A Delaunay refinement algorithm for quality 2-dimensional mesh generation
SODA '93 Selected papers from the fourth annual ACM SIAM symposium on Discrete algorithms
Geodesic Voronoi Diagrams on Parametric Surfaces
CGI '97 Proceedings of the 1997 Conference on Computer Graphics International
Crest Lines for Surface Segmentation and Flattening
IEEE Transactions on Visualization and Computer Graphics
Shortest paths on realistic polyhedra
SCG '07 Proceedings of the twenty-third annual symposium on Computational geometry
Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms
An Optimal-Time Algorithm for Shortest Paths on a Convex Polytope in Three Dimensions
Discrete & Computational Geometry
Computational Geometry: Theory and Applications
Parallel algorithms for approximation of distance maps on parametric surfaces
ACM Transactions on Graphics (TOG)
The Complexity of Bisectors and Voronoi Diagrams on Realistic Terrains
ESA '08 Proceedings of the 16th annual European symposium on Algorithms
Two-site Voronoi diagrams in geographic networks
Proceedings of the 16th ACM SIGSPATIAL international conference on Advances in geographic information systems
Curve and Surface Reconstruction: Algorithms with Mathematical Analysis
Curve and Surface Reconstruction: Algorithms with Mathematical Analysis
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We investigate the combinatorial complexity of geodesic Voronoi diagrams on polyhedral terrains using a probabilistic analysis. Aronov et.al. [abt-cbvdrt-08] 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(n m2/3) on the expected worst-case complexity if these assumptions on the terrain are dropped.