + ] + }, + { + "reference": { + "prob_id": "906b", + "tags": [ + "Probabilities and Expected Values|Probabilities and Expected Values|Math" + ] + }, + "positive": { + "prob_id": "623d", + "tags": [ + "Probabilities and Expected Values|Probabilities and Expected Values|Math" + ] + }, + "negative": [ + { + "prob_id": "1427g", + "tags": [ + "Min Cost Max Flow (MCMF)|Flow and Min Cut|Graph Theory" + ] + }, + { + "prob_id": "961e", + "tags": [ + "BIT / Fenwick Tree|Binary Indexed Tree (BIT)|Data Structures (DS)" + ] + }, + { + "prob_id": "1365d", + "tags": [ + "Flood Fill||" + ] + } + ] + }, + { + "reference": { + "prob_id": "1096f", + "tags": [ + "Probabilities and Expected Values|Probabilities and Expected Values|Math" + ] + }, + "positive": { + "prob_id": "113d", + "tags": [ + "Probabilities and Expected Values|Probabilities and Expected Values|Math" + ] + }, + "negative": [ + { + "prob_id": "893f", + "tags": [ + "Tree Flattening||Advanced Data Structures" + ] + }, + { + "prob_id": "724c", + "tags": [ + "Reflections on a Rectangle|Miscellaneous|Geometry" + ] + }, + { + "prob_id": "600f", + "tags": [ + "Bipartite Matching|Network Flows|Advanced Algorithms" + ] + } + ] + }, + { + "reference": { + "prob_id": "26d", + "tags": [ + "Probabilities and Expected Values|Probabilities and Expected Values|Math" + ] + }, + "positive": { + "prob_id": "843b", + "tags": [ + "Probabilities and Expected Values|Probabilities and Expected Values|Math" + ] + }, + "negative": [ + { + "prob_id": "1011e", + "tags": [ + "Bezout's Identity|Euclidean Algorithm|Number Theory" + ] + }, + { + "prob_id": "873f", + "tags": [ + "Suffix Array|Suffix Structures|Strings" + ] + }, + { + "prob_id": "1105e", + "tags": [ + "Meet in the Middle|Techniques|Miscellaneous" + ] + } + ] + }, + { + "reference": { + "prob_id": "sorted and sorted097", + "tags": [ + "Order Statistics||Intermediate" + ] + }, + "positive": { + "prob_id": "61e", + "tags": [ + "Order Statistics||Intermediate" + ] + }, + "negative": [ + { + "prob_id": "path pass i163", + "tags": [ + "DP on Trees and DAGs|Dp on Different Structures|Dynamic Programming (DP)" + ] + }, + { + "prob_id": "modularness156", + "tags": [ + "Divisibility, Modular Arithmetic|Number Theory|Basics" + ] + }, + { + "prob_id": "payment155", + "tags": [ + "DP on Digits||Intermediate" + ] + } + ] + }, + { + "reference": { + "prob_id": "817d", + "tags": [ + "All Nearest Smaller Values and All Subarray Maximum/Minimum|Miscellaneous|Miscellaneous" + ] + }, + "positive": { + "prob_id": "flip and rectangles081", + "tags": [ + "All Nearest Smaller Values and All Subarray Maximum/Minimum|Miscellaneous|Miscellaneous" + ] + }, + "negative": [ + { + "prob_id": "865d", + "tags": [ + "Slope Trick|More Tricks|Dynamic Programming (DP)" + ] + }, + { + "prob_id": "1132g", + "tags": [ + "Segment Tree with Lazy Propagation|Segment Tree|Data Structures (DS)" + ] + }, + { + "prob_id": "522d", + "tags": [ + "Persistent Segment Tree|Segment Tree|Data Structures (DS)" + ] + } + ] + }, + { + "reference": { + "prob_id": "602d", + "tags": [ + "All Nearest Smaller Values and All Subarray Maximum/Minimum|Miscellaneous|Miscellaneous" + ] + }, + "positive": { + "prob_id": "flip and rectangles081", + "tags": [ + "All Nearest Smaller Values and All Subarray Maximum/Minimum|Miscellaneous|Miscellaneous" + ] + }, + "negative": [ + { + "prob_id": "1073g", + "tags": [ + "String Suffix||", + "Suffix Automaton|Suffix Structures|Strings" + ] + }, + { + "prob_id": "919d", + "tags": [ + "Topological Sorting|Graph Traversal|Graph Theory" + ] + }, + { + "prob_id": "506d", + "tags": [ + "Splitting Objects into Light and Heavy / Case Processing|Square Root Decomposition|Data Structures (DS)" + ] + } + ] + }, + { + "reference": { + "prob_id": "DIVQUERY", + "tags": [ + "Offline Processing of Queries||Advanced Data Structures" + ] + }, + "positive": { + "prob_id": "707d", + "tags": [ + "Offline Processing of Queries||Advanced Data Structures" + ] + }, + "negative": [ + { + "prob_id": "280d", + "tags": [ + "Min Cost Max Flow (MCMF)|Flow and Min Cut|Graph Theory" + ] + }, + { + "prob_id": "10d", + "tags": [ + "Longest Increasing Subsequence (LIS)|Classic DP|Dynamic Programming (DP)" + ] + }, + { + "prob_id": "486a", + "tags": [ + "Sums and Asymptotics||Preliminaries" + ] + } + ] + }, + { + "reference": { + "prob_id": "351d", + "tags": [ + "Mo's Algorithm|Square Root Decomposition|Data Structures (DS)" + ] + }, + "positive": { + "prob_id": "577e", + "tags": [ + "Mo's Algorithm|Square Root Decomposition|Data Structures (DS)" + ] + }, + "negative": [ + { + "prob_id": "585f", + "tags": [ + "Aho-Corasick|String Structures|Advanced Data Structures" + ] + }, + { + "prob_id": "296c", + "tags": [ + "Prefix Sums and Difference Arrays|Range Queries|Basics" + ] + }, + { + "prob_id": "295a", + "tags": [ + "Prefix Sums and Difference Arrays|Range Queries|Basics", + "Prefix Sum|Range Queries Without Updates|Basics" + ] + } + ] + }, + { + "reference": { + "prob_id": "620f", + "tags": [ + "Mo's Algorithm|Square Root Decomposition|Data Structures (DS)" + ] + }, + "positive": { + "prob_id": "577e", + "tags": [ + "Mo's Algorithm|Square Root Decomposition|Data Structures (DS)" + ] + }, + "negative": [ + { + "prob_id": "51c", + "tags": [ + "Two Pointers|Leveraging Monotonicity|Basics" + ] + }, + { + "prob_id": "916e", + "tags": [ + "Binary Lifting||Advanced Data Structures" + ] + }, + { + "prob_id": "860d", + "tags": [ + "Spanning Tree||" + ] + } + ] + }, + { + "reference": { + "prob_id": "696e", + "tags": [ + "Heavy Light Decomposition (HLD) ft Subtrees and Path Query|Heavy Light Decomposition (HLD)|Data Structures (DS)" + ] + }, + "positive": { + "prob_id": "DGCD", + "tags": [ + "Heavy Light Decomposition (HLD) ft Subtrees and Path Query|Heavy Light Decomposition (HLD)|Data Structures (DS)" + ] + }, + "negative": [ + { + "prob_id": "1004e", + "tags": [ + "Tree Diameter|Graph Traversal|Graph Theory" + ] + }, + { + "prob_id": "587e", + "tags": [ + "Vector Space and XOR Basis|Linear Algebra|Math" + ] + }, + { + "prob_id": "756114", + "tags": [ + "Prime Factorization|Number Theory|Basics" + ] + } + ] + }, + { + "reference": { + "prob_id": "276e", + "tags": [ + "Tree Flattening||Advanced Data Structures" + ] + }, + "positive": { + "prob_id": "796c", + "tags": [ + "Tree Flattening||Advanced Data Structures" + ] + }, + "negative": [ + { + "prob_id": "263d", + "tags": [ + "Cycle Detection|Cycles|Graph Theory" + ] + }, + { + "prob_id": "341d", + "tags": [ + "2D BIT with Range Update and Range Query|Binary Indexed Tree (BIT)|Data Structures (DS)" + ] + }, + { + "prob_id": "87e", + "tags": [ + "Minkowski Sum|Convex Polygon|Geometry", + "Point in Convex Polygon in log(N)(2D)|Geometry 2D|Geometry" + ] + } + ] + }, + { + "reference": { + "prob_id": "631d", + "tags": [ + "Knuth Morris Pratt (KMP) and Prefix Automaton|String Matching|Strings" + ] + }, + "positive": { + "prob_id": "808g", + "tags": [ + "Knuth Morris Pratt (KMP) and Prefix Automaton|String Matching|Strings" + ] + }, + "negative": [ + { + "prob_id": "527d", + "tags": [ + "Maximum Clique|Cliques and Independent Sets|Graph Theory" + ] + }, + { + "prob_id": "305b", + "tags": [ + "Continued Fractions|Miscellaneous|Math" + ] + }, + { + "prob_id": "237e", + "tags": [ + "Min Cost Max Flow (MCMF)|Flow and Min Cut|Graph Theory" + ] + } + ] + }, + { + "reference": { + "prob_id": "809d", + "tags": [ + "Splay Trees and Treaps|Augmenting Data Structures|Advanced Data Structures" + ] + }, + "positive": { + "prob_id": "863d", + "tags": [ + "Splay Trees and Treaps|Augmenting Data Structures|Advanced Data Structures" + ] + }, + "negative": [ + { + "prob_id": "1100f", + "tags": [ + "XOR Basis||" + ] + }, + { + "prob_id": "33b", + "tags": [ + "Floyd Warshall|Shortest Paths|Graph Theory" + ] + }, + { + "prob_id": "438e", + "tags": [ + "Fast Fourier Transform (FFT)|Convolution|Math", + "Generating Functions|Polynomial|Math" + ] + } + ] + }, + { + "reference": { + "prob_id": "702f", + "tags": [ + "Splay Trees and Treaps|Augmenting Data Structures|Advanced Data Structures" + ] + }, + "positive": { + "prob_id": "455d", + "tags": [ + "Splay Trees and Treaps|Augmenting Data Structures|Advanced Data Structures" + ] + }, + "negative": [ + { + "prob_id": "869e", + "tags": [ + "Orthogonal Range Queries|Range Queries|Advanced Data Structures" + ] + }, + { + "prob_id": "722c", + "tags": [ + "Disjoint Set Union||" + ] + }, + { + "prob_id": "1240f", + "tags": [ + "Edge Coloring of Bipartite Graph|Graph Coloring|Graph Theory" + ] + } + ] + }, + { + "reference": { + "prob_id": "167d", + "tags": [ + "Splay Trees and Treaps|Augmenting Data Structures|Advanced Data Structures" + ] + }, + "positive": { + "prob_id": "863d", + "tags": [ + "Splay Trees and Treaps|Augmenting Data Structures|Advanced Data Structures" + ] + }, + "negative": [ + { + "prob_id": "852f", + "tags": [ + "Multiplicative Order and Carmichael's Lambda Function|Modular Arithmetic|Number Theory" + ] + }, + { + "prob_id": "1252l", + "tags": [ + "Network Flows||Advanced Algorithms" + ] + }, + { + "prob_id": "455c", + "tags": [ + "Tree Centers and Diameter||Advanced Data Structures" + ] + } + ] + }, + { + "reference": { + "prob_id": "420d", + "tags": [ + "Splay Trees and Treaps|Augmenting Data Structures|Advanced Data Structures" + ] + }, + "positive": { + "prob_id": "762e", + "tags": [ + "Splay Trees and Treaps|Augmenting Data Structures|Advanced Data Structures" + ] + }, + "negative": [ + { + "prob_id": "456e", + "tags": [ + "Tree Diameter|Graph Traversal|Graph Theory" + ] + }, + { + "prob_id": "383e", + "tags": [ + "Bit Manipulation|Bit Manipulation|Basics", + "Sum Over Subsets DP (SOS DP)|Bit Related DP|Dynamic Programming (DP)" + ] + }, + { + "prob_id": "777e", + "tags": [ + "O(n lg n) Longest Monotonic Subsequence|DP Optimization|Advanced Algorithms" + ] + } + ] + }, + { + "reference": { + "prob_id": "546e", + "tags": [ + "Network Flows||Advanced Algorithms" + ] + }, + "positive": { + "prob_id": "lotus leaves074", + "tags": [ + "Network Flows||Advanced Algorithms" + ] + }, + "negative": [ + { + "prob_id": "585b", + "tags": [ + "Grid Graphs|Graphs|Basics" + ] + }, + { + "prob_id": "760f", + "tags": [ + "Binomial Coefficients|Fundamentals|Combinatorics" + ] + }, + { + "prob_id": "329c", + "tags": [ + "Probabilities and Expected Values|Probabilities and Expected Values|Math" + ] + } + ] + }, + { + "reference": { + "prob_id": "434d", + "tags": [ + "Network Flows||Advanced Algorithms" + ] + }, + "positive": { + "prob_id": "lotus leaves074", + "tags": [ + "Network Flows||Advanced Algorithms" + ] + }, + "negative": [ + { + "prob_id": "855d", + "tags": [ + "Level Ancestors and Lowest Common Ancestors||Advanced Data Structures" + ] + }, + { + "prob_id": "594e", + "tags": [ + "Lyndon Factorization / Duval Algorithm|Miscellaneous|Strings" + ] + }, + { + "prob_id": "873e", + "tags": [ + "Sparse Table|Sparse Table|Data Structures (DS)" + ] + } + ] + }, + { + "reference": { + "prob_id": "653d", + "tags": [ + "Network Flows||Advanced Algorithms" + ] + }, + "positive": { + "prob_id": "lotus leaves074", + "tags": [ + "Network Flows||Advanced Algorithms" + ] + }, + "negative": [ + { + "prob_id": "802i", + "tags": [ + "Suffix Array|Suffix Structures|Strings" + ] + }, + { + "prob_id": "598c", + "tags": [ + "Vectors and Applications|Geometry|Basics" + ] + }, + { + "prob_id": "1305c", + "tags": [ + "Pigeonhole Principle|Basic Counting|Basics" + ] + } + ] + }, + { + "reference": { + "prob_id": "1082g", + "tags": [ + "Network Flows||Advanced Algorithms" + ] + }, + "positive": { + "prob_id": "mul085", + "tags": [ + "Network Flows||Advanced Algorithms" + ] + }, + "negative": [ + { + "prob_id": "1381a2", + "tags": [ + "Basic Constructive Algorithms|Greedy and Constructive|Basics" + ] + }, + { + "prob_id": "120j", + "tags": [ + "Closest Pair of Points|Sweep Line|Geometry" + ] + }, + { + "prob_id": "1108e2", + "tags": [ + "Sorted Set||" + ] + } + ] + }, + { + "reference": { + "prob_id": "727d", + "tags": [ + "Bipartite Matching|Network Flows|Advanced Algorithms" + ] + }, + "positive": { + "prob_id": "prime flip080", + "tags": [ + "Bipartite Matching|Network Flows|Advanced Algorithms" + ] + }, + "negative": [ + { + "prob_id": "6e", + "tags": [ + "Two Pointers|Leveraging Monotonicity|Basics" + ] + }, + { + "prob_id": "1148h", + "tags": [ + "[Problem] MEX of all Subarrays|Miscellaneous|Miscellaneous" + ] + }, + { + "prob_id": "407c", + "tags": [ + "Binomial Coefficients|Fundamentals|Combinatorics" + ] + } + ] + }, + { + "reference": { + "prob_id": "387d", + "tags": [ + "Bipartite Matching|Network Flows|Advanced Algorithms" + ] + }, + "positive": { + "prob_id": "prime flip080", + "tags": [ + "Bipartite Matching|Network Flows|Advanced Algorithms" + ] + }, + "negative": [ + { + "prob_id": "1081c", + "tags": [ + "Counting with DP||Intermediate" + ] + }, + { + "prob_id": "852f", + "tags": [ + "Multiplicative Order and Carmichael's Lambda Function|Modular Arithmetic|Number Theory" + ] + }, + { + "prob_id": "398d", + "tags": [ + "Splitting Objects into Light and Heavy / Case Processing|Square Root Decomposition|Data Structures (DS)" + ] + } + ] + }, + { + "reference": { + "prob_id": "1009g", + "tags": [ + "Bipartite Matching|Network Flows|Advanced Algorithms" + ] + }, + "positive": { + "prob_id": "prime flip080", + "tags": [ + "Bipartite Matching|Network Flows|Advanced Algorithms" + ] + }, + "negative": [ + { + "prob_id": "703d", + "tags": [ + "Offline Processing of Queries||Advanced Data Structures" + ] + }, + { + "prob_id": "505d", + "tags": [ + "Strongly Connected Components (SCC)|Graph Connectivity|Graph Theory" + ] + }, + { + "prob_id": "149d", + "tags": [ + "Counting with DP||Intermediate" + ] + } + ] + }, + { + "reference": { + "prob_id": "729f", + "tags": [ + "Skip Useless States|DP Optimization|Advanced Algorithms" + ] + }, + "positive": { + "prob_id": "select half162", + "tags": [ + "Skip Useless States|DP Optimization|Advanced Algorithms" + ] + }, + "negative": [ + { + "prob_id": "78e", + "tags": [ + "Bipartite Matching|Network Flows|Advanced Algorithms" + ] + }, + { + "prob_id": "662a", + "tags": [ + "Vector Space and XOR Basis|Linear Algebra|Math" + ] + }, + { + "prob_id": "348a", + "tags": [ + "Binary Search|Leveraging Monotonicity|Basics" + ] + } + ] + }, + { + "reference": { + "prob_id": "474e", + "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": "362e", + "tags": [ + "Bipartite Matching|Network Flows|Advanced Algorithms" + ] + }, + { + "prob_id": "585d", + "tags": [ + "Meet in the Middle|Techniques|Miscellaneous" + ] + }, + { + "prob_id": "280d", + "tags": [ + "Min Cost Max Flow (MCMF)|Flow and Min Cut|Graph Theory" + ] + } + ] + }, + { + "reference": { + "prob_id": "960f", + "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": "160d", + "tags": [ + "Biconnectivity|DFS Lowlink|Advanced Algorithms" + ] + }, + { + "prob_id": "877f", + "tags": [ + "Square Root Decomposition||" + ] + }, + { + "prob_id": "1176e", + "tags": [ + "Bipartite Graph||", + "Graph Traversal||" + ] + } + ] + }, + { + "reference": { + "prob_id": "930c", + "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": "1109f", + "tags": [ + "Link Cut Tree (LCT)|Link Cut Tree (LCT)|Data Structures (DS)" + ] + }, + { + "prob_id": "715b", + "tags": [ + "Dijkstra's Algorithm|Shortest Paths|Graph Theory" + ] + }, + { + "prob_id": "891e", + "tags": [ + "Generating Functions|Polynomial|Math" + ] + } + ] + }, + { + "reference": { + "prob_id": "314c", + "tags": [ + "O(n lg n) Longest Monotonic Subsequence|DP Optimization|Advanced Algorithms" + ] + }, + "positive": { + "prob_id": "629d", + "tags": [ + "O(n lg n) Longest Monotonic Subsequence|DP Optimization|Advanced Algorithms" + ] + }, + "negative": [ + { + "prob_id": "367b", + "tags": [ + "Sliding Window|Data Structure Techniques|Intermediate" + ] + }, + { + "prob_id": "37c", + "tags": [ + "Trie||" + ] + }, + { + "prob_id": "280d", + "tags": [ + "Min Cost Max Flow (MCMF)|Flow and Min Cut|Graph Theory" + ] + } + ] + }, + { + "reference": { + "prob_id": "772d", + "tags": [ + "Subset Convolution|Bit Related DP|Dynamic Programming (DP)" + ] + }, + "positive": { + "prob_id": "BEAUTY", + "tags": [ + "Subset Convolution|Bit Related DP|Dynamic Programming (DP)" + ] + }, + "negative": [ + { + "prob_id": "587e", + "tags": [ + "Vector Space and XOR Basis|Linear Algebra|Math" + ] + }, + { + "prob_id": "849e", + "tags": [ + "BIT / Fenwick Tree|Binary Indexed Tree (BIT)|Data Structures (DS)" + ] + }, + { + "prob_id": "716e", + "tags": [ + "Centroid Decomposition|Centroid Decomposition|Data Structures (DS)", + "String Hashing|Hashing|Strings" + ] + } + ] + }, + { + "reference": { + "prob_id": "BEAUTY", + "tags": [ + "Subset Convolution|Bit Related DP|Dynamic Programming (DP)" + ] + }, + "positive": { + "prob_id": "or plus max100", + "tags": [ + "Subset Convolution|Bit Related DP|Dynamic Programming (DP)" + ] + }, + "negative": [ + { + "prob_id": "371c", + "tags": [ + "Binary Search|Leveraging Monotonicity|Basics" + ] + }, + { + "prob_id": "DIVQUERY", + "tags": [ + "Offline Processing of Queries||Advanced Data Structures" + ] + }, + { + "prob_id": "149e", + "tags": [ + "Z algorithm|String Matching|Strings" + ] + } + ] + }, + { + "reference": { + "prob_id": "286e", + "tags": [ + "Fast Fourier Transform (FFT)|Convolution|Math" + ] + }, + "positive": { + "prob_id": "unhappy hacking059", + "tags": [ + "Fast Fourier Transform (FFT)|Convolution|Math" + ] + }, + "negative": [ + { + "prob_id": "1439b", + "tags": [ + "Maximum Clique|Cliques and Independent Sets|Graph Theory" + ] + }, + { + "prob_id": "678f", + "tags": [ + "Convex Hull Trick|DP Optimization|Advanced Algorithms" + ] + }, + { + "prob_id": "27d", + "tags": [ + "Bipartite Graph||" + ] + } + ] + }, + { + "reference": { + "prob_id": "553e", + "tags": [ + "Fast Fourier Transform (FFT)|Convolution|Math" + ] + }, + "positive": { + "prob_id": "unhappy hacking059", + "tags": [ + "Fast Fourier Transform (FFT)|Convolution|Math" + ] + }, + "negative": [ + { + "prob_id": "817d", + "tags": [ + "All Nearest Smaller Values and All Subarray Maximum/Minimum|Miscellaneous|Miscellaneous" + ] + }, + { + "prob_id": "678f", + "tags": [ + "Convex Hull Trick|DP Optimization|Advanced Algorithms" + ] + }, + { + "prob_id": "594e", + "tags": [ + "Lyndon Factorization / Duval Algorithm|Miscellaneous|Strings" + ] + } + ] + }, + { + "reference": { + "prob_id": "1398g", + "tags": [ + "Fast Fourier Transform (FFT)|Convolution|Math" + ] + }, + "positive": { + "prob_id": "958f3", + "tags": [ + "Fast Fourier Transform (FFT)|Convolution|Math" + ] + }, + "negative": [ + { + "prob_id": "1000c", + "tags": [ + "Coordinate Compression|Techniques|Miscellaneous", + "Prefix Sum|Range Queries Without Updates|Basics" + ] + }, + { + "prob_id": "1193a", + "tags": [ + "Chromatic Polynomial ft Number of DAGs|Graph Coloring|Graph Theory" + ] + }, + { + "prob_id": "1082g", + "tags": [ + "Network Flows||Advanced Algorithms" + ] + } + ] + }, + { + "reference": { + "prob_id": "850e", + "tags": [ + "Fast Walsh Hadamard Transform (FWHT) and its Variations|Convolution|Math" + ] + }, + "positive": { + "prob_id": "453d", + "tags": [ + "Fast Walsh Hadamard Transform (FWHT) and its Variations|Convolution|Math" + ] + }, + "negative": [ + { + "prob_id": "1106f", + "tags": [ + "Discrete Logarithm||Advanced Mathematics" + ] + }, + { + "prob_id": "76f", + "tags": [ + "O(n lg n) Longest Monotonic Subsequence|DP Optimization|Advanced Algorithms" + ] + }, + { + "prob_id": "318a", + "tags": [ + "Sums and Asymptotics||Preliminaries" + ] + } + ] + }, + { + "reference": { + "prob_id": "633a", + "tags": [ + "Linear Diophantine Equations|Fundamentals|Algebra" + ] + }, + "positive": { + "prob_id": "986f", + "tags": [ + "Linear Diophantine Equations|Fundamentals|Algebra" + ] + }, + "negative": [ + { + "prob_id": "704c", + "tags": [ + "2-SAT|DFS Lowlink|Advanced Algorithms" + ] + }, + { + "prob_id": "1591d", + "tags": [ + "Invariants and Monovariants|Techniques|Miscellaneous", + "Permutations and Permutation Cycles|Permutations|Miscellaneous" + ] + }, + { + "prob_id": "768b", + "tags": [ + "Divide and Conquer DP|DP optimizations|Dynamic Programming" + ] + } + ] + }, + { + "reference": { + "prob_id": "776e", + "tags": [ + "Euler's totient function|Number-theoretic functions|Algebra" + ] + }, + "positive": { + "prob_id": "1036f", + "tags": [ + "Euler's totient function|Number-theoretic functions|Algebra" + ] + }, + "negative": [ + { + "prob_id": "776d", + "tags": [ + "2-SAT|DFS Lowlink|Advanced Algorithms", + "Disjoint Set Union||", + "Strongly Connected Components (SCC)|Graph Connectivity|Graph Theory" + ] + }, + { + "prob_id": "860d", + "tags": [ + "Spanning Tree||" + ] + }, + { + "prob_id": "358c", + "tags": [ + "Simulation||" + ] + } + ] + }, + { + "reference": { + "prob_id": "475d", + "tags": [ + "Sparse Table|Sparse Table|Data Structures (DS)" + ] + }, + "positive": { + "prob_id": "863e", + "tags": [ + "Sparse Table|Sparse Table|Data Structures (DS)" + ] + }, + "negative": [ + { + "prob_id": "585d", + "tags": [ + "Meet in the Middle|Techniques|Miscellaneous" + ] + }, + { + "prob_id": "862b", + "tags": [ + "Bipartite Graph||", + "Graph Traversal||" + ] + }, + { + "prob_id": "518d", + "tags": [ + "Markov Chains|Probability and Expectation|Intermediate" + ] + } + ] + }, + { + "reference": { + "prob_id": "669e", + "tags": [ + "BIT / Fenwick Tree|Binary Indexed Tree (BIT)|Data Structures (DS)" + ] + }, + "positive": { + "prob_id": "597c", + "tags": [ + "BIT / Fenwick Tree|Binary Indexed Tree (BIT)|Data Structures (DS)" + ] + }, + "negative": [ + { + "prob_id": "1353d", + "tags": [ + "Basic Constructive Algorithms|Greedy and Constructive|Basics" + ] + }, + { + "prob_id": "1421a", + "tags": [ + "Bit Manipulation|Bit Manipulation|Basics" + ] + }, + { + "prob_id": "727d", + "tags": [ + "Bipartite Matching|Network Flows|Advanced Algorithms" + ] + } + ] + }, + { + "reference": { + "prob_id": "597c", + "tags": [ + "BIT / Fenwick Tree|Binary Indexed Tree (BIT)|Data Structures (DS)" + ] + }, + "positive": { + "prob_id": "707e", + "tags": [ + "BIT / Fenwick Tree|Binary Indexed Tree (BIT)|Data Structures (DS)" + ] + }, + "negative": [ + { + "prob_id": "442b", + "tags": [ + "Probabilities and Expected Values|Probabilities and Expected Values|Math" + ] + }, + { + "prob_id": "687b", + "tags": [ + "Chinese Remainder Theorem|Modular arithmetic|Algebra" + ] + }, + { + "prob_id": "962f", + "tags": [ + "DFS Tree|Graph Traversal|Graph Theory" + ] + } + ] + }, + { + "reference": { + "prob_id": "849e", + "tags": [ + "BIT / Fenwick Tree|Binary Indexed Tree (BIT)|Data Structures (DS)" + ] + }, + "positive": { + "prob_id": "669e", + "tags": [ + "BIT / Fenwick Tree|Binary Indexed Tree (BIT)|Data Structures (DS)" + ] + }, + "negative": [ + { + "prob_id": "846d", + "tags": [ + "Monotonic Queue 2D|Miscellaneous|Data Structures (DS)" + ] + }, + { + "prob_id": "987e", + "tags": [ + "Permutations and Permutation Cycles|Permutations|Miscellaneous" + ] + }, + { + "prob_id": "1205f", + "tags": [ + "Permutation Tree|More Tree Data Structures|Data Structures (DS)" + ] + } + ] + }, + { + "reference": { + "prob_id": "433b", + "tags": [ + "Sqrt Decomposition|Trees|Data Structures" + ] + }, + "positive": { + "prob_id": "786c", + "tags": [ + "Sqrt Decomposition|Trees|Data Structures" + ] + }, + "negative": [ + { + "prob_id": "427c", + "tags": [ + "Strongly Connected Components (SCC)|Graph Connectivity|Graph Theory" + ] + }, + { + "prob_id": "877d", + "tags": [ + "Grid Graphs|Graphs|Basics" + ] + }, + { + "prob_id": "371d", + "tags": [ + "Disjoint Set Union||" + ] + } + ] + }, + { + "reference": { + "prob_id": "727e", + "tags": [ + "String Hashing|Hashing|Strings" + ] + }, + "positive": { + "prob_id": "7d", + "tags": [ + "String Hashing|Hashing|Strings" + ] + }, + "negative": [ + { + "prob_id": "821e", + "tags": [ + "Matrix Exponentiation|Matrices|Math" + ] + }, + { + "prob_id": "704d", + "tags": [ + "Flow with Lower Bounds||" + ] + }, + { + "prob_id": "1085e", + "tags": [ + "Tree Centers and Diameter||Advanced Data Structures" + ] + } + ] + }, + { + "reference": { + "prob_id": "464e", + "tags": [ + "Dijkstra's Algorithm|Shortest Paths|Graph Theory" + ] + }, + "positive": { + "prob_id": "757f", + "tags": [ + "Dijkstra's Algorithm|Shortest Paths|Graph Theory" + ] + }, + "negative": [ + { + "prob_id": "231e", + "tags": [ + "Biconnectivity|DFS Lowlink|Advanced Algorithms" + ] + }, + { + "prob_id": "878c", + "tags": [ + "Sorted Set||" + ] + }, + { + "prob_id": "607b", + "tags": [ + "Interval DP / Range DP / Matrix Chain Multiplication (MCM)|Classic DP|Dynamic Programming (DP)" + ] + } + ] + }, + { + "reference": { + "prob_id": "843d", + "tags": [ + "Dijkstra's Algorithm|Shortest Paths|Graph Theory" + ] + }, + "positive": { + "prob_id": "757f", + "tags": [ + "Dijkstra's Algorithm|Shortest Paths|Graph Theory" + ] + }, + "negative": [ + { + "prob_id": "505d", + "tags": [ + "Strongly Connected Components (SCC)|Graph Connectivity|Graph Theory" + ] + }, + { + "prob_id": "878c", + "tags": [ + "Sorted Set||" + ] + }, + { + "prob_id": "1213g", + "tags": [ + "Disjoint Set Union||", + "Minimum Spanning Tree||" + ] + } + ] + }, + { + "reference": { + "prob_id": "733f", + "tags": [ + "Lowest Common Ancestor|Lowest common ancestor|Graphs" + ] + }, + "positive": { + "prob_id": "828f", + "tags": [ + "Lowest Common Ancestor|Lowest common ancestor|Graphs" + ] + }, + "negative": [ + { + "prob_id": "567e", + "tags": [ + "Dijkstra's Algorithm|Shortest Paths|Graph Theory" + ] + }, + { + "prob_id": "25c", + "tags": [ + "All-Pairs|Shortest Paths|Intermediate" + ] + }, + { + "prob_id": "700c", + "tags": [ + "Finding Bridges in O(N+M)|Connected components, bridges, articulations points|Graphs" + ] + } + ] + }, + { + "reference": { + "prob_id": "removing coins033", + "tags": [ + "Tree Diameter|Graph Traversal|Graph Theory" + ] + }, + "positive": { + "prob_id": "1434d", + "tags": [ + "Tree Diameter|Graph Traversal|Graph Theory" + ] + }, + "negative": [ + { + "prob_id": "painting graphs with atcodeer062", + "tags": [ + "Biconnectivity|DFS Lowlink|Advanced Algorithms", + "Burnside's Lemma / Polya Enumeration Theorem|Miscellaneous|Combinatorics" + ] + }, + { + "prob_id": "snuke line068", + "tags": [ + "Harmonic Lemma||Intermediate" + ] + }, + { + "prob_id": "travel by car143", + "tags": [ + "All-Pairs|Shortest Paths|Intermediate" + ] + } + ] + }, + { + "reference": { + "prob_id": "1526d", + "tags": [ + "Inversions|Miscellaneous|Miscellaneous" + ] + }, + "positive": { + "prob_id": "papple sort088", + "tags": [ + "Inversions|Miscellaneous|Miscellaneous" + ] + }, + "negative": [ + { + "prob_id": "704e", + "tags": [ + "Heavy Light Decomposition (HLD) ft Subtrees and Path Query|Heavy Light Decomposition (HLD)|Data Structures (DS)" + ] + }, + { + "prob_id": "527d", + "tags": [ + "Maximum Clique|Cliques and Independent Sets|Graph Theory" + ] + }, + { + "prob_id": "1085e", + "tags": [ + "Tree Centers and Diameter||Advanced Data Structures" + ] + } + ] + }, + { + "reference": { + "prob_id": "987e", + "tags": [ + "Permutations and Permutation Cycles|Permutations|Miscellaneous" + ] + }, + "positive": { + "prob_id": "DPOWPERM", + "tags": [ + "Permutations and Permutation Cycles|Permutations|Miscellaneous" + ] + }, + "negative": [ + { + "prob_id": "983b", + "tags": [ + "Prefix Sums and Difference Arrays|Range Queries|Basics" + ] + }, + { + "prob_id": "145e", + "tags": [ + "Segment Tree with Lazy Propagation|Segment Tree|Data Structures (DS)" + ] + }, + { + "prob_id": "1404e", + "tags": [ + "Maximum Flow / Minimum Cut|Flow and Min Cut|Graph Theory" + ] + } + ] + }, + { + "reference": { + "prob_id": "1335b", + "tags": [ + "Basic Constructive Algorithms|Greedy and Constructive|Basics" + ] + }, + "positive": { + "prob_id": "1396a", + "tags": [ + "Basic Constructive Algorithms|Greedy and Constructive|Basics" + ] + }, + "negative": [ + { + "prob_id": "1073e", + "tags": [ + "DP on Digits||Intermediate" + ] + }, + { + "prob_id": "235c", + "tags": [ + "String Suffix||", + "Suffix Automaton|Suffix Structures|Strings" + ] + }, + { + "prob_id": "1225d", + "tags": [ + "Divisibility||" + ] + } + ] + }, + { + "reference": { + "prob_id": "1353d", + "tags": [ + "Basic Constructive Algorithms|Greedy and Constructive|Basics" + ] + }, + "positive": { + "prob_id": "1028b", + "tags": [ + "Basic Constructive Algorithms|Greedy and Constructive|Basics" + ] + }, + "negative": [ + { + "prob_id": "1603f", + "tags": [ + "q Binomial|Linear Algebra|Math" + ] + }, + { + "prob_id": "1444d", + "tags": [ + "Bitsets||" + ] + }, + { + "prob_id": "1439b", + "tags": [ + "Maximum Clique|Cliques and Independent Sets|Graph Theory" + ] + } + ] + }, + { + "reference": { + "prob_id": "1364c", + "tags": [ + "Basic Constructive Algorithms|Greedy and Constructive|Basics" + ] + }, + "positive": { + "prob_id": "1028b", + "tags": [ + "Basic Constructive Algorithms|Greedy and Constructive|Basics" + ] + }, + "negative": [ + { + "prob_id": "1603f", + "tags": [ + "q Binomial|Linear Algebra|Math" + ] + }, + { + "prob_id": "1364e", + "tags": [ + "Interactive Problems|Interactive Problems|Miscellaneous" + ] + }, + { + "prob_id": "1439b", + "tags": [ + "Maximum Clique|Cliques and Independent Sets|Graph Theory" + ] + } + ] + }, + { + "reference": { + "prob_id": "1381a2", + "tags": [ + "Basic Constructive Algorithms|Greedy and Constructive|Basics" + ] + }, + "positive": { + "prob_id": "1028b", + "tags": [ + "Basic Constructive Algorithms|Greedy and Constructive|Basics" + ] + }, + "negative": [ + { + "prob_id": "1603f", + "tags": [ + "q Binomial|Linear Algebra|Math" + ] + }, + { + "prob_id": "1349a", + "tags": [ + "Divisibility||", + "Prime Factorization|Number Theory|Basics" + ] + }, + { + "prob_id": "1325f", + "tags": [ + "DFS Tree|Graph Traversal|Graph Theory" + ] + } + ] + }, + { + "reference": { + "prob_id": "1092f", + "tags": [ + "Rerooting Technique|More Tree Stuff|Graph Theory" + ] + }, + "positive": { + "prob_id": "852e", + "tags": [ + "Rerooting Technique|More Tree Stuff|Graph Theory" + ] + }, + "negative": [ + { + "prob_id": "1328a", + "tags": [ + "Big O Notation and Time and Space Complexity Analysis|Complexity Analysis|Basics" + ] + }, + { + "prob_id": "1175f", + "tags": [ + "XOR Hashing|Hashing|Strings" + ] + }, + { + "prob_id": "120j", + "tags": [ + "Closest Pair of Points|Sweep Line|Geometry" + ] + } + ] + }, + { + "reference": { + "prob_id": "222c", + "tags": [ + "Sieve, Prime Factorization and Divisors|Primes and Divisors|Number Theory" + ] + }, + "positive": { + "prob_id": "1561d2", + "tags": [ + "Sieve, Prime Factorization and Divisors|Primes and Divisors|Number Theory" + ] + }, + "negative": [ + { + "prob_id": "176e", + "tags": [ + "Level Ancestors and Lowest Common Ancestors||Advanced Data Structures" + ] + }, + { + "prob_id": "2c", + "tags": [ + "Hill Climbing and Simulated Annealing|Miscellaneous|Geometry" + ] + }, + { + "prob_id": "280d", + "tags": [ + "Min Cost Max Flow (MCMF)|Flow and Min Cut|Graph Theory" + ] + } + ] + }, + { + "reference": { + "prob_id": "962f", + "tags": [ + "DFS Tree|Graph Traversal|Graph Theory" + ] + }, + "positive": { + "prob_id": "118e", + "tags": [ + "DFS Tree|Graph Traversal|Graph Theory" + ] + }, + "negative": [ + { + "prob_id": "1245f", + "tags": [ + "DP on Digits||Intermediate" + ] + }, + { + "prob_id": "1430e", + "tags": [ + "Inversions|Miscellaneous|Miscellaneous" + ] + }, + { + "prob_id": "915g", + "tags": [ + "Mobius Function and Mobius Inversion|Multiplicative Functions|Number Theory" + ] + } + ] + }, + { + "reference": { + "prob_id": "1508c", + "tags": [ + "Inverse Graph|Graph Traversal|Graph Theory" + ] + }, + "positive": { + "prob_id": "190e", + "tags": [ + "Inverse Graph|Graph Traversal|Graph Theory" + ] + }, + "negative": [ + { + "prob_id": "1213g", + "tags": [ + "Disjoint Set Union||", + "Minimum Spanning Tree||" + ] + }, + { + "prob_id": "1299c", + "tags": [ + "Convex Hull Trick|DP Optimization|Advanced Algorithms" + ] + }, + { + "prob_id": "914e", + "tags": [ + "Centroid Decomposition|Centroid Decomposition|Data Structures (DS)" + ] + } + ] + }, + { + "reference": { + "prob_id": "1510c", + "tags": [ + "Cactus Graph|Special Graphs|Graph Theory" + ] + }, + "positive": { + "prob_id": "980f", + "tags": [ + "Cactus Graph|Special Graphs|Graph Theory" + ] + }, + "negative": [ + { + "prob_id": "1454a", + "tags": [ + "Basic Constructive Algorithms|Greedy and Constructive|Basics" + ] + }, + { + "prob_id": "855d", + "tags": [ + "Level Ancestors and Lowest Common Ancestors||Advanced Data Structures" + ] + }, + { + "prob_id": "1383f", + "tags": [ + "Maximum Flow / Minimum Cut|Flow and Min Cut|Graph Theory" + ] + } + ] + }, + { + "reference": { + "prob_id": "1436f", + "tags": [ + "Mobius Function and Mobius Inversion|Multiplicative Functions|Number Theory" + ] + }, + "positive": { + "prob_id": "920g", + "tags": [ + "Mobius Function and Mobius Inversion|Multiplicative Functions|Number Theory" + ] + }, + "negative": [ + { + "prob_id": "1430g", + "tags": [ + "DP on Trees and DAGs|Dp on Different Structures|Dynamic Programming (DP)", + "Duality in Linear Programming (LP <> Flow)|Mathematical Optimization|Math" + ] + }, + { + "prob_id": "1408e", + "tags": [ + "Minimum Spanning Tree||" + ] + }, + { + "prob_id": "960f", + "tags": [ + "O(n lg n) Longest Monotonic Subsequence|DP Optimization|Advanced Algorithms" + ] + } + ] + }, + { + "reference": { + "prob_id": "915g", + "tags": [ + "Mobius Function and Mobius Inversion|Multiplicative Functions|Number Theory" + ] + }, + "positive": { + "prob_id": "920g", + "tags": [ + "Mobius Function and Mobius Inversion|Multiplicative Functions|Number Theory" + ] + }, + "negative": [ + { + "prob_id": "803d", + "tags": [ + "Basic Greedy Algorithms|Greedy and Constructive|Basics", + "Binary Search|Leveraging Monotonicity|Basics", + "Two Pointers|Leveraging Monotonicity|Basics" + ] + }, + { + "prob_id": "865g", + "tags": [ + "Generating Functions|Polynomial|Math" + ] + }, + { + "prob_id": "1132g", + "tags": [ + "Segment Tree with Lazy Propagation|Segment Tree|Data Structures (DS)" + ] + } + ] + }, + { + "reference": { + "prob_id": "1552i", + "tags": [ + "PQ Tree|More Tree Data Structures|Data Structures (DS)" + ] + }, + "positive": { + "prob_id": "243e", + "tags": [ + "PQ Tree|More Tree Data Structures|Data Structures (DS)" + ] + }, + "negative": [ + { + "prob_id": "1091e", + "tags": [ + "Degree Sequences|Miscellaneous|Graph Theory" + ] + }, + { + "prob_id": "1181d", + "tags": [ + "Binary Search|Leveraging Monotonicity|Basics", + "Point Update Range Sum||" + ] + }, + { + "prob_id": "1240f", + "tags": [ + "Edge Coloring of Bipartite Graph|Graph Coloring|Graph Theory" + ] + } + ] + }, + { + "reference": { + "prob_id": "1619h", + "tags": [ + "Splitting Objects into Light and Heavy / Case Processing|Square Root Decomposition|Data Structures (DS)" + ] + }, + "positive": { + "prob_id": "398d", + "tags": [ + "Splitting Objects into Light and Heavy / Case Processing|Square Root Decomposition|Data Structures (DS)" + ] + }, + "negative": [ + { + "prob_id": "1093e", + "tags": [ + "2D BIT|Binary Indexed Tree (BIT)|Data Structures (DS)", + "CDQ Divide and Conquer|Techniques|Data Structures (DS)" + ] + }, + { + "prob_id": "1205f", + "tags": [ + "Permutation Tree|More Tree Data Structures|Data Structures (DS)" + ] + }, + { + "prob_id": "1401f", + "tags": [ + "XOR Segment Tree|Segment Tree|Data Structures (DS)" + ] + } + ] + }, + { + "reference": { + "prob_id": "543d", + "tags": [ + "Rerooting Technique|More Tree Stuff|Graph Theory" + ] + }, + "positive": { + "prob_id": "1187e", + "tags": [ + "DP on Trees and DAGs|Dp on Different Structures|Dynamic Programming (DP)" + ] + }, + "negative": [ + { + "prob_id": "543b", + "tags": [ + "Breadth First Search|Graph traversal|Graphs" + ] + }, + { + "prob_id": "546e", + "tags": [ + "Network Flows||Advanced Algorithms" + ] + }, + { + "prob_id": "118e", + "tags": [ + "DFS Tree|Graph Traversal|Graph Theory" + ] + } + ] + }, + { + "reference": { + "prob_id": "1006c", + "tags": [ + "Two Pointers|Leveraging Monotonicity|Basics" + ] + }, + "positive": { + "prob_id": "701c", + "tags": [ + "Two Pointers|Leveraging Monotonicity|Basics" + ] + }, + "negative": [ + { + "prob_id": "868f", + "tags": [ + "Divide and Conquer DP|DP optimizations|Dynamic Programming", + "Divide and Conquer DP|DP optimizations|Dynamic Programming" + ] + }, + { + "prob_id": "1132g", + "tags": [ + "Segment Tree with Lazy Propagation|Segment Tree|Data Structures (DS)" + ] + }, + { + "prob_id": "988d", + "tags": [ + "Bit Manipulation|Bit Manipulation|Basics" + ] + } + ] + }, + { + "reference": { + "prob_id": "279b", + "tags": [ + "Two Pointers|Leveraging Monotonicity|Basics" + ] + }, + "positive": { + "prob_id": "702c", + "tags": [ + "Binary Search|Leveraging Monotonicity|Basics", + "Two Pointers|Leveraging Monotonicity|Basics" + ] + }, + "negative": [ + { + "prob_id": "33b", + "tags": [ + "Floyd Warshall|Shortest Paths|Graph Theory" + ] + }, + { + "prob_id": "33c", + "tags": [ + "Classic Problems with Twists|DP|Intermediate" + ] + }, + { + "prob_id": "689b", + "tags": [ + "Breadth First Search|Graph traversal|Graphs" + ] + } + ] + }, + { + "reference": { + "prob_id": "701c", + "tags": [ + "Two Pointers|Leveraging Monotonicity|Basics" + ] + }, + "positive": { + "prob_id": "1006c", + "tags": [ + "Two Pointers|Leveraging Monotonicity|Basics" + ] + }, + "negative": [ + { + "prob_id": "739e", + "tags": [ + "Aliens Trick / WQS Binary Search|DP Optimizations|Dynamic Programming (DP)" + ] + }, + { + "prob_id": "332b", + "tags": [ + "Classic Problems with Twists|DP|Intermediate" + ] + }, + { + "prob_id": "190e", + "tags": [ + "Inverse Graph|Graph Traversal|Graph Theory" + ] + } + ] + }, + { + "reference": { + "prob_id": "702c", + "tags": [ + "Binary Search|Leveraging Monotonicity|Basics", + "Two Pointers|Leveraging Monotonicity|Basics" + ] + }, + "positive": { + "prob_id": "1006c", + "tags": [ + "Two Pointers|Leveraging Monotonicity|Basics" + ] + }, + "negative": [ + { + "prob_id": "546e", + "tags": [ + "Network Flows||Advanced Algorithms" + ] + }, + { + "prob_id": "707b", + "tags": [ + "Single Source|Shortest Paths|Intermediate" + ] + }, + { + "prob_id": "1572f", + "tags": [ + "Segment Tree Beats|Segment Tree|Data Structures (DS)", + "Segment Tree Beats|Segment Tree|Data Structures (DS)" + ] + } + ] + }, + { + "reference": { + "prob_id": "484a", + "tags": [ + "Bit Manipulation|Bit Manipulation|Basics" + ] + }, + "positive": { + "prob_id": "1513b", + "tags": [ + "Bit Manipulation|Bit Manipulation|Basics" + ] + }, + "negative": [ + { + "prob_id": "920f", + "tags": [ + "Segment Tree|Trees|Data Structures" + ] + }, + { + "prob_id": "407c", + "tags": [ + "Binomial Coefficients|Fundamentals|Combinatorics" + ] + }, + { + "prob_id": "602d", + "tags": [ + "All Nearest Smaller Values and All Subarray Maximum/Minimum|Miscellaneous|Miscellaneous" + ] + } + ] + }, + { + "reference": { + "prob_id": "1303e", + "tags": [ + "Interval DP / Range DP / Matrix Chain Multiplication (MCM)|Classic DP|Dynamic Programming (DP)" + ] + }, + "positive": { + "prob_id": "607b", + "tags": [ + "Interval DP / Range DP / Matrix Chain Multiplication (MCM)|Classic DP|Dynamic Programming (DP)" + ] + }, + "negative": [ + { + "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": "202a", + "tags": [ + "All-Subsets / Binary Choice|Complete Search|Basics" + ] + }, + { + "prob_id": "305e", + "tags": [ + "Nim, Sprague-Grundy Theorem|Combinatorial Games|Advanced Mathematics" + ] + } + ] + }, + { + "reference": { + "prob_id": "476e", + "tags": [ + "Interval DP / Range DP / Matrix Chain Multiplication (MCM)|Classic DP|Dynamic Programming (DP)" + ] + }, + "positive": { + "prob_id": "607b", + "tags": [ + "Interval DP / Range DP / Matrix Chain Multiplication (MCM)|Classic DP|Dynamic Programming (DP)" + ] + }, + "negative": [ + { + "prob_id": "204e", + "tags": [ + "Suffix Array|Suffix Structures|Strings" + ] + }, + { + "prob_id": "202a", + "tags": [ + "All-Subsets / Binary Choice|Complete Search|Basics" + ] + }, + { + "prob_id": "528d", + "tags": [ + "Fast Fourier Transform (FFT)|Convolution|Math" + ] + } + ] + }, + { + "reference": { + "prob_id": "1335b", + "tags": [ + "Basic Constructive Algorithms|Greedy and Constructive|Basics" + ] + }, + "positive": { + "prob_id": "1028b", + "tags": [ + "Basic Constructive Algorithms|Greedy and Constructive|Basics" + ] + }, + "negative": [ + { + "prob_id": "1016b", + "tags": [ + "Prefix Sums and Difference Arrays|Range Queries|Basics" + ] + }, + { + "prob_id": "245h", + "tags": [ + "Interval DP / Range DP / Matrix Chain Multiplication (MCM)|Classic DP|Dynamic Programming (DP)" + ] + }, + { + "prob_id": "1234d", + "tags": [ + "Segment Tree|Trees|Data Structures" + ] + } + ] + }, + { + "reference": { + "prob_id": "1272e", + "tags": [ + "Breadth First Search|Graph traversal|Graphs" + ] + }, + "positive": { + "prob_id": "796d", + "tags": [ + "Breadth First Search|Graph traversal|Graphs" + ] + }, + "negative": [ + { + "prob_id": "1154g", + "tags": [ + "Sieve, Prime Factorization and Divisors|Primes and Divisors|Number Theory" + ] + }, + { + "prob_id": "988d", + "tags": [ + "Bit Manipulation|Bit Manipulation|Basics" + ] + }, + { + "prob_id": "1132g", + "tags": [ + "Segment Tree with Lazy Propagation|Segment Tree|Data Structures (DS)" + ] + } + ] + }, + { + "reference": { + "prob_id": "769c", + "tags": [ + "Breadth First Search|Graph traversal|Graphs" + ] + }, + "positive": { + "prob_id": "1272e", + "tags": [ + "Breadth First Search|Graph traversal|Graphs" + ] + }, + "negative": [ + { + "prob_id": "36b", + "tags": [ + "Recursion||Preliminaries" + ] + }, + { + "prob_id": "908b", + "tags": [ + "All-Permutations|Complete Search|Basics" + ] + }, + { + "prob_id": "1451f", + "tags": [ + "Nim, Sprague-Grundy Theorem|Combinatorial Games|Advanced Mathematics" + ] + } + ] + }, + { + "reference": { + "prob_id": "727e", + "tags": [ + "String Hashing|Hashing|Strings" + ] + }, + "positive": { + "prob_id": "7d", + "tags": [ + "String Hashing|Hashing|Strings" + ] + }, + "negative": [ + { + "prob_id": "152c", + "tags": [ + "Rule of Sum, Rule of Product|Counting|Basics" + ] + }, + { + "prob_id": "576a", + "tags": [ + "Prime Factorization|Number Theory|Basics" + ] + }, + { + "prob_id": "114b", + "tags": [ + "All-Subsets / Binary Choice|Complete Search|Basics" + ] + } + ] + }, + { + "reference": { + "prob_id": "1077f2", + "tags": [ + "Monotonic Queue|Miscellaneous|Data Structures (DS)" + ] + }, + "positive": { + "prob_id": "488d", + "tags": [ + "Monotonic Queue|Miscellaneous|Data Structures (DS)" + ] + }, + "negative": [ + { + "prob_id": "1203f1", + "tags": [ + "Exchange Argument|Greedy|Basics" + ] + }, + { + "prob_id": "1081d", + "tags": [ + "Minimum Spanning Tree||", + "Centroid Decomposition|Centroid Decomposition|Data Structures (DS)" + ] + }, + { + "prob_id": "1446a", + "tags": [ + "Knapsack DP||" + ] + } + ] + }, + { + "reference": { + "prob_id": "488d", + "tags": [ + "Monotonic Queue|Miscellaneous|Data Structures (DS)" + ] + }, + "positive": { + "prob_id": "1077f2", + "tags": [ + "Monotonic Queue|Miscellaneous|Data Structures (DS)" + ] + }, + "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": "113d", + "tags": [ + "Probabilities and Expected Values|Probabilities and Expected Values|Math" + ] + }, + "positive": { + "prob_id": "26d", + "tags": [ + "Probabilities and Expected Values|Probabilities and Expected Values|Math" + ] + }, + "negative": [ + { + "prob_id": "832d", + "tags": [ + "Lowest Common Ancestor|Lowest common ancestor|Graphs" + ] + }, + { + "prob_id": "875e", + "tags": [ + "Knapsack DP||" + ] + }, + { + "prob_id": "519e", + "tags": [ + "Level Ancestors and Lowest Common Ancestors||Advanced Data Structures" + ] + } + ] + }, + { + "reference": { + "prob_id": "1418e", + "tags": [ + "Probabilities and Expected Values|Probabilities and Expected Values|Math" + ] + }, + "positive": { + "prob_id": "26d", + "tags": [ + "Probabilities and Expected Values|Probabilities and Expected Values|Math" + ] + }, + "negative": [ + { + "prob_id": "1408e", + "tags": [ + "Minimum Spanning Tree||" + ] + }, + { + "prob_id": "1093d", + "tags": [ + "Depth First Search (DFS)|Graph Traversal|Graph Theory" + ] + }, + { + "prob_id": "704d", + "tags": [ + "Flow with Lower Bounds||" + ] + } + ] + }, + { + "reference": { + "prob_id": "167b", + "tags": [ + "Probabilities and Expected Values|Probabilities and Expected Values|Math" + ] + }, + "positive": { + "prob_id": "540d", + "tags": [ + "Probabilities and Expected Values|Probabilities and Expected Values|Math" + ] + }, + "negative": [ + { + "prob_id": "1103d", + "tags": [ + "Bit Manipulation|Bit Manipulation|Basics", + "Bitmask DP|Bit Related DP|Dynamic Programming (DP)" + ] + }, + { + "prob_id": "653d", + "tags": [ + "Network Flows||Advanced Algorithms" + ] + }, + { + "prob_id": "700c", + "tags": [ + "Finding Bridges in O(N+M)|Connected components, bridges, articulations points|Graphs" + ] + } + ] + }, + { + "reference": { + "prob_id": "26d", + "tags": [ + "Probabilities and Expected Values|Probabilities and Expected Values|Math" + ] + }, + "positive": { + "prob_id": "1096f", + "tags": [ + "Probabilities and Expected Values|Probabilities and Expected Values|Math" + ] + }, + "negative": [ + { + "prob_id": "1011e", + "tags": [ + "Bezout's Identity|Euclidean Algorithm|Number Theory" + ] + }, + { + "prob_id": "1010c", + "tags": [ + "B\u00e9zout's Lemma, Extended Euclidean Algorithm, Chinese Remainder Theorem|Number Theory|Intermediate", + "Abstract Algebra||Advanced Mathematics" + ] + }, + { + "prob_id": "580e", + "tags": [ + "Segment Tree with Lazy Propagation|Segment Tree|Data Structures (DS)" + ] + } + ] + }, + { + "reference": { + "prob_id": "351b", + "tags": [ + "Probabilities and Expected Values|Probabilities and Expected Values|Math" + ] + }, + "positive": { + "prob_id": "1525e", + "tags": [ + "Probabilities and Expected Values|Probabilities and Expected Values|Math" + ] + }, + "negative": [ + { + "prob_id": "785e", + "tags": [ + "2D BIT|Binary Indexed Tree (BIT)|Data Structures (DS)" + ] + }, + { + "prob_id": "214a", + "tags": [ + "Iterative|Complete Search|Basics" + ] + }, + { + "prob_id": "1477d", + "tags": [ + "Inverse Graph|Graph Traversal|Graph Theory" + ] + } + ] + }, + { + "reference": { + "prob_id": "518d", + "tags": [ + "Markov Chains|Probability and Expectation|Intermediate" + ] + }, + "positive": { + "prob_id": "540d", + "tags": [ + "Probabilities and Expected Values|Probabilities and Expected Values|Math" + ] + }, + "negative": [ + { + "prob_id": "535e", + "tags": [ + "Convex Hull Trick|DP Optimization|Advanced Algorithms" + ] + }, + { + "prob_id": "827f", + "tags": [ + "Dijkstra's Algorithm|Shortest Paths|Graph Theory" + ] + }, + { + "prob_id": "114b", + "tags": [ + "All-Subsets / Binary Choice|Complete Search|Basics" + ] + } + ] + }, + { + "reference": { + "prob_id": "440d", + "tags": [ + "DP on Trees and DAGs|Dp on Different Structures|Dynamic Programming (DP)" + ] + }, + "positive": { + "prob_id": "461b", + "tags": [ + "DP on Trees and DAGs|Dp on Different Structures|Dynamic Programming (DP)" + ] + }, + "negative": [ + { + "prob_id": "700c", + "tags": [ + "Finding Bridges in O(N+M)|Connected components, bridges, articulations points|Graphs" + ] + }, + { + "prob_id": "212a", + "tags": [ + "Edge Coloring of Bipartite Graph|Graph Coloring|Graph Theory" + ] + }, + { + "prob_id": "25c", + "tags": [ + "All-Pairs|Shortest Paths|Intermediate" + ] + } + ] + }, + { + "reference": { + "prob_id": "631e", + "tags": [ + "Convex Hull Trick|DP Optimization|Advanced Algorithms" + ] + }, + "positive": { + "prob_id": "311b", + "tags": [ + "Convex Hull Trick|DP Optimization|Advanced Algorithms" + ] + }, + "negative": [ + { + "prob_id": "631c", + "tags": [ + "Combining Data Structures|Data Structure Techniques|Intermediate" + ] + }, + { + "prob_id": "193b", + "tags": [ + "Backtracking|Miscellaneous|Miscellaneous" + ] + }, + { + "prob_id": "1201c", + "tags": [ + "Binary Search|Leveraging Monotonicity|Basics" + ] + } + ] + }, + { + "reference": { + "prob_id": "1097b", + "tags": [ + "Bit Manipulation|Bit Manipulation|Basics" + ] + }, + "positive": { + "prob_id": "550b", + "tags": [ + "All-Subsets / Binary Choice|Complete Search|Basics", + "Backtracking|Miscellaneous|Miscellaneous" + ] + }, + "negative": [ + { + "prob_id": "1203f1", + "tags": [ + "Exchange Argument|Greedy|Basics" + ] + }, + { + "prob_id": "251a", + "tags": [ + "Binary Search|Leveraging Monotonicity|Basics" + ] + }, + { + "prob_id": "1203f2", + "tags": [ + "Presorting|DP Common Tricks|Intermediate" + ] + } + ] + }, + { + "reference": { + "prob_id": "550b", + "tags": [ + "All-Subsets / Binary Choice|Complete Search|Basics", + "Backtracking|Miscellaneous|Miscellaneous" + ] + }, + "positive": { + "prob_id": "96b", + "tags": [ + "All-Subsets / Binary Choice|Complete Search|Basics" + ] + }, + "negative": [ + { + "prob_id": "1100b", + "tags": [ + "Amortized Analysis||Intermediate" + ] + }, + { + "prob_id": "1295f", + "tags": [ + "Probabilities and Expected Values|Probabilities and Expected Values|Math" + ] + }, + { + "prob_id": "877f", + "tags": [ + "Square Root Decomposition||" + ] + } + ] + }, + { + "reference": { + "prob_id": "1280d", + "tags": [ + "DP on Trees and DAGs|Dp on Different Structures|Dynamic Programming (DP)" + ] + }, + "positive": { + "prob_id": "815c", + "tags": [ + "DP on Trees and DAGs|Dp on Different Structures|Dynamic Programming (DP)" + ] + }, + "negative": [ + { + "prob_id": "1059d", + "tags": [ + "Ternary Search|Search Techniques|Miscellaneous" + ] + }, + { + "prob_id": "1252l", + "tags": [ + "Network Flows||Advanced Algorithms" + ] + }, + { + "prob_id": "700c", + "tags": [ + "Finding Bridges in O(N+M)|Connected components, bridges, articulations points|Graphs" + ] + } + ] + }, + { + "reference": { + "prob_id": "735e", + "tags": [ + "DP on Trees and DAGs|Dp on Different Structures|Dynamic Programming (DP)" + ] + }, + "positive": { + "prob_id": "815c", + "tags": [ + "DP on Trees and DAGs|Dp on Different Structures|Dynamic Programming (DP)" + ] + }, + "negative": [ + { + "prob_id": "160d", + "tags": [ + "Biconnectivity|DFS Lowlink|Advanced Algorithms" + ] + }, + { + "prob_id": "1081d", + "tags": [ + "Minimum Spanning Tree||", + "Centroid Decomposition|Centroid Decomposition|Data Structures (DS)" + ] + }, + { + "prob_id": "451e", + "tags": [ + "Principle of Inclusion-Exclusion|Counting|Intermediate", + "Generating Functions|Polynomial|Math" + ] + } + ] + }, + { + "reference": { + "prob_id": "815c", + "tags": [ + "DP on Trees and DAGs|Dp on Different Structures|Dynamic Programming (DP)" + ] + }, + "positive": { + "prob_id": "1280d", + "tags": [ + "DP on Trees and DAGs|Dp on Different Structures|Dynamic Programming (DP)" + ] + }, + "negative": [ + { + "prob_id": "816b", + "tags": [ + "Prefix Sum|Range Queries Without Updates|Basics", + "Prefix Sums and Difference Arrays|Range Queries|Basics" + ] + }, + { + "prob_id": "877f", + "tags": [ + "Square Root Decomposition||" + ] + }, + { + "prob_id": "802c", + "tags": [ + "Min Cost Max Flow (MCMF)|Flow and Min Cut|Graph Theory" + ] + } + ] + }, + { + "reference": { + "prob_id": "1101g", + "tags": [ + "XOR Basis||" + ] + }, + "positive": { + "prob_id": "662a", + "tags": [ + "Vector Space and XOR Basis|Linear Algebra|Math" + ] + }, + "negative": [ + { + "prob_id": "1132g", + "tags": [ + "Segment Tree with Lazy Propagation|Segment Tree|Data Structures (DS)" + ] + }, + { + "prob_id": "1194e", + "tags": [ + "Sweep Line||" + ] + }, + { + "prob_id": "1042d", + "tags": [ + "Global Deltaing||Advanced Data Structures" + ] + } + ] + }, + { + "reference": { + "prob_id": "662a", + "tags": [ + "Vector Space and XOR Basis|Linear Algebra|Math" + ] + }, + "positive": { + "prob_id": "1101g", + "tags": [ + "XOR Basis||" + ] + }, + "negative": [ + { + "prob_id": "650d", + "tags": [ + "O(n lg n) Longest Monotonic Subsequence|DP Optimization|Advanced Algorithms" + ] + }, + { + "prob_id": "729f", + "tags": [ + "Skip Useless States|DP Optimization|Advanced Algorithms" + ] + }, + { + "prob_id": "576a", + "tags": [ + "Prime Factorization|Number Theory|Basics" + ] + } + ] + }, + { + "reference": { + "prob_id": "71e", + "tags": [ + "Bitmask DP|Bit Related DP|Dynamic Programming (DP)" + ] + }, + "positive": { + "prob_id": "11d", + "tags": [ + "DP on Subsets||Intermediate" + ] + }, + "negative": [ + { + "prob_id": "1012b", + "tags": [ + "Bipartite Graph||" + ] + }, + { + "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": "700c", + "tags": [ + "Finding Bridges in O(N+M)|Connected components, bridges, articulations points|Graphs" + ] + } + ] + }, + { + "reference": { + "prob_id": "757d", + "tags": [ + "Bitmask DP|Bit Related DP|Dynamic Programming (DP)" + ] + }, + "positive": { + "prob_id": "8c", + "tags": [ + "DP on Subsets||Intermediate" + ] + }, + "negative": [ + { + "prob_id": "1137b", + "tags": [ + "Z algorithm|String Matching|Strings" + ] + }, + { + "prob_id": "340c", + "tags": [ + "Binomial Coefficients|Fundamentals|Combinatorics" + ] + }, + { + "prob_id": "776f", + "tags": [ + "Centroid Decomposition|Centroid Decomposition|Data Structures (DS)" + ] + } + ] + }, + { + "reference": { + "prob_id": "8c", + "tags": [ + "DP on Subsets||Intermediate" + ] + }, + "positive": { + "prob_id": "757d", + "tags": [ + "Bitmask DP|Bit Related DP|Dynamic Programming (DP)" + ] + }, + "negative": [ + { + "prob_id": "761e", + "tags": [ + "Depth First Search|Graph traversal|Graphs" + ] + }, + { + "prob_id": "877d", + "tags": [ + "Grid Graphs|Graphs|Basics" + ] + }, + { + "prob_id": "827f", + "tags": [ + "Dijkstra's Algorithm|Shortest Paths|Graph Theory" + ] + } + ] + }, + { + "reference": { + "prob_id": "1109f", + "tags": [ + "Link Cut Tree (LCT)|Link Cut Tree (LCT)|Data Structures (DS)" + ] + }, + "positive": { + "prob_id": "117e", + "tags": [ + "Link Cut Tree (LCT)|Link Cut Tree (LCT)|Data Structures (DS)" + ] + }, + "negative": [ + { + "prob_id": "930c", + "tags": [ + "O(n lg n) Longest Monotonic Subsequence|DP Optimization|Advanced Algorithms" + ] + }, + { + "prob_id": "1416f", + "tags": [ + "Flow with Lower Bounds||" + ] + }, + { + "prob_id": "895e", + "tags": [ + "Segment Tree with Lazy Propagation|Segment Tree|Data Structures (DS)" + ] + } + ] + }, + { + "reference": { + "prob_id": "1172e", + "tags": [ + "Link Cut Tree (LCT)|Link Cut Tree (LCT)|Data Structures (DS)" + ] + }, + "positive": { + "prob_id": "1109f", + "tags": [ + "Link Cut Tree (LCT)|Link Cut Tree (LCT)|Data Structures (DS)" + ] + }, + "negative": [ + { + "prob_id": "960e", + "tags": [ + "DP on Trees and DAGs|Dp on Different Structures|Dynamic Programming (DP)" + ] + }, + { + "prob_id": "1394d", + "tags": [ + "DP on Trees and DAGs|Dp on Different Structures|Dynamic Programming (DP)" + ] + }, + { + "prob_id": "599e", + "tags": [ + "Sum Over Subsets DP (SOS DP)|Bit Related DP|Dynamic Programming (DP)" + ] + } + ] + }, + { + "reference": { + "prob_id": "117e", + "tags": [ + "Link Cut Tree (LCT)|Link Cut Tree (LCT)|Data Structures (DS)" + ] + }, + "positive": { + "prob_id": "1109f", + "tags": [ + "Link Cut Tree (LCT)|Link Cut Tree (LCT)|Data Structures (DS)" + ] + }, + "negative": [ + { + "prob_id": "977e", + "tags": [ + "Depth First Search (DFS)|Graph Traversal|Graph Theory" + ] + }, + { + "prob_id": "1176e", + "tags": [ + "Bipartite Graph||", + "Graph Traversal||" + ] + }, + { + "prob_id": "263d", + "tags": [ + "Cycle Detection|Cycles|Graph Theory" + ] + } + ] + }, + { + "reference": { + "prob_id": "348a", + "tags": [ + "Binary Search|Leveraging Monotonicity|Basics" + ] + }, + "positive": { + "prob_id": "1201c", + "tags": [ + "Binary Search|Leveraging Monotonicity|Basics" + ] + }, + "negative": [ + { + "prob_id": "319c", + "tags": [ + "Convex Hull Trick|DP Optimization|Advanced Algorithms" + ] + }, + { + "prob_id": "193b", + "tags": [ + "Backtracking|Miscellaneous|Miscellaneous" + ] + }, + { + "prob_id": "204e", + "tags": [ + "Suffix Array|Suffix Structures|Strings" + ] + } + ] + }, + { + "reference": { + "prob_id": "145e", + "tags": [ + "Segment Tree with Lazy Propagation|Segment Tree|Data Structures (DS)" + ] + }, + "positive": { + "prob_id": "52c", + "tags": [ + "Segment Tree with Lazy Propagation|Segment Tree|Data Structures (DS)" + ] + }, + "negative": [ + { + "prob_id": "121a", + "tags": [ + "All-Subsets / Binary Choice|Complete Search|Basics" + ] + }, + { + "prob_id": "96b", + "tags": [ + "All-Subsets / Binary Choice|Complete Search|Basics" + ] + }, + { + "prob_id": "95e", + "tags": [ + "Knapsack DP||" + ] + } + ] + }, + { + "reference": { + "prob_id": "498c", + "tags": [ + "Bipartite Matching|Network Flows|Advanced Algorithms" + ] + }, + "positive": { + "prob_id": "1139e", + "tags": [ + "Bipartite Matching|Network Flows|Advanced Algorithms", + "Maximum Flow / Minimum Cut|Flow and Min Cut|Graph Theory" + ] + }, + "negative": [ + { + "prob_id": "438d", + "tags": [ + "Segment Tree Beats|Segment Tree|Data Structures (DS)", + "Segment Tree Beats|Segment Tree|Data Structures (DS)" + ] + }, + { + "prob_id": "1408e", + "tags": [ + "Minimum Spanning Tree||" + ] + }, + { + "prob_id": "402e", + "tags": [ + "Matrix Exponentiation|Matrices|Math" + ] + } + ] + }, + { + "reference": { + "prob_id": "150e", + "tags": [ + "Centroid Decomposition|Centroid Decomposition|Data Structures (DS)" + ] + }, + "positive": { + "prob_id": "1140g", + "tags": [ + "Binary Lifting||Advanced Data Structures", + "Lowest Common Ancestor|Lowest common ancestor|Graphs" + ] + }, + "negative": [ + { + "prob_id": "788b", + "tags": [ + "Eulerian Path and Cycle|Eulerian Path|Graph Theory" + ] + }, + { + "prob_id": "794d", + "tags": [ + "Depth First Search|Graph traversal|Graphs" + ] + }, + { + "prob_id": "793d", + "tags": [ + "Directed Acyclic Graphs|Shortest Paths|Intermediate" + ] + } + ] + }, + { + "reference": { + "prob_id": "833d", + "tags": [ + "Centroid Decomposition|Centroid Decomposition|Data Structures (DS)" + ] + }, + "positive": { + "prob_id": "1140g", + "tags": [ + "Binary Lifting||Advanced Data Structures", + "Lowest Common Ancestor|Lowest common ancestor|Graphs" + ] + }, + "negative": [ + { + "prob_id": "721c", + "tags": [ + "Directed Acyclic Graphs|Shortest Paths|Intermediate", + "Rotate the Recurrence|DP Optimization|Advanced Algorithms" + ] + }, + { + "prob_id": "715b", + "tags": [ + "Dijkstra's Algorithm|Shortest Paths|Graph Theory" + ] + }, + { + "prob_id": "584b", + "tags": [ + "Rule of Sum, Rule of Product|Counting|Basics" + ] + } + ] + }, + { + "reference": { + "prob_id": "466e", + "tags": [ + "Level Ancestors and Lowest Common Ancestors||Advanced Data Structures" + ] + }, + "positive": { + "prob_id": "733f", + "tags": [ + "Lowest Common Ancestor|Lowest common ancestor|Graphs" + ] + }, + "negative": [ + { + "prob_id": "727e", + "tags": [ + "String Hashing|Hashing|Strings" + ] + }, + { + "prob_id": "631c", + "tags": [ + "Combining Data Structures|Data Structure Techniques|Intermediate" + ] + }, + { + "prob_id": "1572f", + "tags": [ + "Segment Tree Beats|Segment Tree|Data Structures (DS)", + "Segment Tree Beats|Segment Tree|Data Structures (DS)" + ] + } + ] + }, + { + "reference": { + "prob_id": "832d", + "tags": [ + "Lowest Common Ancestor|Lowest common ancestor|Graphs" + ] + }, + "positive": { + "prob_id": "1328e", + "tags": [ + "Binary Lifting||Advanced Data Structures" + ] + }, + "negative": [ + { + "prob_id": "1500b", + "tags": [ + "Chinese Remainder Theorem|Modular arithmetic|Algebra" + ] + }, + { + "prob_id": "623d", + "tags": [ + "Probabilities and Expected Values|Probabilities and Expected Values|Math" + ] + }, + { + "prob_id": "809d", + "tags": [ + "Splay Trees and Treaps|Augmenting Data Structures|Advanced Data Structures" + ] + } + ] + }, + { + "reference": { + "prob_id": "1225d", + "tags": [ + "Divisibility||" + ] + }, + "positive": { + "prob_id": "154b", + "tags": [ + "Sieve of Eratosthenes|Prime numbers|Algebra" + ] + }, + "negative": [ + { + "prob_id": "1364c", + "tags": [ + "Basic Constructive Algorithms|Greedy and Constructive|Basics" + ] + }, + { + "prob_id": "1176e", + "tags": [ + "Bipartite Graph||", + "Graph Traversal||" + ] + }, + { + "prob_id": "1272e", + "tags": [ + "Breadth First Search|Graph traversal|Graphs" + ] + } + ] + }, + { + "reference": { + "prob_id": "154b", + "tags": [ + "Sieve of Eratosthenes|Prime numbers|Algebra" + ] + }, + "positive": { + "prob_id": "546d", + "tags": [ + "Fast IO||" + ] + }, + "negative": [ + { + "prob_id": "119c", + "tags": [ + "DP with Solution Construction||Intermediate" + ] + }, + { + "prob_id": "1355f", + "tags": [ + "Interactive Problems|Interactive Problems|Miscellaneous" + ] + }, + { + "prob_id": "176e", + "tags": [ + "Level Ancestors and Lowest Common Ancestors||Advanced Data Structures" + ] + } + ] + }, + { + "reference": { + "prob_id": "1561d2", + "tags": [ + "Sieve, Prime Factorization and Divisors|Primes and Divisors|Number Theory" + ] + }, + "positive": { + "prob_id": "154b", + "tags": [ + "Sieve of Eratosthenes|Prime numbers|Algebra" + ] + }, + "negative": [ + { + "prob_id": "1093d", + "tags": [ + "Depth First Search (DFS)|Graph Traversal|Graph Theory" + ] + }, + { + "prob_id": "616c", + "tags": [ + "Grid Graphs|Graphs|Basics" + ] + }, + { + "prob_id": "540c", + "tags": [ + "Grid Graphs|Graphs|Basics" + ] + } + ] + }, + { + "reference": { + "prob_id": "222c", + "tags": [ + "Sieve, Prime Factorization and Divisors|Primes and Divisors|Number Theory" + ] + }, + "positive": { + "prob_id": "154b", + "tags": [ + "Sieve of Eratosthenes|Prime numbers|Algebra" + ] + }, + "negative": [ + { + "prob_id": "177d2", + "tags": [ + "Prefix Sums and Difference Arrays|Range Queries|Basics" + ] + }, + { + "prob_id": "286e", + "tags": [ + "Fast Fourier Transform (FFT)|Convolution|Math" + ] + }, + { + "prob_id": "715c", + "tags": [ + "Centroid Decomposition|Centroid Decomposition|Data Structures (DS)" + ] + } + ] + }, + { + "reference": { + "prob_id": "230b", + "tags": [ + "Prime and Composite, Primality Testing, Prime Sieves, Finding Divisors|Number Theory|Preliminaries" + ] + }, + "positive": { + "prob_id": "154b", + "tags": [ + "Sieve of Eratosthenes|Prime numbers|Algebra" + ] + }, + "negative": [ + { + "prob_id": "161d", + "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" + ] + } + ] + }, + { + "reference": { + "prob_id": "1523d", + "tags": [ + "Sum Over Subsets DP (SOS DP)|Bit Related DP|Dynamic Programming (DP)" + ] + }, + "positive": { + "prob_id": "165e", + "tags": [ + "Bit Manipulation|Bit Manipulation|Basics", + "Sum Over Subsets DP (SOS DP)|Bit Related DP|Dynamic Programming (DP)" + ] + }, + "negative": [ + { + "prob_id": "1446a", + "tags": [ + "Knapsack DP||" + ] + }, + { + "prob_id": "1552i", + "tags": [ + "PQ Tree|More Tree Data Structures|Data Structures (DS)" + ] + }, + { + "prob_id": "1240f", + "tags": [ + "Edge Coloring of Bipartite Graph|Graph Coloring|Graph Theory" + ] + } + ] + }, + { + "reference": { + "prob_id": "165e", + "tags": [ + "Bit Manipulation|Bit Manipulation|Basics", + "Sum Over Subsets DP (SOS DP)|Bit Related DP|Dynamic Programming (DP)" + ] + }, + "positive": { + "prob_id": "1523d", + "tags": [ + "Sum Over Subsets DP (SOS DP)|Bit Related DP|Dynamic Programming (DP)" + ] + }, + "negative": [ + { + "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": { + "prob_id": "600f", + "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": [ + "Greedy Sorting||", + "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": [ + { + "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": { + "prob_id": "553a", + "tags": [ + "Permutations with Repetition, Stars and Bars|Counting|Intermediate" + ] + }, + "negative": [ + { + "prob_id": "1476a", + "tags": [ + "Basic Constructive Algorithms|Greedy and Constructive|Basics" + ] + }, + { + "prob_id": "1364c", + "tags": [ + "Basic Constructive Algorithms|Greedy and Constructive|Basics" + ] + }, + { + "prob_id": "1093d", + "tags": [ + "Depth First Search (DFS)|Graph Traversal|Graph Theory" + ] + } + ] + }, + { + "reference": { + "prob_id": "553a", + "tags": [ + "Permutations with Repetition, Stars and Bars|Counting|Intermediate" + ] + }, + "positive": { + "prob_id": "1288c", + "tags": [ + "Stars and Bars|Basic Counting|Basics" + ] + }, + "negative": [ + { + "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" + ] + }, + { + "prob_id": "446a", + "tags": [ + "Classic Problems with Twists|DP|Intermediate" + ] + }, + { + "prob_id": "877f", + "tags": [ + "Square Root Decomposition||" + ] + } + ] + }, + { + "reference": { + "prob_id": "1547f", + "tags": [ + "Sparse Table|Sparse Table|Data Structures (DS)" + ] + }, + "positive": { + "prob_id": "514d", + "tags": [ + "Sparse Table|Sparse Table|Data Structures (DS)" + ] + }, + "negative": [ + { + "prob_id": "1513d", + "tags": [ + "Minimum Spanning Tree||" + ] + }, + { + "prob_id": "1513b", + "tags": [ + "Bit Manipulation|Bit Manipulation|Basics" + ] + }, + { + "prob_id": "1270c", + "tags": [ + "Bit Manipulation|Bit Manipulation|Basics" + ] + } + ] + }, + { + "reference": { + "prob_id": "1572f", + "tags": [ + "Segment Tree Beats|Segment Tree|Data Structures (DS)" + ] + }, + "positive": { + "prob_id": "453e", + "tags": [ + "Segment Tree Beats|Segment Tree|Data Structures (DS)" + ] + }, + "negative": [ + { + "prob_id": "1245d", + "tags": [ + "Minimum Spanning Tree||" + ] + }, + { + "prob_id": "1240f", + "tags": [ + "Edge Coloring of Bipartite Graph|Graph Coloring|Graph Theory" + ] + }, + { + "prob_id": "1277e", + "tags": [ + "DFS Tree|Graph Traversal|Graph Theory" + ] + } + ] + }, + { + "reference": { + "prob_id": "296c", + "tags": [ + "Prefix Sums and Difference Arrays|Range Queries|Basics" + ] + }, + "positive": { + "prob_id": "816b", + "tags": [ + "Prefix Sums and Difference Arrays|Range Queries|Basics", + "Prefix Sum|Range Queries Without Updates|Basics" + ] + }, + "negative": [ + { + "prob_id": "243a", + "tags": [ + "Bit Manipulation|Bit Manipulation|Basics" + ] + }, + { + "prob_id": "348c", + "tags": [ + "Square-Root Discrimination||Intermediate" + ] + }, + { + "prob_id": "295b", + "tags": [ + "All-Pairs|Shortest Paths|Intermediate" + ] + } + ] + }, + { + "reference": { + "prob_id": "827f", + "tags": [ + "Dijkstra's Algorithm|Shortest Paths|Graph Theory" + ] + }, + "positive": { + "prob_id": "843d", + "tags": [ + "Dijkstra's Algorithm|Shortest Paths|Graph Theory" + ] + }, + "negative": [ + { + "prob_id": "1000e", + "tags": [ + "Articulation Bridges and Bridge Tree|Graph Connectivity|Graph Theory" + ] + }, + { + "prob_id": "1100b", + "tags": [ + "Amortized Analysis||Intermediate" + ] + }, + { + "prob_id": "340c", + "tags": [ + "Binomial Coefficients|Fundamentals|Combinatorics" + ] + } + ] + }, + { + "reference": { + "prob_id": "843d", + "tags": [ + "Dijkstra's Algorithm|Shortest Paths|Graph Theory" + ] + }, + "positive": { + "prob_id": "827f", + "tags": [ + "Dijkstra's Algorithm|Shortest Paths|Graph Theory" + ] + }, + "negative": [ + { + "prob_id": "506d", + "tags": [ + "Splitting Objects into Light and Heavy / Case Processing|Square Root Decomposition|Data Structures (DS)" + ] + }, + { + "prob_id": "891c", + "tags": [ + "Disjoint Set Union||", + "Offline Deletion||" + ] + }, + { + "prob_id": "293e", + "tags": [ + "Centroid Decomposition|Centroid Decomposition|Data Structures (DS)" + ] + } + ] + }, + { + "reference": { + "prob_id": "786c", + "tags": [ + "Sqrt Decomposition|Trees|Data Structures" + ] + }, + "positive": { + "prob_id": "837g", + "tags": [ + "Persistent Segment Tree|Segment Tree|Data Structures (DS)" + ] + }, + "negative": [ + { + "prob_id": "786a", + "tags": [ + "Winning and Losing Positions, Strategy Stealing|Combinatorial Games|Advanced Algorithms" + ] + }, + { + "prob_id": "363b", + "tags": [ + "Prefix Sum|Range Queries Without Updates|Basics" + ] + }, + { + "prob_id": "546e", + "tags": [ + "Network Flows||Advanced Algorithms" + ] + } + ] + }, + { + "reference": { + "prob_id": "25c", + "tags": [ + "All-Pairs|Shortest Paths|Intermediate" + ] + }, + "positive": { + "prob_id": "33b", + "tags": [ + "Floyd Warshall|Shortest Paths|Graph Theory" + ] + }, + "negative": [ + { + "prob_id": "212a", + "tags": [ + "Edge Coloring of Bipartite Graph|Graph Coloring|Graph Theory" + ] + }, + { + "prob_id": "191c", + "tags": [ + "Level Ancestors and Lowest Common Ancestors||Advanced Data Structures" + ] + }, + { + "prob_id": "440d", + "tags": [ + "DP on Trees and DAGs|Dp on Different Structures|Dynamic Programming (DP)" + ] + } + ] + }, + { + "reference": { + "prob_id": "33b", + "tags": [ + "Floyd Warshall|Shortest Paths|Graph Theory" + ] + }, + "positive": { + "prob_id": "25c", + "tags": [ + "All-Pairs|Shortest Paths|Intermediate" + ] + }, + "negative": [ + { + "prob_id": "706c", + "tags": [ + "Common Patterns|DP|Intermediate" + ] + }, + { + "prob_id": "279b", + "tags": [ + "Two Pointers|Leveraging Monotonicity|Basics" + ] + }, + { + "prob_id": "30e", + "tags": [ + "Suffix Array|Suffix Structures|Strings" + ] + } + ] + }, + { + "reference": { + "prob_id": "808g", + "tags": [ + "Knuth Morris Pratt (KMP) and Prefix Automaton|String Matching|Strings" + ] + }, + "positive": { + "prob_id": "346b", + "tags": [ + "Knuth Morris Pratt (KMP) and Prefix Automaton|String Matching|Strings", + "Substring DP|Dp on Different Structures|Dynamic Programming (DP)" + ] + }, + "negative": [ + { + "prob_id": "204e", + "tags": [ + "Suffix Array|Suffix Structures|Strings" + ] + }, + { + "prob_id": "808b", + "tags": [ + "Sliding Window|Data Structure Techniques|Intermediate" + ] + }, + { + "prob_id": "191c", + "tags": [ + "Level Ancestors and Lowest Common Ancestors||Advanced Data Structures" + ] + } + ] + }, + { + "reference": { + "prob_id": "82a", + "tags": [ + "Sums and Asymptotics||Preliminaries" + ] + }, + "positive": { + "prob_id": "486a", + "tags": [ + "Sums and Asymptotics||Preliminaries" + ] + }, + "negative": [ + { + "prob_id": "1446a", + "tags": [ + "Knapsack DP||" + ] + }, + { + "prob_id": "1552i", + "tags": [ + "PQ Tree|More Tree Data Structures|Data Structures (DS)" + ] + }, + { + "prob_id": "727e", + "tags": [ + "String Hashing|Hashing|Strings" + ] + } + ] + }, + { + "reference": { + "prob_id": "318a", + "tags": [ + "Sums and Asymptotics||Preliminaries" + ] + }, + "positive": { + "prob_id": "486a", + "tags": [ + "Sums and Asymptotics||Preliminaries" + ] + }, + "negative": [ + { + "prob_id": "243a", + "tags": [ + "Bit Manipulation|Bit Manipulation|Basics" + ] + }, + { + "prob_id": "204a", + "tags": [ + "DP on Digits||Intermediate" + ] + }, + { + "prob_id": "96b", + "tags": [ + "All-Subsets / Binary Choice|Complete Search|Basics" + ] + } + ] + }, + { + "reference": { + "prob_id": "822a", + "tags": [ + "Greatest Common Divisor and Least Common Multiple|Number Theory|Basics" + ] + }, + "positive": { + "prob_id": "527a", + "tags": [ + "Greatest Common Divisor and Least Common Multiple|Number Theory|Basics" + ] + }, + "negative": [ + { + "prob_id": "822e", + "tags": [ + "Dynamic Programming||", + "Hashing||" + ] + }, + { + "prob_id": "1245f", + "tags": [ + "DP on Digits||Intermediate" + ] + }, + { + "prob_id": "961e", + "tags": [ + "BIT / Fenwick Tree|Binary Indexed Tree (BIT)|Data Structures (DS)" + ] + } + ] + }, + { + "reference": { + "prob_id": "527a", + "tags": [ + "Greatest Common Divisor and Least Common Multiple|Number Theory|Basics" + ] + }, + "positive": { + "prob_id": "822a", + "tags": [ + "Greatest Common Divisor and Least Common Multiple|Number Theory|Basics" + ] + }, + "negative": [ + { + "prob_id": "47d", + "tags": [ + "Backtracking|Miscellaneous|Miscellaneous" + ] + }, + { + "prob_id": "152c", + "tags": [ + "Rule of Sum, Rule of Product|Counting|Basics" + ] + }, + { + "prob_id": "895e", + "tags": [ + "Segment Tree with Lazy Propagation|Segment Tree|Data Structures (DS)" + ] + } + ] + }, + { + "reference": { + "prob_id": "284a", + "tags": [ + "Divisibility, Modular Arithmetic|Number Theory|Basics" + ] + }, + "positive": { + "prob_id": "375a", + "tags": [ + "Divisibility, Modular Arithmetic|Number Theory|Basics" + ] + }, + "negative": [ + { + "prob_id": "468d", + "tags": [ + "Bipartite Matching|Network Flows|Advanced Algorithms" + ] + }, + { + "prob_id": "802c", + "tags": [ + "Min Cost Max Flow (MCMF)|Flow and Min Cut|Graph Theory" + ] + }, + { + "prob_id": "547b", + "tags": [ + "Stack||", + "Stacks||" + ] + } + ] + }, + { + "reference": { + "prob_id": "793a", + "tags": [ + "Divisibility, Modular Arithmetic|Number Theory|Basics" + ] + }, + "positive": { + "prob_id": "742a", + "tags": [ + "Divisibility, Modular Arithmetic|Number Theory|Basics" + ] + }, + "negative": [ + { + "prob_id": "729f", + "tags": [ + "Skip Useless States|DP Optimization|Advanced Algorithms" + ] + }, + { + "prob_id": "1154e", + "tags": [ + "Lists, Stacks, Queues|Basic Data Structures|Basics" + ] + }, + { + "prob_id": "1151c", + "tags": [ + "Sums and Asymptotics||Preliminaries" + ] + } + ] + }, + { + "reference": { + "prob_id": "495b", + "tags": [ + "Divisibility, Modular Arithmetic|Number Theory|Basics" + ] + }, + "positive": { + "prob_id": "375a", + "tags": [ + "Divisibility, Modular Arithmetic|Number Theory|Basics" + ] + }, + "negative": [ + { + "prob_id": "494b", + "tags": [ + "Knuth Morris Pratt (KMP) and Prefix Automaton|String Matching|Strings" + ] + }, + { + "prob_id": "507d", + "tags": [ + "Counting with DP||Intermediate" + ] + }, + { + "prob_id": "214a", + "tags": [ + "Iterative|Complete Search|Basics" + ] + } + ] + }, + { + "reference": { + "prob_id": "276c", + "tags": [ + "Prefix Sums and Difference Arrays|Range Queries|Basics", + "Prefix Sum|Range Queries Without Updates|Basics" + ] + }, + "positive": { + "prob_id": "18c", + "tags": [ + "Prefix Sums and Difference Arrays|Range Queries|Basics" + ] + }, + "negative": [ + { + "prob_id": "204e", + "tags": [ + "Suffix Array|Suffix Structures|Strings" + ] + }, + { + "prob_id": "243a", + "tags": [ + "Bit Manipulation|Bit Manipulation|Basics" + ] + }, + { + "prob_id": "455d", + "tags": [ + "Splay Trees and Treaps|Augmenting Data Structures|Advanced Data Structures" + ] + } + ] + }, + { + "reference": { + "prob_id": "816b", + "tags": [ + "Prefix Sums and Difference Arrays|Range Queries|Basics", + "Prefix Sum|Range Queries Without Updates|Basics" + ] + }, + "positive": { + "prob_id": "18c", + "tags": [ + "Prefix Sums and Difference Arrays|Range Queries|Basics" + ] + }, + "negative": [ + { + "prob_id": "877f", + "tags": [ + "Square Root Decomposition||" + ] + }, + { + "prob_id": "815c", + "tags": [ + "DP on Trees and DAGs|Dp on Different Structures|Dynamic Programming (DP)" + ] + }, + { + "prob_id": "587e", + "tags": [ + "Vector Space and XOR Basis|Linear Algebra|Math" + ] + } + ] + }, + { + "reference": { + "prob_id": "18c", + "tags": [ + "Prefix Sums and Difference Arrays|Range Queries|Basics" + ] + }, + "positive": { + "prob_id": "816b", + "tags": [ + "Prefix Sums and Difference Arrays|Range Queries|Basics", + "Prefix Sum|Range Queries Without Updates|Basics" + ] + }, + "negative": [ + { + "prob_id": "464b", + "tags": [ + "Backtracking|Miscellaneous|Miscellaneous", + "All-Permutations|Complete Search|Basics" + ] + }, + { + "prob_id": "817c", + "tags": [ + "Binary Search|Leveraging Monotonicity|Basics" + ] + }, + { + "prob_id": "849b", + "tags": [ + "Basics|Geometry|Basics" + ] + } + ] + }, + { + "reference": { + "prob_id": "877b", + "tags": [ + "Prefix Sums and Difference Arrays|Range Queries|Basics" + ] + }, + "positive": { + "prob_id": "983b", + "tags": [ + "Prefix Sums and Difference Arrays|Range Queries|Basics" + ] + }, + "negative": [ + { + "prob_id": "149e", + "tags": [ + "Z algorithm|String Matching|Strings" + ] + }, + { + "prob_id": "884f", + "tags": [ + "Min Cost Max Flow (MCMF)|Flow and Min Cut|Graph Theory" + ] + }, + { + "prob_id": "825f", + "tags": [ + "Z algorithm|String Matching|Strings" + ] + } + ] + }, + { + "reference": { + "prob_id": "611c", + "tags": [ + "Prefix Sums and Difference Arrays|Range Queries|Basics" + ] + }, + "positive": { + "prob_id": "983b", + "tags": [ + "Prefix Sums and Difference Arrays|Range Queries|Basics" + ] + }, + "negative": [ + { + "prob_id": "679c", + "tags": [ + "Sliding Window|Data Structure Techniques|Intermediate" + ] + }, + { + "prob_id": "60b", + "tags": [ + "Grid Graphs|Graphs|Basics" + ] + }, + { + "prob_id": "377a", + "tags": [ + "Grid Graphs|Graphs|Basics" + ] + } + ] + }, + { + "reference": { + "prob_id": "983b", + "tags": [ + "Prefix Sums and Difference Arrays|Range Queries|Basics" + ] + }, + "positive": { + "prob_id": "877b", + "tags": [ + "Prefix Sums and Difference Arrays|Range Queries|Basics" + ] + }, + "negative": [ + { + "prob_id": "1270c", + "tags": [ + "Bit Manipulation|Bit Manipulation|Basics" + ] + }, + { + "prob_id": "1176e", + "tags": [ + "Bipartite Graph||", + "Graph Traversal||" + ] + }, + { + "prob_id": "1513d", + "tags": [ + "Minimum Spanning Tree||" + ] + } + ] + }, + { + "reference": { + "prob_id": "340c", + "tags": [ + "Binomial Coefficients|Fundamentals|Combinatorics" + ] + }, + "positive": { + "prob_id": "131c", + "tags": [ + "Binomial Coefficients|Fundamentals|Combinatorics" + ] + }, + "negative": [ + { + "prob_id": "821e", + "tags": [ + "Matrix Exponentiation|Matrices|Math" + ] + }, + { + "prob_id": "689b", + "tags": [ + "Breadth First Search|Graph traversal|Graphs" + ] + }, + { + "prob_id": "1083a", + "tags": [ + "DP on Trees and DAGs|Dp on Different Structures|Dynamic Programming (DP)" + ] + } + ] + }, + { + "reference": { + "prob_id": "467b", + "tags": [ + "Bit Manipulation|Bit Manipulation|Basics" + ] + }, + "positive": { + "prob_id": "893b", + "tags": [ + "Bit Manipulation|Bit Manipulation|Basics" + ] + }, + "negative": [ + { + "prob_id": "467d", + "tags": [ + "Depth-First Traversal, Connectivity, Cycle Detection|Graphs|Basics" + ] + }, + { + "prob_id": "387e", + "tags": [ + "Square-Root Decomposition, Segment Trees, Binary-Indexed Trees|Range Queries|Intermediate" + ] + }, + { + "prob_id": "348a", + "tags": [ + "Binary Search|Leveraging Monotonicity|Basics" + ] + } + ] + }, + { + "reference": { + "prob_id": "988d", + "tags": [ + "Bit Manipulation|Bit Manipulation|Basics" + ] + }, + "positive": { + "prob_id": "424c", + "tags": [ + "Bit Manipulation|Bit Manipulation|Basics" + ] + }, + "negative": [ + { + "prob_id": "1176e", + "tags": [ + "Bipartite Graph||", + "Graph Traversal||" + ] + }, + { + "prob_id": "1446a", + "tags": [ + "Knapsack DP||" + ] + }, + { + "prob_id": "1439b", + "tags": [ + "Maximum Clique|Cliques and Independent Sets|Graph Theory" + ] + } + ] + }, + { + "reference": { + "prob_id": "424c", + "tags": [ + "Bit Manipulation|Bit Manipulation|Basics" + ] + }, + "positive": { + "prob_id": "988d", + "tags": [ + "Bit Manipulation|Bit Manipulation|Basics" + ] + }, + "negative": [ + { + "prob_id": "193b", + "tags": [ + "Backtracking|Miscellaneous|Miscellaneous" + ] + }, + { + "prob_id": "242e", + "tags": [ + "Segment Tree with Lazy Propagation|Segment Tree|Data Structures (DS)" + ] + }, + { + "prob_id": "691e", + "tags": [ + "Matrix Exponentiation|Matrices|Math" + ] + } + ] + }, + { + "reference": { + "prob_id": "243a", + "tags": [ + "Bit Manipulation|Bit Manipulation|Basics" + ] + }, + "positive": { + "prob_id": "484a", + "tags": [ + "Bit Manipulation|Bit Manipulation|Basics" + ] + }, + "negative": [ + { + "prob_id": "193b", + "tags": [ + "Backtracking|Miscellaneous|Miscellaneous" + ] + }, + { + "prob_id": "296c", + "tags": [ + "Prefix Sums and Difference Arrays|Range Queries|Basics" + ] + }, + { + "prob_id": "371c", + "tags": [ + "Binary Search|Leveraging Monotonicity|Basics" + ] + } + ] + }, + { + "reference": { + "prob_id": "193b", + "tags": [ + "Backtracking|Miscellaneous|Miscellaneous" + ] + }, + "positive": { + "prob_id": "47d", + "tags": [ + "Backtracking|Miscellaneous|Miscellaneous" + ] + }, + "negative": [ + { + "prob_id": "242e", + "tags": [ + "Segment Tree with Lazy Propagation|Segment Tree|Data Structures (DS)" + ] + }, + { + "prob_id": "243a", + "tags": [ + "Bit Manipulation|Bit Manipulation|Basics" + ] + }, + { + "prob_id": "348a", + "tags": [ + "Binary Search|Leveraging Monotonicity|Basics" + ] + } + ] + }, + { + "reference": { + "prob_id": "47d", + "tags": [ + "Backtracking|Miscellaneous|Miscellaneous" + ] + }, + "positive": { + "prob_id": "292c", + "tags": [ + "Backtracking|Miscellaneous|Miscellaneous" + ] + }, + "negative": [ + { + "prob_id": "659a", + "tags": [ + "Divisibility, Modular Arithmetic|Number Theory|Basics" + ] + }, + { + "prob_id": "895e", + "tags": [ + "Segment Tree with Lazy Propagation|Segment Tree|Data Structures (DS)" + ] + }, + { + "prob_id": "576a", + "tags": [ + "Prime Factorization|Number Theory|Basics" + ] + } + ] + }, + { + "reference": { + "prob_id": "535b", + "tags": [ + "All-Subsets / Binary Choice|Complete Search|Basics" + ] + }, + "positive": { + "prob_id": "550b", + "tags": [ + "All-Subsets / Binary Choice|Complete Search|Basics", + "Backtracking|Miscellaneous|Miscellaneous" + ] + }, + "negative": [ + { + "prob_id": "535e", + "tags": [ + "Convex Hull Trick|DP Optimization|Advanced Algorithms" + ] + }, + { + "prob_id": "547c", + "tags": [ + "Principle of Inclusion-Exclusion|Counting|Intermediate", + "M\u00f6bius Inversion||Advanced Mathematics" + ] + }, + { + "prob_id": "95e", + "tags": [ + "Knapsack DP||" + ] + } + ] + }, + { + "reference": { + "prob_id": "550b", + "tags": [ + "All-Subsets / Binary Choice|Complete Search|Basics", + "Backtracking|Miscellaneous|Miscellaneous" + ] + }, + "positive": { + "prob_id": "535b", + "tags": [ + "All-Subsets / Binary Choice|Complete Search|Basics" + ] + }, + "negative": [ + { + "prob_id": "1100b", + "tags": [ + "Amortized Analysis||Intermediate" + ] + }, + { + "prob_id": "1295f", + "tags": [ + "Probabilities and Expected Values|Probabilities and Expected Values|Math" + ] + }, + { + "prob_id": "877f", + "tags": [ + "Square Root Decomposition||" + ] + } + ] + }, + { + "reference": { + "prob_id": "812b", + "tags": [ + "All-Subsets / Binary Choice|Complete Search|Basics" + ] + }, + "positive": { + "prob_id": "96b", + "tags": [ + "All-Subsets / Binary Choice|Complete Search|Basics" + ] + }, + "negative": [ + { + "prob_id": "294c", + "tags": [ + "Permutations with Repetition, Stars and Bars|Counting|Intermediate" + ] + }, + { + "prob_id": "821d", + "tags": [ + "Breadth First Search|Graph traversal|Graphs" + ] + }, + { + "prob_id": "147b", + "tags": [ + "Matrix Exponentiation|Matrices|Math" + ] + } + ] + }, + { + "reference": { + "prob_id": "114b", + "tags": [ + "All-Subsets / Binary Choice|Complete Search|Basics" + ] + }, + "positive": { + "prob_id": "244b", + "tags": [ + "All-Subsets / Binary Choice|Complete Search|Basics" + ] + }, + "negative": [ + { + "prob_id": "895e", + "tags": [ + "Segment Tree with Lazy Propagation|Segment Tree|Data Structures (DS)" + ] + }, + { + "prob_id": "107b", + "tags": [ + "Basics|Probability and Expectation|Basics" + ] + }, + { + "prob_id": "727e", + "tags": [ + "String Hashing|Hashing|Strings" + ] + } + ] + }, + { + "reference": { + "prob_id": "431b", + "tags": [ + "All-Permutations|Complete Search|Basics" + ] + }, + "positive": { + "prob_id": "124b", + "tags": [ + "All-Permutations|Complete Search|Basics" + ] + }, + "negative": [ + { + "prob_id": "832d", + "tags": [ + "Lowest Common Ancestor|Lowest common ancestor|Graphs" + ] + }, + { + "prob_id": "321e", + "tags": [ + "Divide and Conquer DP|DP optimizations|Dynamic Programming", + "Divide and Conquer DP|DP optimizations|Dynamic Programming" + ] + }, + { + "prob_id": "191c", + "tags": [ + "Level Ancestors and Lowest Common Ancestors||Advanced Data Structures" + ] + } + ] + }, + { + "reference": { + "prob_id": "908b", + "tags": [ + "All-Permutations|Complete Search|Basics" + ] + }, + "positive": { + "prob_id": "431b", + "tags": [ + "All-Permutations|Complete Search|Basics" + ] + }, + "negative": [ + { + "prob_id": "377a", + "tags": [ + "Grid Graphs|Graphs|Basics" + ] + }, + { + "prob_id": "1063b", + "tags": [ + "0/1 BFS|Shortest Paths|Graph Theory" + ] + }, + { + "prob_id": "1394d", + "tags": [ + "DP on Trees and DAGs|Dp on Different Structures|Dynamic Programming (DP)" + ] + } + ] + }, + { + "reference": { + "prob_id": "124b", + "tags": [ + "All-Permutations|Complete Search|Basics" + ] + }, + "positive": { + "prob_id": "431b", + "tags": [ + "All-Permutations|Complete Search|Basics" + ] + }, + "negative": [ + { + "prob_id": "1476a", + "tags": [ + "Basic Constructive Algorithms|Greedy and Constructive|Basics" + ] + }, + { + "prob_id": "1096g", + "tags": [ + "Fast Fourier Transform (FFT)|Convolution|Math" + ] + }, + { + "prob_id": "1439b", + "tags": [ + "Maximum Clique|Cliques and Independent Sets|Graph Theory" + ] + } + ] + }, + { + "reference": { + "prob_id": "702c", + "tags": [ + "Binary Search|Leveraging Monotonicity|Basics", + "Two Pointers|Leveraging Monotonicity|Basics" + ] + }, + "positive": { + "prob_id": "251a", + "tags": [ + "Binary Search|Leveraging Monotonicity|Basics" + ] + }, + "negative": [ + { + "prob_id": "546e", + "tags": [ + "Network Flows||Advanced Algorithms" + ] + }, + { + "prob_id": "707b", + "tags": [ + "Single Source|Shortest Paths|Intermediate" + ] + }, + { + "prob_id": "1572f", + "tags": [ + "Segment Tree Beats|Segment Tree|Data Structures (DS)", + "Segment Tree Beats|Segment Tree|Data Structures (DS)" + ] + } + ] + }, + { + "reference": { + "prob_id": "251a", + "tags": [ + "Binary Search|Leveraging Monotonicity|Basics" + ] + }, + "positive": { + "prob_id": "702c", + "tags": [ + "Binary Search|Leveraging Monotonicity|Basics", + "Two Pointers|Leveraging Monotonicity|Basics" + ] + }, + "negative": [ + { + "prob_id": "121a", + "tags": [ + "All-Subsets / Binary Choice|Complete Search|Basics" + ] + }, + { + "prob_id": "113b", + "tags": [ + "Hashing||Intermediate" + ] + }, + { + "prob_id": "243a", + "tags": [ + "Bit Manipulation|Bit Manipulation|Basics" + ] + } + ] + }, + { + "reference": { + "prob_id": "701c", + "tags": [ + "Two Pointers|Leveraging Monotonicity|Basics" + ] + }, + "positive": { + "prob_id": "1073c", + "tags": [ + "Two Pointers|Leveraging Monotonicity|Basics" + ] + }, + "negative": [ + { + "prob_id": "739e", + "tags": [ + "Aliens Trick / WQS Binary Search|DP Optimizations|Dynamic Programming (DP)" + ] + }, + { + "prob_id": "332b", + "tags": [ + "Classic Problems with Twists|DP|Intermediate" + ] + }, + { + "prob_id": "190e", + "tags": [ + "Inverse Graph|Graph Traversal|Graph Theory" + ] + } + ] + }, + { + "reference": { + "prob_id": "1073c", + "tags": [ + "Two Pointers|Leveraging Monotonicity|Basics" + ] + }, + "positive": { + "prob_id": "701c", + "tags": [ + "Two Pointers|Leveraging Monotonicity|Basics" + ] + }, + "negative": [ + { + "prob_id": "895e", + "tags": [ + "Segment Tree with Lazy Propagation|Segment Tree|Data Structures (DS)" + ] + }, + { + "prob_id": "152e", + "tags": [ + "DP on Subsets||Intermediate" + ] + }, + { + "prob_id": "1016c", + "tags": [ + "Prefix Sums and Difference Arrays|Range Queries|Basics" + ] + } + ] + }, + { + "reference": { + "prob_id": "922d", + "tags": [ + "Exchange Argument|Greedy|Basics" + ] + }, + "positive": { + "prob_id": "1203f1", + "tags": [ + "Exchange Argument|Greedy|Basics" + ] + }, + "negative": [ + { + "prob_id": "363b", + "tags": [ + "Prefix Sum|Range Queries Without Updates|Basics" + ] + }, + { + "prob_id": "798b", + "tags": [ + "Iterative|Complete Search|Basics" + ] + }, + { + "prob_id": "1303e", + "tags": [ + "Interval DP / Range DP / Matrix Chain Multiplication (MCM)|Classic DP|Dynamic Programming (DP)" + ] + } + ] + }, + { + "reference": { + "prob_id": "1203f1", + "tags": [ + "Exchange Argument|Greedy|Basics" + ] + }, + "positive": { + "prob_id": "922d", + "tags": [ + "Exchange Argument|Greedy|Basics" + ] + }, + "negative": [ + { + "prob_id": "864e", + "tags": [ + "Presorting|DP Common Tricks|Intermediate" + ] + }, + { + "prob_id": "961e", + "tags": [ + "BIT / Fenwick Tree|Binary Indexed Tree (BIT)|Data Structures (DS)" + ] + }, + { + "prob_id": "808b", + "tags": [ + "Sliding Window|Data Structure Techniques|Intermediate" + ] + } + ] + }, + { + "reference": { + "prob_id": "1283e", + "tags": [ + "Greedy Stays Ahead|Greedy|Basics", + "Structural Argument|Greedy|Basics" + ] + }, + "positive": { + "prob_id": "979b", + "tags": [ + "Structural Argument|Greedy|Basics" + ] + }, + "negative": [ + { + "prob_id": "1552i", + "tags": [ + "PQ Tree|More Tree Data Structures|Data Structures (DS)" + ] + }, + { + "prob_id": "500a", + "tags": [ + "Depth-First Traversal, Connectivity, Cycle Detection|Graphs|Basics" + ] + }, + { + "prob_id": "780b", + "tags": [ + "Binary Search|Leveraging Monotonicity|Basics", + "Ternary Search|Search Techniques|Miscellaneous" + ] + } + ] + }, + { + "reference": { + "prob_id": "814d", + "tags": [ + "Trees and Tree Traversals|Graphs|Basics" + ] + }, + "positive": { + "prob_id": "219d", + "tags": [ + "Trees and Tree Traversals|Graphs|Basics" + ] + }, + "negative": [ + { + "prob_id": "704d", + "tags": [ + "Flow with Lower Bounds||" + ] + }, + { + "prob_id": "212a", + "tags": [ + "Edge Coloring of Bipartite Graph|Graph Coloring|Graph Theory" + ] + }, + { + "prob_id": "621c", + "tags": [ + "Basics|Probability and Expectation|Basics" + ] + } + ] + }, + { + "reference": { + "prob_id": "1205b", + "tags": [ + "Breadth First Search|Graph traversal|Graphs" + ] + }, + "positive": { + "prob_id": "601a", + "tags": [ + "Breadth First Search|Graph traversal|Graphs" + ] + }, + "negative": [ + { + "prob_id": "1416c", + "tags": [ + "Trie on Strings and Bits|Trie|Data Structures (DS)" + ] + }, + { + "prob_id": "1132g", + "tags": [ + "Segment Tree with Lazy Propagation|Segment Tree|Data Structures (DS)" + ] + }, + { + "prob_id": "1527a", + "tags": [ + "Bit Manipulation|Bit Manipulation|Basics" + ] + } + ] + }, + { + "reference": { + "prob_id": "689b", + "tags": [ + "Breadth First Search|Graph traversal|Graphs" + ] + }, + "positive": { + "prob_id": "1037d", + "tags": [ + "Breadth First Search|Graph traversal|Graphs" + ] + }, + "negative": [ + { + "prob_id": "340c", + "tags": [ + "Binomial Coefficients|Fundamentals|Combinatorics" + ] + }, + { + "prob_id": "587c", + "tags": [ + "Level Ancestors and Lowest Common Ancestors||Advanced Data Structures" + ] + }, + { + "prob_id": "547c", + "tags": [ + "Principle of Inclusion-Exclusion|Counting|Intermediate", + "M\u00f6bius Inversion||Advanced Mathematics" + ] + } + ] + }, + { + "reference": { + "prob_id": "377a", + "tags": [ + "Grid Graphs|Graphs|Basics" + ] + }, + "positive": { + "prob_id": "329b", + "tags": [ + "Grid Graphs|Graphs|Basics" + ] + }, + "negative": [ + { + "prob_id": "611c", + "tags": [ + "Prefix Sums and Difference Arrays|Range Queries|Basics" + ] + }, + { + "prob_id": "225c", + "tags": [ + "Common Patterns|DP|Intermediate" + ] + }, + { + "prob_id": "1404e", + "tags": [ + "Maximum Flow / Minimum Cut|Flow and Min Cut|Graph Theory" + ] + } + ] + }, + { + "reference": { + "prob_id": "510b", + "tags": [ + "Grid Graphs|Graphs|Basics" + ] + }, + "positive": { + "prob_id": "329b", + "tags": [ + "Grid Graphs|Graphs|Basics" + ] + }, + "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": "31d", + "tags": [ + "Grid Graphs|Graphs|Basics" + ] + }, + "positive": { + "prob_id": "540c", + "tags": [ + "Grid Graphs|Graphs|Basics" + ] + }, + "negative": [ + { + "prob_id": "617b", + "tags": [ + "Permutations with Repetition, Stars and Bars|Counting|Intermediate" + ] + }, + { + "prob_id": "552d", + "tags": [ + "Basics|Geometry|Basics" + ] + }, + { + "prob_id": "1081d", + "tags": [ + "Minimum Spanning Tree||", + "Centroid Decomposition|Centroid Decomposition|Data Structures (DS)" + ] + } + ] + }, + { + "reference": { + "prob_id": "975d", + "tags": [ + "Vectors and Applications|Geometry|Basics" + ] + }, + "positive": { + "prob_id": "850a", + "tags": [ + "Vectors and Applications|Geometry|Basics" + ] + }, + "negative": [ + { + "prob_id": "1394d", + "tags": [ + "DP on Trees and DAGs|Dp on Different Structures|Dynamic Programming (DP)" + ] + }, + { + "prob_id": "1109f", + "tags": [ + "Link Cut Tree (LCT)|Link Cut Tree (LCT)|Data Structures (DS)" + ] + }, + { + "prob_id": "1252l", + "tags": [ + "Network Flows||Advanced Algorithms" + ] + } + ] + }, + { + "reference": { + "prob_id": "371c", + "tags": [ + "Binary Search|Leveraging Monotonicity|Basics" + ] + }, + "positive": { + "prob_id": "448d", + "tags": [ + "Binary Search|Leveraging Monotonicity|Basics" + ] + }, + "negative": [ + { + "prob_id": "243a", + "tags": [ + "Bit Manipulation|Bit Manipulation|Basics" + ] + }, + { + "prob_id": "282e", + "tags": [ + "Trie on Strings and Bits|Trie|Data Structures (DS)" + ] + }, + { + "prob_id": "842b", + "tags": [ + "Basics|Geometry|Basics" + ] + } + ] + }, + { + "reference": { + "prob_id": "580b", + "tags": [ + "Binary Search|Leveraging Monotonicity|Basics" + ] + }, + "positive": { + "prob_id": "701c", + "tags": [ + "Two Pointers|Leveraging Monotonicity|Basics" + ] + }, + "negative": [ + { + "prob_id": "580d", + "tags": [ + "DP on Subsets||Intermediate" + ] + }, + { + "prob_id": "580c", + "tags": [ + "Trees and Tree Traversals|Graphs|Basics" + ] + }, + { + "prob_id": "639a", + "tags": [ + "Priority Queues, Sets, Maps, Counters|Basic Data Structures|Basics" + ] + } + ] + }, + { + "reference": { + "prob_id": "701c", + "tags": [ + "Two Pointers|Leveraging Monotonicity|Basics" + ] + }, + "positive": { + "prob_id": "580b", + "tags": [ + "Binary Search|Leveraging Monotonicity|Basics" + ] + }, + "negative": [ + { + "prob_id": "739e", + "tags": [ + "Aliens Trick / WQS Binary Search|DP Optimizations|Dynamic Programming (DP)" + ] + }, + { + "prob_id": "332b", + "tags": [ + "Classic Problems with Twists|DP|Intermediate" + ] + }, + { + "prob_id": "190e", + "tags": [ + "Inverse Graph|Graph Traversal|Graph Theory" + ] + } + ] + }, + { + "reference": { + "prob_id": "778a", + "tags": [ + "Binary Search|Leveraging Monotonicity|Basics" + ] + }, + "positive": { + "prob_id": "803d", + "tags": [ + "Basic Greedy Algorithms|Greedy and Constructive|Basics", + "Binary Search|Leveraging Monotonicity|Basics", + "Two Pointers|Leveraging Monotonicity|Basics" + ] + }, + "negative": [ + { + "prob_id": "305e", + "tags": [ + "Nim, Sprague-Grundy Theorem|Combinatorial Games|Advanced Mathematics" + ] + }, + { + "prob_id": "785e", + "tags": [ + "2D BIT|Binary Indexed Tree (BIT)|Data Structures (DS)" + ] + }, + { + "prob_id": "204e", + "tags": [ + "Suffix Array|Suffix Structures|Strings" + ] + } + ] + }, + { + "reference": { + "prob_id": "780b", + "tags": [ + "Binary Search|Leveraging Monotonicity|Basics", + "Ternary Search|Search Techniques|Miscellaneous" + ] + }, + "positive": { + "prob_id": "448d", + "tags": [ + "Binary Search|Leveraging Monotonicity|Basics" + ] + }, + "negative": [ + { + "prob_id": "629e", + "tags": [ + "Binary Lifting||Advanced Data Structures" + ] + }, + { + "prob_id": "1283e", + "tags": [ + "Greedy Stays Ahead|Greedy|Basics", + "Structural Argument|Greedy|Basics" + ] + }, + { + "prob_id": "29e", + "tags": [ + "Breadth First Search|Graph traversal|Graphs" + ] + } + ] + }, + { + "reference": { + "prob_id": "1260d", + "tags": [ + "Binary Search|Leveraging Monotonicity|Basics" + ] + }, + "positive": { + "prob_id": "803d", + "tags": [ + "Basic Greedy Algorithms|Greedy and Constructive|Basics", + "Binary Search|Leveraging Monotonicity|Basics", + "Two Pointers|Leveraging Monotonicity|Basics" + ] + }, + "negative": [ + { + "prob_id": "540c", + "tags": [ + "Grid Graphs|Graphs|Basics" + ] + }, + { + "prob_id": "1000e", + "tags": [ + "Articulation Bridges and Bridge Tree|Graph Connectivity|Graph Theory" + ] + }, + { + "prob_id": "1240f", + "tags": [ + "Edge Coloring of Bipartite Graph|Graph Coloring|Graph Theory" + ] + } + ] + }, + { + "reference": { + "prob_id": "803d", + "tags": [ + "Basic Greedy Algorithms|Greedy and Constructive|Basics", + "Binary Search|Leveraging Monotonicity|Basics", + "Two Pointers|Leveraging Monotonicity|Basics" + ] + }, + "positive": { + "prob_id": "1260d", + "tags": [ + "Binary Search|Leveraging Monotonicity|Basics" + ] + }, + "negative": [ + { + "prob_id": "794d", + "tags": [ + "Depth First Search|Graph traversal|Graphs" + ] + }, + { + "prob_id": "191c", + "tags": [ + "Level Ancestors and Lowest Common Ancestors||Advanced Data Structures" + ] + }, + { + "prob_id": "456d", + "tags": [ + "Trie on Strings and Bits|Trie|Data Structures (DS)" + ] + } + ] + }, + { + "reference": { + "prob_id": "578c", + "tags": [ + "Ternary Search|Search Techniques|Miscellaneous" + ] + }, + "positive": { + "prob_id": "201b", + "tags": [ + "Ternary Search|Search Techniques|Miscellaneous" + ] + }, + "negative": [ + { + "prob_id": "486e", + "tags": [ + "O(n lg n) Longest Monotonic Subsequence|DP Optimization|Advanced Algorithms" + ] + }, + { + "prob_id": "280d", + "tags": [ + "Min Cost Max Flow (MCMF)|Flow and Min Cut|Graph Theory" + ] + }, + { + "prob_id": "868f", + "tags": [ + "Divide and Conquer DP|DP optimizations|Dynamic Programming", + "Divide and Conquer DP|DP optimizations|Dynamic Programming" + ] + } + ] + }, + { + "reference": { + "prob_id": "201b", + "tags": [ + "Ternary Search|Search Techniques|Miscellaneous" + ] + }, + "positive": { + "prob_id": "578c", + "tags": [ + "Ternary Search|Search Techniques|Miscellaneous" + ] + }, + "negative": [ + { + "prob_id": "793d", + "tags": [ + "Directed Acyclic Graphs|Shortest Paths|Intermediate" + ] + }, + { + "prob_id": "1407e", + "tags": [ + "Breadth First Search|Graph traversal|Graphs" + ] + }, + { + "prob_id": "66e", + "tags": [ + "Global Deltaing||Advanced Data Structures" + ] + } + ] + }, + { + "reference": { + "prob_id": "241e", + "tags": [ + "Single Source with Negative Weights|Shortest Paths|Intermediate" + ] + }, + "positive": { + "prob_id": "346d", + "tags": [ + "Single Source with Negative Weights|Shortest Paths|Intermediate" + ] + }, + "negative": [ + { + "prob_id": "505d", + "tags": [ + "Strongly Connected Components (SCC)|Graph Connectivity|Graph Theory" + ] + }, + { + "prob_id": "576d", + "tags": [ + "Matrix Exponentiation|Matrices|Math" + ] + }, + { + "prob_id": "330b", + "tags": [ + "Basic Graph Theory|Graphs|Basics" + ] + } + ] + }, + { + "reference": { + "prob_id": "25c", + "tags": [ + "All-Pairs|Shortest Paths|Intermediate" + ] + }, + "positive": { + "prob_id": "295b", + "tags": [ + "All-Pairs|Shortest Paths|Intermediate" + ] + }, + "negative": [ + { + "prob_id": "212a", + "tags": [ + "Edge Coloring of Bipartite Graph|Graph Coloring|Graph Theory" + ] + }, + { + "prob_id": "191c", + "tags": [ + "Level Ancestors and Lowest Common Ancestors||Advanced Data Structures" + ] + }, + { + "prob_id": "440d", + "tags": [ + "DP on Trees and DAGs|Dp on Different Structures|Dynamic Programming (DP)" + ] + } + ] + }, + { + "reference": { + "prob_id": "2b", + "tags": [ + "DP with Solution Construction||Intermediate" + ] + }, + "positive": { + "prob_id": "119c", + "tags": [ + "DP with Solution Construction||Intermediate" + ] + }, + "negative": [ + { + "prob_id": "243e", + "tags": [ + "PQ Tree|More Tree Data Structures|Data Structures (DS)" + ] + }, + { + "prob_id": "616c", + "tags": [ + "Grid Graphs|Graphs|Basics" + ] + }, + { + "prob_id": "1451f", + "tags": [ + "Nim, Sprague-Grundy Theorem|Combinatorial Games|Advanced Mathematics" + ] + } + ] + }, + { + "reference": { + "prob_id": "82d", + "tags": [ + "DP with Solution Construction||Intermediate" + ] + }, + "positive": { + "prob_id": "2b", + "tags": [ + "DP with Solution Construction||Intermediate" + ] + }, + "negative": [ + { + "prob_id": "1500b", + "tags": [ + "Chinese Remainder Theorem|Modular arithmetic|Algebra" + ] + }, + { + "prob_id": "1263c", + "tags": [ + "Square-Root Discrimination||Intermediate" + ] + }, + { + "prob_id": "412d", + "tags": [ + "Conclusion||", + "DFS||" + ] + } + ] + }, + { + "reference": { + "prob_id": "56d", + "tags": [ + "DP with Solution Construction||Intermediate" + ] + }, + "positive": { + "prob_id": "82d", + "tags": [ + "DP with Solution Construction||Intermediate" + ] + }, + "negative": [ + { + "prob_id": "1012d", + "tags": [ + "Tricky Implementation||Intermediate" + ] + }, + { + "prob_id": "1234d", + "tags": [ + "Segment Tree|Trees|Data Structures" + ] + }, + { + "prob_id": "610e", + "tags": [ + "Segment Tree with Lazy Propagation|Segment Tree|Data Structures (DS)" + ] + } + ] + }, + { + "reference": { + "prob_id": "119c", + "tags": [ + "DP with Solution Construction||Intermediate" + ] + }, + "positive": { + "prob_id": "2b", + "tags": [ + "DP with Solution Construction||Intermediate" + ] + }, + "negative": [ + { + "prob_id": "191c", + "tags": [ + "Level Ancestors and Lowest Common Ancestors||Advanced Data Structures" + ] + }, + { + "prob_id": "567e", + "tags": [ + "Dijkstra's Algorithm|Shortest Paths|Graph Theory" + ] + }, + { + "prob_id": "434d", + "tags": [ + "Network Flows||Advanced Algorithms" + ] + } + ] + }, + { + "reference": { + "prob_id": "580d", + "tags": [ + "DP on Subsets||Intermediate" + ] + }, + "positive": { + "prob_id": "8c", + "tags": [ + "DP on Subsets||Intermediate" + ] + }, + "negative": [ + { + "prob_id": "580c", + "tags": [ + "Trees and Tree Traversals|Graphs|Basics" + ] + }, + { + "prob_id": "580e", + "tags": [ + "Segment Tree with Lazy Propagation|Segment Tree|Data Structures (DS)" + ] + }, + { + "prob_id": "580b", + "tags": [ + "Binary Search|Leveraging Monotonicity|Basics" + ] + } + ] + }, + { + "reference": { + "prob_id": "16e", + "tags": [ + "DP on Subsets||Intermediate", + "Markov Chains|Probability and Expectation|Intermediate" + ] + }, + "positive": { + "prob_id": "855e", + "tags": [ + "DP on Digits||Intermediate", + "DP on Subsets||Intermediate" + ] + }, + "negative": [ + { + "prob_id": "402e", + "tags": [ + "Matrix Exponentiation|Matrices|Math" + ] + }, + { + "prob_id": "362e", + "tags": [ + "Bipartite Matching|Network Flows|Advanced Algorithms" + ] + }, + { + "prob_id": "117c", + "tags": [ + "Tournament Graph|Special Graphs|Graph Theory" + ] + } + ] + }, + { + "reference": { + "prob_id": "8c", + "tags": [ + "DP on Subsets||Intermediate" + ] + }, + "positive": { + "prob_id": "11d", + "tags": [ + "DP on Subsets||Intermediate" + ] + }, + "negative": [ + { + "prob_id": "761e", + "tags": [ + "Depth First Search|Graph traversal|Graphs" + ] + }, + { + "prob_id": "877d", + "tags": [ + "Grid Graphs|Graphs|Basics" + ] + }, + { + "prob_id": "827f", + "tags": [ + "Dijkstra's Algorithm|Shortest Paths|Graph Theory" + ] + } + ] + }, + { + "reference": { + "prob_id": "417d", + "tags": [ + "DP on Subsets||Intermediate" + ] + }, + "positive": { + "prob_id": "855e", + "tags": [ + "DP on Digits||Intermediate", + "DP on Subsets||Intermediate" + ] + }, + "negative": [ + { + "prob_id": "639a", + "tags": [ + "Priority Queues, Sets, Maps, Counters|Basic Data Structures|Basics" + ] + }, + { + "prob_id": "535e", + "tags": [ + "Convex Hull Trick|DP Optimization|Advanced Algorithms" + ] + }, + { + "prob_id": "1000e", + "tags": [ + "Articulation Bridges and Bridge Tree|Graph Connectivity|Graph Theory" + ] + } + ] + }, + { + "reference": { + "prob_id": "855e", + "tags": [ + "DP on Digits||Intermediate", + "DP on Subsets||Intermediate" + ] + }, + "positive": { + "prob_id": "11d", + "tags": [ + "DP on Subsets||Intermediate" + ] + }, + "negative": [ + { + "prob_id": "587e", + "tags": [ + "Vector Space and XOR Basis|Linear Algebra|Math" + ] + }, + { + "prob_id": "232d", + "tags": [ + "Suffix Array|Suffix Structures|Strings" + ] + }, + { + "prob_id": "816b", + "tags": [ + "Prefix Sum|Range Queries Without Updates|Basics", + "Prefix Sums and Difference Arrays|Range Queries|Basics" + ] + } + ] + }, + { + "reference": { + "prob_id": "53e", + "tags": [ + "DP on Subsets||Intermediate" + ] + }, + "positive": { + "prob_id": "678e", + "tags": [ + "DP on Subsets||Intermediate" + ] + }, + "negative": [ + { + "prob_id": "118e", + "tags": [ + "DFS Tree|Graph Traversal|Graph Theory" + ] + }, + { + "prob_id": "700c", + "tags": [ + "Finding Bridges in O(N+M)|Connected components, bridges, articulations points|Graphs" + ] + }, + { + "prob_id": "25d", + "tags": [ + "Disjoint Set Union||" + ] + } + ] + }, + { + "reference": { + "prob_id": "152e", + "tags": [ + "DP on Subsets||Intermediate" + ] + }, + "positive": { + "prob_id": "11d", + "tags": [ + "DP on Subsets||Intermediate" + ] + }, + "negative": [ + { + "prob_id": "1500b", + "tags": [ + "Chinese Remainder Theorem|Modular arithmetic|Algebra" + ] + }, + { + "prob_id": "704d", + "tags": [ + "Flow with Lower Bounds||" + ] + }, + { + "prob_id": "659a", + "tags": [ + "Divisibility, Modular Arithmetic|Number Theory|Basics" + ] + } + ] + }, + { + "reference": { + "prob_id": "296b", + "tags": [ + "Counting with DP||Intermediate", + "DP on Digits||Intermediate" + ] + }, + "positive": { + "prob_id": "1245f", + "tags": [ + "DP on Digits||Intermediate" + ] + }, + "negative": [ + { + "prob_id": "33b", + "tags": [ + "Floyd Warshall|Shortest Paths|Graph Theory" + ] + }, + { + "prob_id": "1085e", + "tags": [ + "Tree Centers and Diameter||Advanced Data Structures" + ] + }, + { + "prob_id": "898f", + "tags": [ + "Hashing||Intermediate" + ] + } + ] + }, + { + "reference": { + "prob_id": "1245f", + "tags": [ + "DP on Digits||Intermediate" + ] + }, + "positive": { + "prob_id": "1073e", + "tags": [ + "DP on Digits||Intermediate" + ] + }, + "negative": [ + { + "prob_id": "1362b", + "tags": [ + "Bit Manipulation|Bit Manipulation|Basics" + ] + }, + { + "prob_id": "1420b", + "tags": [ + "Bit Manipulation|Bit Manipulation|Basics" + ] + }, + { + "prob_id": "1304d", + "tags": [ + "Longest Increasing Subsequence (LIS)|Classic DP|Dynamic Programming (DP)" + ] + } + ] + }, + { + "reference": { + "prob_id": "580e", + "tags": [ + "Segment Tree with Lazy Propagation|Segment Tree|Data Structures (DS)" + ] + }, + "positive": { + "prob_id": "558e", + "tags": [ + "Segment Tree with Lazy Propagation|Segment Tree|Data Structures (DS)" + ] + }, + "negative": [ + { + "prob_id": "580d", + "tags": [ + "DP on Subsets||Intermediate" + ] + }, + { + "prob_id": "580c", + "tags": [ + "Trees and Tree Traversals|Graphs|Basics" + ] + }, + { + "prob_id": "243a", + "tags": [ + "Bit Manipulation|Bit Manipulation|Basics" + ] + } + ] + }, + { + "reference": { + "prob_id": "56e", + "tags": [ + "Amortized Analysis||Intermediate" + ] + }, + "positive": { + "prob_id": "47e", + "tags": [ + "Amortized Analysis||Intermediate" + ] + }, + "negative": [ + { + "prob_id": "47d", + "tags": [ + "Backtracking|Miscellaneous|Miscellaneous" + ] + }, + { + "prob_id": "1143f", + "tags": [ + "Convex Hull Trick|DP Optimization|Advanced Algorithms" + ] + }, + { + "prob_id": "1016c", + "tags": [ + "Prefix Sums and Difference Arrays|Range Queries|Basics" + ] + } + ] + }, + { + "reference": { + "prob_id": "1311d", + "tags": [ + "Harmonic Lemma||Intermediate" + ] + }, + "positive": { + "prob_id": "632d", + "tags": [ + "Divisor Sieve|Number Theory|Intermediate", + "Harmonic Lemma||Intermediate" + ] + }, + "negative": [ + { + "prob_id": "1328a", + "tags": [ + "Big O Notation and Time and Space Complexity Analysis|Complexity Analysis|Basics" + ] + }, + { + "prob_id": "1476a", + "tags": [ + "Basic Constructive Algorithms|Greedy and Constructive|Basics" + ] + }, + { + "prob_id": "1439b", + "tags": [ + "Maximum Clique|Cliques and Independent Sets|Graph Theory" + ] + } + ] + }, + { + "reference": { + "prob_id": "633c", + "tags": [ + "Trie||" + ] + }, + "positive": { + "prob_id": "126b", + "tags": [ + "Z algorithm|String Matching|Strings" + ] + }, + "negative": [ + { + "prob_id": "704d", + "tags": [ + "Flow with Lower Bounds||" + ] + }, + { + "prob_id": "739e", + "tags": [ + "Aliens Trick / WQS Binary Search|DP Optimizations|Dynamic Programming (DP)" + ] + }, + { + "prob_id": "114b", + "tags": [ + "All-Subsets / Binary Choice|Complete Search|Basics" + ] + } + ] + }, + { + "reference": { + "prob_id": "1105e", + "tags": [ + "Meet in the Middle|Techniques|Miscellaneous" + ] + }, + "positive": { + "prob_id": "525e", + "tags": [ + "Meet in the Middle|Techniques|Miscellaneous" + ] + }, + "negative": [ + { + "prob_id": "639a", + "tags": [ + "Priority Queues, Sets, Maps, Counters|Basic Data Structures|Basics" + ] + }, + { + "prob_id": "1091e", + "tags": [ + "Degree Sequences|Miscellaneous|Graph Theory" + ] + }, + { + "prob_id": "1552i", + "tags": [ + "PQ Tree|More Tree Data Structures|Data Structures (DS)" + ] + } + ] + }, + { + "reference": { + "prob_id": "272d", + "tags": [ + "Permutations with Repetition, Stars and Bars|Counting|Intermediate" + ] + }, + "positive": { + "prob_id": "57c", + "tags": [ + "Permutations with Repetition, Stars and Bars|Counting|Intermediate" + ] + }, + "negative": [ + { + "prob_id": "367b", + "tags": [ + "Sliding Window|Data Structure Techniques|Intermediate" + ] + }, + { + "prob_id": "177d2", + "tags": [ + "Prefix Sums and Difference Arrays|Range Queries|Basics" + ] + }, + { + "prob_id": "468b", + "tags": [ + "Disjoint Set Union||" + ] + } + ] + }, + { + "reference": { + "prob_id": "547c", + "tags": [ + "M\u00f6bius Inversion||Advanced Mathematics", + "Principle of Inclusion-Exclusion|Counting|Intermediate" + ] + }, + "positive": { + "prob_id": "451e", + "tags": [ + "Generating Functions|Polynomial|Math", + "Principle of Inclusion-Exclusion|Counting|Intermediate" + ] + }, + "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": "474d", + "tags": [ + "Counting with DP||Intermediate" + ] + }, + "positive": { + "prob_id": "118d", + "tags": [ + "Counting with DP||Intermediate" + ] + }, + "negative": [ + { + "prob_id": "474f", + "tags": [ + "Square-Root Decomposition, Segment Trees, Binary-Indexed Trees|Range Queries|Intermediate" + ] + }, + { + "prob_id": "741c", + "tags": [ + "Bipartite Graph||" + ] + }, + { + "prob_id": "802n", + "tags": [ + "Min Cost Max Flow (MCMF)|Flow and Min Cut|Graph Theory" + ] + } + ] + }, + { + "reference": { + "prob_id": "149d", + "tags": [ + "Counting with DP||Intermediate" + ] + }, + "positive": { + "prob_id": "118d", + "tags": [ + "Counting with DP||Intermediate" + ] + }, + "negative": [ + { + "prob_id": "653f", + "tags": [ + "Suffix Automaton|Advanced|String Processing", + "String Suffix||" + ] + }, + { + "prob_id": "223a", + "tags": [ + "Balanced Brackets|Miscellaneous|Strings" + ] + }, + { + "prob_id": "380c", + "tags": [ + "Square-Root Decomposition, Segment Trees, Binary-Indexed Trees|Range Queries|Intermediate" + ] + } + ] + }, + { + "reference": { + "prob_id": "9d", + "tags": [ + "Counting with DP||Intermediate" + ] + }, + "positive": { + "prob_id": "296b", + "tags": [ + "Counting with DP||Intermediate", + "DP on Digits||Intermediate" + ] + }, + "negative": [ + { + "prob_id": "580c", + "tags": [ + "Trees and Tree Traversals|Graphs|Basics" + ] + }, + { + "prob_id": "30e", + "tags": [ + "Suffix Array|Suffix Structures|Strings" + ] + }, + { + "prob_id": "727e", + "tags": [ + "String Hashing|Hashing|Strings" + ] + } + ] + }, + { + "reference": { + "prob_id": "296b", + "tags": [ + "Counting with DP||Intermediate", + "DP on Digits||Intermediate" + ] + }, + "positive": { + "prob_id": "9d", + "tags": [ + "Counting with DP||Intermediate" + ] + }, + "negative": [ + { + "prob_id": "33b", + "tags": [ + "Floyd Warshall|Shortest Paths|Graph Theory" + ] + }, + { + "prob_id": "1085e", + "tags": [ + "Tree Centers and Diameter||Advanced Data Structures" + ] + }, + { + "prob_id": "898f", + "tags": [ + "Hashing||Intermediate" + ] + } + ] + }, + { + "reference": { + "prob_id": "518d", + "tags": [ + "Markov Chains|Probability and Expectation|Intermediate" + ] + }, + "positive": { + "prob_id": "148d", + "tags": [ + "Markov Chains|Probability and Expectation|Intermediate" + ] + }, + "negative": [ + { + "prob_id": "535e", + "tags": [ + "Convex Hull Trick|DP Optimization|Advanced Algorithms" + ] + }, + { + "prob_id": "827f", + "tags": [ + "Dijkstra's Algorithm|Shortest Paths|Graph Theory" + ] + }, + { + "prob_id": "114b", + "tags": [ + "All-Subsets / Binary Choice|Complete Search|Basics" + ] + } + ] + }, + { + "reference": { + "prob_id": "466e", + "tags": [ + "Level Ancestors and Lowest Common Ancestors||Advanced Data Structures" + ] + }, + "positive": { + "prob_id": "609e", + "tags": [ + "Binary Lifting||Advanced Data Structures", + "Lowest Common Ancestor|Lowest common ancestor|Graphs" + ] + }, + "negative": [ + { + "prob_id": "727e", + "tags": [ + "String Hashing|Hashing|Strings" + ] + }, + { + "prob_id": "631c", + "tags": [ + "Combining Data Structures|Data Structure Techniques|Intermediate" + ] + }, + { + "prob_id": "1572f", + "tags": [ + "Segment Tree Beats|Segment Tree|Data Structures (DS)", + "Segment Tree Beats|Segment Tree|Data Structures (DS)" + ] + } + ] + }, + { + "reference": { + "prob_id": "877e", + "tags": [ + "Tree Flattening||Advanced Data Structures" + ] + }, + "positive": { + "prob_id": "383c", + "tags": [ + "Tree Flattening||Advanced Data Structures" + ] + }, + "negative": [ + { + "prob_id": "763d", + "tags": [ + "Tree Isomorphism|Miscellaneous|Graph Theory" + ] + }, + { + "prob_id": "414b", + "tags": [ + "Counting with DP||Intermediate" + ] + }, + { + "prob_id": "1081d", + "tags": [ + "Minimum Spanning Tree||", + "Centroid Decomposition|Centroid Decomposition|Data Structures (DS)" + ] + } + ] + }, + { + "reference": { + "prob_id": "432d", + "tags": [ + "Z algorithm|String Matching|Strings" + ] + }, + "positive": { + "prob_id": "471d", + "tags": [ + "Z algorithm|String Matching|Strings" + ] + }, + "negative": [ + { + "prob_id": "202a", + "tags": [ + "All-Subsets / Binary Choice|Complete Search|Basics" + ] + }, + { + "prob_id": "380c", + "tags": [ + "Square-Root Decomposition, Segment Trees, Binary-Indexed Trees|Range Queries|Intermediate" + ] + }, + { + "prob_id": "1016b", + "tags": [ + "Prefix Sums and Difference Arrays|Range Queries|Basics" + ] + } + ] + }, + { + "reference": { + "prob_id": "471d", + "tags": [ + "Z algorithm|String Matching|Strings" + ] + }, + "positive": { + "prob_id": "432d", + "tags": [ + "Z algorithm|String Matching|Strings" + ] + }, + "negative": [ + { + "prob_id": "547c", + "tags": [ + "Principle of Inclusion-Exclusion|Counting|Intermediate", + "M\u00f6bius Inversion||Advanced Mathematics" + ] + }, + { + "prob_id": "650d", + "tags": [ + "O(n lg n) Longest Monotonic Subsequence|DP Optimization|Advanced Algorithms" + ] + }, + { + "prob_id": "268d", + "tags": [ + "Eliminate Dependent Parameters|DP Optimization|Advanced Algorithms" + ] + } + ] + }, + { + "reference": { + "prob_id": "516d", + "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": { + "prob_id": "78e", + "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" + ] + } + ] + }, + { + "reference": { + "prob_id": "797f", + "tags": [ + "Knapsack DP||" + ] + }, + "positive": { + "prob_id": "115e", + "tags": [ + "Knapsack DP||" + ] + }, + "negative": [ + { + "prob_id": "707b", + "tags": [ + "Single Source|Shortest Paths|Intermediate" + ] + }, + { + "prob_id": "241c", + "tags": [ + "Reflections on a Rectangle|Miscellaneous|Geometry" + ] + }, + { + "prob_id": "832d", + "tags": [ + "Lowest Common Ancestor|Lowest common ancestor|Graphs" + ] + } + ] + }, + { + "reference": { + "prob_id": "18e", + "tags": [ + "Knapsack DP||" + ] + }, + "positive": { + "prob_id": "909c", + "tags": [ + "Knapsack DP||" + ] + }, + "negative": [ + { + "prob_id": "732f", + "tags": [ + "Biconnectivity|DFS Lowlink|Advanced Algorithms" + ] + }, + { + "prob_id": "786c", + "tags": [ + "Sqrt Decomposition|Trees|Data Structures" + ] + }, + { + "prob_id": "25d", + "tags": [ + "Disjoint Set Union||" + ] + } + ] + }, + { + "reference": { + "prob_id": "621e", + "tags": [ + "Matrix Exponentiation|Matrices|Math" + ] + }, + "positive": { + "prob_id": "852b", + "tags": [ + "Matrix Exponentiation|Matrices|Math" + ] + }, + "negative": [ + { + "prob_id": "715c", + "tags": [ + "Centroid Decomposition|Centroid Decomposition|Data Structures (DS)" + ] + }, + { + "prob_id": "621c", + "tags": [ + "Basics|Probability and Expectation|Basics" + ] + }, + { + "prob_id": "269c", + "tags": [ + "Network Flows||Advanced Algorithms" + ] + } + ] + }, + { + "reference": { + "prob_id": "852b", + "tags": [ + "Matrix Exponentiation|Matrices|Math" + ] + }, + "positive": { + "prob_id": "621e", + "tags": [ + "Matrix Exponentiation|Matrices|Math" + ] + }, + "negative": [ + { + "prob_id": "505d", + "tags": [ + "Strongly Connected Components (SCC)|Graph Connectivity|Graph Theory" + ] + }, + { + "prob_id": "330b", + "tags": [ + "Basic Graph Theory|Graphs|Basics" + ] + }, + { + "prob_id": "587c", + "tags": [ + "Level Ancestors and Lowest Common Ancestors||Advanced Data Structures" + ] + } + ] + }, + { + "reference": { + "prob_id": "576d", + "tags": [ + "Matrix Exponentiation|Matrices|Math" + ] + }, + "positive": { + "prob_id": "696d", + "tags": [ + "Aho-Corasick|String Structures|Advanced Data Structures", + "Matrix Exponentiation|Matrices|Math" + ] + }, + "negative": [ + { + "prob_id": "241e", + "tags": [ + "Single Source with Negative Weights|Shortest Paths|Intermediate" + ] + }, + { + "prob_id": "330b", + "tags": [ + "Basic Graph Theory|Graphs|Basics" + ] + }, + { + "prob_id": "505d", + "tags": [ + "Strongly Connected Components (SCC)|Graph Connectivity|Graph Theory" + ] + } + ] + }, + { + "reference": { + "prob_id": "696d", + "tags": [ + "Aho-Corasick|String Structures|Advanced Data Structures", + "Matrix Exponentiation|Matrices|Math" + ] + }, + "positive": { + "prob_id": "576d", + "tags": [ + "Matrix Exponentiation|Matrices|Math" + ] + }, + "negative": [ + { + "prob_id": "706c", + "tags": [ + "Common Patterns|DP|Intermediate" + ] + }, + { + "prob_id": "704d", + "tags": [ + "Flow with Lower Bounds||" + ] + }, + { + "prob_id": "33b", + "tags": [ + "Floyd Warshall|Shortest Paths|Graph Theory" + ] + } + ] + }, + { + "reference": { + "prob_id": "351c", + "tags": [ + "Matrix Exponentiation|Matrices|Math" + ] + }, + "positive": { + "prob_id": "576d", + "tags": [ + "Matrix Exponentiation|Matrices|Math" + ] + }, + "negative": [ + { + "prob_id": "351d", + "tags": [ + "Mo's Algorithm|Square Root Decomposition|Data Structures (DS)" + ] + }, + { + "prob_id": "414b", + "tags": [ + "Counting with DP||Intermediate" + ] + }, + { + "prob_id": "177d2", + "tags": [ + "Prefix Sums and Difference Arrays|Range Queries|Basics" + ] + } + ] + }, + { + "reference": { + "prob_id": "917c", + "tags": [ + "Matrix Exponentiation|Matrices|Math" + ] + }, + "positive": { + "prob_id": "576d", + "tags": [ + "Matrix Exponentiation|Matrices|Math" + ] + }, + "negative": [ + { + "prob_id": "1110e", + "tags": [ + "Invariants and Monovariants|Techniques|Miscellaneous" + ] + }, + { + "prob_id": "1446a", + "tags": [ + "Knapsack DP||" + ] + }, + { + "prob_id": "689b", + "tags": [ + "Breadth First Search|Graph traversal|Graphs" + ] + } + ] + }, + { + "reference": { + "prob_id": "319c", + "tags": [ + "Convex Hull Trick|DP Optimization|Advanced Algorithms" + ] + }, + "positive": { + "prob_id": "643c", + "tags": [ + "Convex Hull Trick|DP Optimization|Advanced Algorithms", + "Divide and Conquer DP|DP optimizations|Dynamic Programming" + ] + }, + "negative": [ + { + "prob_id": "348a", + "tags": [ + "Binary Search|Leveraging Monotonicity|Basics" + ] + }, + { + "prob_id": "653d", + "tags": [ + "Network Flows||Advanced Algorithms" + ] + }, + { + "prob_id": "243a", + "tags": [ + "Bit Manipulation|Bit Manipulation|Basics" + ] + } + ] + }, + { + "reference": { + "prob_id": "1083e", + "tags": [ + "Convex Hull Trick|DP Optimization|Advanced Algorithms" + ] + }, + "positive": { + "prob_id": "319c", + "tags": [ + "Convex Hull Trick|DP Optimization|Advanced Algorithms" + ] + }, + "negative": [ + { + "prob_id": "1084c", + "tags": [ + "Rule of Sum, Rule of Product|Counting|Basics" + ] + }, + { + "prob_id": "1176e", + "tags": [ + "Bipartite Graph||", + "Graph Traversal||" + ] + }, + { + "prob_id": "1242b", + "tags": [ + "Inverse Graph|Graph Traversal|Graph Theory" + ] + } + ] + }, + { + "reference": { + "prob_id": "932f", + "tags": [ + "Convex Hull Trick|DP Optimization|Advanced Algorithms" + ] + }, + "positive": { + "prob_id": "631e", + "tags": [ + "Convex Hull Trick|DP Optimization|Advanced Algorithms" + ] + }, + "negative": [ + { + "prob_id": "506d", + "tags": [ + "Splitting Objects into Light and Heavy / Case Processing|Square Root Decomposition|Data Structures (DS)" + ] + }, + { + "prob_id": "177d2", + "tags": [ + "Prefix Sums and Difference Arrays|Range Queries|Basics" + ] + }, + { + "prob_id": "653e", + "tags": [ + "Breadth First Search|Graph traversal|Graphs" + ] + } + ] + }, + { + "reference": { + "prob_id": "643c", + "tags": [ + "Convex Hull Trick|DP Optimization|Advanced Algorithms", + "Divide and Conquer DP|DP optimizations|Dynamic Programming" + ] + }, + "positive": { + "prob_id": "319c", + "tags": [ + "Convex Hull Trick|DP Optimization|Advanced Algorithms" + ] + }, + "negative": [ + { + "prob_id": "776f", + "tags": [ + "Centroid Decomposition|Centroid Decomposition|Data Structures (DS)" + ] + }, + { + "prob_id": "623d", + "tags": [ + "Probabilities and Expected Values|Probabilities and Expected Values|Math" + ] + }, + { + "prob_id": "839e", + "tags": [ + "Meet in the Middle|Techniques|Miscellaneous" + ] + } + ] + }, + { + "reference": { + "prob_id": "377e", + "tags": [ + "Convex Hull Trick|DP Optimization|Advanced Algorithms" + ] + }, + "positive": { + "prob_id": "631e", + "tags": [ + "Convex Hull Trick|DP Optimization|Advanced Algorithms" + ] + }, + "negative": [ + { + "prob_id": "1016c", + "tags": [ + "Prefix Sums and Difference Arrays|Range Queries|Basics" + ] + }, + { + "prob_id": "417d", + "tags": [ + "DP on Subsets||Intermediate" + ] + }, + { + "prob_id": "1500b", + "tags": [ + "Chinese Remainder Theorem|Modular arithmetic|Algebra" + ] + } + ] + }, + { + "reference": { + "prob_id": "438e", + "tags": [ + "Fast Fourier Transform (FFT)|Convolution|Math", + "Generating Functions|Polynomial|Math" + ] + }, + "positive": { + "prob_id": "848e", + "tags": [ + "Fast Fourier Transform (FFT)|Convolution|Math" + ] + }, + "negative": [ + { + "prob_id": "715b", + "tags": [ + "Dijkstra's Algorithm|Shortest Paths|Graph Theory" + ] + }, + { + "prob_id": "431c", + "tags": [ + "Counting with DP||Intermediate" + ] + }, + { + "prob_id": "1242b", + "tags": [ + "Inverse Graph|Graph Traversal|Graph Theory" + ] + } + ] + }, + { + "reference": { + "prob_id": "553e", + "tags": [ + "Fast Fourier Transform (FFT)|Convolution|Math" + ] + }, + "positive": { + "prob_id": "623e", + "tags": [ + "Fast Fourier Transform (FFT)|Convolution|Math" + ] + }, + "negative": [ + { + "prob_id": "821e", + "tags": [ + "Matrix Exponentiation|Matrices|Math" + ] + }, + { + "prob_id": "567e", + "tags": [ + "Dijkstra's Algorithm|Shortest Paths|Graph Theory" + ] + }, + { + "prob_id": "832d", + "tags": [ + "Lowest Common Ancestor|Lowest common ancestor|Graphs" + ] + } + ] + }, + { + "reference": { + "prob_id": "623e", + "tags": [ + "Fast Fourier Transform (FFT)|Convolution|Math" + ] + }, + "positive": { + "prob_id": "848e", + "tags": [ + "Fast Fourier Transform (FFT)|Convolution|Math" + ] + }, + "negative": [ + { + "prob_id": "414b", + "tags": [ + "Counting with DP||Intermediate" + ] + }, + { + "prob_id": "500b", + "tags": [ + "Depth-First Traversal, Connectivity, Cycle Detection|Graphs|Basics" + ] + }, + { + "prob_id": "272d", + "tags": [ + "Permutations with Repetition, Stars and Bars|Counting|Intermediate" + ] + } + ] + }, + { + "reference": { + "prob_id": "662c", + "tags": [ + "DP on Subsets||Intermediate", + "Fast Walsh Hadamard Transform (FWHT) and its Variations|Convolution|Math" + ] + }, + "positive": { + "prob_id": "850e", + "tags": [ + "Fast Walsh Hadamard Transform (FWHT) and its Variations|Convolution|Math" + ] + }, + "negative": [ + { + "prob_id": "844b", + "tags": [ + "Rule of Sum, Rule of Product|Counting|Basics" + ] + }, + { + "prob_id": "1063b", + "tags": [ + "0/1 BFS|Shortest Paths|Graph Theory" + ] + }, + { + "prob_id": "243e", + "tags": [ + "PQ Tree|More Tree Data Structures|Data Structures (DS)" + ] + } + ] + }, + { + "reference": { + "prob_id": "850e", + "tags": [ + "Fast Walsh Hadamard Transform (FWHT) and its Variations|Convolution|Math" + ] + }, + "positive": { + "prob_id": "662c", + "tags": [ + "DP on Subsets||Intermediate", + "Fast Walsh Hadamard Transform (FWHT) and its Variations|Convolution|Math" + ] + }, + "negative": [ + { + "prob_id": "332b", + "tags": [ + "Classic Problems with Twists|DP|Intermediate" + ] + }, + { + "prob_id": "272d", + "tags": [ + "Permutations with Repetition, Stars and Bars|Counting|Intermediate" + ] + }, + { + "prob_id": "319c", + "tags": [ + "Convex Hull Trick|DP Optimization|Advanced Algorithms" + ] + } + ] + }, + { + "reference": { + "prob_id": "891e", + "tags": [ + "Generating Functions|Polynomial|Math" + ] + }, + "positive": { + "prob_id": "865g", + "tags": [ + "Generating Functions|Polynomial|Math" + ] + }, + "negative": [ + { + "prob_id": "691e", + "tags": [ + "Matrix Exponentiation|Matrices|Math" + ] + }, + { + "prob_id": "920f", + "tags": [ + "Segment Tree|Trees|Data Structures" + ] + }, + { + "prob_id": "459d", + "tags": [ + "Order Statistics||Intermediate" + ] + } + ] + } +] \ No newline at end of file