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Reliable communication over highly connected noisy networks

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Abstract

We consider the task of multiparty computation performed over networks in the presence of random noise. Given an n-party protocol that takes R rounds assuming noiseless communication, the goal is to find a coding scheme that takes \(R'\) rounds and computes the same function with high probability even when the communication is noisy, while maintaining a constant asymptotic rate, i.e., while keeping \(\liminf _{n,R\rightarrow \infty } R/R'\) positive. Rajagopalan and Schulman (STOC ’94) were the first to consider this question, and provided a coding scheme with rate \(O(1/\log (d+1))\), where d is the maximal degree in the network. While that scheme provides a constant rate coding for many practical situations, in the worst case, e.g., when the network is a complete graph, the rate is \(O(1/\log n)\), which tends to 0 as n tends to infinity. We revisit this question and provide an efficient coding scheme with a constant rate for the interesting case of fully connected networks. We furthermore extend the result and show that if a (d-regular) network has mixing time m, then there exists an efficient coding scheme with rate \(O(1/m^3\log m)\). This implies a constant rate coding scheme for any n-party protocol over a d-regular network with a constant mixing time, and in particular for random graphs with n vertices and degrees \(n^{\varOmega (1)}\).

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Notes

  1. In fact, the analysis in [43] measures the wrong quantity: the correctness of transmitted messages, rather than the correctness of received messages. This technicality can easily be solved by re-transmitting received messages. See [36, Appendix A.3] for further details.

  2. Each transmission in [43] is of a symbol taken from a small finite alphabet. Hence, each such symbol can be communicated with O(1) rounds of (bit) neighborhood connectivity.

  3. The adjacency matrix of a graph \(G=(V,E)\) is defined by \(A(i, j) = 1\) if \((i, j) \in E\) and zero otherwise. In d-regular graphs, the normalized matrix is given by \(\hat{A}=\frac{1}{d} A\).

  4. In relay we mean that if party i wants to send a bit to party j, it can send that bit to some party k who will later relay that bit to j. Thus, after sending the bit to k, the communication demand changes so that \(a_{i,j}=0\) and \(a_{k,j}\) increases by one.

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Authors and Affiliations

  1. Sackler School of Mathematics and Blavatnik School of Computer Science, Tel Aviv University, Tel Aviv, 69978, Israel

    Noga Alon

  2. Department of Computer Science, Princeton University, Princeton, NJ, 08540, USA

    Mark Braverman

  3. Department of Computer Science, Tel Aviv University, Tel Aviv, 69978, Israel

    Klim Efremenko

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

    Ran Gelles

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

    Bernhard Haeupler

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  1. Noga Alon
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  4. Ran Gelles
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Correspondence to Ran Gelles.

Additional information

A preliminary extended abstract of this work [4], appeared in the Proceedings of the 2016 ACM Symposium on Principles of Distributed Computing (PODC ’16).

Noga Alon: Research supported in part by a BSF Grant, an ISF Grant, a GIF Grant and the Israeli I-Core program. Mark Braverman: Research supported in part by NSF Awards, DMS-1128155, CCF-1525342, and CCF-1149888, a Packard Fellowship in Science and Engineering, and the Simons Collaboration on Algorithms and Geometry. Any opinions, findings and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation. Klim Efremenko: Research supported in part by: European Communitys Seventh Framework Programme (FP7/2007-2013) under Grant agreement number 257575. Bernhard Haeupler: Research supported in part by NSF Grants CCF-1527110 and CCF-1618280.

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Alon, N., Braverman, M., Efremenko, K. et al. Reliable communication over highly connected noisy networks. Distrib. Comput. 32, 505–515 (2019). https://doi.org/10.1007/s00446-017-0303-5

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