Polynomial Area Bounds for MST Embeddings of Trees

  • Michael Kaufmann
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4875)

Abstract

In their seminal paper on geometric minimum spanning trees, Monma and Suri [6] gave a method to embed any tree of maximal degree 5 as a minimum spanning tree in the Euclidean plane. They derived area bounds of \(O(2^{k^2} \times 2^{k^2})\) for trees of height k and conjectured that an improvement below cn ×cn is not possible for some constant c > 0. We partially disprove this conjecture by giving polynomial area bounds for arbitrary trees of maximal degree 3 and 4.

References

  1. 1.
    Di Battista, G., Eades, P., Tamassia, R., Tollis, I.G.: Graph Drawing: Algorithms for the Visualization of Graphs. Prentice Hall, Englewood Cliffs (1999)MATHGoogle Scholar
  2. 2.
    Di Battista, G., Liotta, G.: Computing proximity drawings of trees in the 3-dimensioanl space. In: Sack, J.-R., Akl, S.G., Dehne, F., Santoro, N. (eds.) WADS 1995. LNCS, vol. 955, pp. 239–250. Springer, Heidelberg (1995)Google Scholar
  3. 3.
    Eades, P., Whitesides, S.: The Realization Problem for Euclidean Minimum Spanning Trees in NP-Hard. Algorithmica 16(1), 60–82, 1–15 (1996)Google Scholar
  4. 4.
    Cheng, H., Liu, Q., Jia, X.: Heuristic algorithms for real-time data aggregation in wireless sensor networks. In: Proceedings of the 2006 International Conference on Communications and Mobile Computing (2006)Google Scholar
  5. 5.
    King, J.: Realization of Degree 10 Minimum Spanning Trees in 3-Space. In: CCCG 2006. Proceedings of the 18th Canadian Conference on Computational Geometry, pp. 39–42 (2006)Google Scholar
  6. 6.
    Clyde, L., Monma, C.L., Suri, S.: Transitions in Geometric Minimum Spanning Trees. Discrete & Computational Geometry 8, 265–293 (1992)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Michael Kaufmann
    • 1
  1. 1.Wilhelm-Schickard-Institut für InformatikUniversität TübingenGermany

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