Amortized Communication Complexity

Abstract

In this work we study the direct-sum problem with respect to communication complexity: Consider a relation f defined over $\{0,1\}^{n} \times \{0,1\}^{n}$. Can the communication complexity of simultaneously computing f on $\ell $ instances $(x_{1}, y_{1}), \dotsc , (x_{\ell}, y_{\ell})$ be smaller than the communication complexity of separately computing f on the $\ell $ instances? Let the amortized communication complexity of f be the communication complexity of simultaneously computing f on $\ell $ instances divided by $\ell $. We study the properties of the amortized communication complexity. We show that the amortized communication complexity of a relation can be smaller than its communication complexity. More precisely, we present a partial function whose (deterministic) communication complexity is $\Theta (\log n)$ and amortized (deterministic) communication complexity is $O(1)$. Similarly, for randomized protocols we present a function whose randomized communication complexity is $\Theta (\log n)$ and amortized randomized communication complexity is $O(1)$. We also give a general lower bound on the amortized communication complexity of any functionf in terms of its communication complexity $C(f)$: for every function f the amortized communication complexity of f is $\Omega (\sqrt{C(f)} - \log n)$.