概率论 (Summer 2013)/Problem Set 5 and 随机算法 (Spring 2014): Difference between pages

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== Problem 1 ==
{{Infobox
|name        = Infobox
|bodystyle    =  
|title        = <font size=3>随机算法
<br>Randomized Algorithms</font>
|titlestyle  =  


Let <math>G(V,E)</math> be an undirected connected graph with <math>V=\{v_1,\dots,v_n\}</math> and <math>|E|=m</math>. Consider the following random process on <math>G</math>: In the beginning, every vertex in <math>V</math> is colored either black or white. Then for each step, we do the following for every vertex <i>simultaneously</i>:
|image        =
* with probability <math>\frac{1}{2}</math> do nothing,
|imagestyle  =
* otherwise, choose an incident vertex uniformly at random and change own color to the color of that vertex.
|caption      =  
|captionstyle =
|headerstyle  = background:#ccf;
|labelstyle  = background:#ddf;
|datastyle    =


Eventually, all vertices are monochromatic and the process terminates. We use <math>X_t\in\{white,black\}^V</math> to denote the color of each vertex at step <math>t</math>.
|header1 =Instructor
|label1  =
|data1  =
|header2 =
|label2  =
|data2  = 尹一通
|header3 =
|label3  = Email
|data3  = yitong.yin@gmail.com  yinyt@nju.edu.cn 
|header4 =
|label4= office
|data4= 计算机系 804
|header5 = Class
|label5  =
|data5  =
|header6 =
|label6  = Class meetings
|data6  = Tuesday, 10am-12pm <br> 仙I-101
|header7 =
|label7  = Place
|data7  =
|header8 =
|label8  = Office hours
|data8  = Wednesday, 2-4pm <br>计算机系 804
|header9 = Textbooks
|label9  =
|data9  =
|header10 =
|label10  =
|data10  = [[File:MR-randomized-algorithms.png|border|100px]]
|header11 =
|label11  =
|data11  = Motwani and Raghavan. <br>''Randomized Algorithms''.<br> Cambridge Univ Press, 1995.
|header12 =
|label12  =
|data12  = [[File:Probability_and_Computing.png|border|100px]]
|header13 =
|label13  =
|data13  =  Mitzenmacher and Upfal. <br>''Probability and Computing: Randomized Algorithms and Probabilistic Analysis''. <br> Cambridge Univ Press, 2005.
|belowstyle = background:#ddf;
|below =
}}


# Let the random variable <math>Y_t</math> denote the sum of the degrees of all the white vertices at time <math>t</math>. Show that <math>Y_t</math> is a martingale with respect to <math>\{X_t\}</math>.
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.  
# Use the optional stopping theorem to compute the probability that the process terminates in the all-white configuration, as a function of the initial configuration.
# (Bonus) Use the optional stopping theorem again to show that the expected duration of the process is at most <math>O(m^2)</math> steps.


== Problem 2 ==
= Announcement =
Let <math>G(V,E)</math> be an undirected connected graph with maximum degree <math>\Delta</math>.
To be added
* Design an efficient, time reversible, ergodic random walk on <math>G</math> whose stationary distribution is the uniform distribution.
* Let <math>\pi</math> be an arbitrary distribution on <math>V</math> such that <math>\pi(v)>0</math> for all <math>v\in V</math>. Design a time reversible, ergodic random walk on <math>G</math> whose stationary distribution is <math>\pi</math>.


== Problem 3 ==
= 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.


Consider the following random walk on <math>n</math>-dimensional hypercube: Assume we are now at the vertex <math>b_1b_2\dots b_n</math> where each <math>b_i\in\{0,1\}</math>, then
= Syllabus =
* With probability <math>\frac{1}{n+1}</math>, do nothing.
* Otherwise, with probability <math>\frac{1}{n+1}</math> for each coordinate <math>i</math>, flip <math>b_i</math>.
Prove by coupling that the mixing time of this markov chain is <math>O(n\ln n)</math>


== Problem 4 ==
=== 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。


Recall the markov chain we used in class to sample proper colorings: Let <math>G(V,E)</math> be an undirected graph with maximum degree <math>\Delta</math>, <math>q\ge\delta+2</math> is the number of colors. Assume we are currently given a proper coloring, then
=== Course materials ===
* Pick a vertex <math>v\in V</math> uniformly at random and a color <math>c\in\{1,2,\dots,q\}</math> uniformly at random.
* [[随机算法 (Spring 2014)/Course materials|<font size=3>教材和参考书</font>]]
* Recolor <math>v</math> with <math>c</math> if this yields a proper coloring, else do nothing.


# Prove that the markov chain is irreducible, aperiodic, time reversible and the stationary distribution is the uniform distribution.
=== 成绩 Grades ===
# Suppose <math>q\le\Delta+1</math>, show that the markov chain is no longer always irreducible.
* 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。


== Problem 5 ==
=== <font color=red> 学术诚信 Academic Integrity </font>===
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。


Let <math>n,k>0</math> be two numbers and <math>k\le\frac{n}{2}</math>. Let <math>\Omega=\binom{[n]}{k}</math>, i.e., the family all subsets of <math>\{1,\dots,n\}</math> of cardinality <math>k</math> and choose a number <math>p\in[0,1)</math>. We run a markov chain on <math>\Omega</math> in the following way: Assume you are now at some <math>S\in\Omega</math>,
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”。
* With probability p, do nothing.
* Otherwise, pick <math>a\in S</math> uniformly at random and pick <math>b\in\{1,2,\dots,n\}-S</math> uniformly at random. Move to <math>S-\{a\}+\{b\}</math>


# Show that this Markov chain is ergodic with uniform stationary distribution. You can choose arbitrary <math>p\in[0,1)</math> to ease your analysis.
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 [http://www.acm.org/publications/policies/plagiarism_policy 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，<font color=red> 抄袭和被抄袭双方的成绩都将被取消</font>。因此请主动防止自己的作业被他人抄袭。
# Using coupling to show that the mixing time is asymptotically <math>O(n\log k)</math> or less.
 
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 =
# [[随机算法 (Spring 2014)/Introduction and Probability Space|Introduction and Probability Space]]: checking matrix multiplication, polynomial identity testing, communication complexity
# [[随机算法 (Spring 2014)/Conditional Probability|Conditional Probability]]: polynomial identity testing, min-cut
# [[随机算法 (Spring 2014)/Random Variables and Expectations|Random Variables and Expectations]]: random quicksort, balls and bins
 
= The Probability Theory Toolkit =
* [http://en.wikipedia.org/wiki/Probability_space Probability space] and [http://en.wikipedia.org/wiki/Probability_axioms probability axioms]
* [http://en.wikipedia.org/wiki/Independence_(probability_theory)#Independent_events Independent events]
* [http://en.wikipedia.org/wiki/Conditional_probability Conditional probability]
* [http://en.wikipedia.org/wiki/Random_variable Random variable] and [http://en.wikipedia.org/wiki/Expected_value expectation]
* [http://en.wikipedia.org/wiki/Expected_value#Linearity Linearity of expectation]
* The [http://en.wikipedia.org/wiki/Law_of_total_probability law of total probability] and the [http://en.wikipedia.org/wiki/Law_of_total_expectation law of total expectation]
* The [http://en.wikipedia.org/wiki/Boole's_inequality union bound]
* [http://en.wikipedia.org/wiki/Bernoulli_trial Bernoulli trials]
* [http://en.wikipedia.org/wiki/Geometric_distribution Geometric distribution]
* [http://en.wikipedia.org/wiki/Binomial_distribution Binomial distribution]
* [http://en.wikipedia.org/wiki/Markov's_inequality Markov's inequality]
* [http://en.wikipedia.org/wiki/Variance Variance] and [http://en.wikipedia.org/wiki/Covariance covariance]
* [http://en.wikipedia.org/wiki/Chebyshev's_inequality Chebyshev's inequality]
* [http://en.wikipedia.org/wiki/Chernoff_bound Chernoff bound]
* [http://en.wikipedia.org/wiki/Martingale_(probability_theory) Martingale]
* [http://en.wikipedia.org/wiki/Azuma's_inequality Azuma's inequality] and [http://en.wikipedia.org/wiki/Hoeffding's_inequality Hoeffding's inequality]
* [http://en.wikipedia.org/wiki/Doob_martingale Doob martingale]
* [http://en.wikipedia.org/wiki/Pairwise_independence k-wise independence]
* The [http://en.wikipedia.org/wiki/Probabilistic_method  probabilistic method]
* The [http://en.wikipedia.org/wiki/Lov%C3%A1sz_local_lemma  Lovász local lemma]  and the [http://en.wikipedia.org/wiki/Algorithmic_Lov%C3%A1sz_local_lemma algorithmic Lovász local lemma]
* [http://en.wikipedia.org/wiki/Markov_chain Markov chain]:
::[http://en.wikipedia.org/wiki/Markov_chain#Reducibility reducibility], [http://en.wikipedia.org/wiki/Markov_chain#Periodicity Periodicity], [http://en.wikipedia.org/wiki/Markov_chain#Steady-state_analysis_and_limiting_distributions stationary distribution], [http://en.wikipedia.org/wiki/Hitting_time hitting time], cover time;
::[http://en.wikipedia.org/wiki/Markov_chain_mixing_time mixing time], [http://en.wikipedia.org/wiki/Conductance_(probability) conductance]

Revision as of 06:39, 3 March 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.
v · d · e

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

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

  1. Introduction and Probability Space: checking matrix multiplication, polynomial identity testing, communication complexity
  2. Conditional Probability: polynomial identity testing, min-cut
  3. Random Variables and Expectations: random quicksort, balls and bins

The Probability Theory Toolkit

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