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 for the purpose of devising constant time algorithms for a number of important computational geometry tasks relevant to image processing, computer graphics, 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 context, we propose constant-time solutions for testing an arbitrary polygon for convexity, solving the point location problem, the supporting lines problem, the stabbing problem, constructing the common tangents for separable convex polygons, deciding whether two convex polygons intersect, 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 are proposed for these problems on this architecture for the “dense” case. The proposed algorithms translate immediately into constant-time algorithms that work on binary images.