Randomized Methods in Computation
Notes for a Self-Study Course [Spring 2011]
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 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This material is believed to superseded the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.
- 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.
Material available on-line
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
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).
Finally, see also the text
A Taste of Randomized Computations.
Additional recommended reading
N. Alon and J.H. Spencer: The Probabilistic Method.
John Wiley Sons, Inc., 1992.
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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