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

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 \(\beta(m,n)=\min\{k\geq 1: \log^{(k)}n\leq\frac{m}{n}\}\).