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 ω(n+m √n) in the worst case, where n denotes the number of triangles that define the terrain and mdenotes 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 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 (nm^2/3) 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.