Kinetic convex hulls, Delaunay triangulations and connectivity structures in the black-box model

Abstract

Over the past decade, the kinetic-data-structures framework has become thestandard 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 d max , 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 d max : there is some constant k such that for any point p in P the disk of radius d max contains at most k points. We analyze our algorithms in terms of Δ 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 Δ 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 2 Δ k 2 ) at each time step.