Lower bounds for number-in-hand multiparty communication complexity, made easy

In this paper we prove lower bounds on randomized mul-tiparty communication complexity, both in the blackboardmodel (where each message is written on a blackboard for allplayers to see) and (mainly) in the message-passing model,where messages are sent player-to-player. We introduce anew technique for proving such bounds, called symmetriza-tion, which is natural, intuitive, and often easy to use.For example, for the problem where each of k playersgets a bit-vector of length n, and the goal is to compute thecoordinate-wise XOR of these vectors, we prove a tight lowerbounds of Ω(nk) in the blackboard model. For the sameproblem with AND instead of XOR, we prove a lower boundsof roughly Ω(nk) in the message-passing model (assumingk ≤ n/3200) and Ω(n log k) in the blackboard model. Wealso prove lower bounds for bit-wise majority, for a graph-connectivity problem, and for other problems; the techniqueseems applicable to a wide range of other problems as well.The obtained communication lower bounds imply new lowerbounds in the functional monitoring model [11] (also calledthe distributed streaming model). All of our lower boundsallow randomized communication protocols with two-sidederror. We also use the symmetrization technique to prove several direct-sum-like results for multiparty communication.