Witnesses for Boolean matrix multiplication and for shortest paths

The subcubic (O(n^w) for w⟨3) algorithms to multiply Boolean matrices do not provide the witnesses; namely, they compute C=A·B but if C_ij=1 they do not find an index k (a witness) such that A_ik=B_kj=1. The authors design a deterministic algorithm for computing the matrix of witnesses that runs in O˜(n^w) time, where here O˜(n^w ) denotes O(n^w(log n)^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 v_i to v _j is an index k such that v_k 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^(3+w)/2) time in the directed case and O˜(n^w) 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