Finding the Convex Hull of Discs in Parallel

Abstract

We present a parallel method for finding the convex hull of planar discs in the EREW PRAM model. We show that the convex hull of n discs in the plane can be computed in O( log 1+ε n) time using O(n/ log ε n) processors or in O( log n log log n) time using O(n log 1+ε n) processors for any positive constant ε. The first result achieves cost optimal and the second one runs faster. We also show that the convex hull of planar discs can be constructed in O( log n) time using O(n) processors when the number of different kinds of radii is restricted to 2O( log α n) for any positive constant α with 0 < α < 1. Finally, we show that our method can be generalized to computing the convex hull of a large class of planar curves. We use a technique called multi-level divide-and-conquer in our algorithm.