Randomized algorithms and probabilistic analysis

Spring, 2019

T Th 11:30-12:50, SAV 166

Instructor: James R. Lee

Office hours: Tues, 3-4pm, CSE 640

Teaching assistant:

  • Farzam Ebrahimnejad (febrahim@cs). Wed, 3-4pm, CSE2 223

Course email list [archives]


Dropbox for Final Exam

Syllabus (approximate):

[Only a subset of topics are covered in a given offering.]
  • Discrete probability
    • The probabilistic method
    • Linearity of expectation
    • Variance and the 2nd moment method
    • Lovasz Local Lemma
    • Martingales
    • Chernoff and Hoeffding-Azuma inequalities
  • High-dimensional geometry and statistics
    • Concentration of measure
    • Dimension reduction
    • Locality sensitive hashing
    • Nearest-neighbor search
    • Restricted isometry and compressive sensing
  • Information and entropy
    • Measures of information
    • Max-entropy approximation
    • Sparsification via sampling
    • Multiplicative weights update and boosting
    • Online stochastic gradient descent
  • Markov chains and convergence to equilibrium
    • Random walks
    • Eigenvalues and rapid mixing
    • Expander graphs
    • Markov chain Monte Carlo
    • Approximate counting


Tue, Apr 02
First moments
Thu, Apr 04
Second moments
  • The $G_{n,p}$ model of random graphs
  • Chebyshev’s inequality
  • The second moment method
  • Unbiased estimators
Tue, Apr 09
More second moments
Thu, Apr 11
Chernoff bounds
  • Randomized rounding of LP solutions
  • Chernoff bounds
  • The method of Laplace transforms

  • See this paper for the matching integrality gap construction and an even stronger result, that the min-congestion disjoint-paths problem is NP-hard to apprixmate within any factor better than .
  • Concentration inequalities: A non-asymptotic theory of independence
Tue, Apr 16
Applications of Chernoff bounds
  • Balls in bins
  • Randomized quicksort
  • Negatively correlated random variables

  • Survey on uniform spanning trees (Pemantle)
    Here you can find the proof of negative correlation
  • Shayan’s class on spanning trees, electrical flows, and ATSP
    In the first eight lectures, you can find a discussion of the Thin Trees Conjecture, the use of large-deviation bounds for negatively dependent random variables, and a new approximation algorithm for the Asymmetric Traveling Salesman Problem (ATSP).
Thu, Apr 18
Martingales and Azuma's inequality
  • Martingales
  • Doob martingales
  • The vertex exposure martingale
  • Azuma’s inequality
Tue, Apr 23
Applications of martingales
  • Proof of Azuma’s inequality
  • Concentration of measure in product spaces
  • Chromatic number of random graphs
Thu, Apr 25
Random tree embeddings
  • Embeddings and distortion
  • Random partitions of metric spaces
  • Memoryless random variables
Tue, Apr 30
Bourgain's embedding
  • Embeddings into Hilbert spaces
  • Random zero-sets
Thu, May 02
Dimensionality reduction
  • Exponential moments for chi-squared random variables
  • Laplace transform trick for independent sub-Gaussian entries
Tue, May 07
Compressive sensing
  • Sparse recovery
  • The restricted isometry property
  • -minimization
Thu, May 09
Sums of independent random matrices
Tue, May 14
Spectral sparsification
Thu, May 16
Random walks and electrical networks
Tue, May 21
Hitting times and cover times
Thu, May 23
Markov chains and mixing times
Tue, May 28
Markov chains and mixing times (cont.)
Thu, May 30
Eigenvalues, expansion, and rapid mixing
Tue, Jun 04
NO CLASS [final exam]
Thu, Jun 06
NO CLASS [final exam]


You may discuss problems with your classmates, but when you write down the solutions, you should do so by yourself. You can use the internet and books for reference material but you must cite every source that you consulted (the name of the book or web page suffices). You should also cite any classmates with whom you discussed solutions.

All homework should be turned in electronically to the Dropbox by the due date for the assignment. Homework should either be typeset (greatly preferred) or written very neatly and scanned.

Late HW policy: I will allow one (slightly) late homework per person (e.g., to accomodate for a paper deadline). This must be arranged beforehand.

Note on Bonus problems: These problems are usually quite challenging. If you do some of them, I will be impressed. But they are in no way required (or even suggested) for obtaining a good grade in the course.