Amortized Communication Complexity

In this work we study the direct-sum problem with respect to communication complexity: Consider a relation f deened over f0; 1g n f0; 1g n. Can the communication complexity of simultaneously computing f onìnstances (x 1 ; y 1); : : : ; (x ` ; y `) be smaller than the communication complexity of computing f on thè instances, separately? Let the amortized communication complexity of f be the communication complexity of simultaneously computing f onìnstances, divided by`. 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 (log n) and its amortized (deterministic) communication complexity is O(1). Similarly, for randomized protocols, we present a function whose randomized communication complexity is (log n) and its 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 p C (f) logn .