Global Synchronization and Consensus Using Beeps in a Fault-Prone MAC

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10050)

Abstract

Global synchronization is an important prerequisite to many distributed tasks. Communication between processors proceeds in synchronous rounds. Processors are woken up in possibly different rounds. The clock of each processor starts in its wakeup round showing local round 0, and ticks once per round, incrementing the value of the local clock by one. The global round 0, unknown to processors, is the wakeup round of the earliest processor. Global synchronization (or establishing a global clock) means that each processor chooses a local clock round such that their chosen rounds all correspond to the same global round t.

We study the task of global synchronization in a Multiple Access Channel (MAC) prone to faults, under a very weak communication model called the beeping model. Some processors wake up spontaneously, in possibly different rounds decided by an adversary. In each round, an awake processor can either listen, i.e., stay silent, or beep, i.e., emit a signal. In each round, a fault can occur in the channel independently with constant probability \(0<p<1\). In a fault-free round, an awake processor hears a beep if it listens in this round and if one or more other processors beep in this round. A processor still dormant in a fault-free round in which some other processor beeps is woken up by this beep and hears it. In a faulty round nothing is heard, regardless of the behaviour of the processors. An algorithm working with error probability at most \(\epsilon \), for a given \(\epsilon >0\), is called \(\epsilon \)-safe. Our main result is the design and analysis, for any constant \(\epsilon >0\), of a deterministic \(\epsilon \)-safe global synchronization algorithm that works in constant time in any fault-prone MAC using beeps.

As an application, we solve the consensus problem in a fault-prone MAC using beeps. Processors have input values from some set V and they have to decide the same value from this set. If all processors have the same input value, then they must all decide this value. Using global synchronization, we give a deterministic \(\epsilon \)-safe consensus algorithm that works in time \(O(\log w)\) in a fault-prone MAC, where w is the smallest input value of all participating processors. We show that this time cannot be improved, even when the MAC is fault-free.

Keywords

Global synchronization Consensus Multiple access channel Fault Beep 

References

  1. 1.
    Afek, Y., Alon, N., Bar-Joseph, Z., Cornejo, A., Haeupler, B., Kuhn, F.: Beeping a maximal independent set. In: Peleg, D. (ed.) DISC 2011. LNCS, vol. 6950, pp. 32–50. Springer, Heidelberg (2011). doi:10.1007/978-3-642-24100-0_3 CrossRefGoogle Scholar
  2. 2.
    Anantharamu, L., Chlebus, B.S., Kowalski, D.R., Rokicki, M.A.: Medium access control for adversarial channels with jamming. In: Kosowski, A., Yamashita, M. (eds.) SIROCCO 2011. LNCS, vol. 6796, pp. 89–100. Springer, Heidelberg (2011). doi:10.1007/978-3-642-22212-2_9 CrossRefGoogle Scholar
  3. 3.
    Attiya, H., Welch, J.: Distributed Computing. John Wiley and Sons Inc., Chichester (2004)CrossRefGoogle Scholar
  4. 4.
    Awerbuch, B., Richa, A.W., Scheideler, C., Schmid, S., Zhang, J.: Principles of robust medium access and an application to leader election. ACM Trans. Algorithms 10(4), 24:1–24:26 (2014)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Bar-Yehuda, R., Goldreich, O., Itai, A.: On the time complexity of broadcast in radio networks: an exponential gap between determinism and randomization. J. Comput. Syst. Sci. 45, 104–126 (1992)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Chlebus, B.S., Gąsieniec, L., Kowalski, D.R., Radzik, T.: On the wake-up problem in radio networks. In: Caires, L., Italiano, G.F., Monteiro, L., Palamidessi, C., Yung, M. (eds.) ICALP 2005. LNCS, vol. 3580, pp. 347–359. Springer, Heidelberg (2005). doi:10.1007/11523468_29 CrossRefGoogle Scholar
  7. 7.
    Chlebus, B.S., Kowalski, D.R., Strojnowski, M.: Scalable quantum consensus for crash failures. In: Lynch, N.A., Shvartsman, A.A. (eds.) DISC 2010. LNCS, vol. 6343, pp. 236–250. Springer, Heidelberg (2010). doi:10.1007/978-3-642-15763-9_24 CrossRefGoogle Scholar
  8. 8.
    Chockler, G., Demirbas, M., Gilbert, S., Lynch, N.A., Newport, C.C., Nolte, T.: Consensus and collision detectors in radio networks. Distrib. Comput. 21, 55–84 (2008)CrossRefMATHGoogle Scholar
  9. 9.
    Cornejo, A., Kuhn, F.: Deploying wireless networks with beeps. In: Lynch, N.A., Shvartsman, A.A. (eds.) DISC 2010. LNCS, vol. 6343, pp. 148–162. Springer, Heidelberg (2010). doi:10.1007/978-3-642-15763-9_15 CrossRefGoogle Scholar
  10. 10.
    Czumaj, A., Davis, P.: Communicating with beeps, arXiv:1505.06107 [cs.DC] (2015)
  11. 11.
    Clementi, A.E.F., Monti, A., Silvestri, R.: Selective families, superimposed codes, and broadcasting on unknown radio networks. In: Proceedings of the 12th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2001), pp. 709–718 (2001)Google Scholar
  12. 12.
    Czumaj, A., Rytter, W.: Broadcasting algorithms in radio networks with unknown topology. In: Proceedings of the 44th IEEE Symposium on Foundations of Computer Science (FOCS 2003), pp. 492–501 (2003)Google Scholar
  13. 13.
    Czyzowicz, J., Gąsieniec, L., Kowalski, D.R., Pelc, A.: Consensus and mutual exclusion in a multiple access channel. In: Keidar, I. (ed.) DISC 2009. LNCS, vol. 5805, pp. 512–526. Springer, Heidelberg (2009). doi:10.1007/978-3-642-04355-0_51 CrossRefGoogle Scholar
  14. 14.
    Förster, K.-T., Seidel, J., Wattenhofer, R.: Deterministic leader election in multi-hop beeping networks. In: Kuhn, F. (ed.) DISC 2014. LNCS, vol. 8784, pp. 212–226. Springer, Heidelberg (2014). doi:10.1007/978-3-662-45174-8_15 Google Scholar
  15. 15.
    Gasieniec, L., Pelc, A., Peleg, D.: The wakeup problem in synchronous broadcast systems. SIAM J. Discrete Math. 14, 207–222 (2001)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Ghaffari, M., Haeupler, B.: Near optimal leader election in multi-hop radio networks. In: Proceedings of the 24th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2013), pp. 748–766 (2013)Google Scholar
  17. 17.
    Gilbert, S., Kowalski, D.R.: Distributed agreement with optimal communication complexity. In: Proceedings of the 21st ACM-SIAM Symposium on Discrete Algorithms (SODA 2010), pp. 965–977 (2010)Google Scholar
  18. 18.
    Gray, J.N.: Notes on data base operating systems. In: Bayer, R., Graham, R.M., Seegmüller, G. (eds.) Operating Systems an Advanced Course. LNCS, vol. 60, pp. 393–481. Springer, Heidelberg (1978). doi:10.1007/3-540-08755-9_9 CrossRefGoogle Scholar
  19. 19.
    Greenberg, A.G., Winograd, S.: A lower bound on the time needed in the worst case to resolve conflicts deterministically in multiple access channels. J. ACM 32, 589–596 (1985)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Huang, B., Moscibroda, T.: Conflict resolution and membership problem in beeping channels. In: Afek, Y. (ed.) DISC 2013. LNCS, vol. 8205, pp. 314–328. Springer, Heidelberg (2013). doi:10.1007/978-3-642-41527-2_22 CrossRefGoogle Scholar
  21. 21.
    Lynch, N.A.: Distributed Algorithms. Morgan Kaufmann Publ. Inc., San Francisco (1996)MATHGoogle Scholar
  22. 22.
    Métivier, Y., Robson, J.M., Zemmari, A.: On distributed computing with beeps, arXiv:1507.02721 [cs.DC] (2015)
  23. 23.
    Moses, Y., Raynal, M.: Revisiting simultaneous consensus with crash failures. J. Parallel Distrib. Comput. 69, 400–409 (2009)CrossRefGoogle Scholar
  24. 24.
    Navlakha, S., Bar-Joseph, Z.: Distributed information processing in biological and computational systems. Commun. ACM 58(1), 94–102 (2014)CrossRefGoogle Scholar
  25. 25.
    Pease, M.C., Shostak, R.E., Lamport, L.: Reaching agreement in the presence of faults. J. ACM 27, 228–234 (1980)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Raynal, M.: Fault-Tolerant Agreement in Synchronous Distributed Systems. Morgan & Claypool Publishers, San Francisco (2010)Google Scholar
  27. 27.
    Santoro, N., Widmayer, P.: Time is not a healer. In: Monien, B., Cori, R. (eds.) STACS 1989. LNCS, vol. 349, pp. 304–313. Springer, Heidelberg (1989). doi:10.1007/BFb0028994 CrossRefGoogle Scholar
  28. 28.
    Santoro, N., Widmayer, P.: Distributed function evaluation in presence of transmission faults. In: Proceedings of the International Symposium on Algorithms (SIGAL 1990), pp. 358–367 (1990)Google Scholar
  29. 29.
    Willard, D.E.: Log-logarithmic selection resolution protocols in a multiple access channel. SIAM J. Comput. 15, 468–477 (1986)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Yu, Y., Jia, L., Yu, D., Li, G., Cheng, X.: Minimum connected dominating set construction in wireless networks under the beeping model. In: Proceedings of the IEEE Conference on Computer Communications (INFOCOM 2015), pp. 972–980 (2015)Google Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Université du Québec en OutaouaisGatineauCanada
  2. 2.University of ManitobaWinnipegCanada

Personalised recommendations