On the expected complexity of voronoi diagrams on terrains


We investigate the combinatorial complexity of geodesic Voronoi diagrams on polyhedral terrains using a probabilistic analysis. Aronov et al. [2008] prove that, if one makes certain realistic input assumptions on the terrain, this complexity is &Theta;(<i>n</i>+<i>m</i>&sqrt;<i>n</i>) in the worst case, where <i>n</i> denotes the number of triangles that define the terrain and <i>m</i> denotes the number of Voronoi sites. We prove that, under a relaxed set of assumptions, the Voronoi diagram has expected complexity <i>O</i>(<i>n</i>+<i>m</i>), given that the sites are sampled uniformly at random from the domain of the terrain (or the surface of the terrain). Furthermore, we present a construction of a terrain that implies a lower bound of &#937;(<i>nm</i><sup>2/3</sup>) on the expected worst-case complexity if these assumptions on the terrain are dropped. As an additional result, we show that the expected fatness of a cell in a random planar Voronoi diagram is bounded by a constant.

DOI: 10.1145/2846099

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@inproceedings{Driemel2012OnTE, title={On the expected complexity of voronoi diagrams on terrains}, author={Anne Driemel and Sariel Har-Peled and Benjamin Raichel}, booktitle={Symposium on Computational Geometry}, year={2012} }