Reachability Problems for Transmission Graphs

Abstract

Let P be a set of n points in the plane where each point p of P is associated with a radius \(r_p>0\). The transmission graph \(G=(P,E)\) of P is defined as the directed graph such that E contains an edge from p to q if and only if \(|pq|\le r_p\) for any two points p and q in P, where |pq| denotes the Euclidean distance between p and q. In this paper, we present a data structure of size \(O(n^{5/3})\) such that for any two points in P, we can check in \(O(n^{2/3})\) time if there is a path in G between the two points. This is the first data structure for answering reachability queries whose performance depends only on n but not on the ratio between the largest and smallest radii.