Fast Sequential and Parallel Algorithms for Finding Extremal Sets

Abstract

We consider the problem of finding the minimal (maximal) sets of a family of sets that have no subset (superset) in the family. Given a family F of k sets with N elements and a domain of size n, first we show that using word-to-bit mapping, a technique of compressing ^ords into bits and processing bits instead of words, we can obtain a simple algorithm tlfiat solves this problem in [ILM0001] time using [ILM0002] space in the worst case. When [ILM0003], our algorithm runs in O(N 2/log2 N) time and O(N 2/log3 N) space, thus improving known results. We then present two fasif parallel algorithms for solving this problem - an O(log N) time algorithm using [ILM0004] processors on a CREW PRAM and a constant-time algorithm using [ILM0005] processors on a combining CRCW PRAM in which concurrent writing is resolved by writing the sum of the individual values to be written. These <kre respectively the first NC algorithm on the CREW and constant-time algorithm on the CRCW for the extremal set problem. Finally we extend the extremal set problem to the case when F contains multisets, and show that in this case the problem can b<b solved in [ILM0006] time and O (N + (k 2/log N)) space when the maximal number ofduplicates of any element within a multiset is m and all duplicates are uniformly distributed.