Fast randomized parallel methods for planar convex hull construction

Abstract

We present a number of efficient parallel algorithms for constructing 2-dimensional convex hulls on a randomized CRCW PRAM. Specifically, we show how to build the convex hull of n presorted points in the plane in O(1) time using O( n log n) work, with n-exponential probability, or, alternately, in O( log ∗ n) time using O( n) work, with n-exponential probability. We also show how to find the convex hull of n unsorted planar points in O(log n) time using O( n log h) work, with n-exponential probability, where h is the number of edges in the convex hull ( h is O( n), but can be as small as O(1)). Our algorithm for unsorted inputs depends on the use of new in-place procedures, that is, procedures that are defined on a subset of elements in the input and that work without reordering the input. In order to achieve our n-exponential confidence bounds we use a new parallel technique called failure sweeping.