Communication and Computation

In computer science, often computational problems can be transformed into communication problems. The most illustrative example is probably the evaluation of a function via a Boolean formula. The correspondence to an equivalent communication game is as follows. A Boolean formula computing the function value g(x1, . . . , xk) can be represented by a binary tree. The leaves are labeled with variables xi ∈ {0, 1} or their negations (the same variable may occur on several leaves). In each inner vertex the two incoming subtrees are combined either via an “and” (∧) gate or an “or” (∨) gate, i. e., in these inner nodes the results g1 and g2 obtained so far, are processed according to g1 ∧ g2 or g1 ∨ g2. The result g(x1, . . . , xk) will finally be found at the root of the tree. So a computation is carried out as a process starting at the leaves and ending at the root. The depth (minimal length of a path from a leaf to a root) of an optimal Boolean formula for the calculation of a function g is an important parameter in theoretical computer science. The equivalent communication game is as follows. Alice holds an input (x1, . . . , xk) with g(x1, . . . , xk) = 1 and Bob holds an input (y1, . . . , yk) with g(y1, . . . , yk) = 0. They exchange bits of information until they find a component i ∈ {1, . . . , k} in which xi = yi. The minimal number of communication bits exchanged by Alice and Bob maximized over all inputs is the communication complexity of this game. The communication protocol can be represented by a binary tree. In this tree the vertices are labeled with the person whose turn it is to send, further an edge to the left successor of a vertex corresponds to a bit 1 and an edge to the right successor correspond to a bit 0. A one–to– one correspondence to a Boolean formula now is obtained by assigning the ∨– gates to Alice and the ∧–gates to Bob (so it is the respective person’s turn to send in these vertices). Hence, we can use the same tree as for the Boolean formula with the difference that communication starts in the root and terminates in the leaves. For recent results on this model we refer to [4]. A similar communication model, based on a decision problem rather than a search problem, is introduced in the next section. Methods from data compression are used to derive lower bounds. Finally, the application to a computation problem will be discussed.