[论文解读] Potent Tree Codes and their applications: Coding for Interactive Communication, revisited

Ran Gelles, Amit Sahai|arXiv (Cornell University)|Apr 5, 2011

Cooperative Communication and Network Coding参考文献 13被引用 10

一句话总结

本文引入了强健树码——一种新型的、可显式构造的树码类别，使得在噪声信道上实现首个完全显式、恒定开销的交互式通信模拟成为可能。通过证明随机生成的树码以极大概率具有强健性且可高效解码，作者实现了失败概率在协议长度上呈指数级小的性能，从而解决了交互编码理论中长期存在的开放性问题。

ABSTRACT

We study the fundamental problem of reliable interactive communication over a noisy channel. In a breakthrough sequence of papers published in 1992 and 1993, Schulman gave non-constructive proofs of the existence of general methods to emulate any two-party interactive protocol such that: (1) the emulation protocol takes a constant-factor longer than the original protocol, and (2) if the emulation protocol is executed over a noisy channel, then the probability that the emulation protocol fails is exponentially small in the total length of the protocol. Unfortunately, Schulman's emulation procedures either only work in a model with a large amount of shared randomness, or are non-constructive in that they rely on the existence of good tree codes. The only known proofs of the existence of good tree codes are non-constructive, and finding an explicit construction remains an important open problem. Indeed, randomly generated tree codes are not good tree codes with overwhelming probability. In this work, we revisit the problem of reliable interactive communication, and obtain the following results: We introduce a new notion of goodness for a tree code, and define the notion of a potent tree code. We believe that this notion is of independent interest. We prove the correctness of an explicit emulation procedure based on any potent tree code. We show that a randomly generated tree code (with suitable constant alphabet size) is a potent tree code with overwhelming probability. Furthermore we are able to partially derandomize this result using only O(n) random bits, where $n$ is the depth of the tree. These results allow us to obtain the first fully explicit emulation procedure for reliable interactive communication over noisy channels with a constant communication overhead, and exponentially small failure probability.

研究动机与目标

• 解决长期存在的开放性问题：为可靠交互式通信构造显式树码。
• 用显式、高效的替代方案取代Schulman 1993年协议中对良好树码存在的非构造性依赖。
• 在原始协议长度上实现恒定因子通信开销，并使失败概率呈指数级小。
• 将结果扩展至多方交互通信，对常数个参与方仅造成O(n)的延迟。
• 通过仅使用O(n)随机位实现去随机化构造，从而实现确定性模拟。

提出的方法

• 引入强健树码的概念，这是一种对可靠交互式通信而言更宽松但已足够充分的条件。
• 证明任意强健树码均可实现两方及多方协议的完全显式且高效的模拟过程。
• 证明使用常数大小字母表随机生成的树码，以极大概率是强健树码。
• 展示即使在二进制对称信道上，此类树码也可通过标准纠错码实现高效解码。
• 通过使用O(n)随机位对构造过程进行部分去随机化，随后利用标准纠错码对这些随机位进行编码与共享。
• 通过确保各方间解码错误在高概率下保持有界，将框架扩展至多方通信。

实验结果

研究问题

• RQ1 能否在不依赖非构造性树码存在性证明的前提下，实现对噪声信道上交互协议的完全显式、恒定开销模拟？
• RQ2 是否存在一种树码优良性的弱化概念，可在保持指数级错误抑制的同时实现高效且显式的交互编码？
• RQ3 能否证明随机生成的树码以高概率足够好（即作为强健树码）？这是否能促成实际构造？
• RQ4 能否将构造此类码所需的随机性减少至O(n)比特，同时保持效率与可靠性？
• RQ5 该框架能否扩展至多方交互协议，并保持有界的通信开销？

主要发现

• 使用常数大小字母表随机生成的树码，以极大概率是强健树码，从而实现显式构造。
• 模拟协议的失败概率在原始协议长度N上呈指数级小，具体上界为2^(-Ω(N))。
• 两方协议的通信开销为常数，n方协议的开销为O(n)，当n为常数时该开销为最优。
• 该构造仅需O(n)随机位，可通过标准纠错码高效共享与去随机化。
• 可通过标准纠错码实现高效解码，当标签错误概率p < |S|^(-2)时，确保解码复杂度较低。
• 在合适参数下，该框架可扩展至多方协议，对固定时序路径的失败概率最多为1/(2(r+1))^N。