Polychromatic coloring for half-planes

Shakhar Smorodinsky; Yelena Yuditsky

doi:10.1007/978-3-642-13731-0_12

Polychromatic coloring for half-planes

Shakhar Smorodinsky, Yelena Yuditsky

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-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 H_p of at least 4k-3 of the half-planes then H_p 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 language	English
Title of host publication	Algorithm Theory - SWAT 2010 - 12th Scandinavian Symposium and Workshops on Algorithm Theory, Proceedings
Pages	118-126
Number of pages	9
DOIs	https://doi.org/10.1007/978-3-642-13731-0_12
State	Published - 21 Jul 2010
Event	12th Scandinavian Symposium and Workshops on Algorithm Theory, SWAT 2010 - Bergen, Norway Duration: 21 Jun 2010 → 23 Jun 2010

Publication series

Name	Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume	6139 LNCS
ISSN (Print)	0302-9743
ISSN (Electronic)	1611-3349

Conference

Conference	12th Scandinavian Symposium and Workshops on Algorithm Theory, SWAT 2010
Country/Territory	Norway
City	Bergen
Period	21/06/10 → 23/06/10

Keywords

Discrete Geometry
Geometric Hypergraphs
Polychromatic Coloring
ε-Nets

ASJC Scopus subject areas

Theoretical Computer Science
General Computer Science

Access to Document

10.1007/978-3-642-13731-0_12

Cite this

Smorodinsky, S., & Yuditsky, Y. (2010). Polychromatic coloring for half-planes. In Algorithm Theory - SWAT 2010 - 12th Scandinavian Symposium and Workshops on Algorithm Theory, Proceedings (pp. 118-126). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 6139 LNCS). https://doi.org/10.1007/978-3-642-13731-0_12

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Smorodinsky, S & Yuditsky, Y 2010, Polychromatic coloring for half-planes. in Algorithm Theory - SWAT 2010 - 12th Scandinavian Symposium and Workshops on Algorithm Theory, Proceedings. Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 6139 LNCS, pp. 118-126, 12th Scandinavian Symposium and Workshops on Algorithm Theory, SWAT 2010, Bergen, Norway, 21/06/10. https://doi.org/10.1007/978-3-642-13731-0_12

Polychromatic coloring for half-planes. / Smorodinsky, Shakhar; Yuditsky, Yelena.
Algorithm Theory - SWAT 2010 - 12th Scandinavian Symposium and Workshops on Algorithm Theory, Proceedings. 2010. p. 118-126 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 6139 LNCS).

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

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N2 - 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.

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Polychromatic coloring for half-planes

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Keywords

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