Single-Source Bottleneck Path Algorithm Faster than Sorting for Sparse Graphs

TL;DR

This paper presents a faster randomized algorithm for finding bottleneck paths from a single source to all destinations in sparse directed graphs, surpassing classic methods by reducing the time complexity.

Contribution

It introduces the first algorithm that improves upon the Fibonacci heap bound for single-source bottleneck path problems in sparse graphs.

Findings

Achieves $O(m\sqrt{ ext{log} hinspace n})$ time complexity.

Breaks the classic Fibonacci heap time bound for certain graph densities.

Uses a Las-Vegas randomized approach.

Abstract

In a directed graph $G=(V,E)$ with a capacity on every edge, a \emph{bottleneck path} (or \emph{widest path}) between two vertices is a path maximizing the minimum capacity of edges in the path. For the single-source all-destination version of this problem in directed graphs, the previous best algorithm runs in $O(m+n\log n)$ ($m=|E|$ and $n=|V|$) time, by Dijkstra search with Fibonacci heap [Fredman and Tarjan 1987]. We improve this time bound to $O(m\sqrt{\log n})$, thus it is the first algorithm which breaks the time bound of classic Fibonacci heap when $m=o(n\sqrt{\log n})$. It is a Las-Vegas randomized approach. By contrast, the s-t bottleneck path has an algorithm with running time $O(m\beta(m,n))$ [Chechik et al. 2016], where $\beta(m,n)=\min\{k\geq 1: \log^{(k)}n\leq\frac{m}{n}\}$.