Capacity-Constrained Voronoi Diagrams in Finite Spaces

Abstract

A Voronoi diagram of a set of n sites partitions a finite set of m points into regions of different areas, called the capacities of the sites. In this paper we are interested in Voronoi diagrams in which the capacities are given as constraints. We compute such capacity-constrained Voronoi diagrams in finite spaces by starting with an arbitrary partition that fulfills the capacity constraint without representing a valid Voronoi diagram. We then iteratively swap the assignment of points to sites guided by an energy minimization that is related to the distance function of the Voronoi diagram. This optimization converges towards a valid Voronoi diagram that fulfills the capacity constraint due to the restriction to swap operations. The straightforward structure of our algorithm makes it easy to implement and flexible. The computational time complexity for each iteration step is O(n2+nmlog m/n). We provide examples for extensions of our algorithm such as applying different energy functions, or generating Voronoi diagrams with centroidal and/or void regions.