All pairs shortest paths in undirected graphs with integer weights

We show that the all pairs shortest paths (APSP) problem for undirected graphs with integer edge weights taken from the range {1, 2, ..., M} can be solved using only a logarithmic number of distance products of matrices with elements in the range (1, 2, ..., M). As a result, we get an algorithm for the APSP problem in such graphs that runs in O¯(Mn^ω) time, where n is the number of vertices in the input graph, M is the largest edge weight in the graph, and ω<2.376 is the exponent of matrix multiplication. This improves, and also simplifies, an O¯(M^(ω+1)/2n^ω) time algorithm of Galil and Margalit (1997)