Depth lower bound for matching

Definition 2.1. Let f : X × Y → V . A subset R of X × Y is a rectangle1 if it is of the form A × B for some A ⊆ X and B ⊆ Y . The rectangle R is said to be monochromatic (wrt. f) if f is constant on R. A monochromatic rectangle R is a 0-rectangle if f(R) = {0}; it is a 1-rectangle if f(R) = {1}. Observation 2.2. A subset S of X × Y is a rectangle iff for all x, x′ ∈ X and y, y′ ∈ Y (x, x′) ∈ S and (y, y′) ∈ S implies (x, y′) ∈ S and (x′, y) ∈ S. Observation 2.3. Any deterministic protocol P on X×Y induces a partition of X×Y . If P computes a function f : X × Y → {0, 1}, then the rectangles of this partition are 0and 1-rectangles. There is one such rectangle for each leaf of P . Similarly, a non-deterministic protocol for f induces a set of 1-rectangles of f whose union (and not necessarily a partition) is f−1(1).