  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.
