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While there are several ways of implementing disjoint-set data structures, in practice they are often identified with a particular implementation called a '''disjoint-set forest'''. This is a specialized type of [[Forest (graph theory)|forest]] which performs unions and finds in near-constant [[Amortized analysis|amortized time]]. To perform a sequence of {{mvar|m}} addition, union, or find operations on a disjoint-set forest with {{mvar|n}} nodes requires total time {{math|[[Big O notation|''O'']](''m''α(''n''))}}, where {{math|α(''n'')}} is the extremely slow-growing [[inverse Ackermann function]]. Disjoint-set forests do not guarantee this performance on a per-operation basis. Individual union and find operations can take longer than a constant times {{math|α(''n'')}} time, but each operation causes the disjoint-set forest to adjust itself so that successive operations are faster. 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 underlie a wide variety of algorithms. In addition, disjoint-set data structures also have applications to symbolic computation, as well as in compilers, especially for [[register allocation]] problems.
== History ==
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