Farthest-Polygon Voronoi Diagrams

TL;DR

This paper studies the farthest-site Voronoi diagram for polygonal sites, proving its linear complexity, providing an efficient algorithm for its construction, and exploring its structural properties.

Contribution

It establishes the linear complexity of the diagram, presents an O(n log^3 n) construction algorithm, and analyzes key structural features.

Findings

The diagram's complexity is O(n).

An O(n log^3 n) algorithm computes the diagram.

Structural properties of the diagram are characterized.

Abstract

Given a family of k disjoint connected polygonal sites in general position and of total complexity n, we consider the farthest-site Voronoi diagram of these sites, where the distance to a site is the distance to a closest point on it. We show that the complexity of this diagram is O(n), and give an O(n log^3 n) time algorithm to compute it. We also prove a number of structural properties of this diagram. In particular, a Voronoi region may consist of k-1 connected components, but if one component is bounded, then it is equal to the entire region.