Physical Expander in Virtual Tree Overlay

  • Taisuke Izumi
  • Maria Gradinariu Potop-Butucaru
  • Mathieu Valero
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6950)


In this paper, we propose a new distributed construction of constant-degree expanders motivated by their application in P2P overlay networks and in particular in the design of robust tree overlays. Our key result can be stated as follows. Consider a complete binary tree T and construct a random pairing Π between leaf nodes and internal nodes. We prove that the graph G Π obtained from T by contracting all pairs (leaf-internal nodes) achieves a constant node expansion with high probability. In the context of P2P overlays our result can be interpreted as follows: if each physical node participating to the tree overlay manages a random pair that couples one virtual internal node and one virtual leaf node then the physical-node layer exhibits a constant expansion with high probability. We encompass the difficulty of obtaining the random tree virtualization by proposing a local, self-organizing and churn resilient uniformly-random pairing algorithm with O(log2 n) running time. Our algorithm has the merit to not modify the original tree overlay (we just control the mapping between physical nodes and virtual nodes). Therefore, our scheme is general and can be easilly extended to a large class of overlays.


Leaf Node Internal Node Overlay Network Virtual Node Physical Node 
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.


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  1. 1.
    Aberer, K., Cudre-Mauroux, P., Datta, A., Despotovic, Z., Hauswith, M., Punceva, M., Schmidt, R.: P-Grid: A self-organizing access structure for p2p information. In: Batini, C., Giunchiglia, F., Giorgini, P., Mecella, M. (eds.) CoopIS 2001. LNCS, vol. 2172, pp. 179–194. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  2. 2.
    Abraham, I., Aspnes, J., Yuan, J.: Skip B-trees. In: Anderson, J.H., Prencipe, G., Wattenhofer, R. (eds.) OPODIS 2005. LNCS, vol. 3974, pp. 366–380. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  3. 3.
    Aspnes, J., Wieder, U.: The expansion and mixing time of skip graphs with applications. Distributed Computing 21(6), 385–393 (2008)CrossRefzbMATHGoogle Scholar
  4. 4.
    Baehni, S., Eugster, P.T., Guerraoui, R.: Data-aware multicast. In: DSN, pp. 233–242 (2004)Google Scholar
  5. 5.
    Bianchi, S., Felber, P., Potop-Butucaru, M.G.: Stabilizing distributed r-trees for peer-to-peer content routing. IEEE Trans. Parallel Distrib. Syst. 21(8), 1175–1187 (2010)CrossRefGoogle Scholar
  6. 6.
    Caron, E., Desprez, F., Fourdrignier, C., Petit, F., Tedeschi, C.: A repair mechanism for fault-tolerance for tree-structured peer-to-peer systems. In: Robert, Y., Parashar, M., Badrinath, R., Prasanna, V.K. (eds.) HiPC 2006. LNCS, vol. 4297, pp. 171–182. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  7. 7.
    Castro, M., Druschel, P., Kermarrec, A.-M., Nandi, A., Rowstron, A.I.T., Singh, A.: Splitstream: High-bandwidth content distribution in cooperative environments. In: SOSP, pp. 298–313 (2003)Google Scholar
  8. 8.
    Cooper, C., Dyer, M., Handley, A.: The flip markov chain and a randomizing p2p protocol. In: PODC, pp. 141–150 (2009)Google Scholar
  9. 9.
    Czumaj, A., Kutylowski, M.: Delayed path coupling and generating random permutations. Random Struct. Algorithms 17(3-4), 238–259 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Dolev, S., Tzachar, N.: Spanders: distributed spanning expanders. In: SAC, pp. 1309–1314 (2010)Google Scholar
  11. 11.
    du Mouza, C., Litwin, W., Rigaux, P.: SD-Rtree: A scalable distributed rtree. In: ICDE, pp. 296–305 (2007)Google Scholar
  12. 12.
    Eugster, P.T., Guerraoui, R., Handurukande, S.B., Kouznetsov, P., Kermarrec, A.-M.: Lightweight probabilistic broadcast. ACM Trans. Comput. Syst. 21(4) (2003)Google Scholar
  13. 13.
    Feder, T., Guetz, A., Mihail, M., Saberi, A.: A local switch Markov chain on given degree graphs with application in connectivity of peer-to-peer networks. In: FOCS, pp. 69–76 (2006)Google Scholar
  14. 14.
    Goyal, N., Rademacher, L., Vempala, S.: Expanders via random spanning trees. In: SODA, pp. 576–585 (2009)Google Scholar
  15. 15.
    Hoory, S., Linial, N., Wigderson, A.: Expendar graphs and their applications. Bull. Amer. Math. Soc. (43), 439–561 (2006)Google Scholar
  16. 16.
    Jagadish, H.V., Ooi, B.C., Vu, Q.H.: Baton: a balanced tree structure for peer-to-peer networks. In: VLDB, pp. 661–671 (2005)Google Scholar
  17. 17.
    Jagadish, H.V., Ooi, B.C., Vu, Q.H., Zhang, R., Zhou, A.: Vbi-tree: A peer-to-peer framework for supporting multi-dimensional indexing schemes. In: ICDE, p. 34 (2006)Google Scholar
  18. 18.
    Law, C., Siu, K.-Y.: Distributed construction of random expander networks. In: IEEE Infocom, pp. 2133–2143 (2003)Google Scholar
  19. 19.
    Pandurangan, G., Trehan, A.: Xheal: Localized Self-healing using Expanders. In: PODC, pp. 301–310 (2011)Google Scholar
  20. 20.
    Paris, C., Kalogeraki, V.: A topologically-aware overlay tree for efficient and low-latency media streaming. In: QShine/AAA-IDEA. LNICST, vol. 22, pp. 383–399 (2009)Google Scholar
  21. 21.
    Reiter, M.K., Samar, A., Wang, C.: Distributed construction of a fault-tolerant network from a tree. In: SRDS, pp. 155–165 (2005)Google Scholar
  22. 22.
    Zhang, C., Krishnamurthy, A., Wang, R.Y.: Brushwood: Distributed trees in peer-to-peer systems. In: ITPTPS, pp. 47–57 (2005)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Taisuke Izumi
    • 1
  • Maria Gradinariu Potop-Butucaru
    • 2
  • Mathieu Valero
    • 2
  1. 1.Graduate School of EngineeringNagoya Institute of TechnologyNagoyaJapan
  2. 2.Université Pierre et Marie Curie - Paris 6, LIP6 CNRS 7606France

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