Succinct Greedy Drawings Do Not Always Exist

  • Patrizio Angelini
  • Giuseppe Di Battista
  • Fabrizio Frati
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5849)

Abstract

A greedy drawing is a graph drawing containing a distance-decreasing path for every pair of nodes. A path (v 0,v 1,...,v m ) is distance-decreasing if d(v i ,v m ) < d(v i − 1,v m ), for i = 1,...,m. Greedy drawings easily support geographic greedy routing. Hence, a natural and practical problem is the one of constructing greedy drawings in the plane using few bits for representing vertex Cartesian coordinates and using the Euclidean distance as a metric. We show that there exist greedy-drawable graphs that do not admit any greedy drawing in which the Cartesian coordinates have less than a polynomial number of bits.

References

  1. 1.
    Angelini, P., Frati, F., Grilli, L.: An algorithm to construct greedy drawings of triangulations. In: Tollis, I.G., Patrignani, M. (eds.) GD 2008. LNCS, vol. 5417, pp. 26–37. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  2. 2.
    Dhandapani, R.: Greedy drawings of triangulations. In: Huang, S.T. (ed.) SODA 2008, pp. 102–111 (2008)Google Scholar
  3. 3.
    Di Battista, G., Lenhart, W., Liotta, G.: Proximity drawability: a survey. In: Tamassia, R., Tollis, I.G. (eds.) GD 1994. LNCS, vol. 894, pp. 328–339. Springer, Heidelberg (1995)Google Scholar
  4. 4.
    Di Battista, G., Tamassia, R., Tollis, I.G.: Area requirement and symmetry display of planar upward drawings. Discrete & Computational Geometry 7, 381–401 (1992)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Eppstein, D., Goodrich, M.T.: Succinct greedy graph drawing in the hyperbolic plane. In: Tollis, I.G., Patrignani, M. (eds.) GD 2008. LNCS, vol. 5417, pp. 14–25. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  6. 6.
    Kaufmann, M.: Polynomial area bounds for MST embeddings of trees. In: Hong, S.-H., Nishizeki, T., Quan, W. (eds.) GD 2007. LNCS, vol. 4875, pp. 88–100. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  7. 7.
    Kleinberg, R.: Geographic routing using hyperbolic space. In: INFOCOM 2007, pp. 1902–1909 (2007)Google Scholar
  8. 8.
    Knaster, B., Kuratowski, C., Mazurkiewicz, C.: Ein beweis des fixpunktsatzes fur n dimensionale simplexe. Fundamenta Mathematicae 14, 132–137 (1929)MATHGoogle Scholar
  9. 9.
    Leighton, T., Moitra, A.: Some results on greedy embeddings in metric spaces. In: FOCS 2008, pp. 337–346 (2008)Google Scholar
  10. 10.
    Monma, C.L., Suri, S.: Transitions in geometric minimum spanning trees. Discrete & Computational Geometry 8, 265–293 (1992)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Papadimitriou, C.H., Ratajczak, D.: On a conjecture related to geometric routing. Theoretical Computer Science 344(1), 3–14 (2005)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Penna, P., Vocca, P.: Proximity drawings in polynomial area and volume. Computational Geometry 29(2), 91–116 (2004)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Rao, A., Papadimitriou, C.H., Shenker, S., Stoica, I.: Geographic routing without location information. In: Johnson, D.B., Joseph, A.D., Vaidya, N.H. (eds.) MOBICOM 2003, pp. 96–108 (2003)Google Scholar
  14. 14.
    Schnyder, W.: Embedding planar graphs on the grid. In: SODA 1990, pp. 138–148 (1990)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Patrizio Angelini
    • 1
  • Giuseppe Di Battista
    • 1
  • Fabrizio Frati
    • 1
  1. 1.Dipartimento di Informatica e AutomazioneRoma Tre UniversityItaly

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