Communication Complexity and Parallel Computing

Transcription

1 Juraj Hromkovic Communication Complexity and Parallel Computing With 40 Figures Springer

2 Table of Contents 1 Introduction Motivation and Aims Concept and Organization How to Read the Book 6 2 Communication Protocol Models Basic Notions Introduction Alphabets, Words, and Languages Boolean Functions and Boolean Matrices Representation of Computing Problems Exercises Communication Complexity According to a Fixed Partition Definitions Methods for Proving Lower Bounds Theoretical Properties of Communication Complexity According to a Fixed Partition Exercises Research Problems Communication Complexity Introduction Definitions Lower Bound Methods Theoretical Properties of Communication Complexity Communication Complexity and Chomsky Hierarchy Exercises Research Problems One-Way Communication Complexity Introduction Definitions Methods for Proving Lower Bounds Communication Complexity Versus One-way Communication Complexity Exercises Research Problems 96

3 viii Table of Contents 2.5 Nondeterministic Communication Complexity and Randomized Protocols Introduction Nondeterministic Protocols Lower Bounds on Nondeterministic Communication Complexity Deterministic Protocols Versus Nondeterministic Protocols Randomized Protocols Randomness Versus Nondeterminism and Determinism Exercises Research problems An Improved Model of Communication Protocols Introduction Definitions Lower Bound Methods Communication Complexity Versus s-communication Complexity Some Properties of s-communication Complexity Exercises Problems Bibliographical Remarks Boolean Circuits Introduction Definitions and Fundamental Properties Introduction Boolean Circuit Models Fundamental Observations Exercises Lower Bounds on the Area of Boolean Circuits Introduction Definitions Lower Bounds on the Area Complexity Measures A Comparison of two Area Complexity Measures Three-Dimensional Layout Exercises Problems Topology of Circuits and Lower Bounds Introduction Separators Lower Bounds on Boolean Circuits with a Sublinear Separator 192

4 Table of Contents ix Circuit Structures for Which Communication Complexity Does Not Help Planar Boolean Circuits Exercises Problems Lower Bounds on the Size of Unbounded Fan-in Circuits Introduction Method of Communication Complexity of Infinite Bases Unbounded Fan-in Circuits with Sublinear Vertex-Separators Exercises Problems Lower Bounds on the Depth of Boolean Circuits Introduction Monotone Boolean Circuits Communication Complexity of Relations Characterizations of Circuit Depth by the Communication Complexity of Relations Exercises Research Problems Bibliographical Remarks 237 VLSI Circuits and Interconnection Networks Introduction Definitions Introduction A VLSI circuit Model Complexity Measures Probabilistic Models Exercises Lower Bounds on VLSI Complexity Measures Introduction Lower Bounds on Area Complexity Lower Bounds on Tradeoffs of Area and Time VLSI circuits with Special Communication Structures Exercises Problems Interconnection Networks Introduction A Model of Interconnection Networks Separators and Lower Bounds Exercises Problems 270

5 x Table of Contents 4.5 Multilective VLSI circuits Introduction and Definitions Multilectivity Versus Semilectivity Lower Bounds on Multilective VLSI programs Exercises Problems Bibliographical Remarks Sequential Computations Introduction Finite Automata Introduction Definitions One-Way Communication Complexity and Lower Bounds on the Size of Finite Automata Uniform Protocols Exercises Research Problems Turing Machines Introduction Time Complexity of Classical Turing Machines Sequential Space and Time-Space Complexity Exercises Research Problems Decision Trees and Branching Programs Introduction Definitions Capacity of Branching Programs Lower Bounds on the Depth of Decision Trees Exercises Research Problems Bibliographical Remarks. 311 References 317 Index 331