## 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 lim inf _{n} _{,} _{R} _{→} _{∞}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^{Ω} ^{(} ^{1} ^{)}.

Original language | English |
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Pages (from-to) | 505-515 |

Number of pages | 11 |

Journal | Distributed Computing |

Volume | 32 |

Issue number | 6 |

DOIs | |

State | Published - 1 Dec 2019 |

### Bibliographical note

Publisher Copyright:© 2017, Springer-Verlag Berlin Heidelberg.

### Funding

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.

Funders | Funder number |
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Israeli I-Core | |

National Science Foundation | NSF-BSF, 1933331, CCF-1149888, CCF-1525342, DMS-1128155 |

Simons Foundation | |

Center for Selective C-H Functionalization, National Science Foundation | |

Seventh Framework Programme | CCF-1618280, 257575, CCF-1527110 |

United States-Israel Binational Science Foundation | |

Israel Science Foundation | |

Seventh Framework Programme |

## Keywords

- Coding theory
- Computation with noise
- Interactive coding
- Random noise