Distributed Computing

, Volume 32, Issue 3, pp 173–191 | Cite as

Minimizing message size in stochastic communication patterns: fast self-stabilizing protocols with 3 bits

  • Lucas BoczkowskiEmail author
  • Amos Korman
  • Emanuele Natale


This paper considers the basic \({\mathcal {PULL}}\) model of communication, in which in each round, each agent extracts information from few randomly chosen agents. We seek to identify the smallest amount of information revealed in each interaction (message size) that nevertheless allows for efficient and robust computations of fundamental information dissemination tasks. We focus on the Majority Bit Dissemination problem that considers a population of n agents, with a designated subset of source agents. Each source agent holds an input bit and each agent holds an output bit. The goal is to let all agents converge their output bits on the most frequent input bit of the sources (the majority bit). Note that the particular case of a single source agent corresponds to the classical problem of Broadcast (also termed Rumor Spreading). We concentrate on the severe fault-tolerant context of self-stabilization, in which a correct configuration must be reached eventually, despite all agents starting the execution with arbitrary initial states. In particular, the specification of who is a source and what is its initial input bit may be set by an adversary. We first design a general compiler which can essentially transform any self-stabilizing algorithm with a certain property (called “the bitwise-independence property”) that uses \(\ell \)-bits messages to one that uses only \(\log \ell \)-bits messages, while paying only a small penalty in the running time. By applying this compiler recursively we then obtain a self-stabilizing Clock Synchronization protocol, in which agents synchronize their clocks modulo some given integer T, within \(\tilde{\mathcal {O}}(\log n\log T)\) rounds w.h.p., and using messages that contain 3 bits only. We then employ the new Clock Synchronization tool to obtain a self-stabilizing Majority Bit Dissemination protocol which converges in \(\tilde{\mathcal {O}}(\log n)\) time, w.h.p., on every initial configuration, provided that the ratio of sources supporting the minority opinion is bounded away from half. Moreover, this protocol also uses only 3 bits per interaction.



The problem of self-stabilizing Bit Dissemination was introduced through discussions with Ofer Feinerman. The authors are also thankful for Omer Angel, Bernhard Haeupler, Parag Chordia, Iordanis Kerenidis, Fabian Kuhn, Uri Feige, and Uri Zwick for helpful discussions regarding that problem. The authors also thank Michele Borassi for his helpful suggestions regarding the Clock Synchronization problem.


  1. 1.
    Afek, Y., Alon, N., Barad, O., Hornstein, E., Barkai, N., Bar-Joseph, Z.: A biological solution to a fundamental distributed computing problem. Science 331, 183–185 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Alistarh, D., Gelashvili, R.: Polylogarithmic-time leader election in population protocols. In: ICALP, pp. 479–491 (2015)Google Scholar
  3. 3.
    Angluin, D., Aspnes, J., Diamadi, Z., Fischer, M.J., Peralta, R.: Computation in networks of passively mobile finite-state sensors. Distrib. Comput. 18(4), 235–253 (2006)CrossRefzbMATHGoogle Scholar
  4. 4.
    Angluin, D., Aspnes, J., Eisenstat, D.: A simple population protocol for fast robust approximate majority. Distrib. Comput. 21(2), 87–102 (2008)CrossRefzbMATHGoogle Scholar
  5. 5.
    Angluin, D., Aspnes, J., Fischer, M.J., Jiang, H.: Self-stabilizing population protocols. TAAS 3(4), 13 (2008)CrossRefGoogle Scholar
  6. 6.
    Angluin, D., Fischer, M.J., Jiang, H.: Stabilizing Consensus in Mobile Networks, pp. 37–50. Springer, Berlin (2006)Google Scholar
  7. 7.
    Aspnes, J., Ruppert, E.: An introduction to population protocols. Bull. EATCS 93, 98–117 (2007)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Attiya, H., Herzberg, A., Rajsbaum, S.: Optimal clock synchronization under different delay assumptions. SIAM J. Comput. 25(2), 369–389 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Beauquier, J., Burman, J., Kutten, S.: A self-stabilizing transformer for population protocols with covering. Theor. Comput. Sci. 412(33), 4247–4259 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Becchetti, L., Clementi, A., Natale, E., Pasquale, F., Posta, G.: Self-stabilizing repeated balls-into-bins. In: SPAA, pp. 332–339 (2015)Google Scholar
  11. 11.
    Becchetti, L., Clementi, A.E.F., Natale, E., Pasquale, F., Silvestri, R.: Plurality consensus in the gossip model. In: SODA, pp. 371–390 (2015)Google Scholar
  12. 12.
    Becchetti, L., Clementi, A.E.F., Natale, E., Pasquale, F., Trevisan, L.: Stabilizing consensus with many opinions. In: SODA, pp. 620–635 (2016)Google Scholar
  13. 13.
    Ben-Or, M., Dolev, D., Hoch, E.N.: Fast self-stabilizing byzantine tolerant digital clock synchronization. In: PODC, pp. 385–394 (2008)Google Scholar
  14. 14.
    Boczkowski, L., Korman, A., Natale, E.: Brief announcement: self-stabilizing clock synchronization with 3-bit messages. In: PODC (2016)Google Scholar
  15. 15.
    Boczkowski, L., Korman, A., Natale, E.: Minimizing message size in stochastic communication patterns: fast self-stabilizing protocols with 3 bits. In: SODA (2017)Google Scholar
  16. 16.
    Censor-Hillel, K., Haeupler, B., Kelner, J.A., Maymounkov, P.: Global computation in a poorly connected world: fast rumor spreading with no dependence on conductance. In: STOC, pp. 961–970 (2012)Google Scholar
  17. 17.
    Chen, H.-L., Cummings, R., Doty, D., Soloveichik, D.: Speed faults in computation by chemical reaction networks. In: Distributed Computing, pp. 16–30 (2014)Google Scholar
  18. 18.
    Chierichetti, F., Lattanzi, S., Panconesi, A.: Rumor spreading in social networks. In: ICALP, pp. 375–386 (2009)Google Scholar
  19. 19.
    Cooper, C., Elsässer, R., Radzik, T., Rivera, N., Shiraga, T.: Fast consensus for voting on general expander graphs. In: DISC, pp. 248–262. Springer, Berlin (2015)Google Scholar
  20. 20.
    Couzin, I., Krause, J., Franks, N., Levin, S.: Effective leadership and decision making in animal groups on the move. Nature 433, 513–516 (2005)CrossRefGoogle Scholar
  21. 21.
    Demers, A.J., Greene, D.H., Hauser, C., Irish, W., Larson, J., Shenker, S., Sturgis, H.E., Swinehart, D.C., Terry, D.B.: Epidemic algorithms for replicated database maintenance. Oper. Syst. Rev. 22(1), 8–32 (1988)CrossRefGoogle Scholar
  22. 22.
    Dijkstra, E.W.: Self-stabilizing systems in spite of distributed control. Commun. ACM 17(11), 643–644 (1974)CrossRefzbMATHGoogle Scholar
  23. 23.
    Doerr, B., Fouz, M.: Asymptotically optimal randomized rumor spreading. Electron. Notes Discrete Math. 38, 297–302 (2011)CrossRefzbMATHGoogle Scholar
  24. 24.
    Doerr, B., Goldberg, L.A., Minder, L., Sauerwald, T., Scheideler, C.: Stabilizing consensus with the power of two choices. In: SPAA, pp. 149–158 (2011)Google Scholar
  25. 25.
    Dolev, D. Hoch, E.N.: On self-stabilizing synchronous actions despite byzantine attacks. In: DISC, pp. 193–207 (2007)Google Scholar
  26. 26.
    Dolev, D., Korhonen, J.H., Lenzen, C., Rybicki, J., Suomela, J.: Synchronous counting and computational algorithm design. In: SSS, pp. 237–250 (2013)Google Scholar
  27. 27.
    Dolev, S.: Possible and impossible self-stabilizing digital clock synchronization in general graphs. Real-Time Syst. 12(1), 95–107 (1997)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Dolev, S., Welch, J.L.: Self-stabilizing clock synchronization in the presence of byzantine faults. J. ACM 51(5), 780–799 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Doty, D., Soloveichik, D.: Stable leader election in population protocols requires linear time. CoRR, abs/1502.04246 (2015)Google Scholar
  30. 30.
    Elsässer, R., Friedetzky, T., Kaaser, D., Mallmann-Trenn, F., Trinker, H.: Efficient k-party voting with two choices. CoRR, abs/1602.04667 (2016)Google Scholar
  31. 31.
    Elsässer, R., Sauerwald, T.: On the runtime and robustness of randomized broadcasting. Theor. Comput. Sci. 410(36), 3414–3427 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Emek, Y., Wattenhofer, R.: Stone age distributed computing. In: PODC, pp. 137–146 (2013)Google Scholar
  33. 33.
    Feinerman, O., Haeupler, B., Korman, A.: Breathe before speaking: efficient information dissemination despite noisy, limited and anonymous communication. In: PODC, pp. 114–123 (2014)Google Scholar
  34. 34.
    Feinerman, O., Korman, A.: Clock synchronization and estimation in highly dynamic networks: an information theoretic approach. In: SIROCCO, pp. 16–30 (2015)Google Scholar
  35. 35.
    Feinerman, O., Korman, A.: Individual versus collective cognition in social insects. Submitted to Journal of Experimental Biology (2016)Google Scholar
  36. 36.
    Harkness, R., Maroudas, N.: Central place foraging by an ant (cataglyphis bicolor fab.): a model of searching. Anim. Behav. 33(3), 916–928 (1985)CrossRefGoogle Scholar
  37. 37.
    Herman, T.: Phase clocks for transient fault repair. IEEE Trans. Parallel Distrib. Syst. 11(10), 1048–1057 (2000)CrossRefGoogle Scholar
  38. 38.
    Karp, R.M., Schindelhauer, C., Shenker, S., Vöcking, B.: Randomized rumor spreading. In: FOCS, pp. 565–574 (2000)Google Scholar
  39. 39.
    Kempe, D., Dobra, A., Gehrke, J.: Gossip-based computation of aggregate information. In: FOCS, pp. 482–491. IEEE (2003)Google Scholar
  40. 40.
    Kravchik, A., Kutten, S.: Time optimal synchronous self stabilizing spanning tree. In: Afek, Y. (ed.) DISC, Jerusalem, Israel, October 14–18, 2013. Proceedings, volume 8205 of Lecture Notes in Computer Science, pp. 91–105. Springer (2013)Google Scholar
  41. 41.
    Lamport, L.: Time, clocks, and the ordering of events in a distributed system. Commun. ACM 21(7), 558–565 (1978)CrossRefzbMATHGoogle Scholar
  42. 42.
    Lenzen, C., Locher, T., Sommer, P., Wattenhofer, R.: Clock synchronization: ppen problems in theory and practice. In: SOFSEM, pp. 61–70 (2010)Google Scholar
  43. 43.
    Lenzen, C., Locher, T., Wattenhofer, R.: Tight bounds for clock synchronization. J. ACM 57(2), 8 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Lenzen, C., Rybicki, J.: Efficient counting with optimal resilience. In: DISC, pp. 16–30 (2015)Google Scholar
  45. 45.
    Lenzen, C., Rybicki, J., Suomela, J.: Towards optimal synchronous counting. In: PODC, pp. 441–450 (2015)Google Scholar
  46. 46.
    McDiarmid, C.: Concentration, pp. 195–248. Springer, New York (1998)zbMATHGoogle Scholar
  47. 47.
    Razin, N., Eckmann, J., Feinerman, O.: Desert ants achieve reliable recruitment across noisy interactions. J. R. Soc. Interface 10(20170079) (2013)Google Scholar
  48. 48.
    Roberts, G.: Why individual vigilance increases as group size increases. Anim. Behav. 51, 1077–1086 (1996)CrossRefGoogle Scholar
  49. 49.
    Sumpter, D.J., et al.: Consensus decision making by fish. Curr. Biol. 22(25), 1773–1777 (2008)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.IRIF, CNRS and University Paris DiderotParisFrance
  2. 2.Max Planck Institute for InformaticsSaarbrückenGermany

Personalised recommendations