KMP DFA前缀功能

Question

I was asked to learn about KMP DFA, and what I've found in my book is that implementation, but our lecturer calls something "the prefix function" all the time. I really can't understand which part is this function here, can someone explain it to me ? I'm sorry if that was asked somewhere, but I couldn't find it.

public class KMP {
private String pat;
private String t;
private int[][] fsm;

public static final int ALPHABET = 256;

public KMP(String pat) {
    this.pat = pat;
    char[] pattern = pat.toCharArray();

    int M = pattern.length;

    fsm = new int[ALPHABET][pattern.length];
    fsm[pattern[0]][0] = 1;

    for(int X = 0, j = 1; j < M; j++) {

        for(int c = 0; c < ALPHABET; c++) {
            fsm[c][j] = fsm[c][X];
        }
        fsm[pattern[j]][j] = j + 1;
        X = fsm[pattern[j]][X];
    }
    display(fsm);
}

public void search(String t) {
    char[] text = t.toCharArray();
    this.t = t;
    int N = text.length;
    int M = pat.length();

    int i, j;
    for(i = 0, j = 0; i < N; i++) {
        j = fsm[t.charAt(i)][j];
        if(j == M) {
            System.out.println("Found at " + (i - M + 1));
            j = 0;
        }
    }
}

Answer 1

The KMP algorithm does not construct a DFA. What you have implemented is looks more like a DFA, which recognizes some string pattern .

The idea behind KMP algorithm is to construct the so called prefix function for the given pattern . And what is this function? It's definition is that for each position i of the string we are interested in the length of the longest suffix of pattern[1..i] , which is also a prefix of the pattern string (0-indexed). This may sound confusing, but here is an example:

The prefix function of pattern = "abacabacada" is pf[] = 0 0 1 0 1 2 3 4 5 0 1 . pf[8] is equal to 5, because the longest suffix of "bacabaca", that is also a prefix of "abacabacada" is "abaca", which has length 5. Analogically, pf[9] = 0 because there is no suffix of bacabacad which is also a prefix of abacabacada (the pattern).

I hope that this explanation makes the prefix function clearer. Some friends call the array, storing the prefix function fl , short for "fail link" because while doing the matching, we use the values in this array only when the characters from text and pattern mismatch.

Here is a clear implementation of the algorithm (in Java).