Foundations and Random Hashing 
1. 1/26 
Wed 
Course overview. Probability review. Example applications: polynomial identity testing, Frievald's algorithm 
Slides. Compressed slides. Reading: Chapters 13 of Mitzenmacher Upfal, with good coverage of probability basics, concentration bounds, and algorithmic applications. Greg Valiant's course notes covering the full analysis of polynomial identity testing (Lec 1). 
2. 2/2 
Wed 
Concentration bounds: Markov's, Chebyshev's. Union bound. Applications to Quicksort analysis, coupon collecting, statistical estimation, balls into bins. 
Slides. Compressed slides. Reading: Chapters 13 of Mitzenmacher Upfal, which cover most of the tools and applications from class. Greg Valiant's course notes covering randomized Quicksort (Lec 2) and concentration bounds (Lec 5). 
3. 2/9 
Wed 
Finish concentration inequalities: exponential tail bounds and application to linear probing analysis. Start on randomized hash functions and fingerprints. 
Slides. Compressed slides. Lecture recording.
Reading: Jelani Nelson's course notes, giving a proof of the Chernoff bound (Lec 3) and the linear probing analysis (Lec 4). Chapter 7 of Motwani Raghavan, with coverage of fingerprinting and its applications.

4. 2/16 
Wed 
Finish fingerprints. RabinKarp pattern matching algorithm. Randomized communication protocols. ℓ_{0} sampling with applications to graph sketching and streaming. 
Slides. Compressed slides. Reading: Chapter 7 of Motwani Raghavan, with coverage of fingerprinting and its applications. Many helpful slides/notes by Prof. McGregor on graph sketching algorithms. A good survey on ℓ_{0} sampling methods.

Random Sketching and Randomized Numerical Linear Algebra 
5. 2/23 
Wed 
Prof away. Class to be held over Zoom. Intro to randomized numerical linear algebra. Approximate matrix multiplication and Hutchinson's trace estimator. 
Slides. Compressed slides. Lecture recording. Reading: Original paper on sampling based approximate matrix multiplication, with the analysis covered in class. General notes on RandNLA, (from Drineas and Mahoney) with good background material and coverage of approximate matrix multiplication. 
6. 3/2 
Wed 
Randomized lowrank approximation. Adaptive sampling and kmeans++. 
Slides. Compressed slides. Reading: Original kmeans++ paper. Some notes on sampling based lowrank approximation (Mahoney at UC Berkeley).

3/9 
Wed 
Midterm 
Study guide and review questions. 
3/16 
Wed 
No Class, Spring Recess. 

7. 3/23 
Wed 
The JohnsonLindenstrauss lemma. Proof via HansonWright. Epsilonnets and subspace embedding. Sketching for linear regression. 
Slides. Compressed slides. Reading: Jelani Nelson's course notes, which include the proof of distributional JL via HansonWright, along with discussion of subspace embeddings and various applications. Chris Muscos's notes on subspace embedding, on which my proof in class was based.

8. 3/30 
Wed 
Importance sampling and leverage scores. Matrix concentration bounds. Spectral graph sparsifiers. 
Slides. Compressed slides.
Reading: Dan Spielman's great book on spectral graph theory, including Chapter 32 on spectral sparsification via sampling, and analysis via matrix Chernoff bounds, which is equivilant to the leverage score analysis shown in class.

Markov Chains 
9. 4/6 
Wed 
Finish up graph sparsification. Introduction to Markov chains and their analysis. Applications to randomized algorithms for satisfiability problems (2SAT and 3SAT). 
Slides. Compressed slides.
Reading: Chapter 7 of Mitzenmacher Upfal on Markov Chains, including the application to 2SAT and gambler's ruin.

10. 4/13 
Wed 
Markov chains continued. Gambler's ruin. Stationary distributions and start on mixing time analysis via coupling.

Slides. Compressed slides.
Reading: Notes by Greg Valiant and Mary Wooters covering stationary distributions and mixing times/coupling. Also Chapters 10/11 of Mitzenmacher Upfal covering mixing time and coupling.

4/20 
Wed 
No Class, Monday Schedule. 

11. 4/27 
Wed 
Finish Markov chains: coupling and Markov chain Monte Carlo methods (MCMC). 
Slides. Compressed slides.
Reading: Notes by Greg Valiant and Mary Wooters covering the MetropolisHastings algorithm and mixing times/coupling. Also Chapters 10/11 of Mitzenmacher Upfal covering MCMC, mixing time, coupling, and counting via sampling. 
Other Topics 
12. 5/4 
Wed 
Probabilistic method and the Lovasz Local Lemma. 
Slides. Compressed slides.
Reading: Chapter 6 of Mitzenmacher Upfal. Greg Valiant's lecture notes on probabilistic method and algortihmic Lovasz local lemma.

5/6 
Fri 
Final Exam. 10:30am  12:30pm 
Study guide and review questions. 