Book contents
- Frontmatter
- Contents
- List of Figures
- Preface
- Organization and Chapter Summaries
- Acknowledgments
- 1 Introduction and Preliminaries
- 2 P, NP, and NP-Completeness
- 3 Variations on P and NP
- 4 More Resources, More Power?
- 5 Space Complexity
- 6 Randomness and Counting
- 7 The Bright Side of Hardness
- 8 Pseudorandom Generators
- 9 Probabilistic Proof Systems
- 10 Relaxing the Requirements
- Epilogue
- Appendix A Glossary of Complexity Classes
- Appendix B On the Quest for Lower Bounds
- Appendix C On the Foundations of Modern Cryptography
- Appendix D Probabilistic Preliminaries and Advanced Topics in Randomization
- Appendix E Explicit Constructions
- Appendix F Some Omitted Proofs
- Appendix G Some Computational Problems
- Bibliography
- Index
5 - Space Complexity
Published online by Cambridge University Press: 05 June 2012
- Frontmatter
- Contents
- List of Figures
- Preface
- Organization and Chapter Summaries
- Acknowledgments
- 1 Introduction and Preliminaries
- 2 P, NP, and NP-Completeness
- 3 Variations on P and NP
- 4 More Resources, More Power?
- 5 Space Complexity
- 6 Randomness and Counting
- 7 The Bright Side of Hardness
- 8 Pseudorandom Generators
- 9 Probabilistic Proof Systems
- 10 Relaxing the Requirements
- Epilogue
- Appendix A Glossary of Complexity Classes
- Appendix B On the Quest for Lower Bounds
- Appendix C On the Foundations of Modern Cryptography
- Appendix D Probabilistic Preliminaries and Advanced Topics in Randomization
- Appendix E Explicit Constructions
- Appendix F Some Omitted Proofs
- Appendix G Some Computational Problems
- Bibliography
- Index
Summary
Open are the double doors of the horizon; unlocked are its bolts.
Philip Glass, Akhnaten, PreludeWhereas the number of steps taken during a computation is the primary measure of its efficiency, the amount of temporary storage used by the computation is also a major concern. Furthermore, in some settings, space is even more scarce than time.
In addition to the intrinsic interest in space complexity, its study provides an interesting perspective on the study of time complexity. For example, in contrast to the common conjecture by which NP ≠ coNP, we shall see that analogous space-complexity classes (e.g., Nℒ) are closed under complementation (e.g., Nℒ = coNℒ).
Summary: This chapter is devoted to the study of the space complexity of computations, while focusing on two rather extreme cases. The first case is that of algorithms having logarithmic space complexity. We view such algorithms as utilizing the naturally minimal amount of temporary storage, where the term “minimal” is used here in an intuitive (but somewhat inaccurate) sense, and note that logarithmic space complexity seems a more stringent requirement than polynomial time. The second case is that of algorithms having polynomial space complexity, which seems a strictly more liberal restriction than polynomial time complexity. Indeed, algorithms utilizing polynomial space can perform almost all the computational tasks considered in this book (e.g., the class PSP ACε contains almost all complexity classes considered in this book). […]
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- Computational ComplexityA Conceptual Perspective, pp. 143 - 183Publisher: Cambridge University PressPrint publication year: 2008