Search Convex Hull Algorithm

This note concerns the computation of the convex hull of a given set P = {p1, p2, . . . , pn} of n points in the plane. Let h denote the size of the convex hull, ie the number of its vertices. The value h is not known beforehand, and it can range anywhere from a small constant to n. We hav e already seen that any convex hull algorithm requires at least Ω(n lg n) time in the worst case, and have studied a number of algorithms, such as Graham’s scan algorithm, whose worst case time complexity is O(n lg n). If n was the only measure of problem size, then these algorithms are optimal. However, we also know that the Jarvis march algorithm requires O(nh) time. The latter can range anywhere from O(n) to O(n) depending on the value of h. Is there an algorithm which is asymptotically superior to both Graham scan and Jarvis march, for all possible values of h? Below, we will describe Kirkpatrick and Seidel’s [KiS86] algorithm that requires O(n lg h) time.