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

Authors:
Yong-Jin Liu;Kai Tang
Affiliations:
Tsinghua National Lab for Information Science and Technology, Department of Computer Science and Technology, Tsinghua University, Beijing, PR China;Department of Mechanical Engineering, The Hong Kong University of Science and Technology, Hong Kong, China
Venue:
Information Processing Letters
Year:
2013

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Abstract

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 V"T(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 V"T(S) is O((m+g)n).