Maximum Plane Trees in Multipartite Geometric Graphs

  • Ahmad Biniaz
  • Prosenjit Bose
  • Kimberly Crosbie
  • Jean-Lou De Carufel
  • David Eppstein
  • Anil Maheshwari
  • Michiel Smid
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10389)

Abstract

A geometric graph is a graph whose vertices are points in the plane and whose edges are straight-line segments between the points. A plane spanning tree in a geometric graph is a spanning tree that is non-crossing. Let R and B be two disjoint sets of points in the plane where the points of R are colored red and the points of B are colored blue, and let \(n=|R\cup B|\). A bichromatic plane spanning tree is a plane spanning tree in the complete bipartite geometric graph with bipartition (RB). In this paper we consider the maximum bichromatic plane spanning tree problem, which is the problem of computing a bichromatic plane spanning tree of maximum total edge length.

  1. 1.

    For the maximum bichromatic plane spanning tree problem, we present an approximation algorithm with ratio 1/4 that runs in \(O(n\log n)\) time.

     
  2. 2.

    We also consider the multicolored version of this problem where the input points are colored with \(k>2\) colors. We present an approximation algorithm that computes a plane spanning tree in a complete k-partite geometric graph, and whose ratio is 1/6 if \(k=3\), and 1/8 if \(k\geqslant 4\).

     
  3. 3.

    We also revisit the special case of the problem where \(k=n\), i.e., the problem of computing a maximum plane spanning tree in a complete geometric graph. For this problem, we present an approximation algorithm with ratio 0.503; this is an extension of the algorithm presented by Dumitrescu and Tóth (2010) whose ratio is 0.502.

     

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References

  1. 1.
    Alon, N., Rajagopalan, S., Suri, S.: Long non-crossing configurations in the plane. Fundamenta Informaticae 22(4), 385–394 (1995). Also in Proceedings of the 9th ACM Symposium on Computational Geometry (SoCG), pp. 257–263 (1993)MathSciNetMATHGoogle Scholar
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    Biniaz, A., Bose, P., Eppstein, D., Maheshwari, A., Morin, P., Smid, M.: Spanning trees in multipartite geometric graphs. CoRR, abs/1611.01661 (2016). Also submitted to AlgorithmicaGoogle Scholar
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    Borgelt, M.G., van Kreveld, M.J., Löffler, M., Luo, J., Merrick, D., Silveira, R.I., Vahedi, M.: Planar bichromatic minimum spanning trees. Journal of Discrete Algorithms 7(4), 469–478 (2009)MathSciNetCrossRefMATHGoogle Scholar
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    Dumitrescu, A., Tóth, C.D.: Long non-crossing configurations in the plane. Discrete & Computational Geometry 44(4), 727–752 (2010). Also in Proceedings of the 27th International Symposium on Theoretical Aspects of Computer Science (STACS), pp. 311–322 (2010)MathSciNetCrossRefMATHGoogle Scholar
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    Monma, C.L., Paterson, M., Suri, S., Yao, F.F.: Computing Euclidean maximum spanning trees. Algorithmica 5(3), 407–419 (1990)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Ahmad Biniaz
    • 1
  • Prosenjit Bose
    • 1
  • Kimberly Crosbie
    • 1
  • Jean-Lou De Carufel
    • 2
  • David Eppstein
    • 3
  • Anil Maheshwari
    • 1
  • Michiel Smid
    • 1
  1. 1.Carleton UniversityOttawaCanada
  2. 2.University of OttawaOttawaCanada
  3. 3.University of CaliforniaIrvineUSA

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