Computing the diameter polynomially faster than APSP

We present a new randomized algorithm for computing the diameter of a weighted directed graph. The algorithm runs in Õ(Mn 2+3)/(ω+1)) time, where ω < 2.376 is the exponent of fast matrix multiplication, n is the number of vertices of the graph, and the edge weights are integers in {−M, . . . , 0, . . . ,M}. For bounded integer weights the running time is O(n) and if ω = 2+ o(1) it is Õ(n). This is the first algorithm that computes the diameter of an integer weighted directed graph polynomially faster than any known All-Pairs Shortest Paths (APSP) algorithm. For bounded integer weights, the fastest algorithm for APSP runs in O(n) time for the present value of ω and runs in Õ(n) time if ω = 2 + o(1). For directed graphs with positive integer weights in {1, . . . ,M} we obtain a deterministic algorithm that computes the diameter in Õ(Mn) time. This extends a simple Õ(n) algorithm for computing the diameter of an unweighted directed graph to the positive integer weighted setting and is the first algorithm in this setting whose time complexity matches that of the fastest known Diameter algorithm for undirected graphs. The diameter algorithms are consequences of a more general result. We construct algorithms that for any given integer d, report all ordered pairs of vertices having distance at most d. The diameter can therefore be computed using binary search for the smallest d for which all pairs are reported.