Advertisement

The Expected Duration of Sequential Gossiping

  • Hans van Ditmarsch
  • Ioannis Kokkinis
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10767)

Abstract

A gossip protocol aims at arriving, by means of point-to-point communications (or telephone calls), at a situation in which every agent knows all the information initially present in the network. If it is forbidden to have more than one call at the same time, the protocol is called sequential. We generalise a method, that originates from the famous coupon collector’s problem and that was proposed by John Haigh in 1981, for bounding the expected duration of sequential gossip protocols. We give two examples of protocols where this method succeeds and two examples of protocols where this method fails to give useful bounds. Our main contribution is that, although Haigh originally applied this method in a protocol where any call is available at any moment, we show that this method can be applied in protocols where the number of available calls is decreasing. Furthermore, for one of the protocols where Haigh’s method fails we were able to obtain lower bounds for the expectation using results from random graph theory.

Keywords

Sequential gossip Networks Coupon collector’s problem Expectation 

Notes

Acknowledgements

We acknowledge financial support from ERC project EPS 313360. We are also grateful to Aris Pagourtzis for useful discussions.

References

  1. 1.
    Apt, K.R., Wojtczak, D.: Decidability of fair termination of gossip protocols. In: Proceedings of the IWIL Workshop and LPAR Short Presentations, pp. 73–85. Kalpa Publications (2017)Google Scholar
  2. 2.
    Apt, K.R., Wojtczak, D.: On the computational complexity of gossip protocols. In: Proceedings of the Twenty-Sixth International Joint Conference on Artificial Intelligence, IJCAI 2017, Melbourne, Australia, 19–25 August 2017 , pp. 765–771 (2017)Google Scholar
  3. 3.
    Apt, K.R., Grossi, D., van der Hoek, W.: Epistemic protocols for distributed gossiping. In: Proceedings of 15th TARK (2015)Google Scholar
  4. 4.
    Attamah, M., van Ditmarsch, H., Grossi, D., van der Hoek, W.: The pleasure of gossip. In: Başkent, C., Moss, L.S., Ramanujam, R. (eds.) Rohit Parikh on Logic, Language and Society. OCL, vol. 11, pp. 145–163. Springer, Cham (2017).  https://doi.org/10.1007/978-3-319-47843-2_9CrossRefGoogle Scholar
  5. 5.
    Attamah, M., Van Ditmarsch, H., Grossi, D., van der Hoek, W.: Knowledge and gossip. In: Proceedings of the Twenty-first European Conference on Artificial Intelligence, pp. 21–26. IOS Press (2014)Google Scholar
  6. 6.
    Berenbrink, P., Elsässer, R., Friedetzky, T., Nagel, L., Sauerwald, T.: Faster coupon collecting via replication with applications in gossiping. In: Murlak, F., Sankowski, P. (eds.) MFCS 2011. LNCS, vol. 6907, pp. 72–83. Springer, Heidelberg (2011).  https://doi.org/10.1007/978-3-642-22993-0_10CrossRefGoogle Scholar
  7. 7.
    Boyd, D.W., Steele, J.M.: Random exchanges of information. J. Appl. Prob. 16, 657–661 (1979)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Deb, S., Médard, M., Choute, C.: Algebraic gossip: a network coding approach to optimal multiple rumor mongering. IEEE Trans. Inf. Theory 52(6), 2486–2507 (2006)MathSciNetCrossRefGoogle Scholar
  9. 9.
    van Ditmarsch, H., van Eijck, J., Pardo, P., Ramezanian, R., Schwarzentruber, F.: Epistemic protocols for dynamic gossip. J. Appl. Log. 20, 1–31 (2017)MathSciNetCrossRefGoogle Scholar
  10. 10.
    van Ditmarsch, H., van Eijck, J., Pardo, P., Ramezanian, R., Schwarzentruber, F.: Dynamic gossip. CoRR abs/1511.00867 (2015)Google Scholar
  11. 11.
    van Ditmarsch, H., Kokkinis, I., Stockmarr, A.: Reachability and expectation in gossiping. In: An, B., Bazzan, A., Leite, J., Villata, S., van der Torre, L. (eds.) PRIMA 2017. LNCS (LNAI), vol. 10621, pp. 93–109. Springer, Cham (2017).  https://doi.org/10.1007/978-3-319-69131-2_6CrossRefGoogle Scholar
  12. 12.
    Doerr, B., Friedrich, T., Sauerwald, T.: Quasirandom rumor spreading. ACM Trans. Algorithms 11(2), 1–35 (2014)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Ferrante, M., Saltalamacchia, M.: The coupon collector’s problem. Materials Matemátics 2014(2), 35 (2014), www.mat.uab.cat/matmat
  14. 14.
    Frieze, A., Karoński, M.: Introduction to Random Graphs. Cambridge University Press, Cambridge (2015)zbMATHGoogle Scholar
  15. 15.
    Haeupler, B.: Simple, fast and deterministic gossip and rumor spreading. J. ACM 62(6), 47 (2015)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Haigh, J.: Random exchanges of information. J. Appl. Probab. 18, 743–746 (1981)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Hedetniemi, S., Hedetniemi, S., Liestman, A.: A survey of gossiping and broadcasting in communication networks. Networks 18, 319–349 (1988)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Hurkens, C.A.: Spreading gossip efficiently. Nieuw Archief voor Wiskunde 1, 208–210 (2000)MathSciNetGoogle Scholar
  19. 19.
    Moon, J.: Random exchanges of information. Nieuw Archief voor Wiskunde 20, 246–249 (1972)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Tijdeman, R.: On a telephone problem. Nieuw Archief voor Wiskunde 3(19), 188–192 (1971)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.CNRS, LORIA, University of LorraineNancyFrance
  2. 2.ReLaX UMI 2000, IMScChennaiIndia
  3. 3.TU Dortmund UniversityDortmundGermany

Personalised recommendations