Time-optimal proximity algorithms on meshes with multiple broadcasting

The all-nearest neighbor problem (ANN) is stated as follows: given a set 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. We propose two time-optimal algorithms to solve the ANN problem for an arbitrary set of points in the plane and also for the special case in which the points are vertices of a convex polygon. Both algorithms run on meshes with multiple broadcasting. We first establish an Ω(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. This lower bound holds for both the CREW-PRAM and for the mesh with multiple broadcasting. Next, we show that the bound is tight by exhibiting algorithms solving the problem in O(log n) time on a mesh with multiple broadcasting of size n×n. The first algorithm is for an arbitrary point-set, while the second solves the problem in the special case when the points are the vertices of a convex polygon