最长公共子串和最长公共序列

1. 最长公共子串

注意子串是连续的。有下列动态转移方程

c[i][j] = c[i-1][j-1] + 1   when X[i] = Y[j]

c[i][j] = 0   when X[i] != Y[j]

 1 c[100][100];
 2
 3 int LCS(char x[], int len_x, char y[], int len_y){
 4
 5        int max_len = 0;
 6
 7        for(int i =0; i < len_x ; i++){
 8            for(int j = 0; j < len_y;  j++){
 9                  if(x[i] == y[j]){
10                       if(x && y)
11                           c[i][j] = c[i-1][j-1] + 1;
12                       if(x == 0 || y == 0)
13                           c[i][j] = 1;
14                       if(c[i][j] > max_len)
15                            max_len = c[i][j];
16                 }
17            }
18       }
19
20       return max_len;
21 }    

2. 最长公共子序列

子序列不要求连续。

有下列状态转移方程

c[i][j] = c[i-1][j-1] + 1 if x[i] == y[j]

c[i][j] = max{c[i-1][j], c[i][j-1]} if x[i] != y[j]

// x = "+AB"   y = "+CD"

int c[100][100] = 0;

int lcs(char x[], char y[]){
     int m = strlen(x);
     int n = strlen(y);

     for(int i = 1; i<=m; i++)
       for(int j = 1; j <= n; j++){
             if(x[i] == y[j])
                   c[i][j] = c[i-1][j-1] + 1;
             else
                   c[i][j] = max(c[i][j-1], c[i-1][j]);
       }

    return c[m][n];
}