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

Authors:

Harold N. Gabow,

Robert Endre TarjanAuthors Info & Claims

STOC '83: Proceedings of the fifteenth annual ACM symposium on Theory of computing

Pages 246 - 251

https://doi.org/10.1145/800061.808753

Published: 01 December 1983 Publication History

Abstract

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 m union and find operations on n elements in 0(m+n) time and 0(n) space. This is a slight but theoretically significant improvement over the fastest known algorithm for the general problem, which runs in 0(mα(m+n, n)+n) time and 0(n) 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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Index Terms

A linear-time algorithm for a special case of disjoint set union
1. Mathematics of computing
  1. Discrete mathematics
    1. Combinatorics
      1. Combinatorial algorithms
2. Theory of computation
  1. Computational complexity and cryptography
    1. Complexity classes
  2. Logic
    1. Type theory

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cover image ACM Conferences

STOC '83: Proceedings of the fifteenth annual ACM symposium on Theory of computing

December 1983

487 pages

ISBN:0897910990

DOI:10.1145/800061

Copyright © 1983 ACM.

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

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Publication History

Published: 01 December 1983

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