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)


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 \(k+\varepsilon \).



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


  1. 1.
    Arora, S.: Polynomial time approximation schemes for Euclidean traveling salesman and other geometric problems. J. ACM 45(5), 753–782 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bastert, O., Fekete, S.P.: Geometric wire routing. Technical report. 96.247, Universität zu Köln (1998).
  3. 3.
    Bereg, S., Fleszar, K., Kindermann, P., Pupyrev, S., Spoerhase, J., Wolff, A.: Colored non-crossing Euclidean steiner forest. CoRR abs/1509.05681 (2015).
  4. 4.
    Borradaile, G., Klein, P., Mathieu, C.: An \({O}(n \log n)\) approximation scheme for Steiner tree in planar graphs. ACM Trans. Algorithms 5(3), 31 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Chan, T.M., Hoffmann, H.-F., Kiazyk, S., Lubiw, A.: Minimum length embedding of planar graphs at fixed vertex locations. In: Wismath, S., Wolff, A. (eds.) GD 2013. LNCS, vol. 8242, pp. 376–387. Springer, Heidelberg (2013) CrossRefGoogle Scholar
  6. 6.
    Chung, F.R.K., Graham, R.L.: A new bound for Euclidean Steiner minimal trees. Ann. New York Acad. Sci. 440(1), 328–346 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Efrat, A., Hu, Y., Kobourov, S.G., Pupyrev, S.: MapSets: visualizing embedded and clustered graphs. In: Duncan, C., Symvonis, A. (eds.) GD 2014. LNCS, vol. 8871, pp. 452–463. Springer, Heidelberg (2014) Google Scholar
  8. 8.
    Efrat, A., Kobourov, S.G., Lubiw, A.: Computing homotopic shortest paths efficiently. Comput. Geom. Theory Appl. 35(3), 162–172 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Erickson, J., Nayyeri, A.: Shortest non-crossing walks in the plane. In: Proceedings of ACM-SIAM Symposium on Discrete Algorithms (SODA 2011), pp. 297–308 (2011)Google Scholar
  10. 10.
    Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman & Co., New York (1979) zbMATHGoogle Scholar
  11. 11.
    Hurtado, F., Korman, M., van Kreveld, M., Löffler, M., Sacristán, V., Silveira, R.I., Speckmann, B.: Colored spanning graphs for set visualization. In: Wismath, S., Wolff, A. (eds.) GD 2013. LNCS, vol. 8242, pp. 280–291. Springer, Heidelberg (2013) CrossRefGoogle Scholar
  12. 12.
    Kusakari, Y., Masubuchi, D., Nishizeki, T.: Finding a noncrossing Steiner forest in plane graphs under a 2-face condition. J. Combin. Optim. 5, 249–266 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Liebling, T.M., Margot, F., Müller, D., Prodon, A., Stauffer, L.: Disjoint paths in the plane. ORSA J. Comput. 7(1), 84–88 (1995)CrossRefzbMATHGoogle Scholar
  14. 14.
    Löffler, M.: Existence and computation of tours through imprecise points. Int. J. Comput. Geom. Appl. 21(1), 1–24 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Mitchell, J.S.: Guillotine subdivisions approximate polygonal subdivisions: a simple polynomial-time approximation scheme for geometric TSP, \(k\)-MST, and related problems. SIAM J. Comput. 28(4), 1298–1309 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Mitchell, J.S.: Geometric shortest paths and network optimization. In: Urrutia, J., Sack, J.R. (eds.) Handbook of Computational Geometry, chap. 15, pp. 633–701. North-Holland (2000)Google Scholar
  17. 17.
    Müller-Hannemann, M., Tazari, S.: A near linear time approximation scheme for Steiner tree among obstacles in the plane. Comput. Geom. Theory Appl. 43(4), 395–409 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Papadopoulou, E.: \(k\)-pairs non-crossing shortest paths in a simple polygon. Int. J. Comput. Geom. Appl. 9(6), 533–552 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Polishchuk, V., Mitchell, J.S.B.: Thick non-crossing paths and minimum-cost flows in polygonal domains. In: Proceedings of the ACM Symposium on Computational Geometry (SoCG 2007), pp. 56–65 (2007)Google Scholar
  20. 20.
    Verbeek, K.: Homotopic \(\cal {C}\)-oriented routing. In: Didimo, W., Patrignani, M. (eds.) GD 2012. LNCS, vol. 7704, pp. 272–278. Springer, Heidelberg (2013) CrossRefGoogle Scholar

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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