Space-Optimal Proportion Consensus with Population Protocols

  • Gennaro CordascoEmail author
  • Luisa Gargano
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10616)


Population protocols provide a distributed computing model in which a set of finite-state identical agents cooperate through random interactions, between neighbors in the interaction graph, to collectively carry out a computation in a distributed setting. Population protocols have become very popular in various research areas, such as distributed computing, sensor or social networks, as well as chemistry and biology. A central task in this model is majority computation, in which agents need to reach an agreement on the leading one of two possible initial opinions. In this paper we consider a generalization of the majority problem, named proportion consensus, which asks for an agreement on the proportion of one opinion, between two possible views (say \(\mathcal A\) or \(\mathcal B\)). The objective is to reach a configuration where all the agents agree on a range \(\gamma _A \subseteq [0,1]\) which contains the value of the fraction \(\rho _A\) of agents that started with view \(\mathcal A\); the goal is to get the size of \(\gamma _A\) as small as possible while also minimizing the number of states adopted by agents. We provide a lower bound on the trade-off between precision \(\epsilon \) (the size of \(\gamma _A\)) and the number of states required by any population protocol that solves the proportion consensus problem. In particular, we show that in any population protocol that solves the proportion consensus problem with precision \(\epsilon \), any agent must have at least \(\lceil 2/\epsilon \rceil \) states. We also provide a population protocol that exactly solves the proportion consensus problem with precision \(\epsilon \) and \(6\lceil 1/(2\epsilon ) \rceil -1\) states. We show that in case of an arbitrary interaction graph our protocol requires \(O(n^6/\epsilon )\) interactions (which corresponds to the number of rounds in the sequential communication model) on any network with n agents. On complete interaction networks, the expected number of required interactions is \(O(n^2 \log n)\). Using the random matching communication model, the expected number of rounds, required to reach a consensus, decreases to \(O(\varDelta n^4/\epsilon )\) in case of arbitrary interaction networks (where \(\varDelta \) denotes the maximum degree among the agents in the network) and \(O(n \log n)\) for complete networks.


  1. 1.
    Alistarh, D., Aspnes, J., Eisenstat, D., Gelashvili, R., Rivest, R.L.: Time-space trade-offs in population protocols. In: Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2017, Barcelona, Spain, Hotel Porta Fira, 16–19 January, pp. 2560–2579 (2017)Google Scholar
  2. 2.
    Alistarh, D., Aspnes, J., Gelashvili, R.: Space-Optimal Majority in Population Protocols. ArXiv e-prints arXiv:1704.04947, April 2017
  3. 3.
    Alistarh, D., Dudek, B., Kosowski, A., Soloveichik, D., Uznanski, P.: Robust detection in leak-prone population protocols. arXiv arXiv:1706.09937 (2017)zbMATHGoogle Scholar
  4. 4.
    Alistarh, D., Gelashvili, R., Vojnović, M.: Fast and exact majority in population protocols. In: Proceedings of the 2015 ACM Symposium on Principles of Distributed Computing, PODC 2015, New York, NY, USA, pp. 47–56 (2015)Google Scholar
  5. 5.
    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
  6. 6.
    Angluin, D., Aspnes, J., Eisenstat, D.: Stably computable predicates are semilinear. In: Ruppert, E., Malkhi, D. (eds.) PODC, pp. 292–299. ACM (2006)Google Scholar
  7. 7.
    Angluin, D., Aspnes, J., Eisenstat, D.: A simple population protocol for fast robust approximate majority. Distrib. Comput. 21(2), 87–102 (2008)CrossRefGoogle Scholar
  8. 8.
    Angluin, D., Fischer, M.J., Jiang, H.: Stabilizing consensus in mobile networks. In: Gibbons, P.B., Abdelzaher, T., Aspnes, J., Rao, R. (eds.) DCOSS 2006. LNCS, vol. 4026, pp. 37–50. Springer, Heidelberg (2006). doi: 10.1007/11776178_3CrossRefGoogle Scholar
  9. 9.
    Aspnes, J., Ruppert, E.: An introduction to population protocols. Bull. Eur. Assoc. Theoret. Comput. Sci. 93, 98–117 (2007)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Beauquier, J., Blanchard, P., Burman, J.: Self-stabilizing leader election in population protocols over arbitrary communication graphs. In: Baldoni, R., Nisse, N., Steen, M. (eds.) OPODIS 2013. LNCS, vol. 8304, pp. 38–52. Springer, Cham (2013). doi: 10.1007/978-3-319-03850-6_4CrossRefGoogle Scholar
  11. 11.
    Becchetti, L., Clementi, A., Natale, E., Pasquale, F., Raghavendra, P., Trevisan, L.: Friend or Foe? Population Protocols can perform Community Detection. ArXiv e-prints arXiv:1703.05045, March 2017
  12. 12.
    Berenbrink, P., Friedetzky, T., Kling, P., Mallmann-Trenn, F., Wastell, C.: Plurality consensus in arbitrary graphs: lessons learned from load balancing. In: 24th Annual European Symposium on Algorithms (ESA 2016), pp. 10:1–10:18 (2016)Google Scholar
  13. 13.
    Chen, Y.-J., Dalchau, N., Srinivas, N., Phillips, A., Cardelli, L., Soloveichik, D., Seelig, G.: Programmable chemical controllers made from DNA. Nat. Nanotechnol. 8, 755–762 (2013)CrossRefGoogle Scholar
  14. 14.
    Cordasco, G., Gargano, L.: Label propagation algorithm: a semi-synchronous approach. Int. J. Soc. Netw. Mining (IJSNM) 1(1), 3–26 (2012)CrossRefGoogle Scholar
  15. 15.
    Delporte-Gallet, C., Fauconnier, H., Guerraoui, R., Ruppert, E.: When birds die: making population protocols fault-tolerant. In: Gibbons, P.B., Abdelzaher, T., Aspnes, J., Rao, R. (eds.) DCOSS 2006. LNCS, vol. 4026, pp. 51–66. Springer, Heidelberg (2006). doi: 10.1007/11776178_4CrossRefGoogle Scholar
  16. 16.
    Draief, M., Vojnovi, M.: Convergence speed of binary interval consensus. SIAM J. Control Optim. 50(3), 1087–1109 (2012)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Gasieniec, L., Hamilton, D., Martin, R., Spirakis, P.G., Stachowiak, G.: Deterministic population protocols for exact majority and plurality. In: 20th International Conference on Principles of Distributed Systems (OPODIS 2016), vol. 70, pp. 14:1–14:14 (2017)Google Scholar
  18. 18.
    Gasieniec, L., Stachowiak, G.: Fast space optimal leader election in population protocols. arXiv e-prints arXiv:1704.07649
  19. 19.
    Mertzios, G.B., Nikoletseas, S.E., Raptopoulos, C.L., Spirakis, P.G.: Stably computing order statistics with arithmetic population protocols. In: 41st International Symposium on Mathematical Foundations of Computer Science, MFCS, pp. 68:1–68:14 (2016)Google Scholar
  20. 20.
    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
  21. 21.
    Mizoguchi, R., Ono, H., Kijima, S., Yamashita, M.: On space complexity of self-stabilizing leader election in mediated population protocol. Distrib. Comput. 25(6), 451–460 (2012)CrossRefGoogle Scholar
  22. 22.
    Mocquard, Y., Anceaume, E., Aspnes, J., Busnel, Y., Sericola, B.: Counting with population protocols. In: 2015 IEEE 14th International Symposium on Network Computing and Applications (NCA), pp. 35–42, September 2015Google Scholar
  23. 23.
    Mocquard, Y., Anceaume, E., Sericola, B.: Optimal proportion computation with population protocols. In: 2016 IEEE 15th International Symposium on Network Computing and Applications (NCA), pp. 216–223, October 2016Google Scholar
  24. 24.
    Perron, E., Vasudevan, D., Vojnovic, M.: Using three states for binary consensus on complete graphs. IEEE INFOCOM 2009, 2527–2535 (2009)Google Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.University of Campania “L.Vanvitelli”CasertaItaly
  2. 2.University of SalernoFiscianoItaly

Personalised recommendations