In the standard two-party model the input (x, y) is partitioned in a fixed way. That is, Alice always gets x and Bob always gets y. In this chapter we discuss models in which the partition of the input among the players is not fixed. The main motivation for these models is that in many cases we wish to use communication complexity lower bounds to obtain lower bounds in other models of computation. This would typically require finding a communication complexity problem “hidden” somewhere in the computation that the model under consideration must perform. Because in such a model the input usually is not partitioned into two distinct sets x1, …, xn and y1, …, yn, such a partition must be given by the reduction. In some cases the partition can be figured out and fixed. In some other cases we must use arguments regarding any partition (of a certain kind). That is, we require a model where the partition is not fixed beforehand but the protocol determines the partition (independently of the particular input). Several such “variable partition models” are discussed in this chapter.

Throughout this chapter the input will be m Boolean variables x1, …, xm, and we consider functions f: {0, l}m → {0, 1}. We will talk about the communication complexity of f between two disjoint sets of variables S and T. That is, one player gets all bits in S and the other all bits in T

Worst-Case Partition

The simplest variable partition model we may consider is the “worst-case” partition: split the input into two sets in the way that maximizes the communication complexity.