Probability and Computing Randomized Algorithms and Probabilistic Analysis 1st editon by Michael Mitzenmacher, Eli Upfal – Ebook PDF Instant Download/Delivery: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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