CS6170 Randomized Algorithms

Aug-Nov 2021

The lecture videos and notes written during the lectures are available in the links below.


Lecture 27, Oct 14
FPRAS for #SAT (contd.); Connection between sampling and counting: FPAUS implies FPRAS.
  • References: [MU] - Chapter 11
Lecture 26, Oct 13
FPRAS for #DNF; FPRAS for #SAT with black-box access to SAT.
  • References: [MU] - Chapter 11
Lecture 25, Oct 11
Monte-Carlo method; FPRAS for #DNF; Importance sampling.
  • References: [MU] - Chapter 11
Lecture 24, Oct 7
Random walks on expander; Probability amplification using expander random walks; Counting problems.
Lecture 23, Oct 6
Random walks on undirected graphs - s-t connectivity; convergence of random walks - connections with spectrum of a graph.
Lecture 22, Oct 4
Markov chains - more definitions, properties, stationary distributions; Random walks on undirected graphs.
  • References: [MU] - Chapter 7
Lecture 21, Sep 30
Randomized 3-SAT - extending the idea of the 2-SAT algorithm, Modifications and a better analysis - Schoening’s algorithm.
  • References: [MU] - Chapter 7
Lecture 20, Sep 29
Randomized 2-SAT, Markov chains - definition and basics, Randomized 3-SAT.
  • References: [MU] - Chapter 7
Lecture 19, Sep 27
Random-walk based algorithm for 2-SAT.
  • References: [MU] - Chapter 7
Tutorial, Sep 22
Discussion of Pset 2
Lecture 18, Sep 20
Cuckoo hashing - analysis (contd.) and bounds.
  • References: [MU] - Chapter 17
  • Additional reading - Cuckoo Hashing - Pagh and Rodler.
Lecture 17, Sep 16
Cuckoo hashing - bounds and analysis using random graphs.
  • References: [MU] - Chapter 17
Lecture 16, Sep 13
FKS hashing - construction and bounds; Hashing with open addressing - linear probing.
Lecture 15, Sep 9
Universal hash families - constructions, applications, bounds; Perfect hash families.
  • References: [MU] - Chapter 15
Lecture 14, Sep 8
Bloom filters - false positives; Negatively associated random variables and their properties - concentration bounds and application to analyzing the Bloom filter.
  • References: [MU] - Chapter 5; [DP] - Chapter 3.1
Lecture 13, Sep 6
Poisson approximation - Coupon collector problem.
  • References: [MU] - Chapter 5
Lecture 12, Sep 2
Balls and bins - Poisson approximation; lower bound on the maximum load.
  • References: [MU] - Chapter 5
Lecture 11, Sep 1
Balls and bins - birthday paradox, maximum load, Poisson approximation.
  • References: [MU] - Chapter 5
  • Additional reading: Tight bounds for balls and bins for different values of \(m\) and \(n\)
Tutorial, Aug 30
Discussion of Pset 1
Lecture 10, Aug 26
Probability amplification of one-sided error algorithms; Chernoff-Hoeffding bounds - outline of the proof and applications.
  • References: [MU] - Chapter 4, [DP] - Chapters 1, 2
Lecture 9, Aug 25
Chebyshev’s inequality - probability amplification of one-sided error algorithms; pairwise independent hash families.
  • References: Salil Vadhan’s notes - Section 3.5, [MU] - Chapter 3
Lecture 8, Aug 19
Analysis of the Coupon-collector problem; Markov’s inequality - analyis of randomized Quicksort.
  • References: [MU] - Chapters 2, 3
Lecture 7, Aug 16
Random variables - Bernoulli, Binomial, Geometric; Probability Mass functions; Coupon-collector problem.
  • References: [MU] - Chapter 2
Lecture 6, Aug 12
Randomized Maxcut - pairwise independent random bits and derandomization; Randomized Quicksort and its analysis.
  • References: [MU] - Chapter 2
Lecture 5, Aug 11
Randomized Maxcut - analysis of the algorithm, properties of the random variables.
  • References: [MU] - Chapter 2
Lecture 4, Aug 9
Mincut - Karger’s algorithm and analysis.
  • References: [MU] - Chapter 1
Lecture 3, Aug 5
Proof of DeMillo-Lipton-Schwartz-Zippel lemma - recall of basic probability definitions and concepts; Verifying matrix multiplication - Frievald’s algorithm.
  • References: [MU] - Chapter 1
Lecture 2, Aug 4
Polynomial Identity Testing - connections to perfect matching in graphs; statement of the DeMillo-Lipton-Schwartz-Zippel lemma.
  • References: [MU] - Chapter 1
Lecture 1, Aug 2
Introduction to the course; Toy example - verifying polynomial factorization; extending the idea to multivariate polynomials - Polynomial identity testing.
  • References: [MU] - Chapter 1