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 |pq|≤ 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.