**Pseudorandomness and combinatorial constructions (CS286C)**

Instructor: Chris Umans

Office: Jorgensen 286

Times: Tu/Th 1:00-2:25 in Jorgensen 287

Office hours: email me

Announcements:

- Paper presentations will be 1-2:30 on Tuesday May 30 and Thursday June 1 in Jorgensen 287.

Handouts:

- Syllabus (pdf)

Lectures:

- Lecture 1: Introduction, polynomial identity testing, Schwartz-Zippel Lemma.
- Lecture 2: error correcting codes: existence, Reed-Solomon, Reed-Muller, Hadamard, concatenated codes, k-wise independent sample spaces.
- Madhu Sudan's coding theory course.
- Mike Luby and Avi Wigderson's notes on pairwise independence and derandomization .
- Lecture 3: epsilon-bias spaces and k-wise epsilon-bias spaces, k-wise "delta-dependent" spaces
- J. Naor and M. Naor: Small Bias Probability Spaces: efficient constructions and applications
- N. Alon, O. Goldreich, J. Hastad, R. Peralta: Simple Constructions of Almost k-wise Independent Random Variables
- Lecture 4: expander graphs, existence, mixing, spectral -> vertex expansion, operations on graphs, Zig-Zag construction
- O. Reingold, S. Vadhan, A. Wigderson: Entropy Waves, The Zig-Zag Graph Product, and New Constant-Degree Expanders and Extractors
- See also: Gabber-Galil construction and analysis (notes from a course by Umesh Vazirani)
- See also: N. Alon, A. Lubotzky, A. Wigderson: Semi-direct product in groups and Zig-zag product in graphs: connections and applications
- Lecture 5: proof of the zig-zag theorem, Chernoff bound for expander walks, application to error reduction, hitting version with proof
- D. Gillman: A Chernoff bound for random walks on expander graphs
- See also: A. Wigderson, D. Xiao: A randomness-efficient sampler for matrix-valued functions and applications
- Lecture 6: extractors, simulation using weak random sources, existence, the SZ construction
- N. Nisan: Extracting randomness: How and why (an older survey)
- A. Srinivasan, D. Zuckerman: Computing with very weak random sources
- See also: J. Radhakrishnan, A. Ta-Shma: Bounds for dispersers, extractors, and depth-two superconcentrators.
- Lecture 7: Yao's Lemma, Trevisan's construction
- L. Trevisan: Extractors and pseudorandom generators
- Lecture 8: Weak designs, q-ary extractors and conversion to ordinary extractors, "reconstruction" framework, SU construction
- R. Raz, O. Reingold, S. Vadhan: Extracting all the randomness and reducing the error in Trevisan's Extractors
- R. Shaltiel, C. Umans: Simple extractors for all min-entropies and a new pseudo-random generator
- See also: A. Ta-Shma, D. Zuckerman, S. Safra: Extractors from Reed-Muller codes
- Lecture 9: Analysis of the SU construction, condensers.
- A. Ta-Shma, C. Umans, D. Zuckerman: Loss-less condensers, unbalanced expanders, and extractors
- Lecture 10: elements of the analysis for condensers, PRGs: existence and consequences for hard functions, NW construction
- N. Nisan, A. Wigderson: Hardness vs. randomness
- Lecture 11: converting worst-case to average-case hardness, efficient list-decoding of Reed-Solomon codes
- Lecture 10 of my CS151 course
- M. Sudan, L. Trevisan, S. Vadhan: Pseudorandom generators without the XOR lemma
- M. Sudan: Decoding of Reed-Solomon codes beyond the error-correction bound
- see also: V. Guruswami, M. Sudan: Improved decoding of Reed-Solomon and algebraic-geometric codes
- see also: R. Impagliazzo, A. Wigderson: P=BPP unless E has subexponential circuits: derandomizing the XOR lemma (the original proof, using different methods)
- Lecture 12: an algebraic PRG construction
- Lecture 13: PRGs against higher classes, PRGs against limited space, Nisan's generator
- A. Klivans, D. van Melkebeek: Graph nonisomorphism has subexponential size proofs unless the polynomial-time hierarchy collapses
- see also: R. Shaltiel, C. Umans: Pseudorandomness for approximate counting and sampling
- N. Nisan: Pseudorandom generators for space-bounded computation (restricted access)
- see also: Pseudorandomness in space-bounded computation (lecture notes from a course by Oded Goldreich)
- Lecture 14: analysis of Nisan's generator, applications
- see also: M. Saks, S. Zhou: BP_H SPACE(S) \subseteq DSPACE(S^{3/2}) (restricted access)
- Lectures 15,16: out of town
- Lecture 17: Nisan/Zuckerman generator, the class SL, SL in RL
- N. Nisan, D. Zuckerman: Randomness is Linear in Space
- Lecture 18: proof of "SL = L"
- O. Reingold: Undirected ST-Connectivity in Log-Space
- see also: E. Rozenman, S. Vadhan: Derandomized Squaring of Graphs

Similar courses (with online lecture notes):

- David Zuckerman: Pseudorandomness and combinatorial constructions
- Salil Vadhan: Pseudorandomness
- Luca Trevisan: Pseudorandomness and combinatorial constructions
- Valentine Kabanets: Pseudorandomness
- Nati Linial and Avi Wigderson: Expander graphs and their applications

Surveys:

- R. Shaltiel: Recent developments in extractors
- V. Kabanets: Derandomization: a brief overview

Possible papers for presentation:

- Any of the "see also" papers listed above
- A. Ta-Shma, D. Zuckerman: Extractor codes
- V. Kabanets: Easiness assumptions and hardness tests: trading time for zero error
- O. Goldreich, A. Wigderson: Derandomization that is rarely wrong from short advice that is typically good
- J. Buresh-Oppenheim, R. Santhanam: Making hard problems harder
- C.-J. Lu, O. Reingold, S. Vadhan, A. Wigderson: Extractors: optimal up to constant factors
- M. Capalbo, O. Reingold, S. Vadhan, A. Wigderson Randomness conductors and constant-degree lossless expanders
- D. Zuckerman: Linear degree extractors and the inapproximability of max clique and chromatic number
- D. Gutfreund, R. Shaltiel, A. Ta-Shma: Uniform hardness vs. randomness for Arthur-Merlin games
- R. Impagliazzo, A. Wigderson: Randomness vs. time: de-randomization under a uniform assumption
- C. Umans: Reconstructive dispersers and hitting set generators
- O. Reingold, L. Trevisan, S. Vadhan: Pseudorandom walks on regular digraphs and the RL vs. L problem
- R. Impagliazzo, N. Nisan, A. Wigderson: Pseudorandomness for network algorithms
- R. Impagliazzo, V. Kabanets, A. Wigderson: In Search of an Easy Witness: Exponential Time vs. Probabilistic Polynomial Time
- V. Kabanets, R. Impagliazzo: Derandomizing Polynomial Identity Tests Means Proving Circuit Lower Bounds