Divide and Conquer for Llyear Emecfed Time

Divide-and-conquer is one of the most frequently used methods for the design or” fast algorithms. The most common application of the technique involves breaking a problem of size N into two subproblems of size N/2, solving these subproblems, then doing work proportional to N to “marry” the partial answers into a solution for the entire problem; this scheme leads to algorithms of O(N log N) worstsasc time complexity. In this paper we investigate a similar divide-and-conquer technique which can be used to construct algorithms with linear average-case time complexity. The problem of determining the convex hull of a set of points in two and three dimensions has produced a rash of recent papers [4,8,15,16], all containing algorithms with o(N logA!) worst-case performance. That this is optimal follows from the fact that in the worst case all N points may be vertices of the convex hull, and since the vertices of a convex palygon occur in sorted angular order about each interior point, any convex hull dgorit.?m must be able to sort [ 14,8] e If the boundary of the convex hull contains very few points, however, this lower bound does not apply, and a faster algorithm ma:{ be possible. The algorithm of Jarvis [S] runs in time Q&N), where it is the number of actual hull vertices,