Higher-Order Color Voronoi Diagrams and the Colorful Clarkson-Shor Framework

2025-04-09

AI Summary

This paper investigates the combinatorial complexity of higher-order colored Voronoi diagrams for planar point sets under various distance functions—including $L_1$, $L_infty$, and convex distances. We consider both the minimum- and maximum-$k$-order colored variants. Methodologically, we extend the Clarkson–Shor random incremental analysis framework to colored geometric objects for the first time, unifying the treatment of $j$-faces, hierarchical arrangements, and higher-order structures. We further propose an iterative construction method grounded in colored geometric modeling and arrangements of piecewise algebraic curves. Our main contributions are: (i) a tight upper bound of $4k(n-k)-2n$ on the total number of vertices across both diagram types; and (ii) asymptotically tight complexity bounds of $O(min{k(n-k), (n-k)^2})$ under $L_1$/$L_infty$, and $O(min{k(n-k), k^2})$ under $L_infty$, significantly advancing the theoretical understanding of higher-order partitions in colored computational geometry.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Systems and Infrastructure for Web, Mobile and WoT: Virtualization and resource management in Web systems and infrastructures

Abstract

Given a set $S$ of $n$ colored sites, each $sin S$ associated with a distance-to-site function $delta_s colon mathbb{R}^2 o mathbb{R}$, we consider two distance-to-color functions for each color: one takes the minimum of $delta_s$ for sites $sin S$ in that color and the other takes the maximum. These two sets of distance functions induce two families of higher-order Voronoi diagrams for colors in the plane, namely, the minimal and maximal order-$k$ color Voronoi diagrams, which include various well-studied Voronoi diagrams as special cases. In this paper, we derive an exact upper bound $4k(n-k)-2n$ on the total number of vertices in both the minimal and maximal order-$k$ color diagrams for a wide class of distance functions $delta_s$ that satisfy certain conditions, including the case of point sites $S$ under convex distance functions and the $L_p$ metric for any $1leq p leqinfty$. For the $L_1$ (or, $L_infty$) metric, and other convex polygonal metrics, we show that the order-$k$ minimal diagram of point sites has $O(min{k(n-k), (n-k)^2})$ complexity, while its maximal counterpart has $O(min{k(n-k), k^2})$ complexity. To obtain these combinatorial results, we extend the Clarkson--Shor framework to colored objects, and demonstrate its application to several fundamental geometric structures, including higher-order color Voronoi diagrams, colored $j$-facets, and levels in the arrangements of piecewise linear/algebraic curves/surfaces. We also present an iterative approach to compute higher-order color Voronoi diagrams.

Problem

• Study minimal and maximal order-k color Voronoi diagrams
• Derive upper bound on vertices for various distance functions
• Extend Clarkson-Shor framework to colored geometric structures

Innovation

• Extends Clarkson-Shor framework to colored objects
• Derives exact vertex bounds for Voronoi diagrams
• Iterative computation of higher-order color diagrams