Randomized parallel 3D convex hull algorithm for coarse grained multicomputers

Abstract

We present a randomized parallel algorithm for constructing the 3D convex hull on a generic p-processor coarse grained multicomputer with arbitrary interconnection network and n/p local memory per processor, where n/p ≥ p2+ε (for some arbitrarily small ε > 0). For any given set of n points in 3-space, the algorithm computes the 3D convex hull, with high probability, in O(n log n÷p) local computation time and O(1) communication phases with at most O(n÷p) data sent/received by each processor. That is, with high probability, the algorithm computes the 3D convex hull of an arbitrary point set in time O(n log n÷p + Γn,p), where Γn,p denotes the time complexity of one communication phase. In the terminology of the BSP model, our algorithm requires, with high probability, O(1) supersteps and a synchronization period Θ(n log n÷p). In the LogP model, the execution time of our algorithm is asymptotically optimal for several architectures.