Disjoint-set data structure

In computer science, a disjoint-set data structure, also called a union–find data structure or merge–find set, is a data structure that stores a collection of disjoint (non-overlapping) sets. Equivalently, it stores a partition of a set into disjoint subsets. It provides operations for adding new sets, merging sets (replacing them with their union), and finding a representative member of a set. The last operation makes it possible to determine efficiently whether any two elements belong to the same set or to different sets.

While there are several ways of implementing disjoint-set data structures, in practice they are often identified with a particular implementation known as a disjoint-set forest. This specialized type of forest performs union and find operations in near-constant amortized time. For a sequence of m addition, union, or find operations on a disjoint-set forest with n nodes, the total time required is O(mα(n)), where α(n) is the extremely slow-growing inverse Ackermann function. Although disjoint-set forests do not guarantee this time per operation, each operation rebalances the structure (via tree compression) so that subsequent operations become faster. As a result, disjoint-set forests are both asymptotically optimal and practically efficient.

Disjoint-set data structures play a key role in Kruskal's algorithm for finding the minimum spanning tree of a graph. The importance of minimum spanning trees means that disjoint-set data structures support a wide variety of algorithms. In addition, these data structures find applications in symbolic computation and in compilers, especially for register allocation problems.