Polychromatic coloring for half-planes

Shakhar Smorodinsky, Yelena Yuditsky

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

2 Scopus citations

Abstract

We prove that for every integer k, every finite set of points in the plane can be k-colored so that every half-plane that contains at least 2k-1 points, also contains at least one point from every color class. We also show that the bound 2k-1 is best possible. This improves the best previously known lower and upper bounds of 4/3k and 4k-1 respectively. As a corollary, we also show that every finite set of half-planes can be k colored so that if a point p belongs to a subset Hp of at least 4k-3 of the half-planes then Hp contains a half-plane from every color class. This improves the best previously known upper bound of 8k-3. Another corollary of our first result is a new proof of the existence of small size ε-nets for points in the plane with respect to half-planes.

Original languageEnglish
Title of host publicationAlgorithm Theory - SWAT 2010 - 12th Scandinavian Symposium and Workshops on Algorithm Theory, Proceedings
Pages118-126
Number of pages9
DOIs
StatePublished - 21 Jul 2010
Event12th Scandinavian Symposium and Workshops on Algorithm Theory, SWAT 2010 - Bergen, Norway
Duration: 21 Jun 201023 Jun 2010

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume6139 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

Conference

Conference12th Scandinavian Symposium and Workshops on Algorithm Theory, SWAT 2010
Country/TerritoryNorway
CityBergen
Period21/06/1023/06/10

Keywords

  • Discrete Geometry
  • Geometric Hypergraphs
  • Polychromatic Coloring
  • ε-Nets

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Computer Science (all)

Fingerprint

Dive into the research topics of 'Polychromatic coloring for half-planes'. Together they form a unique fingerprint.

Cite this