### Abstract

Given a set P of n points in ℝ ^{d} and >0, we consider the problem of constructing weak -nets for P. We show the following: pick a random sample Q of size O(1/εlog(1/ε)) from P. Then, with constant probability, a weak ε-net of P can be constructed from only the points of Q. This shows that weak -nets in ℝ ^{d} can be computed from a subset of P of size O(1/εlog(1/ε)) with only the constant of proportionality depending on the dimension, unlike all previous work where the size of the subset had the dimension in the exponent of 1/ε. However, our final weak -nets still have a large size (with the dimension appearing in the exponent of 1/).

Original language | English (US) |
---|---|

Pages (from-to) | 84-91 |

Number of pages | 8 |

Journal | Computational Geometry: Theory and Applications |

Volume | 40 |

Issue number | 1 |

DOIs | |

State | Published - May 1 2008 |

### Fingerprint

### Keywords

- Combinatorial geometry
- Hitting convex sets
- Weak -nets

### ASJC Scopus subject areas

- Geometry and Topology
- Computer Science Applications
- Control and Optimization
- Computational Theory and Mathematics
- Computational Mathematics

### Cite this

**Weak ε-nets have basis of size O (1 /ε log (1 / ε)) in any dimension.** / Mustafa, Nabil H.; Ray, Saurabh.

Research output: Contribution to journal › Article

*Computational Geometry: Theory and Applications*, vol. 40, no. 1, pp. 84-91. https://doi.org/10.1016/j.comgeo.2007.02.006

}

TY - JOUR

T1 - Weak ε-nets have basis of size O (1 /ε log (1 / ε)) in any dimension

AU - Mustafa, Nabil H.

AU - Ray, Saurabh

PY - 2008/5/1

Y1 - 2008/5/1

N2 - Given a set P of n points in ℝ d and >0, we consider the problem of constructing weak -nets for P. We show the following: pick a random sample Q of size O(1/εlog(1/ε)) from P. Then, with constant probability, a weak ε-net of P can be constructed from only the points of Q. This shows that weak -nets in ℝ d can be computed from a subset of P of size O(1/εlog(1/ε)) with only the constant of proportionality depending on the dimension, unlike all previous work where the size of the subset had the dimension in the exponent of 1/ε. However, our final weak -nets still have a large size (with the dimension appearing in the exponent of 1/).

AB - Given a set P of n points in ℝ d and >0, we consider the problem of constructing weak -nets for P. We show the following: pick a random sample Q of size O(1/εlog(1/ε)) from P. Then, with constant probability, a weak ε-net of P can be constructed from only the points of Q. This shows that weak -nets in ℝ d can be computed from a subset of P of size O(1/εlog(1/ε)) with only the constant of proportionality depending on the dimension, unlike all previous work where the size of the subset had the dimension in the exponent of 1/ε. However, our final weak -nets still have a large size (with the dimension appearing in the exponent of 1/).

KW - Combinatorial geometry

KW - Hitting convex sets

KW - Weak -nets

UR - http://www.scopus.com/inward/record.url?scp=84867924611&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84867924611&partnerID=8YFLogxK

U2 - 10.1016/j.comgeo.2007.02.006

DO - 10.1016/j.comgeo.2007.02.006

M3 - Article

AN - SCOPUS:84867924611

VL - 40

SP - 84

EP - 91

JO - Computational Geometry: Theory and Applications

JF - Computational Geometry: Theory and Applications

SN - 0925-7721

IS - 1

ER -