Round optimal parallel algorithms for the convex hull of sorted points

In this paper, we present deterministic parallel algorithms for the convex hull of sorted points and their application to a related problem. The algorithms are proposed for the coarse grained multicomputer (CGM) model. We first propose a cost optimal parallel algorithm for computing the problem with a constant number of communication rounds for n/p greater than or equal to P-2, where n is the size of an input and p is the number of processors. Next we propose a cost optimal algorithm, which is more complicated, for n/q greater than or equal to p(epsilon). where 0 < epsilon < 2. From the above two results, we can compute the convex hull of sorted points with O(n/p) computation time and a constant number of communication rounds for n/p greater than or equal to P-epsilon. where epsilon > 0. Finally we show an application of our convex hull algorithms. We solve the convex layers for d lines in O(n log n/p) computation time with a constant number of communication rounds. The algorithm is also cost optimal for the problem.