We investigate the combinatorial complexity of geodesic Voronoi diagrams on polyhedral terrains using a probabilistic analysis. Aronov et al.  prove that, if one makes certain realistic input assumptions on the terrain, this complexity is Θ(<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 Ω(<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.
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