CS6025 Sublinear Algorithms
Jan - May 2022
The lecture videos and notes written during the lectures are available in the links below.
- Week 1, Jan 17-21
- Outline of the course; logistics; reservoir sampling; parameter estimation; basic probability and concentration bounds; Probabilistic counting - Morris counter.
- References:
- Additional reading: Optimal bounds for approximate counting - Nelson and Yu
- Week 2, Jan 24-28
- Frequency moments - counting distinct elements; Flajolet-Martin algorithm; BJKST’s algorithm and analysis; Pairwise-independent hash families.
- References:
- Additional reading:
- HyperLogLog - the analysis of a near-optimal cardinality estimation algorithm - Flajolet, Fusy, Gandouet, Meunier
- Wikipedia entry for HyperLogLog
- Counting distinct elements in a data stream - Bar-Yossef, Jayram, Kumar, Sivakumar, Trevisan.
- Week 3, Jan 31-Feb 4
- Counting distinct elements in turstile streams; Linear sketching; Nisan’s PRG; Frequency moments - AMS algorithm and sketch for \(F_2\).
- Week 4, Feb 7-11
- Heavy-hitters problem - CountMin sketch and Count sketch; \(\ell_0\)-sampling and sparse recovery.
- Videos: Feb 7, Feb 9, Feb 10
- References:
- Additional reading:
- An Improved Data Stream Summary: The Count-Min Sketch and its Applications - Cormode, Muthukrishnan.
- Finding Frequent Items in Data Streams - Charikar, Chen, Farch-Colton.
- Week 5, Feb 14-18
- \(\ell_0\)-sampling and sparse recovery (contd.), Graph streams - connectivity, spanners, sparsifiers, triangle counting.
- Videos: Feb 14, Feb 16, Feb 17
- References:
- Jelani Nelson’s notes
- Notes from a previous offering of this course (incomplete and unedited)
- Graph Stream Algorithms: A Survey - McGregor
- Week 6, Feb 21-25
- Graph streams - random walks; Graph sketches - connectivity in fully dynamic streams; Lower bounds for streaming algorithms - intro to communication complexity.
- Videos: Feb 21, Feb 23, Feb 24
- References:
- Notes from a previous offering of this course (incomplete and unedited)
- Graph Stream Algorithms: A Survey - McGregor
- [RY]
- Additional reading:
- Estimating PageRank on Graph Streams - Das Sarma, Gollapudi, Panigrahy.
- Week 7, Feb 28-Mar 4
- Lower bounds for streaming algorithms using communication complexity.
- Videos: Feb 28, Mar 2, Mar 3
- References:
- Notes from a previous offering of this course (incomplete and unedited)
- [RY]
- Week 8, Mar 7-11
- Introduction to property testing; Testing linearity - the BLR test; Fourier analysis of Boolean functions.
- Videos: Mar 7, Mar 9
- References:
- Chapter 1 - Analysis of Boolean Functions by O’Donnell
- Week 9, Mar 14-18
- PAC learning and Fourier concentration - decision trees; Dictatorship testing.
- References:
- Chapters 2, 3, 7 - Analysis of Boolean Functions by O’Donnell
- Week 10, Mar 21-25
- Testing monotonicity of Boolean functions; Lower bounds for monotonicity testing; Graph property testing in the dense model - biclique and bipartiteness.
- References:
- Servedio’s lecture notes
- Chapter 8 - Introduction to Property Testing by Goldreich
- Week 11, Mar 28-Apr 1
- Canonical testers for graph properties; Triangle-freeness testing; Szemeredi regularity lemma and characterization of constant-query testable properties; Testing properties of sparse graphs
- References
- Chapter 8,9 - Introduction to Property Testing by Goldreich
- Chapter 17 - The Probabilistic Method by Alon, Spencer
- Lecture notes on Regularity lemma and graph removal lemmas by Zhao
- Three Theorems Regarding Testing Graph Properties by Goldreich, Trevisan
- Week 12, Apr 4-8
- Connectivity testing; estimating the number of connected components; Cycle-freeness - one-sided and two-sided testers; One-sided tester for bipartiteness; Lower bound for bipartiteness testing
- References
- Chapter 9 - Introduction to Property Testing by Goldreich
- Property Testing in Bounded Degree Graphs - Goldreich and Ron
- Assadi’s lecture notes
- Additional reading:
- A Sublinear Bipartiteness Tester for Bounded Degree Graphs - Goldreich and Ron
- Week 13, Apr 11-15
- Partition oracles and applications - testing minor-closed properties
- References
- Chapter 9 - Introduction to Property Testing by Goldreich
- Local Graph Partitions for Approximation and Testing - Hassidim, Kelner, Nguyen and Onak