CS6170 Randomized Algorithms

July - Nov 2021

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

Lectures
Scribbles and Notes (from Aug 9), from Sep 29

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.

References: Pseudorandomness by Vadhan - Sections 4.1, 4.2

Lecture 23, Oct 6: Random walks on undirected graphs - s-t connectivity; convergence of random walks - connections with spectrum of a graph.

References: [MU] - Chapter 7, Pseudorandomness by Vadhan - Section 2.4

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.

References: [MU] - Chapter 15, [MR] - Chapter 8.5
Additional reading: Knuth’s note on the analysis of 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