We show how to efficiently simulate the sending of a message M to a receiver who has partial information about the message, so that the expected number of bits communicated in the simulation is close to the amount of additional information that the message reveals to the receiver. This is a generalization and strengthening of the Slepian-Wolf theorem, which shows how to carry out such a simulation with low amortized communication in the case that M is a deterministic function of X. A caveat is that our simulation is interactive.
As a consequence, we prove that the internal information cost (namely the information revealed to the parties) involved in computing any relation or function using a two party interactive protocol is exactly equal to the amortized communication complexity of computing independent copies of the same relation or function. We also show that the only way to prove a strong direct sum theorem for randomized communication complexity is by solving a particular variant of the pointer jumping problem that we define. Our work implies that a strong direct sum theorem for communication complexity holds if and only if efficient compression of communication protocols is possible.

We show how to efficiently simulate the sending of a message M to a receiver who has partial information about the message, so that the expected number of bits communicated in the simulation is close to the amount of additional information that the message reveals to the receiver.

We use our simulation method to obtain several results in communication complexity.

- We prove a new direct sum theorem for bounded round communication protocols. For every k, if the two party communication complexity of computing a function f is C>k, then the complexity of computing n copies of f using a k round protocol is at least Omega(n(C-O(k+\sqrt{Ck}))). This is true in the distributional setting under any distribution on inputs, and also in the setting of worst case randomized computation.

-We prove that the internal information cost (namely the information revealed to the parties) involved in computing any functionality using a two party interactive protocol on any input distribution is exactly equal to the amortized communication complexity of computing independent copies of the same functionality. Here by amortized communication complexity we mean the average per copy communication in the best protocol for computing multiple copies of a functionality, with a fixed error in each copy of the functionality.

Finally, we show that the only way to prove a strong direct sum theorem for communication complexity is by solving a particular variant of the pointer jumping problem that we define. If this problem has a cheap communication protocol, then a strong direct sum theorem must hold. On the other hand, if it does not, then the problem itself gives a counterexample for the direct sum question. In the process we show that a strong direct sum theorem for communication complexity holds if and only if efficient compression of communication protocols is possible.