Advertisement

Global Versus Local Computations: Fast Computing with Identifiers

  • Mikaël RabieEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10641)

Abstract

This paper studies what can be computed by using probabilistic local interactions with agents with a very restricted power in polylogarithmic parallel time.

It is known that if agents are only finite state (corresponding to the Population Protocol model by Angluin et al.), then only semilinear predicates over the global input can be computed. In fact, if the population starts with a unique leader, these predicates can even be computed in a polylogarithmic parallel time.

If identifiers are added (corresponding to the Community Protocol model by Guerraoui and Ruppert), then more global predicates over the input multiset can be computed. Local predicates over the input sorted according to the identifiers can also be computed, as long as the identifiers are ordered. The time of some of those predicates might require exponential parallel time.

In this paper, we consider what can be computed with Community Protocol in a polylogarithmic number of parallel interactions. We introduce the class CPPL corresponding to protocols that use \(O(n\log ^kn)\), for some k, expected interactions to compute their predicates, or equivalently a polylogarithmic number of parallel expected interactions.

We provide some computable protocols, some boundaries of the class, using the fact that the population can compute its size. We also prove two impossibility results providing some arguments showing that local computations are no longer easy: the population does not have the time to compare a linear number of consecutive identifiers. The Linearly Local languages, such that the rational language \((ab)^*\), are not computable.

References

  1. 1.
    Angluin, D., Aspnes, J., Eisenstat, D., Ruppert, E.: The computational power of population protocols. Distrib. Comput. DISC 20, 279–304 (2007)CrossRefzbMATHGoogle Scholar
  2. 2.
    Angluin, D., Aspnes, J., Chan, M., Fischer, M.J., Jiang, H., Peralta, R.: Stably computable properties of network graphs. In: Prasanna, V.K., Iyengar, S.S., Spirakis, P.G., Welsh, M. (eds.) DCOSS 2005. LNCS, vol. 3560, pp. 63–74. Springer, Heidelberg (2005).  https://doi.org/10.1007/11502593_8 CrossRefGoogle Scholar
  3. 3.
    Angluin, D., Aspnes, J., Diamadi, Z., Fischer, M.J., Peralta, R.: Computation in networks of passively mobile finite-state sensors. In: Principles of Distributed Computing, PODC, July 2004Google Scholar
  4. 4.
    Angluin, D., Aspnes, J., Eisenstat, D.: Fast computation by population protocols with a leader. Distrib. Comput. DISC 21, 183–199 (2008)CrossRefzbMATHGoogle Scholar
  5. 5.
    Angluin, D., Aspnes, J., Fischer, M.J., Jiang, H.: Self-stabilizing population protocols. In: Anderson, J.H., Prencipe, G., Wattenhofer, R. (eds.) OPODIS 2005. LNCS, vol. 3974, pp. 103–117. Springer, Heidelberg (2006).  https://doi.org/10.1007/11795490_10 CrossRefGoogle Scholar
  6. 6.
    Beauquier, J., Blanchard, P., Burman, J., Delaët, S.: Computing time complexity of population protocols with cover times - the zebranet example. In: Défago, X., Petit, F., Villain, V. (eds.) SSS 2011. LNCS, vol. 6976, pp. 47–61. Springer, Heidelberg (2011).  https://doi.org/10.1007/978-3-642-24550-3_6 CrossRefGoogle Scholar
  7. 7.
    Beauquier, J., Burman, J., Kutten, S.: Making population protocols self-stabilizing. In: Guerraoui, R., Petit, F. (eds.) SSS 2009. LNCS, vol. 5873, pp. 90–104. Springer, Heidelberg (2009).  https://doi.org/10.1007/978-3-642-05118-0_7 CrossRefGoogle Scholar
  8. 8.
    Beauquier, J., Burman, J., Rosaz, L., Rozoy, B.: Non-deterministic population protocols. In: Baldoni, R., Flocchini, P., Binoy, R. (eds.) OPODIS 2012. LNCS, vol. 7702, pp. 61–75. Springer, Heidelberg (2012).  https://doi.org/10.1007/978-3-642-35476-2_5 CrossRefGoogle Scholar
  9. 9.
    Bournez, O., Cohen, J., Rabie, M.: Homonym population protocols. In: Bouajjani, A., Fauconnier, H. (eds.) NETYS 2015. LNCS, vol. 9466, pp. 125–139. Springer, Cham (2015).  https://doi.org/10.1007/978-3-319-26850-7_9 CrossRefGoogle Scholar
  10. 10.
    Chatzigiannakis, I., Michail, O., Nikolaou, S., Pavlogiannis, A., Spirakis, P.G.: Passively mobile communicating machines that use restricted space. In: International Workshop on Foundations of Mobile Computing, FOMC 2011 (2011)Google Scholar
  11. 11.
    Daley, D.J., Kendall, D.G.: Stochastic rumours. IMA J. Appl. Math. 1, 42–55 (1965)MathSciNetCrossRefGoogle Scholar
  12. 12.
    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).  https://doi.org/10.1007/11776178_4 CrossRefGoogle Scholar
  13. 13.
    Doty, D., Soloveichik, D.: Stable leader election in population protocols requires linear time. In: Moses, Y. (ed.) DISC 2015. LNCS, vol. 9363, pp. 602–616. Springer, Heidelberg (2015).  https://doi.org/10.1007/978-3-662-48653-5_40 CrossRefGoogle Scholar
  14. 14.
    Fraigniaud, P., Korman, A., Lebhar, E.: Local MST computation with short advice. In: Symposium on Parallelism in Algorithms and Architectures, SPAA (2007)Google Scholar
  15. 15.
    Gillespie, D.T.: A rigorous derivation of the chemical master equation. Phys. A 188, 404–425 (1992)CrossRefGoogle Scholar
  16. 16.
    Guerraoui, R., Ruppert, E.: Names trump malice: tiny mobile agents can tolerate byzantine failures. In: Albers, S., Marchetti-Spaccamela, A., Matias, Y., Nikoletseas, S., Thomas, W. (eds.) ICALP 2009. LNCS, vol. 5556, pp. 484–495. Springer, Heidelberg (2009).  https://doi.org/10.1007/978-3-642-02930-1_40 CrossRefGoogle Scholar
  17. 17.
    Hethcote, H.W.: The mathematics of infectious diseases. SIAM Rev. 42, 599–653 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Mertzios, G.B., Nikoletseas, O.E., Raptopoulos, C.L., Spirakis, P.G.: Stably computing order statistics with arithmetic population protocols. In: Mathematical Foundations of Computer Science, MFCS (2016)Google Scholar
  19. 19.
    Michail, O., Chatzigiannakis, I., Spirakis, P.G.: Mediated population protocols. Theor. Comput. Sci. 412, 2434–2450 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Murray, J.D.: Mathematical Biology. I: An Introduction, 3rd edn. Springer, Heidelberg (2002).  https://doi.org/10.1007/b98868 zbMATHGoogle Scholar
  21. 21.
    Schönhage, A.: Storage modification machines. SIAM J. Comput. 9, 490–508 (1980)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.LIP, ENS de LyonLyonFrance

Personalised recommendations