Alpha Skip Search algorithm

The preprocessing phase of the Alpha Skip Search algorithm consists in building a trie T(x) of all the factors of the length ell =logm occurring in the word x. The leaves of T(x) represent all the factors of length ell of x. There is then one bucket for each leaf of T(x) in which is stored the list of positions where the factor, associated to the leaf, occurs in x.

The worst case time of this preprocessing phase is linear if the alphabet size is considered to be a constant.

The searching phase consists in looking into the buckets of the text factors y[j .. j+ ell -1] for all j = k.(m- ell +1)-1 with the integer k in the interval y[1, lfloor (n- ell ) / m rfloor ].

The worst case time complexity of the searching phase is quadratic but the expected number of text character comparisons is O(log(m).(n / (m-log(m)))).

The description of a linked list List can be found in the introduction section implementation.

List *z;
 
#define getZ(i) z[(i)]
 
void setZ(int node, int i) {
   List cell;
 
   cell = (List)malloc(sizeof(struct _cell));
   if (cell == NULL)
      error("ALPHASKIP/setZ");
   cell->element = i;
   cell->next = z[node];
   z[node] = cell;
}
 
 
/* Create the transition labelled by the
    character c from node node.
   Maintain the suffix links accordingly. */
int addNode(Graph trie, int art, int node, char c) {
   int childNode, suffixNode, suffixChildNode;
 
   childNode = newVertex(trie);
   setTarget(trie, node, c, childNode);
   suffixNode = getSuffixLink(trie, node);
   if (suffixNode == art)
      setSuffixLink(trie, childNode, node);
   else {
      suffixChildNode = getTarget(trie, suffixNode, c);
      if (suffixChildNode == UNDEFINED)
         suffixChildNode = addNode(trie, art,
                                   suffixNode, c);
      setSuffixLink(trie, childNode, suffixChildNode);
   }
   return(childNode);
}


void ALPHASKIP(char *x, int m, char *y, int n, int a) {
   int b, i, j, k, logM, temp, shift, size, pos;
   int art, childNode, node, root, lastNode;
   List current;
   Graph trie;
 
   logM = 0;
   temp = m;
   while (temp > a) {
      ++logM;
      temp /= a;
   }
   if (logM == 0) logM = 1;
   
 
   /* Preprocessing */
   size = 2 + (2*m - logM + 1)*logM;
   trie = newTrie(size, size*ASIZE);
   z = (List *)calloc(size, sizeof(List));
   if (z == NULL)
      error("ALPHASKIP");
 
   root = getInitial(trie);
   art = newVertex(trie);
   setSuffixLink(trie, root, art);
   node = newVertex(trie);
   setTarget(trie, root, x[0], node);
   setSuffixLink(trie, node, root);
   for (i = 1; i < logM; ++i)
      node = addNode(trie, art, node, x[i]);
   pos = 0;
   setZ(node, pos);
   pos++;
   for (i = logM; i < m - 1; ++i) {
      node = getSuffixLink(trie, node);
      childNode = getTarget(trie, node, x[i]);
      if (childNode == UNDEFINED)
         node = addNode(trie, art, node, x[i]);
      else
         node = childNode;
      setZ(node, pos);
      pos++;
   }
   node = getSuffixLink(trie, node);
   childNode = getTarget(trie, node, x[i]);
   if (childNode == UNDEFINED) {
      lastNode = newVertex(trie);
      setTarget(trie, node, x[m - 1], lastNode);
      node = lastNode;
   }
   else
      node = childNode;
   setZ(node, pos);
 
   /* Searching */
   shift = m - logM + 1;
   for (j = m + 1 - logM; j < n - logM; j += shift) {
      node = root;
      for (k = 0; node != UNDEFINED && k < logM; ++k)
         node = getTarget(trie, node, y[j + k]);
      if (node != UNDEFINED)
         for (current = getZ(node);
              current != NULL;
              current = current->next) {
            b = j - current->element;
            if (x[0] == y[b] &&
                memcmp(x + 1, y + b + 1, m - 1) == 0)
               OUTPUT(b);
         }
   }
   free(z);
}