Networking Science

, Volume 3, Issue 1–4, pp 2–12 | Cite as

On the choice of a spanning tree for greedy embedding of network graphs

Research Article


Greedy embedding is a graph embedding that makes the simple greedy geometric packet forwarding successful for every source-destination pair. It is desirable that graph embeddings also yield low hop overhead (stretch) of the greedy paths over the corresponding shortest paths. In this paper we study how topological and geometric properties of embedded graphs influence the hop stretch. Based on the obtained insights, we design embedding heuristics that yield minimal hop stretch greedy embeddings and verify their effectiveness on models of synthetic graphs. Finally, we verify the effectiveness of our heuristics on instances of several classes of large, real-world network graphs.


greedy routing shortest path hyperbolic geometry network graph AS graph hop stretch 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    D. J. Aldous, “The random walk construction of uniform spanning trees and uniform labelled trees,” SIAM J. Discrete Math., vol. 3, no. 4, pp. 450–465, Nov. 1990.CrossRefMATHMathSciNetGoogle Scholar
  2. [2]
    J. W. Anderson, Hyperbolic Geometry, 2nd ed. London: Springer, 2007.Google Scholar
  3. [3]
    M. Bui, F. Butelle, and C. Lavault, “A distributed algorithm for constructing a minimum diameter spanning tree,” J. Parallel Distrib. Comput., vol. 64, no. 5, pp. 571–577, May 2004.CrossRefMATHGoogle Scholar
  4. [4]
    P. M. Camerini, G. Galbiati, and F. Maffioli, “Complexity of spanning tree problems: Part I,” Eur. J. Oper. Res, vol. 5, no. 5, pp. 346–352, Nov. 1980.CrossRefMATHMathSciNetGoogle Scholar
  5. [5]
    A. Cvetkovski and M. Crovella, “Hyperbolic embedding and routing for dynamic graphs,” in Proc. IEEE Infocom 2009, pp. 1647–1655.Google Scholar
  6. [6]
    A. Cvetkovski and M. Crovella, “Low-stretch greedy embedding heuristics,” in 4th Int. Workshop Network Science for Communication Networks (NetSciCom’12), 2012, pp. 232–237. Orlando, USA.Google Scholar
  7. [7]
    D. Eppstein, “Manhattan orbifolds,” Topology and its Applications, vol. 157, no. 2, pp. 494–507, Feb. 2010.CrossRefMATHMathSciNetGoogle Scholar
  8. [8]
    D. Eppstein and M. T. Goodrich, “Succinct greedy geometric routing using hyperbolic geometry,” IEEE Trans. Comput., vol. 60, no. 11, pp. 1571–1580, Nov. 2011.CrossRefMathSciNetGoogle Scholar
  9. [9]
    G. G. Finn, “Routing and addressing problems in large metropolitan-scale internetworks,” Technical Report ISI/RR-87-180, University of Southern California, Information Sciences Institute, 1987.Google Scholar
  10. [10]
    S. Giordano, I. Stojmenovic, and L. Blazevic, “Position based routing algorithms for ad hoc networks: A taxonomy,” in Ad Hoc Wireless Networking, X. Chen, X. Huang, and D. Du, Eds. Boston: Kluwer, 2003, pp. 103–136.Google Scholar
  11. [11]
    R. Hassin and A. Tamir, “On the minimum diameter spanning tree problem,” Inform. Process. Lett., vol. 53, no. 2, pp. 109–111. Jan. 1995.CrossRefMATHMathSciNetGoogle Scholar
  12. [12]
    R. Kleinberg, “Geographic routing using hyperbolic space,” in Proc. IEEE Infocom 2007, pp. 1902–1909.Google Scholar
  13. [13]
    B. Klimt and Y. Yang, “Introducing the Enron corpus,” presented at the 1st Conf. Email and Anti-Spam (CEAS), Mountain View, CA, USA, 2004.Google Scholar
  14. [14]
    J. B. Kruskal, “On the shortest spanning subtree of a graph and the traveling salesman problem,” Proc. Amer. Math. Soc, vol. 7, no. 1, pp. 48–50, Feb. 1956.CrossRefMATHMathSciNetGoogle Scholar
  15. [15]
    B. Leong, “New techniques for geographic routing,” Ph.D. thesis, Massachusetts Institute of Technology, Cambridge, USA, 2006.Google Scholar
  16. [16]
    J. Leskovec, J. Kleinberg, and C. Faloutsos, “Graph evolution: Densification and shrinking diameters,” ACM Trans. Knowl. Discov. Data, vol. 1, no. 1, Article no. 2, Mar. 2007.Google Scholar
  17. [17]
    M. Mauve, J. Widmer, and H. Hartenstein, “A survey on position-based routing in mobile ad hoc networks,” IEEE Network, vol. 15, no. 6, pp. 30–39, Nov.–Dec. 2001.CrossRefGoogle Scholar
  18. [18]
    P. Maymounkov, “Greedy embeddings, trees, and Euclidean vs. Lobachevsky geometry,” [online]. Available:
  19. [19]
    R. Nelson and L. Kleinrock, “The spatial capacity of a slotted aloha multihop packet radio network with capture,” IEEE Trans. Commun., vol. 32, no. 6, pp. 684–694, Jun. 1984.CrossRefGoogle Scholar
  20. [20]
    C. H. Papadimitriou and D. Ratajczak, “On a conjecture related to geometric routing,” Theor. Comput. Sci., vol. 344, no. 1, pp. 3–14, Nov. 2005.CrossRefMATHMathSciNetGoogle Scholar
  21. [21]
    R. C. Prim, “Shortest connection networks and some generalizations,” Bell System Tech. J., vol. 36, no. 6, pp. 1389–1401, Nov. 1957.CrossRefGoogle Scholar
  22. [22]
    A. Rao, S. Ratnasamy, C. Papadimitriou, S. Shenker, and I. Stoica, “Geographic routing without location information,” in Proc. 9th Annu. Int. Conf. Mobile Computing and Networking, MobiCom’ 03, New York: ACM, 2003, pp. 96–108.CrossRefGoogle Scholar
  23. [23]
    M. Ripeanu, A. Iamnitchi, and I. Foster, “Mapping the Gnutella network,” IEEE Internet Comput., Vol. 6, no. 1, pp. 50–57, Jan. 2002.CrossRefGoogle Scholar
  24. [24]
    H. Takagi and L. Kleinrock, “Optimal transmission ranges for randomly distributed packet radio terminals,” IEEE Trans. Commun., vol. 32, no. 3, pp. 246–257, Mar. 1984.CrossRefGoogle Scholar
  25. [25]
    J. Y. Yen, “Finding the K shortest loopless paths in a network,” Manage. Sci., vol. 17, no. 11, pp. 712–716, Jul. 1971.CrossRefMATHGoogle Scholar
  26. [26]
    B. Zhang, R. Liu, D. Massey, and L. Zhang, “Collecting the internet as-level topology,” ACM SIGCOMM Comput. Commun. Rev. vol. 35, no. 1, pp. 53–61, Jan. 2005.CrossRefGoogle Scholar

Copyright information

© Tsinghua University Press and Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Department of Computer ScienceBoston UniversityBostonUSA

Personalised recommendations