AN EFFICIENT ALGORITHM FOR A SPECIAL CASE OF THE SET PARTITION PROBLEM

In this paper, we study the Two Functions Set Partition Problem, which is defined as follows: Given a set S of n elements, functions f1and f2 from S to S, and an initial partition B = (B1,B2,…,BS) of S, find the coarsest refinement E = (E1 E2,… Et)of B such that for each i, i = 1,2, and j, 1≦j≦t,fi (Ej)⊑ Ek for some k. For the special case when f1consists of a single cycle, we present an 0(nβ(n)) algorithm, where β(n) is the number of distinct prime factors of n. β(n) is loglogn + 0(loglogn) for almost all n, and is Θ(logn/loglogn) in the worst case. This algorithm represents an improvement over the previously known O(n log n) algorithm. © 1990, Taylor & Francis Group, LLC. All rights reserved.