# Communication complexity with synchronized clocks

## Abstract

We consider two natural extensions of the communication complexity model that are inspired by distributed computing. In both models, two parties are equipped with synchronized discrete clocks, and we assume that a bit can be sent from one party to another in one step of time. Both models allow implicit communication, by allowing the parties to choose whether to send a bit during each step. We examine trade-offs between time (total number of possible time steps elapsed) and communication (total number of bits actually sent). In the synchronized bit model, we measure the total number of bits sent between the two parties (e.g., email). We show that, in this model, communication costs can differ from the usual communication complexity by a factor roughly logarithmic in the number of time steps, and no more than such a factor. In the synchronized connection model, both parties choose whether or not to open their end of the communication channel at each time step. An exchange of bits takes place only when both ends of the channel are open (e.g., instant messaging), in which case we say that a connection has occurred. If a party does not open its end, it does not learn whether the other party opened its channel. When we restrict the number of time steps to be polynomial in the input length, and the number of connections to be polylogarithmic in the input length, the class of problems solved with this model turns out to be roughly equivalent to the communication complexity analogue of PNP ([BFS86]). Using our new model, we give what we believe to be the first lower bounds for this class, separating P NP from Σ2 ∩ Π2 in the communication complexity setting. Although these models are both quite natural, they have unexpected power, and lead to a refinement of problem classifications in communication complexity. © 2010 IEEE.