The hardness of Median in the synchronized bit communication model

The synchronized bit communication model, defined recently by Impagliazzo and Williams in \emph{Communication complexity with synchronized clocks}, CCC '10, is a communication model which allows the participants to share a common clock. The main open problem posed in this paper was the following: does the synchronized bit model allow a logarithmic speed-up for all functions over the standard deterministic model of communication? We resolve this question in the negative by showing that the Median function, whose communication complexity is $O(\log n)$, does not admit polytime synchronized bit protocol with communication complexity $O\left(\log^{1-ε} n\right)$ for any $ε> 0$. Our results follow by a new round-communication trade-off for the Median function in the standard model, which easily translates to its hardness in the synchronized bit model.

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