Stable leader election in population protocols requires linear time

Article

Abstract

A population protocol stably elects a leader if, for all n, starting from an initial configuration with n agents each in an identical state, with probability 1 it reaches a configuration \(\mathbf {y}\) that is correct (exactly one agent is in a special leader state \(\ell \)) and stable (every configuration reachable from \(\mathbf {y}\) also has a single agent in state \(\ell \)). We show that any population protocol that stably elects a leader requires \(\varOmega (n)\) expected “parallel time”—\(\varOmega (n^2)\) expected total pairwise interactions—to reach such a stable configuration. Our result also informs the understanding of the time complexity of chemical self-organization by showing an essential difficulty in generating exact quantities of molecular species quickly.

Keywords

Population protocols Leader election Time lower bound Chemical reaction network 

References

  1. 1.
    Alistarh, D., Aspnes, J., Eisenstat, D., Gelashvili, R., Rivest, R.L.: Time-space trade-offs in molecular computation. Technical Report, arXiv:1602.08032 (2016)
  2. 2.
    Alistarh, D., Gelashvili, R.: Polylogarithmic-time leader election in population protocols. In: ICALP 2015: Proceedings of the 42nd International Colloquium on Automata, Languages, and Programming, Kyoto, Japan (2015)Google Scholar
  3. 3.
    Alistarh, D., Gelashvili, R., Vojnović, M.: Fast and exact majority in population protocols. In: PODC 2015: Proceedings of the 2015 ACM Symposium on Principles of Distributed Computing, pp. 47–56. ACM, New York (2015)Google Scholar
  4. 4.
    Angluin, D., Aspnes, J., Diamadi, Z., Fischer, M., Peralta, R.: Computation in networks of passively mobile finite-state sensors. Distrib. Comput. 18, 235–253 (2006). Preliminary version appeared in PODC (2004)CrossRefMATHGoogle Scholar
  5. 5.
    Angluin, D., Aspnes, J., Diamadi, Z., Fischer, M.J., Peralta, R.: Urn automata. Technical Report YALEU/DCS/TR-1280, Yale University, November (2003)Google Scholar
  6. 6.
    Angluin, D., Aspnes, J., Eisenstat, D.: Fast computation by population protocols with a leader. Distrib. Comput. 21(3), 183–199 (2008). Preliminary version appeared in DISC (2006)CrossRefMATHGoogle 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)CrossRefMATHGoogle Scholar
  8. 8.
    Angluin, D., Aspnes, J., Eisenstat, D., Ruppert, E.: The computational power of population protocols. Distrib. Comput. 20(4), 279–304 (2007)CrossRefMATHGoogle Scholar
  9. 9.
    Angluin, D., Aspnes, J., Fischer, M.J., Jiang, H.: Self-stabilizing population protocols. In: Anderson, J.H., Prencipe, G., Wattenhofer, R. (eds.) Principles of Distributed Systems, pp. 103–117. Springer, Berlin (2006)Google Scholar
  10. 10.
    Aspnes, J., Beauquier, J., Burman, J., Sohier, D.: Time and space optimal counting in population protocols. http://www.cs.yale.edu/homes/aspnes/papers/one-bit-counting-abstract.html (2016)
  11. 11.
    Beauquier, J., Burman, J., Clavière, S., Sohier, D.: Space-optimal counting in population protocols. In: DISC 2015: Proceedings of the 29th International Symposium on Distributed Computing, pp. 631–646 (2015)Google Scholar
  12. 12.
    Bower, J.M., Bolouri, H.: Computational Modeling of Genetic and Biochemical Networks. MIT Press, Cambridge (2004)Google Scholar
  13. 13.
    Chen, H.-L., Cummings, R., Doty, D., Soloveichik, D.: Speed faults in computation by chemical reaction networks. In: DISC 2014: Proceedings of the 28th International Symposium on Distributed Computing, Austin, TX, USA, pp. 16–30 (2014)Google Scholar
  14. 14.
    Chen, Y.-J., Dalchau, N., Srinivas, N., Phillips, A., Cardelli, L., Soloveichik, D., Seelig, G.: Programmable chemical controllers made from DNA. Nat. Nanotechnol. 8(10), 755–762 (2013)CrossRefGoogle Scholar
  15. 15.
    Cunha-Ferreira, I., Bento, I., Bettencourt-Dias, M.: From zero to many: control of centriole number in development and disease. Traffic 10(5), 482–498 (2009)Google Scholar
  16. 16.
    Dickson, L.E.: Finiteness of the odd perfect and primitive abundant numbers with \(n\) distinct prime factors. Am. J. Math. 35(4), 413–422 (1913)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Doty, D.: Timing in chemical reaction networks. In: SODA 2014: Proceedings of the 25th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 772–784 (2014)Google Scholar
  18. 18.
    Doty, D., Soloveichik, D.: Stable leader election in population protocols requires linear time. In: DISC 2015: Proceedings of the 29th International Symposium on Distributed Computing. Lecture Notes in Computer Science, pp. 602–616. Springer, Berlin (2015)Google Scholar
  19. 19.
    Gillespie, D.T.: Exact stochastic simulation of coupled chemical reactions. J. Phys. Chem. 81(25), 2340–2361 (1977)CrossRefGoogle Scholar
  20. 20.
    Izumi, T., Kinpara, K., Izumi, T., Wada, K.: Space-efficient self-stabilizing counting population protocols on mobile sensor networks. Theor. Comput. Sci. 552, 99–108 (2014)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Karp, R.M., Miller, R.E.: Parallel program schemata. J. Comput. Syst. Sci. 3(2), 147–195 (1969)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Petri, C.A.: Communication with automata. Technical report, DTIC Document (1966)Google Scholar
  23. 23.
    Soloveichik, D., Seelig, G., Winfree, E.: DNA as a universal substrate for chemical kinetics. Proc. Natl. Acad. Sci. 107(12), 5393 (2010). Preliminary version appeared in DNA 2008CrossRefGoogle Scholar
  24. 24.
    Volterra, V.: Variazioni e fluttuazioni del numero dindividui in specie animali conviventi. Mem. Acad. Lincei Roma 2, 31–113 (1926)MATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Department of Computer ScienceUniversity of California, DavisDavisUSA
  2. 2.Department of Electrical and Computer EngineeringUniversity of Texas, AustinAustinUSA

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