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. [AdBT08] prove that, if one makes certain realistic input assumptions on the terrain, this complexity is Θ(n+m√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 Ω(nm ^2/3) on the expected worst-case complexity if these assumptions on the terrain are dropped.