Probability and Computing Randomization and Probabilistic Techniques in Algorithms and Data Analysis 2nd edition by Mitzenmacher Michael, Upfal Eli ISBN 1108105958 9781108105958

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ISBN 10: 1108105958
ISBN 13: 9781108105958
Author: Mitzenmacher Michael, Upfal Eli

Greatly expanded, this new edition requires only an elementary background in discrete mathematics and offers a comprehensive introduction to the role of randomization and probabilistic techniques in modern computer science. Newly added chapters and sections cover topics including normal distributions, sample complexity, VC dimension, Rademacher complexity, power laws and related distributions, cuckoo hashing, and the Lovasz Local Lemma. Material relevant to machine learning and big data analysis enables students to learn modern techniques and applications. Among the many new exercises and examples are programming-related exercises that provide students with excellent training in solving relevant problems. This book provides an indispensable teaching tool to accompany a one- or two-semester course for advanced undergraduate students in computer science and applied mathematics.

Probability and Computing Randomization and Probabilistic Techniques in Algorithms and Data Analysis 2nd 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: Naïve Bayesian Classifier

1.5 Application: A Randomized Min-Cut Algorithm

1.6 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 Median and Mean

3.5 Application: A Randomized Algorithm for Computing the Median

3.5.1 The Algorithm

3.5.2 Analysis of the Algorithm

3.6 Exercises

4 Chernoff and Hoeffding 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 The Hoeffding Bound

4.6* Application: Packet Routing in Sparse Networks

4.6.1 Permutation Routing on the Hypercube

4.6.2 Permutation Routing on the Butterfly

4.7 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 Lovász 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 Lovász Local Lemma: The General Case

6.10* The Algorithmic Lovász Local Lemma

6.11 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 s–t 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/∞ Queue

8.7 Exercises

9 The Normal Distribution

9.1 The Normal Distribution

9.1.1 The Standard Normal Distribution

9.1.2 The General Univariate Normal Distribution

9.1.3 The Moment Generating Function

9.2* Limit of the Binomial Distribution

9.3 The Central Limit Theorem

9.4* Multivariate Normal Distributions

9.4.1 Properties of the Multivariate Normal Distribution

9.5 Application: Generating Normally Distributed Random Values

9.6 Maximum Likelihood Point Estimates

9.7 Application: EM Algorithm For a Mixture of Gaussians

9.8 Exercises

10 Entropy, Randomness, and Information

10.1 The Entropy Function

10.2 Entropy and Binomial Coefficients

10.3 Entropy: A Measure of Randomness

10.4 Compression

10.5* Coding: Shannon’s Theorem

10.6 Exercises

11 The Monte Carlo Method

11.1 The Monte Carlo Method

11.2 Application: The DNF Counting Problem

11.2.1 The Naïve Approach

11.2.2 A Fully Polynomial Randomized Scheme for DNF Counting

11.3 From Approximate Sampling to Approximate Counting

11.4 The Markov Chain Monte Carlo Method

11.4.1 The Metropolis Algorithm

11.5 Exercises

11.6 An Exploratory Assignment on Minimum Spanning Trees

12 Coupling of Markov Chains

12.1 Variation Distance and Mixing Time

12.2 Coupling

12.2.1 Example: Shuffling Cards

12.2.2 Example: Random Walks on the Hypercube

12.2.3 Example: Independent Sets of Fixed Size

12.3 Application: Variation Distance Is Nonincreasing

12.4 Geometric Convergence

12.5 Application: Approximately Sampling Proper Colorings

12.6 Path Coupling

12.7 Exercises

13 Martingales

13.1 Martingales

13.2 Stopping Times

13.2.1 Example: A Ballot Theorem

13.3 Wald’s Equation

13.4 Tail Inequalities for Martingales

13.5 Applications of the Azuma–Hoeffding Inequality

13.5.1 General Formalization

13.5.2 Application: Pattern Matching

13.5.3 Application: Balls and Bins

13.5.4 Application: Chromatic Number

13.6 Exercises

14 Sample Complexity, VC Dimension, and Rademacher Complexity

14.1 The Learning Setting

14.2 VC Dimension

14.2.1 Additional Examples of VC Dimension

14.2.2 Growth Function

14.2.3 VC dimension component bounds

14.2.4 ε-nets and ε-samples

14.3 The ε-net Theorem

14.4 Application: PAC Learning

14.5 The ε-sample Theorem

14.5.1 Application: Agnostic Learning

14.5.2 Application: Data Mining

14.6 Rademacher Complexity

14.6.1 Rademacher Complexity and Sample Error

14.6.2 Estimating the Rademacher Complexity

14.6.3 Application: Agnostic Learning of a Binary Classification

14.7 Exercises

15 Pairwise Independence and Universal Hash Functions

15.1 Pairwise Independence

15.1.1 Example: A Construction of Pairwise Independent Bits

15.1.2 Application: Derandomizing an Algorithm for Large Cuts

15.1.3 Example: Constructing Pairwise Independent Values Modulo a Prime

15.2 Chebyshev’s Inequality for Pairwise Independent Variables

15.2.1 Application: Sampling Using Fewer Random Bits

15.3 Universal Families of Hash Functions

15.3.1 Example: A 2-Universal Family of Hash Functions

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

15.3.3 Application: Perfect Hashing

15.4 Application: Finding Heavy Hitters in Data Streams

15.5 Exercises

16 Power Laws and Related Distributions

16.1 Power Law Distributions: Basic Definitions and Properties

16.2 Power Laws in Language

16.2.1 Zipf’s Law and Other Examples

16.2.2 Languages via Optimization

16.2.3 Monkeys Typing Randomly

16.3 Preferential Attachment

16.3.1 A Formal Version

16.4 Using the Power Law in Algorithm Analysis

16.5 Other Related Distributions

16.5.1 Lognormal Distributions

16.5.2 Power Law with Exponential Cutoff

16.6 Exercises

17 Balanced Allocations and Cuckoo Hashing

17.1 The Power of Two Choices

17.1.1 The Upper Bound

17.2 Two Choices: The Lower Bound

17.3 Applications of the Power of Two Choices

17.3.1 Hashing

17.3.2 Dynamic Resource Allocation

17.4 Cuckoo Hashing

17.5 Extending Cuckoo Hashing

17.5.1 Cuckoo Hashing with Deletions

17.5.2 Handling Failures

17.5.3 More Choices and Bigger Bins

17.6 Exercises

Further Reading

Index

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Tags: Mitzenmacher Michael, Upfal Eli, Computing Randomization, Data Analysis