Constructing strongly convex hulls using exact or rounded arithmetic

Abstract

One useful generalization of the convex hull of a setS ofn points is the ź-strongly convex ź-hull. It is defined to be a convex polygon with vertices taken fromS such that no point inS lies farther than ź outside and such that even if the vertices of are perturbed by as much as ź, remains convex. It was an open question as to whether an ź-strongly convexO(ź)-hull existed for all positive ź. We give here anO(n logn) algorithm for constructing it (which thus proves its existence). This algorithm uses exact rational arithmetic. We also show how to construct an ź-strongly convexO(ź + μ)-hull inO(n logn) time using rounded arithmetic with rounding unit μ. This is the first rounded-arithmetic convex-hull algorithm which guarantees a convex output and which has error independent ofn.