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Randomized Distributed Decision

  • Pierre Fraigniaud
  • Amos Korman
  • Merav Parter
  • David Peleg
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7611)

Abstract

The paper tackles the power of randomization in the context of locality by analyzing the ability to “boost” the success probability of deciding a distributed language. The main outcome of this analysis is that the distributed computing setting contrasts significantly with the sequential one as far as randomization is concerned. Indeed, we prove that in some cases, the ability to increase the success probability for deciding distributed languages is rather limited.

We focus on the notion of a (p,q)-decider for a language \(\mathcal{L}\), which is a distributed randomized algorithm that accepts instances in \(\mathcal{L}\) with probability at least p and rejects instances outside of \(\mathcal{L}\) with probability at least q. It is known that every hereditary language that can be decided in t rounds by a (p,q)-decider, where p 2 + q > 1, can be decided deterministically in O(t) rounds. One of our results gives evidence supporting the conjecture that the above statement holds for all distributed languages and not only for hereditary ones, by proving the conjecture for the restricted case of path topologies.

For the range below the aforementioned threshold, namely, p 2 + q ≤ 1, we study the class B k (t) (for k ∈ ℕ* ∪ { ∞ }) of all languages decidable in at most t rounds by a (p,q)-decider, where \(p^{1+\frac{1}{k}}+q>1\). Since every language is decidable (in zero rounds) by a (p,q)-decider satisfying p + q = 1, the hierarchy B k provides a spectrum of complexity classes between determinism (k = 1, under the above conjecture) and complete randomization (k = ∞). We prove that all these classes are separated, in a strong sense: for every integer k ≥ 1, there exists a language \(\mathcal{L}\) satisfying \(\mathcal{L}\in B_{k+1}(0)\) but \(\mathcal{L}\notin B_k(t)\) for any t = o(n). In addition, we show that B  ∞ (t) does not contain all languages, for any t = o(n). In other words, we obtain the hierarchy B 1(t) ⊂ B 2 (t) ⊂ ⋯ ⊂ B  ∞ (t) ⊂ All.

Finally, we show that if the inputs can be restricted in certain ways, then the ability to boost the success probability becomes almost null, and in particular, derandomization is not possible even beyond the threshold p 2 + q = 1.

Keywords

Local distributed algorithms local decision randomized algorithms 

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References

  1. 1.
    Afek, Y., Kutten, S., Yung, M.: The local detection paradigm and its applications to self stabilization. Theoretical Computer Science 186(1-2), 199–230 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Alon, N., Babai, L., Itai, A.: A fast and simple randomized parallel algorithm for the maximal independent set problem. J. Algorithms 7(4), 567–583 (1986)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Amit, A., Linial, N., Matousek, J., Rozenman, E.: Random lifts of graphs. In: Proc. 12th ACM-SIAM Symp. on Discrete Algorithms (SODA), pp. 883–894 (2001)Google Scholar
  4. 4.
    Awerbuch, B., Patt-Shamir, B., Varghese, G.: Self-Stabilization By Local Checking and Correction. In: Proc. IEEE Symp. on the Foundations of Computer Science (FOCS), pp. 268–277 (1991)Google Scholar
  5. 5.
    Barenboim, L., Elkin, M.: Distributed (Δ + 1)-coloring in linear (in delta) time. In: Proc. 41st ACM Symp. on Theory of computing (STOC), pp. 111–120 (2009)Google Scholar
  6. 6.
    Das Sarma, A., Holzer, S., Kor, L., Korman, A., Nanongkai, D., Pandurangan, G., Peleg, D., Wattenhofer, R.: Distributed Verification and Hardness of Distributed Approximation. In: Proc. 43rd ACM Symp. on Theory of Computing, STOC (2011)Google Scholar
  7. 7.
    Dereniowski, D., Pelc, A.: Drawing maps with advice. Journal of Parallel and Distributed Computing 72, 132–143 (2012)zbMATHCrossRefGoogle Scholar
  8. 8.
    Dijkstra, E.W.: Self-stabilization in spite of distributed control. Comm. ACM 17(11), 643–644 (1974)zbMATHCrossRefGoogle Scholar
  9. 9.
    Dolev, S., Gouda, M., Schneider, M.: Requirements for silent stabilization. Acta Informatica 36(6), 447–462 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Fraigniaud, P., Gavoille, C., Ilcinkas, D., Pelc, A.: Distributed Computing with Advice: Information Sensitivity of Graph Coloring. In: Arge, L., Cachin, C., Jurdziński, T., Tarlecki, A. (eds.) ICALP 2007. LNCS, vol. 4596, pp. 231–242. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  11. 11.
    Fraigniaud, P.: D Ilcinkas, and A. Pelc. Communication algorithms with advice. J. Comput. Syst. Sci. 76(3-4), 222–232 (2008)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Fraigniaud, P.: A Korman, and E. Lebhar. Local MST computation with short advice. In: Proc. 19th ACM Symp. on Parallelism in Algorithms and Architectures (SPAA), pp. 154–160 (2007)Google Scholar
  13. 13.
    Fraigniaud, P., Korman, A., Peleg, D.: Local Distributed Decision. In: Proc. 52nd Annual IEEE Symposium on Foundations of Computer Science (FOCS), pp. 708–717 (2011)Google Scholar
  14. 14.
    Fraigniaud, P., Pelc, A.: Decidability Classes for Mobile Agents Computing. In: Fernández-Baca, D. (ed.) LATIN 2012. LNCS, vol. 7256, pp. 362–374. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  15. 15.
    Fraigniaud, P., Rajsbaum, S., Travers, C.: Locality and Checkability in Wait-Free Computing. In: Peleg, D. (ed.) DISC 2011. LNCS, vol. 6950, pp. 333–347. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  16. 16.
    Fraigniaud, P., Rajsbaum, S., Travers, C.: Universal Distributed Checkers and Orientation-Detection Tasks (submitted, 2012)Google Scholar
  17. 17.
    Göös, M., Suomela, J.: Locally checkable proofs. In: Proc. 30th ACM Symp. on Principles of Distributed Computing, PODC (2011)Google Scholar
  18. 18.
    Kor, L., Korman, A., Peleg, D.: Tight Bounds For Distributed MST Verification. In: Proc. 28th Int. Symp. on Theoretical Aspects of Computer Science, STACS (2011)Google Scholar
  19. 19.
    Korman, A., Kutten, S.: Distributed verification of minimum spanning trees. Distributed Computing 20, 253–266 (2007)CrossRefGoogle Scholar
  20. 20.
    Korman, A., Kutten, S., Masuzawa, T.: Fast and Compact Self-Stabilizing Verification, Computation, and Fault Detection of an MST. In: Proc. 30th ACM Symp. on Principles of Distributed Computing, PODC (2011)Google Scholar
  21. 21.
    Korman, A., Kutten, S.: D Peleg. Proof labeling schemes. Distributed Computing 22, 215–233 (2010)CrossRefGoogle Scholar
  22. 22.
    Korman, A., Sereni, J.S., Viennot, L.: Toward More Localized Local Algorithms: Removing Assumptions Concerning Global Knowledge. In: Proc. 30th ACM Symp. on Principles of Distributed Computing (PODC), pp. 49–58 (2011)Google Scholar
  23. 23.
    Kuhn, F.: Weak graph colorings: distributed algorithms and applications. In: Proc. 21st ACM Symp. on Parallel Algorithms and Architectures (SPAA), pp. 138–144 (2009)Google Scholar
  24. 24.
    Luby, M.: A simple parallel algorithm for the maximal independent set problem. SIAM J. Comput. 15, 1036–1053 (1986)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Naor, M.: A Lower Bound on Probabilistic Algorithms for Distributive Ring Coloring. SIAM J. Discrete Math. 4(3), 409–412 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Naor, M., Stockmeyer, L.: What can be computed locally? SIAM J. Comput. 24(6), 1259–1277 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Panconesi, A., Srinivasan, A.: On the Complexity of Distributed Network Decomposition. J. Algorithms 20(2), 356–374 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Peleg, D.: Distributed Computing: A Locality-Sensitive Approach. SIAM (2000)Google Scholar
  29. 29.
    Schneider, J., Wattenhofer, R.: A new technique for distributed symmetry breaking. In: Proc. 29th ACM Symp. on Principles of Distributed Computing (PODC), pp. 257–266 (2010)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Pierre Fraigniaud
    • 1
  • Amos Korman
    • 1
  • Merav Parter
    • 2
  • David Peleg
    • 2
  1. 1.CNRS and University Paris DiderotFrance
  2. 2.The Weizmann Institute of ScienceRehovotIsrael

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