Succinct Greedy Graph Drawing in the Hyperbolic Plane

  • David Eppstein
  • Michael T. Goodrich
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5417)


We describe a method for producing a greedy embedding of any n-vertex simple graph G in the hyperbolic plane, so that a message M between any pair of vertices may be routed by having each vertex that receives M pass it to a neighbor that is closer to M’s destination. Our algorithm produces succinct drawings, where vertex positions are represented using O(logn) bits and distance comparisons may be performed efficiently using these representations.


Planar Graph Buffer Zone Binary Tree Hyperbolic Plane Tree Distance 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    Alstrup, S., Lauridsen, P.W., Sommerlund, P., Thorup, M.: Finding cores of limited length. In: Rau-Chaplin, A., Dehne, F., Sack, J.-R., Tamassia, R. (eds.) WADS 1997. LNCS, vol. 1272, pp. 45–54. Springer, Heidelberg (1997)CrossRefGoogle Scholar
  2. 2.
    Bose, P., Morin, P., Stojmenović, I., Urrutia, J.: Routing with Guaranteed Delivery in Ad Hoc Wireless Networks. Wireless Networks 6(7), 609–616 (2001)CrossRefGoogle Scholar
  3. 3.
    Chen, M.B., Gotsman, C., Wormser, C.: Distributed computation of virtual coordinates. In: Proc. 23rd Symp. Computational Geometry (SoCG 1997), pp. 210–219 (1997)Google Scholar
  4. 4.
    Dhandapani, R.: Greedy drawings of triangulations. In: Proc. 19th ACM-SIAM Symp. Discrete Algorithms (SODA 2008), pp. 102–111 (2008)Google Scholar
  5. 5.
    de Fraysseix, H., Pach, J., Pollack, R.: How to draw a planar graph on a grid. Combinatorica 10(1), 41–51 (1990)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Gilbert, E.N., Moore, E.F.: Variable-Length binary encodings. Bell System Tech. J. 38, 933–968 (1959)CrossRefMathSciNetGoogle Scholar
  7. 7.
    Karp, B., Kung, H.T.: GPSR: greedy perimeter stateless routing for wireless networks. In: Proc. 6th ACM Mobile Computing and Networking (MobiCom), pp. 243–254 (2000)Google Scholar
  8. 8.
    Kleinberg, R.: Geographic Routing Using Hyperbolic Space. In: Proc. 26th IEEE Int. Conf. Computer Communications (INFOCOM 2007), pp. 1902–1909. IEEE Press, Los Alamitos (2007)CrossRefGoogle Scholar
  9. 9.
    Kranakis, E., Singh, H., Urrutia, J.: Compass routing on geometric networks. In: Proc. 11th Canad. Conf. Computational Geometry (CCCG), pp. 51–54 (1999)Google Scholar
  10. 10.
    Kuhn, F., Wattenhofer, R., Zhang, Y., Zollinger, A.: Geometric ad-hoc routing: of theory and practice. In: Proc. 22nd ACM Symp. Principles of Distributed Computing (PODC), pp. 63–72 (2003)Google Scholar
  11. 11.
    Kuhn, F., Wattenhofer, R., Zollinger, A.: Asymptotically optimal geometric mobile ad-hoc routing. In: Proc. 6th ACM Discrete Algorithms and Methods for Mobile Computing and Communications (DIALM), pp. 24–33 (2002)Google Scholar
  12. 12.
    Kuhn, F., Wattenhofer, R., Zollinger, A.: Worst-Case optimal and average-case efficient geometric ad-hoc routing. In: Proc. 4th ACM Symp. Mobile Ad Hoc Networking & Computing (MobiHoc), pp. 267–278 (2003)Google Scholar
  13. 13.
    Leighton, T., Moitra, A.: Some results on greedy embeddings in metric spaces. In: Proc. 49th IEEE Symp. Foundations of Computer Science (FOCS) (2008)Google Scholar
  14. 14.
    Lillis, K.M., Pemmaraju, S.V.: On the Efficiency of a Local Iterative Algorithm to Compute Delaunay Realizations. In: McGeoch, C.C. (ed.) WEA 2008. LNCS, vol. 5038, pp. 69–86. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  15. 15.
    Maymounkov, P.: Greedy Embeddings, Trees, and Euclidean vs. Lobachevsky Geometry. M.I.T (manuscript, 2006),
  16. 16.
    Muhammad, R.B.: A distributed geometric routing algorithm for ad hoc wireless networks. In: Proc. IEEE Conf. Information Technology (ITNG), pp. 961–963 (2007)Google Scholar
  17. 17.
    Papadimitriou, C.H., Ratajczak, D.: On a conjecture related to geometric routing. Theor. Comput. Sci. 344(1), 3–14 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Rao, A., Ratnasamy, S., Papadimitriou, C.H., Shenker, S., Stoica, I.: Geographic routing without location information. In: Proc. 9th Int. Conf. Mobile Computing and Networking (MobiCom 2003), pp. 96–108. ACM, New York (2003)CrossRefGoogle Scholar
  19. 19.
    Schieber, B., Vishkin, U.: On finding lowest common ancestors: simplification and parallelization. SIAM J. Comput. 17(6), 1253–1262 (1988)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Schnyder, W.: Embedding planar graphs on the grid. In: Proc. 1st ACM-SIAM Sympos. Discrete Algorithms, pp. 138–148 (1990)Google Scholar
  21. 21.
    Sleator, D.D., Tarjan, R.E.: A data structure for dynamic trees. J. Comp. and Sys. Sci. 26(3), 362–391 (1983)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • David Eppstein
    • 1
  • Michael T. Goodrich
    • 1
  1. 1.Computer Science DepartmentUniversity of CaliforniaIrvineUSA

Personalised recommendations