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Improved Bounds for Discrete Voronoi Games

Mark de Berg, Geert van Wordragen

TL;DR

Lower bounds on the number of voters Player 1 wins under optimal play of the game in R 2 with ℓ = 1 are presented and better bounds are proved with an algorithm based on the quadtree of V.

Abstract

In the planar one-round discrete Voronoi game, two players $\mathcal{P}$ and $\mathcal{Q}$ compete over a set $V$ of $n$ voters represented by points in $\mathbb{R}^2$. First, $\mathcal{P}$ places a set $P$ of $k$ points, then $\mathcal{Q}$ places a set $Q$ of $\ell$ points, and then each voter $v\in V$ is won by the player who has placed a point closest to $v$. It is well known that if $k=\ell=1$, then $\mathcal{P}$ can always win $n/3$ voters and that this is worst-case optimal. We study the setting where $k>1$ and $\ell=1$. We present lower bounds on the number of voters that $\mathcal{P}$ can always win, which improve the existing bounds for all $k\geq 4$. As a by-product, we obtain improved bounds on small $\varepsilon$-nets for convex ranges. These results are for the $L_2$ metric. We also obtain lower bounds on the number of voters that $\mathcal{P}$ can always win when distances are measured in the $L_1$ metric.

Improved Bounds for Discrete Voronoi Games

TL;DR

Lower bounds on the number of voters Player 1 wins under optimal play of the game in R 2 with ℓ = 1 are presented and better bounds are proved with an algorithm based on the quadtree of V.

Abstract

In the planar one-round discrete Voronoi game, two players and compete over a set of voters represented by points in . First, places a set of points, then places a set of points, and then each voter is won by the player who has placed a point closest to . It is well known that if , then can always win voters and that this is worst-case optimal. We study the setting where and . We present lower bounds on the number of voters that can always win, which improve the existing bounds for all . As a by-product, we obtain improved bounds on small -nets for convex ranges. These results are for the metric. We also obtain lower bounds on the number of voters that can always win when distances are measured in the metric.
Paper Structure (8 sections, 5 theorems, 1 equation, 3 figures, 1 table, 1 algorithm)

This paper contains 8 sections, 5 theorems, 1 equation, 3 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Background and motivation.
  3. Previous work.
  4. Our results.
  5. Better ε-nets for convex ranges
  6. A quadtree-based strategy for player P
  7. The algorithm.
  8. An analysis of the number of voters player $\mathcal{Q}$ can win.

Key Result

Theorem 1

Let $\mu$ be a finite, absolutely continuous measure on $\mathbb{R}^2$. For any given $\alpha,\beta,\gamma \geqslant 0$ such that $2\alpha+2\beta+2\gamma=\mu(\mathbb{R}^2)$, we can find a set of three concurrent lines that partitions the plane into six wedges with measure $\alpha, \beta, \gamma, \al

Figures (3)

  • Figure 1: Lower bounds on the fraction of voters that $\mathcal{P}$ can win as a function of $k$ (the number of points of $\mathcal{P}$) when $\mathcal{Q}$ has a single point, for the $L_2$-metric. The red and green graphs do not intersect, so for large $k$ the quadtree method gives the best solution.
  • Figure 2: Illustration for the proof of Theorem \ref{['thm:l2lowk']}.
  • Figure 3: (i) The 13 points (in red) placed in $P$ for a region $R\in\mathcal{R}$. (ii) A region $R$ (shown in green) and its blocks (that is, its child regions). The white area is covered by regions that were created before $R$. Since $\hbox{\sc sw}(\sigma(R))$ has already been fully covered, $\hbox{\sc sw}(R)$ does not exist. (iii) The eight points placed in $P$ for a type-II block $B\in\mathcal{B}$.

Theorems & Definitions (5)

  • Theorem 1: Theorem 3 of Ceder64
  • Lemma 2
  • Theorem 4
  • Corollary 5
  • Lemma 7