| Summary: | 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 <em>d</em><sub>max</sub>, the maximum displacement of any point in one time step.<p>We study the maintenance of the convex hull and the Delaunay triangulation of a planar point set <em>P</em> in the black-box model, under the following assumption on <em>d</em><sub>max</sub>: there is some constant <em>k</em> such that for any point <em>p</em> in <em>P</em> the disk of radius <em>d</em><sub>max</sub> contains at most <em>k</em> points. We analyze our algorithms in terms of Δ<sub><em>k</em></sub>, the so-called <em>k</em>-spread of <em>P</em>. We show how to update the convex hull at each time step in <em>O</em>(min(<em>n</em>, <em>k</em>Δ<sub><em>k</em></sub>log <em>n</em>)log <em>n</em>) 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 <em>O</em>(k<sup>2</sup>Δ<sub><em>k</em></sub><sup>2</sup>) at each time step.</p>
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