# Randomized Methods in Computation

## Notes for a Self-Study Course [Spring 2011]

### Oded Goldreich

A variety of randomized methods are being employed in the study of computation. The aim of the suggested (self-study) course is to make the students familiar with some of these methods. We wish to stress three aspects regarding this course:

• This course focuses on methods (i.e., tools and techniques) and not on concepts.
• These methods are randomized and so the result is often not ``(fully) explicit''.
• The focus is on applications to the study of computation.
This webpage provides access to a new set of notes that are suggested as a basis for the self-study course. This material is believed to superseded the lecture notes taken in 2001 by students attending a course with a similar topic.

#### Material available on-line

• A highly tentative version, July 2011.

The material was compiled based on several older texts of mine. In particular, much of it appears in my book Computational Complexity: A Conceptual Perspective (e.g., see Appendix D on Probabilistic Preliminaries and Advanced Topics in Randomization). A list of topics (+ pointers to parts of the CC book) follows.

• Basic probability and Laws of large numbers: Appendix D.1.
Two examples of an application of the probability method appear in Appendix E (see Prop. E.1 and Footnote 14).
• Pairwise independence and hashing: Section 8.5.1 and Appendix D.2.
• Random walks on expanders: Section 8.5.3 and Appendix E.2.
• Samplers and hitters: Appendix D.3 (or a more detailed survey).
• Randomness Extractors: Appendix D.4.
• Small bias spaces: Section 8.5.2
• Approximate counting and uniform generation: Sections 6.2.2 and 6.2.4, see also related Section 6.2.3 (optional).

• N. Alon and J.H. Spencer: The Probabilistic Method. John Wiley Sons, Inc., 1992.
• O. Goldreich, Computational Complexity: A Conceptual Perspective, Cambridge University Press, 2008.
• R. Motwani and P. Raghavan: Randomized Algorithms. Cambridge University Press, 1995.
• R. Shaltiel: Recent Developments in Explicit Constructions of Extractors. In Current Trends in Theoretical Computer Science: The Challenge of the New Century, Vol 1: Algorithms and Complexity, World scietific, 2004. (Editors: G. Paun, G. Rozenberg and A. Salomaa.) Preliminary version in Bulletin of the EATCS 77, pages 67--95, 2002.

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