Direct product results and the GCD problem, in old and new communication models
STOC '97 Proceedings of the twenty-ninth annual ACM symposium on Theory of computing
SIAM Journal on Computing
Towards proving strong direct product theorems
Computational Complexity
The Communication Complexity of Correlation
CCC '07 Proceedings of the Twenty-Second Annual IEEE Conference on Computational Complexity
A Direct Product Theorem for Discrepancy
CCC '08 Proceedings of the 2008 IEEE 23rd Annual Conference on Computational Complexity
Probabilistic computations: Toward a unified measure of complexity
SFCS '77 Proceedings of the 18th Annual Symposium on Foundations of Computer Science
FOCS '08 Proceedings of the 2008 49th Annual IEEE Symposium on Foundations of Computer Science
Numerical linear algebra in the streaming model
Proceedings of the forty-first annual ACM symposium on Theory of computing
How to compress interactive communication
Proceedings of the forty-second ACM symposium on Theory of computing
A strong direct product theorem for disjointness
Proceedings of the forty-second ACM symposium on Theory of computing
Information Equals Amortized Communication
FOCS '11 Proceedings of the 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science
Interactive information complexity
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
A Direct Product Theorem for the Two-Party Bounded-Round Public-Coin Communication Complexity
FOCS '12 Proceedings of the 2012 IEEE 53rd Annual Symposium on Foundations of Computer Science
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We obtain a strong direct product theorem for two-party bounded round communication complexity. Let sucr(μ,f,C) denote the maximum success probability of an r-round communication protocol that uses at most C bits of communication in computing f(x,y) when (x,y)~μ. Jain et al. [12] have recently showed that if ${\sf suc}_{r}(\mu,f,C) \leq \frac{2}{3}$ and $T\ll (C-\Omega (r^2)) \cdot\frac{n}{r}$, then ${\sf suc}_r(\mu^n,f^n,T)\leq \exp(-\Omega(n/r^2))$. Here we prove that if ${\sf suc}_{7r}(\mu,f,C) \leq \frac{2}{3}$ and T≪(C−Ω(r logr)) ·n then ${\sf suc}_{r}(\mu^n,f^n,T)\leq\exp(-\Omega(n))$. Up to a logr factor, our result asymptotically matches the upper bound on suc7r(μn,fn,T) given by the trivial solution which applies the per-copy optimal protocol independently to each coordinate. The proof relies on a compression scheme that improves the tradeoff between the number of rounds and the communication complexity over known compression schemes.