On the Expected Complexity of Voronoi Diagrams on Terrains

Anne Driemel; Sariel Har-Peled; Benjamin Raichel

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arXiv · 1112.0695

On the Expected Complexity of Voronoi Diagrams on Terrains

Anne Driemel·Sariel Har-Peled·Benjamin Raichel

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Abstract

We investigate the combinatorial complexity of geodesic Voronoi diagrams on polyhedral terrains using a probabilistic analysis. Aronov etal [ABT08] prove that, if one makes certain realistic input assumptions on the terrain, this complexity is Θ(n + m \sqrt n) in the worst case, where n denotes the number of triangles that define the terrain and m denotes the number of Voronoi sites. We prove that under a relaxed set of assumptions the Voronoi diagram has expected complexity O(n+m), given that the sites have a uniform distribution on the domain of the terrain(or the surface of the terrain). Furthermore, we present a worst-case construction of a terrain which implies a lower bound of Vmega(n m2/3) on the expected worst-case complexity if these assumptions on the terrain are dropped. As an additional result, we can show that the expected fatness of a cell in a random planar Voronoi diagram is bounded by a constant.

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Anne Driemel, Sariel Har-Peled, Benjamin Raichel. 2011-12-03. On the Expected Complexity of Voronoi Diagrams on Terrains. https://arxiv.org/abs/1112.0695

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