Reverse Shortest Path Problem for Unit-Disk Graphs

Given a set P of n points in the plane, the unit-disk graph G_{r}(P) with respect to a parameter 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 weight of the edge is 1 in the unweighted case and is the distance between p and q in the weighted case). Given a value \lambda>0 and two points s and t of P, we consider the following reverse shortest path problem: computing the smallest r such that the shortest path length between s and t in G_r(P) is at most \lambda. 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^{7/4} n) time for the unweighted case, as well as an O(n^{5/4} \log^{5/2} n) time algorithm for the weighted case. We also consider the L_1 version of the problem where the distance of two points is measured by the L_1 metric; we solve the problem in O(n \log^3 n) time for both the unweighted and weighted cases.