On the expected behavior of disjoint set union algorithms

Authors:
B Bollobás;I Simon
Affiliations:
Universtiy of Cambridge, Department of Pure Mathematics and Mathematical Statistics, 16 Mill Lane, Cambridge, England CB2 1SB;Universidade Estadual de Campinas, Instituto de Matemática, Estatística, e Ciência da Computação, Caixa Postal 1170, 13.100 Campinas, SP, Brasil
Venue:
STOC '85 Proceedings of the seventeenth annual ACM symposium on Theory of computing
Year:
1985

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Abstract

We show that the expected time of the Weighted Quickfind (QFW) disjoint set union and find algorithm to perform (n - 1) randomly chosen unions is cn + o(n/log n), where c = 2.0847 …. This implies, through an observation of Tarjan and Van Leeuwen, linear expected time bounds to perform O(n) unions and finds for a class of other union -find algorithms. We also prove that the expected time of the unweighted Quickfind (QF) algorithm is n2/8 + o(n(log n)2), and set the several related open questions of Knuth and Schönhage.