Dog Bites Postman
In this paper, we discuss two variations of the two-dimensional post-office problem that arise when the post-offices are n postmen moving with constant velocities. The first variation addresses the question: given a point q0and time t0who is the nearest postman to q0at time t0? 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 vdcould reach first. While it is quite straightforward to design a data structure for the first problem, designing one for the second appears more difficult. We show that if the dog is quicker than all of the postmen then there is a nice correspondence between the problems. This correspondence will permit us to use the data structure developed for the first problem to solve the second one in O( log2n) time as well.The proposed structure is semi-dynamic, that is the set of postmen can be modified by inserting new postmen. A fully dynamic structure that also supports deletions can be obtained, but in that case the query time becomes O( log3n).