### Abstract

It is an old problem of Danzer and Rogers to decide whether it is possible arrange O(1/ε) points in the unit square so that every rectangle of area ε contains at least one of them. We show that the answer to this question is in the negative if we slightly relax the notion of rectangles, as follows. Let δ be a fixed small positive number. A quasi-rectangle is a region swept out by a continuously moving segment s, with no rotation, so that throughout the motion the angle between the trajectory of the center of s and its normal vector remains at most δ. We show that the smallest number of points needed to pierce all quasi-rectangles of area ε is Θ (1/ε log 1/ε).

Original language | English |
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Title of host publication | Algorithms and Data Structures - 12th International Symposium, WADS 2011, Proceedings |

Publisher | Springer Verlag |

Number of pages | 1 |

ISBN (Print) | 9783642222993 |

DOIs | |

Publication status | Published - Jan 1 2011 |

Event | 12th International Symposium on Algorithms and Data Structures, WADS 2011 - New York, United States Duration: Aug 15 2011 → Aug 17 2011 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 6844 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Other

Other | 12th International Symposium on Algorithms and Data Structures, WADS 2011 |
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Country | United States |

City | New York |

Period | 8/15/11 → 8/17/11 |

### Fingerprint

### ASJC Scopus subject areas

- Theoretical Computer Science
- Computer Science(all)

### Cite this

*Algorithms and Data Structures - 12th International Symposium, WADS 2011, Proceedings*(Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 6844 LNCS). Springer Verlag. https://doi.org/10.1007/978-3-642-22300-6_55

**Piercing quasi-rectangles : On a problem of danzer and rogers.** / Pach, János; Tardos, G.

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

*Algorithms and Data Structures - 12th International Symposium, WADS 2011, Proceedings.*Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 6844 LNCS, Springer Verlag, 12th International Symposium on Algorithms and Data Structures, WADS 2011, New York, United States, 8/15/11. https://doi.org/10.1007/978-3-642-22300-6_55

}

TY - GEN

T1 - Piercing quasi-rectangles

T2 - On a problem of danzer and rogers

AU - Pach, János

AU - Tardos, G.

PY - 2011/1/1

Y1 - 2011/1/1

N2 - It is an old problem of Danzer and Rogers to decide whether it is possible arrange O(1/ε) points in the unit square so that every rectangle of area ε contains at least one of them. We show that the answer to this question is in the negative if we slightly relax the notion of rectangles, as follows. Let δ be a fixed small positive number. A quasi-rectangle is a region swept out by a continuously moving segment s, with no rotation, so that throughout the motion the angle between the trajectory of the center of s and its normal vector remains at most δ. We show that the smallest number of points needed to pierce all quasi-rectangles of area ε is Θ (1/ε log 1/ε).

AB - It is an old problem of Danzer and Rogers to decide whether it is possible arrange O(1/ε) points in the unit square so that every rectangle of area ε contains at least one of them. We show that the answer to this question is in the negative if we slightly relax the notion of rectangles, as follows. Let δ be a fixed small positive number. A quasi-rectangle is a region swept out by a continuously moving segment s, with no rotation, so that throughout the motion the angle between the trajectory of the center of s and its normal vector remains at most δ. We show that the smallest number of points needed to pierce all quasi-rectangles of area ε is Θ (1/ε log 1/ε).

UR - http://www.scopus.com/inward/record.url?scp=85037718568&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85037718568&partnerID=8YFLogxK

U2 - 10.1007/978-3-642-22300-6_55

DO - 10.1007/978-3-642-22300-6_55

M3 - Conference contribution

SN - 9783642222993

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

BT - Algorithms and Data Structures - 12th International Symposium, WADS 2011, Proceedings

PB - Springer Verlag

ER -