Fast Randomized Parallel Methods for Planar Convex Hull Construction

We present a number of eecient parallel algorithms for constructing 2-dimensional convex hulls on a randomized CRCW PRAM. Speciically, we show how to build the convex hull of n pre-sorted 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 nd the convex hull of n unsorted planar points in 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 deened on a subset of elements in the input and that work without reordering the input. In order to achieve our n-exponential conndence bounds we use a new parallel technique called failure sweeping.