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

1/21— Introduction and course announcements

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

1/26— Balls and Bins I: The birthday paradox (Notes)

Reading: Chapter 1.1

1/28— Balls and Bins II: The coupon collector problem (Notes)

Reading: Chapter 1.2

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

Reading: Chapter 1.3.1, 1.3.2, 1.3.4

2/4— Balls and Bins IV: Proof of the Chernoff bound (Notes)

Reading: Chapter 1.3.3

2/9— The Hoeffding bound and its applications (Notes)

Reading: Chapter 1.4

2/11— Hashing I: Definitions (Notes)

Reading: Chapter 2.1-2.3

2/18— Hashing II: 2-universal hash constructions, randomized dictionaries (Notes)

Reading: Chapter 2.4

2/23— Streaming I: Estimating distinct elements (Notes)

Reading: Chapter 3.2

2/25— Streaming II: Misra-Gries (Notes)

Reading: Chapter 3.1

3/2— Streaming IV: Count-Min Sketch (Notes)

Reading: Chapter 3.3

3/4— Streaming V: Count Sketch (Notes)

Reading: Chapter 3.3

3/9— Streaming VI: Quantile Estimation (Notes)

Reading: Chapter 3.4

3/11— Random Graphs I: Definitions, isolated vertices, connectivity (Notes)

Reading: Chapter 4.1, 4.2.1

3/16— Random Graphs II: Diameter, expansion, probabilistic method (Notes)

Reading: Chapter 4.2.2-4.2.3

3/18— Markov Chains I: Definitions (Notes)

Reading: Chapter 6.1

3/23— Markov Chains II: Metropolis-Hastings and Mixing Times (Notes)

Reading: Chapter 6.2-6.3
Markov chain simulations presented in class: Schelling model and Ising model.

3/25— Markov Chains III: Couplings (Notes)

Reading: Chapter 6.4