Lecture 1 Lecturer : Troy Lee Scribe : Troy Lee 1 Deterministic Communication Complexity

Two parties, Alice and Bob, wish to compute some function f : X × Y → Z, where X, Y, Z are finite sets. Alice receives x ∈ X, Bob y ∈ Y . Informally, in a deterministic communication protocol Alice and Bob send messages back and forth to each other, stopping when both parties know the answer f(x, y). The main measure of interest is how many bits of information they have to exchange to do this in a worst case sense, maximized over all (x, y) ∈ X × Y . No computational restrictions are placed on Alice and Bob—we are really just measuring how much information needs to be exchanged. More formally, a communication protocol is modeled as a binary tree. The leaves of the tree are labeled by outputs z ∈ Z, and internal nodes are designated Alice nodes or Bob nodes, indicating who will speak at that point in the protocol. An Alice node v is further labeled by a function av : X → {0, 1} which dictates Alice’s communication. Similarly a Bob node is labeled by a function bv : Y → {0, 1}. An input (x, y) determines a path in this tree in the following way. Starting from the root, look at the function labeling the current node v. If it is an Alice node, av(x), continuing to the left child if av(x) = 0 and to the right child otherwise. Similarly for Bob nodes. Continue in this way until a leaf is reached. The label of this leaf is the output of the protocol on input (x, y). A protocol is correct if for every (x, y) ∈ X×Y the path determined by (x, y) arrives at a leaf labeled by f(x, y). The communication complexity of f , denoted D(f), is the minimum over all correct protocols for f of the depth of the protocol tree (length of a longest path from root to leaf). Another useful complexity measure we will use is C (f), the protocol partition number, which is the minimum over all correct protocols for f of the number of leaves in the protocol. Note the obvious relationship log C (f) ≤ D(f). One protocol that always works—we will call it the naive protocol—is where Alice simply sends her entire input to Bob, Bob evaluates f(x, y) and sends the answer back to Alice. This shows that D(f) ≤ min{log |X|, log |Y |}+ log |Z|. We will see that for many common functions of interest the naive protocol is optimal.