# Witnesses for Boolean matrix multiplication and for shortest paths

## Abstract

The subcubic (O(nw) for w(3) algorithms to multiply Boolean matrices do not provide the witnesses; namely, they compute C=A.B but if Cij=1 they do not find an index k (a witness) such that Aik=Bkj=1. The authors design a deterministic algorithm for computing the matrix of witnesses that runs in O(nw) time, where here O(nw) denotes O(nw(log n)/sup O(1)/). The subcubic methods to compute the shortest distances between all pairs of vertices also do not provide for witnesses; namely they compute the shortest distances but do not generate information for computing quickly the paths themselves. A witness for a shortest path from vi to vj is an index k such that vk is the first vertex on such a path. They describe subcubic methods to compute such witnesses for several versions of the all pairs shortest paths problem. As a result, they derive shortest paths algorithms that provide characterization of the shortest paths in addition to the shortest distances in the same time (up to a polylogarithmic factor) needed for computing the distances; namely O(n/sup (3+w)/2/) time in the directed case and O(nw) time in the undirected case. They also design an algorithm that computes witnesses for the transitive closure in the same time needed to compute witnesses for Boolean matrix multiplication.