Constant-Time Convexity Problems on Reconfigurable Meshes

Abstract

The purpose of this paper is to demonstrate that the versatility of the reconfigurable mesh can be exploited to devise constant-time algorithms for a number of important computational tasks relevant to robotics, computer graphics, image processing, and computer vision. In all our algorithms, we assume that one or two n-vertex (convex) polygons are pretiled, one vertex per processor, onto a reconfigurable mesh of size √ n × √ n. In this setup, we propose constant-time solutions for testing an arbitrary polygon for convexity, solving the point location problem, solving the supporting lines problem, solving the stabbing problem, determining the minimum area/perimeter corner triangle for a convex polygon, determining the k-maximal vertices of a restricted class of convex polygons, constructing the common tangents of two separable convex polygons, deciding whether two convex polygons intersect, answering queries concerning two convex polygons, and computing the smallest distance between the boundaries of two convex polygons. To the best of our knowledge, this is the first time that O(1) time algorithms to solve dense instances of these problems are proposed on reconfigurable meshes.