On the Necessary Memory to Compute the Plurality in Multi-agent Systems

  • Emanuele Natale
  • Iliad RamezaniEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11485)


We consider the Relative-Majority Problem (also known as Plurality), in which, given a multi-agent system where each agent is initially provided an input value out of a set of k possible ones, each agent is required to eventually compute the input value with the highest frequency in the initial configuration. We consider the problem in the general Population Protocols model in which, given an underlying undirected connected graph whose nodes represent the agents, edges are selected by a globally fair scheduler.

The state complexity that is required for solving the Plurality Problem (i.e., the minimum number of memory states that each agent needs to have in order to solve the problem), has been a long-standing open problem. The best protocol so far for the general multi-valued case requires polynomial memory: Salehkaleybar et al. (2015) devised a protocol that solves the problem by employing \(\mathcal O(k 2^k)\) states per agent, and they conjectured their upper bound to be optimal. On the other hand, under the strong assumption that agents initially agree on a total ordering of the initial input values, Gąsieniec et al. (2017), provided an elegant logarithmic-memory plurality protocol.

In this work, we refute Salehkaleybar et al.’s conjecture, by providing a plurality protocol which employs \(\mathcal O(k^{11}) \) states per agent. Central to our result is an ordering protocol which allows to leverage on the plurality protocol by Gąsieniec et al., of independent interest. We also provide a \(\varOmega (k^2)\)-state lower bound on the necessary memory to solve the problem, proving that the Plurality Problem cannot be solved within the mere memory necessary to encode the output.


  1. 1.
    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)CrossRefGoogle Scholar
  2. 2.
    Angluin, D., Aspnes, J., Eisenstat, D., Ruppert, E.: The computational power of population protocols. Distrib. Comput. 20(4), 279–304 (2007)CrossRefGoogle Scholar
  3. 3.
    Aspnes, J., Beauquier, J., Burman, J., Sohier, D.: Time and Space optimal counting in population protocols. In: 20th International Conference on Principles of Distributed Systems (OPODIS 2016), Leibniz International Proceedings in Informatics (LIPIcs), vol. 70, pp. 13:1–13:17. Dagstuhl, Germany (2017)Google Scholar
  4. 4.
    Beauquier, J., Burman, J., Clavière, S., Sohier, D.: Space-optimal counting in population protocols. In: Moses, Y. (ed.) DISC 2015. LNCS, vol. 9363, pp. 631–646. Springer, Heidelberg (2015). Scholar
  5. 5.
    Becchetti, L., Clementi, A.E.F., Natale, E., Pasquale, F., Silvestri, R.: Plurality consensus in the gossip model. In: Proceedings of the 26th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2015, pp. 371–390 (2015)Google Scholar
  6. 6.
    Becchetti, L., Clementi, A.E.F., Natale, E., Pasquale, F., Trevisan, L.: Stabilizing consensus with many opinions. In: Proceedings of the 27th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016, pp. 620–635 (2016)Google Scholar
  7. 7.
    Ben-Shahar, O., Dolev, S., Dolgin, A., Segal, M.: Direction election in flocking swarms. Ad Hoc Netw. 12, 250–258 (2014)CrossRefGoogle Scholar
  8. 8.
    Benezit, F., Thiran, P., Vetterli, M.: The distributed multiple voting problem. IEEE J. Sel. Top. Signal Process. 5(4), 791–804 (2011)CrossRefGoogle Scholar
  9. 9.
    Bénézit, F., Thiran, P., Vetterli, M.: Interval consensus: from quantized gossip to voting. In: Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing, ICASSP 2009, pp. 3661–3664 (2009)Google Scholar
  10. 10.
    Boyd, S., Ghosh, A., Prabhakar, B., Shah, D.: Randomized gossip algorithms. IEEE/ACM Trans. Netw. 14(SI), 2508–2530 (2006)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Couzin, I.D., Krause, J., Franks, N.R., Levin, S.A.: Effective leadership and decision-making in animal groups on the move. Nature 433(7025), 513–516 (2005)CrossRefGoogle Scholar
  12. 12.
    Levin, D.A., Peres, Y.: Markov Chains and Mixing Times, 1st edn. American Mathematical Society, Providence (2008)CrossRefGoogle Scholar
  13. 13.
    Doty, D.: Timing in chemical reaction networks. In: Proceedings of the 25th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014, pp. 772–784 (2014)Google Scholar
  14. 14.
    Elsässer, R., Friedetzky, T., Kaaser, D., Mallmann-Trenn, F., Trinker, H.: Brief announcement: rapid asynchronous plurality consensus. In: Proceedings of the ACM Symposium on Principles of Distributed Computing, PODC 2017, pp. 363–365. ACM, New York (2017)Google Scholar
  15. 15.
    Ghaffari, M., Lengler, J.: Tight analysis for the 3-majority consensus dynamics. CoRR, abs/1705.05583 (2017)Google Scholar
  16. 16.
    Ghaffari, M., Parter, M.: A polylogarithmic gossip algorithm for plurality consensus. In: Proceedings of the 2016 ACM Symposium on Principles of Distributed Computing, PODC 2016, pp. 117–126 (2016)Google Scholar
  17. 17.
    Gąsieniec, L., Hamilton, D., Martin, R., Spirakis, P.G., Stachowiak, G.: Deterministic population protocols for exact majority and plurality. In: LIPIcs-Leibniz International Proceedings in Informatics, vol. 70 (2016)Google Scholar
  18. 18.
    Gmyr, R., Hinnenthal, K., Kostitsyna, I., Kuhn, F., Rudolph, D., Scheideler, C.: Shape recognition by a finite automaton robot. In: 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018), Leibniz International Proceedings in Informatics (LIPIcs), vol. 117, pp. 52:1–52:15, Dagstuhl, Germany (2018)Google Scholar
  19. 19.
    Holzer, M., Kutrib, M.: Descriptional and computational complexity of finite automata-a survey. Inf. Comput. 209(3), 456–470 (2011)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Ma, Q., Johansson, A., Tero, A., Nakagaki, T., Sumpter, D.J.T.: Current-reinforced random walks for constructing transport networks. J. R. Soc. Interface 10(80), 20120864 (2012)CrossRefGoogle Scholar
  21. 21.
    Mertzios, G.B., Nikoletseas, S.E., Raptopoulos, C.L., Spirakis, P.G.: Determining majority in networks with local interactions and very small local memory. Distrib. Comput. 30(1), 1–16 (2017)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Pitoni, V.: Memory management with explicit time in resource-bounded agents. In: Proceedings of the 32nd AAAI Conference on Artificial Intelligence, New Orleans, Louisiana, USA, 2–7 February 2018Google Scholar
  23. 23.
    Ranjbar-Sahraei, B., Ammar, H.B., Bloembergen, D., Tuyls, K., Weiss, G.: Theory of cooperation in complex social networks. In: Proceedings of the 28th AAAI Conference on Artificial Intelligence, AAAI 2014, pp. 1471–1477. AAAI Press, Québec City (2014)Google Scholar
  24. 24.
    Salehkaleybar, S., Sharif-Nassab, A., Golestani, S.J.: Distributed voting/ranking with optimal number of states per node. IEEE Trans. Signal Inf. Process. Netw. 1(4), 259–267 (2015)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Santoro, N.: Design and Analysis of Distributed Algorithms, 1st edn. Wiley, Hoboken (2006)CrossRefGoogle Scholar
  26. 26.
    Sumpter, D.J.T., Krause, J., James, R., Couzin, I.D., Ward, A.J.W.: Consensus decision making by fish. Curr. Biol. 18(22), 1773–1777 (2008)CrossRefGoogle Scholar
  27. 27.
    Temkin, O.N., Zeigarnik, A.V., Bonchev, D.G.: Chemical Reaction Networks: A Graph-Theoretical Approach, 1st edn. CRC Press, Boca Raton (1996)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Max Planck Institute for InformaticsSaarbrückenGermany
  2. 2.Université Côte d’Azur, CNRS, I3S, InriaSophia AntipolisFrance
  3. 3.Sharif University of TechnologyTehranIran

Personalised recommendations