Reverse Shortest Path Problem for Unit-Disk Graphs

04/29/2021

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by   Haitao Wang, et al.

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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 ∈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 λ>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 λ. 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(⌊λ⌋·n logn) time and another algorithm of O(n^5/4 log^2 n) time.