Some Extensions of the All Pairs Bottleneck Paths Problem

TL;DR

This paper extends the bottleneck paths problem to compute network bottlenecks efficiently and introduces algorithms for shortest paths considering all flow amounts, with applications in network optimization.

Contribution

It presents new algorithms for bottleneck computation and shortest paths for all flows, expanding the problem domain and improving computational efficiency.

Findings

Bottleneck of entire network computed in $O(n^{ ext{omega}} ext{log}n)$ time.

Single source shortest paths for all flows solved in $O(mn)$ time.

All pairs shortest paths for all flows achieved in $O(\u221a{d}n^{( ext{omega}}+9)/4})$ time.

Abstract

We extend the well known bottleneck paths problem in two directions for directed unweighted (unit edge cost) graphs with positive real edge capacities. Firstly we narrow the problem domain and compute the bottleneck of the entire network in $O(n^{\omega}\log{n})$ time, where $O(n^{\omega})$ is the time taken to multiply two $n$-by-$n$ matrices over ring. Secondly we enlarge the domain and compute the shortest paths for all possible flow 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 $O(\sqrt{d}n^{(\omega+9)/4})$ time, where $d$ is the number of distinct edge capacities. We also discuss real life applications for these new problems.