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International Symposium on Algorithms and Computation

ISAAC 2015: Algorithms and Computation pp 429–441Cite as

Colored Non-crossing Euclidean Steiner Forest

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9472))

Abstract

Given a set of k-colored points in the plane, we consider the problem of finding k trees such that each tree connects all points of one color class, no two trees cross, and the total edge length of the trees is minimized. For \(k = 1\), this is the well-known Euclidean Steiner tree problem. For general k, a \(k\rho \)-approximation algorithm is known, where \(\rho \le 1.21\) is the Steiner ratio.

We present a PTAS for \(k=2\), a \((5/3+\varepsilon )\)-approximation for \(k=3\), and two approximation algorithms for general k, with ratios \(O(\sqrt{n} \log k)\) and \(k+\varepsilon \).

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Acknowledgments

We are grateful to Alon Efrat, Jackson Toeniskoetter, and Thomas van Dijk for the initial discussion of the problem.

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Authors and Affiliations

  1. University of Texas, Dallas, USA

    Sergey Bereg

  2. Lehrstuhl für Informatik I, Universität Würzburg, Würzburg, Germany

    Krzysztof Fleszar, Philipp Kindermann, Joachim Spoerhase & Alexander Wolff

  3. Department of Computer Science, University of Arizona, Tucson, AZ, USA

    Sergey Pupyrev

  4. Institute of Mathematics and Computer Science, Ural Federal University, Yekaterinburg, Russia

    Sergey Pupyrev

Authors
  1. Sergey Bereg
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  2. Krzysztof Fleszar
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  3. Philipp Kindermann
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  4. Sergey Pupyrev
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  5. Joachim Spoerhase
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  6. Alexander Wolff
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Corresponding author

Correspondence to Krzysztof Fleszar .

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Editors and Affiliations

  1. Masdar Institute, Abu Dhabi, United Arab Emirates

    Khaled Elbassioni

  2. Kyoto University, Kyoto, Japan

    Kazuhisa Makino

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Bereg, S., Fleszar, K., Kindermann, P., Pupyrev, S., Spoerhase, J., Wolff, A. (2015). Colored Non-crossing Euclidean Steiner Forest. In: Elbassioni, K., Makino, K. (eds) Algorithms and Computation. ISAAC 2015. Lecture Notes in Computer Science(), vol 9472. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-48971-0_37

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  • DOI: https://doi.org/10.1007/978-3-662-48971-0_37

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-662-48970-3

  • Online ISBN: 978-3-662-48971-0

  • eBook Packages: Computer ScienceComputer Science (R0)

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