String Algorithm

KMP

char	a	b	a	b	a	b	c	a
index	0	1	2	3	4	5	6	7
value	0	0	1	2	3	4	0	1

"slash"

proper profix: s, sl, sla, slas

proper suffix: h, sh, ash, lash

partial match table（也有人叫失配函数，也有人叫next数组）:

每个index表示字符串是一个子串，子串的下标∈(0, index)

value = "proper profix"和"proper suffix"相同的最长字符串

举例：
index = 3处, proper profix: a, ab, aba;　　proper suffix:b, ab, bab;

value = "ab"的长度 = 2

实现：

void computeLPSArray(char *pat, int M, int *lps)
{
    // length of the previous longest prefix suffix
    int len = 0;
 
    lps[0] = 0; // lps[0] is always 0
 
    // the loop calculates lps[i] for i = 1 to M-1
    int i = 1;
    while (i < M)
    {
        if (pat[i] == pat[len])
        {
            len++;
            lps[i] = len;
            i++;
        }
        else // (pat[i] != pat[len])
        {
            // This is tricky. Consider the example.
            // AAACAAAA and i = 7. The idea is similar 
            // to search step.
            if (len != 0)
            {
                len = lps[len-1];
 
                // Also, note that we do not increment
                // i here
            }
            else // if (len == 0)
            {
                lps[i] = 0;
                i++;
            }
        }
    }
}

模式字符串移动:

当table[partial_match_length] > 1, 移动partial_match_length - table[partial_match_length - 1]


KMP search实现：

void KMPSearch(char *pat, char *txt)
{
    int M = strlen(pat);
    int N = strlen(txt);
 
    // create lps[] that will hold the longest prefix suffix
    // values for pattern
    int lps[M];
 
    // Preprocess the pattern (calculate lps[] array)
    computeLPSArray(pat, M, lps);
 
    int i = 0;  // index for txt[]
    int j  = 0;  // index for pat[]
    while (i < N)
    {
        if (pat[j] == txt[i])
        {
            j++;
            i++;
        }
 
        if (j == M)
        {
            printf("Found pattern at index %d 
", i-j);
            j = lps[j-1];
        }
 
        // mismatch after j matches
        else if (i < N && pat[j] != txt[i])
        {
            // Do not match lps[0..lps[j-1]] characters,
            // they will match anyway
            if (j != 0)
                j = lps[j-1];
            else
                i = i+1;
        }
    }
}

紧凑的实现：

void preparation(char *P, int *f) {
    int m = strlen(P);
    f[0] = f[1] = 0;
    for (int i = 1; i < m; i++) {
        int j = f[i];
        while (j && P[i] != P[j])
              j = f[j];
        f[i + 1] = (P[i] == P[j])? j+1 : 0;
    }
}
void KMP(char *T, char *P, int *f) {
    int n = strlen(T), m = strlen(P);
    preparation(P, f);
    int j = 0;
    for (int i = 0; i < n; i++) {
        while (j && P[i] != T[i]) j = f[j];
        if (P[j] == T[i]) j++;
        if (j == m) answer(i - m + 1);
    }
}

参考：

http://jakeboxer.com/blog/2009/12/13/the-knuth-morris-pratt-algorithm-in-my-own-words/

 http://www.inf.fh-flensburg.de/lang/algorithmen/pattern/kmpen.htm

AC 自动机

背景：基于有限状态自动机

KMP的partial match table那么多叫法什么失败指针，失配函数，就来源于AC自动机。

参考：http://www.cs.uku.fi/~kilpelai/BSA05/lectures/slides04.pdf

http://www.cnblogs.com/en-heng/p/5247903.html