The complexity of geodesic Voronoi diagrams on triangulated 2-manifold surfaces

摘要

We study the combinatorial complexity of Voronoi diagram of point sites on a general triangulated 2-manifold surface, based on the geodesic metric. Given a triangulated 2- manifold T of n faces and a set of m point sites S = {S_1,S_2.....S_m) ∈ T, we prove that the complexity of Voronoi diagram VXS) of S on I is 0(mn) if the genus of T is zero. For a genus-g manifold T in which the samples in S are dense enough and the resulting Voronoi diagram satisfies the closed ball property, we prove that the complexity of Voronoi diagram V_T(S) is O((m + g)n).