Skip to main content
SpringerLink
Log in

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

  • Research Article
  • Published:
Networking Science

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  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.

    Article  MATH  MathSciNet  Google Scholar 

  2. J. W. Anderson, Hyperbolic Geometry, 2nd ed. London: Springer, 2007.

    Google Scholar 

  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.

    Article  MATH  Google Scholar 

  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.

    Article  MATH  MathSciNet  Google Scholar 

  5. A. Cvetkovski and M. Crovella, “Hyperbolic embedding and routing for dynamic graphs,” in Proc. IEEE Infocom 2009, pp. 1647–1655.

  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. D. Eppstein, “Manhattan orbifolds,” Topology and its Applications, vol. 157, no. 2, pp. 494–507, Feb. 2010.

    Article  MATH  MathSciNet  Google Scholar 

  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.

    Article  MathSciNet  Google Scholar 

  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. 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. R. Hassin and A. Tamir, “On the minimum diameter spanning tree problem,” Inform. Process. Lett., vol. 53, no. 2, pp. 109–111. Jan. 1995.

    Article  MATH  MathSciNet  Google Scholar 

  12. R. Kleinberg, “Geographic routing using hyperbolic space,” in Proc. IEEE Infocom 2007, pp. 1902–1909.

  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. 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.

    Article  MATH  MathSciNet  Google Scholar 

  15. B. Leong, “New techniques for geographic routing,” Ph.D. thesis, Massachusetts Institute of Technology, Cambridge, USA, 2006.

    Google Scholar 

  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. 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.

    Article  Google Scholar 

  18. P. Maymounkov, “Greedy embeddings, trees, and Euclidean vs. Lobachevsky geometry,” [online]. Available: http://pdos.csail.mit.edu/~petar/papers/maymounkov-greedy-prelim.pdf

  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.

    Article  Google Scholar 

  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.

    Article  MATH  MathSciNet  Google Scholar 

  21. R. C. Prim, “Shortest connection networks and some generalizations,” Bell System Tech. J., vol. 36, no. 6, pp. 1389–1401, Nov. 1957.

    Article  Google Scholar 

  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.

    Chapter  Google Scholar 

  23. M. Ripeanu, A. Iamnitchi, and I. Foster, “Mapping the Gnutella network,” IEEE Internet Comput., Vol. 6, no. 1, pp. 50–57, Jan. 2002.

    Article  Google Scholar 

  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.

    Article  Google Scholar 

  25. J. Y. Yen, “Finding the K shortest loopless paths in a network,” Manage. Sci., vol. 17, no. 11, pp. 712–716, Jul. 1971.

    Article  MATH  Google Scholar 

  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.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science, Boston University, 111 Cummington Mall, Boston, MA, 02215, USA

    Andrej Cvetkovski & Mark Crovella

Authors
  1. Andrej Cvetkovski
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mark Crovella
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Andrej Cvetkovski.

Additional information

A preliminary version of this work appears in the Fourth International Workshop on Network Science for Communication Networks (NetSciCom’12) [6].

Rights and permissions

Reprints and permissions

About this article

Cite this article

Cvetkovski, A., Crovella, M. On the choice of a spanning tree for greedy embedding of network graphs. Netw.Sci. 3, 2–12 (2013). https://doi.org/10.1007/s13119-013-0013-7

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13119-013-0013-7

Keywords

Search

Navigation