Probability and Algorithms, Caltech CS150, Winter 2004

Leonard J. Schulman

Office hours by appointment. Feel free to email me.
Lectures: MW 10:30-12:00. New location: Steele 125 (instead of Jorgensen 287).

TAs:

Xiaojie Gao. Office hours: Thursdays 9pm-10:30pm in Jorgensen 160H.
Piyush Prakash. Office hours: Mondays 4:30pm-6pm in the Intel Lab, Jorgensen 154.

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:

Adams and Guillemin, Measure theory and probability

Motwani & Raghavan, Randomized Algorithms

Alon & Spencer, The Probabilistic Method

Feller, Probability Theory

Cover & Thomas, Information Theory

Grimmett & Stirzaker, Probability and Random Processes. This is also the source of the following two quotes. The first says something instructive about probability: "To understand the theory of chance thoroughly, requires a great knowledge of numbers, and a pretty competent one of Algebra." (John Arbuthnot, An essay on the usefulness of mathematical learning, 25 November 1700.) The second quote is taken totally out of context and provides dubious instruction about probabilists: "Besides gambling, many probabilists have been interested in reproduction." (Grimmett & Stirzaker p. 171.)

Artin, Algebra

Herstein, Topics in Algebra

Lectures:

Lecture 1 (January 5): Deviation bounds. (Typos corrected, January 7.)
Lecture 2 (January 7): Deviation bounds. Johnson-Lindenstrauss.
Lecture 3 (January 21): Linear error-correcting codes, k-wise independent sample spaces.
Lecture 4 (January 26): Upper and lower bounds on the size of k-wise independent sample spaces.
Lecture 5 (January 28): 4-wise independence and L_2 to L_p embedding.
Lecture 6 (February 2): 2-wise independence: hash functions, max-cut, Fredman-Komlos-Szemeredi. P,RP,BPP,NP.
Lecture 7 (February 4): "Uniquely-solving" NP in RP is enough to imply NP=RP. BPP vs. PH.
Lecture 8 (February 9): BPP is in Sigma_2 intersect Pi_2.
Lecture 9 (February 11): Deterministic amplification using Nisan's pseudorandom number generator.
Lecture 10 (February 18): Randomized and distributional complexity (minimax).
Lecture 11 (February 23): Game tree evaluation. Algebraic methods: Frievalds. (Not yet posted.)
Lecture 12 (February 25): Algebraic methods: Perfect matching. (Not yet posted.)
Lecture 13 (March 1): Algebraic methods, minimax: Associativity testing. (Not yet posted.)
Lecture 14 (March 3): Algebraic methods: finding a perfect matching (MVV). (Not yet posted.)
Lecture 15 (March 8): Lovasz local lemma. Applications: lower bound on van der Waerden function; existence of a path of length divisible by k in a digraph.
Lecture 16 (March 10): Learning theory: PAC model, VC dimension.

Assignments:

Problem set 1, due Friday January 23. There was a typo in problem 1 which has now been fixed. The due date is deferred to Monday January 26.
Problem set 2, due Friday February 13.
Problem set 3, due Friday February 27. There was a mistake in problem 2b which has now been fixed.
Problem set 4, due Monday March 15.

Announcements:

On January 12 and 14 I'll be out of town and class will be cancelled.

Links:

The page for the previous offering of this class is here.