Given a set P of n points in the plane, a unit-disk graph G_{r}(P) with respect to a radius r is an undirected graph whose vertex set is P such that an edge connects two points p, q \in P if the Euclidean distance between p and q is at most r. The length of any path in G_r(P) is the number of edges of the path. Given a value \lambda>0 and two points s and t of P, we consider the following reverse shortest path problem: finding the smallest r such that the shortest path length between s and t in G_r(P) is at most \lambda. It was known previously that the problem can be solved in O(n^{4/3} \log^3 n) time. In this paper, we present an algorithm of O(\lfloor \lambda \rfloor \cdot n \log n) time and another algorithm of O(n^{5/4} \log^2 n) time.

中文翻译：

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单位磁盘图的反向最短路径问题

给定平面中n个点的集合P，相对于半径r的单位圆图G_ {r}（P）是无向图，其顶点集为P使得边连接两个点p，q \ in如果p和q之间的欧几里得距离最多为r，则为P。G_r（P）中任何路径的长度是路径的​​边数。给定值\ lambda> 0以及P的两个点s和t，我们考虑以下反向最短路径问题：找到最小的r，以使G_r（P）中s和t之间的最短路径长度最大为\ lambda。以前已知可以在O（n ^ {4/3} \ log ^ 3 n）的时间内解决该问题。在本文中，我们提出了一种O（\ lfloor \ lambda \ rfloor \ cdot n \ log n）时间的算法，以及另一种O（n ^ {5/4} \ log ^ 2 n）时间的算法。