Article

Conflict-free coloring for rectangle ranges using O(n^.382) colors

Authors:

Deepak Ajwani,

Khaled Elbassioni,

Sathish Govindarajan,

Saurabh RayAuthors Info & Claims

SPAA '07: Proceedings of the nineteenth annual ACM symposium on Parallel algorithms and architectures

Pages 181 - 187

https://doi.org/10.1145/1248377.1248406

Published: 09 June 2007 Publication History

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Abstract

Given a set of points P ⊆ R², a conflict-free coloring of P w.r.t. rectangle ranges is an assignment of colors to points of P, such that each non-empty axis-parallel rectangle T in the plane contains a point whose color is distinct from all other points in P ∩ T. This notion has been the subject of recent interest, and is motivated by frequency assignment in wireless cellular networks: one naturally would like to minimize the number of frequencies (colors) assigned to bases stations (points), such that within any range (for instance, rectangle), there is no interference. We show that any set of n points in R² can be conflict-free colored with Õ(n^β+ε) colors in expected polynomial time, for any arbitrarily small ε > 0 and β = 3?√5<over>2 < 0.382. This improves upon the previously known bound of O(√nlog log n/ log n).

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Cited By

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Singhal RSengar APrem Prakash VAgrawal M(2021)A Novel Heuristic for Estimating Two-Colorability for Conflict-Free Coloring in Simple GraphsAdvances in Systems Engineering10.1007/978-981-15-8025-3_33(327-335)Online publication date: 24-Jan-2021
https://doi.org/10.1007/978-981-15-8025-3_33
Abel ZAlvarez VDemaine EFekete SGour AHesterberg AKeldenich PScheffer CKlein P(2017)Three colors sufficeProceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms10.5555/3039686.3039813(1951-1963)Online publication date: 16-Jan-2017
https://dl.acm.org/doi/10.5555/3039686.3039813
Keszegh BLemons NPálvölgyi D(2013)Online and Quasi-online Colorings of Wedges and IntervalsSOFSEM 2013: Theory and Practice of Computer Science10.1007/978-3-642-35843-2_26(292-306)Online publication date: 2013
https://doi.org/10.1007/978-3-642-35843-2_26
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Index Terms

Conflict-free coloring for rectangle ranges using O(n^.382) colors
1. Mathematics of computing
  1. Discrete mathematics
    1. Combinatorics
      1. Combinatorial algorithms

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Online Conflict-Free Coloring for Intervals

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Published In

SPAA '07: Proceedings of the nineteenth annual ACM symposium on Parallel algorithms and architectures

June 2007

376 pages

ISBN:9781595936677

DOI:10.1145/1248377

General Chair:
Phillip B. Gibbons
Intel Research, USA
,
Program Chair:
Christian Scheideler
Technische Universität München, Germany

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

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Association for Computing Machinery

New York, NY, United States

Publication History

Published: 09 June 2007

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SPAA07

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SPAA07: 19th ACM Symposium on Parallelism in Algorithms and Architectures

June 9 - 11, 2007

California, San Diego, USA

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Overall Acceptance Rate 447 of 1,461 submissions, 31%

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Cited By

View all

Singhal RSengar APrem Prakash VAgrawal M(2021)A Novel Heuristic for Estimating Two-Colorability for Conflict-Free Coloring in Simple GraphsAdvances in Systems Engineering10.1007/978-981-15-8025-3_33(327-335)Online publication date: 24-Jan-2021
https://doi.org/10.1007/978-981-15-8025-3_33
Abel ZAlvarez VDemaine EFekete SGour AHesterberg AKeldenich PScheffer CKlein P(2017)Three colors sufficeProceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms10.5555/3039686.3039813(1951-1963)Online publication date: 16-Jan-2017
https://dl.acm.org/doi/10.5555/3039686.3039813
Keszegh BLemons NPálvölgyi D(2013)Online and Quasi-online Colorings of Wedges and IntervalsSOFSEM 2013: Theory and Practice of Computer Science10.1007/978-3-642-35843-2_26(292-306)Online publication date: 2013
https://doi.org/10.1007/978-3-642-35843-2_26
Ajwani DElbassioni KGovindarajan SRay S(2012)Conflict-Free Coloring for Rectangle Ranges Using O(n.382) ColorsDiscrete & Computational Geometry10.5555/3116652.311692048:1(39-52)Online publication date: 1-Jul-2012
https://dl.acm.org/doi/10.5555/3116652.3116920
Chan TDey TWhitesides S(2012)Conflict-free coloring of points with respect to rectangles and approximation algorithms for discrete independent setProceedings of the twenty-eighth annual symposium on Computational geometry10.1145/2261250.2261293(293-302)Online publication date: 17-Jun-2012
https://dl.acm.org/doi/10.1145/2261250.2261293
Keszegh B(2012)Coloring half-planes and bottomless rectanglesComputational Geometry: Theory and Applications10.1016/j.comgeo.2011.09.00445:9(495-507)Online publication date: 1-Nov-2012
https://dl.acm.org/doi/10.1016/j.comgeo.2011.09.004
Ajwani DElbassioni KGovindarajan SRay S(2012)Conflict-Free Coloring for Rectangle Ranges Using O(n .382) ColorsDiscrete & Computational Geometry10.1007/s00454-012-9425-548:1(39-52)Online publication date: 28-Apr-2012
https://doi.org/10.1007/s00454-012-9425-5
Pira E(2012)Unique-Minimum Conflict-Free Coloring for a Chain of RingsAdvances in Computer Science and Information Technology. Computer Science and Engineering10.1007/978-3-642-27308-7_35(319-324)Online publication date: 2012
https://doi.org/10.1007/978-3-642-27308-7_35
Cheilaris PSmorodinsky SSulovský MHurtado Fvan Kreveld M(2011)The potential to improve the choiceProceedings of the twenty-seventh annual symposium on Computational geometry10.1145/1998196.1998266(424-432)Online publication date: 13-Jun-2011
https://dl.acm.org/doi/10.1145/1998196.1998266
Pach JTardos G(2010)NotesJournal of Combinatorial Theory Series A10.1016/j.jcta.2009.04.007117:6(776-782)Online publication date: 1-Aug-2010
https://dl.acm.org/doi/10.1016/j.jcta.2009.04.007
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