## 210 Citations

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

- 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…

### Relative (p,epsilon)-Approximations in Geometry

- MathematicsArXiv
- 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.

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

- 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…

### Epsilon-Nets for Halfspaces Revisited

- Mathematics, Computer ScienceArXiv
- 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.

### Semialgebraic Range Reporting and Emptiness Searching with Applications

- Computer ScienceSIAM 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.

### Approximate Halfspace Range Counting

- Computer ScienceSIAM 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…

### On Geometric Range Searching, Approximate Counting and Depth Problems

- Computer Science
- 2008

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

- Computer ScienceACM 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.

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

- Computer ScienceDiscrete & 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.

### Output-Sensitive Tools for Range Searching in Higher Dimensions

- MathematicsArXiv
- 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.