Convex hull: Incremental variations on the Akl-Toussaint heuristics Simple, optimal and space-saving convex hull algorithms

Abstract

Convex hulls are a fundamental geometric tool used in a number of algorithms. A famous paper by Akl and Toussaint in 1978 described a way to reduce the number of points involved in the computation, which is since known as the Akl-Toussaint heuristics. This paper first studies what this heurstics really represents in terms of reduction of points and demonstrates that the optimum selection is reached using an octogon as the remaining number of points is in O(sqrt(N)) rather than the usual O(N). Then it focuses on optimising the overall computational efficiency in a convex hull computation. Although the heuristics is usually used as a first step in computations one can obtain the convex hull directly from the heuristics's basis. First a simple incremental implementation is described, and if the number of characteristic points of the Akl-Toussaint heuristics p is taken as a parametre the convex hull is then computed in a O(N(p+h/p)) average complexity or O(Nh) asymptotic complexity. Given the relative constant factor of 1/p however experimental results show that this algorithm should be considered linear in average. Worst-case complexity is in O(N^2) and space complexity is O(h) but could be O(1) if the required output is the array of convex vertices's indexes. Then a remark on why the basic incremental method should be preferred for average cases is made. Finally an optimal linear algorithm both in average and worst-case and using a minimal space complexity in O(sqrt(N)) in average (or O(1) if in-place computation is allowed) is presented.