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

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 qo and time to who is the nearest postman to qo at time to? We present a randomized incremental data structure that answers the query in expected O(log2n) time. The second variation views a query point as a dog searching for a postman to bite and finds the postman that a dog running with speed vo 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(log2n) time as well.