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Broadcasting vs. Mixing and Information Dissemination on Cayley Graphs

  • Robert Elsässer
  • Thomas Sauerwald
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4393)

Abstract

One frequently studied problem in the context of information dissemination in communication networks is the broadcasting problem. In this paper, we study the following randomized broadcasting protocol: At some time t an information r is placed at one of the nodes of a graph G. In the succeeding steps, each informed node chooses one neighbor, independently and uniformly at random, and informs this neighbor by sending a copy of r to it.

First, we consider the relationship between randomized broadcasting and random walks on graphs. In particular, we prove that the runtime of the algorithm described above is upper bounded by the corresponding mixing time, up to a logarithmic factor. One key ingredient of our proofs is the analysis of a continuous-type version of the afore mentioned algorithm, which might be of independent interest. Then, we introduce a general class of Cayley graphs, including (among others) Star graphs, Transposition graphs, and Pancake graphs. We show that randomized broadcasting has optimal runtime on all graphs belonging to this class. Finally, we develop a new proof technique by combining martingale tail estimates with combinatorial methods. Using this approach, we show the optimality of our algorithm on another Cayley graph and obtain new knowledge about the runtime distribution on several Cayley graphs.

Keywords

Markov Chain Cayley Graph Information Dissemination Star Graph Broadcasting Algorithm 
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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References

  1. 1.
    Akers, S., Harel, D., Krishnamurthy, B.: The star graph: An attractive alternative to the n-cube. In: Proc. of ICPP’87, pp. 393–400 (1987)Google Scholar
  2. 2.
    Akers, S., Krishnamurthy, B.: A group-theoretic model for symmetric innterconnection networks. In: Proc. of ICPP’86, pp. 555–565 (1986)Google Scholar
  3. 3.
    Alon, N., Spencer, J.H.: The Probabilistic Method. Wiley-Interscience Series in Discrete Mathematics and Optimization. Wiley, Chichester (2000)zbMATHGoogle Scholar
  4. 4.
    Benjamini, I., et al.: Mixing times of the biased card shuffling and the asymmetric exclusion process. Transactions of the American Mathematical Society 357, 3013–3029 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Biggs, N.: Algebraic Graph Theory. Cambridge University Press, Cambridge (1993)Google Scholar
  6. 6.
    Chernoff, H.: A measure of asymptotic efficiency for tests of a hypothesis based on the sum of observations. Ann. Math. Stat. 23, 493–507 (1952)CrossRefMathSciNetGoogle Scholar
  7. 7.
    Chung, F.: Spectral Graph Theory. CBMS Regional conference series in mathematics, vol. 92. American Mathematical Society, Providence (1997)zbMATHGoogle Scholar
  8. 8.
    Chung, F., Lu, L.: Concentration inequalities and martingale inequalities — a survey. In: Internet Mathematics (to appear)Google Scholar
  9. 9.
    Demers, A., et al.: Epidemic algorithms for replicated database maintenance. In: Proc. of PODC’87, pp. 1–12 (1987)Google Scholar
  10. 10.
    Diaconis, P.: Group Representations in Probability and Statistics. Lecture notes-Monograph Series, vol. 11 (1988)Google Scholar
  11. 11.
    Diekmann, R., Frommer, A., Monien, B.: Efficient schemes for nearest neighbor load balancing. Parallel Computing 25(7), 789–812 (1999)CrossRefMathSciNetGoogle Scholar
  12. 12.
    Elsässer, R., Sauerwald, T.: On randomized broadcasting in star graphs. In: Kratsch, D. (ed.) WG 2005. LNCS, vol. 3787, pp. 307–318. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  13. 13.
    Elsässer, R., Sauerwald, T.: On the runtime and robustness of randomized broadcasting. In: Asano, T. (ed.) ISAAC 2006. LNCS, vol. 4288, pp. 349–358. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  14. 14.
    Feige, U., et al.: Randomized broadcast in networks. Random Structures and Algorithm 1(4), 447–460 (1990)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Gasieniec, L., Pelc, A.: Adaptive broadcasting with faulty nodes. Parallel Computing 22, 903–912 (1996)zbMATHCrossRefGoogle Scholar
  16. 16.
    Habib, M., et al.: Probabilistic Methods for Algorithmic Discrete Mathematics. In: Algorithms and Combinatorics (1991)Google Scholar
  17. 17.
    Hagerup, T., Rüb, C.: A guided tour of chernoff bounds. Information Processing Letters 36(6), 305–308 (1990)CrossRefGoogle Scholar
  18. 18.
    Hromkovič, J., et al.: Dissemination of Information in Communication Networks. Springer, Heidelberg (2005)Google Scholar
  19. 19.
    Leighton, F., Maggs, B., Sitamaran, R.: On the fault tolerance of some popular bounded-degree networks. In: Proc. of FOCS’92, pp. 542–552 (1992)Google Scholar
  20. 20.
    Lubotzky, A., Phillips, R., Sarnak, P.: Ramanujan graphs. Combinatorica 8(3), 261–277 (1988)zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Pittel, B.: On spreading rumor. SIAM Journal on Applied Mathematics 47(1), 213–223 (1987)zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Sinclair, A., Jerrum, M.: Approximate counting, uniform generation, and rapidly mixing markov chains. Inform. and Comput. 82, 93–113 (1989)zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Wilson, D.: Mixing times of lozenge tiling and card shuffling markov chains. Annals of Applied Probability 14, 274–325 (2004)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Berlin Heidelberg 2007

Authors and Affiliations

  • Robert Elsässer
    • 1
  • Thomas Sauerwald
    • 1
  1. 1.University of Paderborn, Institute for Computer Science, 33102 PaderbornGermany

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