KMP Skip Search algorithm

It is possible to make the Skip Search algorithm (see chapter Skip Search algorithm) linear using the two shift tables of Morris-Pratt (see chapter Morris and Pratt algorithm) and Knuth-Morris-Pratt (see chapter Knuth, Morris and Pratt algorithm).

For 1  i  m, mpNext[i] is equal to the length of the longest border of x[0 .. i-1] and mpNext[0]=-1.

For 1  i < m, kmpNext[i] is equal to length of the longest border of x[0 .. i-1] followed by a character different from x[i], kmpNext[0]=-1 and kmpNext[m]=m-per(x).

The lists in the buckets are explicitly stored in a table list.

The preprocessing phase of the KmpSkip Search algorithm is in O(m+) time and space complexity.

A general situation for an attempt during the searching phase is the following ((see figure 30.1):

j is the current text position;

x[i] = y[j];

start = j-i is the possible starting position of an occurrence of x in y;

wall is the rightmost scanned text position;

x[0 .. wall-start-1] = y[start .. wall-1];

 
Figure 30.1: General situation during the searching phase of the linear algorithm.

The comparisons are performed from left to right between x[wall-start .. m-1] and y[wall .. start+m-1] until a mismatch or a whole match occurs. Let k  wall-start be the smallest integer such that x[k]  y[start+k] or k = m if an occurrence of x starts at position start in y.

Then wall takes the value of start+k.

After that the algorithm KmpSkip computes two shifts (two new starting positions): the first one according to the skip algorithm (see algorithm AdvanceSkip for details), this gives us a starting position skipStart, the second one according to the shift table of Knuth-Morris-Pratt, which gives us another starting position kmpStart.

Several cases can arise:

skipStart < kmpStart then a shift according to the skip algorithm is applied which gives a new value for skipStart, and we have to compare again skipStart and kmpStart;

kmpStart < skipStart < wall then a shift according to the shift table of Morris-Pratt is applied. This gives a new value for kmpStart. We have to compare again skipStart and kmpStart;

skipStart = kmpStart then another attempt can be performed with start = skipStart;

kmpStart < wall < skipStart then another attempt can be performed with start = skipStart.

The searching phase of the KmpSkip Search algorithm is in O(n) time.

int attempt(char *y, char *x, int m, int start, int wall) {
   int k;

   k = wall - start;
   while (k < m && x[k] == y[k + start])
      ++k;
   return(k);
}


void KMPSKIP(char *x, int m, char *y, int n) {
   int i, j, k, kmpStart, per, start, wall;
   int kmpNext[XSIZE], list[XSIZE], mpNext[XSIZE],
       z[ASIZE];

   /* Preprocessing */
   preMp(x, m, mpNext);
   preKmp(x, m, kmpNext);
   memset(z, -1, ASIZE*sizeof(int));
   memset(list, -1, m*sizeof(int));
   z[x[0]] = 0;
   for (i = 1; i < m; ++i) {
      list[i] = z[x[i]];
      z[x[i]] = i;
   }

   /* Searching */
   wall = 0;
   per = m - kmpNext[m];
   i = j = -1;
   do {
      j += m;
   } while (j < n && z[y[j]] < 0);
   if (j >= n)
     return;
   i = z[y[j]];
   start = j - i;
   while (start <= n - m) {
      if (start > wall)
         wall = start;
      k = attempt(y, x, m, start, wall);
      wall = start + k;
      if (k == m) {
         OUTPUT(start);
         i -= per;
      }
      else
         i = list[i];
      if (i < 0) {
         do {
            j += m;
         } while (j < n && z[y[j]] < 0);
         if (j >= n)
            return;
         i = z[y[j]];
      }
      kmpStart = start + k - kmpNext[k];
      k = kmpNext[k];
      start = j - i;
      while (start < kmpStart ||
             (kmpStart < start && start < wall)) {
         if (start < kmpStart) {
            i = list[i];
            if (i < 0) {
               do {
                  j += m;
               } while (j < n && z[y[j]] < 0);
               if (j >= n)
                  return;
               i = z[y[j]];
            }
            start = j - i;
         }
         else {
            kmpStart += (k - mpNext[k]);
            k = mpNext[k];
         }
      }
   }
}