### Abstract

Given a set P of n points in R^{d} and ε> 0, we consider the problem of constructing weak ε-nets for P. We show the following: pick a random sample Q of size O(1/ε log(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 ε-nets in Rd can be computed from a subset of P of size O(1/ε log(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 ε-nets still have a large size (with the dimension appearing in the exponent of 1/ε).

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

Pages (from-to) | 565-571 |

Number of pages | 7 |

Journal | Computational Geometry: Theory and Applications |

Volume | 43 |

Issue number | 6-7 |

DOIs | |

State | Published - Aug 1 2010 |

### Fingerprint

### Keywords

- Combinatorial geometry
- Hitting convex sets
- Weak ε-nets

### ASJC Scopus subject areas

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

### Cite this

*Computational Geometry: Theory and Applications*,

*43*(6-7), 565-571. https://doi.org/10.1016/j.comgeo.2007.02.007

**Reprint of : Weak ε-nets have basis of size.** / Mustafa, Nabil H.; Ray, Saurabh.

Research output: Contribution to journal › Article

*Computational Geometry: Theory and Applications*, vol. 43, no. 6-7, pp. 565-571. https://doi.org/10.1016/j.comgeo.2007.02.007

}

TY - JOUR

T1 - Reprint of

T2 - Weak ε-nets have basis of size

AU - Mustafa, Nabil H.

AU - Ray, Saurabh

PY - 2010/8/1

Y1 - 2010/8/1

N2 - Given a set P of n points in Rd and ε> 0, we consider the problem of constructing weak ε-nets for P. We show the following: pick a random sample Q of size O(1/ε log(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 ε-nets in Rd can be computed from a subset of P of size O(1/ε log(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 ε-nets still have a large size (with the dimension appearing in the exponent of 1/ε).

AB - Given a set P of n points in Rd and ε> 0, we consider the problem of constructing weak ε-nets for P. We show the following: pick a random sample Q of size O(1/ε log(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 ε-nets in Rd can be computed from a subset of P of size O(1/ε log(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 ε-nets still have a large size (with the dimension appearing in the exponent of 1/ε).

KW - Combinatorial geometry

KW - Hitting convex sets

KW - Weak ε-nets

UR - http://www.scopus.com/inward/record.url?scp=77951879621&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77951879621&partnerID=8YFLogxK

U2 - 10.1016/j.comgeo.2007.02.007

DO - 10.1016/j.comgeo.2007.02.007

M3 - Article

AN - SCOPUS:77951879621

VL - 43

SP - 565

EP - 571

JO - Computational Geometry: Theory and Applications

JF - Computational Geometry: Theory and Applications

SN - 0925-7721

IS - 6-7

ER -