高级算法 (Fall 2016)/Nonconstructive Proof of Lovász Local Lemma and 高级算法 (Fall 2018): Difference between pages

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== Lovász Local Lemma==
{{Infobox
Given a sequence of events <math>A_1,A_2,\ldots,A_n</math>, we use the '''dependency graph''' to describe the dependencies between these events.
|name        = Infobox
|bodystyle    =  
|title        = <font size=3>高级算法
<br>Advanced Algorithms</font>
|titlestyle  =


{{Theorem
|image        =
|Definition|
|imagestyle  =
:Let <math>A_1,A_2,\ldots,A_n</math> be a sequence of events. A graph <math>D=(V,E)</math> on the set of vertices <math>V=\{1,2,\ldots,n\}</math> is called a '''dependency graph''' for the events <math>A_1,\ldots,A_n</math> if for each <math>i</math>, <math>1\le i\le n</math>, the event <math>A_i</math> is mutually independent of all the events <math>\{A_j\mid (i,j)\not\in E\}</math>.
|caption      =
}}
|captionstyle =
|headerstyle  = background:#ccf;
|labelstyle  = background:#ddf;
|datastyle    =  


The notion of mutual independence between an event and a set of events is formally defined as follows.
|header1 =Instructor
{{Theorem|Definition|
|label1  =
:An event <math>A</math> is said to be '''mutually independent''' of events <math>B_1,B_2,\ldots, B_k</math>, if for any disjoint <math>I^+,I^-\subseteq\{1,2,lots,k\}</math>, it holds that
|data1  =
::<math>\Pr\left[A\mid \bigwedge_{i\in I^+}B_i\wedge \bigwedge_{i\in I^-}\overline{B_i}\right]=\Pr[A]</math>.
|header2 =
|label2  =
|data2  = 尹一通<br>郑朝栋
|header3 =
|label3  = Email
|data3  = yinyt@nju.edu.cn chaodong@nju.edu.cn 
|header4 =
|label4= office
|data4= 计算机系 804
|header5 = Class
|label5  =
|data5  =
|header6 =
|label6  = Class meetings
|data6  = Wednesday, 8am-10am <br> 仙I-319
|header7 =
|label7  = Place
|data7  =
|header8 =
|label8  = Office hours
|data8  = Wednesday, 10am-12pm <br>计算机系 804（尹一通）、302（郑朝栋）
|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:Approximation_Algorithms.jpg|border|100px]]
|header13 =
|label13  =
|data13  =  Vazirani. <br>''Approximation Algorithms''. <br> Springer-Verlag, 2001.
|belowstyle = background:#ddf;
|below =  
}}
}}


;Example
This is the webpage for the ''Advanced Algorithms'' class of fall 2018. Students who take this class should check this page periodically for content updates and new announcements.  
:Let <math>X_1,X_2,\ldots,X_m</math> be a set of ''mutually independent'' random variables. Each event <math>A_i</math> is a predicate defined on a number of variables among <math>X_1,X_2,\ldots,X_m</math>. Let <math>v(A_i)</math> be the unique smallest set of variables which determine <math>A_i</math>. The dependency graph <math>D=(V,E)</math> is defined by
:::<math>(i,j)\in E</math> iff <math>v(A_i)\cap v(A_j)\neq \emptyset</math>.


The following lemma, known as the Lovász local lemma, first proved by Erdős and Lovász in 1975, is an extremely powerful tool, as it supplies a way for dealing with rare events.
= Announcement =
* (2018/9/5) 新学期第一次上课。


{{Theorem
= Course info =
|Lovász Local Lemma (symmetric case)|
* '''Instructor ''': 尹一通、郑朝栋
:Let <math>A_1,A_2,\ldots,A_n</math> be a set of events, and assume that the following hold:
:*email: yinyt@nju.edu.cn, chaodong@nju.edu.cn
:#for all <math>1\le i\le n</math>, <math>\Pr[A_i]\le p</math>;
* '''Class meeting''': Wednesday 8am-10am, 仙I-319.
:#the maximum degree of the dependency graph for the events <math>A_1,A_2,\ldots,A_n</math> is <math>d</math>, and
* '''Office hour''': Wednesday 10am-12pm, 计算机系 804.
:::<math>ep(d+1)\le 1</math>.
:Then
::<math>\Pr\left[\bigwedge_{i=1}^n\overline{A_i}\right]>0</math>.
}}


We will prove a general version of the local lemma, where the events <math>A_i</math> are not symmetric. This generalization is due to Spencer.
= Syllabus =
{{Theorem
|Lovász Local Lemma (general case)|
:Let <math>D=(V,E)</math> be the dependency graph of events <math>A_1,A_2,\ldots,A_n</math>. Suppose there exist real numbers <math>x_1,x_2,\ldots, x_n</math> such that <math>0\le x_i<1</math> and for all <math>1\le i\le n</math>,
::<math>\Pr[A_i]\le x_i\prod_{(i,j)\in E}(1-x_j)</math>.
:Then
::<math>\Pr\left[\bigwedge_{i=1}^n\overline{A_i}\right]\ge\prod_{i=1}^n(1-x_i)</math>.
}}


To see that the general LLL implies symmetric LLL, we set <math>x_i=\frac{1}{d+1}</math> for all <math>i=1,2,\ldots,n</math>. Then we have <math>\left(1-\frac{1}{d+1}\right)^d>\frac{1}{\mathrm{e}}</math>.
=== 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。


Assume the condition in the symmetric LLL:
=== Course materials ===
:#for all <math>1\le i\le n</math>, <math>\Pr[A_i]\le p</math>;
* [[高级算法 (Fall 2018) / Course materials|<font size=3>教材和参考书</font>]]
:#<math>ep(d+1)\le 1</math>;
then it is easy to verify that for all <math>1\le i\le n</math>,
:<math>\Pr[A_i]\le p\le\frac{1}{e(d+1)}<\frac{1}{d+1}\left(1-\frac{1}{d+1}\right)^d\le x_i\prod_{(i,j)\in E}(1-x_j)</math>.
Due to the general LLL, we have
:<math>\Pr\left[\bigwedge_{i=1}^n\overline{A_i}\right]\ge\prod_{i=1}^n(1-x_i)=\left(1-\frac{1}{d+1}\right)^n>0</math>.
This proves the symmetric LLL.


Now we prove the general LLL by the original induction proof.
=== 成绩 Grades ===
{{Proof|
* 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。
First, apply the chain rule. We have
* 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。
:<math>\Pr\left[\bigwedge_{i=1}^n\overline{A_i}\right]=\prod_{i=1}^n\Pr\left[\overline{A_i}\mid \bigwedge_{j=1}^{i-1}\overline{A_{j}}\right]=\prod_{i=1}^n\left(1-\Pr\left[{A_i}\mid \bigwedge_{j=1}^{i-1}\overline{A_{j}}\right]\right)</math>.


Next we prove by induction on <math>m</math> that for any set of <math>m</math> events <math>i_1,\ldots,i_m</math>,
=== <font color=red> 学术诚信 Academic Integrity </font>===
:<math>\Pr\left[A_{i_1}\mid \bigwedge_{j=2}^m\overline{A_{i_j}}\right]\le x_{i_1}</math>.
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。
The local lemma follows immediately by the above chain rule.


For <math>m=1</math>, this is obvious because
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”。
:<math>\Pr[A_{i_1}]\le x_{i_1}\prod_{(i_1,j)\in E}(1-x_j)\le x_{i_1}</math>.


For general <math>m</math>, let <math>i_2,\ldots,i_k</math> be the set of vertices adjacent to  <math>i_1</math> in the dependency graph, i.e. event <math>A_{i_1}</math> is mutually independent of <math>A_{i_{k+1}},A_{i_{k+2}},\ldots, A_{i_{m}}</math>.
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。
By conditional probability, we have
:<math>
\Pr\left[A_{i_1}\mid \bigwedge_{j=2}^m\overline{A_{i_j}}\right]
=\frac{\Pr\left[ A_i\wedge \bigwedge_{j=2}^k\overline{A_{i_j}}\mid \bigwedge_{j=k+1}^m\overline{A_{i_j}}\right]}
{\Pr\left[\bigwedge_{j=2}^k\overline{A_{i_j}}\mid \bigwedge_{j=k+1}^m\overline{A_{i_j}}\right]}
</math>.
First, we bound the numerator. Due to that <math>A_{i_1}</math> is mutually independent of <math>A_{i_{k+1}},A_{i_{k+2}},\ldots, A_{i_{m}}</math>, we have
:<math>
\begin{align}
\Pr\left[ A_{i_1}\wedge \bigwedge_{j=2}^k\overline{A_{i_j}}\mid \bigwedge_{j=k+1}^m\overline{A_{i_j}}\right]
&\le\Pr\left[ A_{i_1}\mid \bigwedge_{j=k+1}^m\overline{A_{i_j}}\right]\\
&=\Pr[A_{i_1}]\\
&\le x_{i_1}\prod_{(i_1,j)\in E}(1-x_j).
\end{align}
</math>


Next, we bound the denominator. Applying the chain rule, we have
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。
:<math>
\Pr\left[\bigwedge_{j=2}^k\overline{A_{i_j}}\mid \bigwedge_{j=k+1}^m\overline{A_{i_j}}\right]
=\prod_{j=2}^k\Pr\left[\overline{A_{i_j}}\mid \bigwedge_{\ell=j+1}^m\overline{A_{i_\ell}}\right]
</math>
which by the induction hypothesis, is at least
:<math>
\prod_{j=2}^k(1-x_{i_j})=\prod_{\{i_1,i_j\}\in E}(1-x_j)
</math>
where <math>E</math> is the set of edges in the dependency graph.


Altogether, we prove the induction hypothesis
= Assignments =
:<math>
* TBA
\Pr\left[A_{i_1}\mid \bigwedge_{j=2}^m\overline{A_{i_j}}\right]
\le\frac{x_{i_1}\prod_{(i_1,j)\in E}(1-x_j)}{\prod_{\{i_1,i_j\}\in E}(1-x_j)}\le x_{i_1}.
</math>


Due to the chain rule, it holds that
= Lecture Notes =
:<math>
# [[高级算法 (Fall 2018)/Min-Cut and Max-Cut|Min-Cut and Max-Cut]]
\begin{align}
#:  [[高级算法 (Fall 2018)/Probability Basics|Probability basics]]
\Pr\left[\bigwedge_{i=1}^n\overline{A_i}\right]
&=\prod_{i=1}^n\Pr\left[\overline{A_i}\mid \bigwedge_{j=1}^{i-1}\overline{A_{j}}\right]\\
&=\prod_{i=1}^n\left(1-\Pr\left[A_i\mid \bigwedge_{j=1}^{i-1}\overline{A_{j}}\right]\right)\\
&\ge\prod_{i=1}^n\left(1-x_i\right).
\end{align}
</math>
}}

Revision as of 14:15, 4 September 2018

高级算法
Advanced Algorithms
Instructor
尹一通
郑朝栋
Email yinyt@nju.edu.cn chaodong@nju.edu.cn
office 计算机系 804
Class
Class meetings Wednesday, 8am-10am
仙I-319
Office hours Wednesday, 10am-12pm
计算机系 804（尹一通）、302（郑朝栋）
Textbooks
Motwani and Raghavan.
Randomized Algorithms.
Cambridge Univ Press, 1995.
Vazirani.
Approximation Algorithms.
Springer-Verlag, 2001.
v · d · e

This is the webpage for the Advanced Algorithms class of fall 2018. Students who take this class should check this page periodically for content updates and new announcements.

Announcement

  • (2018/9/5) 新学期第一次上课。

Course info

  • Instructor : 尹一通、郑朝栋
  • email: yinyt@nju.edu.cn, chaodong@nju.edu.cn
  • Class meeting: Wednesday 8am-10am, 仙I-319.
  • Office hour: Wednesday 10am-12pm, 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

  • TBA

Lecture Notes

  1. Min-Cut and Max-Cut
    Probability basics
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