The Nature of ComputationComputational complexity is one of the most beautiful fields of modern mathematics, and it is increasingly relevant to other sciences ranging from physics to biology. But this beauty is often buried underneath layers of unnecessary formalism, and exciting recent results like interactive proofs, phase transitions, and quantum computing are usually considered too advanced for the typical student. This book bridges these gaps by explaining the deep ideas of theoretical computer science in a clear and enjoyable fashion, making them accessible to non-computer scientists and to computer scientists who finally want to appreciate their field from a new point of view. The authors start with a lucid and playful explanation of the P vs. NP problem, explaining why it is so fundamental, and so hard to resolve. They then lead the reader through the complexity of mazes and games; optimization in theory and practice; randomized algorithms, interactive proofs, and pseudorandomness; Markov chains and phase transitions; and the outer reaches of quantum computing. At every turn, they use a minimum of formalism, providing explanations that are both deep and accessible. The book is intended for graduate and undergraduate students, scientists from other areas who have long wanted to understand this subject, and experts who want to fall in love with this field all over again. |
Contents
Chapter 1 Prologue | 1 |
Chapter 2 The Basics | 15 |
Chapter 3 Insights and Algorithms | 41 |
the Class NP | 95 |
Chapter 5 Who is the Hardest One of All? NPCompleteness | 127 |
P vs NP | 173 |
Chapter 7 The Grand Unified Theory of Computation | 223 |
Chapter 8 Memory Paths and Games | 301 |
Chapter 11 Interaction and Pseudorandomness | 507 |
Chapter 12 Random Walks and Rapid Mixing | 563 |
Chapter 13 Counting Sampling and Statistical Physics | 651 |
Phase Transitions in Computation | 723 |
Chapter 15 Quantum Computation | 819 |
Appendix A Mathematical Tools | 911 |
References | 945 |
| 974 | |
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Common terms and phrases
algorithm apply approximation Arthur assignments assume bits bound called choose clauses color complexity computation connected consider consists constant constraints corresponding cover cycle define denote described discussed distribution edges equations exactly Exercise exists expected exponentially fact factor Figure flip flow formula function given gives graph hard independent input instance integer least length linear machine matchings mathematical matrix mixing move neighbors Note NP-complete optimal pair path perfect physics polynomial position possible prime probability problem proof prove quantum question random recursive reduction result running satisfying shown simple solution solve spanning square step string Suppose tell Theorem tiles tree true variables vector vertex vertices walk weight write