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 MinCut
 Randomized Quicksort Analsysis
 MaxE3SAT
 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
 2coloring 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 4clique property
 Edgedisjoint paths
 kSAT


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,tconnectivity in logspace
 randomized algorithm for 2sat
 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, #PCompleteness)
 Approximate Counting
 "Equivalence" between Approximate Counting and
Uniform Sampling
Examples
 #DNF
 01Permanent
 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. 