CS225: Pseudorandomness
Prof. Salil Vadhan
Lecture 1: Introduction
February 1, 2007 Based on scribe notes by Saurabh Sanghvi.
1
Course Overview
Over the past few decades, randomization has become one of the most pervasive paradigms in computer science. It
CS 225: Pseudorandomness
Prof. Salil Vadhan
Problem Set 5
Assigned: Tue. Apr. 14, 2009
Due: Wed. Apr. 29, 2009(1 PM)
Recall that your problem set solutions must be typed. You can email your solutions to
cs225-hw@eecs.harvard.edu, or turn in it to MD138.
CS 225: Pseudorandomness
Prof. Salil Vadhan
Problem Set 3
Assigned: Tue. Mar. 10, 2009
Due: Wed. Apr. 1, 2009(1 PM)
Recall that your problem set solutions must be typed. You can email your solutions to
cs225-hw@eecs.harvard.edu, or turn in it to MD138. Y
CS 225: Pseudorandomness
Prof. Salil Vadhan
Problem Set 6
Assigned: Tue. Apr. 28, 2009
Due: Wed. May. 13, 2009(1 PM)
This problem set is a substitute for the nal exam. You must work alone (but you still may
come to oce hours with questions about the mate
CS 225: Pseudorandomness Problem Set 1
Assigned: Tue. Feb. 3, 2009
Prof. Salil Vadhan
Due: Wed. Feb. 18, 2009(1 PM)
Recall that your problem set solutions must be typed. You can email your solutions to cs225-hw@eecs.harvard.edu, or turn in it to MD138. Y
Foundations and Trends R in
sample
Vol. xx, No xx (xxxx) 171
c xxxx xxxxxxxxx
DOI: xxxxxx
Pseudorandomness II
Salil P. Vadhan1
1
Harvard UniversityCambridge, MA02138, USA, salil@eecs.harvard.edu
Abstract
This is the second volume of a 2-part survey on pse
2
The Power of Randomness
2.1
Polynomial Identity Testing
Before we study the derandomization of randomized algorithms, we will need some algorithms to
derandomize. This section introduces one such algorithm. It solves the following computational
problem.
Contents
5 Randomness Extractors
5.1
5.2
5.3
5.4
5.5
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Motivation and Denition
Connections with Hash Functions and Expanders
Constructing Extractors
More Connections with Expanders
Exercises
References
1
8
12
19
22
25
ii
5
Randomness Extractors
We now mov
Foundations and Trends R in
sample
Vol. xx, No xx (xxxx) 182
c xxxx xxxxxxxxx
DOI: xxxxxx
Pseudorandomness I
Salil P. Vadhan1
1
Harvard UniversityCambridge, MA02138, USA, salil@eecs.harvard.edu
Abstract
This is the rst volume of a 2-part survey on pseudor
CS 225: Pseudorandomness
Prof. Salil Vadhan
Problem Set 4
Assigned: Thus. Apr. 2, 2009
Due: Wed. Apr. 15, 2009(1 PM)
Recall that your problem set solutions must be typed. You can email your solutions to
cs225-hw@eecs.harvard.edu, or turn in it to MD138.
CS225: Pseudorandomness
Prof. Salil Vadhan
Lecture 6: Basic Derandomization Techniques II
February 20, 2007 Based on scribe notes by Chun-Yun Hsiao and Vinod Vaikuntanathan. Gutfreund. Lecture given by Dan
1
Recap
In the previous lecture, we saw several d
CS225: Pseudorandomness
Prof. Salil Vadhan
Lecture 12: Constructing Extractors
March 20, 2007 Based on scribe notes by Adam Kirsch. In the previous lectures, we have seen that very good extractors exist - extracting almost all of the min-entropy from a so
CS225: Pseudorandomness
Prof. Salil Vadhan
Lecture 13: More Connections with Expanders
March 22, 2007 Based on scribe notes by Adam Kirsch and Alexandr Andoni.
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Lossless Condensers vs. Expanders
Last time we saw the notion of a k k condenser Con : cfw_0,
CS225: Pseudorandomness
Prof. Salil Vadhan
Lecture 14: Error-Correcting Codes
April 3, 2007 Based on scribe notes by Sasha Schwartz and Adi Akavia.
1
Basic Denitions
The eld of coding theory is motivated by the problem of communicating reliably over noisy
CS225: Pseudorandomness
Prof. Salil Vadhan
Lecture 7: Expander Graphs
February 22, 2007
Based on scribe notes by Kartik Venkatram and Mihai Ptracu. a s
1
Expander Graphs
Now that we have seen a variety of basic derandomization techniques, we will move on
CS225: Pseudorandomness
Prof. Salil Vadhan
Lecture 8: Random Walks on Expanders
March 1, 2007 Based on scribe notes by Mihai Ptracu. a s
1
Rapid Mixing of Random Walks
From the previous lecture, we know that one way of characterizing an expander graph G i
CS225: Pseudorandomness
Prof. Salil Vadhan
Lecture 2: Randomized Algorithms and Complexity Classes
February 3, 2007 Based on scribe notes by Grant Schoenebeck.
1
Polynomial Identity Testing
Before we study the derandomization of randomized algorithms, we
CS225: Pseudorandomness
Prof. Salil Vadhan
Lecture 3: Sampling and Approximation Problems
February 8, 2007
Based on scribe notes by David Troiano, Grant Schoenebeck, and Brian Greenberg.
1
Sampling Problem
The power of randomization is well-known to stati
CS225: Pseudorandomness
Prof. Salil Vadhan
Lecture 5: Basic Derandomization Techniques
February 15, 2007 Based on scribe notes by Arthur Rudolph and Chun-Yun Hsiao.
1
Recap
Over the past few lectures, we have discussed some striking examples of the power
4
Expander Graphs
Now that we have seen a variety of basic derandomization techniques, we will move on to study
the rst major pseudorandom object in this survey, expander graphs. These are graphs that are
sparse yet very well-connected.
4.1
Measures of Ex