Com S 631: Lower bounds and Separation Results Lecture 16 Scribe: Sanghack Lee 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 = Collections: Computer Technologies and Information Sciences