Advertisement

On the Message Complexity of Global Computations

  • Doron Nussbaum
  • Nicola Santoro
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6490)

Abstract

It is well known that, for most non-trivial problems (such as Election, Spanning-Tree Construction, Traversal, Broadcast, etc.), any generic solution requires at least Ω(m) messages in the worst case, where m is the number of links among the n entities. However, all the existing proofs of this fact assume that the network size (i.e., the parameters n and m) are not known to the protocol.

A natural question arises whether this rather strong assumption, which is crucial for the proofs, is truly necessary for establishing a lower bound to these problems.

In this paper we answer this question and prove that the Ω(m) bound is inherent for all these problems, as well as many more. In fact, we consider the class of global problems, that is those whose solution requires the involvement of every entity in the communication (sending or receiving messages). The relationship between n and m plays an important role in establishing the lower bound. We show that for most networks (where \(m \le \frac{1}{2}(n-2)(n-3)+1\)) a generic solution for any problem in this class requires at least m messages even if n, m, and the degree of each node are known. This result holds for almost all values of m (e.g., when \( \frac{1}{2}(n-2)(n-3)+ 1 < m \le \frac{1}{2}(n-1)(n-2)+1\) the number of required messages is m − 1), even if there is a single initiator and the entities have distinct identifiers, and both these facts are known. Moreover, the results hold even if the protocol can maintain a global view of the network.

As the networks become more dense, namely the network approaches a complete graph, the number of required messages is gradually reduced. For extreme values of m ( i.e., \(m = \frac{1}{2}n(n-1) - c > \frac{1}{2}(n-1)(n-2)+1\)), where c ≥ 0 is constant, the lower bound gradually approaches Ω(n); this is understandable since we establish it for the single initiator scenario. However, we prove that in networks of such a size, single initiators problems such as Broadcast and Traversal can be solved with precisely that order of magnitude. This means that for those problems the knowledge of n and m generates a significant and sudden complexity drop from Θ(n 2) to Θ(n).

Keywords

distributed algorithms global computations generic protocols asynchronous systems message complexity 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Awerbuch, B.: A new distributed depth-first search algorithm. Information Processing Letters 20, 147–150 (1985)CrossRefMATHGoogle Scholar
  2. 2.
    Awerbuch, B., Goldreich, O., Peleg, D., Vainish, R.: A Trade-off between Information and Communication in Broadcast Protocols. Journal of the ACM 37(2), 238–256 (1990)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Chang, E.J.H.: Echo algorithms: Depth parallel operations on general graphs. IEEE Transactions on Software Engineering SE-8(4), 391–401 (1982)CrossRefGoogle Scholar
  4. 4.
    Cheung, T.Y.: Graph traversal techniques and the maximum flow problem in distributed computation. IEEE Transactions on Software Engineering 9, 504–512 (1983)CrossRefMATHGoogle Scholar
  5. 5.
    Cidon, I.: Yet another distributed depth-first search algorithm. Information Processing Letters 26, 301–305 (1987)CrossRefGoogle Scholar
  6. 6.
    Dobrev, S.: Leader election using any sense of direction. In: 6th International Colloquium on Structural Information and Communication Complexity, Lacanau, pp. 93–104 (July 1999)Google Scholar
  7. 7.
    Erdös, P., Gallai, T.: On maximal paths and circuits of graphs. Acta Mathemtica Hunarica 10(3-4), 337–356 (1959)MathSciNetMATHGoogle Scholar
  8. 8.
    Gafni, E.: Improvements in the time complexity of two message-optimal election algorithms. In: 4th ACM Symposium on Principles of Distributed Computing, Minaki, pp. 175–185 (August 1985)Google Scholar
  9. 9.
    Gallager, R.G., Humblet, P.A., Spira, P.M.: A distributed algorithm for minimum spanning tree. ACM Transactions on Programming Languages and Systems 5(1), 66–77 (1983)CrossRefMATHGoogle Scholar
  10. 10.
    Garay, J.A., Kutten, S., Peleg, D.: A sublinear time distributed algorithm for minimum-weight spanning trees. SIAM Journal on Computing 27(1), 302–316 (1998)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Israeli, A., Kranakis, E., Krizanc, D., Santoro, N.: Time-Messages tradeoffs for the weak unison problem. Nordic Journal of Computing 4, 317–329 (1997)MathSciNetMATHGoogle Scholar
  12. 12.
    Korach, E., Kutten, S., Moran, S.: A modular technique for the design of efficient distributed leader finding algorithms. ACM Transactions on Programming Languages and Systems 12(1), 84–101 (1990)CrossRefGoogle Scholar
  13. 13.
    Lynch, N.: A hundred impossibility proofs for distributed computing. In: Proceedings of the Eighth Annual ACM Symposium on Principles of Distributed Computing, pp. 1–28 (1989)Google Scholar
  14. 14.
    Santoro, N.: On the message complexity of distributed problems. Int. Journal of Computer and Information Sciences 13(3), 131–147 (1984)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Santoro, N.: Design and Analysis of Distributed Algorithms. John Wiley, Chichester (2007)MATHGoogle Scholar
  16. 16.
    Segall, A.: Distributed network protocols. IEEE Transactions on Information Theory IT-29(1), 23–35 (1983)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Tel, G.: Introduction to Distributed Algorithms. Cambridge University Press, Cambridge (1994)CrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Doron Nussbaum
    • 1
  • Nicola Santoro
    • 1
  1. 1.School of Computer ScienceCarleton UniversityOttawaCanada

Personalised recommendations