On Slepian--Wolf Theorem with Interaction

TL;DR

This paper explores interactive communication protocols for transmitting correlated data efficiently, improving bounds on communication complexity and rounds compared to previous results, with implications for information theory and distributed computing.

Contribution

It introduces new bounds for one-shot interactive Slepian-Wolf coding, reducing the number of bits and rounds needed for reliable transmission, and discusses tightness of these bounds.

Findings

Expected communication close to H(X|Y) + 2√H(X|Y)

One-round protocol compression with expected bits I + 2√I

Multi-round protocol achieving 3H(X|Y) + O(log(1/ε)) bits

Abstract

In this paper we study interactive "one-shot" analogues of the classical Slepian-Wolf theorem. Alice receives a value of a random variable $X$, Bob receives a value of another random variable $Y$ that is jointly distributed with $X$. Alice's goal is to transmit $X$ to Bob (with some error probability $\varepsilon$). Instead of one-way transmission, which is studied in the classical coding theory, we allow them to interact. They may also use shared randomness. We show, that Alice can transmit $X$ to Bob in expected $H(X|Y) + 2\sqrt{H(X|Y)} + O(\log_2\left(\frac{1}{\varepsilon}\right))$ number of bits. Moreover, we show that every one-round protocol $\pi$ with information complexity $I$ can be compressed to the (many-round) protocol with expected communication about $I + 2\sqrt{I}$ bits. This improves a result by Braverman and Rao \cite{braverman2011information}, where they had $5\sqrt{I}$. Further, we show how to solve this problem (transmitting $X$) using $3H(X|Y) + O(\log_2\left(\frac{1}{\varepsilon}\right))$ bits and $4$ rounds on average. This improves a result of~\cite{brody2013towards}, where they had $4H(X|Y) + O(\log1/\varepsilon)$ bits and 10 rounds on average. In the end of the paper we discuss how many bits Alice and Bob may need to communicate on average besides $H(X|Y)$. The main question is whether the upper bounds mentioned above are tight. We provide an example of $(X, Y)$, such that transmission of $X$ from Alice to Bob with error probability $\varepsilon$ requires $H(X|Y) + \Omega\left(\log_2\left(\frac{1}{\varepsilon}\right)\right)$ bits on average.