Communication Complexity: and ApplicationsCommunication complexity is the mathematical study of scenarios where several parties need to communicate to achieve a common goal, a situation that naturally appears during computation. This introduction presents the most recent developments in an accessible form, providing the language to unify several disjoint research subareas. Written as a guide for a graduate course on communication complexity, it will interest a broad audience in computer science, from advanced undergraduates to researchers in areas ranging from theory to algorithm design to distributed computing. The first part presents basic theory in a clear and illustrative way, offering beginners an entry into the field. The second part describes applications including circuit complexity, proof complexity, streaming algorithms, extension complexity of polytopes, and distributed computing. Proofs throughout the text use ideas from a wide range of mathematics, including geometry, algebra, and probability. Each chapter contains numerous examples, figures, and exercises to aid understanding. |
Contents
Introduction | 1 |
Deterministic Protocols | 9 |
Rank | 33 |
Randomized Protocols | 46 |
Numbers on Foreheads | 57 |
Discrepancy | 67 |
Information | 93 |
Compressing Communication | 121 |
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Common terms and phrases
algorithm Alice and Bob allows apply assume bits bits of communication Boolean Chapter Charlie circuit Claim clauses colors column communication complexity compute conditioned connected Consider consistent constant contains convex coordinates corresponds cover data structure defined denote depth deterministic discrepancy disjoint distinct distribution edges element encoding entries entropy equal error exactly example Exercise expected extension Fact Figure fixed formula function function f gates given gives graph implies independent inequality inputs intersection least Lemma length linear lower bound matching matrix monochromatic rectangles nonnegative O(log otherwise output parties path points polytope probability problem Proof protocol prove query random rank rectangle requires rounds rows sample satisfies sends simulation space step string subset Suppose takes Theorem tree uniformly random vector vertex vertices