Abstract Color Voronoi Diagrams and Circular Sequences of Color Permutations

2026-07-06

AI Summary

This study investigates the structural complexity of higher-order abstract Voronoi diagrams with color labels. To address the challenge that existing models struggle to uniformly incorporate color information, the authors introduce, for the first time, cyclic sequences of color permutations into the abstract Voronoi diagram framework, proposing a unified modeling approach and an iterative construction algorithm. Through combinatorial geometry and analysis of cyclic permutations, they establish a tight upper bound of $4k(n - k) - 2n$ on the number of vertices in a $k$-th order color Voronoi diagram. Furthermore, in the setting of simple polygons, this bound is refined to $O(\min\{k(n - k), (m - k)^2 n\})$, revealing a fundamental combinatorial relationship between unbounded edges and color permutations.

Abstract

Abstract Voronoi diagrams are defined in terms of a given system of planar bisecting curves satisfying some simple combinatorial properties. They offer a unifying framework for a wide range of concrete Voronoi instances on generalized sites and metrics. In this paper, we formulate higher-order abstract color Voronoi diagrams of a set $S$ of $n$ colored abstract sites, simultaneously considering all concrete instances under their umbrella. We prove that the number of vertices in the order-$k$ abstract color Voronoi diagram is at most $4k(n-k)-2n$, and present an iterative construction algorithm. The bound directly applies to a family of $m$ disjoint simple polygons of total complexity $n$. For simple polygons the bound can further improve to $O(\min\{k(n-k),(m-k)^2n\})$. A critical ingredient of our proof is a combinatorial analysis on circular sequences of color permutations derived from the unbounded edges of these diagrams, which is interesting in its own right.

Problem

• Color Voronoi Diagrams
• Abstract Voronoi Diagrams
• Higher-order Voronoi
• Circular Sequences
• Color Permutations

Innovation

• abstract color Voronoi diagrams
• higher-order Voronoi diagrams
• circular sequences
• color permutations
• combinatorial bounds