Witnesses for Boolean Matrix Multiplication and for Shortest Paths

The subcubic (O(n) for ω < 3) algorithms to multiply Boolean matrices do not provide the witnesses; namely, they compute C = AB but if Cij = 1 they do not find an index k (a witness) such that Aik = Bkj = 1. We design a deterministic algorithm for computing the matrix of witnesses that runs in Õ(n) time, where here Õ(n) denotes O(n(log n)). 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. We describe subcubic methods to compute such witnesses for several versions of the all pairs shortest paths problem. As a result, we derive shortest paths algorithms that provide characterization of the shortest paths in addition to the shortest distances in essentially the same time needed for computing the distances; namely Õ(n) in the directed case and Õ(n) time in the undirected case. We also design an algorithm that computes witnesses for the transitive closure in the same time needed to compute witnesses for Boolean matrix multiplication.