A Practical Algorithm for Boolean Matrix Multiplication

Research on fast methods for multiplying two n x n Boolean matrices has followed two directions. On the one hand, there are the asymptotically fast methods derived from algorithms to multiply matrices with integer entries. On the other hand, it has been of interest to devise algorithms which, while inferior in the asymptotic sense, are simple enough to be advantageously implemented. This paper addresses the latter class of algorithms. The first such algorithm (apart from the classical straightforward 0(n3) algorithm) was given in [3] and is known, at least in Western literature, as the Four Russians’ algorithm. It requires 0(n3/log n) time and auxiliary storage of 0(n3/log n) bits; simple modifications can reduce the amount of auxiliary storage to O(n’). More recent work on the problem has focussed on reducing the amount of auxiliary storage [5] and on exploiting properties of the arrays to be multiplied [4,6]. The purpose of this paper is to point out that a rather simple and known technique can be used to derive a Boolean matrix multiplication algorithm which has a lower time complexity and requires less auxiliary storage than the Four Russians’