高级算法 (Fall 2019)/Min-Cut and Max-Cut and General Circulation(Fall 2021): Difference between pages

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= Graph Cut =
{{Infobox
Let <math>G(V, E)</math> be an undirected graph. A subset <math>C\subseteq E</math> of edges is a '''cut''' of graph <math>G</math> if <math>G</math> becomes ''disconnected'' after deleting all edges in <math>C</math>.
|name        = Infobox
|bodystyle    =  
|title        = 大气环流 <br>
General Circulation of the Atmosphere
|titlestyle  =


Let <math>\{S,T\}</math> be a '''bipartition''' of <math>V</math> into nonempty <math>S,T\subseteq V</math>, where <math>S\cap T=\emptyset</math> and <math>S\cup T=V</math>.  A cut <math>C</math> is specified by this bipartition as
|headerstyle  = background:#ccf;
:<math>C=E(S,T)\,</math>,
|labelstyle  = background:#ddf;
where <math>E(S,T)</math> denotes the set of "crossing edges" with one endpoint in each of <math>S</math> and <math>T</math>, formally defined as
|datastyle    =  
:<math>E(S,T)\triangleq\{uv\in E\mid u\in S, v\in T\}.</math>


Given a graph <math>G</math>, there might be many cuts in <math>G</math>, and we are interested in finding the '''minimum''' or '''maximum''' cut.
|header1 =Instructor
 
|label1  =
= Min-Cut =
|data1  =  
The '''min-cut problem''', also called the '''global minimum cut problem''', is defined as follows.
|header2 =  
{{Theorem|Min-cut problem|
|label2  =
*'''Input''': an undirected graph <math>G(V,E)</math>;
|data2  = 张洋
*'''Output''': a cut <math>C</math> in <math>G</math> with the smallest size <math>|C|</math>.
|header3 =  
}}
|label3  = Email
 
|data3  = yangzhang@nju.edu.cn
Equivalently, the problem asks to find a bipartition of <math>V</math> into disjoint non-empty subsets <math>S</math> and <math>T</math> that minimizes <math>|E(S,T)|</math>.
|header4 =
 
|label4= office
We consider the problem in a slightly more generalized setting, where the input graphs <math>G</math> can be '''multi-graphs''', meaning that there could be multiple edges between two vertices <math>u</math> and <math>v</math>. We call such edges the '''parallel edges'''. The cuts in multi-graphs are defined in the same way as before, and the cost of a cut <math>C</math> is given by the total number of edges (including parallel edges) in <math>C</math>. Equivalently, one may think of a multi-graph as a graph with integer edge weights, and the cost of a cut <math>C</math> is the total weights of all edges in <math>C</math>.
|data4= 仙林大气楼 B410
 
|header5 = Class
A canonical deterministic algorithm for this problem is through the [http://en.wikipedia.org/wiki/Max-flow_min-cut_theorem max-flow min-cut theorem]. The max-flow algorithm finds us a minimum '''<math>s</math>-<math>t</math> cut''', which disconnects a '''source''' <math>s\in V</math> from a '''sink''' <math>t\in V</math>, both specified as part of the input. A global min cut can be found by exhaustively finding the minimum <math>s</math>-<math>t</math> cut for an arbitrarily fixed source <math>s</math> and all possible sink <math>t\neq s</math>. This takes <math>(n-1)\times</math>max-flow time where <math>n=|V|</math> is the number of vertices.
|label5  =
 
|data5  =
The fastest known deterministic algorithm for the minimum cut problem on multi-graphs is the [https://en.wikipedia.org/wiki/Stoer–Wagner_algorithm Stoer–Wagner algorithm], which achieves an <math>O(mn+n^2\log n)</math> time complexity where <math>m=|E|</math> is the total number of edges (counting the parallel edges).
|header6 =
 
|label6  = Class meetings
If we restrict the input to be '''simple graphs''' (meaning there is no parallel edges) with no edge weight, there are better algorithms. A deterministic algorithm of [https://dl.acm.org/citation.cfm?id=2746588 Ken-ichi Kawarabayashi and Mikkel Thorup] published in STOC 2015, achieves the near-linear (in the number of edges) time complexity.
|data6  = 周四 下午 2:00-4:00，线上课程腾讯会议号：5966863228
 
|header7 =
== Karger's ''Contraction'' algorithm ==
|label7  = Place
We will describe a simple and elegant randomized algorithm for the min-cut problem. The algorithm is due to [http://people.csail.mit.edu/karger/ David Karger].
|data7  =
 
|header8 =
Let <math>G(V, E)</math> be a '''multi-graph''', which allows more than one '''parallel edges''' between two distinct vertices <math>u</math> and <math>v</math> but does not allow any '''self-loops''': the edges that adjoin a vertex to itself. A multi-graph <math>G</math> can be represented by an adjacency matrix <math>A</math>, in the way that each non-diagonal entry <math>A(u,v)</math> takes nonnegative integer values instead of just 0 or 1, representing the number of parallel edges between <math>u</math> and <math>v</math> in <math>G</math>, and all diagonal entries <math>A(v,v)=0</math> (since there is no self-loop).
|label8  = Office hours
 
|data8  = 周四 下午4:00-4:30 <br>线上答疑
Given a multi-graph <math>G(V,E)</math> and an edge <math>e\in E</math>, we define the following '''contraction''' operator Contract(<math>G</math>, <math>e</math>), which transform <math>G</math> to a new multi-graph.
|header9 = Reference book
{{Theorem|The contraction operator ''Contract''(<math>G</math>, <math>e</math>)|
|label9  =
:say <math>e=uv</math>:
|data9  =
:*replace <math>\{u,v\}</math> by a new vertex <math>x</math>;
|header10 =
:*for every edge (no matter parallel or not) in the form of <math>uw</math> or <math>vw</math> that connects one of <math>\{u,v\}</math> to a vertex <math>w\in V\setminus\{u,v\}</math> in the graph other than <math>u,v</math>, replace it by a new edge <math>xw</math>;
|label10  =  
:*the reset of the graph does not change.
|data10  = {{Infobox
}}
|name        =
 
|bodystyle  =
In other words, the <math>Contract(G,uv)</math> merges the two vertices <math>u</math> and <math>v</math> into a new vertex <math>x</math> whose incident edges preserves the edges incident to <math>u</math> or <math>v</math> in the original graph <math>G</math> except for the parallel edges between them. Now you should realize why we consider multi-graphs instead of simple graphs, because even if we start with a simple graph without parallel edges, the contraction operator may create parallel edges.
|title        =  
 
|titlestyle  =  
The contraction operator is illustrated by the following picture:
|image        = [[File:James-Circulating.jpg|border|100px]]
[[Image:Contract.png|600px|center]]
|imagestyle  =  
 
|caption      = Introduction to Circulating Atmospheres, <br>''I. James'', Cambridge Press, 1995
Karger's algorithm uses a simple idea:
|captionstyle = }}
*At each step we randomly select an edge in the current multi-graph to contract until there are only two vertices left.
|header11 =
*The parallel edges between these two remaining vertices must be a cut of the original graph.
|label11  =  
*We return this cut and hope that with good chance this gives us a minimum cut.
|data11  = {{Infobox
The following is the pseudocode for Karger's algorithm.
|name        =  
{{Theorem|''RandomContract'' (Karger 1993)|
|bodystyle  =  
:'''Input:''' multi-graph <math>G(V,E)</math>;
|title        =  
----
|titlestyle  =  
:while <math>|V|>2</math> do
|image        = [[File:Oort.jpg|border|100px]]
:* choose an edge <math>uv\in E</math> uniformly at random;
|imagestyle  =
:* <math>G=Contract(G,uv)</math>;
|caption      =Physics of Climate, ''Peixoto, J. P.'' and ''A. H. Oort'', Springer-Verlag New York, 1992
:return <math>C=E</math> (the parallel edges between the only two vertices in <math>V</math>);
|captionstyle =  
}}
 
Another way of looking at the contraction operator Contract(<math>G</math>,<math>e</math>) is that we are dealing with classes of vertices. Let <math>V=\{v_1,v_2,\ldots,v_n\}</math> be the set of all vertices. We start with <math>n</math> vertex classes <math>S_1,S_2,\ldots, S_n</math> with each class <math>S_i=\{v_i\}</math> contains one vertex. By calling <math>Contract(G,uv)</math>, where <math>u\in S_i</math> and <math>v\in S_j</math> for distinct <math>i\neq j</math>, we take union of <math>S_i</math> and <math>S_j</math>. The edges in the contracted multi-graph are the edges that cross between different vertex classes.
 
This view of contraction is illustrated by the following picture:
[[Image:Contract_class.png|600px|center]]
 
The following claim is left as an exercise for the class:
:{|border="2" width="100%" cellspacing="4" cellpadding="3" rules="all" style="margin:1em 1em 1em 0; border:solid 1px #AAAAAA; border-collapse:collapse;empty-cells:show;"
|
*With suitable choice of data structures, each operation <math>Contract(G,e)</math> can be implemented within running time <math>O(n)</math> where <math>n=|V|</math> is the number of vertices.
|}
 
In the above '''''RandomContract''''' algorithm, there are precisely <math>n-2</math> contractions. Therefore, we have the following time upper bound.
{{Theorem|Theorem|
: For any multigraph with <math>n</math> vertices, the running time of the '''''RandomContract''''' algorithm is <math>O(n^2)</math>.
}}
We emphasize that it's the time complexity of a "single running" of the algorithm: later we will see we may need to run this algorithm for many times to guarantee a desirable accuracy.
 
== Analysis of accuracy ==
We now analyze the performance of the above algorithm. Since the algorithm is '''''randomized''''', its output cut is a random variable even when the input is fixed, so ''the output may not always be correct''. We want to give a theoretical guarantee of the chance that the algorithm returns a correct answer on an arbitrary input.
 
More precisely, on an arbitrarily fixed input multi-graph <math>G</math>, we want to answer the following question rigorously:
:<math>p_{\text{correct}}=\Pr[\,\text{a minimum cut is returned by }RandomContract\,]\ge ?</math>
 
To answer this question, we prove a stronger statement: for arbitrarily fixed input multi-graph <math>G</math> and a particular minimum cut <math>C</math> in <math>G</math>,
:<math>p_{C}=\Pr[\,C\mbox{ is returned by }RandomContract\,]\ge ?</math>
Obviously this will imply the previous lower bound for <math>p_{\text{correct}}</math> because the event in <math>p_{C}</math> implies the event in <math>p_{\text{correct}}</math>.
:{|border="2" width="100%" cellspacing="4" cellpadding="3" rules="all" style="margin:1em 1em 1em 0; border:solid 1px #AAAAAA; border-collapse:collapse;empty-cells:show;"
|
*In above argument we use the simple law in probability that <math>\Pr[A]\le \Pr[B]</math> if <math>A\subseteq B</math>, i.e. event <math>A</math> implies event <math>B</math>.
|}
 
We introduce the following notations:
*Let <math>e_1,e_2,\ldots,e_{n-2}</math> denote the sequence of random edges chosen to contract in a running of ''RandomContract'' algorithm.  
*Let <math>G_1=G</math> denote the original input multi-graph. And for <math>i=1,2,\ldots,n-2</math>, let <math>G_{i+1}=Contract(G_{i},e_i)</math> be the multigraph after <math>i</math>th contraction.
Obviously <math>e_1,e_2,\ldots,e_{n-2}</math> are random variables, and they are the ''only'' random choices used in the algorithm: meaning that they along with the input <math>G</math>, uniquely determine the sequence of multi-graphs <math>G_1,G_2,\ldots,G_{n-2}</math> in every iteration as well as the final output.
 
We now compute the probability <math>p_C</math> by decompose it into more elementary events involving <math>e_1,e_2,\ldots,e_{n-2}</math>. This is due to the following proposition.
{{Theorem
|Proposition 1|
:If <math>C</math> is a minimum cut in a multi-graph <math>G</math> and <math>e\not\in C</math>, then <math>C</math> is still a minimum cut in the contracted graph <math>G'=contract(G,e)</math>.
}}
{{Proof|
We first observe that contraction will never create new cuts: every cut in the contracted graph <math>G'</math> must also be a cut in the original graph <math>G</math>.
 
We then observe that a cut <math>C</math> in <math>G</math> "survives" in the contracted graph <math>G'</math> if and only if the contracted edge <math>e\not\in C</math>.
 
Both observations are easy to verify by the definition of contraction operator (in particular, easier to verify if we take the vertex class interpretation). The detailed proofs are left as an exercise.
}}
}}
 
|header12 =
Recall that <math>e_1,e_2,\ldots,e_{n-2}</math> denote the sequence of random edges chosen to contract in a running of ''RandomContract'' algorithm.
|label12  =
 
|data12  = {{Infobox
By Proposition 1, the event <math>\mbox{``}C\mbox{ is returned by }RandomContract\mbox{''}\,</math> is equivalent to the event <math>\mbox{``}e_i\not\in C\mbox{ for all }i=1,2,\ldots,n-2\mbox{''}</math>. Therefore:
|name        =  
:<math>
|bodystyle  =  
\begin{align}
|title        =  
p_C
|titlestyle  =  
&=
|image        = [[File:geoff.jpg|border|100px]]
\Pr[\,C\mbox{ is returned by }{RandomContract}\,]\\
|imagestyle  =
&=
|caption      =Atmospheric and Oceanic Fluid Dynamics: Fundamentals and Large-scale Circulation, ''Vallis, G. K.'', Cambridge University Press, 2006
\Pr[\,e_i\not\in C\mbox{ for all }i=1,2,\ldots,n-2\,]\\
|captionstyle =  
&=
\prod_{i=1}^{n-2}\Pr[e_i\not\in C\mid \forall j<i, e_j\not\in C].
\end{align}
</math>
The last equation is due to the so called '''chain rule''' in probability.
:{|border="2" width="100%" cellspacing="4" cellpadding="3" rules="all" style="margin:1em 1em 1em 0; border:solid 1px #AAAAAA; border-collapse:collapse;empty-cells:show;"
|
*The '''chain rule''', also known as the '''law of progressive conditioning''', is the following proposition: for a sequence of events (not necessarily independent) <math>A_1,A_2,\ldots,A_n</math>,
::<math>\Pr[\forall i, A_i]=\prod_{i=1}^n\Pr[A_i\mid \forall j<i, A_j]</math>.
:It is a simple consequence of the definition of conditional probability. By definition of conditional probability,
::<math>\Pr[A_n\mid \forall j<n]=\frac{\Pr[\forall i, A_i]}{\Pr[\forall j<n, A_j]}</math>,
:and equivalently we have
::<math>\Pr[\forall i, A_i]=\Pr[\forall j<n, A_j]\Pr[A_n\mid \forall j<n]</math>.
:Recursively apply this to <math>\Pr[\forall j<n, A_j]</math> we obtain the chain rule.
|}
 
Back to the analysis of probability <math>p_C</math>.  
 
Now our task is to give lower bound to each <math>p_i=\Pr[e_i\not\in C\mid \forall j<i, e_j\not\in C]</math>. The condition <math>\mbox{``}\forall j<i, e_j\not\in C\mbox{''}</math> means the min-cut <math>C</math> survives all first <math>i-1</math> contractions <math>e_1,e_2,\ldots,e_{i-1}</math>, which due to Proposition 1 means that <math>C</math> is also a min-cut in the multi-graph <math>G_i</math> obtained from applying the first <math>(i-1)</math> contractions.
 
Then the conditional probability <math>p_i=\Pr[e_i\not\in C\mid \forall j<i, e_j\not\in C]</math> is the probability that no edge in <math>C</math> is hit when a uniform random edge in the current multi-graph is chosen assuming that <math>C</math> is a minimum cut in the current multi-graph. Intuitively this probability should be bounded from below, because as a min-cut <math>C</math> should be sparse among all edges. This intuition is justified by the following proposition.
 
{{Theorem
|Proposition 2|
:If <math>C</math> is a min-cut in a multi-graph <math>G(V,E)</math>, then <math>|E|\ge \frac{|V||C|}{2}</math>.
}}
}}
{{Proof|  
|belowstyle = background:#ddf;
:It must hold that the degree of each vertex <math>v\in V</math> is at least <math>|C|</math>, or otherwise the set of edges incident to <math>v</math> forms a cut of size smaller than <math>|C|</math> which separates <math>\{v\}</math> from the rest of the graph, contradicting that <math>C</math> is a min-cut. And the bound <math>|E|\ge \frac{|V||C|}{2}</math> follows directly from applying the [https://en.wikipedia.org/wiki/Handshaking_lemma handshaking lemma] to the fact that every vertex in <math>G</math> has degree at least <math>|C|</math>.
|below =
}}
}}


Let <math>V_i</math> and <math>E_i</math> denote the vertex set and edge set of the multi-graph <math>G_i</math> respectively, and recall that <math>G_i</math> is the multi-graph obtained from applying first <math>(i-1)</math> contractions. Obviously <math>|V_{i}|=n-i+1</math>. And due to Proposition 2, <math>|E_i|\ge \frac{|V_i||C|}{2}</math> if <math>C</math> is still a min-cut in <math>G_i</math>.
The probability <math>p_i=\Pr[e_i\not\in C\mid \forall j<i, e_j\not\in C]</math> can be computed as
:<math>
\begin{align}
p_i
&=1-\frac{|C|}{|E_i|}\\
&\ge1-\frac{2}{|V_i|}\\
&=1-\frac{2}{n-i+1}
\end{align},</math>
where the inequality is due to Proposition 2.


We now can put everything together. We arbitrarily fix the input multi-graph <math>G</math> and any particular minimum cut <math>C</math> in <math>G</math>.
This is the page for the class ''General Circulation of the Atmosphere (大气环流)'' for the Fall 2021 semester. Students who take this class should check this page periodically for content updates and new announcements.
:<math>\begin{align}
p_{\text{correct}}
&=\Pr[\,\text{a minimum cut is returned by }RandomContract\,]\\
&\ge
\Pr[\,C\mbox{ is returned by }{RandomContract}\,]\\
&=
\Pr[\,e_i\not\in C\mbox{ for all }i=1,2,\ldots,n-2\,]\\
&=
\prod_{i=1}^{n-2}\Pr[e_i\not\in C\mid \forall j<i, e_j\not\in C]\\
&\ge
\prod_{i=1}^{n-2}\left(1-\frac{2}{n-i+1}\right)\\
&=
\prod_{k=3}^{n}\frac{k-2}{k}\\
&= \frac{2}{n(n-1)}.
\end{align}</math>


This gives us the following theorem.
= Announcement =
{{Theorem
* 由于疫情影响，本学期前段大气环流课采取线上直播课的形式，  <font color="red" size="2>直播软件为腾讯会议，课程会议号为：5966863228；课程交流QQ群：924139004 </font>。【2021.9.2】
|Theorem|
* 由于国庆调休，下一次大气环流课的时间为10月9日（周日）。【2021.9.30】
: For any multigraph with <math>n</math> vertices, the ''RandomContract'' algorithm returns a minimum cut with probability at least <math>\frac{2}{n(n-1)}</math>.
}}
At first glance this seems to be a miserable chance of success. However, notice that there may be exponential many cuts in a graph (because potentially every nonempty subset <math>S\subset V</math> corresponds to a cut <math>C=E(S,\overline{S})</math>), and Karger's algorithm effectively reduce this exponential-sized space of feasible solutions to a quadratic size one, an exponential improvement!


We can run ''RandomContract'' independently for <math>t=\frac{n(n-1)\ln n}{2}</math> times and return the smallest cut ever returned. The probability that a minimum cut is found is at least:
= Course info =
* '''Instructor ''': 张洋，
:*office: 仙林气象楼 B410
:*email: yangzhang@nju.edu.cn
* '''Class meeting''': 周四 下午 2:00-4:00
* '''Office hour''': 周四 <font color="red" size="2>下午4:00-4:30，线上答疑</font>
* '''Prerequisites''': 动力气象，天气学，气候学
* '''Grading''': 平时作业（50％）＋ 期末考试（50％）
本课程将大致布置4次作业，每次作业一二道题目左右。题目将选择每个课题最具有代表性、需要一定思维强度和动手能力的训练用题目,意在使学生通过顺利完成作业来建立环流系统的物理模型、以对课程内容得到深刻全面地理解和掌握。期末考试题目数量将会比平时作业多,覆盖面更广,但会比作业题目简单,只涉及对基本内容的掌握和对环流理论的直接应用。


:<math>\begin{align}
= Course Slides =
&\quad 1-\Pr[\,\mbox{all }t\mbox{ independent runnings of } RandomContract\mbox{ fails to find a min-cut}\,] \\
''Many figures in the course slides are adapted from the reference books, NOAA and other educational sources. These figures are <font color="red" size="2">for class use only.</font>''
&= 1-\Pr[\,\mbox{a single running of }{RandomContract}\mbox{ fails}\,]^{t} \\
*[2021.9.2]  [http://tcs.nju.edu.cn/yzhang/Course_intro_2021.pdf Course Introduction]
&\ge 1- \left(1-\frac{2}{n(n-1)}\right)^{\frac{n(n-1)\ln n}{2}} \\
*[2021.9.9]  [http://tcs.nju.edu.cn/yzhang/Chap_1_1_2021.pdf Chapter 1_1] (reference reading: PO Chapter 5.1-5.2; James Chapter 2.2, 2.4)
&\ge 1-\frac{1}{n}.
*[2021.9.16]  [http://tcs.nju.edu.cn/yzhang/Chap_1_2_2021.pdf Chapter 1_2] and [http://tcs.nju.edu.cn/yzhang/Chap_2_1_2021.pdf Chapter 2_1] (reference reading: PO Chapter 4.1, 6.3)
\end{align}</math>
*[2021.9.23]  [http://tcs.nju.edu.cn/yzhang/Chap_2_2_2021.pdf Chapter 2_2] (reference reading: PO Chapter 6.3, 6.7-6.8; Lindzen 2005, [http://tcs.nju.edu.cn/yzhang/lINDZEN_CHAP2.pdf Chapter 2])
*[2021.9.30]  [http://tcs.nju.edu.cn/yzhang/Chap_2_3_2021.pdf Chapter 2_3] and [http://tcs.nju.edu.cn/yzhang/Chap_3_1_2021.pdf Chapter 3_1] (reference reading: Lindzen 2005, [http://tcs.nju.edu.cn/yzhang/lINDZEN_CHAP2.pdf Chapter 2]; Lindzen and Farrell 1977, [http://tcs.nju.edu.cn/yzhang/Lindzen_1977.pdf JAS];Vallis Chapter 11.1-11.2; Lindzen 2005, [http://tcs.nju.edu.cn/yzhang/Lindzen_chap7.pdf Chapter 7]; Held and Hou, 1980, [http://tcs.nju.edu.cn/yzhang/held_hou_1980.pdf JAS])
*[2021.10.9]  [http://tcs.nju.edu.cn/yzhang/Chap_3_2_2021.pdf Chapter 3_2] (reference reading: Vallis Chapter 11.1-11.2; Lindzen 2005, [http://tcs.nju.edu.cn/yzhang/Lindzen_chap7.pdf Chapter 7]; Held and Hou, 1980, [http://tcs.nju.edu.cn/yzhang/held_hou_1980.pdf JAS])
*[2021.10.14]  [http://tcs.nju.edu.cn/yzhang/Chap_3_3_2021.pdf Chapter 3_3] (reference reading: Vallis Chapter 11.1-11.2; Lindzen 2005, [http://tcs.nju.edu.cn/yzhang/Lindzen_chap7.pdf Chapter 7]; Lindzen and Hou, 1988, [http://tcs.nju.edu.cn/yzhang/lindzen_hou_1988.pdf JAS])
*[2021.10.21]  [http://tcs.nju.edu.cn/yzhang/Chap_4_1_2021.pdf Chapter 4_1] (reference reading: Vallis Chapter 11.4-11.7; PO Chapter 7.5)


Recall that a running of ''RandomContract'' algorithm takes <math>O(n^2)</math> time. Altogether this gives us a randomized algorithm running in time <math>O(n^4\log n)</math> and find a minimum cut [https://en.wikipedia.org/wiki/With_high_probability '''with high probability'''].
= Assignments =
# [[Assignment 1, Fall 2021|Reanalysis data and the earth's climatology]] [Due:2021.10.21] <font color="red" size="1.5">请在本次作业的截止日期前将本次作业的图片部分和文字部分打包作为附件发送至邮箱 circulation_nju@126.com </font>
# [[Assignment 2, Fall 2021|Simple energy balance climate model]] [Due:2021.11.4]
# [[Assignment 3, Fall 2021|Hadley Cell]] [Due:2021.11.18]


== A Corollary by the Probabilistic Method ==
= Course intro =
The analysis of Karger's algorithm implies the following combinatorial proposition for the number of distinct minimum cuts in a graph.
“大气环流”常指地球大气较大空间范围、较长时间尺度上的空气流动，及其对地球大气热量、动量、能量和水汽的全球输送。虽然从十七、十八世纪起人们就开始研究大尺度的大气运动(如Hadley在1735年提出的信风理论)，但大气环流真正发展成为一门较完备的学科方向却是近半个世纪的事情。随着四五十年代探空资料等高空气象要素的取得，以及六十年代卫星等覆盖全球的观测资料的加入，大气环流的空间结构和时间变化开始被系统、全面地揭示。与此同时，大气环流的数值模拟，也开始成为研究大气环流的一个主要方法，并发展至今成为了解和预估未来气候变化的主要手段。随着观测和模拟手段的进步，大气环流的理论研究也在近三十年开始快速地发展，人们对各种环流系统的维持和变化有了更全面、更深刻、也更为现代的理解。
{{Theorem|Corollary|
:For any graph <math>G(V,E)</math> of <math>n</math> vertices, the number of distinct minimum cuts in <math>G</math> is at most <math>\frac{n(n-1)}{2}</math>.
}}
{{Proof|
Let <math>\mathcal{C}</math> denote the set of all minimum cuts in <math>G</math>. For each min-cut <math>C\in\mathcal{C}</math>, let <math>A_C</math> denote the event "<math>C</math> is returned by ''RandomContract''", whose probability is given by
:<math>p_C=\Pr[A_C]\,</math>.


Clearly we have:
现代的大气环流是大气动力学、天气学和气候学相结合的产物。大气环流，既是各种天气现象产生的背景流场，又是各种气候状态形成的动力机制。大气环流在低频、季节、年际、年代际等时间尺度的变化，不但会引起天气现象的变化，也影响着气候状态的形成。而在大气科学领域面临着诸如全球暖化、气候变化、环流异常等重大科学问题的今天，大气环流研究的重要性被推到了前所未有的高度,大气环流也成为活跃发展又充满挑战的学科方向。
* for any distinct <math>C,D\in\mathcal{C}</math>, <math>A_C\,</math> and <math>A_{D}\,</math> are '''disjoint events'''; and
* the union <math>\bigcup_{C\in\mathcal{C}}A_C</math> is precisely the event "a minimum cut is returned by ''RandomContract''", whose probability is given by
::<math>p_{\text{correct}}=\Pr[\,\text{a minimum cut is returned by } RandomContract\,]</math>.
Due to the [https://en.wikipedia.org/wiki/Probability_axioms#Third_axiom '''additivity of probability'''], it holds that
:<math>
p_{\text{correct}}=\sum_{C\in\mathcal{C}}\Pr[A_C]=\sum_{C\in\mathcal{C}}p_C.
</math>


By the analysis of Karger's algorithm, we know <math>p_C\ge\frac{2}{n(n-1)}</math>. And since <math>p_{\text{correct}}</math> is a well defined probability, due to the [https://en.wikipedia.org/wiki/Probability_axioms#Second_axiom '''unitarity of probability'''], it must hold that <math>p_{\text{correct}}\le 1</math>. Therefore,
本课程将讲述在过去几十年里大气环流在观测、理论和模拟上取得的进展。希望学生借此课程能熟悉大气环流的基本分布和形态，掌握各主要环流系统的维持和变化机制，建立各环流系统形成的物理模型，了解现阶段的大气环流模式，知道大气环流方向有待解决的科学问题。
:<math>1\ge p_{\text{correct}}=\sum_{C\in\mathcal{C}}p_C\ge|\mathcal{C}|\frac{2}{n(n-1)}</math>,
which means <math>|\mathcal{C}|\le\frac{n(n-1)}{2}</math>.
}}


Note that the statement of this theorem has no randomness at all, while the proof consists of a randomized procedure. This is an example of [http://en.wikipedia.org/wiki/Probabilistic_method the probabilistic method].
作为一门课程，大气环流内容的讲述常可以有两条线索。一条是全球尺度上大气热量、动量、能量和水汽的分布与输送，Lorenz（1967）和 Peixoto and Oort（1992）是按此线索介绍大气环流的优秀教材；另一条线索，是各纬度、各区域内大气环流系统的形成、维持和变化机制，James（1995）和 Vallis（2006）是按此线索介绍大气环流的经典讲义。根据现阶段大气环流方向的研究特点，本课程的讲述将主要按照后一种方式来展开，并辅以介绍各环流系统对大气各要素场的输送。在介绍各环流系统时，本课程将以观测、理论和模拟为顺序，从各大气环流系统的观测事实入手，介绍大气环流系统的分布特征和时空变化特征；着重介绍关于环流系统的各种动力学模型和现阶段对环流系统的理解；辅以对环流系统模拟研究的介绍；最后通过三者的对比，讨论各环流系统有待研究的问题。


== Fast Min-Cut ==
= Syllabus =
In the analysis of ''RandomContract'' algorithm, recall that we lower bound the probability <math>p_C</math> that a min-cut <math>C</math> is returned by ''RandomContract'' by the following '''telescopic product''':
本课程具体的内容安排如下：第一章为大气环流的概述，介绍大气环流发展的历史、包含的内容以及大气环流研究的常用观测资料和分析方法。第二章介绍大气环流产生的外部强迫：辐射强迫和下界面过程。第三至六章介绍大气环流中的各个环流系统及它们的动力机制。第七章详细介绍各复杂度的大气环流模式。第八章介绍大气环流领域现阶段最大的一个开放课题：全球暖化背景下的大气环流。这一章既是对前几章所介绍的大气环流理论的应用与检验，又是对未来大气环流研究方向的探讨。借此让学生熟悉并理解大气环流领域亟需解决的课题。具体课程安排和参考书目如下。
:<math>p_C\ge\prod_{i=1}^{n-2}\left(1-\frac{2}{n-i+1}\right)</math>.
== Course schedule ==
Here the index <math>i</math> corresponds to the <math>i</math>th contraction. The factor <math>\left(1-\frac{2}{n-i+1}\right)</math> is decreasing in <math>i</math>, which means:
*大气环流概述 (Introduction) （4课时）
* The probability of success is only getting bad when the graph is getting "too contracted", that is, when the number of remaining vertices is getting small.
*大气环流的外部强迫（3课时）
This motivates us to consider the following alternation to the algorithm: first using random contractions to reduce the number of vertices to a moderately small number, and then recursively finding a min-cut in this smaller instance. This seems just a restatement of exactly what we have been doing. Inspired by the idea of boosting the accuracy via independent repetition, here we apply the recursion on ''two'' smaller instances generated independently.
**辐射强迫 (Radiative forcing)
**下界面过程 (Surface boundaries)
*经向环流系统 (Zonally-averaged circulations)
**Hadley 环流（4课时）
**Ferrel 环流，急流，中纬度的波流相互作用（8课时）
*纬向环流系统（Non-zonal circulations）（6课时）
**Storm tracks
**Monsoon
**ENSO and Walker circulation
*能量和水汽循环 (Angular momentum, energy and water vapor)（2课时）
*不同复杂度的大气环流模式 (General circulation in a hierarchy of models)（2课时）
*全球暖化背景下的大气环流 (General circulation in the global warming scenario)（2课时）


The algorithm obtained in this way is called ''FastCut''. We first define a procedure to randomly contract edges until there are <math>t</math> number of vertices left.
[[点击此处看详细课程安排 (click for more)]]


{{Theorem|''RandomContract''<math>(G, t)</math>|
== References ==
:'''Input:''' multi-graph <math>G(V,E)</math>, and integer <math>t\ge 2</math>;
*观测部分：Peixoto, J. P. and A. H. Oort, 1992: Physics of Climate. Springer-Verlag New York, Inc., 520 pp. 中文译本：气候物理学，1995，吴国雄、刘辉等译校，气象出版社。
----
*综合介绍：James, I., 1995: Introduction to circulating atmospheres. Cambridge University Press, 448 pp.  
:while <math>|V|>t</math> do
*理论部分：Vallis, G. K., 2006: Atmospheric and Oceanic Fluid Dynamics: Fundamentals and Large-scale Circulation. Cambridge University Press. 745 pp.
:* choose an edge <math>uv\in E</math> uniformly at random;
* [[其它参考书目（点击看详情）]]
:* <math>G=Contract(G,uv)</math>;
:return <math>G</math>;
}}
 
The ''FastCut'' algorithm is recursively defined as follows.
{{Theorem|''FastCut''<math>(G)</math>|
:'''Input:''' multi-graph <math>G(V,E)</math>;
----
:if <math>|V|\le 6</math> then return a mincut by brute force;
:else let <math>t=\left\lceil1+|V|/\sqrt{2}\right\rceil</math>;
:: <math>G_1=RandomContract(G,t)</math>;
:: <math>G_2=RandomContract(G,t)</math>;
::return the smaller one of <math>FastCut(G_1)</math> and <math>FastCut(G_2)</math>;
}}
 
As before, all <math>G</math> are multigraphs.
 
Fix a min-cut <math>C</math> in the original multigraph <math>G</math>. By the same analysis as in the case of ''RandomContract'', we have
:<math>
\begin{align}
&\Pr[C\text{ survives all contractions in }RandomContract(G,t)]\\
=
&\prod_{i=1}^{n-t}\Pr[C\text{ survives the }i\text{-th contraction}\mid C\text{ survives the first }(i-1)\text{-th contractions}]\\
\ge
&\prod_{i=1}^{n-t}\left(1-\frac{2}{n-i+1}\right)\\
=
&\prod_{k=t+1}^{n}\frac{k-2}{k}\\
=
&\frac{t(t-1)}{n(n-1)}.
\end{align}
</math>
When <math>t=\left\lceil1+n/\sqrt{2}\right\rceil</math>, this probability is at least <math>1/2</math>. The choice of <math>t</math> is due to our purpose to make this probability at least <math>1/2</math>. You will see this is crucial in the following analysis of accuracy.
 
We denote by <math>A</math> and <math>B</math> the following events:
:<math>
\begin{align}
A:
&\quad C\text{  survives all contractions in }RandomContract(G,t);\\
B:
&\quad\text{size of min-cut is unchanged after }RandomContract(G,t);
\end{align}
</math>
Clearly, <math>A</math> implies <math>B</math> and by above analysis <math>\Pr[B]\ge\Pr[A]\ge\frac{1}{2}</math>.
 
We denote by <math>p(n)</math> the lower bound on the probability that <math>FastCut(G)</math> succeeds for a multigraph of <math>n</math> vertices, that is
:<math>
p(n)
=\min_{G: |V|=n}\Pr[\,FastCut(G)\text{ returns a min-cut in }G\,].
</math>
Suppose that <math>G</math> is the multigraph that achieves the minimum in above definition. The following recurrence holds for <math>p(n)</math>.
:<math>
\begin{align}
p(n)
&=
\Pr[\,FastCut(G)\text{ returns a min-cut in }G\,]\\
&=
\Pr[\,\text{ a min-cut of }G\text{ is returned by }FastCut(G_1)\text{ or }FastCut(G_2)\,]\\
&\ge
1-\left(1-\Pr[B\wedge FastCut(G_1)\text{ returns a min-cut in }G_1\,]\right)^2\\
&\ge
1-\left(1-\Pr[A\wedge FastCut(G_1)\text{ returns a min-cut in }G_1\,]\right)^2\\
&=
1-\left(1-\Pr[A]\Pr[ FastCut(G_1)\text{ returns a min-cut in }G_1\mid A]\right)^2\\
&\ge
1-\left(1-\frac{1}{2}p\left(\left\lceil1+n/\sqrt{2}\right\rceil\right)\right)^2,
\end{align}
</math>
where <math>A</math> and <math>B</math> are defined as above such that <math>\Pr[A]\ge\frac{1}{2}</math>.
 
The base case is that  <math>p(n)=1</math> for <math>n\le 6</math>. By induction it is easy to prove that
:<math>
p(n)=\Omega\left(\frac{1}{\log n}\right).
</math>
 
Recall that we can implement an edge contraction in <math>O(n)</math> time, thus it is easy to verify the following recursion of time complexity:
:<math>
T(n)=2T\left(\left\lceil1+n/\sqrt{2}\right\rceil\right)+O(n^2),
</math>
where <math>T(n)</math> denotes the running time of <math>FastCut(G)</math> on a multigraph <math>G</math> of <math>n</math> vertices.
 
By induction with the base case <math>T(n)=O(1)</math> for <math>n\le 6</math>, it is easy to verify that <math>T(n)=O(n^2\log n)</math>.
 
{{Theorem
|Theorem|
: For any multigraph with <math>n</math> vertices, the ''FastCut'' algorithm returns a minimum cut with probability <math>\Omega\left(\frac{1}{\log n}\right)</math> in time <math>O(n^2\log n)</math>.
}}
 
At this point, we see the name ''FastCut'' is misleading because it is actually slower than the original ''RandomContract'' algorithm, only the chance of successfully finding a min-cut is much better (improved from an <math>\Omega(1/n^2)</math> to an <math>\Omega(1/\log n)</math>).
 
Given any input multi-graph, repeatedly running the ''FastCut'' algorithm independently for some <math>O((\log n)^2)</math> times and returns the smallest cut ever returned, we have an algorithm which runs in time <math>O(n^2\log^3n)</math> and returns a min-cut with probability <math>1-O(1/n)</math>, i.e. with high probability.
 
Recall that the running time of best known deterministic algorithm for min-cut on multi-graph is <math>O(mn+n^2\log n)</math>. On dense graph, the randomized algorithm outperforms the best known deterministic algorithm.
 
Finally, Karger further improves this and obtains a near-linear (in the number of edges) time [https://arxiv.org/abs/cs/9812007 randomized algorithm] for minimum cut in multi-graphs.
 
= Max-Cut=
The '''maximum cut problem''', in short the '''max-cut problem''', is defined as follows.
{{Theorem|Max-cut problem|
*'''Input''': an undirected graph <math>G(V,E)</math>;
*'''Output''': a bipartition of <math>V</math> into disjoint subsets <math>S</math> and <math>T</math> that maximizes <math>|E(S,T)|</math>.
}}
 
The problem is a typical MAX-CSP, an optimization version of the [https://en.wikipedia.org/wiki/Constraint_satisfaction_problem constraint satisfaction problem]. An instance of CSP consists of:
* a set of variables <math>x_1,x_2,\ldots,x_n</math> usually taking values from some finite domain;
* a sequence of constraints (predicates) <math>C_1,C_2,\ldots, C_m</math> defined on those variables.
The MAX-CSP asks to find an assignment of values to variables <math>x_1,x_2,\ldots,x_n</math> which maximizes the number of satisfied constraints.
 
In particular, when the variables <math>x_1,x_2,\ldots,x_n</math> takes Boolean values <math>\{0,1\}</math> and every constraint is a binary constraint <math>\cdot\neq\cdot</math> in the form of <math>x_1\neq x_j</math>, then the MAX-CSP is precisely the max-cut problem.
 
Unlike the min-cut problem, which can be solved in polynomial time, the max-cut is known to be [https://en.wikipedia.org/wiki/NP-hardness '''NP-hard''']. Its decision version is among the [https://en.wikipedia.org/wiki/Karp%27s_21_NP-complete_problems 21 '''NP-complete''' problems found by Karp]. This means we should not hope for a polynomial-time algorithm for solving the problem if [https://en.wikipedia.org/wiki/P_versus_NP_problem a famous conjecture in computational complexity] is correct. And due to another [https://en.wikipedia.org/wiki/BPP_(complexity)#Problems less famous conjecture in computational complexity], randomization alone probably cannot help this situation either.
 
We may compromise our goal and allow algorithm to ''not always find the optimal solution''. However, we still want to guarantee that the algorithm ''always returns a relatively good solution on all possible instances''. This notion is formally captured by '''approximation algorithms''' and '''approximation ratio'''.
 
== Greedy algorithm ==
A natural heuristics for solving the max-cut is to sequentially join the vertices to one of the two disjoint subsets <math>S</math> and <math>T</math> to ''greedily'' maximize the ''current'' number of edges crossing between <math>S</math> and <math>T</math>.
 
To state the algorithm, we overload the definition <math>E(S,T)</math>. Given an undirected graph <math>G(V,E)</math>, for any disjoint subsets <math>S,T\subseteq V</math> of vertices, we define
:<math>E(S,T)=\{uv\in E\mid u\in S, v\in T\}</math>.
 
We also assume that the vertices are ordered arbitrarily as <math>V=\{v_1,v_2,\ldots,v_n\}</math>.
 
The greedy heuristics is then described as follows.
{{Theorem|''GreedyMaxCut''|
:'''Input:''' undirected graph <math>G(V,E)</math>,
:::with an arbitrary order of vertices <math>V=\{v_1,v_2,\ldots,v_n\}</math>;
----
:initially <math>S=T=\emptyset</math>;
:for <math>i=1,2,\ldots,n</math>
::<math>v_i</math> joins one of <math>S,T</math> to maximize the current <math>|E(S,T)|</math> (breaking ties arbitrarily);
}}
 
The algorithm certainly runs in polynomial time.
 
Without any guarantee of how good the solution returned by the algorithm approximates the optimal solution, the algorithm is only a heuristics, not an '''approximation algorithm'''.
 
=== Approximation ratio ===
For now we restrict ourselves to the max-cut problem, although the notion applies more generally.
 
Let <math>G</math> be an arbitrary instance of max-cut problem. Let <math>OPT_G</math> denote the size of the of max-cut in graph <math>G</math>. More precisely,
:<math>OPT_G=\max_{S\subseteq V}|E(S,\overline{S})|</math>.
Let <math>SOL_G</math> be the size of of the cut <math>|E(S,T)|</math> returned by the ''GreedyMaxCut'' algorithm on input graph <math>G</math>.
 
As a maximization problem it is trivial that <math>SOL_G\le OPT_G</math> for all <math>G</math>. To guarantee that the ''GreedyMaxCut'' gives good approximation of optimal solution, we need the other direction:
{{Theorem|Approximation ratio|
:We say that the '''approximation ratio''' of the ''GreedyMaxCut'' algorithm is <math>\alpha</math>, or ''GreedyMaxCut'' is an '''<math>\alpha</math>-approximation''' algorithm, for some <math>0<\alpha\le 1</math>, if
::<math>\frac{SOL_G}{OPT_G}\ge \alpha</math> for every possible instance <math>G</math> of max-cut.
}}
 
With this notion, we now try to analyze the approximation ratio of the ''GreedyMaxCut'' algorithm.
 
A dilemma to apply this notion in our analysis is that in the definition of approximation ratio, we compare the solution returned by the algorithm with the '''optimal solution'''. However, in the analysis we can hardly conduct similar comparisons to the optimal solutions. A fallacy in this logic is that the optimal solutions are '''NP-hard''', meaning there is no easy way to calculate them (e.g. a closed form).
 
A popular step (usually the first step of analyzing approximation ratio) to avoid this dilemma is that instead of directly comparing to the optimal solution, we compare to an '''upper bound''' of the optimal solution (for minimization problem, this needs to be a lower bound), that is, we compare to something which is even better than the optimal solution (which means it cannot be realized by any feasible solution).
 
For the max-cut problem, a simple upper bound to <math>OPT_G</math> is <math>|E|</math>, the number of all edges. This is a trivial upper bound of max-cut since any cut is a subset of edges.
 
Let <math>G(V,E)</math> be the input graph and <math>V=\{v_1,v_2,\ldots,v_n\}</math>. Initially <math>S_1=T_1=\emptyset</math>. And for <math>i=1,2,\ldots,n</math>, we let <math>S_{i+1}</math> and <math>T_{i+1}</math> be the respective <math>S</math> and <math>T</math> after <math>v_i</math> joins one of <math>S,T</math>. More precisely,
* <math>S_{i+1}=S_i\cup\{v_i\}</math> and <math>T_{i+1}=T_i\,</math> if <math>E(S_{i}\cup\{v_i\},T_i)>E(S_{i},T_i\cup\{v_i\})</math>;
* <math>S_{i+1}=S_i\,</math> and <math>T_{i+1}=T_i\cup\{v_i\}</math>  if otherwise.
Finally, the max-cut is given by
:<math>
SOL_G=|E(S_{n+1},T_{n+1})|.
</math>
 
We first observe that we can count the number of edges <math>|E|</math> by summarizing the contributions of individual <math>v_i</math>'s.
{{Theorem|Proposition 1|
:<math>|E| = \sum_{i=1}^n\left(|E(S_i,\{v_i\})|+|E(T_i,\{v_i\})|\right)</math>.
}}
{{Proof|
Note that <math>S_i\cup T_i=\{v_1,v_2,\ldots,v_{i-1}\}</math>, i.e. <math>S_i</math> and <math>T_i</math> together contain precisely those vertices preceding <math>v_i</math>. Therefore, by taking the sum
:<math>\sum_{i=1}^n\left(|E(S_i,\{v_i\})|+|E(T_i,\{v_i\})|\right)</math>,
we effectively enumerate all <math>(v_j,v_i)</math> that <math>v_jv_i\in E</math> and <math>j<i</math>. The total number is precisely <math>|E|</math>.
}}
 
We then observe that the <math>SOL_G</math> can be decomposed into contributions of individual <math>v_i</math>'s in the same way.
{{Theorem|Proposition 2|
:<math>SOL_G = \sum_{i=1}^n\max\left(|E(S_i, \{v_i\})|,|E(T_i, \{v_i\})|\right)</math>.
}}
{{Proof|
It is east to observe that <math>E(S_i,T_i)\subseteq E(S_{i+1},T_{i+1})</math>, i.e. once an edge joins the cut between current <math>S,T</math> it will never drop from the cut in the future.
 
We then define
:<math>\Delta_i= |E(S_{i+1},T_{i+1})|-|E(S_i,T_i)|=|E(S_{i+1},T_{i+1})\setminus E(S_i,T_i)|</math>
to be the contribution of <math>v_i</math> in the final cut.
 
It holds that
:<math>\sum_{i=1}^n\Delta_i=|E(S_{n+1},T_{n+1})|-|E(S_{1},T_{1})|=|E(S_{n+1},T_{n+1})|=SOL_G</math>.
On the other hand, due to the greedy rule:
* <math>S_{i+1}=S_i\cup\{v_i\}</math> and <math>T_{i+1}=T_i\,</math> if <math>E(S_{i}\cup\{v_i\},T_i)>E(S_{i},T_i\cup\{v_i\})</math>;
* <math>S_{i+1}=S_i\,</math> and <math>T_{i+1}=T_i\cup\{v_i\}</math>  if otherwise;
it holds that
:<math>\Delta_i=|E(S_{i+1},T_{i+1})\setminus E(S_i,T_i)| = \max\left(|E(S_i, \{v_i\})|,|E(T_i, \{v_i\})|\right)</math>.
Together the proposition follows.
}}
 
Combining the above Proposition 1 and Proposition 2, we have
:<math>
\begin{align}
SOL_G
&= \sum_{i=1}^n\max\left(|E(S_i, \{v_i\})|,|E(T_i, \{v_i\})|\right)\\
&\ge \frac{1}{2}\sum_{i=1}^n\left(|E(S_i, \{v_i\})|+|E(T_i, \{v_i\})|\right)\\
&=\frac{1}{2}|E|\\
&\ge\frac{1}{2}OPT_G.
\end{align}
</math>
 
{{Theorem|Theorem|
:The ''GreedyMaxCut'' is a <math>0.5</math>-approximation algorithm for the max-cut problem.
}}
 
This is not the best approximation ratio achieved by polynomial-time algorithms for max-cut.
* The best known approximation ratio achieved by any polynomial-time algorithm is achieved by the [http://www-math.mit.edu/~goemans/PAPERS/maxcut-jacm.pdf Goemans-Williamson algorithm], which relies on rounding an [https://en.wikipedia.org/wiki/Semidefinite_programming SDP] relaxation of the max-cut, and achieves an approximation ratio <math>\alpha^*\approx 0.878</math>, where <math>\alpha^*</math> is an irrational whose precise value is given by <math>\alpha^*=\frac{2}{\pi}\inf_{x\in[-1,1]}\frac{\arccos(x)}{1-x}</math>.
* Assuming the [https://en.wikipedia.org/wiki/Unique_games_conjecture unique game conjecture], there does not exist any polynomial-time algorithm for max-cut with approximation ratio <math>\alpha>\alpha^*</math>.
 
== Derandomization by conditional expectation ==
There is a probabilistic interpretation of the greedy algorithm, which may explains why we use greedy scheme for max-cut and why it works for finding an approximate max-cut.
 
Given an undirected graph <math>G(V,E)</math>, let us calculate the average size of cuts in <math>G</math>. For every vertex <math>v\in V</math> let <math>X_v\in\{0,1\}</math> be a ''uniform'' and ''independent'' random bit which indicates whether <math>v</math> joins <math>S</math> or <math>T</math>. This gives us a uniform random bipartition of <math>V</math> into <math>S</math> and <math>T</math>.
 
The size of the random cut <math>|E(S,T)|</math> is given by
:<math>
|E(S,T)| = \sum_{uv\in E} I[X_u\neq X_v],
</math>
where <math>I[X_u\neq X_v]</math> is the Boolean indicator random variable that indicates whether event <math>X_u\neq X_v</math> occurs.
 
Due to '''linearity of expectation''',
:<math>
\mathbb{E}[|E(S,T)|]=\sum_{uv\in E} \mathbb{E}[I[X_u\neq X_v]] =\sum_{uv\in E} \Pr[X_u\neq X_v]=\frac{|E|}{2}.
</math>
Recall that <math>|E|</math> is a trivial upper bound for the max-cut <math>OPT_G</math>. Due to the above argument, we have
:<math>
\mathbb{E}[|E(S,T)|]\ge\frac{OPT_G}{2}.
</math>
:{|border="2" width="100%" cellspacing="4" cellpadding="3" rules="all" style="margin:1em 1em 1em 0; border:solid 1px #AAAAAA; border-collapse:collapse;empty-cells:show;"
|
*In above argument we use a few probability propositions.
: '''linearity of expectation:'''
:: Let <math>\boldsymbol{X}=(X_1,X_2,\ldots,X_n)</math> be a random vector. Then
:::<math>\mathbb{E}\left[\sum_{i=1}^nc_iX_i\right]=\sum_{i=1}^nc_i\mathbb{E}[X_i]</math>,
::where <math>c_1,c_2,\ldots,c_n</math> are scalars.
::That is, the order of computations of expectation and linear (affine) function of a random vector can be exchanged.
::Note that this property ignores the dependency between random variables, and hence is very useful.
:'''Expectation of indicator random variable:'''
::We usually use the notation <math>I[A]</math> to represent the Boolean indicator random variable that indicates whether the event <math>A</math> occurs: i.e. <math>I[A]=1</math> if event <math>A</math> occurs and <math>I[A]=0</math> if otherwise.
::It is easy to see that <math>\mathbb{E}[I[A]]=\Pr[A]</math>. The expectation of an indicator random variable equals the probability of the event it indicates.
|}
 
By above analysis, the average (under uniform distribution) size of all cuts in any graph <math>G</math> must be at least <math>\frac{OPT_G}{2}</math>. Due to '''the probabilistic method''', in particular '''the averaging principle''', there must exists a bipartition of <math>V</math> into <math>S</math> and <math>T</math> whose cut <math>E(S,T)</math> is of size at least <math>\frac{OPT_G}{2}</math>. Then next question is how to find such a bipartition <math>\{S,T\}</math> ''algorithmically''.
 
We still fix an arbitrary order of all vertices as <math>V=\{v_1,v_2,\ldots,v_n\}</math>. Recall that each vertex <math>v_i</math> is associated with a uniform and independent random bit <math>X_{v_i}</math> to indicate whether <math>v_i</math> joins <math>S</math> or <math>T</math>. We want to fix the value of <math>X_{v_i}</math> one after another to construct a bipartition <math>\{\hat{S},\hat{T}\}</math> of <math>V</math> such that
:<math>|E(\hat{S},\hat{T})|\ge\mathbb{E}[|E(S,T)|]\ge\frac{OPT_G}{2}</math>.
 
We start with the first vertex <math>v_i</math> and its random variable <math>X_{v_1}</math>. By the '''law of total expectation''',
:<math>
\mathbb{E}[E(S,T)]=\frac{1}{2}\mathbb{E}[E(S,T)\mid X_{v_1}=0]+\frac{1}{2}\mathbb{E}[E(S,T)\mid X_{v_1}=1].
</math>
There must exist an assignment <math>x_1\in\{0,1\}</math> of <math>X_{v_1}</math> such that
:<math>\mathbb{E}[E(S,T)\mid X_{v_1}=x_1]\ge \mathbb{E}[E(S,T)]</math>.
We can continuously applying this argument. In general, for any <math>i\le n</math> and any particular partial assignment <math>x_1,x_2,\ldots,x_{i-1}\in\{0,1\}</math> of <math>X_{v_1},X_{v_2},\ldots,X_{v_{i-1}}</math>, by the law of total expectation
:<math>
\begin{align}
\mathbb{E}[E(S,T)\mid X_{v_1}=x_1,\ldots, X_{v_{i-1}}=x_{i-1}]
=
&\frac{1}{2}\mathbb{E}[E(S,T)\mid X_{v_1}=x_1,\ldots, X_{v_{i-1}}=x_{i-1}, X_{v_{i}}=0]\\
&+\frac{1}{2}\mathbb{E}[E(S,T)\mid X_{v_1}=x_1,\ldots, X_{v_{i-1}}=x_{i-1}, X_{v_{i}}=1].
\end{align}
</math>
There must exist an assignment <math>x_{i}\in\{0,1\}</math> of <math>X_{v_i}</math> such that
:<math>
\mathbb{E}[E(S,T)\mid X_{v_1}=x_1,\ldots, X_{v_{i}}=x_{i}]\ge \mathbb{E}[E(S,T)\mid X_{v_1}=x_1,\ldots, X_{v_{i-1}}=x_{i-1}].
</math>
By this argument, we can find a sequence <math>x_1,x_2,\ldots,x_n\in\{0,1\}</math> of bits which forms a ''monotone path'':
:<math>
\mathbb{E}[E(S,T)]\le \cdots \le \mathbb{E}[E(S,T)\mid X_{v_1}=x_1,\ldots, X_{v_{i-1}}=x_{i-1}] \le \mathbb{E}[E(S,T)\mid X_{v_1}=x_1,\ldots, X_{v_{i}}=x_{i}] \le \cdots \le  \mathbb{E}[E(S,T)\mid X_{v_1}=x_1,\ldots, X_{v_{n}}=x_{n}].
</math>
We already know the first step of this monotone path <math>\mathbb{E}[E(S,T)]\ge\frac{OPT_G}{2}</math>. And for the last step of the monotone path <math>\mathbb{E}[E(S,T)\mid X_{v_1}=x_1,\ldots, X_{v_{n}}=x_{n}]</math> since all random bits have been fixed, a bipartition <math>(\hat{S},\hat{T})</math> is determined by the assignment <math>x_1,\ldots, x_n</math>, so the expectation has no effect except just retuning the size of that cut <math>|E(\hat{S},\hat{T})|</math>. We found the cut <math>E(\hat{S},\hat{T})</math> such that <math>|E(\hat{S},\hat{T})|\ge \frac{OPT_G}{2}</math>.
 
We translate the procedure of constructing this monotone path of conditional expectation to the following algorithm.
{{Theorem|''MonotonePath''|
:'''Input:''' undirected graph <math>G(V,E)</math>,
:::with an arbitrary order of vertices <math>V=\{v_1,v_2,\ldots,v_n\}</math>;
----
:initially <math>S=T=\emptyset</math>;
:for <math>i=1,2,\ldots,n</math>
::<math>v_i</math> joins one of <math>S,T</math> to maximize the average size of cut conditioning on the choices made so far by the vertices <math>v_1,v_2,\ldots,v_i</math>;
}}
We leave as an exercise to verify that the choice of each <math>v_i</math> (to join which one of <math>S,T</math>) in the ''MonotonePath'' algorithm (which maximizes the average size of cut conditioning on the choices made so far by the vertices <math>v_1,v_2,\ldots,v_i</math>) must be the same choice made by <math>v_i</math> in the ''GreedyMaxCut'' algorithm (which maximizes the current <math>|E(S,T)|</math>).
 
Therefore, the greedy algorithm for max-cut is actually due to a derandomization of average-case.
 
== Derandomization by pairwise independence ==
We still construct a random bipartition of <math>V</math> into <math>S</math> and <math>T</math>. But this time the random choices have '''bounded independence'''.
 
For each vertex <math>v\in V</math>, we use a Boolean random variable <math>Y_v\in\{0,1\}</math> to indicate whether <math>v</math> joins <math>S</math> and <math>T</math>. The dependencies between <math>Y_v</math>'s are to be specified later.
 
By linearity of expectation, regardless of the dependencies between <math>Y_v</math>'s, it holds that:
:<math>
\mathbb{E}[|E(S,T)|]=\sum_{uv\in E} \Pr[Y_u\neq Y_v].
</math>
In order to have the average cut <math>\mathbb{E}[|E(S,T)|]=\frac{|E|}{2}</math> as the fully random case, we need <math>\Pr[Y_u\neq Y_v]=\frac{1}{2}</math>. This only requires that the Boolean random variables <math>Y_v</math>'s are uniform and '''pairwise independent''' instead of being '''mutually independent'''.
 
The <math>n</math> pairwise independent random bits <math>\{Y_v\}_{v\in V}</math> can be constructed by at most <math>k=\lceil\log (n+1)\rceil</math> mutually independent random bits <math>X_1,X_2,\ldots,X_k\in\{0,1\}</math> by the following standard routine.
 
{{Theorem|Theorem|
:Let <math>X_1, X_2, \ldots, X_k\in\{0,1\}</math> be mutually independent uniform random bits.
:Let <math>S_1, S_2, \ldots, S_{2^k-1}\subseteq \{1,2,\ldots,k\}</math> enumerate the <math>2^k-1</math> nonempty subsets of <math>\{1,2,\ldots,k\}</math>.
:For each <math>i\le i\le2^k-1</math>, let
::<math>Y_i=\bigoplus_{j\in S_i}X_j=\left(\sum_{j\in S_i}X_j\right)\bmod 2.</math>
:Then <math>Y_1,Y_2,\ldots,Y_{2^k-1}</math> are pairwise independent uniform random bits.
}}
 
If <math>Y_v</math> for each vertex <math>v\in V</math> is constructed in this way by at most <math>k=\lceil\log (n+1)\rceil</math> mutually independent random bits <math>X_1,X_2,\ldots,X_k\in\{0,1\}</math>, then they are uniform and pairwise independent, which by the above calculation, it holds for the corresponding bipartition <math>\{S,T\}</math> of <math>V</math> that
:<math>
\mathbb{E}[|E(S,T)|]=\sum_{uv\in E} \Pr[Y_u\neq Y_v]=\frac{|E|}{2}.
</math>
Note that the average is taken over the random choices of <math>X_1,X_2,\ldots,X_k\in\{0,1\}</math> (because they are the only random choices used to construct the bipartition <math>\{S,T\}</math>). By the probabilistic method, there must exist an assignment of <math>X_1,X_2,\ldots,X_k\in\{0,1\}</math> such that the corresponding <math>Y_v</math>'s and the bipartition <math>\{S,T\}</math> of <math>V</math> indicated by the <math>Y_v</math>'s have that
:<math>|E(S,T)|\ge \frac{|E|}{2}\ge\frac{OPT}{2}</math>.
 
This gives us the following algorithm for exhaustive search in a smaller solution space of size <math>2^k-1=O(n^2)</math>.
{{Theorem|Algorithm|
:Enumerate vertices as <math>V=\{v_1,v_2,\ldots,v_n\}</math>;
:let <math>k=\lceil\log (n+1)\rceil</math>;
:for all <math>\vec{x}\in\{0,1\}^k</math>
::initialize <math>S_{\vec{x}}=T_{\vec{x}}=\emptyset</math>;
::for <math>i=1, 2, \ldots, n</math>
:::if <math>\bigoplus_{j:\lfloor i/2^j\rfloor\bmod 2=1}x_j=1</math> then <math>v_i</math> joins <math>S_{\vec{x}}</math>;
:::else <math>v_i</math> joins <math>T_{\vec{x}}</math>;
:return the <math>\{S_{\vec{x}},T_{\vec{x}}\}</math> with the largest <math>|E(S_{\vec{x}},T_{\vec{x}})|</math>;
}}
The algorithm has approximation ratio 1/2 and runs in polynomial time.

Revision as of 09:58, 21 October 2021

大气环流
General Circulation of the Atmosphere
Instructor
张洋
Email yangzhang@nju.edu.cn
office 仙林大气楼 B410
Class
Class meetings 周四 下午 2:00-4:00，线上课程腾讯会议号：5966863228
Office hours 周四 下午4:00-4:30
线上答疑
Reference book
File:James-Circulating.jpg
Introduction to Circulating Atmospheres,
I. James, Cambridge Press, 1995
File:Oort.jpg
Physics of Climate, Peixoto, J. P. and A. H. Oort, Springer-Verlag New York, 1992
File:Geoff.jpg
Atmospheric and Oceanic Fluid Dynamics: Fundamentals and Large-scale Circulation, Vallis, G. K., Cambridge University Press, 2006
v · d · e


This is the page for the class General Circulation of the Atmosphere (大气环流) for the Fall 2021 semester. Students who take this class should check this page periodically for content updates and new announcements.

Announcement

  • 由于疫情影响，本学期前段大气环流课采取线上直播课的形式， 直播软件为腾讯会议，课程会议号为：5966863228；课程交流QQ群：924139004 。【2021.9.2】
  • 由于国庆调休，下一次大气环流课的时间为10月9日（周日）。【2021.9.30】

Course info

  • Instructor : 张洋，
  • office: 仙林气象楼 B410
  • email: yangzhang@nju.edu.cn
  • Class meeting: 周四 下午 2:00-4:00
  • Office hour: 周四 下午4:00-4:30，线上答疑
  • Prerequisites: 动力气象，天气学，气候学
  • Grading: 平时作业（50％）＋ 期末考试（50％）

本课程将大致布置4次作业，每次作业一二道题目左右。题目将选择每个课题最具有代表性、需要一定思维强度和动手能力的训练用题目,意在使学生通过顺利完成作业来建立环流系统的物理模型、以对课程内容得到深刻全面地理解和掌握。期末考试题目数量将会比平时作业多,覆盖面更广,但会比作业题目简单,只涉及对基本内容的掌握和对环流理论的直接应用。

Course Slides

Many figures in the course slides are adapted from the reference books, NOAA and other educational sources. These figures are for class use only.

Assignments

  1. Reanalysis data and the earth's climatology [Due:2021.10.21] 请在本次作业的截止日期前将本次作业的图片部分和文字部分打包作为附件发送至邮箱 circulation_nju@126.com
  2. Simple energy balance climate model [Due:2021.11.4]
  3. Hadley Cell [Due:2021.11.18]

Course intro

“大气环流”常指地球大气较大空间范围、较长时间尺度上的空气流动，及其对地球大气热量、动量、能量和水汽的全球输送。虽然从十七、十八世纪起人们就开始研究大尺度的大气运动(如Hadley在1735年提出的信风理论)，但大气环流真正发展成为一门较完备的学科方向却是近半个世纪的事情。随着四五十年代探空资料等高空气象要素的取得，以及六十年代卫星等覆盖全球的观测资料的加入，大气环流的空间结构和时间变化开始被系统、全面地揭示。与此同时，大气环流的数值模拟，也开始成为研究大气环流的一个主要方法，并发展至今成为了解和预估未来气候变化的主要手段。随着观测和模拟手段的进步，大气环流的理论研究也在近三十年开始快速地发展，人们对各种环流系统的维持和变化有了更全面、更深刻、也更为现代的理解。

现代的大气环流是大气动力学、天气学和气候学相结合的产物。大气环流，既是各种天气现象产生的背景流场，又是各种气候状态形成的动力机制。大气环流在低频、季节、年际、年代际等时间尺度的变化，不但会引起天气现象的变化，也影响着气候状态的形成。而在大气科学领域面临着诸如全球暖化、气候变化、环流异常等重大科学问题的今天，大气环流研究的重要性被推到了前所未有的高度,大气环流也成为活跃发展又充满挑战的学科方向。

本课程将讲述在过去几十年里大气环流在观测、理论和模拟上取得的进展。希望学生借此课程能熟悉大气环流的基本分布和形态，掌握各主要环流系统的维持和变化机制，建立各环流系统形成的物理模型，了解现阶段的大气环流模式，知道大气环流方向有待解决的科学问题。

作为一门课程，大气环流内容的讲述常可以有两条线索。一条是全球尺度上大气热量、动量、能量和水汽的分布与输送，Lorenz（1967）和 Peixoto and Oort（1992）是按此线索介绍大气环流的优秀教材；另一条线索，是各纬度、各区域内大气环流系统的形成、维持和变化机制，James（1995）和 Vallis（2006）是按此线索介绍大气环流的经典讲义。根据现阶段大气环流方向的研究特点，本课程的讲述将主要按照后一种方式来展开，并辅以介绍各环流系统对大气各要素场的输送。在介绍各环流系统时，本课程将以观测、理论和模拟为顺序，从各大气环流系统的观测事实入手，介绍大气环流系统的分布特征和时空变化特征；着重介绍关于环流系统的各种动力学模型和现阶段对环流系统的理解；辅以对环流系统模拟研究的介绍；最后通过三者的对比，讨论各环流系统有待研究的问题。

Syllabus

本课程具体的内容安排如下：第一章为大气环流的概述，介绍大气环流发展的历史、包含的内容以及大气环流研究的常用观测资料和分析方法。第二章介绍大气环流产生的外部强迫：辐射强迫和下界面过程。第三至六章介绍大气环流中的各个环流系统及它们的动力机制。第七章详细介绍各复杂度的大气环流模式。第八章介绍大气环流领域现阶段最大的一个开放课题：全球暖化背景下的大气环流。这一章既是对前几章所介绍的大气环流理论的应用与检验，又是对未来大气环流研究方向的探讨。借此让学生熟悉并理解大气环流领域亟需解决的课题。具体课程安排和参考书目如下。

Course schedule

  • 大气环流概述 (Introduction) （4课时）
  • 大气环流的外部强迫（3课时）
    • 辐射强迫 (Radiative forcing)
    • 下界面过程 (Surface boundaries)
  • 经向环流系统 (Zonally-averaged circulations)
    • Hadley 环流（4课时）
    • Ferrel 环流，急流，中纬度的波流相互作用（8课时）
  • 纬向环流系统（Non-zonal circulations）（6课时）
    • Storm tracks
    • Monsoon
    • ENSO and Walker circulation
  • 能量和水汽循环 (Angular momentum, energy and water vapor)（2课时）
  • 不同复杂度的大气环流模式 (General circulation in a hierarchy of models)（2课时）
  • 全球暖化背景下的大气环流 (General circulation in the global warming scenario)（2课时）

点击此处看详细课程安排 (click for more)

References

  • 观测部分：Peixoto, J. P. and A. H. Oort, 1992: Physics of Climate. Springer-Verlag New York, Inc., 520 pp. 中文译本：气候物理学，1995，吴国雄、刘辉等译校，气象出版社。
  • 综合介绍：James, I., 1995: Introduction to circulating atmospheres. Cambridge University Press, 448 pp.
  • 理论部分：Vallis, G. K., 2006: Atmospheric and Oceanic Fluid Dynamics: Fundamentals and Large-scale Circulation. Cambridge University Press. 745 pp.
  • 其它参考书目（点击看详情）
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