Incremental Convex Hull Algorithms Are Not Output Sensitive

A polytope is the bounded intersection of a finite set of half-spaces of . Every polytope can also be represented as the convex hull conv of its vertices (or extreme points) . The convex hull problem is to convert from the vertex representation to the half-space 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 half-space descriptions _n-k . . . _n where _i is the half-space description of conv {v ₁ . . . v _i } . Let m _i denote | _i |, and let m denote m _n . Let φ (d) denote ; in this paper we give families of polytopes for which m _n-1 ∈ Ω m ^φ(d) ) 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 that the same families of polytopes are also hard for the other main types of convex hull algorithms known. Received November 13, 1996, and in revised form April 28, 1997.