Derandomization, witnesses for Boolean matrix multiplication and construction of perfect hash functions

Abstract

Small sample spaces with almost independent random variables are applied to design efficient sequential deterministic algorithms for two problems. The first algorithm, motivated by the attempt to design efficient algorithms for the All Pairs Shortest Path problem using fast matrix multiplication, solves the problem of computingwitnesses for the Boolean product of two matrices. That is, ifA andB are twon byn matrices, andC=AB is their Boolean product, the algorithm finds for every entryC ij =1 a witness: an indexk so thatA ik =B kj =1. Its running time exceeds that of computing the product of twon byn matrices with small integer entries by a polylogarithmic factor. The second algorithm is a nearly linear time deterministic procedure for constructing a perfect hash function for a givenn-subset of {1,...,m}.