Amortized Communication Complexity

We thank Mauricio Karchmer and Avi Wigderson for raising the question and for helpful discussions, and Amos Beimel, Benny Chor, Alon Orlitsky and Steve Ponzio for many interesting comments on earlier versions of this paper. Finally, we thank an anonymous referee for his very helpful comments and criticism. 17 For values of`which are around log n we would like to replace the term p ` log n with p `+log n. This can be done by sampling the b i 's via a random walk in an expander a la Ajtai, Komll os and Szemer edi 2] (in such a case the b i 's are not independent): The elements of B n are mapped to nodes of a constant degree expander G. Then, a random walk of length k in G is generated, and the vectors b 1 ; b 2 ; : : :; b k are the vectors corresponding to the nodes of the walk. The number of bits required to specify the walk is O(log jB n j + k) which is O(log n + k). (See e.g. 13] for details.) As before, P 1 selects the random bits and sends them to P 2 , so that they both agree on the same sequence. If x 6 = y then the probability that hb i ; xi = hb i ; yi for all 1 i k goes down exponentially in k. The strings c 1 ; c 2 ; : : :; c k are selected similarly in B ` using O(k + log`) bits. To conclude, we have a randomized protocol, in the private coins model, for computing the identity function onìnstances with probability of error at most 2 ?(p `) and expected complexity of O(`+log n), which is O(`), for`suuciently large. With a \small" additional error the protocol can be converted to a protocol that uses O(`) bits in the worst case. This gives a constructive proof for Theorem 10. 7 Open Problems We conclude this work by mentioning some open problems: In 7] it was conjectured that for any relation f, the communication complexity, C(f), can not be smaller than C(f) by more than an additive factor of O(log n). The examples given in our paper do not contradict this conjecture. On the other hand, according to the best lower bound we are able to prove (Corollary 8), even for (non-partial) …