KMP算法通俗理解

客官°小女子只卖身不卖艺 2022-06-13 00:26 236阅读 0赞

# 有关字符串的基本知识 #

串（string或字符串）是由零个或多个字符组成的有限序列，一般记为![这里写图片描述][20160420204547026] 其中s是串的名，用单引号括起来的字符序列是串的值；ai（1<=i<=n）可以是字母、数值或其他字符；串中字符的数组 n称为串的长度。零个字符的串称为空串，它的长度为0

串中任意个连续的字符组成的子序列称为该串的子串。包含子串的串相应的称为主串。通常称字符在序列中的序号为该字符在串中的位置。子串在主串中的位置则以子串的第一个字符在主串中的位置来表示。

下面主要说一下串的模式匹配算法

# 传统的串匹配法 #

算法的基本思想是：从主串S的第pos个字符起和模式的第一个字符比较，若相等，则继续逐个比较后续字符；否则从主串的下一个字符起再重新和模式的字符比较。依次类推，直至模式T中的每个字符依次和主串S中的一个连续的字符序列相等，则匹配成功，函数值为和模式T中第一个字符相等的字符在主串S中的序号，否则称匹配不成功，函数值为零。   
此算法在最坏情况下的时间复杂度为O(m\*n)

# 模式匹配的一种改进算法（KMP算法） #

## 网上一比较易懂的讲解 ##

[字符串匹配的KMP算法][KMP]

字符串匹配是计算机的基本任务之一。   
举例来说，有一个字符串”BBC ABCDAB ABCDABCDABDE”，我想知道，里面是否包含另一个字符串”ABCDABD”？

许多算法可以完成这个任务，Knuth-Morris-Pratt算法（简称KMP）是最常用的之一。它以三个发明者命名，起头的那个K就是著名科学家Donald Knuth。

下面，我用自己的语言，试图写一篇比较好懂的KMP算法解释。   
1.![这里写图片描述][20160420212633989]

首先，字符串”BBC ABCDAB ABCDABCDABDE”的第一个字符与搜索词”ABCDABD”的第一个字符，进行比较。因为B与A不匹配，所以搜索词后移一位。   
2.   
![这里写图片描述][20160420212704099]   
因为B与A不匹配，搜索词再往后移。   
3.![这里写图片描述][20160420213627024]

就这样，直到字符串有一个字符，与搜索词的第一个字符相同为止。   
4.![这里写图片描述][20160420213638462]

接着比较字符串和搜索词的下一个字符，还是相同。   
5.![这里写图片描述][20160420213652079]

直到字符串有一个字符，与搜索词对应的字符不相同为止。   
6.   
![这里写图片描述][20160420213702016]   
这时，最自然的反应是，将搜索词整个后移一位，再从头逐个比较。这样做虽然可行，但是效率很差，因为你要把”搜索位置”移到已经比较过的位置，重比一遍。   
7.   
![这里写图片描述][20160420213720947]   
一个基本事实是，当空格与D不匹配时，你其实知道前面六个字符是”ABCDAB”。KMP算法的想法是，设法利用这个已知信息，不要把”搜索位置”移回已经比较过的位置，继续把它向后移，这样就提高了效率。   
8.   
![这里写图片描述][20160420213745103]   
怎么做到这一点呢？可以针对搜索词，算出一张《部分匹配表》（Partial Match Table）。这张表是如何产生的，后面再介绍，这里只要会用就可以了。   
9.   
![这里写图片描述][20160420213755650]   
已知空格与D不匹配时，前面六个字符”ABCDAB”是匹配的。查表可知，最后一个匹配字符B对应的”部分匹配值”为2，因此按照下面的公式算出向后移动的位数：   
　　移动位数 = 已匹配的字符数 - 对应的部分匹配值   
因为 6 - 2 等于4，所以将搜索词向后移动4位。   
10.   
![这里写图片描述][20160420213817713]   
因为空格与Ｃ不匹配，搜索词还要继续往后移。这时，已匹配的字符数为2（”AB”），对应的”部分匹配值”为0。所以，移动位数 = 2 - 0，结果为 2，于是将搜索词向后移2位。   
11.   
![这里写图片描述][20160420213841135]   
因为空格与A不匹配，继续后移一位。   
12.   
![这里写图片描述][20160420213851619]   
逐位比较，直到发现C与D不匹配。于是，移动位数 = 6 - 2，继续将搜索词向后移动4位。   
13.   
![这里写图片描述][20160420213904284]   
逐位比较，直到搜索词的最后一位，发现完全匹配，于是搜索完成。如果还要继续搜索（即找出全部匹配），移动位数 = 7 - 0，再将搜索词向后移动7位，这里就不再重复了。   
14.   
![这里写图片描述][20160420213915760]   
下面介绍《部分匹配表》是如何产生的。   
首先，要了解两个概念：”前缀”和”后缀”。 “前缀”指除了最后一个字符以外，一个字符串的全部头部组合；”后缀”指除了第一个字符以外，一个字符串的全部尾部组合。   
15.   
![这里写图片描述][20160420213927972]   
“部分匹配值”就是”前缀”和”后缀”的最长的共有元素的长度。以”ABCDABD”为例，   
　　－　“A”的前缀和后缀都为空集，共有元素的长度为0；   
　　－　“AB”的前缀为\[A\]，后缀为\[B\]，共有元素的长度为0；   
　　－　“ABC”的前缀为\[A, AB\]，后缀为\[BC, C\]，共有元素的长度0；   
　　－　“ABCD”的前缀为\[A, AB, ABC\]，后缀为\[BCD, CD, D\]，共有元素的长度为0；   
　　－　“ABCDA”的前缀为\[A, AB, ABC, ABCD\]，后缀为\[BCDA, CDA, DA, A\]，共有元素为”A”，长度为1；   
　　－　“ABCDAB”的前缀为\[A, AB, ABC, ABCD, ABCDA\]，后缀为\[BCDAB, CDAB, DAB, AB, B\]，共有元素为”AB”，长度为2；   
　　－　“ABCDABD”的前缀为\[A, AB, ABC, ABCD, ABCDA, ABCDAB\]，后缀为\[BCDABD, CDABD, DABD, ABD, BD, D\]，共有元素的长度为0。   
16.   
![这里写图片描述][20160420214001604]   
“部分匹配”的实质是，有时候，字符串头部和尾部会有重复。比如，”ABCDAB”之中有两个”AB”，那么它的”部分匹配值”就是2（”AB”的长度）。搜索词移动的时候，第一个”AB”向后移动4位（字符串长度-部分匹配值），就可以来到第二个”AB”的位置。

--------------------

## 小例子 ##

求字符串 ‘ababaabab’的next和nextval，结果如下

--------------------

## （1）计算next ##

> 计算next的时候需要遵循的规则如下
> 
> > A、 j = 0 next\[j\] = 0   
> > B、如果存在 Max\{ k | k 属于 （1，j-1） 且‘P1…Pk-1’= ‘Pj-k+1…Pj-1’ \} （next\[j\] = k）   
> > C、其他情况 next\[j\] = 1

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   <th align="left" style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> j</th> 
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   <th align="center" style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> str(1,k-1)</th> 
   <th style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> ( j-k+1 , j-1 )</th> 
   <th style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> str( j-k+1, j-1)</th> 
   <th style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> 规则</th> 
   <th style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> next[ j ]</th> 
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   <td align="left" style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> 2</td> 
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   <td align="center" style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> NULL</td> 
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   <td style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> C</td> 
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   <td align="left" style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> 3</td> 
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   <td align="center" style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> a</td> 
   <td style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> (2,2)</td> 
   <td style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> b</td> 
   <td style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> C</td> 
   <td style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> 1</td> 
  </tr> 
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   <td align="left" style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> 4</td> 
   <td align="right" style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> (1,2)</td> 
   <td align="center" style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> ab</td> 
   <td style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> (2,3)</td> 
   <td style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> ba</td> 
   <td style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> &nbsp;</td> 
   <td style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> &nbsp;</td> 
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   <td align="left" style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> 4</td> 
   <td align="right" style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> (1,1)</td> 
   <td align="center" style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> a</td> 
   <td style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> (3,3)</td> 
   <td style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> a</td> 
   <td style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> B</td> 
   <td style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> 2</td> 
  </tr> 
  <tr style=""> 
   <td align="left" style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> 5</td> 
   <td align="right" style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> (1,3)</td> 
   <td align="center" style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> aba</td> 
   <td style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> (2,4)</td> 
   <td style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> bab</td> 
   <td style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> &nbsp;</td> 
   <td style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> &nbsp;</td> 
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  <tr style=""> 
   <td align="left" style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> 5</td> 
   <td align="right" style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> (1,2)</td> 
   <td align="center" style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> ab</td> 
   <td style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> (3,4)</td> 
   <td style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> ab</td> 
   <td style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> B</td> 
   <td style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> 3</td> 
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   <td align="left" style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> 6</td> 
   <td align="right" style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> (1,4)</td> 
   <td align="center" style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> abab</td> 
   <td style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> (2,5)</td> 
   <td style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> baba</td> 
   <td style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> &nbsp;</td> 
   <td style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> &nbsp;</td> 
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   <td align="left" style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> 6</td> 
   <td align="right" style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> (1,3)</td> 
   <td align="center" style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> aba</td> 
   <td style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> (3,5)</td> 
   <td style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> aba</td> 
   <td style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> B</td> 
   <td style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> 4</td> 
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  <tr style=""> 
   <td align="left" style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> 7</td> 
   <td align="right" style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> (1,5)</td> 
   <td align="center" style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> ababa</td> 
   <td style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> (2,6)</td> 
   <td style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> babaa</td> 
   <td style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> &nbsp;</td> 
   <td style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> &nbsp;</td> 
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  <tr style=""> 
   <td align="left" style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> 7</td> 
   <td align="right" style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> (1,4)</td> 
   <td align="center" style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> abab</td> 
   <td style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> (3,6)</td> 
   <td style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> abaa</td> 
   <td style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> &nbsp;</td> 
   <td style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> &nbsp;</td> 
  </tr> 
  <tr style=""> 
   <td align="left" style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> 7</td> 
   <td align="right" style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> (1,3)</td> 
   <td align="center" style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> aba</td> 
   <td style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> (4,6)</td> 
   <td style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> baa</td> 
   <td style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> &nbsp;</td> 
   <td style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> &nbsp;</td> 
  </tr> 
  <tr style=""> 
   <td align="left" style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> 7</td> 
   <td align="right" style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> (1,2)</td> 
   <td align="center" style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> ab</td> 
   <td style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> (5,6)</td> 
   <td style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> aa</td> 
   <td style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> &nbsp;</td> 
   <td style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> &nbsp;</td> 
  </tr> 
  <tr style=""> 
   <td align="left" style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> 7</td> 
   <td align="right" style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> (1,1)</td> 
   <td align="center" style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> a</td> 
   <td style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> (6,6)</td> 
   <td style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> a</td> 
   <td style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> B</td> 
   <td style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> 2</td> 
  </tr> 
  <tr style=""> 
   <td align="left" style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> 8</td> 
   <td align="right" style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> (1,6)</td> 
   <td align="center" style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> ababaa</td> 
   <td style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> (2,7)</td> 
   <td style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> babaab</td> 
   <td style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> &nbsp;</td> 
   <td style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> &nbsp;</td> 
  </tr> 
  <tr style=""> 
   <td align="left" style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> 8</td> 
   <td align="right" style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> (1,5)</td> 
   <td align="center" style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> ababa</td> 
   <td style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> (3,7)</td> 
   <td style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> abaab</td> 
   <td style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> &nbsp;</td> 
   <td style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> &nbsp;</td> 
  </tr> 
  <tr style=""> 
   <td align="left" style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> 8</td> 
   <td align="right" style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> (1,4)</td> 
   <td align="center" style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> abab</td> 
   <td style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> (4,7)</td> 
   <td style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> baab</td> 
   <td style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> &nbsp;</td> 
   <td style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> &nbsp;</td> 
  </tr> 
  <tr style=""> 
   <td align="left" style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> 8</td> 
   <td align="right" style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> (1,3)</td> 
   <td align="center" style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> aba</td> 
   <td style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> (5,7)</td> 
   <td style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> aab</td> 
   <td style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> &nbsp;</td> 
   <td style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> &nbsp;</td> 
  </tr> 
  <tr style=""> 
   <td align="left" style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> 8</td> 
   <td align="right" style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> (1,2)</td> 
   <td align="center" style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> ab</td> 
   <td style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> (6,7)</td> 
   <td style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> ab</td> 
   <td style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> B</td> 
   <td style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> 3</td> 
  </tr> 
  <tr style=""> 
   <td align="left" style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> 9</td> 
   <td align="right" style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> (1,7)</td> 
   <td align="center" style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> ababaab</td> 
   <td style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> (2,8)</td> 
   <td style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> babaaba</td> 
   <td style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> &nbsp;</td> 
   <td style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> &nbsp;</td> 
  </tr> 
  <tr style=""> 
   <td align="left" style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> 9</td> 
   <td align="right" style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> (1,6)</td> 
   <td align="center" style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> ababaa</td> 
   <td style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> (3,8)</td> 
   <td style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> abaaba</td> 
   <td style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> &nbsp;</td> 
   <td style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> &nbsp;</td> 
  </tr> 
  <tr style=""> 
   <td align="left" style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> 9</td> 
   <td align="right" style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> (1,5)</td> 
   <td align="center" style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> ababa</td> 
   <td style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> (4,8)</td> 
   <td style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> baaba</td> 
   <td style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> &nbsp;</td> 
   <td style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> &nbsp;</td> 
  </tr> 
  <tr style=""> 
   <td align="left" style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> 9</td> 
   <td align="right" style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> (1,4)</td> 
   <td align="center" style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> abab</td> 
   <td style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> (5,8)</td> 
   <td style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> aaba</td> 
   <td style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> &nbsp;</td> 
   <td style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> &nbsp;</td> 
  </tr> 
  <tr style=""> 
   <td align="left" style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> 9</td> 
   <td align="right" style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> (1,3)</td> 
   <td align="center" style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> aba</td> 
   <td style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> (6,8)</td> 
   <td style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> aba</td> 
   <td style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> B</td> 
   <td style="padding:8px; line-height:20px; vertical-align:top; border:1px solid rgb(238,238,238)"> 4</td> 
  </tr> 
 </tbody> 
</table>

--------------------

## （2）计算nextval ##

nextval\[i\]的求解需要比较s中next\[i\]所在位置的字符是否与s\[i\]的字符一致，如果一致则用s\[next\[i\]\]的nextval的值作为nextval\[i\]，如果不一致，则用next\[i\]做为nextval\[i\]。

--------------------

## 代码 ##

该代码参考 [数据结构][Link 1]（[C语言][C]版） 严蔚敏 吴伟民 编著。。。

/** * @filename kmp.cc * @Synopsis KMP algorithm * @author XIU * @version 1 * @date 2016-04-21 */
    // 此代码中所用的数组，或者是字符串都是从下标1开始
    
    #include<iostream>
    #include<string.h>
    
    using namespace std;
    
    /* ============================================================================*/
    /** * @Synopsis the next index data of the model string s_mode * * @Param s_mode: the string * @Param next : the next array * @Param len : the length of the string */
    /* ============================================================================*/
    void get_next( string s_mode, int *next, int len )
    {
        int i = 1;
        int j = 0;
        next[1] = 0;
    
        //cout << len << endl;
        while( i<len )
        {
    
            if( j==0 || s_mode[i] == s_mode[j] )
            {
                ++i;
                ++j;
                if( i>len ) break;
                next[i] = j;
                if( j>len ) break;
                //下面的是修正的next算法（nextval）。
                /* if( s_mode[i] != s_mode[j]) next[i] = j; else next[i] = next[j]; */
            }
            else
            {
                j = next[j];
            }
        }
        for( int i=1; i<len; i++ )
        {
            cout << next[i] << " ";
        }
    
        cout << endl; 
    
    }
    
    /* ============================================================================*/
    /** * @Synopsis 利用模式串s_mode中的next函数求s_model在主串 s_primary中第pos个字符之后的位置 * * @Param s_primary * @Param s_mode * @Param pos * @Param next * * @Returns */
    /* ============================================================================*/
    int Index_KMP( string s_primary, string s_mode, int pos, int *next )
    {
        int i = pos;
        int j = 1;
    
        int len_p = s_primary.size();
        int len_m = s_mode.size();
    
        while( i < len_p && j < len_m )
        {
    
            if( j == 0 || s_primary[i] == s_mode[j] )
            {
                ++i;
                ++j;
            }
            else
                j = next[j];
        }
    
        if( j >= len_m )
            return i - len_m;
        else
            return 0;
    }
    
    /* ============================================================================*/
    /** * @Synopsis output function to check the result * * @Param s_primary * @Param s_mode * @Param len * @Param next * @Param index */
    /* ============================================================================*/
    void output( string s_primary, string s_mode, int len, int *next, int index )
    {
        cout << "s_primary = " << s_primary << endl;
        cout << "s_mode = " << s_mode << endl;
    
        for( int i=1; i<len; i++ )
        {
            cout << next[i] << " ";
        }
    
        cout << endl; 
    
        cout << "index = " << index << endl;
    
    }
    int main()
    {
        string s_primary = " acabaabaabcacaabc";
        string s_mode    = " abaabcac";
        int len = s_mode.size() ;
    
        int *next = new int[len];
    
        get_next( s_mode, next, len );
    
        int tmp = Index_KMP( s_primary, s_mode, 1, next );
    
        output( s_primary, s_mode, len, next, tmp );
    
    
        delete [] next;
        return 0;
    }
     
      
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--------------------

参考网址   
【1】字符串匹配的KMP算法 - 阮一峰的网络日志   
[http://www.ruanyifeng.com/blog/2013/05/Knuth%E2%80%93Morris%E2%80%93Pratt\_algorithm.html][KMP]  
【2】KMP算法详解 - joylnwang的专栏 - 博客频道 - CSDN[.NET][]   
[http://blog.csdn.net/joylnwang/article/details/6778316][http_blog.csdn.net_joylnwang_article_details_6778316]   
【3】经典算法研究系列：六、教你初步了解KMP算法、updated - 结构之法 算法之道 - 博客频道 - CSDN[.net][.NET]   
[http://blog.csdn.net/v\_JULY\_v/article/details/6111565][http_blog.csdn.net_v_JULY_v_article_details_6111565]

[20160420204547026]: /images/20220613/7dc4f5a0b1e5416daa8fce8c511ad360.png
[KMP]: http://www.ruanyifeng.com/blog/2013/05/Knuth%E2%80%93Morris%E2%80%93Pratt_algorithm.html
[20160420212633989]: /images/20220613/1cd1cdf8514d4f35a44074ef67604db8.png
[20160420212704099]: /images/20220613/c509e9fd85454e3b9f5dced30cfaaa59.png
[20160420213627024]: /images/20220613/cf4e8cc423a54c87bd290b8cfc4098cd.png
[20160420213638462]: /images/20220613/a1eb12046fcb49a48e55f836ee832f82.png
[20160420213652079]: /images/20220613/7933b2830a2440e08aa7c3cac672963c.png
[20160420213702016]: /images/20220613/92252fbb0b154feb945d29291a47c49a.png
[20160420213720947]: /images/20220613/d0f9b49de91446af91757346a6657037.png
[20160420213745103]: /images/20220613/96fa222f7b3d4d839b3c2f952bcd7c89.png
[20160420213755650]: /images/20220613/53f71059917d40a4bb5962c7eb97493a.png
[20160420213817713]: /images/20220613/d4c5894c50bb49d98fe3d569097391dd.png
[20160420213841135]: /images/20220613/058374196eee4fef8a9017598c6f02ce.png
[20160420213851619]: /images/20220613/0d2cca4f961d405d9c8e6291d8fc5863.png
[20160420213904284]: /images/20220613/22f9db1cc233473bbd4e9d40020060b3.png
[20160420213915760]: /images/20220613/cae11737736340f690d9823559e54cca.png
[20160420213927972]: /images/20220613/017f0da0e4b245378663959b907b4262.png
[20160420214001604]: /images/20220613/bda8e3b5931d4b84b5938efcdb0da3b7.png
[Link 1]: http://lib.csdn.net/base/datastructure
[C]: http://lib.csdn.net/base/c
[.NET]: http://lib.csdn.net/base/dotnet
[http_blog.csdn.net_joylnwang_article_details_6778316]: http://blog.csdn.net/joylnwang/article/details/6778316
[http_blog.csdn.net_v_JULY_v_article_details_6111565]: http://blog.csdn.net/v_JULY_v/article/details/6111565