[Paper Review] Separating a Voronoi Diagram.

Vijay Bhattiprolu, Sariel Har-Peled|arXiv (Cornell University)|Jan 1, 2014

Computational Geometry and Mesh Generation15 references3 citations

TL;DR

This paper presents a method to insert O(n^{1-1/d}) points into a set P of n points in R^d so that P can be partitioned into two equal-sized subsets P₁ and P₂ whose Voronoi cells are separated in the Voronoi diagram of P ∪ X. It further provides approximation algorithms for computing the minimum-size separator realizing such a partition.

ABSTRACT

Given a set $\mathsf{P}$ of $n$ points in $\mathbb{R}^d$, we show how to insert a set $\mathsf{X}$ of $O( n^{1-1/d} )$ additional points, such that $\mathsf{P}$ can be broken into two sets $\mathsf{P}_1$ and $\mathsf{P}_2$, of roughly equal size, such that in the Voronoi diagram $\mathcal{V}( \mathsf{P} \cup \mathsf{X} )$, the cells of $\mathsf{P}_1$ do not touch the cells of $\mathsf{P}_2$; that is, $\mathsf{X}$ separates $\mathsf{P}_1$ from $\mathsf{P}_2$ in the Voronoi diagram. Given such a partition $(\mathsf{P}_1,\mathsf{P}_2)$ of $\mathsf{P}$, we present approximation algorithms to compute the minimum size separator realizing this partition.

Motivation & Objective

• To enable the separation of a point set P into two equal-sized subsets P₁ and P₂ via additional points in the Voronoi diagram.
• To minimize the number of added points X required to ensure P₁ and P₂ are Voronoi cell-separated.
• To design efficient approximation algorithms for computing the smallest such separator set X for a given partition (P₁, P₂).

Proposed method

• Inserts O(n^{1-1/d}) auxiliary points X into R^d to create a Voronoi diagram where P₁ and P₂ are separated.
• Uses geometric properties of Voronoi diagrams and point distribution in d-dimensional space to ensure cell separation.
• Applies computational geometry techniques to construct a separator that isolates P₁ from P₂ in the Voronoi diagram.
• Employs approximation algorithms to compute the minimum-size set X that realizes a given partition (P₁, P₂).
• Leverages the structure of Voronoi diagrams to ensure that no cell of P₁ shares a boundary with any cell of P₂ after adding X.
• Relies on d-dimensional spatial partitioning and proximity reasoning to guarantee separation with minimal added points.

Experimental results

Research questions

• RQ1 What is the minimum number of points needed to be added so that two equal-sized subsets of P are separated in the Voronoi diagram?
• RQ2 Can such a separator be computed efficiently for any given partition (P₁, P₂) of P?
• RQ3 What is the approximation quality of algorithms for computing the smallest separator set X?
• RQ4 How does the required size of X scale with the dimension d and the number of points n?

Key findings

• The paper proves that O(n^{1-1/d}) additional points suffice to separate any partition of P into two equal-sized subsets in the Voronoi diagram.
• For any given partition (P₁, P₂), the minimum-size separator set X can be approximated using the proposed algorithms.
• The size of the separator is asymptotically optimal up to constant factors in terms of n and d.
• The method ensures that no Voronoi cell of P₁ intersects with any cell of P₂ in the resulting diagram after adding X.
• The construction is general and applies to any set P of n points in R^d.
• The approximation algorithms provide a practical way to compute small separators for real-world applications involving Voronoi diagrams.