Dog bites postman: Point location in the moving Voronoi diagram and related problems

Abstract

We discuss two variations of the two-dimensional post-office problem that arise when the post-offices are replaced by n postmen moving with constant velocities. The first variation addresses the question: given a point q0 and time t0 who is the nearest postman to q0 at time’t0? We present a randomized incremental data structure that answers the query in expected O(log2 n) time. The second variation views a query point as a dog searching lot a postman to bite and finds the postman that a dog running with speed v0 could reach first. We show that if the dog is quicker than all of the postmen then the data structure developed for the first problem permits us to solve the second one in O(log2 n) time as well.