Randomness and Computation
Spring 2019
One of the most remarkable developments in Computer Science over the
past 30 years has been the realization that the ability of computers
to toss coins can lead to algorithms that are more efficient, conceptually
simpler and more elegant that their best known deterministic counterparts.
Randomization has by now become a ubiquitous tool in computation.
This course will survey several of the most widely used techniques in
this context, illustrating them with examples taken from algorithms,
random structures and combinatorics. Our goal is to provide a solid
background in the key ideas used in the design and analysis of randomized
algorithms and probabilistic processes.
Students taking this course should have already completed a good
Algorithms courses (with theoretical underpinnings), and have
excellent Maths.
 If you have not taken the
Algorithms
and Data Structures (ADS) course (or similar in another University),
you should take that first unless you are a very strong student.
 You can take a look at the
2016/17 exam paper and/or the 2017/18 exam paper.
 All prospective students should do the
Self
test I have created. When you have finished (it will take an hour and
half or maybe two) drop me a message asking for solutions. You
should be able to do about 70% of the questions without Googling
or needing help  say 10 of the 15 questions.
Course Information
 Instructor: Mary
Cryan
 Lectures: 11:1012:00, Tuesdays and Fridays, in
50 George Square, G.05.
 Course
catalogue entry
 Textbook: The required textbook for the course is ``Probability
and Computing: Randomized Algorithms and Probabilistic Analysis" by
Mitzenmacher and Upfal.
 Assessment: A written examination contributes 80% of the
final grade. The remaining 20% will be based on the second (pencil and
paper) homework exercise.
Tutorials
We will have 5 tutorials during the semester.
Lectures
Lecture recording will appear in Learn; also I have add
direct links to the lecture schedule).
 Lectures 14 (January 16, 19, 23, 26) Introduction. Elementary
examples: identity testing, verifying matrix multiplication, randomized
mincut and maxcut. (Chapter 1 of [MU]).
 lecture1.pdf,
lecture14.pdf, board photos (jpg)
1, 2,
3, 4 on
testing polynomial identity.
 lecture2.pdf,
lecture24.pdf on Matrix
Multiplication Verification.
 lecture3.pdf,
lecture34.pdf,
board photo (jpg) continuing Matrix
Multiplication verification, introducing Min Cut.
 lecture4.pdf,
lecture44.pdf,
mp4(slides+voice), analyzing Karger's
Min Cut; simple 2appx for Max Cut.
 Lectures 5 and 6 (January 30, February 2)
Discrete Probability
(random variables, independence, expectation, variance, moments).
Markov's inequality, Jensen's inequality, Chebyshev's inequality.
Application: Coupon collector's problem. (Sections 2.1, 2.2, 2.4
and 3.13.3 of [MU]).
 Lectures 7 and 8 (February 6, 9) Chernoff Bounds
and applications
(from Chapter 4 of [MU]):
 Lecture 9 (February 13) Balls in Bins
(from Chapter 5 of [MU]):
 Lectures 1012 (February 16 and thereafter)
``The Probabilistic Method, and the Lovasz Local Lemma"
(rearranged and typos corrected)
 Lectures 13 and 14 (March 27 and April 3)
Markov chain basics, application to 2SAT (first half Chapter 7)
 Lectures 1518: (untaught due to the strike)
Mixingtime of Markov chains, relationship of random sampling to
approximate counting.
Coursework
There will be two courseworks in total: the
first will be marked and (formative) feedback returned, but will
not contribute to the final mark. Deadlines will be:
Course Outline
Here is a rough outline of the course material:
 Introduction: Las Vegas and Monte Carlo algorithms
 Elementary Examples: checking identities, fingerprinting
 Moments, Deviations and Tail Inequalities
 Balls and Bins, Coupon Collecting, stable marriage, routing
 Randomization in Sequential Computation
 Data Structures, Graph Algorithms
 Randomization in Parallel and Distributed Computation
 Algebraic techniques, matching, sorting, independent sets
 Randomization in Online Computation
 Online model, adversary models, paging, kserver
 The Probabilistic Method
 Threshold phenomena in random graphs, Lovasz Local Lemma
 Random Walks and Markov Chains
 Hitting and cover times, Markov chain Monte Carlo, mixing times
Guidelines
Please read the following guidelines regarding coursework:

Academic Conduct Policy: Students are expected to adhere to
the academic conduct policy of the University; this policy can be found in full
here.

Late Coursework Policy: Please see here
for the late coursework policy of the School of Informatics.