In-place techniques for parallel convex hull algorithms

Abstract

We present a number of efficient parallel algorithms for constructing 2- and 3-dimensional convex hulls on a randomized CRCW PRAM. Specifically, we show how to build the convex hull of n pre-sorted points in the plane almost surely in O(1) time using O(n log n) processors, or, alternately, almost surely in O (log∗ n) time using an optimal number of processors. We also show how to find the convex hull of n unsorted points in R2 (resp., R3) in O(log2 n) time using O(n log h) work (resp., O(log2n) time using O(min{nlog2 h, n log n}) work), with very high probability, where h is the number of edges in the convex hull (h is O(n), but can be as small as O(l)). Our algorithms for unsorted input depend 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 re-ordering the input. For the pre-sorted case we also exploit a technique that allows one to modify an algorithm that assumes it is given points so that it can be used on hulls; we call such algorithms point-hull invariant.