Amortized Communication Complexity

In this work we study the direct-sum problem with respect to communication complexity: Consider a relation f defined over (0, 1)(n) x 0, 1(n). Can the communication complexity of simultaneously computing f on l instances (x(1), y(1)),..., (x(l), y(l)) be smaller than the communication complexity of separately computing f on the l instances? Let the amortized communication complexity of f be the communication complexity of simultaneously computing f on l instances divided by l. We study the properties of the amortized communication complexity. We show that the amortized communication complexity of a relation can be smaller than its communication complexity. More precisely, we present a partial function whose (deterministic) communication complexity is Theta (log n) and amortized (deterministic) communication complexity is O(1). Similarly, for randomized protocols we present a function whose randomized communication complexity is Theta (log n) and amortized randomized communication complexity is O(1). We also give a general lower bound on the amortized communication complexity of any function f in terms of its communication complexity C(f): for every function f the amortized communication complexity of f is Omega(root C(f)-log n).