Efficient Document Exchange and Error Correcting Codes with Asymmetric Information

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.