Incremental convex hull algorithms are not output sensitive

Abstract

A polytope is the bounded intersection of a finite set of half-spaces of ℝ^d. Every polytope can also be represented as the convex hull conv ν of its vertices (extreme points) ν. The convex hull problem is to convert from the vertex representation to the halfspace representation or (equivalently by geometric duality) vice-versa. Given an ordering v ₁ ... v _n of the input vertices, after some initialization an incremental convex hull algorithm constructs halfspace descriptions H _n-k ... H _n where H _n is the halfspace description of conv{v ₁ ... v _i}. Let m _i denote |H _i|, and let m denote m _n. In this paper we give families of polytopes for which $m_{n - 1} \in \Omega \left( {m^{\sqrt {{d \mathord{\left/{\vphantom {d 2}} \right.\kern-\nulldelimiterspace} 2}} } } \right)$ for any ordering of the input. We also give a family of 0/1-polytopes with a similar blowup in intermediate size. Since m _n−1 is not bounded by any polynomial in m, n, and d, incremental convex hull algorithms cannot in any reasonable sense be considered output sensitive. It turns out the same families of polytopes are also hard for the other main types of convex hull algorithms known.

This research supported by FCAR Québec and NSERC Canada