Reporting Points in Halfspaces

  title={Reporting Points in Halfspaces},
  author={Jiř{\'i} Matou{\vs}ek},
  journal={Comput. Geom.},
  • J. Matoušek
  • Published 1 November 1992
  • Computer Science, Mathematics
  • Comput. Geom.

Ja n 20 10 Relative ( p , ε )-Approximations in Geometry ∗

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

Relative (p,epsilon)-Approximations in Geometry

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.

C G ] 1 2 O ct 2 01 4 ε-Nets for Halfspaces Revisited ∗

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

Epsilon-Nets for Halfspaces Revisited

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.

Semialgebraic Range Reporting and Emptiness Searching with Applications

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.

Approximate Halfspace Range Counting

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

On Geometric Range Searching, Approximate Counting and Depth Problems

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.

On Finding Rank Regret Representatives

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.

Optimal Deterministic Algorithms for 2-d and 3-d Shallow Cuttings

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.

Output-Sensitive Tools for Range Searching in Higher Dimensions

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.