Lower Bounds in Communication Complexity

The communication complexity of a function f(x, y) measures the number of bits that two players, one who knows x and the other who knows y, must exchange to determine the value f(x, y). Communication complexity is a fundamental measure of complexity of functions. Lower bounds on this measure lead to lower bounds on many other measures of computational complexity. This monograph survey lower bounds in the eld of communication complexity. Our focus is on lower bounds that work by rst representing the communication complexity measure in Euclidean space. That is to say, the rst step in these lower bound techniques is to nd a geometric complexity measure, such as rank or trace norm, that serves as a lower bound to the underlying communication complexity measure. Lower bounds on this geometric complexity measure are then found using algebraic and geometric tools.