Queries on Voronoi diagrams of moving points

Suppose we are given n moving postmen described by their motion equations p_i(t) = S_i + v_it, i = 1,...,n, where S_i ∈ ℝ² is the position of the ith postman at time t = 0, and v_i ∈ ℝ² is his velocity. The problem we address is how to preprocess the postmen data so as to be able to efficiently answer two types of nearest-neighbor queries. The first one asks "who is the nearest postman at time t_q to a dog located at point s_q. In the second type a query dog is located at point s_q at time t_q, its speed is v_q > |v_i| (for all i = 1,...,n), and we want to know which postman the dog can catch first. The first type of query is relatively simple to address, the second type at first seems much more complicated. We show that the problems are very closely related, with efficient solutions to the first type of query leading to efficient solutions to the second. We then present two solutions to these problems, with tradeoff between preprocessing time and query time. Both solutions use deterministic data structures.