Coloring points with respect to squares

Eyal Ackerman; Balázs Keszegh; Máté Vizer

doi:10.4230/LIPIcs.SoCG.2016.5

Coloring points with respect to squares

Eyal Ackerman, Balázs Keszegh, Máté Vizer

Faculty of Natural Sciences

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

Abstract

We consider the problem of 2-coloring geometric hypergraphs. Specifically, we show that there is a constant m such that any finite set S of points in the plane can be 2-colored such that every axis-parallel square that contains at least m points from S contains points of both colors. Our proof is constructive, that is, it provides a polynomial-time algorithm for obtaining such a 2-coloring. By affine transformations this result immediately applies also when considering homothets of a fixed parallelogram.

Original language	English
Title of host publication	32nd International Symposium on Computational Geometry, SoCG 2016
Editors	Sandor Fekete, Anna Lubiw
Publisher	Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
Pages	5.1-5.16
ISBN (Electronic)	9783959770095
DOIs	https://doi.org/10.4230/LIPIcs.SoCG.2016.5
State	Published - 1 Jun 2016
Event	32nd International Symposium on Computational Geometry, SoCG 2016 - Boston, United States Duration: 14 Jun 2016 → 17 Jun 2016

Publication series

Name	Leibniz International Proceedings in Informatics, LIPIcs
Volume	51
ISSN (Print)	1868-8969

Conference

Conference	32nd International Symposium on Computational Geometry, SoCG 2016
Country/Territory	United States
City	Boston
Period	14/06/16 → 17/06/16

Bibliographical note

Funding Information:
Most of this work was done during a visit of the first author to the Rényi Institute that was partially supported by Hungarian National Science Fund (OTKA), under grant PD 108406 and by ERC Advanced Research Grant no. 267165 (DISCONV). Second author supported by Hungarian National Science Fund (OTKA), under grant PD 108406 and by the János Bolyai Research Scholarship of the Hungarian Academy of Sciences. Third author supported by Hungarian National Science Fund (OTKA), under grant SNN 116095.

Publisher Copyright:
© Eyal Ackerman, Balázs Keszegh, and Máté Vizer.

Keywords

Cover-decomposability
Geometric hypergraph coloring
Homothets
Polychromatic coloring

ASJC Scopus subject areas

Software

Access to Document

10.4230/LIPIcs.SoCG.2016.5

Cite this

Ackerman, E., Keszegh, B., & Vizer, M. (2016). Coloring points with respect to squares. In S. Fekete, & A. Lubiw (Eds.), 32nd International Symposium on Computational Geometry, SoCG 2016 (pp. 5.1-5.16). (Leibniz International Proceedings in Informatics, LIPIcs; Vol. 51). Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. https://doi.org/10.4230/LIPIcs.SoCG.2016.5

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title = "Coloring points with respect to squares",

abstract = "We consider the problem of 2-coloring geometric hypergraphs. Specifically, we show that there is a constant m such that any finite set S of points in the plane can be 2-colored such that every axis-parallel square that contains at least m points from S contains points of both colors. Our proof is constructive, that is, it provides a polynomial-time algorithm for obtaining such a 2-coloring. By affine transformations this result immediately applies also when considering homothets of a fixed parallelogram.",

keywords = "Cover-decomposability, Geometric hypergraph coloring, Homothets, Polychromatic coloring",

author = "Eyal Ackerman and Bal{\'a}zs Keszegh and M{\'a}t{\'e} Vizer",

note = "Funding Information: Most of this work was done during a visit of the first author to the R{\'e}nyi Institute that was partially supported by Hungarian National Science Fund (OTKA), under grant PD 108406 and by ERC Advanced Research Grant no. 267165 (DISCONV). Second author supported by Hungarian National Science Fund (OTKA), under grant PD 108406 and by the J{\'a}nos Bolyai Research Scholarship of the Hungarian Academy of Sciences. Third author supported by Hungarian National Science Fund (OTKA), under grant SNN 116095. Publisher Copyright: {\textcopyright} Eyal Ackerman, Bal{\'a}zs Keszegh, and M{\'a}t{\'e} Vizer.; 32nd International Symposium on Computational Geometry, SoCG 2016 ; Conference date: 14-06-2016 Through 17-06-2016",

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doi = "10.4230/LIPIcs.SoCG.2016.5",

language = "English",

series = "Leibniz International Proceedings in Informatics, LIPIcs",

publisher = "Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing",

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booktitle = "32nd International Symposium on Computational Geometry, SoCG 2016",

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Ackerman, E, Keszegh, B & Vizer, M 2016, Coloring points with respect to squares. in S Fekete & A Lubiw (eds), 32nd International Symposium on Computational Geometry, SoCG 2016. Leibniz International Proceedings in Informatics, LIPIcs, vol. 51, Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing, pp. 5.1-5.16, 32nd International Symposium on Computational Geometry, SoCG 2016, Boston, United States, 14/06/16. https://doi.org/10.4230/LIPIcs.SoCG.2016.5

Coloring points with respect to squares. / Ackerman, Eyal; Keszegh, Balázs; Vizer, Máté.
32nd International Symposium on Computational Geometry, SoCG 2016. ed. / Sandor Fekete; Anna Lubiw. Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing, 2016. p. 5.1-5.16 (Leibniz International Proceedings in Informatics, LIPIcs; Vol. 51).

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

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AU - Ackerman, Eyal

AU - Keszegh, Balázs

AU - Vizer, Máté

N1 - Funding Information: Most of this work was done during a visit of the first author to the Rényi Institute that was partially supported by Hungarian National Science Fund (OTKA), under grant PD 108406 and by ERC Advanced Research Grant no. 267165 (DISCONV). Second author supported by Hungarian National Science Fund (OTKA), under grant PD 108406 and by the János Bolyai Research Scholarship of the Hungarian Academy of Sciences. Third author supported by Hungarian National Science Fund (OTKA), under grant SNN 116095. Publisher Copyright: © Eyal Ackerman, Balázs Keszegh, and Máté Vizer.

PY - 2016/6/1

Y1 - 2016/6/1

N2 - We consider the problem of 2-coloring geometric hypergraphs. Specifically, we show that there is a constant m such that any finite set S of points in the plane can be 2-colored such that every axis-parallel square that contains at least m points from S contains points of both colors. Our proof is constructive, that is, it provides a polynomial-time algorithm for obtaining such a 2-coloring. By affine transformations this result immediately applies also when considering homothets of a fixed parallelogram.

AB - We consider the problem of 2-coloring geometric hypergraphs. Specifically, we show that there is a constant m such that any finite set S of points in the plane can be 2-colored such that every axis-parallel square that contains at least m points from S contains points of both colors. Our proof is constructive, that is, it provides a polynomial-time algorithm for obtaining such a 2-coloring. By affine transformations this result immediately applies also when considering homothets of a fixed parallelogram.

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KW - Geometric hypergraph coloring

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KW - Polychromatic coloring

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T2 - 32nd International Symposium on Computational Geometry, SoCG 2016

Y2 - 14 June 2016 through 17 June 2016

ER -

Coloring points with respect to squares

Abstract

Publication series

Conference

Bibliographical note

Keywords

ASJC Scopus subject areas

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