Reverse Shortest Path Problem for Unit-Disk Graphs

Abstract

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.

This research was supported in part by NSF under Grant CCF-2005323. A full version of this paper is available at https://arxiv.org/abs/2104.14476.