Time-optimal nearest-neighbor computations on enhanced meshes

Abstract

A simple polygon P is said to be unimodal if for every vertex of P, the Euclidian distance function to the other vertices of P is unimodal. The study of unimodal polygons has emerged as a fruitful area of computational and discrete geometry. Is the well-known that nearest and furthest neighbor computations are a recurring theme in pattern recognition, VLSI design, computer graphics, and image processing, among others. Our contribution is to propose time-optimal algorithms for constructing the Euclidian Minimum Spanning Tree, the Relative Neighborhood Graph, as well as the Symmetric Further Neighbor Graph of an n-vertex unimodal polygon on meshes with multiple broadcasting. We begin by establishing a Ω(log n) time lower bound for solving arbitrary instances of size n of these problems. This lower bound holds for both the CREW-PRAM and for the mesh with multiple broadcasting. 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 well-known 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.

Next, we show that the time lower bound is tight by exhibiting algorithms for these tasks running in O(log n) time on a mesh with multiple broadcasting of size n×n.