Communication Complexity (for Algorithm Designers)

Preface The best algorithm designers prove both possibility and impossibility results — both upper and lower bounds. For example, every serious computer scientist knows a collection of canonical NP-complete problems and how to reduce them to other problems of interest. Communication complexity offers a clean theory that is extremely useful for proving lower bounds for lots of different fundamental problems. Many of the most significant algorithmic consequences of the theory follow from its most elementary aspects. This document collects the lecture notes from my course " Communication Complexity (for Algorithm Designers), " taught at Stanford in the winter quarter of 2015. The two primary goals of the course are: (1) Learn several canonical problems in communication complexity that are useful for proving lower bounds for algorithms (Disjointness, Index, Gap-Hamming, etc.). (2) Learn how to reduce lower bounds for fundamental algorithmic problems to communication complexity lower bounds. Along the way, we'll also: (3) Get exposure to lots of cool computational models and some famous results about them — data streams and linear sketches, compressive sensing, space-query time trade-offs in data structures, sublinear-time algorithms, and the extension complexity of linear programs. (4) Scratch the surface of techniques for proving communication complexity lower bounds (fooling sets, corruption arguments, etc.). Readers are assumed to be familiar with undergraduate-level algorithms, as well as the statements of standard large deviation inequalities (Markov, Chebyshev, and Chernoff-Hoeffding). The course begins in Lectures 1–3 with the simple case of one-way communication protocols — where only a single message is sent — and their relevance to algorithm design. Each of these lectures depends on the previous one. Many of the " greatest hits " of communication complexity applications, including lower bounds for small-space streaming algorithms and compressive sensing, are already implied by lower bounds for one-way ii Preface iii protocols. Reasoning about one-way protocols also provides a gentle warm-up to the standard model of general two-party communication protocols, which is the subject of Lecture 4. Lectures 5–8 translate communication complexity lower bounds into lower bounds in several disparate problem domains: the extension complexity of polytopes, data structure design, algorithmic game theory, and property testing. Each of these final four lectures depends only on Lecture 4. The course Web page (http://theory.stanford.edu/~tim/w15/w15.html) contains links to relevant large deviation inequalities, links to many of the papers cited in these notes, and a partial list of exercises. Lecture notes and videos on several other …