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

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 = {s1, s2, · · · , sm} ∈ T , we prove that the complexity of Voronoi diagram VT (S) of S on T is O(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 VT (S) is O((m+ g)n).