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Probability and Computing Randomized Algorithms and Probabilistic AnalysisPDF电子书下载

外文

• 作 者：
• 出 版 社：CAMBRIDGE UNIVERSITY PRESS
• 出版年份：2005
• ISBN：0521835402
• 页数：352 页

图书介绍：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… 查看图书目录点击购买PDF全本电子书 上一篇：TRIGONOMETRY下一篇：MAJORITY MINORITY RELATIONS THIRD EDITION 《Probability and Computing Randomized Algorithms and Probabilistic Analysis》目录 标签：

1 Events and Probability1

1.1 Application: Verifying Polynomial Identities1

1.2 Axioms of Probability3

1.3 Application: Verifying Matrix Multiplication8

1.4 Application: A Randomized Min-Cut Algorithm12

1.5 Exercises14

2 Discrete Random Variables and Expectation20

2.1 Random Variables and Expectation20

2.1.1 Linearity of Expectations22

2.1.2 Jensen’s Inequality23

2.2 The Bernoulli and Binomial Random Variables25

2.3 Conditional Expectation26

2.4 The Geometric Distribution30

2.4.1 Example: Coupon Collector’s Problem32

2.5 Application: The Expected Run-Time of Quicksort34

2.6 Exercises38

3 Moments and Deviations44

3.1 Markov’s Inequality44

3.2 Variance and Moments of a Random Variable45

3.2.1 Example: Variance of a Binomial Random Variable48

3.3 Chebyshev’s Inequality48

3.3.1 Example: Coupon Collector’s Problem50

3.4 Application: A Randomized Algorithm for Computing the Median52

3.4.1 The Algorithm53

3.4.2 Analysis of the Algorithm54

3.5 Exercises57

4 Chernoff Bounds61

4.1 Moment Generating Functions61

4.2 Deriving and Applying Chernoff Bounds63

4.2.1 Chernoff Bounds for the Sum of Poisson Trials63

4.2.2 Example: Coin Flips67

4.2.3 Application: Estimating a Parameter67

4.3 Better Bounds for Some Special Cases69

4.4 Application: Set Balancing71

4.5 Application: Packet Routing in Sparse Networks72

4.5.1 Permutation Routing on the Hypercube73

4.5.2 Permutation Routing on the Butterfly78

4.6 Exercises83

5 Balls, Bins, and Random Graphs90

5.1 Example: The Birthday Paradox90

5.2 Balls into Bins92

5.2.1 The Balls-and-Bins Model92

5.2.2 Application: Bucket Sort93

5.3 The Poisson Distribution94

5.3.1 Limit of the Binomial Distribution98

5.4 The Poisson Approximation99

5.4.1 Example: Coupon Collector’s Problem, Revisited104

5.5 Application: Hashing106

5.5.1 Chain Hashing106

5.5.2 Hashing: Bit Strings108

5.5.3 Bloom Filters109

5.5.4 Breaking Symmetry112

5.6 Random Graphs112

5.6.1 Random Graph Models112

5.6.2 Application: Hamiltonian Cycles in Random Graphs113

5.7 Exercises118

5.8 An Exploratory Assignment124

6 The Probabilistic Method126

6.1 The Basic Counting Argument126

6.2 The Expectation Argument128

6.2.1 Application: Finding a Large Cut129

6.2.2 Application: Maximum Satisfiability130

6.3 Derandomization Using Conditional Expectations131

6.4 Sample and Modify133

6.4.1 Application: Independent Sets133

6.4.2 Application: Graphs with Large Girth134

6.5 The Second Moment Method134

6.5.1 Application: Threshold Behavior in Random Graphs135

6.6 The Conditional Expectation Inequality136

6.7 The Lovasz Local Lemma138

6.7.1 Application: Edge-Disjoint Paths141

6.7.2 Application: Satisfiability142

6.8 Explicit Constructions Using the Local Lemma142

6.8.1 Application: A Satisfiability Algorithm143

6.9 Lovasz Local Lemma: The General Case146

6.10 Exercises148

7 Markov Chains and Random Walks153

7.1 Markov Chains: Definitions and Representations153

7.1.1 Application: A Randomized Algorithm for 2-Satisfiability156

7.1.2 Application: A Randomized Algorithm for 3-Satisfiability159

7.2 Classification of States163

7.2.1 Example: The Gambler’s Ruin166

7.3 Stationary Distributions167

7.3.1 Example: A Simple Queue173

7.4 Random Walks on Undirected Graphs174

7.4.1 Application: An s-t Connectiviry Algorithm176

7.5 Parrondo’s Paradox177

7.6 Exercises182

8 Continuous Distributions and the Poisson Process188

8.1 Continuous Random Variables188

8.1.1 Probability Distributions in R188

8.1.2 Joint Distributions and Conditional Probability191

8.2 The Uniform Distribution193

8.2.1 Additional Properties of the Uniform Distribution194

8.3 The Exponential Distribution196

8.3.1 Additional Properties of the Exponential Distribution197

8.3.2 Example: Balls and Bins with Feedback199

8.4 The Poisson Process201

8.4.1 Interarrival Distribution204

8.4.2 Combining and Splitting Poisson Processes205

8.4.3 Conditional Arrival Time Distribution207

8.5 Continuous Time Markov Processes210

8.6 Example: Markovian Queues212

8.6.1 M/M/1 Queue in Equilibrium213

8.6.2 M/M/1/K Queue in Equilibrium216

8.6.3 The Number of Customers in an M/M/∞ Queue216

8.7 Exercises219

9 Entropy, Randomness, and Information225

9.1 The Entropy Function225

9.2 Entropy and Binomial Coefficients228

9.3 Entropy: A Measure of Randomness230

9.4 Compression234

9.5 Coding: Shannon’s Theorem237

9.6 Exercises245

10 The Monte Carlo Method252

10.1 The Monte Carlo Method252

10.2 Application: The DNF Counting Problem255

10.2.1 The Naive Approach255

10.2.2 A Fully Polynomial Randomized Scheme for DNF Counting257

10.3 From Approximate Sampling to Approximate Counting259

10.4 The Markov Chain Monte Carlo Method263

10.4.1 The Metropolis Algorithm265

10.5 Exercises267

10.6 An Exploratory Assignment on Minimum Spanning Trees270

11 Coupling of Markov Chains271

11.1 Variation Distance and Mixing Time271

11.2 Coupling274

11.2.1 Example: Shuffling Cards275

11.2.2 Example: Random Walks on the Hypercube276

11.2.3 Example: Independent Sets of Fixed Size277

11.3 Application: Variation Distance Is Nonincreasing278

11.4 Geometric Convergence281

11.5 Application: Approximately Sampling Proper Colorings282

11.6 Path Coupling286

11.7 Exercises289

12 Martingales295

12.1 Martingales295

12.2 Stopping Times297

12.2.1 Example: A Ballot Theorem299

12.3 Wald’s Equation300

12.4 Tail Inequalities for Martingales303

12.5 Applications of the Azuma-Hoeffding Inequality305

12.5.1 General Formalization305

12.5.2 Application: Pattern Matching307

12.5.3 Application: Balls and Bins308

12.5.4 Application: Chromatic Number308

12.6 Exercises309

13 Pairwise Independence and Universal Hash Functions314

13.1 Pairwise Independence314

13.1.1 Example: A Construction of Pairwise Independent Bits315

13.1.2 Application: Derandomizing an Algorithm for Large Cuts316

13.1.3 Example: Constructing Pairwise Independent Values Moduloa Prime317

13.2 Chebyshev’s Inequality for Pairwise Independent Variables318

13.2.1 Application: Sampling Using Fewer Random Bits319

13.3 Families of Universal Hash Functions321

13.3.1 Example: A 2-Universal Family of Hash Functions323

13.3.2 Example: A Strongly 2-Universal Family of Hash Functions324

13.3.3 Application: Perfect Hashing326

13.4 Application: Finding Heavy Hitters in Data Streams328

13.5 Exercises333

14 Balanced Allocations336

14.1 The Power of Two Choices336

14.1.1 The Upper Bound336

14.2 Two Choices: The Lower Bound341

14.3 Applications of the Power of Two Choices344

14.3.1 Hashing344

14.3.2 Dynamic Resource Allocation345

14.4 Exercises345

Further Reading349

Index350

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摘要：本文以《Probability and Computing Randomized Algorithms and Probabilistic Analysis.pdf》电子书版文档下载为中心，从四个方面对其进行了详细阐述，包括概率论基础、随机算法、概率分析以及应用实例。通过对该文档的深入分析，旨在为读者提供对该领域全面而深入的理解。

1、概率论基础

《Probability and Computing Randomized Algorithms and Probabilistic Analysis.pdf》首先介绍了概率论的基本概念，如随机变量、概率分布、期望、方差等。这些基础概念是理解和分析随机算法和概率分析的基础。通过详细阐述这些概念，读者可以更好地理解随机算法的设计和性能分析。

此外，文档还介绍了条件概率、全概率公式、贝叶斯定理等概率论中的重要定理。这些定理在随机算法的设计和概率分析中发挥着关键作用。通过对这些定理的深入探讨，读者可以掌握概率论的核心思想和方法。

最后，文档还介绍了随机过程和马尔可夫链等高级概念。这些概念在随机算法和概率分析中有着广泛的应用，如排队论、网络优化等。通过对这些概念的介绍，读者可以拓宽视野，了解概率论在各个领域的应用。

2、随机算法

文档重点介绍了随机算法的基本原理和设计方法。随机算法是一种利用随机性来解决问题的算法，其核心思想是在算法执行过程中引入随机性，以期望提高算法的效率或解决某些特定问题。

文档详细阐述了随机算法的分类，如概率算法、蒙特卡洛算法、拉斯维加斯算法等。通过对不同类型随机算法的介绍，读者可以了解各种算法的特点和适用场景。

此外，文档还介绍了随机算法的性能分析，如概率正确性、时间复杂度、空间复杂度等。通过对这些性能指标的分析，读者可以评估随机算法的优劣，为实际应用提供参考。

3、概率分析

概率分析是随机算法和概率论研究的重要分支。文档详细介绍了概率分析的基本方法，如大数定律、中心极限定理、切比雪夫不等式等。这些方法在随机算法的设计和性能分析中发挥着关键作用。

文档还介绍了概率分析在各个领域的应用，如金融工程、机器学习、通信系统等。通过对这些应用的介绍，读者可以了解概率分析在实际问题中的重要性。

此外，文档还探讨了概率分析中的挑战和难题，如随机算法的收敛性、概率分析中的不确定性等。通过对这些问题的探讨，读者可以深入理解概率分析的理论和方法。

4、应用实例

文档通过多个应用实例展示了概率论、随机算法和概率分析在实际问题中的应用。这些实例包括数据挖掘、图像处理、网络优化等。通过对这些实例的分析，读者可以了解概率论、随机算法和概率分析在各个领域的应用价值。

文档还介绍了如何将概率论、随机算法和概率分析应用于实际问题中，如如何设计高效的随机算法、如何进行概率分析等。这些内容对于实际应用具有重要的指导意义。

最后，文档还讨论了概率论、随机算法和概率分析在未来的发展趋势，如人工智能、大数据等。通过对这些趋势的探讨，读者可以了解该领域的发展方向和前景。

总结：

本文通过对《Probability and Computing Randomized Algorithms and Probabilistic Analysis.pdf》电子书版文档下载的详细阐述，全面介绍了概率论、随机算法和概率分析的基本概念、设计方法、性能分析以及应用实例。通过对该文档的深入分析，读者可以对该领域有更全面而深入的理解。

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