2023.4.16 讲课

Posted 2023-04-16 dingxutong

Hall 定理

描述

一个二分图存在完美匹配（每个左部点都匹配了一个右部点）当且仅当，对于任意左部点构成的集合 \\(W\\)，它的邻域 \\(N_G(W)\\) 总是比它本身大。

必要性显然。

充分性证明：反证法。找到一个未匹配的点，走增广路即可。

推论：二分图的最大匹配为 \\(|X|-(|W|-|N_G(W)|)\\) 的最大值，\\(X\\) 为左部。

在右部加 \\(\\max(|W|-|N_G(W)|)\\) 个连向所有左部点的点，转化为 Hall 定理。

应用

对于一般二分图，用这个定理的复杂度显然难以接受。这个定理一般用于一些特殊二分图的最大匹配计算。

ARC076F Exhausted?

计算机英语讲课笔记06

文章目录

• I. The Result of Sign-in
• II. Numbers
• III. Software Categories

I. The Result of Sign-in

• Let us take a look at the screenshot of the sign-in result
  
• There are thirty nine students in our class. Thirty one students have signed in and eight students have not. Six students asked for leave and two students are late for the class.

II. Numbers

• Numbers are very important in our learning, work and life.
• When we talk about nubmers, there are two ways of using numbers. In one case, we only care about “how many” or “quantity”, and we use “cardinal numbers”. For example, I ask you how many apples you have eaten. You answer that you have eaten six apples. – one, two, three, four, five, six… In the other case, we care about the “order” of the things, and we use “ordinal numbers”. For example, I ask you which season in a year summer is. You answer that summer is the second season in a year. – first, second, third, fourth, fifth, sixth…
• I want to say that he is No. 1 in his learning in our class. There are several ways to express this fact.
• He is the first one in our class.
• He comes first in our class.
• He is second to none in our class.

III. Software Categories

• Software can fall into two categories: System Software and Application Software.
  $Software\begin{array}{l}SystemSoftware\\ ApplicationSoftware\end{array}$
• System software can fall into three categories: OS (Operating System), Language Processors, DBMS (DataBase Management System).
  $SystemSoftware\{\begin{array}{l}OS:OperatingSystem\{\begin{array}{l}Windows\\ Linux\begin{array}{l}Ubuntu\\ CentOS\end{array}\\ MacOS\end{array}\\ LanguageProcessors\{\begin{array}{l}Compiler\\ Interepter\\ Assembler\end{array}\\ DBMS\{\begin{array}{l}SQLDBMS\{\begin{array}{l}MySQL\\ Oracle\\ SQLServer\\ DB2\\ OceanBase\end{array}\\ NoSQLDBMS\{\begin{array}{l}HBase\\ Redis\\ MongoDB\end{array}\end{array}\end{array}$
• OceanBase是阿里巴巴和蚂蚁金服 100% 自主研发的金融级分布式关系数据库，在普通硬件上实现金融级高可用，在金融行业首创“三地五中心”城市级故障自动无损容灾新标准，同时具备在线水平扩展能力，创造了 6100万次/秒处理峰值的业内纪录，在功能、稳定性、可扩展性、性能方面都经历过严格的检验。
• Next time we will talk about Operating Systems.