From 3ebaeeb191674848d0425942e1b67b285c1973ba Mon Sep 17 00:00:00 2001 From: Ir1dXD Date: Thu, 22 Nov 2018 06:25:49 +0800 Subject: [PATCH] =?utf8?q?=E4=BF=AE=E6=AD=A3=E6=A0=BC=E5=BC=8F=20(#776)?= MIME-Version: 1.0 Content-Type: text/plain; charset=utf8 Content-Transfer-Encoding: 8bit * fix: change
to ` ` * fix: move images to local folder --- docs/ds/bit.md | 26 +++++++++++++------------- docs/geometry/2d.md | 2 +- docs/graph/bridge.md | 6 +++--- docs/graph/images/bridge1.png | Bin 0 -> 19607 bytes docs/graph/images/bridge2.png | Bin 0 -> 17398 bytes docs/graph/images/bridge3.png | Bin 0 -> 15227 bytes docs/math/combination.md | 20 ++++++++++---------- docs/math/stirling.md | 4 ++-- 8 files changed, 29 insertions(+), 29 deletions(-) create mode 100644 docs/graph/images/bridge1.png create mode 100644 docs/graph/images/bridge2.png create mode 100644 docs/graph/images/bridge3.png diff --git a/docs/ds/bit.md b/docs/ds/bit.md index 22de5a9d..c73af8fd 100644 --- a/docs/ds/bit.md +++ b/docs/ds/bit.md @@ -2,8 +2,8 @@ * * * -树状数组和下面的线段树可是亲兄弟了,但他俩毕竟还有一些区别:
-树状数组能有的操作,线段树一定有;
+树状数组和下面的线段树可是亲兄弟了,但他俩毕竟还有一些区别: +树状数组能有的操作,线段树一定有; 线段树有的操作,树状数组不一定有。 这么看来选择线段树不就**「得天下了」**? @@ -22,9 +22,9 @@ 他们上面的参差不齐的剩下的方块就代表 $a$ 的上级——$c$ 数组。 -很显然看出:
-$c[2]$ 管理的是 $a[1]$ & $a[2]$ ;
-$c[4]$ 管理的是 $a[1]$ & $a[2]$ & $a[3]$ & $a[4]$ ;
+很显然看出: +$c[2]$ 管理的是 $a[1]$ & $a[2]$ ; +$c[4]$ 管理的是 $a[1]$ & $a[2]$ & $a[3]$ & $a[4]$ ; $c[6]$ 管理的是 $a[5]$ & $a[6]$ ;$c[8]$ 则管理全部 $8$ 个数。 所以,如果你要算区间和的话,比如说要算 $a[51]$ ~ $a[91]$ 的区间和,暴力算当然可以,那上百万的数,那就 RE 喽。 @@ -47,10 +47,10 @@ int lowbit(int x) { } ``` -`lowbit` 的意思注释说明了,咱们就用这个说法来证明一下 $a[88]$:
-$88_{(10)}=1011000_{(2)}$
-发现第一个 $1$ 以及他后面的 $0$ 组成的二进制是 $1000$
-$1000_{(2)} = 8_{(10)}$
+`lowbit` 的意思注释说明了,咱们就用这个说法来证明一下 $a[88]$: +$88_{(10)}=1011000_{(2)}$ +发现第一个 $1$ 以及他后面的 $0$ 组成的二进制是 $1000$ +$1000_{(2)} = 8_{(10)}$ $1000$ 对应的十进制是 $8$,所以 $c$ 一共管理 $8$ 个 $a$。 这就是 `lowbit` 的用处,仅此而已(但也相当有用)。 @@ -59,9 +59,9 @@ $1000$ 对应的十进制是 $8$,所以 $c$ 一共管理 $8$ 个 $a$。 > $-x$ 代表 $x$ 的负数,计算机中负数使用对应的正数的补码来表示。 -例如 :
-$x =88_{(10)}=1011000_{(2)}$;
-$-x = -88_{(10)} = (0100111_{(2)} + 1_{(2)}) =101000_{(2)}$;
+例如 : +$x =88_{(10)}=1011000_{(2)}$; +$-x = -88_{(10)} = (0100111_{(2)} + 1_{(2)}) =101000_{(2)}$; $x\ \& \ (-x) = 1000_{(2)} = 8_{(10)}$。 神奇吧,我也觉得神奇! @@ -96,5 +96,5 @@ int getsum(int x) // a[1]……a[x]的和 ## 例题 -[传送门](https://www.luogu.org/problemnew/show/P3374)
+[传送门](https://www.luogu.org/problemnew/show/P3374) [传送门 2](https://www.luogu.org/problemnew/show/P3368) diff --git a/docs/geometry/2d.md b/docs/geometry/2d.md index ca9c6c7a..ff5b7778 100644 --- a/docs/geometry/2d.md +++ b/docs/geometry/2d.md @@ -236,7 +236,7 @@ $$ ### 3.7. 极角序 !!! 例题 - [「JOI Spring Camp 2014 Day4」两个人的星座](https://www.ioi-jp.org/camp/2014/2014-sp-tasks/2014-sp-d4.pdf)
+ [「JOI Spring Camp 2014 Day4」两个人的星座](https://www.ioi-jp.org/camp/2014/2014-sp-tasks/2014-sp-d4.pdf) 平面内有 $n$ 个点,有三种颜色,每个点的颜色是三种中的一种。求不相交的三色三角形对数。$6\le n\le 3000$。 如果两个三角形不相交,则一定可以做出两条内公切线,如果相交或内含是做不出内公切线的。三角形的公切线可以类比圆的公切线。 diff --git a/docs/graph/bridge.md b/docs/graph/bridge.md index 523affd7..47153ce3 100644 --- a/docs/graph/bridge.md +++ b/docs/graph/bridge.md @@ -10,13 +10,13 @@ 首先,我们上一个图: -![](https://i.loli.net/2018/08/28/5b853fc3b27bd.png) +![](images/bridge1.png) 很容易的看出割点是 2,而且这个图仅有这一个割点。 首先,我们按照 $DFS$ 序给他打上时间戳(访问的顺序)。 -![](https://i.loli.net/2018/08/28/5b8540a00f313.png) +![](images/bridge2.png) 这些信息被我们保存在一个叫做 `num` 的数组中。 @@ -28,7 +28,7 @@ 另外,如果搜到了自己(在环中),如果他有两个及以上的儿子,那么他一定是割点了,如果只有一个儿子,那么把它删掉,不会有任何的影响。比如下面这个图,此处形成了一个环,从树上来讲它有 2 个儿子: -![](https://i.loli.net/2018/08/28/5b85450b597b5.png) +![](images/bridge3.png) 我们在访问 1 的儿子时候,假设先 $DFS$ 到了 2,然后标记用过,然后递归往下,来到了 4, 4 又来到了 3,当递归回溯的时候,会发现 3 已经被访问过了,所以不是割点。 diff --git a/docs/graph/images/bridge1.png b/docs/graph/images/bridge1.png 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+** 全排列 **: $n$ 个人全部来排队,队长为 $n$。第一个位置可以选 $n$ 个,第二位置可以选 $n-1$ 个,以此类推得: $$ A_n^n = n(n-1)(n-2) \cdots 3 × 2 × 1 = n! $$ -** 部分排列 **:
+** 部分排列 **: $n$ 个人选 $m$ 个来排队 ($m \le n$)。第一个位置可以选 $n$ 个,第二位置可以选 $n-1$ 个,以此类推,第 $m$ 个(最后一个)可以选 $n-m+1$ 个,得: $$ @@ -74,7 +74,7 @@ $$ ### 圆排列 -$n$ 个人全部来围成一圈为 $Q_n^n$,其中已经排好的一圈,从不同位置断开,又变成不同的队列。
+$n$ 个人全部来围成一圈为 $Q_n^n$,其中已经排好的一圈,从不同位置断开,又变成不同的队列。 所以: $$ @@ -123,15 +123,15 @@ $$ ### 不相邻的排列 -$1 \sim n$ 这 $n$ 个自然数中选 $k$ 个,这 $k$ 个数中任何两个数不相邻数的组合有 $C_{n-k+1}^{k}$ 种。
+$1 \sim n$ 这 $n$ 个自然数中选 $k$ 个,这 $k$ 个数中任何两个数不相邻数的组合有 $C_{n-k+1}^{k}$ 种。 证明和上面的相同(其实就是懒得写),请自行证明 XD ### 错位排列(错排) -先看一个小问题:
+先看一个小问题: $5$ 本书,编号分别是 $1,2,3,4,5$,现在要把这 5 本书是放在编号 $1,2,3,4,5$ 的书架上,要求书的编号和书架的编号不一样,请问有多少种不一样的放置方法? -再看一个小问题:
+再看一个小问题: 胸口贴着编号为 $1,2,\cdots,n$ 的 $n$ 个球员分别住在编号为 $1,2,\cdots,n$ 的 $n$ 个房间里面。现规定每个人住一个房间,自己的编号不能和房间的编号一样。 这就是错排问题。当 $n=3$ 时,只能为 312 或 231 这两种。 @@ -141,10 +141,10 @@ $5$ 本书,编号分别是 $1,2,3,4,5$,现在要把这 5 本书是放在编 刚开始所有球员都住在和自己编号一样的房间里面。然后错排开始了,第 $n$ 个球员从第 $n$ 个房间出来。 -第一种情况:
+第一种情况: $n$ 想和 $i(1 \le i \le n-1)$ 其中任何一个球员换房间,其他 $n-2$ 个人换房间的事情,他们就不管了。其他 $n-2$ 个球员的的错排数为 $d[n-2]$,$n$ 可以和前面 $1 \sim n-1$ 对换,所以有 $n-1$ 个 $d[n-2]$。 -第二种情况:
+第二种情况: $n$ 想和 $i(1 \le i \le n-1)$ 其中任何一个球员换房间,但是 $n$ 只想 $i$ 住在第 $N$ 个房间,而 $n$ 不想住第 $I$ 个房间。 可能你会这样想:那么 $n$ 可以让 $j$ 住在第 $I$ 号房间里面,然后 $n$ 住在房间 $J$。抱歉,$j(1 \le j \le n-1,j\neq i)$ 生气 $n$ 为什么一开始就去找 $i$ 不直接来找 $j$。没办法,$n$ 把自己胸口的编码 $N$ 换成了 $I$,他假装自己是 $i$,然后错排 $1 \sim n-1$(也就是 $d[n-2]$)的时候参与进去,这样自己就不会呆在第 $I$ 号房间了。所以有 $n-1$ 个 $d[n-1]$。 @@ -167,12 +167,12 @@ $$ ### 加法原理 -完成一个工程可以有 $n$ 类办法,$a[i](1 \le i \le n)$ 代表第 $i$ 类方法的数目。
+完成一个工程可以有 $n$ 类办法,$a[i](1 \le i \le n)$ 代表第 $i$ 类方法的数目。 那么完成这件事共有 $S=a[1]+a[2]+\cdots +a[n]$ 种不同的方法。 ### 乘法原理 -完成一个工程需要分 $n$ 个步骤,$a[i](1 \le i \le n)$ 代表第 $i$ 个步骤的不同方法数目。
+完成一个工程需要分 $n$ 个步骤,$a[i](1 \le i \le n)$ 代表第 $i$ 个步骤的不同方法数目。 那么完成这件事共有 $S = a[1] \times a[2] \times \cdots \times a[n]$ 种不同的方法。 ### 两原理的区别 diff --git a/docs/math/stirling.md b/docs/math/stirling.md index f81f0e83..fa67e653 100644 --- a/docs/math/stirling.md +++ b/docs/math/stirling.md @@ -1,7 +1,7 @@ ## Stirling 数(子集划分) -根据例题来讲解:
-(2007 普及)将$n$个数$(1,2,…,n)$分成$r$个部分。每个部分至少一个数。将不同划分方法的总数记为$S_n^r$。例如,$S_4^2=7$,这 7 种不同的划分方法依次为 $\{\ (1) , (234) \}\,\{\ (2) , (134) \}\,\{\ (3) , (124) \}\,\{\ (4) , (123) \}\,\{\ (12) , (34) \}\,\{\ (13) , (24) \}\,\{\ (14) , (23) \}$。当$n=6,r=3$时,$S_6^3$=( )
+根据例题来讲解: +(2007 普及)将$n$个数$(1,2,…,n)$分成$r$个部分。每个部分至少一个数。将不同划分方法的总数记为$S_n^r$。例如,$S_4^2=7$,这 7 种不同的划分方法依次为 $\{\ (1) , (234) \}\,\{\ (2) , (134) \}\,\{\ (3) , (124) \}\,\{\ (4) , (123) \}\,\{\ (12) , (34) \}\,\{\ (13) , (24) \}\,\{\ (14) , (23) \}$。当$n=6,r=3$时,$S_6^3$=( ) > 提示:先固定一个数,对于其余的 5 个数考虑$S_5^3$与$S_5^2$,再分这两种情况对原固定的数进行分析。 -- 2.11.0