Tentative Schedule of Lectures

Week	Topic	Readings
Week 1: Jan 6 –12	Monte Carlo Algorithms: Global Min-Cut	From "Algorithm Design" by Kleinberg and Tardos: Min Cut
Week 2: Jan 13–19	Finish Min Cut Las Vegas Algorithms: Closest Pair of Points	Karger-Stein Algorithm From "Algorithm Design" by Kleinberg and Tardos: Closest Pair of Points Finding triangles
Week 3: Jan 20–Jan 26	Approximate Near Neighbours Fingerprinting for String Matching	Approximate Near Neighbour Search CLRS Section 32.2. Lecture Notes on Karp-Rabin
Week 4: Jan 27–Feb 2	Finish Approximate Near Neighbours	Approximate Near Neighbour Search Streaming Algorithms
Week 5: Feb 3–9	Streaming Algorithms	Variance and Chebyshev Worksheet Streaming Algorithms
Week 6: Feb 10–16	Finish Streaming Algorithns Random Walks and Markov Chains	Streaming Algorithms Section 3.3. of this book chapter cover Randomized Selection. Markov Chains You can also read the intro section of Chapter 4 of this book. Section 4.8 of the book covers PageRank. Linear Algebra Review Sheet
Feb 17–23	NO CLASS: Reading Week
Week 7: Feb 24–March 1	Random Walks and Markov Chains	Markov Chains Couplings Worksheet
Week 8: March 2–8	Linear Programming	Linear Programming
Week 9: March 9–15	LP Duality and Complementary Slackness	Linear Programming Examples of LP formulations.
Week 10: March 16–22	Matchings and the Hungarian Algorithm	Goemans's lecture notes. Slides
Week 11: March 23–March 29	Finish Matchings Deterministic and Randomized Rounding Algorithms	Goemans's matchings lecture notes. Matchings slides Sections 1.7, 5.1 of the Williamson and Shmoys book. All of Chapter 1 of the book is recommended. Rounding slides
Week 12: March 30–April 3	Derandomization and Chernoff bounds	Section 5.2, 5.10-5.12 of the Williamson and Shmoys book. Derandomization slides Chernoff bounds slides Chapter 4 of the Motwani-Raghavan book has more information on tail inequalities.

Suggested Exercises

In addition to the exercises below, always attempt the exercises in the posted readings and lecture notes above.

• Suggested exercises from Williamson and Shmoys: 1.4, 1.5, 5.1, 5.2, 5.3, 5.6, 5.7, 5.14
• Do the exercises from Goemans's matchings lecture notes. -->
• For Markov Chains, try exercises 4.1, 4.3, 4.4, 4.7 after Chapter 4 of this book.
• For Karp Rabin fingerprints, try the exercise after Lecture 7 of Jeff Erickson's notes, and the relevant problems in CLRS section 32.2.
• Suggested exercises for Min Cut:
  • From Motwani and Raghavan's Randomized Algorithms (available through the U of T library at http://go.utlib.ca/cat/8230181) try the exercises 10.13 - 10.15 after Chapter 10.
  • From Jeff Erickson's lecture notes, try the exercises after Lecture 13.
  • After doing exercise 1 from Erickson's notes, try to give an O(m) time algorithm to execute the contraction algorithm, without using the Klein-Karger-Tarjan MST algorithm.

Learning Objectives

By the end of this course, you should be able to:

• Distinguish between Monte Carlo and Las Vegas algorithms.
• Design and analyze randomized algorithms for basic problems in graph theory (e.g. minimum cut). In particular, you should have a deep understanding of Karger's Contraction algorithm, including its analysis and how to implement it efficiently.
• Design and analyze locality sensitive hash functions for different distance metrics, and use them for approximate near neighbour search and related problems.
• Use sampling to estimate sizes of sets, and apply sampling in streaming and other algorithms.
• Define basic concepts in Markov Chains: stationary distribution, irreducible and aperiodic Markov chains, time-reversible Markov chains. You should be able to determine when a distribution, or, respectively, a Markov chain satisfies these definitions.
• State the Fundamental Theorem of Markov Chains, and use coupling arguments to bound the mixing time of Markov chains.
• Use the Metropolis Hastings algorithm to design Markov chains with a given stationary distribution.
• Define basic terms in polyhedral geometry, like vertex, face, facet, polytope.
• Model optimization problems as linear programs, and derive the dual of a linear program.
• State the complementary slackness theorem, and understand the analysis of primal-dual algorithms, like the Hungarian algorithm.
• Design and analyze approximation algorithms via deterministic and randomized rounding of linear programming relaxations.
• Derandomize randomized approximation algorithms via the method of conditional expectations.
• State the Chernoff bound and Hoeffding's Inequality, and use them to analyze rounding and sampling algorithms.