Colored Non-crossing Euclidean Steiner Forest

  • Sergey Bereg
  • Krzysztof Fleszar
  • Philipp Kindermann
  • Sergey Pupyrev
  • Joachim Spoerhase
  • Alexander Wolff
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, 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 \(k+\varepsilon \).

Notes

Acknowledgments

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

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Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Sergey Bereg
    • 1
  • Krzysztof Fleszar
    • 2
  • Philipp Kindermann
    • 2
  • Sergey Pupyrev
    • 3
    • 4
  • Joachim Spoerhase
    • 2
  • Alexander Wolff
    • 2
  1. 1.University of TexasDallasUSA
  2. 2.Lehrstuhl für Informatik IUniversität WürzburgWürzburgGermany
  3. 3.Department of Computer ScienceUniversity of ArizonaTucsonUSA
  4. 4.Institute of Mathematics and Computer ScienceUral Federal UniversityYekaterinburgRussia

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