[Paper Review] Small Strong Epsilon Nets

Pradeesha Ashok, Azmi, Umair+1 more|NOT FOUND REPOSITORY (Indian Institute of Science Bangalore)|Aug 14, 2012

Optics and Image Analysis6 citations

TL;DR

This paper introduces and analyzes small strong $ε$-nets for geometric families including axis-parallel rectangles, halfspaces, and disks in $×^d$. It establishes the existence of strong centerpoints for axis-parallel boxes and provides tight upper and lower bounds on $\epsilon_i^\mathcal{S}$, the minimal $\epsilon$ for which an $i$-sized subset of $P$ forms an $\epsilon$-net with respect to $\mathcal{S}$, showing $\epsilon_i^\mathcal{R} \geq 1/i$ for $i \geq 4$ and $\epsilon_i^\mathcal{D} \geq 1/i$ for $i \geq 2$. The key contribution is a systematic study of small strong $\epsilon$-nets with exact bounds for natural geometric families.

ABSTRACT

Let P be a set of n points in $\mathbb{R}^d$. A point x is said to be a centerpoint of P if x is contained in every convex object that contains more than $dn\over d+1$ points of P. We call a point x a strong centerpoint for a family of objects $\mathcal{C}$ if $x \in P$ is contained in every object $C \in \mathcal{C}$ that contains more than a constant fraction of points of P. A strong centerpoint does not exist even for halfspaces in $\mathbb{R}^2$. We prove that a strong centerpoint exists for axis-parallel boxes in $\mathbb{R}^d$ and give exact bounds. We then extend this to small strong $ε$-nets in the plane and prove upper and lower bounds for $ε_i^\mathcal{S}$ where $\mathcal{S}$ is the family of axis-parallel rectangles, halfspaces and disks. Here $ε_i^\mathcal{S}$ represents the smallest real number in $[0,1]$ such that there exists an $ε_i^\mathcal{S}$-net of size i with respect to $\mathcal{S}$.

Motivation & Objective

• To investigate the existence and bounds of strong $\epsilon$-nets, where the net must be a subset of the point set $P$, for geometric families such as axis-parallel rectangles, halfspaces, and disks.
• To determine the minimal $\epsilon$-value, denoted $\epsilon_i^\mathcal{S}$, for which an $i$-point subset of $P$ forms a strong $\epsilon$-net with respect to a family $\mathcal{S}$ of geometric objects.
• To extend the concept of centerpoints by requiring the centerpoint to be a point in $P$, thus introducing the notion of a strong centerpoint.
• To provide tight upper and lower bounds on $\epsilon_i^\mathcal{S}$ for small $i$, particularly for $i \leq 3$, and to explore the asymptotic behavior for larger $i$.
• To resolve open questions on the exact value of $\epsilon_i^\mathcal{S}$ for small $i$ by proving new bounds using geometric and combinatorial arguments.

Proposed method

• Proving that a strong centerpoint exists for axis-parallel boxes in $\mathbb{R}^d$ by showing that for any $n$-point set, there exists a point in $P$ contained in every axis-parallel box containing more than $\frac{2d-1}{2d}n$ points.
• Using geometric constructions involving symmetric point arrangements and partitioning of the plane to derive lower bounds on $\epsilon_i^\mathcal{S}$, particularly for disks and rectangles.
• Applying induction and known bounds for $i=2$ and $i=3$ to prove $\epsilon_i^\mathcal{D} \geq \frac{1}{i}$ for $i \geq 2$, leveraging the fact that $\epsilon_2^\mathcal{D} \geq \frac{1}{2}$.
• Constructing $i$ non-overlapping axis-parallel rectangles that each cover $k$ consecutive points on a circle, showing that $i$ points can cover at most $ik$ rectangles, thus proving $\epsilon_i^\mathcal{R} \geq \frac{1}{i}$ for $i \geq 4$.
• Using duality and geometric covering arguments to show that for any $i$-point set $Q$, there exists a disk or rectangle avoiding $Q$ but containing $\frac{n}{i}$ points, implying $\epsilon_i^\mathcal{S} \geq \frac{1}{i}$.
• Leveraging known results on weak $\epsilon$-nets and VC-dimension to contextualize the bounds and justify the necessity of strong nets in low-complexity geometric families.

Experimental results

Research questions

• RQ1 Does a strong centerpoint exist for axis-parallel boxes in $\mathbb{R}^d$, where the centerpoint must be a point in the input set $P$?
• RQ2 What is the minimal $\epsilon$ such that there exists an $i$-point subset of $P$ that intersects every object in $\mathcal{S}$ containing more than $\epsilon n$ points, for $\mathcal{S}$ being rectangles, halfspaces, or disks?
• RQ3 Can tight lower and upper bounds be established for $\epsilon_i^\mathcal{S}$ for small $i$, particularly $i=1,2,3$, and how do they compare to known weak $\epsilon$-net bounds?
• RQ4 Is $\epsilon_i^\mathcal{R} \geq \frac{1}{i}$ for $i \geq 4$, and can this bound be achieved via a construction of non-overlapping rectangles covering $k$-point groups on a circle?
• RQ5 What is the exact value of $\epsilon_i^\mathcal{S}$ for $i \geq 2$ in the case of disks, and does the bound $\epsilon_i^\mathcal{D} \geq \frac{1}{i}$ hold with equality?

Key findings

• A strong centerpoint exists for axis-parallel boxes in $\mathbb{R}^d$, where every such box containing more than $\frac{2d-1}{2d}n$ points of $P$ contains a point from $P$, and this bound is tight.
• For axis-parallel rectangles in $\mathbb{R}^2$, $\epsilon_1^\mathcal{R} = \frac{3}{4}$, meaning every rectangle containing more than $\frac{3}{4}n$ points must contain one of the $n$ points.
• For halfspaces in $\mathbb{R}^2$, $\epsilon_1^\mathcal{H} = 1$, indicating that no single point in $P$ can be guaranteed to lie in all halfspaces containing more than $n$ points, hence no strong centerpoint exists.
• For disks in $\mathbb{R}^2$, $\epsilon_1^\mathcal{D} = \frac{3}{4}$, meaning every disk containing more than $\frac{3}{4}n$ points must contain one of the $n$ points.
• For $i=2$, $\epsilon_2^\mathcal{R} \geq \frac{5}{9}$ and $\epsilon_2^\mathcal{H} \geq \frac{3}{5}$, with upper bounds $\frac{5}{8}$ and $\frac{2}{3}$ respectively, showing that two points suffice to hit all rectangles or halfspaces containing more than $\frac{5}{9}n$ or $\frac{3}{5}n$ points.
• For $i=3$, $\epsilon_3^\mathcal{R} \geq \frac{2}{5}$ and $\epsilon_3^\mathcal{H} \geq \frac{1}{2}$, with upper bounds $\frac{9}{16}$ and $\frac{1}{2}$, respectively, indicating that three points can form a strong $\epsilon$-net for rectangles with $\epsilon \geq \frac{2}{5}$.