Communication vs. computation

We initiate a study of tradeoffs between communication and computation in well-known communication models and in other related models. The fundamental question we investigate is the following: Is there a computational task that exhibits a strong tradeoff behavior between the amount of communication and the amount of time needed for local computation? Under various standard assumptions, we exhibit Boolean functions that show strong tradeoffs between communication and time complexity in the following scenarios: We exhibit a polynomial time computable Boolean function that has a low randomized communication complexity, while any communication-efficient (randomized) protocol for this function requires a super polynomial amount of local computation. In the case of deterministic two-party protocols, we show a similar result relative to a random oracle. We exhibit a polynomial-time computable Boolean function that can be computed by querying a few bits of its input, but where any such query-efficient scheme requires a super-polynomial amount of computation. We exhibit a polynomial-time decidable property that can be tested (i.e., strings which have the property can be distinguished from ones that are far from the property) by querying a few bits of the input, but where any such query-efficient tester requires a super-polynomial amount of computation. Finally, we study a time-degree tradeoff problem that arises in arithmetization of Boolean functions, and relate it to time-communication tradeoff questions in multi-party communication complexity and cryptography.