Distributed Computing

, Volume 21, Issue 2, pp 87–102 | Cite as

A simple population protocol for fast robust approximate majority

  • Dana Angluin
  • James Aspnes
  • David Eisenstat


We describe and analyze a 3-state one-way population protocol to compute approximate majority in the model in which pairs of agents are drawn uniformly at random to interact. Given an initial configuration of x’s, y’s and blanks that contains at least one non-blank, the goal is for the agents to reach consensus on one of the values x or y. Additionally, the value chosen should be the majority non-blank initial value, provided it exceeds the minority by a sufficient margin. We prove that with high probability n agents reach consensus in O(n log n) interactions and the value chosen is the majority provided that its initial margin is at least \({\omega(\sqrt{n} \,{\rm log}\, n)}\). This protocol has the additional property of tolerating Byzantine behavior in \({o(\sqrt{n})}\) of the agents, making it the first known population protocol that tolerates Byzantine agents.


Population protocols Majority Epidemics Byzantine faults 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Alon N. and Spencer J.H. (1992). The Probabilistic Method. Wiley, New York MATHGoogle Scholar
  2. 2.
    Angluin D., Aspnes J., Diamadi Z., Fischer M.J. and Peralta R. (2006). Computation in networks of passively mobile finite-state sensors. Distrib. Comput. 18(4): 235–253 CrossRefGoogle Scholar
  3. 3.
    Angluin, D., Aspnes, J., Eisenstat, D.: Fast computation by population protocols with a leader. In: Distributed Computing: 20th International Symposium, DISC 2006: Stockholm, Sweden, September 2006: Proceedings, pp. 61–75 (2006)Google Scholar
  4. 4.
    Angluin D., Aspnes J., Eisenstat D. and Ruppert E. (2007). The computational power of population protocols. Distrib. Comput. 20(4): 279–304 CrossRefGoogle Scholar
  5. 5.
    Aspnes J. and Ruppert E. (2007). An introduction to population protocols. Bull. Eur. Assoc. Theor. Comput. Sci. 93: 98–117 MathSciNetGoogle Scholar
  6. 6.
    Chow Y.S., Robbins H. and Siegmund D. (1991). The Theory of Optimal Stopping. Dover, New York MATHGoogle Scholar
  7. 7.
    Ezhilchelvan, P., Mostefaoui, A., Raynal, M.: Randomized multivalued consensus. In: ISORC ’01: Proceedings of the Fourth International Symposium on Object-Oriented Real-Time Distributed Computing, p. 195. IEEE Computer Society, Washington (2001)Google Scholar
  8. 8.
    Feller W. (1958). An Introduction to Probability and its Applications, vol. 1, 3rd edn. Wiley, New York Google Scholar
  9. 9.
    Gillespie D.T. (1977). Exact stochastic simulation of coupled chemical reactions. J. Phys. Chem. 81(25): 2340–2361 CrossRefGoogle Scholar
  10. 10.
    Gillespie D.T. (1992). A rigorous derivation of the chemical master equation. Physica A 188: 404–425 CrossRefGoogle Scholar
  11. 11.
    Grimmet G.R. and Stirzaker D.R. (1992). Probability and Random Processes, 2nd edn. Oxford Science Publications, Oxford Google Scholar
  12. 12.
    Kurtz, T.G.: Approximation of Population Processes. No. 36 in CBMS-NSF Regional Conference Series in Applied Mathematics. Society for Industrial and Applied Mathematics, Philadelphia (1981)Google Scholar
  13. 13.
    Mostefaoui A., Raynal M. and Tronel F. (2000). From binary consensus to multivalued consensus in asynchronous message-passing systems. Inform. Proces. Lett. 73(5–6): 2007–212 MathSciNetGoogle Scholar
  14. 14.
    Wormald N.C. (1995). Differential equations for random processes and random graphs. Ann. Appl. Probab. 5(4): 1217–1235 MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Department of Computer ScienceYale UniversityNew HavenUSA
  2. 2.New HavenUSA

Personalised recommendations