Lecture 16: Communication Complexity

Definition 1.1. Formally, a protocol P is a rooted binary tree with each internal node v labelled by either “A” or “B” and two out-edges, one labelled 0 and the other labelled 1. Each leaf has an output label (typically in {0, 1}. There is a function fv associated with node v; if v is labelled A then fv : X → {0, 1} and if v is labelled B then fv : Y → {0, 1}. The bit sent at node v is sent by the corresponding player and the value is fv(x) if v has label A and fv(y) if v has label B. The output P (x, y) of the protocol on input (x, y) ∈ X × Y is the label of the leaf reached on input x. The cost of the protocol is the height of P .