Voronoi diagrams with overlapping regions

Abstract

Voronoi diagrams are a tessellation of the plane based on a given set of points (called Voronoi points) into polygons so that all the points inside a polygon are closest to the Voronoi point in that polygon. All the regions of existing Voronoi diagrams are mutually exclusive. There are numerous applications of Voronoi diagrams for the solution of many optimization problems. In this paper, we define overlapping Voronoi diagrams so that the Voronoi regions do not partition the plane into disjoint regions. Rather, we allow for points to belong to several regions as long as the distance to a Voronoi point does not exceed p % over the shortest distance to all Voronoi points. The concept is illustrated on a case study of delineating overlapping service areas for public universities. We also apply overlapping Voronoi diagrams to solve the problem of the minimum possible p % required to achieve equal load assigned to each facility. Customers (a fraction of all customers residing at a demand point) can be assigned to a facility within p % of the minimum distance. Computational results are presented. This model is the multiplicative model. The additive model replaces p % over the minimum distance with an extra distance Δd and minimizing Δd.