ARTICLE NO. 0095 Time-Optimal Nearest-Neighbor Computations on Enhanced Meshes 1

as follows: given a set S of points in the plane, determine for every point in S, a point that lies closest to it. The ANN problem is central to VLSI design, computer graphics, pattern recognition, and image processing, among others. In this paper we propose time-optimal algorithms to solve the ANN problem for an arbitrary set of points in the plane and also for the special case where the points are vertices of a convex polygon. Both our algorithms run on meshes with multiple broadcasting. Our first main contribution is to establish an �(log n) time lower bound for the task of solving an arbitrary n-point instance of the ANN problem, even if the points are the vertices of a convex polygon. We obtain our time lower bound results for the CREW-PRAM by using a novel technique involving geometric constructions. These constructions allow us to reduce the wellknown OR problem to each of the geometric problems of interest. We then port these time lower bounds to the mesh with multiple broadcasting using simulation results. Our second main contribution is to show that the time lower bound obtained is tight, by exhibiting algorithms solving the problem in O(log n) time on a mesh with multiple broadcasting of size n � n. © 1996 Academic Press, Inc. 1