Matrix Rank in Communication Complexity

This lecture focuses on proving communication lower bounds using matrix rank. Similar to fooling sets and rectangle size bounds, the matrix rank technique also gives a lower bound on the number of monochromatic rectangles in any partition of X × Y but it does so in an algebraic way[1]. This makes algebraic tools available for proving communication lower bounds. We begin by solving the problem about matrix rank from last lecture. We then present the rank technique and reprove tight lower bounds on the communication complexity of equality, greater-than, disjointness, and inner product. Finally, we look into the comparative strength of fooling sets, rectangle size bounds, and matrix rank as techniques for proving communication lower bounds.