Two theorems on the parallel construction of convex hulls

The parallel complexity of the problem of constructing the convex hull of a sorted planar point set is studied. For any point p in the plane, let x(p) and y(p) denote the xand y-coordinate of p. A planar point set S = {p1, p2, . . . , pN} is said to be x-sorted if the points of S are given by increasing x-coordinate, i.e., x(pi) ≤ x(pi+1) for all i ∈ {1, 2, . . . , N − 1}. The following two results are proved: (1) Given an x-sorted set S ofN points, the convex hull of S can be found inO(logN) time and O(N) space with ⌈N/ logN⌉ processors on an EREW PRAM. (2) Given an x-sorted set S of N points, a padded representation of the convex hull of S can be computed in O(log logN) time and O(N) space with ⌈N/ log logN⌉ processors on a WEAK CRCW PRAM. (If the number of points in the convex hull is h, then the full representation of the convex hull is output in an array of size O(min{h, N}), for any fixed ε > 0.) It is also shown that these algorithms are asymptotically fastest possible in their respective machine model if we insist on cost-optimality, i.e., that the product of the time complexity and the number of processors used is linear.