User:Erel Segal/Probability and Computing 2006

An index to:

1. Events and Probability

 * 1) Application: random algorithm for Polynomial identity testing
 * 2) Axioms of probability
 * 3) Application: random algorithm for verifying matrix multiplication.
 * 4) Application: random algorithm for finding a minimum cut.

2. Discrete Random Variables and Expectation

 * 1) Random variables and expectation
 * 2) * Linearity of Expectations
 * 3) * Jensen's inequality
 * 4) The Bernouli random variable and Binomial random variable.
 * 5) Conditional expectation
 * 6) The Geometric distribution
 * 7) * Example: Coupon collector's problem
 * 8) Application: the expected run-time of quicksort

3. Moments and Deviations

 * 1) Markov's inequality
 * 2) Variance and moments of a random variable.
 * 3) * Example: variance of a binomial random variable
 * 4) Chebyshev's inequality
 * 5) * Example: Coupon collector's problem
 * 6) Application: random algorithm for computing median

4. Chernoff bounds

 * 1) Moment-generating functions
 * 2) Deriving and applying Chernoff bounds
 * 3) * Chernoff bound for the sum of Poisson trials
 * 4) * Example: coin flips
 * 5) * Application: parameter estimation
 * 6) Better bounds for some special cases
 * 7) Application: set balancing
 * 8) Application: Routing protocol for packet-routing in sparse networks.
 * 9) * Permutation routing on a Hypercube graph network
 * 10) * Permutation routing on a Butterfly graph network

5. Balls, bins and random graphs

 * 1) Example: the birthday paradox
 * 2) Balls into bins
 * 3) * The balls-and-bins model
 * 4) * Application: bucket sort
 * 5) The Poisson distribution
 * 6) * Limit of the binomial distribution
 * 7) The Poisson approximation
 * 8) * Example: Coupon collector's problem, revisited
 * 9) Application: Hash table
 * 10) * Chain hashing
 * 11) * Hashing: bit strings
 * 12) * Bloom filters
 * 13) * Breaking symmetry
 * 14) Random graphs
 * 15) * Random graph models
 * 16) * Application: Hamiltonian cycles in random graphs

6. The Probabilistic method

 * 1) The basic counting argument
 * 2) The expectation argument
 * 3) * Application: finding a large cut
 * 4) * Application: maximum satisfiability
 * 5) Derandomisation using conditional expectations
 * 6) Sample and Modify
 * 7) * Application: independent sets
 * 8) * Application: graphs with large girth
 * 9) The Second moment method
 * 10) * Application: threshold behaviour in random graphs
 * 11) The conditional expectation inequality
 * 12) The Lovász local lemma
 * 13) * Application: edge-disjoint paths
 * 14) * Application: Boolean satisfiability problem
 * 15) Explicit constructions using the Local Lemma
 * 16) * Application: a satisfiability algorithm
 * 17) Lovász local lemma: the general case

7. Markov chains and Random walks

 * 1) Markov chains: definitions and representation
 * 2) * Application: a randomized algorithm for 2-satisfiability
 * 3) * Application: a randomized algorithm for 3-satisfiability
 * 4) Classification of states
 * 5) * Example: the gambler's ruin
 * 6) Stationary distributions
 * 7) * Example: a simple queue
 * 8) Random walks on undirected graphs
 * 9) * Application: an s-t connectivity algorithm
 * 10) Parrondo's paradox

8. Continuous distributions and the Poisson process

 * 1) Continuous Random Variables
 * 2) * Probability distributions in R
 * 3) * Joint probability distributions and conditional probability
 * 4) The Continuous uniform distribution
 * 5) * Additional properties of the uniform distribution
 * 6) The Exponential distribution
 * 7) * Additional properties of the exponential distribution
 * 8) * Example: Balls into bins with feedback
 * 9) The Poisson point process
 * 10) * Interarrival distribution
 * 11) * Combining and splitting Poisson processes
 * 12) * Conditional arrival time distribution
 * 13) Continuous time Markov processes
 * 14) Example: Markovian Queues
 * 15) * M/M/1 queue in equilibrium
 * 16) * M/M/1/K queue in equilibrium
 * 17) * The number of customers in an M/M/∞ queue

9. Entropy, randomness and information

 * 1) The entropy function
 * 2) Entropy and binomial coefficients
 * 3) Entropy: a measure of randomness
 * 4) coding: Shannon's theorem

10. The Monte Carlo Method

 * 1) The Monte Carlo method; FPRAS (Fully Polynomial Randomized Approximation Scheme).
 * 2) Application: the DNF counting problem - counting the number of satisfying assignments to a DNF formula
 * 3) * The Naive approach - might be exponential-time.
 * 4) * A FPRAS for DNF counting - always polynomial-time.
 * 5) From approxiamte sampling to approximate counting
 * 6) The Markov chain Monte Carlo method
 * 7) * The Metropolis–Hastings algorithm

11. Coupling of Markov chains

 * 1) Variation distance and Mixing time
 * 2) Coupling
 * 3) * Example: Shuffling cards
 * 4) * Example: random walks on the hypercube graphs
 * 5) * Example: independent sets of fixed size
 * 6) Application: Variation distance is nonincreasing
 * 7) Geometric convergence
 * 8) Application: Approximately sampling proper colorings
 * 9) Path coupling

12. Martingales

 * 1) Martingales and Doob martingale
 * 2) Stopping time
 * 3) * Example: a ballot theorem
 * 4) Wald's equation
 * 5) Tail inequalities for martingales
 * 6) Applications of the Azuma-Hoeffding inequality
 * 7) * General formalization
 * 8) * Application: pattern matching
 * 9) * Application: Balls into bins
 * 10) * Application: Chromatic number

13. Pairwise independence and Universal hash function

 * 1) Pairwise independence
 * 2) * Example: a construction of Pairwise Independent Bits
 * 3) * Application: Derandomizing of Algorithm for Large Cuts
 * 4) * Example: constructing pairwise independent values modulu a prime
 * 5) Chebyshev's inequality for Pairwise independent variables
 * 6) * Application: sampling using fewer random bits
 * 7) Families of Universal hash functions
 * 8) * Example: a 2-universal family of hash functions
 * 9) * Example: a strongly 2-universal family of hash functions
 * 10) * Application: perfect hashing
 * 11) Application: finding heavy hitters in data streams

14. Balanced allocations

 * 1) The power of two choices
 * 2) * The upper bound
 * 3) Two choices: the lower bound
 * 4) Applications of the power of two choices
 * 5) * Hashing
 * 6) * Dynamic resource allocation

Links

 * Randomized algorithm
 * Poisson distribution
 * Bloom filter
 * Chebyshev's inequality
 * Martingale (betting system)
 * Maximum cut
 * Concentration inequality
 * Chernoff bound
 * Set balancing
 * Eli Upfal