随机算法 \ 高级算法 (Fall 2016)/Problem Set 1 and 高级算法 (Fall 2018): Difference between pages

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== Problem 1==
{{Infobox
For any <math>\alpha\ge 1</math>, a cut <math>C</math> in an undirected (multi)graph <math>G(V,E)</math>is called an <math>\alpha</math>-min-cut if <math>|C|\le\alpha|C^*|</math> where <math>C^*</math> is a min-cut in <math>G</math>.
|name        = Infobox
|bodystyle    =  
|title        = <font size=3>高级算法
<br>Advanced Algorithms</font>
|titlestyle  =


# Give a lower bound to the probability that Karger's Random Contraction algorithm returns an <math>\alpha</math>-min-cut in a graph <math>G(V,E)</math> of <math>n</math> vertices.
|image        =
# Use the above bound to estimate the number of distinct <math>\alpha</math>-min cuts in <math>G</math>.
|imagestyle  =
|caption      =
|captionstyle =
|headerstyle  = background:#ccf;
|labelstyle  = background:#ddf;
|datastyle    =


== Problem 2==
|header1 =Instructor
Let <math>G(V,E)</math> be an undirected graph with positive edge weights <math>w:E\to\mathbb{Z}^+</math>. Given a partition of <math>V</math> into <math>k</math> disjoint subsets <math>S_1,S_2,\ldots,S_k</math>, we define
|label1  =  
:<math>w(S_1,S_2,\ldots,S_k)=\sum_{uv\in E\atop \exists i\neq j: u\in S_i,v\in S_j}w(uv)</math>
|data1  =  
as the cost of the '''<math>k</math>-cut''' <math>\{S_1,S_2,\ldots,S_k\}</math>. Our goal is to find a <math>k</math>-cut with maximum cost.
|header2 =
# Give a poly-time greedy algorithm for finding the weighted max <math>k</math>-cut. Prove that the approximation ratio is <math>(1-1/k)</math>.
|label2  =  
# Consider the following local search algorithm for the weighted max cut (max 2-cut).
|data2  = 尹一通<br>郑朝栋
  start with an arbitrary bipartition of <math>V</math> into disjoint <math>S_0,S_1</math>;
|header3 =
  while (true) do
|label3  = Email
    if <math>\exists i\in\{0,1\}</math> and <math>v\in S_i</math> such that <font color=red>(______________)</font>
|data3  = yinyt@nju.edu.cn chaodong@nju.edu.cn 
      then <math>v</math> leaves <math>S_i</math> and joins <math>S_{1-i}</math>;
|header4 =
      continue;
|label4= office
    end if
|data4= 计算机系 804
    break;
|header5 = Class
end
|label5  =
:Fill in the blank parenthesis. Give an analysis of the running time of the algorithm. And prove that the approximation ratio is 0.5.
|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 =
}}


== Problem 3==
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.  
Given <math>m</math> subsets <math>S_1,S_2,\ldots, S_m\subseteq U</math> of a universe <math>U</math> of size <math>n</math>, we want to find a <math>C\subseteq\{1,2,\ldots, n\}</math> of fixed size <math>k=|C|</math> with the maximum '''coverage''' <math>\left|\bigcup_{i\in C}S_i\right|</math>.


* Give a poly-time greedy algorithm for the problem. Prove that the approximation ratio is <math>1-(1-1/k)^k>1-1/e</math>.
= Announcement =
* (2018/9/5) 新学期第一次上课。


== Problem 4==
= Course info =
We consider minimum makespan scheduling on parallel identical machines when jobs are subject to '''precedence constraints'''.  
* '''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.


We still want to schedule <math>n</math> jobs <math>j=1,2,\ldots, n</math> on <math>m</math> identical machines, where job <math>j</math> has  processing time <math>p_j</math>. But now a partial order <math>\preceq</math> is defined on jobs, so that if <math>j\prec k</math> then job <math>j</math> must be completely finished before job <math>k</math> begins. The following is a variant of the ''List'' algorithm for this problem.
= Syllabus =
Input: a list of <math>n</math> jobs with processing times <math>p_1,p_2,\ldots, p_n</math>;


whenever a machine becomes idle
=== 先修课程 Prerequisites ===
    assign the next ''available'' job on the list to the machine;
* 必须：离散数学，概率论，线性代数。
* 推荐：算法设计与分析。


Here a job <math>k</math> is available if all jobs <math>j\prec k</math> have already been completely processed.
=== Course materials ===
* [[高级算法 (Fall 2018) / Course materials|<font size=3>教材和参考书</font>]]


* Prove that the approximation ratio is 2.
=== 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。


== Problem 5 ==
=== <font color=red> 学术诚信 Academic Integrity </font>===
For a '''hypergraph''' <math>H(V,E)</math> with vertex set <math>V</math>, every '''hyperedge''' <math>e\in E</math> is a subset <math>e\subset V</math> of vertices, not necessarily of size 2. A hypergraph <math>H(V,E)</math> is '''<math>k</math>-uniform''' if every hyperedge <math>e\in V</math> is of size <math>k=|e|</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。


A hypergraph <math>H(V,E)</math> is said to have '''property B''' (named after Bernstein) if <math>H</math> is 2-coloable; that is, if there is a '''proper 2-coloring''' <math>f:V\to\{{\color{red}R},{\color{blue}B}\}</math> which assigns each vertex one of the two colors <font color=red>Red</font> or <font color=blue>Blue</font>, such that none of the hyperedge is ''monochromatic''.
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”。


# Let <math>H(V,E)</math> be a <math>k</math>-uniform hypergraph in which every hyperedge <math>e\in E</math> shares vertices with at most <math>d</math> other hyperedges.
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。
#*Show that if <math>2\mathrm{e}\cdot (d+1)\le 2^{k}</math>, then <math>H</math> has property B.
 
#*Describe how to use Moser's recursive Fix algorithm to find a proper 2-coloring of <math>H</math>. Give the pseudocode. Prove the condition under which the algorithm can find a 2-coloring of <math>H</math> with high probability.
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。
#*Describe how to use Moser-Tardos random solver to find a proper 2-coloring of <math>H</math>. Give the pseudocode. Prove the condition under which the algorithm can find a 2-coloring of <math>H</math> within bounded expected time.
 
# Let <math>H(V,E)</math> be a hypergraph (not necessarily uniform) with at least <math>n\ge 2</math> vertices satisfying that
= Assignments =
#:<math>\forall v\in V, \sum_{e\ni v}(1-1/k)^{-|e|}2^{-|e|+1}\le \frac{1}{n}</math>.
* TBA
#*Show that <math>H</math> has property B.
 
#*Describe how to use Moser-Tardos random solver to find a proper 2-coloring of <math>H</math>. Give an upper bound on the expected running time.
= Lecture Notes =
# [[高级算法 (Fall 2018)/Min-Cut and Max-Cut|Min-Cut and Max-Cut]]
#:  [[高级算法 (Fall 2018)/Probability Basics|Probability basics]]

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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