随机算法 (Fall 2011)

随机算法 Randomized Algorithms

Randomized Algorithms by Motwani and Raghavan
Instructor
尹一通
Email	yitong.yin@gmail.com yinyt@nju.edu.cn yinyt@lamda.nju.edu.cn
office	蒙民伟楼 406
Class
Class meetings	10 am-12 am, Tuesday, 馆III-101
Office hours	2pm-5pm, Saturday, MMW 406
Textbook
Motwani and Raghavan, Randomized Algorithms. Cambridge Univ Press, 1995.

This is the page for the class Randomized Algorithms for the Spring 2010 semester. Students who take this class should check this page periodically for content updates and new announcements.

There is also a backup page for off-campus users. The URL is http://lamda.nju.edu.cn/yinyt/random2010wiki/

Announcement

• (07/01/2010) 期末考试：时间7月6日，上午9点至11点；地点：教202。可携带一页A4打印纸（可双面）的笔记。
• (06/26/2010) 由于要统计考试人数，请参加期末考试的同学给我发email，把名字和学号告诉我。请看到通知的同学互相告知一下。
• (06/24/2010) 第四次作业答案公布。第三次作业答案已经补完。Sorry for the delay!
• (06/11/2010) 由于端午节轮休，6月15日的课改在13日星期日。
• (06/01/2010) The fifth homework is out. Due on Tuesday June 15, in class.
• (05/26/2010) 第十三课中的Estimator Theorem中有一处笔误，[math]\displaystyle{ \epsilon }[/math]应为[math]\displaystyle{ \epsilon^2 }[/math]。感谢钱超同学发现这个问题。

(older announcements...)

Course info

• Instructor : 尹一通，

• email: yitong.yin@gmail.com, yinyt@nju.edu.cn, yinyt@lamda.nju.edu.cn
• office: MMW 406.

• Class meeting: 10am-12 am, Tue; 馆III-101.
• Office hour: 2-5pm, Sat; MMW 406.

Syllabus

随机化（randomization）是现代计算机科学最重要的方法之一，近二十年来被广泛的应用于计算机科学的各个领域。在这些应用的背后，是一些共通的随机化原理。在随机算法这门课程中，我们将用数学的语言描述这些原理，将会介绍以下内容：

• 一些重要的随机算法的设计思想和理论分析；
• 概率论工具及其在算法分析中的应用，包括常用的概率不等式，以及数学证明的概率方法 (the probabilistic method);
• 随机算法的概率模型，包括典型的随机算法模型，以及概率复杂度模型。

作为一门理论课程，这门课的内容偏重数学上的分析和证明。这么做的目的不单纯是为了追求严格性，而是因为用更聪明的方法去解决问题往往需要具备有一定深度的数学思维和数学洞察力。

先修课程 Prerequisites

• 必须：离散数学，概率论。
• 推荐：算法设计与分析。

Course materials

• 教材和参考书清单

Policies

• Policies

Assignments

• (06/29/2010) Problem Set 6 makeup assignment, hand in before the final exam.
• (06/01/2010) Problem Set 5 due on June 15 , Tuesday, in class. 中英文不限。
• (05/18/2010) Problem Set 4 due on June 1 , Tuesday, in class. 中英文不限。
• (04/20/2010) Problem Set 3 due on May 4, Tuesday, in class. 中英文不限。
• (03/30/2010) Problem Set 2 due on April 13, Tuesday, in class. 中英文不限。
• (03/16/2010) Problem Set 1 due on March 30, Tuesday, in class. 中英文不限。

Lecture Notes

1. Introduction
   • Randomized Algorithms: an Introduction
   • Complexity Classes
2. Probability Basics
   • Probability Space
   • Verifying Matrix Multiplication
   • Conditional Probability
   • Randomized Min-Cut
   • Random Variables and Expectations
   • Randomized Quicksort
3. Balls and Bins
   • Distributions of Coin Flipping
   • Birthday Problem
   • Coupon Collector
   • Balls-into-balls Occupancy Problem
   • Bloom Filter
   • Stable Marriage
4. Moment and Deviation
   • Markov's Inequality
   • Chebyshev's Inequality
   • Median Selection
   • Random Graphs
5. Hashing and Fingerprinting
   • Pair-wise Independence
   • Derandomization: Two-Point Sampling
6. Chernoff Bound
   • Chernoff Bound
   • Set Balancing
   • DNF Counting
   • Routing in a Parallel Network
7. Concentration of Measure
   • Martingales
   • Azuma's Inequality
   • The Method of Bounded Differences
8. Dimension Reduction
   • Johnson-Lindenstrauss Theorem
   • Locality Sensitive Hashing
9. The Probabilistic Method
   • The Probabilistic Method
   • Lovász Local Lemma
   • Derandomization: Conditional Expectation
   • Derandomization: Color-coding
10. Approximation Algorithms, On-line Algorithms
    • Max-SAT
    • Linear Programming
    • Randomized Rounding
    • On-line Algorithms
11. Markov Chain and Random Walk
    • Markov Chains
    • Random Walks on Undirected Graphs
    • Electrical Network
    • Cover Time
    • Graph Connectivity
12. Random Walk Algorithms
    • Randomized 2SAT
    • Randomized 3SAT
    • Perfect Matching in Regular Bipartite Graph
    • The Metropolis Algorithm
    • Dynamics on Spins
13. Coupling and Mixing Time
    • Mixing Time
    • Coupling
    • Card Shuffling
    • Path Coupling
    • Graph Coloring
14. Expander Graphs I
    • Expander Graphs
    • Graph Spectrum
    • The Spectral Gap
    • Random Walk on Expander Graph
15. Expander Graphs II
    • Expander Mixing Lemma
    • Chernoff Bound for Expander Walks
    • The Zig-Zag Product
    • USTCON in LOGSPACE
16. Sampling and Counting
    • The #P Class and Approximation
    • DNF Counting
    • Canonical Paths
    • Count Matchings
    • Sampling and Counting
17. MCMC
    • Spin Systems
    • Simulated Annealing
    • Volume Estimation
18. Complexity

The Probability Theory Toolkit

• Linearity of expectation
• Independent events and conditional independence
• Conditional probability and conditional expectation
• The law of total probability and the law of total expectation
• The union bound
• Bernoulli trials
• Geometric distribution
• Binomial distribution
• Markov's inequality
• Chebyshev's inequality
• Chernoff bound
• k-wise independence
• Martingale
• Azuma's inequality and Hoeffding's inequality
• Doob martingale
• The probabilistic method
• The Lovász local lemma and the algorithmic Lovász local lemma
• Markov chain:

reducibility, Periodicity, stationary distribution, hitting time, cover time;

mixing time, conductance