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 $O(m+n\log n)$ ($m=\mathrm{\mid }E\mathrm{\mid }$ and $n=\mathrm{\mid }V\mathrm{\mid }$) 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 $\beta (m,n)=\min \{k\ge 1:{\log }^{(k)}n\le \frac{m}{n}\}$.