Knuth-Morris-Pratt algorithm

[acid]

Main features

• performs the comparisons from left to right;
• preprocessing phase in O(m) space and time complexity;
• searching phase in O(n+m) time complexity (independent from the alphabet size);
• delay bounded by log(m) where is the golden ratio ( ).

Description


The design of the Knuth-Morris-Pratt algorithm follows a tight analysis of the Morris and Pratt algorithm. Let us look more closely at the Morris-Pratt algorithm. It is possible to improve the length of the shifts.
Consider an attempt at a left position j, that is when the the window is positioned on the text factor y[j .. j+m-1]. Assume that the first mismatch occurs between x[i] and y[i+j] with 0 < i < m. Then, x[0 .. i-1] = y[j .. i+j-1] =u and a = x[i] y[i+j]=b.
When shifting, it is reasonable to expect that a prefix v of the pattern matches some suffix of the portion u of the text. Moreover, if we want to avoid another immediate mismatch, the character following the prefix v in the pattern must be different from a. The longest such prefix v is called the tagged border of u (it occurs at both ends of u followed by different characters in x).
This introduces the notation: let kmpNext[i] be the length of the longest border of x[0 .. i-1] followed by a character c different from x[i] and -1 if no such tagged border exits, for 0 < i m. Then, after a shift, the comparisons can resume between characters x[kmpNext[i]] and y[i+j] without missing any occurrence of x in y, and avoiding a backtrack on the text (see figure 7.1). The value of kmpNext[0] is set to -1.

Figure 7.1: Shift in the Knuth-Morris-Pratt algorithm (v border of u and c b).

The table kmpNext can be computed in O(m) space and time before the searching phase, applying the same searching algorithm to the pattern itself, as if x=y.
The searching phase can be performed in O(m+n) time. The Knuth-Morris-Pratt algorithm performs at most 2n-1 text character comparisons during the searching phase. The delay (maximal number of comparisons for a single text character) is bounded by log(m) where is the golden ratio ( ).



The C code
void preKmp(char *x, int m, int kmpNext[]) {
int i, j;

i = 0;
j = kmpNext[0] = -1;
while (i < m) {
while (j > -1 && x[i] != x[j])
j = kmpNext[j];
i++;
j++;
if (x[i] == x[j])
kmpNext[i] = kmpNext[j];
else
kmpNext[i] = j;
}
}


void KMP(char *x, int m, char *y, int n) {
int i, j, kmpNext[XSIZE];

/* Preprocessing */
preKmp(x, m, kmpNext);

/* Searching */
i = j = 0;
while (j < n) {
while (i > -1 && x[i] != y[j])
i = kmpNext[i];
i++;
j++;
if (i >= m) {
OUTPUT(j - i);
i = kmpNext[i];
}
}
}


The example


Preprocessing phase

The kmpNext table

First attempt G C A T C G C A G A G A G T A T A C A G T A C G 1 2 3 4 G C A G A G A G Shift by: 4 (i-kmpNext[i]=3- -1)
Second attempt G C A T C G C A G A G A G T A T A C A G T A C G 1 G C A G A G A G Shift by: 1 (i-kmpNext[i]=0- -1)
Third attempt G C A T C G C A G A G A G T A T A C A G T A C G 1 2 3 4 5 6 7 8 G C A G A G A G Shift by: 7 (i-kmpNext[i]=8-1)
Fourth attempt G C A T C G C A G A G A G T A T A C A G T A C G 1 G C A G A G A G Shift by: 1 (i-kmpNext[i]=1-0)
Fifth attempt G C A T C G C A G A G A G T A T A C A G T A C G 1 G C A G A G A G Shift by: 1 (i-kmpNext[i]=0- -1)
Sixth attempt G C A T C G C A G A G A G T A T A C A G T A C G 1 G C A G A G A G Shift by: 1 (i-kmpNext[i]=0- -1)
Seventh attempt G C A T C G C A G A G A G T A T A C A G T A C G 1 G C A G A G A G Shift by: 1 (i-kmpNext[i]=0- -1)
Eighth attempt G C A T C G C A G A G A G T A T A C A G T A C G 1 G C A G A G A G Shift by: 1 (i-kmpNext[i]=0- -1)
The Knuth-Morris-Pratt algorithm performs 18 character comparisons on the example.

http://www-igm.univ-mlv.fr/~lecroq/string/examples/exp8.html