Tentative List of Topics for the Semester

• Balls and bins, hashing
• Data sketching and stream processing
• Random walks and Markov chains
• Random graphs
• Probability and geometry in high-dimensional vector spaces
• Singular value decomposition
• Dimensionality reduction via random projections
• Detecting sparse and low-rank structures in data

Lecture Schedule

1/22— Introduction and course announcements

Reading: (not required) Python notebook used for introductory lecture

1/27— Balls and Bins I: The birthday paradox (Notes)

Reading: Lecture notes on randomized algorithms, §1.1

1/29— Balls and Bins II: The coupon collector problem (Notes)

Reading: Lecture notes on randomized algorithms, §1.2

2/3— Balls and Bins III: Load balancing and the Chernoff bound (Notes)

Reading: Lecture notes on randomized algorithms, §1.3.1-1.3.3

2/5— Balls and Bins IV: Proof of the Chernoff bound (Notes)

Reading: Lecture notes on randomized algorithms, §1.3.4

2/10— The Hoeffding bound and its applications (Notes)

Reading: Lecture notes on randomized algorithms, §1.3.5-1.4

2/12— Hashing I: Dictionaries and hash tables (Notes)

Reading: Lecture notes on randomized algorithms, §2.1-2.3

2/19— Hashing II: Pairwise independence (Notes)

Reading: Lecture notes on randomized algorithms, §2.4, 3.2

2/24— Streaming I: Estimating distinct elements (Notes)

Reading: Lecture notes on randomized algorithms, §2.4, 3.2

2/26— Streaming II: Estimating distinct elements: improving accuracy (Notes)

Reading: Lecture notes on randomized algorithms, §3.2

3/3— Streaming III: Misra-Gries and Count-Min Sketch (Notes)

Reading: Lecture notes on randomized algorithms, §3.1 and §3.3

3/5— Streaming IV: Count Sketch (Notes)

Reading: Lecture notes on randomized algorithms, §3.3

3/10— Streaming V: Quantile Estimation (Notes)

Reading: Lecture notes on randomized algorithms, §3.4

3/12— Random Graphs I: Definitions, estimating isolated vertices (Notes)

Reading: Lecture notes on random graphs, §1-2.1

3/17— Random Graphs II: Connectivity, diameter, and expansion (Notes)

Reading: Lecture notes on random graphs, §2.2

3/19— Random Graphs III: Ramsey Theory (Notes)

Reading: Lecture notes on random graphs, §3

3/24— Random Graphs IV: The Probabilistic Method (Notes)

Reading: Lecture notes on random graphs, §3

3/26— Probability in Vector Spaces I: Gaussian Distributions (Notes)

Reading: Lecture notes on probability in vector spaces, §2-2.1

4/7— Probability in Vector Spaces II: Multivariate Gaussians (Notes)

Reading: Lecture notes on probability in vector spaces, §2.2

4/9— Singular Value Decomposition I (Notes)

Reading: Blum, Hopcroft, and Kannan, “Foundations of Data Science” Chapter 3

4/14— Singular Value Decomposition II (Notes)

Reading: Blum, Hopcroft, and Kannan, “Foundations of Data Science” Chapter 3

4/16— Markov Chains I: Definitions (Notes)

Reading: Lecture notes on Markov chains, §1-2

4/21— Markov Chains II: Metropolis-Hastings and Mixing Times (Notes)

Reading: Lecture notes on Markov chains, §3-4

4/23— Markov Chains III: Coupling (Notes)

Reading: Lecture notes on Markov chains, §5

4/28— Random Matrices I: Matrix Chernoff Bounds (also finish coupling lecture) (Notes)

Reading: Lecture notes on probability in vector spaces, §3

4/30— Random Matrices II: Random Projections and Applications (Notes)

Reading: Lecture notes on probability in vector spaces, §4-4.1

5/5— Random Matrices III: Sparse Recovery (Notes)

Reading: Lecture notes on probability in vector spaces, §4.2