We study the reverse shortest path problem on disk graphs in the plane. In this problem we consider the proximity graph of a set of $n$ disks in the plane of arbitrary radii: In this graph two disks are connected if the distance between them is at most some threshold parameter $r$. The case of intersection graphs is a special case with $r=0$. We give an algorithm that, given a target length $k$, computes the smallest value of $r$ for which there is a path of length at most $k$ between some given pair of disks in the proximity graph. Our algorithm runs in $O^*(n^{5/4})$ randomized expected time, which improves to $O^*(n^{6/5})$ for unit disk graphs, where all the disks have the same radius. Our technique is robust and can be applied to many variants of the problem. One significant variant is the case of weighted proximity graphs, where edges are assigned real weights equal to the distance between the disks or between their centers, and $k$ is replaced by a target weight $w$; that is, we seek a path whose length is at most $w$. In other variants, we want to optimize a parameter different from $r$, such as a scale factor of the radii of the disks. The main technique for the decision version of the problem (determining whether the graph with a given $r$ has the desired property) is based on efficient implementations of BFS (for the unweighted case) and of Dijkstra's algorithm (for the weighted case), using efficient data structures for maintaining the bichromatic closest pair for certain bicliques and several distance functions. The optimization problem is then solved by combining the resulting decision procedure with enhanced variants of the interval shrinking and bifurcation technique of [4].

翻译：我们研究了平面上圆盘图的逆向最短路径问题。在此问题中，我们考虑平面上任意半径的$n$个圆盘的邻近图：在该图中，若两个圆盘之间的距离至多为某个阈值参数$r$，则它们相连。其中，交图是$r=0$的特殊情况。我们提出了一种算法，给定目标长度$k$，该算法计算使得邻近图中某对给定圆盘之间存在长度不超过$k$的路径的最小$r$值。该算法的随机期望运行时间为$O^*(n^{5/4})$，对于单位圆盘图（所有圆盘半径相同）可改进至$O^*(n^{6/5})$。我们的技术具有鲁棒性，可应用于该问题的多种变体。一个重要的变体是加权邻近图，其中边被赋予等于圆盘间距离或圆心间距离的实权重，且$k$被替换为目标权重$w$，即寻找长度不超过$w$的路径。在其他变体中，我们需优化不同于$r$的参数，例如圆盘半径的比例因子。该问题判定版本（确定具有给定$r$的图是否满足期望性质）的主要技术基于广度优先搜索（无权情况）和迪杰斯特拉算法（加权情况）的高效实现，并利用高效数据结构维护某些双团和多种距离函数下的双色最近点对。优化问题则通过结合所得的判定过程与文献[4]中区间缩小及分叉技术的增强变体来求解。