 
Summary: Com S 631: Lower bounds and Separation Results
Lecture 16 Scribe: Sanghack Lee
1. Communication Complexity (continued)
In the last lecture, we've learned that for a given function f : n
× n
{0, 1}, the
communication complexity of the function, cc (f), is the length of bits to be transmitted
between two parties. A protocol is a binary tree whose leaves are labeled 0 or 1 and the length
of the longest path from the root to the leaves defines cc (f). And f is the least number of
monochromatic rectangles to partition Mf where Mf (x, y) = f (x, y). Each monochromatic
rectangle associates with each leaf in the protocol. Hence, we obtain cc (f) log f since
there are at most 2cc(f)
leaves. Today we will figure out the lower bound of f in two ways
by examining the rank and discrepancy of the matrix Mf .
2. The Matrix Rank bound
First, we will show the bound of communication complexity using the rank of Mf which
is called the matrix rank bound. The rank of a matrix is the maximum number of linearly
independent rows (or columns). For example, the communication matrix for the equality
function EQ,
MEQ =
