# Reporting Points in Halfspaces

@article{Matouek1992ReportingPI,
title={Reporting Points in Halfspaces},
author={Jiř{\'i} Matou{\vs}ek},
journal={Comput. Geom.},
year={1992},
volume={2},
pages={169-186}
}
• J. Matoušek
• Published 1 November 1992
• Computer Science, Mathematics
• Comput. Geom.
210 Citations
• Mathematics
• 2010
We re-examine the notion of relative (p, ε)-approximations, recently introduced in [CKMS06], and establish upper bounds on their size, in general range spaces of finite VC-dimension, using the
• Mathematics
ArXiv
• 2009
The notion of relative $(p,\eps)$-approximations is re-examine, recently introduced in [CKMS06], and upper bounds on their size are established, in general range spaces of finite VC-dimension, using the sampling theory developed in [LLS01] and in several earlier studies.
• Mathematics
• 2014
Given a set P of n points in R3, we show that, for any ε > 0, there exists an ε-net of P for halfspace ranges, of size O(1/ε). We give five proofs of this result, which are arguably simpler than
• Mathematics, Computer Science
ArXiv
• 2014
It is shown that, for any $\varepsilon >0$, there exists an $\vARpsilon-net of$P$for halfspace ranges, of size$O(1/\varpsilon)$, and five proofs of this result are given. • Computer Science SIAM J. Comput. • 2011 The analysis is extended to general semialgebraic ranges and shown how to adapt Matousek's technique without the need to linearize the ranges into a higher-dimensional space, which yields more efficient solutions to several useful problems. • Computer Science SIAM J. Comput. • 2010 We present a simple scheme extending the shallow partitioning data structures of Matousek, which supports efficient approximate halfspace range-counting queries in$\mathbb{R}^d\$ with relative error
This thesis obtains the first optimal data structure for approximate halfspace range counting in 3D and provides two nontrivial methods to approximate the simplicial depth of a given point in higher dimension.
• Computer Science
ACM Trans. Database Syst.
• 2022
The rank-regret representative is proposed as the minimal subset of the data containing at least one of the top-k of any possible ranking function, which is polynomial time solvable in two-dimensional space but is NP-hard on three or more dimensions.
• Computer Science
Discrete & Computational Geometry
• 2016
An optimal deterministic algorithm for constructing standard (non-shallow) cuttings in two dimensions, which is arguably simpler than the known optimal algorithms by Matoušek and Chazelle.
• Mathematics
ArXiv
• 2013
The construction of Har-Peled and Sharir [HS11] is extended to three and higher dimensions, to obtain a data structure for halfspace range counting, which uses O(nloglogn) space (and somewhat higher preprocessing cost), and answers a query in half space range counting.