On the Expected Complexity of Voronoi Diagrams on Terrains

Abstract

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 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 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.