COMPSCI 721 : Randomised Algorithms and Probabilistic Methods

Science

2025 Semester One (1253) (15 POINTS)

Course Prescription

Randomised algorithms are algorithms that “flip coins” to make decisions. In many cases, such algorithms are faster, simpler, or more elegant than the classical, deterministic ones. Covers basic principles and techniques used to design and analyse randomised algorithms, and applications of randomised methods in mathematics and computer science. Recommended preparation: STATS 125, COMPSCI 225 or MATHS 254, COMPSCI 320

Course Overview

Randomness is at core of many areas of computer science, from complexity theory and design of efficient algorithms to distributed computing and machine learning. In this course we learn the basics of probabilistic analysis by studying two closely related topics: randomised algorithms and probabilistic method.

Use of a “coin flip” to make decisions makes it possible for algorithms to avoid worst-case scenarios. Consequently, if one is willing to accept a small probability of an error, such randomised algorithms are often faster, simpler, or simply more elegant than deterministic ones. Examples include Welzl’s algorithm for finding a smallest enclosing ball, Karger’s min-cut algorithm, Luby’s algorithm for maximal independent set in a distributed setting, and colour-coding paradigm for finding long paths.

Rather than being solely a concept of study of probability theory, randomness has also become a core tool in many other areas of mathematics. Use of “coin flips” to construct graphs with complex properties or to unveil patterns constitutes a technique known as the probabilistic method. We cover examples such as constructions of graphs with large girth and large chromatic number, and constructions of the so-called expander graphs, one of the central objects in theoretical computer science.

We aim to cover a range of tools and techniques used in probabilistic analysis through study of randomised algorithms, applications of linearity of expectation in analysis of randomised algorithms and probabilistic method, basic concentration inequalities such as Markov’s, Chebyshev’s, and Chernoff’s inequality, as well as more advanced ones such as inequalities of Azuma and Talagrand, analysis of hashing through balls-into-bins problem, compression argument used in analysis of Moser’s LLL algorithm, random walks on graphs and Markov chains, combinatorial properties of random structures and probabilistic construction of expanders, applications of expanders in theoretical computer science, and so on.

This is a fast paced theoretical course with no programming assignments! The emphasis is on mathematically rigorous analysis and proofs. Previous experience in probability and discrete mathematics is necessary. Experience in design of algorithms is recommended.

To help you decide whether this course would be interesting for you, consider the following two problems:

1. Design efficient (randomised) algorithm which finds a path of length log(n) in a graph G.

2. Construct a graph with n vertices such that there is no subset of vertices of size log(n)/2 with the following property: either all vertices are connected by an edge, or none is.

In both problems, flipping a coin is crucial! Don’t worry if you do not manage to solve them – these are difficult problems with surprisingly elegant solutions that we cover in the first week.

Course Requirements

No pre-requisites or restrictions

Capabilities Developed in this Course

Capability 3: Knowledge and Practice
Capability 4: Critical Thinking
Capability 5: Solution Seeking
Capability 7: Collaboration
Graduate Profile: Master of Science

Learning Outcomes

By the end of this course, students will be able to:
  1. Develop and analyse randomized algorithm (Capability 3, 4 and 5)
  2. Estimate expected running time of an algorithm (Capability 3, 4 and 5)
  3. Show that algorithms are correct “with high probability” (Capability 3, 4 and 5)
  4. Apply probabilistic method to construct discrete structures with complex properties. (Capability 3, 4 and 5)
  5. Identify important combinatorial properties of random discrete structures (Capability 3, 4 and 5)
  6. Understand latest research papers presented at prime conferences dedicated to theoretical computer science, such as FOCS, STOC, and SODA. (Capability 3, 4, 5 and 7)

Assessments

Assessment Type Percentage Classification
Assignments 25% Individual Coursework
Test 25% Individual Test
Final Exam 50% Individual Examination
Assessment Type Learning Outcome Addressed
1 2 3 4 5 6
Assignments
Test
Final Exam

Key Topics

  • Randomised algorithms
  • Concentration inequalities and tail bounds for discrete random variables
  • Probabilistic method
  • Random walks and Markov chains
  • Expander graphs

Special Requirements

No special requirements.

Workload Expectations

This course is a standard 15 point course and students are expected to spend 10 hours per week involved in each 15 point course that they are enrolled in.

For this course, you can expect 3 hours of lectures, 2 hours of reading and thinking about the content and 5 hours of work on assignments and/or test preparation.

Delivery Mode

Campus Experience

Lectures will be available as recordings, but attending in person is strongly encouraged. 
Attendance on campus is required for the test and exam.

Learning Resources

Course materials are made available in a learning and collaboration tool called Canvas which also includes reading lists and lecture recordings (where available).

Please remember that the recording of any class on a personal device requires the permission of the instructor.

Lecture notes covering all the material of the crouse will be provided. Additional literature:
  1. “The Probabilistic Method”, Joel Spencer and Noga Alon
  2. “Randomized Algorithms”, Rajeev Motwani and Prabhakar Raghavan
  3. “Probability and Computing”, Michael Mitzenmacher and Eli Upfal

Similar courses at other universities:

  •  Randomized Algorithms and Probabilistic Methods, ETH Zurich https://cadmo.ethz.ch/education/lectures/HS22/RandAlg/index.html
  •  Randomized Algorithms and Probabilistic Analysis, Stanford University https://web.stanford.edu/class/cs265/
  •  Randomized Algorithms, MIT https://courses.csail.mit.edu/6.856/current/
  • Randomized Algorithms and Probabilistic Analysis, University of Washington https://courses.cs.washington.edu/courses/cse525/21wi/

Student Feedback

During the course Class Representatives in each class can take feedback to the staff responsible for the course and staff-student consultative committees.

At the end of the course students will be invited to give feedback on the course and teaching through a tool called SET or Qualtrics. The lecturers and course co-ordinators will consider all feedback.

Your feedback helps to improve the course and its delivery for all students.

During the course, Class Representatives can take feedback to the staff responsible for the course and staff-student consultative committees.

At the end of the course students will be invited to give feedback on the course and teaching through a tool called SET. The lecturers and course co-ordinators will consider all feedback.

Your feedback helps to improve the course and its delivery for all students.

Academic Integrity

The University of Auckland will not tolerate cheating, or assisting others to cheat, and views cheating in coursework, tests and examinations as a serious academic offence. The work that a student submits for grading must be the student's own work, reflecting their learning. Where work from other sources is used, it must be properly acknowledged and referenced. A student's assessed work may be reviewed against electronic source material using computerised detection mechanisms. Upon reasonable request, students may be required to provide an electronic version of their work for computerised review.

Class Representatives

Class representatives are students tasked with representing student issues to departments, faculties, and the wider university. If you have a complaint about this course, please contact your class rep who will know how to raise it in the right channels. See your departmental noticeboard for contact details for your class reps.

Copyright

The content and delivery of content in this course are protected by copyright. Material belonging to others may have been used in this course and copied by and solely for the educational purposes of the University under license.

You may copy the course content for the purposes of private study or research, but you may not upload onto any third party site, make a further copy or sell, alter or further reproduce or distribute any part of the course content to another person.

Inclusive Learning

All students are asked to discuss any impairment related requirements privately, face to face and/or in written form with the course coordinator, lecturer or tutor.

Student Disability Services also provides support for students with a wide range of impairments, both visible and invisible, to succeed and excel at the University. For more information and contact details, please visit the Student Disability Services’ website http://disability.auckland.ac.nz

Special Circumstances

If your ability to complete assessed coursework is affected by illness or other personal circumstances outside of your control, contact a member of teaching staff as soon as possible before the assessment is due.

If your personal circumstances significantly affect your performance, or preparation, for an exam or eligible written test, refer to the University’s aegrotat or compassionate consideration page https://www.auckland.ac.nz/en/students/academic-information/exams-and-final-results/during-exams/aegrotat-and-compassionate-consideration.html.

This should be done as soon as possible and no later than seven days after the affected test or exam date.

Learning Continuity

In the event of an unexpected disruption we undertake to maintain the continuity and standard of teaching and learning in all your courses throughout the year. If there are unexpected disruptions the University has contingency plans to ensure that access to your course continues and your assessment is fair, and not compromised. Some adjustments may need to be made in emergencies. You will be kept fully informed by your course co-ordinator, and if disruption occurs you should refer to the University Website for information about how to proceed.

The delivery mode may change depending on COVID restrictions. Any changes will be communicated through Canvas.

Student Charter and Responsibilities

The Student Charter assumes and acknowledges that students are active participants in the learning process and that they have responsibilities to the institution and the international community of scholars. The University expects that students will act at all times in a way that demonstrates respect for the rights of other students and staff so that the learning environment is both safe and productive. For further information visit Student Charter https://www.auckland.ac.nz/en/students/forms-policies-and-guidelines/student-policies-and-guidelines/student-charter.html.

Disclaimer

Elements of this outline may be subject to change. The latest information about the course will be available for enrolled students in Canvas.

In this course you may be asked to submit your coursework assessments digitally. The University reserves the right to conduct scheduled tests and examinations for this course online or through the use of computers or other electronic devices. Where tests or examinations are conducted online remote invigilation arrangements may be used. The final decision on the completion mode for a test or examination, and remote invigilation arrangements where applicable, will be advised to students at least 10 days prior to the scheduled date of the assessment, or in the case of an examination when the examination timetable is published.

Published on 02/11/2024 08:23 a.m.