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Learning ToolsAlgorithm Complexity ReferenceAlgorithm Learning Paths
Data StructuresArrayLinked ListStackQueue and DequeBinary Search TreeBinary HeapHash TableGraphTrieDisjoint Set UnionLRU CacheSkip ListSegment TreeB+ TreeBloom FilterFenwick Tree
SortingBubble SortCocktail Shaker SortBitonic SortSelection SortInsertion SortBinary Insertion SortShell SortMerge SortTop-Down Merge SortQuick SortThree-Way Quick SortDual-Pivot Quick SortHeap SortCounting SortRadix SortBucket Sort
Graph AlgorithmsDijkstra's Shortest PathKruskal's Minimum Spanning TreePrim's Minimum Spanning TreeBellman-Ford Shortest PathsTopological SortFloyd-WarshallStrongly Connected Components2-SATMaximum FlowBipartite MatchingLowest Common AncestorEulerian Path
Dynamic ProgrammingEdit Distance0/1 KnapsackUnbounded KnapsackLongest Common SubsequenceLongest Increasing SubsequenceCoin ChangeStone MergingTraveling Salesperson DPTree Dynamic ProgrammingDigit DPRerooting DP
Backtracking and SearchN-QueensSubsetsPermutationsCombination SumMaze Solving with DFSNumber of IslandsWord SearchSudoku SolverA* Search
StringsKMP String MatchingRabin-Karp String MatchingBoyer-Moore String MatchingManacher's Longest Palindromic SubstringSuffix ArrayLCP ArrayAho-Corasick AutomatonZ Function
Math and Number TheorySieve of EratosthenesLinear SieveEuclidean AlgorithmBinary ExponentiationExtended Euclidean AlgorithmChinese Remainder TheoremEuler's Totient FunctionMiller-Rabin Primality TestFast Fourier TransformPollard's Rho Factorization
Computational GeometryConvex HullRotating CalipersClosest Pair of PointsLine Segment IntersectionBentley-Ottmann Sweep Line
SearchingBinary SearchLower and Upper BoundSearch in a Rotated Sorted ArrayBinary Search on the AnswerTernary Search

Disjoint Set Union

Maintain dynamic connectivity with parent forests

Represent each component by a root

Disjoint set union (DSU), also called union-find, partitions elements into non-overlapping sets. find(x) follows parent links to a representative root, and union(a, b) connects two roots when their sets differ.

Try it
01234567

Current components: 8

Choose two elements to unite, find a root, or test connectivity. Each starts alone.

Flatten paths as they are discovered

Path compression makes every node seen by find point directly toward the root. Union by rank or size attaches the smaller tree below the larger one. Together, these rules make a long sequence of operations take almost constant amortized time, O(alpha(n)) per operation.

Invariant: two elements are connected exactly when their finds return the same root.

DSU is the cycle detector inside Kruskal's minimum spanning tree and is widely used for offline connectivity problems.