Problem instances[ edit ] A computational problem can be viewed as an infinite collection of instances together with a solution for every instance. The input string for a computational problem is referred to as a problem instance, and should not be confused with the problem itself. In computational complexity theory, a problem refers to the abstract question to be solved. In contrast, an instance of this problem is a rather concrete utterance, which can serve as the input for a decision problem. For example, consider the problem of primality testing. The instance is a number e.
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Basic Complexity Classes: 1. NP and NP completeness; 3. Diagonalization; 4. Space complexity; 5. The polynomial hierarchy and alternations; 6. Boolean circuits; 7. Randomized computation; 8. Interactive proofs; 9.
Cryptography; Quantum computation; Lower Bounds for Concrete Computational Models: Decision trees; Communication complexity; Circuit lower bounds; Proof complexity; Algebraic computation models; Part III. Advanced Topics: Complexity of counting; Hardness amplification and error correcting codes; Derandomization; Pseudorandom constructions: expanders and extractors; Proofs of PCP theorems and the Fourier transform technique; Why are circuit lower bounds so difficult?
It will serve the needs of a wide audience, ranging from experienced researchers to graduate students and ambitious undergraduates seeking an introduction to the mathematical foundations of computer science. I will keep it at my side as a useful reference for my own teaching and research. Student and researchers alike will find it to be an immensely useful resource. This book contains essentially all of the many exciting developments of the last two decades, with high level intuition and detailed technical proofs.
It is a must for everyone interested in this field. He holds a Ph. Boaz Barak is an assistant professor in the department of computer science at Princeton University.