We study two fundamental problems in communication, Document Exchange (DE) and Error Correcting Code (ECC). In the first problem, two parties hold two strings, and one party tries to learn the other party's string through communication. In the second problem, one party tries to send a message to another party through a noisy channel, by adding some redundant information to protect the message. Two important goals in both problems are to minimize the communication complexity or redundancy, and to design efficient protocols or codes. Both problems have been studied extensively. In this paper we study whether asymmetric partial information can help in these two problems. We focus on the case of Hamming distance/errors, and the asymmetric partial information is modeled by one party having a vector of disjoint subsets $\vec{S}=(S_1, \cdots, S_t)$ of indices and a vector of integers $\vec{k}=(k_1, \cdots, k_t)$, such that in each $S_i$ the Hamming distance/errors is at most $k_i$. We establish both lower bounds and upper bounds in this model, and provide efficient randomized constructions that achieve a $\min\lbrace O(t^2), O\left((\log \log n)^2\right) \rbrace $ factor within the optimum, with almost linear running time. We further show a connection between the above document exchange problem and the problem of document exchange under edit distance, and use our techniques to give an efficient randomized protocol with optimal communication complexity and \emph{exponentially} small error for the latter. This improves the previous result by Haeupler \cite{haeupler2018optimal} (FOCS'19) and that by Belazzougui and Zhang \cite{BelazzouguiZ16} (FOCS'16). Our techniques are based on a generalization of the celebrated expander codes by Sipser and Spielman \cite{sipser1996expander}, which may be of independent interests.

中文翻译：

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信息不对称的高效文档交换和纠错码

我们研究通信中的两个基本问题，文档交换 (DE) 和纠错码 (ECC)。在第一个问题中，两方持有两根弦，一方试图通过通信了解对方的弦。在第二个问题中，一方试图通过嘈杂的通道向另一方发送消息，通过添加一些冗余信息来保护消息。这两个问题的两个重要目标是最小化通信复杂性或冗余，以及设计有效的协议或代码。这两个问题都得到了广泛的研究。在本文中，我们研究了不对称部分信息是否可以帮助解决这两个问题。我们专注于汉明距离/误差的情况，非对称部分信息由具有不相交子集向量的一方建模 $\vec{S}=(S_1, \cdots, S_t)$ 索引和整数向量 $\vec{k}=(k_1, \cdots, k_t)$，这样在每个 $S_i$ 中，汉明距离/误差最多为 $k_i$。我们在这个模型中建立了下界和上界，并提供了高效的随机结构，实现了 $\min\lbrace O(t^2), O\left((\log \log n)^2\right) \rbrace $因素在最佳范围内，运行时间几乎是线性的。我们进一步展示了上述文档交换问题与编辑距离下的文档交换问题之间的联系，并使用我们的技术给出了一个有效的随机协议，该协议具有最佳的通信复杂度和\emph {exponentially} 对后者的小误差。这改进了 Haeupler \cite{haeupler2018optimal} (FOCS'19) 和 Belazzougui 和 Zhang \cite{BelazzouguiZ16} (FOCS'16) 的先前结果。