Answering distance queries in directed graphs using fast matrix multiplication

Let G = (V, E, w) be a weighted directed graph, where w : E → {-M,..., 0,..., M}. We show that G can be preprocessed in Õ(Mn ^ω) time, where ω < 2.376 is the exponent of fast matrix multiplication, such that subsequently, each distance δ(u, v) in the graph, where u, v ∈ V, can be computed exactly in O(n) time. We also present a tradeoff between the processing time and the query answering time. As a very special case, we obtain an Õ(Mn ^ω) time algorithm for the Single Source Shortest Paths (SSSP) problem for directed graphs with integer weights of absolute value at most M. For sufficiently dense graphs, with small enough edge weights, this improves upon the O(m√n log M) time algorithm of Goldberg. We note that even the case M = 1, in which all the edge weights are in {-1, 0, +1}, is an interesting case for which no improvement over Goldberg's O(m√n) algorithm was known. Our new Õ(n ^ω) algorithm is faster whenever m > n ^ω-1/2 ≃ n ^1.876.