Relative $(p,\epsilon)$-Approximations in Geometry

TL;DR

This paper investigates the size bounds of relative $(p, heta)$-approximations in geometric range spaces, introduces new constructions with smaller sizes, and applies these to improve approximate range counting in multiple dimensions.

Contribution

It provides new upper bounds on the size of relative $(p, heta)$-approximations and introduces novel structures like spanning trees with small crossing numbers for better geometric approximations.

Findings

Established upper bounds on approximation sizes in finite VC-dimension spaces.

Constructed smaller relative $(p, heta)$-approximations for points and halfspaces.

Applied new structures to improve approximate range counting in 3D.

Abstract

We re-examine the notion of relative $(p,\eps)$-approximations, recently introduced in [CKMS06], and establish upper bounds on their size, in general range spaces of finite VC-dimension, using the sampling theory developed in [LLS01] and in several earlier studies [Pol86, Hau92, Tal94]. We also survey the different notions of sampling, used in computational geometry, learning, and other areas, and show how they relate to each other. We then give constructions of smaller-size relative $(p,\eps)$-approximations for range spaces that involve points and halfspaces in two and higher dimensions. The planar construction is based on a new structure--spanning trees with small relative crossing number, which we believe to be of independent interest. Relative $(p,\eps)$-approximations arise in several geometric problems, such as approximate range counting, and we apply our new structures to obtain efficient solutions for approximate range counting in three dimensions. We also present a simple solution for the planar case.