随机算法 (Spring 2014)

随机算法
Randomized Algorithms

Instructor
尹一通
Email	yitong.yin@gmail.com yinyt@nju.edu.cn
office	计算机系 804
Class
Class meetings	Tuesday, 10am-12pm 仙I-101
Office hours	Wednesday, 2-4pm 计算机系 804
Textbooks
Motwani and Raghavan. Randomized Algorithms. Cambridge Univ Press, 1995.
Mitzenmacher and Upfal. Probability and Computing: Randomized Algorithms and Probabilistic Analysis. Cambridge Univ Press, 2005.

This is the page for the class Randomized Algorithms for the Spring 2014 semester. Students who take this class should check this page periodically for content updates and new announcements.

Announcement

To be added

Course info

• Instructor : 尹一通，

• email: yitong.yin@gmail.com, yinyt@nju.edu.cn
• office: 计算机系 804.

• Class meeting: Tuesday 10am-12pm, 仙I-101.
• Office hour: Wednesday 2-4pm, 计算机系 804.

Syllabus

先修课程 Prerequisites

• 必须：离散数学，概率论，线性代数。
• 推荐：算法设计与分析。

Course materials

• 教材和参考书

成绩 Grades

• 课程成绩：本课程将会有若干次作业和一次期末考试。最终成绩将由平时作业成绩和期末考试成绩综合得出。
• 迟交：如果有特殊的理由，无法按时完成作业，请提前联系授课老师，给出正当理由。否则迟交的作业将不被接受。

学术诚信 Academic Integrity

学术诚信是所有从事学术活动的学生和学者最基本的职业道德底线，本课程将不遗余力的维护学术诚信规范，违反这一底线的行为将不会被容忍。

作业完成的原则：署你名字的工作必须由你完成。允许讨论，但作业必须独立完成，并在作业中列出所有参与讨论的人。不允许其他任何形式的合作——尤其是与已经完成作业的同学“讨论”。

本课程将对剽窃行为采取零容忍的态度。在完成作业过程中，对他人工作（出版物、互联网资料、其他人的作业等）直接的文本抄袭和对关键思想、关键元素的抄袭，按照 ACM Policy on Plagiarism的解释，都将视为剽窃。剽窃者成绩将被取消。如果发现互相抄袭行为， 抄袭和被抄袭双方的成绩都将被取消。因此请主动防止自己的作业被他人抄袭。

学术诚信影响学生个人的品行，也关乎整个教育系统的正常运转。为了一点分数而做出学术不端的行为，不仅使自己沦为一个欺骗者，也使他人的诚实努力失去意义。让我们一起努力维护一个诚信的环境。

Assignments

• Problem Set 1, due on March 18, Tuesday, in class.
• Problem Set 2, due on April 8, Tuesday, in class.
• Problem Set 3, due on May 20, Tuesday, in class.
• Problem Set 4, due on June 17, Tuesday, in class.

Lecture Notes

1. Introduction and Probability Space: checking matrix multiplication, polynomial identity testing, communication complexity
2. Conditional Probability: polynomial identity testing, min-cut
3. Random Variables: balls and bins, stable marriage
4. Moment and Deviation: Markov's inequality, Chebyshev's inequality, median selection, random graphs, threshold phenomenon
5. Second Moment: threshold phenomenon for random graphs, 2-point sampling
6. Chernoff Bound: random graphs, threshold phenomenon, Chernoff bound, error reduction, set balancing, packet routing
7. Martingales: martingales, Azuma's inequality, Doob martingales, chromatic number of random graphs
8. Concentration of Measure: bounded difference method, Johnson-Lindenstrauss Theorem
9. Random Recurrence: random quicksort, Karp-Upfal-Wigderson bound
10. The Probabilistic Method: MAX-SAT, conditional probability method, Lovász Local Lemma
11. Universal Hashing: [math]\displaystyle{ k }[/math]-wise independence, universal hash families, perfect hashing
12. The Monte Carlo Method: counting DNF, the union of sets problem, the coverage algorithm, counting matchings
13. Markov Chain and Random Walk: Markov chain, random walk, stationary distribution, convergence of Markov chain, hitting/cover time
14. Expander Graphs and Mixing: expander graphs, graph spectrum, spectral gap, Cheeger's inequality, rapid mixing of expander walk
15. Mixing Time and Coupling: mixing time, coupling lemma, coupling of Markov chain, rapid mixing by coupling

The Probability Theory Toolkit

• Probability space and probability axioms
• Independent events
• Conditional probability
• Random variable and expectation
• Linearity of expectation
• The law of total probability and the law of total expectation
• The union bound
• Bernoulli trials
• Geometric distribution
• Binomial distribution
• Markov's inequality
• Variance and covariance
• Chebyshev's inequality
• Chernoff bound
• Martingale
• Azuma's inequality and Hoeffding's inequality
• Doob martingale
• k-wise independence
• 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