| Week |
Topics |
Notable Events |
| 1. Jan
14 |
A Gentle
Introduction to Probability Theory, Probabilistic Method, and
Randomized Algorithms
Concepts:
- Independence
- Random variables and Expectation, Linearity of
Expectation
- Conditional Probability and Expectation
- Basic distributions: Bernoulli, Binomial, Geometric
- Derandomization with the method of conditional
probability
- Moments
- Tail bounds: Markov, Chebyshev, Chernoff
Examples:
- Extremal
Combinatorics:
- Ramsey Numbers
- Sperner Lemma
- Randomized
Algorithms:
- Randomized Min-Cut
- Randomized Quicksort Analsysis
- Max-E3SAT
- Networking:
- Probabilistic Packet Marking
- Permuation routing on the Hypercube
- Coding Theory
and Group Testing:
- Expander code
- Disjunct matrices
|
Monday, Jan 14
- Homework 1 out. Elementary practice exercises on
discrete probabilities and the probabilistic method |
| 2. Jan
21 |
No class on Monday Jan 21. Martin Luther King
day.
|
| 3. Jan
28 |
|
| 4. Feb 04 |
Balls,
Bins, and Random Graphs
Concepts
- Balls into Bins Model
- Poisson Distribution and Approximation
Examples
- Hashing, Bloom Filter
- Random Graphs and some basic properties
|
Monday,
Feb 04
- Homework 1 due.
-
Homework 2 out. A more serious set of homework problems on the basic
probabilistic methods, applications of tail bounds
and conditional
probabilities and expectations.
|
| 5. Feb
11 |
The
Probabilistic Method
Concepts
- Union bound
- Argument from expectation
- Alteration
- Second moment method
- Lovasz local lemma
Examples
- Tournaments
- 2-coloring of uniform hypergraphs
- Finding large cuts, Max SAT
- Linear combinations of vectors
- Unbalancing lights
- Dominating sets
- Independent Sets
- Distinct subset sums
- The G(n,p) random graph model and 4-clique property
- Edge-disjoint paths
- k-SAT
|
|
| 6. Feb
18 |
|
| 7.
Feb 25 |
Discrete
Time Markov Chains
Concepts
- DTMC basics
- Classification of states
- Ergodic theorem
Examples
- Birth and death process
- Gambler ruin problem
- Random walks on Z and Z^2
- ...
|
Monday,
Feb 25
- Homework 2 due.
-
Homework 3 out. Some problems on Balls, Bins, and Random Graphs |
| 8. Mar
03 |
|
| 9.
Mar 10 |
No class this entire week. Spring
Recess! |
| 10. Mar
17 |
Random
Walks on Graphs and Expanders
Concepts
- Random walks on graphs
- Eigenvalue Connection
- Expanders
Examples
- randomized algorithm for s,t-connectivity in log-space
- randomized algorithm for 2-sat
- saving random bits
- ...
|
Monday,
Mar 17
- Homework 3 due.
-
Homework 4 out. Problems on DTMC and Random Walks
|
| 11. Mar
24 |
|
| 12. Mar
31 |
Monte
Carlo Method, Approximate Counting
- Reading: Chaper 9, Part of Chapter 11
- Lecture Notes in PDF (Almost Complete. Could be printed. 2 more examples will be updated).
Concepts
- Basic Monte Carlo Method
- Complexity of Counting (#P, #P-Completeness)
- Approximate Counting
- "Equivalence" between Approximate Counting and
Uniform Sampling
Examples
- #DNF
- 01-Permanent
- Network Reliability
- (May be) Volume Estimation
|
|
| 13. Apr 07 |
Monday,
Apr 07
- Homework 4 due.
-
Homework 5 out. Problems on the Monte Carlo Method and Approximate
Counting |
| 14. Apr 14 |
|
| 15. Apr 21 |
|
| 16. Apr 28 |
Student presentations to be arranged.
|
Monday Apr 28 is the last day of
classes. Homework 5 due. |