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