A geometric approach to boolean matrix multiplication

Abstract

For a Boolean matrix D, let rD be the minimum number of rectangles sufficient to cover exactly the rectilinear region formed by the 1-entries in D. Next, let mD be the minimum of the number of 0-entries and the number of 1-entries in D. Suppose that the rectilinear regions formed by the 1-entries in two n×n Boolean matrices A and B totally with q edges are given. We show that in time Õ(q + min{rArB,n(n + rA),n(n + rB)})1 one can construct a data structure which for any entry of the Boolean product of A and B reports whether or not it is equal to 1, and if so, reports also the so called witness of the entry, in time O(log q). Asa corollary, we infer that if the matrices A and B are given as input, their product and the witnesses of the product can be computed in time Õ(n (n+min{r A, rB })). This implies in particular that the product of A and B and its witnesses can be computed in time O(n(n+ min{mA, mB})). In contrast to the known sub-cubic algorithms for Boolean matrix multiplication based on arithmetic 0-1-matrix multiplication, our algorithms do not involve large hidden constants in their running time and are easy to implement. © Springer-Verlag Berlin Heidelberg 2002.