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Amortized Communication Complexity

### Tomas Feder Eyal Kushilevitz Moni
Naor Noam Nisan

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### Abstract:

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 $\cal l$ instances $(x_1,y_1),\ldots,(x_{\cal l},y_{\cal l})$ be smaller than the communication complexity of computing $f$ on the $\cal l$ instances, separately?

Let the amortized communication complexity of $f$ be the communication complexity of simultaneously computing $f$ on $\cal l$ instances, divided by $\cal l$. 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 its amortized (deterministic) communication complexity is $O(1)$. Similarly, for randomized protocols, we present a function whose randomized communication complexity is $\Theta(\log n)$ and its amortized randomized communication complexity is $O(1)$.

We also give a general lower bound on the amortized communication complexity of any function $f$ in terms of its communication complexity $C(f)$: for every function $f$ the amortized communication complexity of $f$ is $\Omega \left (\sqrt{C(f)} - \log n \right)$.

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