# On Slepian–Wolf Theorem with Interaction

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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 *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 \(\left (1 + \frac {1}{r}\right )H(X|Y) + r + O(\log _{2}\left (\frac {1}{\varepsilon }\right ))\) bits on average in \(\frac {2H(X|Y)}{r} + 2\) rounds on average. Setting \(r = \lceil \sqrt {H(X|Y)}\rceil \) 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 \(I + 2\sqrt {I}\) bits. This improves a result by Braverman and Rao (3), where they had \(I+5\sqrt {I}\). Further, we show (by setting *r* = ⌈*H*(*X*|*Y*)⌉) how to solve this problem (transmitting *X*) using \(2H(X|Y) + O(\log _{2}\left (\frac {1}{\varepsilon }\right ))\) bits and 4 rounds on average. This improves a result of Brody et al. (4), where they had \(4H(X|Y) + O(\log 1/\varepsilon )\) 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 *ε* requires \(H(X|Y) + {\Omega }\left (\log _{2}\left (\frac {1}{\varepsilon }\right )\right )\) bits on average.

## Keywords

Slepian–Wolf theorem Communication complexity Information complexity Interactive compression## References

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