Kinetic Convex Hulls, Delaunay Triangulations and Connectivity Structures in the Black-Box Model

Over the past decade, the kinetic-data-structures framework has become the standard in computational geometry for dealing with moving objects. A fundamental assumption underlying the framework is that the motions of the objects are known in advance. This assumption severely limits the applicability of KDSs. We study KDSs in the black-box model, which is a hybrid of the KDS model and the traditional time-slicing approach. In this more practical model we receive the position of each object at regular time steps and we have an upper bound on dmax, the maximum displacement of any point in one time step. We study the maintenance of the convex hull and the Delaunay triangulation of a planar point set P in the black-box model, under the following assumption on dmax: there is some constant k such that for any point p ∈ P the disk of radius dmax contains at most k points. We analyze our algorithms in terms of ∆k, the so-called k-spread of P . We show how to update the convex hull at each time step in O(min(n, k∆k log n) log n) amortized time. For the Delaunay triangulation our main contribution is an analysis of the standard edge-flipping approach; we show that the number of flips is O(k∆k) at each time step.