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arXiv:1506.00617

On Slepian--Wolf Theorem with Interaction

Alexander Kozachinskiy

Published Jun 1, 2015• 0 citations
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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 $π$ 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 $H(X|Y) + Ω\left(\log_2\left(\frac{1}{\varepsilon}\right)\right)$ bits on average.

DOI
10.48550/arxiv.1506.00617

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Two parties observing correlated data seek to exchange their data using interactive communication. How many bits must they communicate? We propose a new interactive protocol for data exchange, which increases the communication size in steps until the task is done. We also derive a lower bound on the minimum number of bits that is based on relating the data exchange problem to the secret key agreement problem. Our single-shot analysis applies to all discrete random variables and yields upper and lower bounds of a similar form. In fact, the bounds are asymptotically tight and lead to a characterization of the optimal rate of communication needed for data exchange for a general source sequence, such as a mixture of independent and identically distributed (IID) random variables as well as the optimal second-order asymptotic term in the length of communication needed for data exchange for IID random variables, when the probability of error is fixed. This gives a precise characterization of the asymptotic reduction in the length of optimal communication due to interaction; in particular, two-sided Slepian-Wolf compression is strictly suboptimal.
Two parties observing correlated data seek to exchange their data using interactive communication. How many bits must they communicate? We propose a new interactive protocol for data exchange which increases the communication size in steps until the task is done. Next, we derive a lower bound on the minimum number of bits that is based on relating the data exchange problem to the secret key agreement problem. Our single-shot analysis applies to all discrete random variables and yields upper and lower bound of a similar form. In fact, the bounds are asymptotically tight and lead to a characterization of the optimal rate of communication needed for data exchange for a general sequence such as mixture of IID random variables as well as the optimal second-order asymptotic term in the length of communication needed for data exchange for the IID random variables, when the probability of error is fixed. This gives a precise characterization of the asymptotic reduction in the length of optimal communication due to interaction; in particular, two-sided Slepian-Wolf compression is strictly suboptimal.
Say Alice and Bob have n-bit strings which are identical except that Bob's contains t "erasures": the corresponding bit may be either zero or one. Alice will send Bob a message he can use to fill in his erased bits. The Slepian-Wolf bound suggests Alice may send as few as t bits, even when Alice does not know which t bits of are erased. We present several protocols acheiving this bound with varying degrees of success. A binary, linear, and deterministic protocol exists only for t ϵ {0, 1, n-1, n}. A nonbinary protocol exists for any t ≤ n ≤ 2m-1, but requires that erasures occur in m-bit blocks. A nonlinear protocol seems promising, but the theory to prove it in general is difficult to explore. Finally, a probabilistic protocol has no parameter constraints, but has suboptimal communication complexity based on an additional parameter c, and will fail with an error-rate Є dependent on c and t.
Data compression is a fundamental problem in quantum and classical information theory. A typical version of the problem is that the sender Alice receives a (classical or quantum) state from some known ensemble and needs to transmit them to the receiver Bob with average error below some specified bound. We consider the case in which the message can have a variable length and the goal is to minimize its expected length. For classical messages this problem has a well-known solution given by Huffman coding. In this scheme, the expected length of the message is equal to the Shannon entropy of the source (with a constant additive factor) and the scheme succeeds with zero error. This is a single-shot result which implies the asymptotic result, viz. Shannon's source coding theorem, by encoding each state sequentially. For the quantum case, the asymptotic compression rate is given by the von-Neumann entropy. However, we show that there is no one-shot scheme which is able to match this rate, even if interactive communication is allowed. This is a relatively rare case in quantum information theory when the cost of a quantum task is significantly different than the classical analogue. Our result has implications for direct sum theorems in quantum communication complexity and one-shot formulations of Quantum Reverse Shannon theorem.
Two parties observing correlated data seek to exchange their data using interactive communication. How many bits must they communicate? We derive a lower bound on the minimum number of bits that is based on relating the data exchange problem to the secret key agreement problem. Furthermore, we propose an interactive protocol for data exchange which increases the communication size in steps until the task is done and matches the performance of our lower bound. Our single-shot analysis applies to all discrete random variables and yields upper and lower bound of a similar form. In fact, the bounds are asymptotically tight and lead to a characterization of the optimal rate of communication needed for data exchange for a general sequence such as mixture of IID random variables as well as the optimal second-order asymptotic term in the length of communication needed for data exchange for the IID random variables, when the probability of error is fixed. This gives a precise characterization of the asymptotic reduction in the length of optimal communication due to interaction; in particular, two-sided Slepian-Wolf compression is strictly suboptimal.