Computation of the Boolean Matrix-Vector, AND/OR-Produkt in Average Time O(m + nlnn)

Abstract

Let A = [a _i,j ]₁≤i≤n be a boolean matrix and let b = (b ₁,...,b _m )^T be a boolean vector. The boolean AND/OR-product c = Ab, c = (c ₁,...,c _n )^T is given by c _i = V _{j =1,...,m} (a _i,j ⋀ b _j ) for i = 1,...,m. We consider random boolean n × m matrices A where the probability for A is preserved by arbitrary permutations of the elements in each column. We describe an algorithm that first preprocesses the matrix A using O(nm) steps and then computes the product Ab using an expected number of O(m + nlnn) steps. As a consequence the boolean matrix product C = AB of the random boolean n × m-matrix A and an arbitrary boolean m × k matrix B can be computed in average time O(nm + km + kn lnn).