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 log n) time and another algorithm of O(n 5/4 log 2 n) time.

No abstract

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 ∈ 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 λ > 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 λ. In this paper, we present an algorithm of O( λ • 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.

Let [Formula: see text] be a path graph of [Formula: see text] vertices embedded in a metric space. We consider the problem of adding a new edge to [Formula: see text] so that the radius of the resulting graph is minimized, where any center is constrained to be one of the vertices of [Formula: see text]. Previously, the “continuous” version of the problem where a center may be a point in the interior of an edge of the graph was studied and a linear-time algorithm was known. Our “discrete” version of the problem has not been studied before. We present a linear-time algorithm for the problem.

No abstract