Dog Bites Postman: Point Location in the Moving Voronoi Diagram and Related Problems

In this paper, we discuss two variations of the two-dimensional posto ce problem that arise when the post-o ces are n postmen moving with constant velocities. The rst 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(log n) time. The second variation views a query point as a dog searching for a postman to bite and nds the postman that a dog running with speed vd could reach rst. We show that if the dog is quicker than all of the postmen then the data structure developed for the rst problem permits us to solve this one in O(log n) time as well. The proposed structure is semi-dynamic, that is the set of postmen can be modi ed by inserting new postmen. A fully dynamic structure supporting also deletions can be obtained, but in that case the query time becomes O(log n). Key-words: Computational geometry, moving points, Voronoi, Delaunay, randomized algorithms, dynamic algorithms, localisation.