### Abstract

We show the existence of e-nets of size O (1/ε log log 1/ε) for planar point sets and axis-parallel rectangular ranges. The same bound holds for points in the plane with "fat" triangular ranges, and for point sets in R^{3} and axis-parallel boxes; these are the first known non-trivial bounds for these range spaces. Our technique also yields improved bounds on the size of ε-nets in the more general context considered by Clarkson and Varadarajan. For example, we show the existence of e-nets of size O (1/εe log log log 1/ε) for the dual range space of "fat" regions and planar point sets (where the regions are the ground objects and the ranges are subsets stabbed by points). Plugging our bounds into the technique of Brönnimann and Goodrich, we obtain improved approximation factors (computable in randomized polynomial time) for the hitting set or the set cover problems associated with the corresponding range spaces.

Original language | English (US) |
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Title of host publication | STOC'09 - Proceedings of the 2009 ACM International Symposium on Theory of Computing |

Pages | 639-648 |

Number of pages | 10 |

DOIs | |

State | Published - 2009 |

Event | 41st Annual ACM Symposium on Theory of Computing, STOC '09 - Bethesda, MD, United States Duration: May 31 2009 → Jun 2 2009 |

### Other

Other | 41st Annual ACM Symposium on Theory of Computing, STOC '09 |
---|---|

Country | United States |

City | Bethesda, MD |

Period | 5/31/09 → 6/2/09 |

### Fingerprint

### Keywords

- E-nets
- Geometric range spaces
- Hitting set
- Randomized algorithms
- Set cover

### ASJC Scopus subject areas

- Software

### Cite this

*STOC'09 - Proceedings of the 2009 ACM International Symposium on Theory of Computing*(pp. 639-648) https://doi.org/10.1145/1536414.1536501

**Small-size ε-nets for axis-parallel rectangles and boxes.** / Aronov, Boris; Ezra, Esther; Sharir, Micha.

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

*STOC'09 - Proceedings of the 2009 ACM International Symposium on Theory of Computing.*pp. 639-648, 41st Annual ACM Symposium on Theory of Computing, STOC '09, Bethesda, MD, United States, 5/31/09. https://doi.org/10.1145/1536414.1536501

}

TY - GEN

T1 - Small-size ε-nets for axis-parallel rectangles and boxes

AU - Aronov, Boris

AU - Ezra, Esther

AU - Sharir, Micha

PY - 2009

Y1 - 2009

N2 - We show the existence of e-nets of size O (1/ε log log 1/ε) for planar point sets and axis-parallel rectangular ranges. The same bound holds for points in the plane with "fat" triangular ranges, and for point sets in R3 and axis-parallel boxes; these are the first known non-trivial bounds for these range spaces. Our technique also yields improved bounds on the size of ε-nets in the more general context considered by Clarkson and Varadarajan. For example, we show the existence of e-nets of size O (1/εe log log log 1/ε) for the dual range space of "fat" regions and planar point sets (where the regions are the ground objects and the ranges are subsets stabbed by points). Plugging our bounds into the technique of Brönnimann and Goodrich, we obtain improved approximation factors (computable in randomized polynomial time) for the hitting set or the set cover problems associated with the corresponding range spaces.

AB - We show the existence of e-nets of size O (1/ε log log 1/ε) for planar point sets and axis-parallel rectangular ranges. The same bound holds for points in the plane with "fat" triangular ranges, and for point sets in R3 and axis-parallel boxes; these are the first known non-trivial bounds for these range spaces. Our technique also yields improved bounds on the size of ε-nets in the more general context considered by Clarkson and Varadarajan. For example, we show the existence of e-nets of size O (1/εe log log log 1/ε) for the dual range space of "fat" regions and planar point sets (where the regions are the ground objects and the ranges are subsets stabbed by points). Plugging our bounds into the technique of Brönnimann and Goodrich, we obtain improved approximation factors (computable in randomized polynomial time) for the hitting set or the set cover problems associated with the corresponding range spaces.

KW - E-nets

KW - Geometric range spaces

KW - Hitting set

KW - Randomized algorithms

KW - Set cover

UR - http://www.scopus.com/inward/record.url?scp=70350668803&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=70350668803&partnerID=8YFLogxK

U2 - 10.1145/1536414.1536501

DO - 10.1145/1536414.1536501

M3 - Conference contribution

AN - SCOPUS:70350668803

SN - 9781605585062

SP - 639

EP - 648

BT - STOC'09 - Proceedings of the 2009 ACM International Symposium on Theory of Computing

ER -