Lower Bounds for Number-in-Hand Multiparty Communication Complexity, Made Easy

In this paper we prove lower bounds on randomized multiparty communication complexity, both in the \emph{blackboard model} (where each message is written on a blackboard for all players to see) and (mainly) in the \emph{message-passing model}, where messages are sent player-to-player. We introduce a new technique for proving such bounds, called \emph{symmetrization}, which is natural, intuitive, and often easy to use. For example, for the problem where each of $k$ players gets a bit-vector of length $n$, and the goal is to compute the coordinate-wise XOR of these vectors, we prove a tight lower bounds of $\Omega(nk)$ in the blackboard model. For the same problem with AND instead of XOR, we prove a lower bounds of roughly $\Omega(nk)$ in the message-passing model (assuming $k \le n/3200$) and $\Omega(n \log k)$ in the blackboard model. We also prove lower bounds for bit-wise majority, for a graph-connectivity problem, and for other problems; the technique seems applicable to a wide range of other problems as well. All of our lower bounds allow randomized communication protocols with two-sided error. We also use the symmetrization technique to prove several direct-sum-like results for multiparty communication.