Reverse Shortest Path Problem for Unit-Disk Graphs

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.