Randomized Algorithms (semester 2023A)

[Course description] [Announcements] [Lectures] [Assignments] [Reading material]

Course description

Instructors: Robert Krauthgamer and Moni Naor

Grader: TBA 

When and where: Monday 14:15-17:00, Ziskind Lecture Room 1 (no zoom/hybrid)

First class: Nov. 7, 2022 

FGS code: 20234051

Syllabus: Algorithms that are randomized, i.e., use coin tosses during their execution, are central in many computational scenarios, and have become a key tool in Theoretical Computer Science and in application areas. The course will cover the development and use of such probabilistic techniques, including some of the following topics: Concentration results; Simulating randomness, in particular limited independence; Load balancing; Routing algorithms; Randomized rounding; Sampling and sublinear algorithms; Algorithms in the data stream model; Communication protocols and the sketching model.

Prerequisites: Students are expected to be familiar with algorithms, complexity theory, probability theory, and linear algebra, at an undergraduate level.

Final: The final assignment will be a take-home exam.

Webpage: http://www.wisdom.weizmann.ac.il/~robi/teaching/2023a-RandomizedAlgorithms

Previous offerings:
http://www.wisdom.weizmann.ac.il/~robi/teaching/2021a-RandomizedAlgorithms
http://www.wisdom.weizmann.ac.il/~robi/teaching/2019a-RandomizedAlgorithms
http://www.wisdom.weizmann.ac.il/~robi/teaching/2017a-RandomizedAlgorithms
http://www.wisdom.weizmann.ac.il/~robi/teaching/2015a-RandomizedAlgorithms
http://www.wisdom.weizmann.ac.il/~robi/teaching/2013a-RandomizedAlgorithms
http://www.wisdom.weizmann.ac.il/~robi/teaching/2011a-RandomizedAlgorithms

Announcements

• [posted 2023-01-31]: Take-home exam schedule: distributed Feb. 27, due within 3 days. The format/instructions will be similar to previous offerings of this course.
  
• [posted 2023-01-12]: Problem set 2 was corrected (in Q.#2, the absolute value was changed and also the hint was expanded)
• [posted 2023-01-10]: lecture notes #6 was slightly updated to fix typos.
  

Lectures

1. 2022-11-07, Moni: Minimum Cut via Randomized Contractions.
   Class material: lecture notes #1, see also [MR, chapters 1.1].
   
2. 2022-11-14, Moni: 2-SAT and 3-SAT.
   Class material: lecture notes #2,see also [MU, chapter 7].
   
3. 2022-11-21, Moni: Existential Proofs via Probabilistic Constructions, Large Deviation Bounds and Communication.
   Class material: lecture notes #3, see also [AS, appendix A] and [RY].
   
4. 2022-11-28, Robi: Dimension Reduction in l_2, JL and fast JL.
   Class material: lecture notes #4. 
   Further Reading: [Dasgupta and Gupta], [Jiri Matousek (lecture notes), chapter 2].
5. 2022-12-05, Robi: JL Transform and Oblivious Subspace Embedding.
   Class material: lecture notes #5.
   Further Reading: [Woodruff, section 2, also as arXiv:1411.4357].
   
6. 2022-12-12, Robi: Least Squares Regression and Probabilistic Embedding into Dominating Trees.
   Class material: lecture notes #6.
   Further Reading: [Fakcharoenphol, Rao, and Talwar].
7. 2022-12-19, Moni: Card Guessing, Hat Guessing and the Local Lemma.
   
8. 2022-12-26, Moni: Card Guessing, Hat Guessing and the Local Lemma (cont'd).
   Class material: lecture notes #8.
   
9. 2023-01-02, Moni: Cuckoo Hashing
   
10. 2023-01-09, Robi: Probabilistic Embedding into Dominating Trees (cont'd) and Importance Sampling.
    Class material: lecture notes #10, see also [MR, chapter 11.2].
    
11. 2023-01-16, Robi: Coresets for Clustering.
    Class material: lecture notes #11, see also [Munteanu and Schwiegelshohn, survey on coresets].
12. 2023-01-23, Moni: Bloom Filters, Random Graphs, and Limited Randomness and the Braverman's Result concerning AC0
    Class material: lecture notes #12.
13. 2023-01-30, Robi: Concentration Bounds and Edge-Sparsification of Hypergraphs
    Class material: lecture notes #13, see also [MR, chapter 4.1], [DP, chapter 1.6]
14. 2023-02-06, Robi: Graph Laplacians and Spectral Sparsification
    Class material: lecture notes #14. 

Assignments

• Problem set 1 (posted Dec. 5)
  
• Problem set 2 (posted Jan. 2)
• Problem set 3 (posted Jan. 31) 

Reading material

Relevant textbooks:

• [MR] Randomized Algorithms by Motwani and Raghavan (errata) -- eBook available inside Weizmann.
  
• [MU] Probability and Computing: Randomized Algorithms and Probabilistic Analysis by Mitzenmacher and Upfal (errata).
• [DP] Concentration of Measure for the Analysis of Randomized Algorithms by Dubhashi and Panconesi (errata) -- eBook available inside Weizmann.
• [AS] The Probabilistic Method by Alon and Spencer (4th Edition) -- eBook available inside Weizmann (2nd Edition).
• [RY] Communication Complexity and Applications by Rao and Yehudayoff -- eBook available inside Weizmann.
  

Similar courses (many with lecture notes):

• Randomized Algorithms by Nicholas Harvey at UBC (2015)
  
• Randomized Algorithms by Sariel Har-Peled at UIUC (2014)
• Randomized Algorithms and Probabilistic Analysis by James R. Lee at UW (2016)
• Randomized Algorithms by Avrim Blum and Anupam Gupta at CMU (2011)
  
• Randomized Algorithms by David Karger at MIT (2015)

Useful references:

• Concentration by Colin McDiarmid. Probabilistic Methods for Algorithmic Discrete Mathematics, 1998, 1-46.
• Probability and Random Processes (2nd ed.) by Grimmett and Stirzaker.