On Slepian–Wolf Theorem with Interaction
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 ε). Instead of one-way transmission we allow them to interact. They may also use shared randomness. We show, that for every natural r Alice can transmit X to Bob using (1+1r)H(X|Y)+r+O(log2(1ε))(1+1r)H(X|Y)+r+O(log2(1ε)) bits on average in 2H(X|Y)r+22H(X|Y)r+2 rounds on average. Setting r=⌈H(X|Y)−−−−−−−√⌉r=⌈H(X|Y)⌉ and using a result of Braverman and Garg (2) we conclude that every one-round protocol π with information complexity I can be compressed to a (many-round) protocol with expected communication about I+2I–√I+2I bits. This improves a result by Braverman and Rao (3), where they had I+5I–√I+5I. Further, we show (by setting r = ⌈H(X|Y)⌉) how to solve this problem (transmitting X) using 2H(X|Y)+O(log2(1ε))2H(X|Y)+O(log2(1ε)) bits and 4 rounds on average. This improves a result of Brody et al. (4), where they had 4H(X|Y)+O(log1/ε)4H(X|Y)+O(log1/ε) 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 ε requires H(X|Y)+Ω(log2(1ε))H(X|Y)+Ω(log2(1ε)) bits on average.