Abstract
Fast algorithms are presented for performing computations in a probabilistic population model. This is a variant of the standard population protocol model—in which finite-state agents interact in pairs under the control of an adversary scheduler—where all pairs are equally likely to be chosen for each interaction. It is shown that when a unique leader agent is provided in the initial population, the population can simulate a virtual register machine in which standard arithmetic operations like comparison, addition, subtraction, and multiplication and division by constants can be simulated in O(n log4 n) interactions with high probability. Applications include a reduction of the cost of computing a semilinear predicate to O(n log4 n) interactions from the previously best-known bound of O(n 2 logn) interactions and simulation of a LOGSPACE Turing machine using the same O(n log4 n) interactions per step. These bounds on interactions translate into O(log4 n) time per step in a natural parallel model in which each agent participates in an expected Θ(1) interactions per time unit. The central method is the extensive use of epidemics to propagate information from and to the leader, combined with an epidemic-based phase clock used to detect when these epidemics are likely to be complete.
Access provided by Autonomous University of Puebla. Download to read the full chapter text
Chapter PDF
Similar content being viewed by others
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
References
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)
Angluin, D., Aspnes, J., Diamadi, Z., Fischer, M.J., Peralta, R.: Urn automata. Technical Report YALEU/DCS/TR-1280, Yale University Department of Computer Science (November 2003)
Angluin, D., Aspnes, J., Diamadi, Z., Fischer, M.J., Peralta, R.: Computation in networks of passively mobile finite-state sensors. In: PODC 2004: Proceedings of the Twenty-Third Annual ACM Symposium on Principles of Distributed Computing, pp. 290–299. ACM Press, New York (2004)
Angluin, D., Aspnes, J., Diamadi, Z., Fischer, M.J., Peralta, R.: Computation in networks of passively mobile finite-state sensors. Distributed Computing 18(4), 235–253 (2006)
Angluin, D., Aspnes, J., Eisenstat, D.: Stably computable predicates are semilinear. In: PODC 2006 (July 2006) (to appear)
Angluin, D., Aspnes, J., Eisenstat, D., Ruppert, E.: On the power of anonymous one-way communication. In: Anderson, J.H., Prencipe, G., Wattenhofer, R. (eds.) OPODIS 2005. LNCS, vol. 3974, pp. 396–411. Springer, Heidelberg (2006)
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)
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)
Arora, A., Dolev, S., Gouda, M.G.: Maintaining digital clocks in step. In: Toueg, S., Kirousis, L.M., Spirakis, P.G. (eds.) WDAG 1991. LNCS, vol. 579, pp. 71–79. Springer, Heidelberg (1992)
Bailey, N.T.J.: The Mathematical Theory of Infectious Diseases, 2nd edn. Charles Griffin & Co., London and High Wycombe (1975)
Birman, K.P., Hayden, M., Ozkasap, O., Xiao, Z., Budiu, M., Minsky, Y.: Bimodal multicast. ACM Trans. Comput. Syst. 17(2), 41–88 (1999)
Daley, D.J., Kendall, D.G.: Stochastic rumours. Journal of the Institute of Mathematics and its Applications 1, 42–55 (1965)
Daliot, A., Dolev, D., Parnas, H.: Self-stabilizing pulse synchronization inspired by biological pacemaker networks. In: Huang, S.-T., Herman, T. (eds.) SSS 2003. LNCS, vol. 2704, pp. 32–48. Springer, Heidelberg (2003)
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. Springer, Heidelberg (2006)
Diamadi, Z., Fischer, M.J.: A simple game for the study of trust in distributed systems. Wuhan University Journal of Natural Sciences 6(1–2), 72–82 (2001); Also appears as Yale Technical Report TR–1207 (January 2001)
Dolev, S., Welch, J.L.: Self-stabilizing clock synchronization in the presence of Byzantine faults. Journal of the ACM 51(5), 780–799 (2004)
Gibson, M.A., Bruck, J.: Efficient exact stochastic simulation of chemical systems with many species and many channels. Journal of Physical Chemistry A 104, 1876–1880 (2000)
Gillespie, D.T.: Exact stochastic simulation of coupled chemical reactions. Journal of Physical Chemistry 81(25), 2340–2361 (1977)
Gillespie, D.T.: A rigorous derivation of the chemical master equation. Physica A 188, 404–425 (1992)
Herman, T.: Phase clocks for transient fault repair. IEEE Transactions on Parallel and Distributed Systems 11(10), 1048–1057 (2000)
Kamath, A.P., Motwani, R., Palem, K., Spirakis, P.: Tail bounds for occupancy and the satisfiability threshold conjecture. Random Structures and Algorithms 7, 59–80 (1995)
Minsky, M.L.: Computation: Finite and Infinite Machines. Prentice-Hall Series in Automatic Computation. Prentice-Hall, Inc., Englewood Cliffs (1967)
Presburger, M.: Über die Vollständigkeit eines gewissen Systems der Arithmetik ganzer Zahlen, in welchem die Addition als einzige Operation hervortritt. In: Comptes-Rendus du I Congrès de Mathématiciens des Pays Slaves, Warszawa, pp. 92–101 (1929)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2006 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Angluin, D., Aspnes, J., Eisenstat, D. (2006). Fast Computation by Population Protocols with a Leader. In: Dolev, S. (eds) Distributed Computing. DISC 2006. Lecture Notes in Computer Science, vol 4167. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11864219_5
Download citation
DOI: https://doi.org/10.1007/11864219_5
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-44624-8
Online ISBN: 978-3-540-44627-9
eBook Packages: Computer ScienceComputer Science (R0)