In this chapter we consider several, more advanced, topics related to the two-party communication model.

Direct Sum

The direct-sum problem is the following: Alice gets two inputs xf ∈ Xf and xg ∈ Xg. Bob gets two inputs yf ∈ Yf and yg ∈ Yg. They wish to compute both f(xf, yf) and g(xg, yg). The obvious solution would be for Alice and Bob to use the best protocol for f to compute the first value, f(xf, yf) and the best protocol for g to compute the second value, g(xg, yg). We stress that the two subproblems are totally independent. Thus one would tend to conjecture that nothing better than the obvious solution can be done: Alice and Bob cannot “save” any communication over the obvious protocol. As we shall see, in some cases and for some measures of complexity, this intuition is wrong.

Denote by D(f, g) the (deterministic) communication complexity of this computation. Similarly, we define all other complexity measures such as R(f, g), N(f, g), and so forth. We also use the notation D(fℓ) as the (deterministic) communication complexity of computing f and ℓ instances; that is, computing f(x1, y1), f(x2, y2), …, f(xℓ, yℓ

Open Problem 4.1: Can D(f, g) be smaller than D(f) + D(g)? How much smaller can it be? How much smaller can D(fℓ) be compared to ℓ · D(f)?

In some cases we are not interested in computing both f and g but rather some function of the two.