An Iterative Algorithm for Computing Measures of Generalized Voronoi Regions

We present and analyze a fast algorithm for directly computing the measure of a generalized Voronoi region associated with generators of arbitrary co-dimension. The algorithm is based upon solving one Eikonal equation to generate a kernel-based operator whose iteration accumulates “mass” along the closest generator. In particular, the algorithm does not require the computation of the ridge set (Voronoi diagram) nor the gradient of the solution to the Eikonal equation. The algorithm is shown to be first order and converge very quickly. Several illustrations are presented including the computation of measures of influence associated with the Los Angeles County highway system. The method can also be used for the fast computation of the centroid and higher moments of the generalized Voronoi regions.