On the Termination of Flooding

Authors Walter Hussak, Amitabh Trehan



PDF
Thumbnail PDF

File

LIPIcs.STACS.2020.17.pdf
  • Filesize: 0.89 MB
  • 13 pages

Document Identifiers

Author Details

Walter Hussak
  • Computer Science, Loughborough University, UK
Amitabh Trehan
  • Computer Science, Loughborough University, UK
  • www.amitabhtrehan.net

Acknowledgements

We would like to thank the anonymous reviewers for their comments, and Saket Saurabh, Jonas Lefèvre, Chhaya Trehan, Gary Bennett, Valerie King, Shay Kutten, Paul Spirakis, Abhinav Aggarwal for the useful discussions and insights and to all others in our network who attempted to solve this rather easy to state puzzle.

Cite AsGet BibTex

Walter Hussak and Amitabh Trehan. On the Termination of Flooding. In 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 154, pp. 17:1-17:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
https://doi.org/10.4230/LIPIcs.STACS.2020.17

Abstract

Flooding is among the simplest and most fundamental of all graph/network algorithms. Consider a (distributed network in the form of a) finite undirected graph G with a distinguished node v that begins flooding by sending copies of the same message to all its neighbours and the neighbours, in the next round, forward the message to all and only the neighbours they did not receive the message from in that round and so on. We assume that nodes do not keep a record of the flooding event, thus, raising the possibility that messages may circulate infinitely even on a finite graph. We call this history-less process amnesiac flooding (to distinguish from a classic distributed implementation of flooding that maintains a history of received messages to ensure a node never sends the same message again). Flooding will terminate when no node in G sends a message in a round, and, thus, subsequent rounds. As far as we know, the question of termination for amnesiac flooding has not been settled - rather, non-termination is implicitly assumed. In this paper, we show that surprisingly synchronous amnesiac flooding always terminates on any arbitrary finite graph and derive exact termination times which differ sharply in bipartite and non-bipartite graphs. In particular, synchronous flooding terminates in e rounds, where e is the eccentricity of the source node, if and only if G is bipartite, and, otherwise, in j rounds where e < j ≤ e+d+1 and d is the diameter of G. Since e is bounded above by d, this implies termination times of at most d and of at most 2d + 1 for bipartite and non-bipartite graphs respectively. This suggests that if communication/broadcast to all nodes is the motivation, the history-less amnesiac flooding is asymptotically time optimal and obviates the need for construction and maintenance of spanning structures like spanning trees. Moreover, the clear separation in the termination times of bipartite and non-bipartite graphs may suggest possible mechanisms for distributed discovery of the topology/distances in an arbitrary graph. For comparison, we also show that, for asynchronous networks, however, an adversary can force the process to be non-terminating.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph algorithms
  • Theory of computation → Graph algorithms analysis
  • Theory of computation → Distributed algorithms
Keywords
  • Flooding algorithm
  • Network algorithms
  • Distributed algorithms
  • Graph theory
  • Termination
  • Bipartiteness
  • Communication
  • Broadcast

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads

References

  1. James Aspnes. Flooding, February 2019. URL: http://www.cs.yale.edu/homes/aspnes/pinewiki/Flooding.html.
  2. Hagit Attiya and Jennifer Welch. Distributed Computing: Fundamentals, Simulations and Advanced Topics. John Wiley & Sons, 2004. Google Scholar
  3. JOSHUA N. COOPER and JOEL SPENCER. Simulating a random walk with constant error. Combinatorics, Probability and Computing, 15(6):815?822, 2006. URL: https://doi.org/10.1017/S0963548306007565.
  4. Atish Das Sarma, Danupon Nanongkai, and Gopal Pandurangan. Fast distributed random walks. In PODC, pages 161-170, 2009. Google Scholar
  5. Atish Das Sarma, Danupon Nanongkai, Gopal Pandurangan, and Prasad Tetali. Efficient distributed random walks with applications. In PODC, 2010. Google Scholar
  6. Benjamin Doerr, Mahmoud Fouz, and Tobias Friedrich. Social networks spread rumors in sublogarithmic time. Electronic Notes in Discrete Mathematics, 38:303-308, 2011. The Sixth European Conference on Combinatorics, Graph Theory and Applications, EuroComb 2011. Google Scholar
  7. Robert Elsässer and Thomas Sauerwald. The power of memory in randomized broadcasting. In Proceedings of the Nineteenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA '08, pages 218-227, Philadelphia, PA, USA, 2008. Society for Industrial and Applied Mathematics. Google Scholar
  8. C. Gkantsidis, M. Mihail, and A. Saberi. Random walks in peer-to-peer networks: Algorithms and evaluation. Performance Evaluation, 63(3):241-263, 2006. Google Scholar
  9. Ajei S. Gopal, Inder S. Gopal, and Shay Kutten. Fast broadcast in high-speed networks. IEEE/ACM Trans. Netw., 7(2):262-275, 1999. URL: https://doi.org/10.1109/90.769773.
  10. Walter Hussak and Amitabh Trehan. On termination of a flooding process. In Peter Robinson and Faith Ellen, editors, Proceedings of the 2019 ACM Symposium on Principles of Distributed Computing, PODC 2019, Toronto, ON, Canada, July 29 - August 2, 2019., pages 153-155. ACM, 2019. URL: https://doi.org/10.1145/3293611.3331586.
  11. Adrian Kosowski and Dominik Pajak. Does adding more agents make a difference? A case study of cover time for the rotor-router. J. Comput. Syst. Sci., 106:80-93, 2019. URL: https://doi.org/10.1016/j.jcss.2019.07.001.
  12. Shay Kutten, Gopal Pandurangan, David Peleg, Peter Robinson, and Amitabh Trehan. On the complexity of universal leader election. J. ACM, 62(1):7:1-7:27, 2015. URL: https://doi.org/10.1145/2699440.
  13. Shay Kutten, Gopal Pandurangan, David Peleg, Peter Robinson, and Amitabh Trehan. Sublinear bounds for randomized leader election. Theoretical Computer Science, 561(0):134-143, 2015. Special Issue on Distributed Computing and Networking. URL: http://www.sciencedirect.com/science/article/pii/S0304397514001029.
  14. N. Lynch. Distributed Algorithms. Morgan Kaufmann Publishers, San Mateo, CA, 1996. Google Scholar
  15. Othon Michail and Paul G. Spirakis. Terminating population protocols via some minimal global knowledge assumptions. Journal of Parallel and Distributed Computing, 81-82:1-10, 2015. URL: https://doi.org/10.1016/j.jpdc.2015.02.005.
  16. David Peleg. Distributed Computing: A Locality Sensitive Approach. SIAM, 2000. Google Scholar
  17. A. Rahman, W. Olesinski, and P. Gburzynski. Controlled flooding in wireless ad-hoc networks. In In Proceedings of IWWAN'04, pages 73-78, 2004. Google Scholar
  18. Andrew Tanenbaum. Computer networks. Pearson Prentice Hall, Boston, 2011. Google Scholar
  19. Gerard Tel. Introduction to distributed algorithms. Cambridge University Press, New York, NY, USA, 1994. Google Scholar