An optimal algorithm for generating equivalence relations on a linear array of processors

Abstract

We describe a cost-optimal parallel algorithm for enumerating all partitions (equivalence relations) of the set {1, ...,n}, in lexicographic order. The algorithm is designed to be executed on a linear array of processors. It usesn processors, each having constant size memory and each being responsible for producing one element of a given set partition. Set partitions are generated with constant delay leading to anO(Bn) time solution, whereBn is the total number of set partitions. The same method can be generalized to enumerate some other combinatorial objects such as variations. The algorithm can be made adaptive, i.e. to run on any prespecified number of processors. To illustrate the model of parallel computation, a simple case of enumerating subsets of the set {1, ...,n}, having at mostm (≤n) elements is also described.