CS 761: Randomized Algorithms (Winter 2025)
CS 761: Randomized Algorithms (Winter 2025) |
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| Lectures | Tuesdays and Thursdays, 2:30 PM - 3:50PM in DC 2568 |
| Instructor | Sepehr Assadi (userid: sassadi) |
| Instructor office hours | Mondays, 5 PM - 6 PM in DC 2334 |
| Textbook | There is no official textbook. Lecture notes and other required materials will be posted on this webpage. |
| Course outline | The course outline is available here. This page contains the highlights of course outline that can be updated as the term progresses. |
| Communication | The lectures will be delivered on the blackboard. For any questions, concerns, or comments regarding your experience in the course, email me (sassadi); just please prefix your email subject with [CS761-W25] for a timely reply. |
This is an advanced graduate algorithm course on the topic of Randomized Algorithms in particular and Theoretical Computer Science (TCS) in general.
The following is a tentative list of the topics that we will cover in this course (not necessarily in this particular order):
| • Probabilistic analysis: | Randomized algorithms, Concentration inequalities, Balls-and-bins experiments |
| • Minimum cuts: | Karger's algorithm, Cut sparsifiers, Vertex connectivity via maximum flow |
| • Minimum spanning trees (MSTs): | Linear-time randomized algorithm |
| • Sample and prune: | Luby's Distributed MIS algorithm, Maximal matching, Approximate matching |
| • Hash functions: | Basics, Limited independence hash functions, Cuckoo hashing |
| • Shortest paths: | Low diameter decompositions, Tree embeddings |
| • Graph coloring: | (Delta+1) coloring, Sampling colorings |
| • Multiplicative weight update (MWU): | Fast packing/covering linear programs |
| • Lovasz Local Lemma (LLL): | Basics, Algorithmic LLL |
| • Random walks: | Connectivity in RL, Perfect matchings in regular bipartite graphs, Markov Chains |
| • Information theory: | Basics, Communication complexity |
| • Sublinear time algorithms: | Connected components, Subgraph counting, Graph coloring |
| • Streaming algorithms: | Distinct element problem, Approximate matching, Graph sketches |
| • Online algorithms: | Ski rental, Karp-Vazirani-Vazirani (KVV) algorithm for bipartite matching |
| • Parallel algorithms: | Matching in RNC, Isolation lemma |
Official course description: Introduction to the design and analysis of algorithms that make use of randomization. Topics include review of basic probabiloity and introduction to randomized algorithms; game theoretic techniques; uses of Markov and Chebyshev inequalities; tail inequalities; Markov chains and random walks; algebraic techniques; data structures and graph algorithms.
I am hoping that by the end of the course, (1) you have a good understanding of randomness and its role in computation and algorithm design, (2) you are able to design and analyze algorithms for a wide range of canonical problems studied in this course, and even more importantly, for new problems that you may have not encountered directly in this course, and (3) you know a wide range of algorithmic and probabilistic tools commonly used in design and analysis of randomized algorithms.
Participation in lectures is not mandatory but I strongly encourage it -- this is a fast-moving course, so if you miss a lecture you may find it more difficult to keep up.
The course outline contains more details on the grading scheme of this course and collaboration policies and groups for the assignments.| # | Date | Topics | Lecture notes & References | Assignments |
|---|---|---|---|---|
| 1 | Tuesday January 7 |
Introduction, Probabilistic analysis, Fast (Delta+1) vertex coloring | [lec1] ACK19, AY25 | HW0 release: [HW0] |
| 2 | Tuesday January 9 |
Karger's minimum cut algorithm, Randomized maximum cut, Derandomization | [lec2], K93 | |
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| * | Tuesday January 14 |
No class: a make-up class will be arranged later in the term | ||
| * | Thursday January 16 |
No class: a make-up class will be arranged later in the term | ||
| 3 | Tuesday January 21 |
Balls and bins: Concentration inequalities | ||
| 4 | Thursday January 23 |
Balls and bins: Power of two choices via random graph theory | HW1 release | |
| 5 | Tuesday January 28 |
Communication complexity: Equality testing, Error correcting codes | ||
| 6 | Thursday January 30 |
Communication complexity: Information theory tools | ||
| 7 | Tuesday February 4 |
Lovász Local Lemma (LLL): Motivation and applications | ||
| 8 | Thursday February 6 |
Algorithmic LLL: k-satisfiability, Entropy compression | HW1 due | |
| 9 | Tuesday February 11 |
Probabilistic method: Basics, Frugal coloring | ||
| 10 | Thursday February 13 |
Probabilistic method: Coloring graphs with sparse neighborhood | ||
| * | Tuesday February 18 |
No class: Reading week | ||
| * | Thursday February 20 |
No class: Reading week | ||
| 11 | Tuesday February 25 |
Dimensionality reduction: Johnson-Lindenstrauss (JL) theorem | ||
| 12 | Thursday February 27 |
Dimensionality reduction: Subspace embedding, eps-Nets, Linear regression | HW2 release | |
| 13 | Tuesday March 3 |
Polynomial methods: Schwartz–Zippel lemma, Polynomial identity testing | ||
| 14 | Thursday March 5 |
Polynomial methods: Parallel matching, Isolation lemma | ||
| 15 | Tuesday March 10 |
Graph Algorithms: Shortest paths, Tree embeddings, Low diameter decomposition | ||
| 16 | Thursday March 12 |
Graph Algorithms: Negative-weight shortest paths | HW2 due | |
| 17 | Tuesday March 17 |
Hashing: Basics, Various types | ||
| 18 | Thursday March 19 |
Hashing: MinHash, Bloom filters | HW3 release | |
| 19 | Tuesday March 24 |
Streaming algorithms: Distinct Elements | ||
| 20 | Thursday March 26 |
Streaming algorithms: AMS Sketches | ||
| 21 | Tuesday March 31 |
Graph streaming algorithms: Spanners | ||
| 22 | Thursday April 2 |
Graph streaming algorithms: Approximate matching | HW3 due | |
| 23 | * * |
Random walks: Basics | ||
| 24 | * * |
Random walks: sampling |
There is no official textbook for the course (most of our material is not in textbooks yet anyway). The following resources are all optional but they can be quite helpful and I encourage you to refer to them throughout the term for more details on the topics we cover in the class.
For some related textbooks and surveys, you can refer to:
| H21a | Nick Harvey, A First Course in Randomized Algorithms. © Copyright 2021 Nick Harvey. |
| H21b | Nick Harvey, A Second Course in Randomized Algorithms. © Copyright 2021 Nick Harvey. |
| MR13 | Rajeev Motwani and Prabhakar Raghavan, Randomized Algorithms. Cambridge University Press, 2013. |
| DP09 | Devdatt P. Dubhashi and Alessandro Panconesi, Concentration of Measure for the Analysis of Randomised Algorithms. Cambridge University Press, 2009. |
| AS16 | Noga Alon and Joel H. Spencer, The Probabilistic Method, 4th Edition. Wiley, 2016. |
| E19 | Jeff Erickson, Algorithms textbook. © Copyright 2019 Jeff Erickson. |
This is a (rather incomprehensive) list of the papers related to the topics discussed in the lectures. The list will be updated frequently throughout the term.
| ACK19 | Sepehr Assadi, Yu Chen, Sanjeev Khanna, Sublinear Algorithms for (Δ + 1) Vertex Coloring. SODA 2019. |
| AY25 | Sepehr Assadi, Helia Yazdanyar, Simple Sublinear Algorithms for (Δ + 1) Vertex Coloring via Asymmetric Palette Sparsification. SOSA 2025. |
| K93 | David R. Karger, Global Min-cuts in RNC, and Other Ramifications of a Simple Min-Cut Algorithm. SODA 1993. |
This is a list of some other closely related courses taught elsewhere:
You can download LaTeX for free. For the purpose of this course, you do not even need to install LaTeX and can instead use an online LaTeX editor such as Overleaf.
Two great introductory resources for LaTeX are Getting started with TeX, LaTeX, and friends by Allin Cottrell (for general purpose LaTeX) and LaTeX for Undergraduates by Jim Hefferon (for undergraduates mathematics) accompanied by the following cheatsheet. You can also use this wonderful tool Detexify by Daniel Kirsch for finding the LaTeX commands of a symbol (just draw the symbol!).
If you are interested in learning more about LaTeX (beyond what is needed for this course), check the Wikibook on LaTeX and the Wikibook on LaTeX for Mathematics.
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Email is fast and tends to give the illusion of constant and immediate access. But that cannot realistically happen. I am available to you in general during this course via email but it may take up 24 hours (48 hours over the weekends) before you hear back from me. Also, please use email only to communicate specific concerns, questions, or comments regarding your experience in the course, or setting up appointments for in-person meetings outside office hours in case necessary. Questions directly related to the materials of the course and clarifications about lectures should be posted on Piazza or discussed during office hours instead.
Important note: Please start the subject of all your emails in this course with '[CS761-W25]' (like "[CS761-W25] discuss my final grade") -- I am using bunch of different email filters to manage my (unmanageable) inbox and I may not get to see emails without this format at all or there is no guarantee on response time for such emails.
You are encouraged to discuss with me any appropriate accommodations that we might make on your behalf following the guidelines of the AccessAbility Services.
AccessAbility Services, located in Needles Hall, Room 1401, collaborates with all academic departments/schools to arrange appropriate accommodations for students with disabilities without compromising the academic integrity of the curriculum. If you require academic accommodations to lessen the impact of your disability, please register with AccessAbility Services at the beginning of each academic term.
This is a challenging course with various advanced topics and not-so-easy assignments and exams. All of these are going to be time and energy consuming. But, this is also an elective course so one of my main goals is for you to enjoy taking this course and learning these cool materials as much as I enjoy teaching them. Part of making sure you have fun involves taking care of yourself. Do your best to maintain a healthy lifestyle and work-life balance this term by eating well, exercising, getting enough sleep, and taking some time to relax -- all of these will tremendously help you to achieve your goals in the course and to enjoy the process in the meantime.
You can find more resources to help you with your health & well-being with the Campus Wellness and Student Success Office -- they have tons of resources on helping you to succeed, including very good tips on time management techniques.
Faculty of Math Statement on Mental Health: The Faculty of Math encourages students to seek out mental health support if needed:
On-campus Resources:Every member of this class---instructor, TA, and students---has rights and responsibilities toward having a pleasant, fair, supportive, and free of discrimination and micro-aggression environment in this course, and we are all answerable to the University policies governing ethical behaviour (Policy 33).
In addition, academic dishonesty and plagiarism is considered a serious offense in this course. I expect that any assignment or exam you submit in this course will be your own product and follows the collaboration and external resources policies specified in the course outline. If an assignment is too hard, start earlier, ask for help, or simply do not answer the question --- academic dishonesty is never the right answer. If you have any concerns or questions about these policies, please discuss them with me.
Finally, a reminder that all the course content including lecture notes, presentations, and other materials prepared for the course, are the intellectual property (IP) of the instructor. These course materials are available to you to enhance your educational experience, and sharing them without permission and proper citation is a violation of intellectual property rights.
University Policies: It is your job to know the university policies that govern your behaviour in this course. Some pointers are:Course materials and the intellectual property contained therein, are used to enhance a student’s educational experience. However, sharing this intellectual property without the intellectual property owner’s permission is a violation of intellectual property rights. For this reason, it is necessary to ask the instructor, TA and/or the University of Waterloo for permission before uploading and sharing the intellectual property of others online (e.g., to an online repository).
Permission from an instructor, TA or the University is also necessary before sharing the intellectual property of others from completed courses with students taking the same/similar courses in subsequent terms/years. In many cases, instructors might be happy to allow distribution of certain materials. However, doing so without expressed permission is considered a violation of intellectual property rights.
Please alert the instructor if you become aware of intellectual property belonging to others (past or present) circulating, either through the student body or online. The intellectual property rights owner deserves to know (and may have already given their consent).
Use of Generative Artificial Intelligence: The following statement is prepared by the Office of Academic Integrity with input from the Centre for Teaching Excellence, Library, and consultations with Associate Deans and members of the Standing Committee on New Technologies, Pedagogy, and Academic Integrity (Last Updated: August 2023):Generative artificial intelligence (GenAI) trained using large language models (LLM) or other methods to produce text, images, music, or code, like Chat GPT, DALL-E, or GitHub CoPilot, may be used for assignments in this class with proper documentation, citation, and acknowledgement. Recommendations for how to cite GenAI in student work at the University of Waterloo may be found through the Library.
Please be aware that generative AI is known to falsify references to other work and may fabricate facts and inaccurately express ideas. GenAI generates content based on the input of other human authors and may therefore contain inaccuracies or reflect biases. In addition, you should be aware that the legal/copyright status of generative AI inputs and outputs is unclear. Exercise caution when using large portions of content from AI sources, especially images. More information is available from the Copyright Advisory Committee.
You are accountable for the content and accuracy of all work you submit in this class, including any supported by generative AI.
Created & maintained by Sepehr Assadi