非对称信息下的高效文档交换与纠错码
Efficient Document Exchange and Error Correcting Codes with Asymmetric Information
摘要：
本文研究了通信中的两个基本问题：文件交换和纠错码。在第一个问题中，两方持有两个字符串，一方试图通过交流学习另一方的字符串。在第二个问题中，一方试图通过噪声信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度或冗余度，并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，我们研究是否不对称的部分信息可以帮助在这两个问题。我们专注于汉明距离/错误的情况下，和不对称的部分信息是由一方具有不相交的索引的子集$\vec{S}=(S_1, \cdots, S_t)$的向量和整数$\vec{k}=(k_1, \cdots, k_t)$的向量，使得在每个$S_i$的汉明距离/错误是最多$k_i$。我们建立在这个模型中的下限和上限，并提供有效的随机结构，实现了$\min\lbrace O(t^2), O\left((\log \log n)^2\right) \rbrace $因素的最佳范围内，几乎线性的运行时间。我们进一步显示了上述文件交换问题和编辑距离下的文件交换问题之间的联系，并使用我们的技术，给一个有效的随机协议，最佳的通信复杂性和\emph{exponentially}小错误，后者。这改进了Haeupler \cite{haeupler2018optimal}（FOCS'19）和Belazzougui和Zhang \cite{BelazzouguiZ16}（FOCS'16）的结果。我们的技术是基于由Sipser和Spielman \cite{sipser1996expander}，这可能是独立的利益，著名的扩展码的推广。
英文摘要：
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.
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