AI 中文总结
针对两类几何图的瓶颈路径问题，提出近线性判定算法，结合该判定过程得到O*(n^(8/7))时间的瓶颈路径算法，还可求解受限跳数版本，相关算法效率较高。
AI 中文摘要
针对应用中自然产生的两种几何场景下的瓶颈路径问题，我们提出了高效算法：平面中天线角度下界为常数的定向天线图，以及顶点位于1.5维地形上或地形上方的可见性图，两种场景均以欧氏距离作为边权重。我们为对应的判定问题提供了近线性算法，即确定保留所有权重不超过阈值bn的边所得子图中是否存在从s到t的路径。随后利用这些判定过程，得到瓶颈路径问题的算法，其期望运行时间为O*(n^(8/7))，其中n为输入规模，O*(·)符号隐藏了亚多项式因子。在相同性能范围内，我们还可求解受限跳数版本，即对于给定整数k<n，仅考虑边数不超过k的s-t路径。
英文摘要
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 $k < n$.
