Time-Optimal Nearest-Neighbor Computations on Enhanced Meshes

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.