Reachability Problems for Transmission Graphs

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 vertical bar pq vertical bar <= 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 radius ratio.