We present efficient algorithms for the bottleneck path problem in two geometric settings that arise naturally in applications: directional-antenna graphs in the plane with antenna angles bounded from below by a constant, and visibility graphs whose vertices lie on or above a 1.5-dimensional terrain, both with Euclidean distances as edge weights. We provide near-linear algorithms for the corresponding decision problems, namely, determining whether the subgraph obtained by retaining all edges with weight at most some threshold ${\bf bn}$ contains a path from $s$ to $t$. We then use the decision procedures to obtain algorithms for the bottleneck path problem that run in $O^*(n^{8/7})$ randomized expected time, where $n$ is the input size and the $O^*(\cdot)$ notation hides subpolynomial factors. Within the same performance bounds, we can also solve the bounded-hop version, in which we only consider $s$-$t$ paths with at most $k$ edges, for a given integer $k < n$.

翻译：暂无翻译