Constant-Rate Coding for Multiparty Interactive Communication Is Impossible

M. Braverman, K. Efremenko, R. Gelles, Bernhard Haeupler
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引用次数: 30

Abstract

We study coding schemes for multiparty interactive communication over synchronous networks that suffer from stochastic noise, where each bit is independently flipped with probability ε. We analyze the minimal overhead that must be added by the coding scheme to succeed in performing the computation despite the noise. Our main result is a lower bound on the communication of any noise-resilient protocol over a synchronous star network with n parties (where all parties communicate in every round). Specifically, we show a task that can be solved by communicating T bits over the noise-free network, but for which any protocol with success probability of 1-o(1) must communicate at least Ω (T /log n log log n) bits when the channels are noisy. By a 1994 result of Rajagopalan and Schulman, the slowdown we prove is the highest one can obtain on any topology, up to a log log n factor. We complete our lower bound with a matching coding scheme that achieves the same overhead; thus, the capacity of (synchronous) star networks is Θ (log log n/log n). Our bounds prove that, despite several previous coding schemes with rate Ω (1) for certain topologies, no coding scheme with constant rate Ω (1) exists for arbitrary n-party noisy networks.
恒速率编码对多方交互通信是不可能的
研究了存在随机噪声的同步网络中每个比特以ε概率独立翻转的多方交互通信编码方案。我们分析了在不受噪声影响的情况下,编码方案为成功执行计算而必须增加的最小开销。我们的主要结果是在具有n方的同步星型网络(其中所有各方在每轮通信)上的任何抗噪声协议的通信的下界。具体来说,我们展示了一个可以通过在无噪声网络上通信T位来解决的任务,但是当信道有噪声时,任何成功概率为1- 0(1)的协议必须至少通信Ω (T /log n log log n)位。根据1994年Rajagopalan和Schulman的结果,我们证明了在任何拓扑上可以得到的最大的减速,直到一个log log n的因子。我们用一个匹配的编码方案来完成下界,实现相同的开销;因此,(同步)星型网络的容量为Θ (log log n/log n)。我们的界证明,尽管对于某些拓扑有几种速率为Ω(1)的编码方案,但对于任意n方噪声网络不存在恒定速率为Ω(1)的编码方案。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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