Dilation, smoothed distance, and minimization diagrams of convex functions

TL;DR

This paper investigates Voronoi diagrams based on sums of convex functions, establishing their properties, complexity, and applications to smoothed distance metrics and network dilation analysis.

Contribution

It introduces new theoretical results on the structure and complexity of Voronoi diagrams for convex function-based distances and applies these to smoothed distance metrics and network analysis.

Findings

Voronoi diagrams have linear complexity under certain growth conditions.

Level sets form pseudocircles, and bisectors form pseudoline arrangements.

Efficient algorithms exist for constructing these diagrams and applying them to network dilation.

Abstract

We study Voronoi diagrams for distance functions that add together two convex functions, each taking as its argument the difference between Cartesian coordinates of two planar points. When the functions do not grow too quickly, then the Voronoi diagram has linear complexity and can be constructed in near-linear randomized expected time. Additionally, the level sets of the distances from the sites form a family of pseudocircles in the plane, all cells in the Voronoi diagram are connected, and the set of bisectors separating any one cell in the diagram from each of the others forms an arrangement of pseudolines in the plane. We apply these results to the smoothed distance or biotope transform metric, a geometric analogue of the Jaccard distance whose Voronoi diagrams can be used to determine the dilation of a star network with a given hub. For sufficiently closely spaced points in the plane, the Voronoi diagram of smoothed distance has linear complexity and can be computed efficiently. We also experiment with a variant of Lloyd's algorithm, adapted to smoothed distance, to find uniformly spaced point samples with exponentially decreasing density around a given point.