A new upper bound on the complexity of the all pairs shortest path problem

Abstract

The all pairs shortest path (APSP) problem is to compute shortest paths between all pairs of vertices of a directed graph with non-negative edge costs. We present an algorithm that computes shortest distances between all pairs of vertices, since shortest paths can be computed easily as by-products as in most other algorithms. It is well-known that the time complexity of (n, n)-distance matrix multiplication (DMM) is asymptotically equal to that of the APSP problem for a graph with n vertices. See Aho, ttopcroft and Ullman [1] for example. Based on this fact, Fredma~ [5] invented an algorithm for DMM of O(n3(log log n/log n) 1/3) time, which is o(n3), meaning that the APSP problem can be solved with the same time complexity. In the average case, Moffat and Takaoka [6] solved this problem with O(n 9 log n) expected time. Our algorithm in this paper solves DMM in O(n3(loglog n/log n) ~/2) time, meaning that the APSP problem can be solved with the same time complexity. This is an asymptotic improvement of Fredman's algorithm by the factor of (logn/loglogn) 1Is. Another merit of our algorithm is that it is simple and easy to implement, whereas Fredman's algorithm is complicated and difficult to implement. Also a possible parallel implementation is mentioned. The base of logarithm is assumed to be two in this paper and fractions are rounded up if necessary.