All Pairs Shortest Paths for Graphs with Small Integer Length Edges

The authors have solved the all pairs shortest distances (APSD) problem for graphs with integer edge lengths. Our algorithm is subcubic for edge lengths of small (?M) absolute value. In this paper we show how to transform these algorithms to solve the all pairs shortest paths (APSP), in the same time complexity, up to a polylogarithmic factor. Forn=|V| the number of vertices,Mthe bound on edge length, and?the exponent of matrix multiplication, we get the following results: 1. A directed nonnegative APSP(n, M) algorithm which runs inO(T(n, M)) time, where T(n, m)=\big\{\begin{align}M^{\omega -1)/2} n^{3+\omega )/2}, & 1\le M\le n^{3-\omega )/(\omega +1)}\\ Mn^{5\omega -3)/(\omega +1)}, & n^{(3-\omega )/(\omega +1)}\le M\le n^2(3-\omega )/(\omega +1)}.\end{align} 2. An undirected APSP(n, M) algorithm which runs inO(M(?+1)/2n?log(Mn)) time. 3. A general APSP(n, M) algorithm which runs inO((Mn)(3+?)/2).