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Simple and optimal randomized fault-tolerant rumor spreading

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Abstract

We revisit the classic problem of spreading a piece of information in a group of \(n\) fully connected processors. By suitably adding a small dose of randomness to the protocol of Gasieniec and Pelc (Parallel Comput 22:903–912, 1996), we derive for the first time protocols that (i) use a linear number of messages, (ii) are correct even when an arbitrary number of adversarially chosen processors does not participate in the process, and (iii) with high probability have the asymptotically optimal runtime of \(O(\log n)\) when at least an arbitrarily small constant fraction of the processors are working. In addition, our protocols do not require that the system is synchronized nor that all processors are simultaneously woken up at time zero, they are fully based on push-operations, and they do not need an a priori estimate on the number of failed nodes. Our protocols thus overcome the typical disadvantages of the two known approaches, algorithms based on random gossip (typically needing a large number of messages due to their unorganized nature) and algorithms based on fair workload splitting (which are either not time-efficient or require intricate preprocessing steps plus synchronization).

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Notes

  1. The simple workload splitting protocol of Gasieniec and Pelc which we describe and use below appeared only in the preliminary version of [9] and not in the journal paper itself. As indicated in the reference section, this preliminary version is available online at http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.49.3838.

  2. We distinguish between opening phase, which is repeated each time a rumor is spread, and a preprocessing phase, which is performed only once, when the network is established.

  3. Using, e.g., [18], it is not hard to see that in a push-pull gossip-based algorithm, in each round in which the fraction of informed processors is bounded away from 0 and from 1, with high probability some informed processor is contacted by \(\varOmega (\log (n) /\log \log n)\) uninformed neighbors each of which wants to receive the rumor from the informed node. This node therefore has to forward the rumor to \(\varOmega (\log (n) /\log \log n)\) neighbors in one round.

  4. Some models assume that an informed processor \(u\) always sends the rumor on the selected edge \((u,v)\), even if \(v\) is already informed.

  5. For a formal definition of asynchronous protocols see, e.g., [13, Chapter 14].

  6. The probability statement in this definition is with respect to the random choice of the permutation index \(i \in [t]\).

  7. To ease the presentation of the algorithm we assume here that an upper bound \(cn\) on the number of failed processors is known. In this case, Theorem 6 suggests to choose \(k=\frac{10 \log n}{1-c-\sqrt{\ln (n)/n}}\). Estimating an a prior bound on \(c\) can be avoided by setting \(k=\omega (\log n)\).

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Acknowledgments

The authors are happy to have received very helpful comments from anonymous reviewers, in particular the suggestion to use \(k\)-wise \(\delta \)-dependent distributions for the result presented in Sect. 6. We would also like to thank Ariel Gabizon for very interesting discussions and for suggesting the use of adaptive distinguishers. Part of this work have been done while Benjamin Doerr was with the Max Planck Institute for Informatics (MPII) in Saarbrücken, Germany, Carola Doerr was with the MPII and the LIAFA, Université Paris Diderot (Paris 7), France, and Shlomo Moran was a visitor at the MPII. Carola Doerr gratefully acknowledges support from a Feodor Lynen postdoctoral research fellowship of the Alexander von Humboldt Foundation and from the Agence Nationale de la Recherche under the Project ANR-09-JCJC-0067-01. Shlomo Moran is supported by the Bernard Elkin Chair in Computer Science.

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Authors and Affiliations

  1. LIX, École Polytechnique, Palaiseau, France

    Benjamin Doerr

  2. Sorbonne Universités, UPMC Univ Paris 06, UMR 7606, LIP6, 75005, Paris, France

    Carola Doerr

  3. CNRS, UMR 7606, LIP6, 75005, Paris, France

    Carola Doerr

  4. Computer Science Department, Technion - Israel Institute of Technology, 32000, Haifa, Israel

    Shay Moran & Shlomo Moran

  5. Max Planck Institute for Informatics, Saarbrücken, Germany

    Shay Moran

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  1. Benjamin Doerr
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  2. Carola Doerr
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  3. Shay Moran
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  4. Shlomo Moran
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Correspondence to Carola Doerr.

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Doerr, B., Doerr, C., Moran, S. et al. Simple and optimal randomized fault-tolerant rumor spreading. Distrib. Comput. 29, 89–104 (2016). https://doi.org/10.1007/s00446-014-0238-z

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