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Reliable communication over highly connected noisy networks. (English) Zbl 1452.68016

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 rate, i.e., while keeping \(\liminf _{n,R\rightarrow \infty } R/R'\) positive. S. Rajagopalan and L. Schulman [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; Zbl 1344.68035)] 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 \(n^{\varOmega (1)} \).

MSC:

68M10 Network design and communication in computer systems
68M12 Network protocols
68M14 Distributed systems
68P30 Coding and information theory (compaction, compression, models of communication, encoding schemes, etc.) (aspects in computer science)
68R10 Graph theory (including graph drawing) in computer science

Citations:

Zbl 1344.68035
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Full Text: DOI

References:

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