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

TL;DR

This work introduces higher-order color Voronoi diagrams based on two distance-to-color functions and develops a colorful Clarkson–Shor framework to bound their combinatorial complexity. It proves an exact bound of $4k(n-k)-2n$ on the total number of vertices for minimal and maximal order-$k$ color diagrams under convex distance functions (including all $L_p$ metrics) and derives $O(\min\{k(n-k),(n-k)^2\})$ and $O(2k^2)$ bounds for the $L_1$/$L_∞$ cases, respectively. The authors also provide an iterative algorithm to compute these diagrams and extend the framework to a wide range of geometric structures, such as envelopes of hyperplanes, colored $j$-facets, and levels in arrangements of piecewise linear/algebraic functions and convex polyhedra. These results unify and extend classical bounds for ordinary higher-order Voronoi diagrams to colored settings and general distance functions, offering new insights and tools for colored geometric configurations with potential applications in computational geometry and related domains.

Abstract

Given a set $S$ of $n$ colored sites, each $s\in S$ associated with a distance-to-site function $δ_s \colon \mathbb{R}^2 \to \mathbb{R}$, we consider two distance-to-color functions for each color: one takes the minimum of $δ_s$ for sites $s\in S$ 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 $δ_s$ that satisfy certain conditions, including the case of point sites $S$ under convex distance functions and the $L_p$ metric for any $1\leq p \leq\infty$. 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.

Figures (6)

• Figure 1: Special cases of $\mathsf{CVD}_k(S)$ and $\overline{\mathsf{CVD}}_k(S)$ for colored points $S$ in the Euclidean plane.
• Figure 2: The minimal color Voronoi diagrams $\mathsf{CVD}_k(S)$ and the refined diagrams $\mathsf{CVD}^*_k(S)$.
• Figure 3: The maximal color diagrams $\overline{\mathsf{CVD}}_k(S)$ and the refined diagrams $\overline{\mathsf{CVD}}^*_k(S)$.
• Figure 4: Colored $j$-facets in colored points in $\mathbb{R}^2$: a $1$-chromatic $2$-facet $\sigma_1$ and a $2$-chromatic $2$-facet $\sigma_2$ are shown, which choose half-planes $\sigma_1^+$ and $\sigma_2^+$ on their right side.
• Figure 5: Selected new vertices (small squares) in $\mathsf{CVD}^*_2(S)$ and $\overline{\mathsf{CVD}}^*_2(S)$ and their corresponding circles. Green vertices are $2$-chromatic, while orange ones are $3$-chromatic. Observe that exactly one color is intersected by the interior $\hat{C}$ of each circle $C$ in (a) and the exterior $\overline{C}$ of each circle $C$ in (b).

Theorems & Definitions (40)

• Lemma 1
• Lemma 2
• Lemma 3
• Lemma 4
• Theorem 5
• Theorem 6
• Corollary 7
• Lemma 8
• Theorem 9
• Lemma 10