The hardness of Median in the synchronized bit communication model

TL;DR

This paper proves that the Median function cannot be significantly sped up using the synchronized bit communication model, establishing a fundamental limit on the model's efficiency for this problem.

Contribution

It introduces a new round-communication trade-off for Median in the standard model, demonstrating its hardness in the synchronized bit model.

Findings

Median requires logarithmic communication in the standard model

Synchronized bit model does not provide logarithmic speed-up for Median

Hardness results translate from standard to synchronized bit model

Abstract

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 $O\left(\log^{1-\epsilon} n\right)$ for any $\epsilon > 0$. 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.