Convexity Problems on Meshes with Multiple Broadcasting

Abstract Our contribution is twofold. First, we show that Ω(log n) is a time lower bound on the CREW-PRAM and the mesh with multiple broadcasting for the tasks of computing the perimeter, the area, the diameter, the width, the modality, the smallest-area enclosing rectangle, and the largest-area inscribed triangle of a convex n-gon. We show that the same time lower bound holds for the tasks of detecting whether a convex n-gon lies inside another as well as for computing the maximum distance between two convex n-gons. 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. Our second contribution is to show that the Ω(log n) time lower bound is tight by providing O(log n) time algorithms to solve these problems on a mesh with multiple broadcasting of size n × n. Finally, we show that for two separable convex n-gons P and Q, the task of computing the minimum distance between P and Q can be performed in O(1) time on a mesh with multiple broadcasting of size n × n.