A linear-time algorithm for a special case of disjoint set union

This paper presents a linear-time algorithm for the special case of the disjoint set union problem in which the structure of the unions (defined by a “union tree”) is known in advance. The algorithm executes an intermixed sequence of <italic>m</italic> union and find operations on <italic>n</italic> elements in 0(<italic>m</italic>+<italic>n</italic>) time and 0(<italic>n</italic>) space. This is a slight but theoretically significant improvement over the fastest known algorithm for the general problem, which runs in 0(<italic>m</italic>α(<italic>m</italic>+<italic>n, n</italic>)+<italic>n</italic>) time and 0(<italic>n</italic>) space, where α is a functional inverse of Ackermann's function. Used as a subroutine, the algorithm gives similar improvements in the efficiency of algorithms for solving a number of other problems, including two-processor scheduling, the off-line min problem, matching on convex graphs, finding nearest common ancestors off-line, testing a flow graph for reducibility, and finding two disjoint directed spanning trees. The algorithm obtains its efficiency by combining a fast algorithm for the general problem with table look-up on small sets, and requires a random access machine for its implementation. The algorithm extends to the case in which single-node additions to the union tree are allowed. The extended algorithm is useful in finding maximum cardinality matchings on nonbipartite graphs.

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