How to compress interactive communication

  • Authors:

  • Boaz Barak;Mark Braverman;Xi Chen;Anup Rao

  • Affiliations:

  • Princeton University, Princeton, NJ, USA;Microsoft Research New England, Cambridge, MA, USA;University of Southern California, Los Angeles, CA, USA;University of Washington, Seattle, WA, USA

  • Venue:
  • Proceedings of the forty-second ACM symposium on Theory of computing
  • Year:
  • 2010

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Abstract

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 most ~O(√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 with ~O(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.