最大公共子字符串(Longest Common Substring)

最大公共子字符串(Longest Common Substring)

博客分類：  algorithms

Longest Common Substring和Longest Common Subsequence是有區別的

X = <a, b, c, f, b, c>

Y = <a, b, f, c, a, b>

X和Y的Longest Common Sequence爲<a, b, c, b>，長度爲4

X和Y的Longest Common Substring爲 <a, b>長度爲2

其實Substring問題是Subsequence問題的特殊情況，也是要找兩個遞增的下標序列

<i1, i2, ...ik> 和 <j1, j2, ..., jk>使

xi1 == yj1

xi2 == yj2

......

xik == yjk

與Subsequence問題不同的是，Substring問題不光要求下標序列是遞增的，還要求每次

遞增的增量爲1， 即兩個下標序列爲：

<i, i+1, i+2, ..., i+k-1> 和 <j, j+1, j+2, ..., j+k-1> 

 

類比Subquence問題的動態規劃解法，Substring也可以用動態規劃解決，令

c[i][j]表示以X[i]和Y[i]結尾的公共子串的長度，如果X[i]不等於Y[i]，則c[i][j]等於0, 比如

X = <y, e, d, f>

Y = <y, e, k, f>

c[1][1] = 1

c[2][2] = 2

c[3][3] = 0

c[4][4] = 1

動態轉移方程爲：

如果xi == yj， 則 c[i][j] = c[i-1][j-1]+1

如果xi ! = yj,  那麼c[i][j] = 0

 

最後求Longest Common Substring的長度等於

max{c[i][j], 1<=i<=n, 1<=j<=m}

 

#include <stdio.h>
#include <string.h>

//#define DEBUG

#ifdef DEBUG
#define debug(...) printf( __VA_ARGS__)
#else
#define debug(...)
#endif

#define N 250

int     c[N][N];

void print_str(char *s1, char *s2, int i, int j)
{
    if (s1[i] == s2[j]) {
        print_str(s1, s2, i-1, j-1);
        putchar(s1[i]);
    }
}

int common_str(char *s1, char *s2)
{
    int     i, j, n, m, max_c;
    int     x, y;

    n = strlen(s1);
    m = strlen(s2);

    max_c = -1;
    for (i = 1; i <= n; i++) {
        for (j = 1; j <= m; j++) {
            if (s1[i-1] == s2[j-1]) {
                c[i][j] = c[i-1][j-1] + 1;
            }
            else {
                c[i][j] = 0;
            }
            if (c[i][j] > max_c) {
                max_c = c[i][j];
                x = i;
                y = j;
            }
            debug("c[%d][%d] = %d\n", i, j, c[i][j]);
        }
    }

    print_str(s1, s2, x-1, y-1);
    printf("\n");
    return max_c;
}

int main()
{
    char    s1[N], s2[N];

    while (scanf("%s%s", s1, s2) != EOF) {
        debug("%s %s\n", s1, s2);
        printf("%d\n", common_str(s1, s2));
    }
    return 0;
}