Probability and Computing Randomized Algorithms and Probabilistic Analysis 1st editon by Michael Mitzenmacher, Eli Upfal ISBN 0521835402 978-0521835404

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ISBN 10: 0521835402
ISBN 13: 978-0521835404
Author: Michael Mitzenmacher, Eli Upfal

Assuming only an elementary background in discrete mathematics, this textbook is an excellent introduction to the probabilistic techniques and paradigms used in the development of probabilistic algorithms and analyses. It includes random sampling, expectations, Markov’s and Chevyshev’s inequalities, Chernoff bounds, balls and bins models, the probabilistic method, Markov chains, MCMC, martingales, entropy, and other topics. The book is designed to accompany a one- or two-semester course for graduate students in computer science and applied mathematics.

Probability and Computing Randomized Algorithms and Probabilistic Analysis 1st Table of contents:

1 Events and Probability

1.1 Application: Verifying Polynomial Identities

1.2 Axioms of Probability

1.3 Application: Verifying Matrix Multiplication

1.4 Application: A Randomized Min-Cut Algorithm

1.5 Exercises

2 Discrete Random Variables and Expectation

2.1 Random Variables and Expectation

2.1.1 Linearity of Expectations

2.1.2 Jensen’s Inequality

2.2 The Bernoulli and Binomial Random Variables

2.3 Conditional Expectation

2.4 The Geometric Distribution

2.4.1 Example: Coupon Collector’s Problem 2.5 Application: The Expected Run-Time of Quicksort

2.6 Exercises

3 Moments and Deviations

3.1 Markov’s Inequality

3.2 Variance and Moments of a Random Variable

3.2.1 Example: Variance of a Binomial Random Variable

3.3 Chebyshev’s Inequality

3.3.1 Example: Coupon Collector’s Problem

3.4 Application: A Randomized Algorithm for Computing the Median

3.4.1 The Algorithm

3.4.2 Analysis of the Algorithm

3.5 Exercises

4 Chernoff Bounds

4.1 Moment Generating Functions

4.2 Deriving and Applying Chernoff Bounds

4.2.1 Chernoff Bounds for the Sum of Poisson Trials

4.2.2 Example: Coin Flips

4.2.3 Application: Estimating a Parameter

4.3 Better Bounds for Some Special Cases

4.4 Application: Set Balancing

4.5* Application: Packet Routing in Sparse Networks

4.5.1 Permutation Routing on the Hypercube 4.5.2 Permutation Routing on the Butterfly

4.6 Exercises

5 Balls, Bins, and Random Graphs

5.1 Example: The Birthday Paradox

5.2 Balls into Bins

5.2.1 The Balls-and-Bins Model

5.2.2 Application: Bucket Sort

5.3 The Poisson Distribution

5.3.1 Limit of the Binomial Distribution

5.4 The Poisson Approximation

5.4.1 Example: Coupon Collector’s Problem, Revisited

5.5 Application: Hashing

5.5.1 Chain Hashing

5.5.2 Hashing: Bit Strings

5.5.3 Bloom Filters

5.5.4 Breaking Symmetry

5.6 Random Graphs

5.6.1 Random Graph Models

5.6.2 Application: Hamiltonian Cycles in Random Graphs

5.7 Exercises

5.8 An Exploratory Assignment

6 The Probabilistic Method

6.1 The Basic Counting Argument

6.2 The Expectation Argument

6.2.1 Application: Finding a Large Cut

6.2.2 Application: Maximum Satisfiability

6.3 Derandomization Using Conditional Expectations

6.4 Sample and Modify

6.4.1 Application: Independent Sets

6.4.2 Application: Graphs with Large Girth

6.5 The Second Moment Method

6.5.1 Application: Threshold Behavior in Random Graphs

6.6 The Conditional Expectation Inequality

6.7 The Lovasz Local Lemma

6.7.1 Application: Edge-Disjoint Paths

6.7.2 Application: Satisfiability

6.8 Explicit Constructions Using the Local Lemma

6.8.1 Application: A Satisfiability Algorithm

6.9 Lovasz Local Lemma: The General Case

6.10 Exercises

7 Markov Chains and Random Walks

7.1 Markov Chains: Definitions and Representations

7.1.1 Application: A Randomized Algorithm for 2-Satisfiability

7.1.2 Application: A Randomized Algorithm for 3-Satisfiability

7.2 Classification of States

7.2.1 Example: The Gambler’s Ruin

7.3 Stationary Distributions

7.3.1 Example: A Simple Queue

7.4 Random Walks on Undirected Graphs

7.4.1 Application: An 5-1 Connectivity Algorithm

7.5 Parrondo’s Paradox

7.6

Exercises

8 Continuous Distributions and the Poisson Process

8.1 Continuous Random Variables

8.1.1 Probability Distributions in R

8.1.2 Joint Distributions and Conditional Probability

8.2 The Uniform Distribution

8.2.1 Additional Properties of the Uniform Distribution

8.3 The Exponential Distribution

8.3.1 Additional Properties of the Exponential Distribution

8.3.2 Example: Balls and Bins with Feedback

8.4 The Poisson Process

8.4.1 Interarrival Distribution

8.4.2 Combining and Splitting Poisson Processes

8.4.3 Conditional Arrival Time Distribution

8.5 Continuous Time Markov Processes

8.6 Example: Markovian Queues

8.6.1 M/M/1 Queue in Equilibrium

8.6.2 M/M/1/K Queue in Equilibrium

8.6.3 The Number of Customers in an M/M/s Queue

8.7 Exercises

9 Entropy, Randomness, and Information

9.1 The Entropy Function

9.2 Entropy and Binomial Coefficients

9.3 Entropy: A Measure of Randomness

9.4 Compression

9.5 Coding: Shannon’s Theorem

9.6 Exercises

10 The Monte Carlo Method

10.1 The Monte Carlo Method

10.2 Application: The DNF Counting Problem

10.2.1 The Naïve Approach

10.2.2 A Fully Polynomial Randomized Scheme for DNF Counting

10.3 From Approximate Sampling to Approximate Counting

10.4 The Markov Chain Monte Carlo Method

10.4.1 The Metropolis Algorithm

10.5 Exercises

10.6 An Exploratory Assignment on Minimum Spanning Trees

11 Coupling of Markov Chains

11.1 Variation Distance and Mixing Time

11.2 Coupling

11.2.1 Example: Shuffling Cards

11.2.2 Example: Random Walks on the Hypercube

11.2.3 Example: Independent Sets of Fixed Size

11.3 Application: Variation Distance Is Nonincreasing

11.4 Geometric Convergence

11.5 Application: Approximately Sampling Proper Colorings

11.6 Path Coupling

11.7 Exercises

12 Martingales

12.1 Martingales

12.2 Stopping Times

12.2.1 Example: A Ballot Theorem

12.3 Wald’s Equation

12.4 Tail Inequalities for Martingales

12.5 Applications of the Azuma-Hoeffding Inequality

12.5.1 General Formalization

12.5.2 Application: Pattern Matching

12.5.3 Application: Balls and Bins

12.5.4 Application: Chromatic Number

12.6 Exercises

13 Pairwise Independence and Universal Hash Functions

13.1 Pairwise Independence

13.1.1 Example: A Construction of Pairwise Independent Bits

13.1.2 Application: Derandomizing an Algorithm for Large Cuts

13.1.3 Example: Constructing Pairwise Independent Values Modulo a Prime

13.2 Chebyshev’s Inequality for Pairwise Independent Variables 13.2.1 Application: Sampling Using Fewer Random Bits

13.3 Families of Universal Hash Functions

13.3.1 Example: A 2-Universal Family of Hash Functions

13.3.2 Example: A Strongly 2-Universal Family of Hash Functions

13.3.3 Application: Perfect Hashing

13.4 Application: Finding Heavy Hitters in Data Streams

13.5 Exercises

14* Balanced Allocations

14.1 The Power of Two Choices

14.1.1 The Upper Bound

14.2 Two Choices: The Lower Bound

14.3 Applications of the Power of Two Choices

14.3.1 Hashing

14.3.2 Dynamic Resource Allocation

14.4 Exercises

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