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Making asynchronous distributed computations robust to noise

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Abstract

We consider the problem of making distributed computations robust to noise, in particular to worst-case (adversarial) corruptions of messages. We give a general distributed interactive coding scheme which simulates any asynchronous distributed protocol while tolerating an optimal corruption of a \(\varTheta (1/n)\) fraction of all messages and incurring a moderate blowup of \(O(n\log ^2 n)\) in the communication complexity. Our result is the first fully distributed interactive coding scheme in which the topology of the communication network is not known in advance. Prior work required either a coordinating node to be connected to all other nodes in the network or assumed a synchronous network in which all nodes already know the complete topology of the network.

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Notes

  1. To ease the readability, we write in parenthesis the functionality of each sent message, but we emphasize that messages in our construction contain no content at all, and the labels of Explore and Ack are given only for the analysis.

  2. Throughout this work, all logarithms are taken to base 2.

  3. While we focus here on deterministic protocols, our results also apply to randomized Monte-Carlo protocols.

  4. This type of noise, commonly called insertion and deletion noise, is known to be more difficult to deal with in the interactive setting [8] and may be destructive for asynchronous protocols [14].

  5. Note that additional messages may arrive from a sibling node as part of the BFS construction. Still, the next message arriving from the parent belongs to the coding scheme rather than to the BFS construction.

  6. Later, in Sect. 5, we apply our root-triggered synchronizer to an input protocol on G which is fully-utilized on a spanning subgraph S of G.

  7. E.g., when using an \(O(\log n)\)-spanner of size \(s=O(n)\) towards Corollary 13.

  8. In Theorem 12 we allow \(\pi _{spanner}\) to be an asynchronous protocol. Note, however, that the same applies to synchronous protocols given as an inputs to our coding scheme: let every party speak at every round (sending dummies as needed); then redefine the protocol in an asynchronous model where each party sends the messages of a given round after receiving all the messages of a previous round. It follows that any synchronous protocol with time complexity \(\tau \) sending messages of size \(\sigma \) can be converted into an equivalent asynchronous protocol with message complexity \(O(\tau m)\) and communication complexity \(O(\tau m \sigma )\).

  9. A different approach would be to add another bit that contains the parity of the message number. Then, the receiver will be able to distinguish whether a received bit is the continuation of a previous message, or the first bit of a new message.

  10. By balancing the parts \(C_1\) and \(C_2\) in a weighted way one can obtain a slightly improved resilience of \(\mu _1\mu _2/(\mu _1+\mu _2)= \varTheta (1/(n+s))\) which is, however, asymptotically equivalent to \(\varTheta (1/s)\).

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Acknowledgements

We are grateful to Merav Parter for bringing [11] to our attention. We thank the anonymous reviewers for their very helpful comments and suggestions.

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Authors and Affiliations

  1. Department of Computer Science, Technion – Israel Institute of Technology, 32000, Haifa, Israel

    Keren Censor-Hillel

  2. Faculty of Engineering, Bar-Ilan University, 52900, Ramat Gan, Israel

    Ran Gelles

  3. Department of Computer Science, Carnegie Mellon University, Pittsburgh, PA, 15213, USA

    Bernhard Haeupler

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  1. Keren Censor-Hillel
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  2. Ran Gelles
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  3. Bernhard Haeupler
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Corresponding author

Correspondence to Ran Gelles.

Additional information

Supported in part by the Israel Science Foundation (Grant No. 1696/14) and the Binational Science Foundation (Grant No. 2015803). Supported in part by the Israel Science Foundation (Grant No. 1078/17). Supported in part by NSF Grants CCF-1527110 and CCF-1618280.

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Censor-Hillel, K., Gelles, R. & Haeupler, B. Making asynchronous distributed computations robust to noise. Distrib. Comput. 32, 405–421 (2019). https://doi.org/10.1007/s00446-018-0343-5

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