Document Zbl 1452.68016

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Comput.</a> <a href="/?q=in%3A442834" title="Articles in this Issue">32, No. 6, 505-515 (2019)</a>. </div> <div class="abstract">Summary: 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 <span class="zbmathjax-textit">rate</span>, i.e., while keeping $\liminf _{n,R\rightarrow \infty } R/R'$ positive. <span class="zbmathjax-textit">S. Rajagopalan</span> and <span class="zbmathjax-textit">L. Schulman</span> [in: Proceedings of the 26th annual ACM symposium on theory of computing, STOC’94. New York, NY: Association for Computing Machinery (ACM). 790–799 (1994; <a href="/1344.68035">Zbl 1344.68035</a>)] 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)} $.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A7137739">Cited in <strong>4</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A68M10" title="MSC2020">68M10</a> </td> <td class="space"> Network design and communication in computer systems </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A68M12" title="MSC2020">68M12</a> </td> <td class="space"> Network protocols </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A68M14" title="MSC2020">68M14</a> </td> <td class="space"> Distributed systems </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A68P30" title="MSC2020">68P30</a> </td> <td class="space"> Coding and information theory (compaction, compression, models of communication, encoding schemes, etc.) (aspects in computer science) </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A68R10" title="MSC2020">68R10</a> </td> <td class="space"> Graph theory (including graph drawing) in computer science </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Acoding+theory">coding theory</a>; <a href="/?q=ut%3Arandom+noise">random noise</a>; <a href="/?q=ut%3Ainteractive+coding">interactive coding</a>; <a href="/?q=ut%3Acomputation+with+noise">computation with noise</a></div><div class="keywords"> <h3>Citations:</h3><a href="/1344.68035">Zbl 1344.68035</a></div>  <div class="modal fade" id="metadataModal" tabindex="-1" role="dialog" aria-labelledby="myModalLabel"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span></button> <h4 class="modal-title" id="myModalLabel">Cite</h4> </div> <div class="modal-body"> <div class="form-group"> <label for="select-output" class="control-label">Format</label> <select id="select-output" class="form-control" aria-label="Select Metadata format"></select> </div> <div class="form-group"> <label for="metadataText" class="control-label">Result</label> <textarea class="form-control" id="metadataText" rows="10" style="min-width: 100%;max-width: 100%"></textarea> </div> <div id="metadata-alert" class="alert alert-danger" role="alert" style="display: none;">  </div> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary" onclick="copyMetadata()">Copy to clipboard</button> <button type="button" class="btn btn-default" data-dismiss="modal">Close</button> </div> </div> </div> </div> <div class="functions clearfix"> <div class="function">  <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 1452.68016" data-ciurl="/ci/07137739" data-biburl="/bibtex/07137739.bib" data-amsurl="/amsrefs/07137739.bib" data-xmlurl="/xml/07137739.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/07137739.pdf" title="Zbl 1452.68016 as PDF">Review PDF</a> </div> <div class="fulltexts"> <span class="fulltext">Full Text:</span> <a class="btn btn-default btn-xs" type="button" href="https://doi.org/10.1007/s00446-017-0303-5" aria-label="DOI for “Reliable communication over highly connected noisy networks”" title="10.1007/s00446-017-0303-5">DOI</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">Agrawal, S., Gelles, R., Sahai, A.: Adaptive protocols for interactive communication. 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