Degenerate convex hulls on-line in any fixed dimension

Abstract

Randomized algorithms have, since the pioneering work of Clarkson and Shor, occupied the front stage of computational geometry. Yet despite their simplicity, they often cannot handle or be extended to cope with degenerate cases. The main goal of this paper is to show how, for the special case of convex hulls, a simple modification of the formulation of an on-line algorithm by Boissonnat et al., based on that of Clarkson and Shor, can handle the case of degenerate sets of points for computing convex hulls on-line in Rd , for a fixed \(d\geq 2\) . At the same time, a standard randomized analysis allows us to prove the expected time complexity to be within \(O(n\log n+n^{{\lfloor{k/2}\rfloor}})\) , where k is the dimension of the convex hull. The latter bound is always \(O(n\log n+n^{{\lfloor{d/2}\rfloor}})\) . The analysis uses connections between regions, pivots, flags, and barycentric triangulations.