Rounds in communication complexity revisited

The $k$-round two-party communication complexity was studied in the deterministic model by [14] and [4] and in the probabilistic model by [20] and [6]. We present new lower bounds that give (1) randomization is more powerful than determinism in $k$-round protocols, and (2) an explicit function which exhibits an exponential gap between its $k$ and $(k-1)$-round randomized complexity. We also study the three party communication model, and exhibit an exponential gap in 3-round protocols that differ in the starting player. Finally, we show new connections of these questions to circuit complexity, that motivate further work in this direction.