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 obtain new relationships between the randomized amortized communication complexity of a function, and its information complexity. We prove that for any given distribution on inputs, 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. Here by amortized communication complexity we mean the average per copy communication in the best protocol for computing multiple copies, with a bound on the error in each copy. This significantly simplifies the relationships between the various measures of complexity for average case communication protocols, and proves that if a function's information cost is smaller than its communication complexity, then multiple copies of the function can be computed more efficiently in parallel than sequentially.Finally, we 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. 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.

We describe new ways to simulate 2-party communication protocols to get protocols with potentially smaller communication. We show that every communication protocol that communicates C bits and reveals I bits of information about the inputs to the participating parties can be simulated by a new protocol involving at mostÕ( √ CI) bits of communication. If the protocol reveals I bits of information about the inputs to an observer that watches the communication in the protocol, we show how to carry out the simulation withÕ(I) bits of communication.These results lead to a direct sum theorem for randomized communication complexity. Ignoring polylogarithmic factors, we show that for worst case computation, computing n copies of a function requires √ n times the communication required for computing one copy of the function. For average case complexity, given any distribution µ on inputs, computing n copies of the function on n inputs sampled independently according to µ requires √ n times the communication for computing one copy. If µ is a product distribution, computing n copies on n independent inputs sampled according to µ requires n times the communication required for computing the function. We also study the complexity of computing the sum (or parity) of n evaluations of f , and obtain results analogous to those above.To the best of our knowledge, our results give the first compression schemes for general randomized protocols and the first direct sum results in the general setting. Previous results applied only when the protocols were restricted to running in a bounded number of rounds, where each message can be compressed in turn, and only applied when the parties are given independent inputs.

We describe new ways to simulate two-party communication protocols to get protocols with potentially less communication. We show that every communication protocol that communicates C bits and reveals I bits of information about the inputs to the participating parties can be simulated by a new protocol involving at mostÕ( √ CI) bits of communication. If the protocol reveals I bits of information about the inputs to an observer that watches the communication in the protocol, we show how to carry out the simulation withÕ(I) bits of communication. These results lead to a direct sum theorem for randomized communication complexity. Ignoring polylogarithmic factors, we show that for worst-case computation, computing n copies of a function requires √ n times the communication required for computing one copy of the function. For average case complexity, given any distribution μ on inputs, computing n copies of the function on n inputs sampled independently according to μ requires √ n times the communication for computing one copy. If μ is a product distribution, computing n copies on n independent inputs sampled according to μ requires n times the communication required for computing the function. We also study the complexity of computing the sum (or parity) of n evaluations of f , and obtain results analogous to those above. Our results give the first compression schemes for general randomized protocols and the first direct sum results in the general setting of randomized and distributional communication complexity, without requiring bound on the number of rounds in the protocol or that the distribution of inputs is independent.

We show that it is possible to encode any communication protocol between two parties so that the protocol succeeds even if a (1/4 − ǫ) fraction of all symbols transmitted by the parties are corrupted adversarially, at a cost of increasing the communication in the protocol by a constant factor (the constant depends on epsilon). This encoding uses a constant sized alphabet. This improves on an earlier result of Schulman, who showed how to recover when the fraction of errors is bounded by 1/240. We also show how to simulate an arbitrary protocol with a protocol using the binary alphabet, a constant factor increase in communication and tolerating a 1/8 − ǫ fraction of errors.

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.