Computer Science and Engineering
SUNY at Buffalo

### CSE 694: Topics in Algorithms - Probabilistic Analysis and Randomized Algorithms

Instructor: Hung Q. Ngo

### Spring 2008

MWF, 2--2:50pm
Talbert 106 (map)

General Information | Schedule and Notes | Assignments | SyllabusHelpful Links
There will be (hopefully light) changes to the following schedule throughout the semester. Please check back regularly!
Week Topics Notable Events
1. Jan 14 A Gentle Introduction to Probability Theory, Probabilistic Method, and Randomized Algorithms
• Reading: Chapers 1 to 4

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

• Reading: Chaper 5

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

• Reading: Chaper 7

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.