### Abstract

Given a set P of n points in R^{d} and > 0, we consider the problemof constructing weak -nets for P.We show the following: pick a random sample Q of size O(1/ log (1/)) from P. Then, with constant probability, a weak -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/ log(1/)) 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/. However, our final weak -nets still have a large size (with the dimension appearing in the exponent of 1/).

Original language | English (US) |
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Title of host publication | Proceedings of the Twenty-third Annual Symposium on Computational Geometry, SCG'07 |

Pages | 239-244 |

Number of pages | 6 |

DOIs | |

State | Published - Oct 22 2007 |

Event | 23rd Annual Symposium on Computational Geometry, SCG'07 - Gyeongju, Korea, Republic of Duration: Jun 6 2007 → Jun 8 2007 |

### Other

Other | 23rd Annual Symposium on Computational Geometry, SCG'07 |
---|---|

Country | Korea, Republic of |

City | Gyeongju |

Period | 6/6/07 → 6/8/07 |

### Fingerprint

### Keywords

- Combinatorial geometry
- Discrete geometry
- Hitting convex sets
- Weak epsilon nets

### ASJC Scopus subject areas

- Theoretical Computer Science
- Geometry and Topology
- Computational Mathematics

### Cite this

*Proceedings of the Twenty-third Annual Symposium on Computational Geometry, SCG'07*(pp. 239-244) https://doi.org/10.1145/1247069.1247113

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

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Proceedings of the Twenty-third Annual Symposium on Computational Geometry, SCG'07.*pp. 239-244, 23rd Annual Symposium on Computational Geometry, SCG'07, Gyeongju, Korea, Republic of, 6/6/07. https://doi.org/10.1145/1247069.1247113

}

TY - GEN

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

AU - Mustafa, Nabil

AU - Ray, Saurabh

PY - 2007/10/22

Y1 - 2007/10/22

N2 - Given a set P of n points in Rd and > 0, we consider the problemof constructing weak -nets for P.We show the following: pick a random sample Q of size O(1/ log (1/)) from P. Then, with constant probability, a weak -net of P can be constructed from only the points of Q. This shows that weak -nets in Rd can be computed from a subset of P of size O(1/ log(1/)) 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/. 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 Rd and > 0, we consider the problemof constructing weak -nets for P.We show the following: pick a random sample Q of size O(1/ log (1/)) from P. Then, with constant probability, a weak -net of P can be constructed from only the points of Q. This shows that weak -nets in Rd can be computed from a subset of P of size O(1/ log(1/)) 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/. However, our final weak -nets still have a large size (with the dimension appearing in the exponent of 1/).

KW - Combinatorial geometry

KW - Discrete geometry

KW - Hitting convex sets

KW - Weak epsilon nets

UR - http://www.scopus.com/inward/record.url?scp=35348925348&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=35348925348&partnerID=8YFLogxK

U2 - 10.1145/1247069.1247113

DO - 10.1145/1247069.1247113

M3 - Conference contribution

SN - 1595937056

SN - 9781595937056

SP - 239

EP - 244

BT - Proceedings of the Twenty-third Annual Symposium on Computational Geometry, SCG'07

ER -