A Note on Finding Convex Hulls Via Maximal Vectors

The problem of :‘inding the convex huh of n points has received widespread attention in the past decade. In particular, if Xr, .,., X, are independent identically distributed random vectors from Rd with common density f, the following questions were investigated: if C is the complexity of the convex hull algorithm for X1, . . . . X, (thus, C is a random variable), then how do ess sup C (the ‘worst-case complexity’) and E(C) (the ‘average complexity’) increase with n for particular densities f? There are algorithms that have worst-case complexity O(n log n) for all densities f [ 1,5,10,1 1 ] on R*. The algorithms of Jarvis [6] and Eddy Ed.1 have worst-case complexity O(n*). Recently, several algorithms were shown to exhibit linear average complexities (E(C) = O(n)) for certain classes of densities on R*: (1) The ‘divide and conquer’ method of Bentley and Shamos does so whenever E(N), the expected number of points on the convex hull, satisfies E(N) = O(nP), p < 1. The latter condition 3s fulfilled, for example, when f is the uniform density on a convex r-gon [9] or when f is normal [8]. (2) The elimination method of Toussaint [l l] is known to do so for uniform densities on the unit square, and for all radial densities with a monotone and slow-varying tail [3]. (3) The recent method of Bentley et al. (21 that is based upon first finding the set of maximal vectors, has E(C) = O(n) whenever f can be written as a d-fold product of densities: