随机算法 (Spring 2013)

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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
仙逸B-207
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.
v · d · e

This is the page for the class Randomized Algorithms for the Spring 2013 semester. Students who take this class should check this page periodically for content updates and new announcements.

Announcement

  • 所有slides已经上传。
  • The last homework assignment is out, due on the date of the final exam.
  • 前几堂课的讲义已经补上，足够完成第三次作业。
  • The third homework assignment is out, due in two weeks.
  • The second homework assignment is out, due in two weeks.
  • 第1次作业第3题新增一问。由于是在作业发布之后修改，是否做这一问题不会影响分数，但增加此问会使该题目更有意义。
  • The first homework assignment is out, due in two weeks.

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, 仙逸B-207.
  • 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

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

Lecture Notes

Slides: 1| 3| 4| 5| 6| 7| 8| 9| 10| 11| 12| 13| 14| 15| 16| 17

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

The Probability Theory Toolkit

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