References
We will be teaching materials from multiple books/sources. Some of them are the following.
- [M-U] Michael Mitzenmacher and Eli Upfal. Probability and computing.
Cambridge university press, 2017.
- [MR] Rajeev Motwani, Prabhakar Raghavan. Randomized Algorithms, Cambridge university press.
- [RK] R.M. Karp, An introduction to randomized algorithms, Discrete Applied Mathematics, 34, pp. 165-201, 1991.
- [BHK] Avrim Blum, John Hopcroft, and Ravindran Kannan. Foundations of Data Science, 2020.
- [RV] Roman Vershynin, High-Dimensional Probability.
- [DP] D.B. Dubhashi, A. Panconesi, Concentration of Measure for the Analysis of Randomized Algorithms, Cambridge University Press, 2009.
- [LPW] David A. Levin, Yuval Peres, Elizabeth L. Wilmer. Markov Chains and Mixing Times.
- [MIT-YZ] Yufei Zhao. Lecture Notes (The Probabilistic Methods in Combinatorics), MIT, 2019.
- [A-S] Noga Alon and Joel Spencer, The probabilistic method, John Wiley & Sons, 2004.
- [UW-TR] Thomas Rothvoss, Lecture Notes (Probabilistic Combinatorics), U Washington, 2019.
- [AC] Amit Chakrabarti, Data Stream Algorithms, 2020.
- [SM] S. Muthukrishnan. Data streams: Algorithms and applications. Now Publishers Inc, 2005.
- Various surveys and lecture notes.
Similar courses elsewhere:
- IISc, 2016 , by Arnab Bhattacharyya and Deeparnab Chakrabarty.
- [JA-YALE] Yale, 2020 , by James Aspnes.
- [SH-UIUC] UIUC, 2018 , by Sariel Har-Peled.
- MIT, 2002 , by David Karger.
- UT Austin, 2020 , by Eric Price.
- UC berkeley, 2003 , by Luca Trevisan.
- Columbia, 2019 , by Tim Roughgarden.
- Stanford, 2020 , by Mary Wooters.
- CMU, 1997 , by Avrim Blum.
- Wiezmann, 2013 , by Robert Krauthgamer and Moni Naor.
- UMCP, 2017 , by Aravind Srinivasan.
- U Iowa, 2018 , by Sriram V. Pemmaraju.
- UBC, 2012 , by Nick Harvey.
- EPFL 2014 , by Friedrich Eisenbrand.
- NUS, 2019 , by Seth Gilbert.
- Duke, 2013 , by Kamesh Munagala.
- NTHU, 2012 , by Wing Kai Hon.
- U Waterloo, 2019, by Gautam Kamath.
- U Waterloo, 2018 , by Lap Chi Lau.
- U Washington, 2016 , by James Lee.
Fun links:
Assignments
Project Topics
For projects you need to select a project topic from the list given below.
Some reference papers are given below.
You are expected to do a survey of the results and techniques in the topic area.
You can form a group of two students and send the project topic and group details
by 20th April.
You need to submit a report by the last_day_of_class.
Finally, you need to make a presentation on the topic on the last week of classes (May 25 and May 27).
- Derandomization
- Sublinear time algorithms
- Sketching
- PCP
- Streaming Algorithms for Coin Tossing
- VC dimension
- Random Order Arrival (Online Algorithms)
- Stochastic analysis of bin packing
- Counting knapsack solutions
- Expanders
- Solving sparse linear systems
- Minimax Theorem
- Singular Value Decompostion (SVD)
- Vempala, Santosh, and Grant Wang. "A spectral algorithm for learning mixtures of distributions." IEEE FOCS 2002.
- The paper shows that a simple spectral algorithm learns mixtures of Gaussians with provable guarantees, and extends the results the mixtures of weakly isotropic distributions.
- k-means clustering
- Kumar, A., Sabharwal, Y., & Sen, S. "A simple linear time (1+ epsilon)-approximation algorithm for k-means clustering in any dimensions."" IEEE FOCS 2004.
- This paper gives the first linear time algorithm for k means, using random subsampling of point sets.
- Anti-concentration
- Lovett, Shachar. "An elementary proof of anti-concentration of polynomials in Gaussian variables." Electronic Colloquium on Computational Complexity (ECCC). Vol. 17. 2010.
- This gives us an elementary proof of anti-concentration for multilinear polynomials for a large class of distributions, which includes polynomials over Gaussian distributions.
- Approximate Matrix Multiplication
- Drineas, P., Kannan, R. and Mahoney, M.W., 2006. Fast Monte Carlo algorithms for matrices I: Approximating matrix multiplication. SIAM Journal on Computing, 36(1), pp.132-157.
- This paper gives fast approximation algorithms for matrix multiplication via matrix sub-sampling, with several applications.
- Nearest Neighbor via LSH and p-Stable distributions
- Indyk, P., Motwani, R., Raghavan, P., & Vempala, S. Locality-preserving hashing in multidimensional spaces. STOC 1997.
- This paper introduces locality preserving hashing which has several applications in geometric optimization problems in high dimensions. It gives explicit constructions and lower bounds for these hash families.
- Computing Gaussian Width
- Meka, R. A PTAS for computing the supremum of Gaussian processes. IEEE FOCS 2012.
- This gives a PTAS for computing the supremum of Gaussian processes.
- Optimality of JL Lemma
- Larsen, Kasper Green, and Jelani Nelson. "Optimality of the Johnson-Lindenstrauss lemma." IEEE FOCS 2017.
- This paper shows bounds on dimensions that are necessary for any approximate isometry preserving maps.
- PIT
Intended audience: Graduate students in computer science
and mathematics with theoretical interests.
Interested undergraduate students with very strong mathematical
skills.
Prerequisites: Mathematical maturity and a solid background in math
(elementary combinatorics, graph theory, discrete probability, algebra, calculus)
and theoretical computer science (big-O/Omega/Theta, P/NP, basic fundamental algorithms).
Grading: 40% HW, 30% Projects, 30% Final.
Prepared and Maintained by Arindam Khan