Direct Sums in Randomized Communication Complexity

We prove a direct sum theorem for randomized communication complexity. Ignoring logarithmic factors, our results show that: • Computing n copies of a function requires √n times the communication. • For average case complexity, given any distribution μ on inputs, computing n copies of the function on n independent inputs sampled 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. We also study the complexity of computing the parity of n evaluations of f , and obtain results analogous to those above. Our results are obtained by designing new compression schemes that can compress the communication in interactive processes that do not reveal too much information about their inputs. This generalizes the notion of traditional compression, which can be viewed as compressing protocols that involve only one way communication. Department of Computer Science, Princeton University, [email protected]. Supported by NSF grants CNS-0627526, CCF-0426582 and CCF-0832797, US-Israel BSF grant 2004288 and Packard and Sloan fellowships. 1 Electronic Colloquium on Computational Complexity, Report No. 44 (2009)