Piercing quasi-rectangles

On a problem of danzer and rogers

János Pach, G. Tardos

Research output: Chapter in Book/Report/Conference proceedingConference contribution

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 languageEnglish
Title of host publicationAlgorithms and Data Structures - 12th International Symposium, WADS 2011, Proceedings
PublisherSpringer Verlag
Number of pages1
ISBN (Print)9783642222993
DOIs
Publication statusPublished - Jan 1 2011
Event12th International Symposium on Algorithms and Data Structures, WADS 2011 - New York, United States
Duration: Aug 15 2011Aug 17 2011

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume6844 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

Other

Other12th International Symposium on Algorithms and Data Structures, WADS 2011
CountryUnited States
CityNew York
Period8/15/118/17/11

Fingerprint

Piercing
Rectangle
Trajectories
Normal vector
Sweep
Trajectory
Angle
Unit
Motion

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Computer Science(all)

Cite this

Pach, J., & Tardos, G. (2011). Piercing quasi-rectangles: On a problem of danzer and rogers. In 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.

Algorithms and Data Structures - 12th International Symposium, WADS 2011, Proceedings. Springer Verlag, 2011. (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 6844 LNCS).

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Pach, J & Tardos, G 2011, Piercing quasi-rectangles: On a problem of danzer and rogers. in 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
Pach J, Tardos G. Piercing quasi-rectangles: On a problem of danzer and rogers. In Algorithms and Data Structures - 12th International Symposium, WADS 2011, Proceedings. Springer Verlag. 2011. (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)). https://doi.org/10.1007/978-3-642-22300-6_55
Pach, János ; Tardos, G. / Piercing quasi-rectangles : On a problem of danzer and rogers. Algorithms and Data Structures - 12th International Symposium, WADS 2011, Proceedings. Springer Verlag, 2011. (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)).
@inproceedings{5ee3f6c7855e440da2b685d735a43936,
title = "Piercing quasi-rectangles: On a problem of danzer and rogers",
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).",
author = "J{\'a}nos Pach and G. Tardos",
year = "2011",
month = "1",
day = "1",
doi = "10.1007/978-3-642-22300-6_55",
language = "English",
isbn = "9783642222993",
series = "Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)",
publisher = "Springer Verlag",
booktitle = "Algorithms and Data Structures - 12th International Symposium, WADS 2011, Proceedings",

}

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 -