Revisiting Hyperbolic Voronoi Diagrams from Theoretical, Applied and Generalized Viewpoints

Voronoi diagram in the hyperbolic space, hyperbolic Voronoi diagrams for short, as well as that with respect to information-geometric divergences has been investigated since mid 1990´s by Onishi et al. This paper revisits the hyperbolic Voronoi diagram from three standpoints, background theory, new applications, and geometric extensions. First, viewed from statistical estimation and information geometry, in the parametric space of normal distributions, the hyperbolic Voronoi diagram is induced by the Fisher metric while the divergence diagram is given by the Kullback-Leibler divergence on a dually flat structure. We show that the hyperbolic Voronoi diagram becomes flat by a linearization similar to one of two flat coordinates in the dually flat space, and it is a part of some power diagram. This gives another proof for the result recently obtained by Nielsen and Nock on the hyperbolic Klein model, completely different linearized model. Our result is interesting in view of the linearization having information geometric interpretations. Second, from the viewpoint of new applications, we discuss the relation between the hyperbolic Voronoi diagram and the greedy embedding in the hyperbolic plane. Kleinberg proved that in the hyperbolic plane the greedy routing is always possible. We point out that results of previous studies about the greedy embedding use a property that any tree is realized as a hyperbolic Delaunay graph easily. Finally, we generalize sites of hyperbolic Voronoi diagrams. Specifically, we consider Voronoi diagrams of geodesic segments in the upper half-plane and propose an algorithm for them.