Single Source Shortest Paths for All Flows with Integer Costs

We consider a shortest path problem for a directed graph with edges labeled with a cost and a capacity. The problem is to push an unsplittable flow f from a specified source to all other vertices with the minimum cost for all f values. Let G = (V,E) with |V | = n and |E| = m. If there are t different capacity values, we can solve the single source shortest path problem t times for all f in O(tm + tn logn) time, which is O(m2) when t = m. We improve this time to O(min(t, cn)m+ cn2), which is less than O(cmn) if edge costs are non-negative integers bounded by c. Our algorithm performs better for denser graphs. 1998 ACM Subject Classification E.1 Data Structures, F.2.2 Nonnumerical Algorithms and Problems, G.2.2 Graph Theory