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

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