Disjoint Set Union Simulator

Disjoint Set Union—also called Union-Find—maintains a collection of non-overlapping sets. It answers which component an element belongs to and efficiently merges two components using path compression and union by rank.

Make SetO(1)

FindAmortized O(α(n))

UnionAmortized O(α(n))

OptimizationsRank + compression

Union-Find operations

New element

First element

Second element

Element to find

Animation speed 400 ms

Example Union-Find forest loaded. Choose an operation.

Elements0

Components0

Maximum depth0

Parent rewrites0

Find path

—

Last operation

—

Current components

Parent, rank, and size table

Element	Parent	Representative	Rank	Component size

Optimized operations

find(x):
  if parent[x] != x:
    parent[x] = find(parent[x])
  return parent[x]

union(a, b):
  rootA = find(a)
  rootB = find(b)

  attach lower-rank root
  below higher-rank root

  if ranks are equal:
    choose one root
    increase its rank

Union-Find concepts

Disjoint components

Every element belongs to exactly one component. Each component is represented by a rooted parent tree.

Path compression

Find rewrites parent links along the search path to point directly to the representative, flattening future searches.

Union by rank

The lower-rank root is attached below the higher-rank root. Equal ranks require one rank increment.

Typical applications

Union-Find is used in Kruskal's minimum spanning tree algorithm, connectivity queries, image segmentation, and dynamic grouping.

Terminology

Term	Meaning
Representative	The root element that identifies a component.
Parent	The next element followed toward the representative.
Rank	An upper-bound estimate of tree height used for merging.
Component size	The number of elements represented by a root.
α(n)	The inverse Ackermann function, which grows extremely slowly.