Some extensions of the bottleneck paths problem

Abstract

We extend the well known bottleneck paths problem in two directions for directed unweighted graphs with positive real edge capacities. Firstly we narrow the problem domain and compute the bottleneck of the entire network in O(mlogn) time, where m and n are the number of edges and vertices in the graph, respectively. Secondly we enlarge the domain and compute the shortest paths for all possible bottleneck amounts. We present a combinatorial algorithm to solve the Single Source Shortest Paths for All Flows (SSSP-AF) problem in O(mn) worst case time, followed by an algorithm to solve the All Pairs Shortest Paths for All Flows (APSP-AF) problem in time, where t is the number of distinct edge capacities and O(n ω ) is the time taken to multiply two n-by-n matrices over a ring. We also discuss practical applications for these new problems. © 2014 Springer International Publishing Switzerland.