Skip to main content
SpringerLink
Log in

Parsimonious flooding in dynamic graphs

  • Published:
Distributed Computing Aims and scope Submit manuscript

Abstract

An edge-Markovian process with birth-rate p and death-rate q generates infinite sequences of graphs (G 0, G 1, G 2,…) with the same node set [n] such that G t is obtained from G t-1 as follows: if \({e\notin E(G_{t-1})}\) then \({e\in E(G_{t})}\) with probability p, and if \({e\in E(G_{t-1})}\) then \({e\notin E(G_{t})}\) with probability q. In this paper, we establish tight bounds on the complexity of flooding in edge-Markovian graphs, where flooding is the basic mechanism in which every node becoming aware of an information at step t forwards this information to all its neighbors at all forthcoming steps t′ > t. These bounds complete previous results obtained by Clementi et al. Moreover, we also show that flooding in dynamic graphs can be implemented in a parsimonious manner, so that to save bandwidth, yet preserving efficiency in term of simplicity and completion time. For a positive integer k, we say that the flooding protocol is k-active if each node forwards an information only during the k time steps immediately following the step at which the node receives that information for the first time. We define the reachability threshold for the flooding protocol as the smallest integer k such that, for any source \({s\in [n]}\) , the k-active flooding protocol from s completes (i.e., reaches all nodes), and we establish tight bounds for this parameter. We show that, for a large spectrum of parameters p and q, the reachability threshold is by several orders of magnitude smaller than the flooding time. In particular, we show that it is even constant whenever the ratio p/(p + q) exceeds log n/n. Moreover, we also show that being active for a number of steps equal to the reachability threshold (up to a multiplicative constant) allows the flooding protocol to complete in optimal time, i.e., in asymptotically the same number of steps as when being perpetually active. These results demonstrate that flooding can be implemented in a practical and efficient manner in dynamic graphs. The main ingredient in the proofs of our results is a reduction lemma enabling to overcome the time dependencies in edge-Markovian dynamic graphs.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Alon N., Spencer J.H.: The Probabilistic Method. Wiley, Colorado (2000)

    Book  MATH  Google Scholar 

  2. Avin, C., Koucký, M., Lotker, Z.: How to explore a fast-changing world. In: Proceedings of 35th International Colloquium on Automata, Languages and Programming (ICALP), pp. 121–132 (2008)

  3. Bollobás B.: Random Graphs. Cambridge University Press, Cambridge (2001)

    Book  MATH  Google Scholar 

  4. Brauer, F., van den Driessche, P., Wu, J. (Eds): Mathematical epidemiology. Lecture Notes in Mathematics (Mathematical Biosciences Subseries) 1945, (2008)

  5. Chakrabarti, D., Leskovec, J., Faloutsos, C., Madden, S., Guestrin, C., Faloutsos, M.: Information survival threshold in sensor and P2P networks. In: Proceedings of 26th IEEE Conference on Computer Communications (INFOCOM), pp. 1316–1324 (2007)

  6. Chang N., Liu M.: Controlled flooding search in a large network. IEEE/ACM Trans. Netw. 15(2), 436–449 (2007)

    Article  MathSciNet  Google Scholar 

  7. Chung F., Lu L.: The diameter of sparse random graphs. Adv. Appl. Math. 26(4), 257–279 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  8. Clementi A., Macci C., Monti A., Pasquale F., Silvestri R.: Flooding time in edge-Markovian dynamic graphs. SIAM J. Discret. Math. 24(4), 1694–1712 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  9. Clementi A., Monti A., Pasquale F., Silvestri R.: Broadcasting in dynamic radio networks. J. Comput. Syst. Sci. 75(4), 213–230 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  10. Clementi, A., Monti, A., Pasquale, F., Silvestri, R.: Information spreading in stationary Markovian evolving graphs. In: Proceedings of 23rd IEEE International Parallel and Distributed Processing Symposium (IPDPS), pp. 1–12 (2009)

  11. Clementi, A., Pasquale, F., Silvestri, R.: MANETS: High mobility can make up for low transmission power. In: Proceedings of 36th International Colloquium on Automata, Languages and Programming (ICALP), pp. 387–398 (2009)

  12. Demers A., Greene D., Hauser C., Irish W., Larson J., Shenker S., Sturgis H., Swinehart D., Terry D.: Epidemic algorithms for replicated database maintenance. Oper. Syst. Rev. 22(1), 8–32 (1988)

    Article  Google Scholar 

  13. Draief M., Ganesh A., Massoulié L.: Thresholds for virus spread on networks. Ann. Appl. Probab. 18(2), 359–378 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  14. Feige U., Peleg D., Raghavan P., Upfal E.: Randomized broadcast in networks. Random Struct. Alg. 1(4), 447–460 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  15. Fraigniaud P., Lazard E.: Methods and problems of communication in usual networks. Discret. Appl. Math. 53, 79–133 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  16. Frieze A., Grimmett G.: The shortest-path problem for graphs with random arc-lengths. Discret. Appl. Math. 10, 57–77 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  17. Gnutella RFC. http://rfc-gnutella.sourceforge.net/

  18. Grossglauser M., Tse D.: Mobility increases the capacity of ad-hoc wireless networks. IEEE/ACM Trans. Netw. 10(4), 477–486 (2002)

    Article  Google Scholar 

  19. Gupta I., Kermarrec A.-M., Ganesh A.: Efficient and adaptive epidemic-style protocols for reliable and scalable multicast. IEEE Trans. Parallel Dist. Syst. 17(7), 593–605 (2006)

    Article  Google Scholar 

  20. Hedetniemi S., Hedetniemi S., Liestman A.: A survey of gossiping and broadcasting in communication networks. Networks 18, 319–349 (1986)

    Article  MathSciNet  Google Scholar 

  21. Hromković J., Klasing R., Monien B., Peine R.: Dissemination of information in interconnection networks (broadcasting and gossiping). In: Du, D.-Z., Hsu, D. (eds) Combinatorial Network Theory, pp. 125–212. Kluwer Academic, Dordrecht (1995)

    Google Scholar 

  22. Janson S., Luczak T., Rucinski A.: Random Graphs. Wiley, Dordrecht (2000)

    MATH  Google Scholar 

  23. Kempe, D., Dobra, A., Gehrke, J.: Computing aggregate information using gossip. In: Proceedings of 44th IEEE Symposium on Foundations of Computer Science (FOCS), pp. 482–491 (2003)

  24. Kempe D., Kleinberg J., Demers A.: Spatial gossip and resource location protocols. J. ACM 51(6), 943–967 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  25. Kempe, D., Kleinberg, J.: Protocols and impossibility results for gossip-based communication mechanisms. In: Proceedings of 43rd IEEE Symposium on Foundations of Computer Science (FOCS), pp. 471–480 (2002)

  26. Luo, J., Eugster, P., Hubaux, J.-P.: Route driven gossip: probabilistic reliable multicast in ad hoc networks. In: Proceedings of 22nd IEEE Conference on Computer Communications (INFOCOM), pp. 2229–2239 (2003)

  27. Lv, Q., Cao, P., Cohen, E., Li, K., Shenker, S.: Search and replication in unstructured peer-to-peer networks. In: Proceedings of 16th ACM International Conference on Supercomputing (ICS), pp. 84–95 (2002)

  28. Pittel B.: On spreading a rumour. SIAM J. Appl. Math. 47, 213–223 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  29. Sarwate, A., Dimakis, A.: The impact of mobility on gossip algorithms. In: Proceedings of 28th IEEE Conference on Computer Communications (INFOCOM), pp. 2088–2096 (2009)

  30. Sripanidkulchai, K., Maggs, B., Zhang, H.: Efficient content location using interest-based locality in peer-to-peer systems. In: Proceedings of 22nd IEEE Conference on Computer Communications (INFOCOM), pp. 2166–2176 (2003)

Download references

Author information

Authors and Affiliations

  1. Laboratoire LIAFA, CNRS and University Paris Diderot, 75205, Paris Cedex 13, France

    Hervé Baumann & Pierre Fraigniaud

  2. Dipartimento di Sistemi e Informatica, University of Florence, 50134, Florence, Italy

    Pierluigi Crescenzi

Authors
  1. Hervé Baumann
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Pierluigi Crescenzi
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Pierre Fraigniaud
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Pierre Fraigniaud.

Additional information

A preliminary version of this paper appeared in the proceedings of the 28th Annual ACM SIGACT-SIGOPS Symposium on Principles of Distributed Computing (PODC), Calgary, Alberta, Canada, August 10–12, 2009. A part of this work was done during the stay of the second author at University Paris Diderot as Invited Professor. All authors were supported by COST Action 295 “DYNAMO”. The first and third authors received additional supports from the ANR projects “ALADDIN” and “PROSE”, and from the INRIA project “GANG”. The second author received additional supports from the EU under the EU/IST Project 15964 “AEOLUS”, and from the Italian PRIN project “DISCO”.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Baumann, H., Crescenzi, P. & Fraigniaud, P. Parsimonious flooding in dynamic graphs. Distrib. Comput. 24, 31–44 (2011). https://doi.org/10.1007/s00446-011-0133-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00446-011-0133-9

Keywords

Search

Navigation