Probability and Algorithms, Caltech CS150, Winter 2003

Leonard J. Schulman

MW 10:30-12:00, Jorgensen 287
TA: Sidharth Jaggi

Catalog listing:
Elementary randomized algorithms and algebraic bounds in communication, hashing, and identity testing. Game tree evaluation. Topics may include randomized parallel computation; independence, k-wise independence and derandomization; rapidly mixing Markov chains; expander graphs and their applications; clustering algorithms.

Some useful books:
Motwani & Raghavan, Randomized Algorithms
Alon & Spencer, The Probabilistic Method
Feller, Probability Theory
Cover & Thomas, Information Theory

Template for scribe notes

On January 13 and 15 I'll be out of town and class will be cancelled. There will be a make-up class on Friday January 24 at 10:30.

Scribe notes:
  1. Lecture 1, January 6 (Cheng Jin): Course overview, communication complexity.
  2. Lecture 2, January 8 (Isaac See): Course overview, game tree evaluation.
  3. Lecture 3, January 22 (Sidharth Jaggi): #DNF approximation.
  4. Lecture 4, Janary 24 (Dylan Simon): Chernoff bound, FPRAS.
  5. Lecture 5, January 27 (Xiaojie Gao): Min cut, network reliability.
  6. Lecture 6 (part a / part b), January 29: 3SAT
  7. Lecture 7, February 3 (Geoffrey Irving): Approximating the permanent.
  8. Lecture 8, February 5 (Anthony Kirilusha): Approximate counting vs. approximate sampling.
  9. Lecture 9, February 10 (Rebecca Schulman): Markov Chain Monte Carlo. Mixing times.
  10. Lecture 10, February 12 (Frank Moradi): Markov Chain Monte Carlo. Strong stationary times, coupling.
  11. Lecture 11, February 19 (Robert Forster): Strong stationary time for riffle shuffle.
  12. Lecture 12, February 24 (Wonjin Jang): Sampling graph colorings.
  13. Lecture 13, February 26 (Sidharth Jaggi): Dimer models: sampling lozenge tilings.
  14. Lecture 14 (old notes / comments), March 3: Randomized and distributional complexity.
  15. Lecture 15, March 5 (Xiaojie Gao): Randomized and distributional complexity. Algebraic methods.
  16. Lecture 16, March 10: Algebraic methods: Testing for a perfect matching (Rabin-Vazirani); Verifying associativity (Rajagopalan-Schulman) including lower bound through distributional complexity.
  17. Lecture 17, March 12: Finding a perfect matching: Rabin-Vazirani and Mulmuley-Vazirani-Vazirani.
Problem sets:
  1. Problem set 1
  2. Problem set 2 (There was an error in problem 1b. I should have specified "at most k coupons each with probability at least p", and asked for a bound of O((1/p) log(1/p)).)
  3. Problem set 3; Additional question
Note: In problem set 3, problem 2b, a coupling argument actually shows the better mixing time of O(k log k).

"Anyone who considers arithmetical methods of producing random numbers is, of course, in a state of sin." -- John von Neumann