On Generalization of Voronoi Diagram

—Consideration on Space Non-uniformity due to Network

Abstract

Voronoi diagram is a typical partitioning of plane according to a number of given points on the plane referred to as generators, based on the Euclidean distances from the points. In the current study, a generalization of such voronoi diagram is discussed from the viewpoint of various consideration on distance. On the basis of discrete voronoi decomposition approach, we take into account the various distance metrics other than the conventional Euclidean distance. The existence of a pathway network to shorten the distance gives the space a non-uniformity in distance. We propose an approach for voronoi decomposition under this non-uniformity. Different weights of generators as well as various evaluations of the distance for the voronoi decomposition are also taken into consideration. A number of calculated examples demonstrate the significance of these various conditions on the obtained voronoi diagrams.