Substring 匹配和最长公共子序列作为编辑距离的变化问题——Skiena

Question

In Algorithm Design Manual, edit distance is solved by the following algorithm

#define INSERT    1       /* enumerated type symbol for insert */
#define DELETE    2       /* enumerated type symbol for delete */

int string_compare(char *s, char *t, int i, int j)
{
        int k;                  /* counter */
        int opt[3];             /* cost of the three options */
        int lowest_cost;        /* lowest cost */

        if (i == 0) return(j * indel(' '));
        if (j == 0) return(i * indel(' '));

        opt[MATCH] = string_compare(s,t,i-1,j-1) + match(s[i],t[j]);
        opt[INSERT] = string_compare(s,t,i,j-1) + indel(t[j]);
        opt[DELETE] = string_compare(s,t,i-1,j) + indel(s[i]);

        lowest_cost = opt[MATCH];
        for (k=INSERT; k<=DELETE; k++)
                if (opt[k] < lowest_cost) lowest_cost = opt[k];

        return( lowest_cost );
}

I understand everything up to this point but am struggling to understand the following section where substring matching and longest common subsequence are solved as variations of the edit distance problem. I believe I kind of understand the intuition behind them, where the least amount of edits means preserving the "sequences of interest". In the case of substring matching, it is the substring; in the case of the longest common subsequence, it is that common subsequence. However, I don't understand how exactly each problem is solved.

For substring matching, following changes are made:

row_init(int i)
{
    m[0][i].cost = 0; /* note change */
    m[0][i].parent = -1; /* note change */
}
goal_cell(char *s, char *t, int *i, int *j)
{
    int k; /* counter */
    *i = strlen(s) - 1;
    *j = 0;
    for (k=1; k<strlen(t); k++)
        if (m[*i][k].cost < m[*i][*j].cost) *j = k;
    }
}

For longest common subsequence, the following change is made:

int match(char c, char d)
{
    if (c == d) return(0);
    else return(MAXLEN);
}

Would someone care to explain and help me understand this better?

Answer 1

There is an explanation of the substring matching problem in Section 21.4 of the book.

Explanation Screenshot -> https://i.stack.imgur.com/N56LP.png