The Complete List of Algorithms

Skip Search algorithm

Main features

uses buckets of positions for each character of the alphabet;
preprocessing phase in O(m+) time and space complexity;
searching phase in O(mn) time complexity;
O(n) expected text character comparisons.

Description

For each character of the alphabet, a bucket collects all the positions of that character in x. When a character occurs k times in the pattern, there are k corresponding positions in the bucket of the character. When the word is much shorter than the alphabet, many buckets are empty.

The preprocessing phase of the Skip Search algorithm consists in computing the buckets for all the characters of the alphabet: for c in z[c] = {i : 0 i m-1 and x[i] = c} The space and time complexity of this preprocessing phase is O(m+).

The main loop of the search phase consists in examining every m-th text character, y[j] (so there will be n / m main iterations). For y[j], it uses each position in the bucket z[y[j]] to obtain a possible starting position p of x in y. It performs a comparison of x with y beginning at position p, character by character, until there is a mismatch, or until all match.

The Skip Search algorithm has a quadratic worst case time complexity but the expected number of text character inspections is O(n).

The C code

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

void SKIP(char *x, int m, char *y, int n) {
   int i, j;
   List ptr, z[ASIZE];

   /* Preprocessing */
   memset(z, NULL, ASIZE*sizeof(List));
   for (i = 0; i < m; ++i) {
      ptr = (List)malloc(sizeof(struct _cell));
      if (ptr == NULL)
         error("SKIP");
      ptr->element = i;
      ptr->next = z[x[i]];
      z[x[i]] = ptr;
   }

   /* Searching */
   for (j = m - 1; j < n; j += m)
      for (ptr = z[y[j]]; ptr != NULL; ptr = ptr->next)
         if (memcmp(x, y + j - ptr->element, m) == 0) {
            if (j - ptr->element <= n - m)
               OUTPUT(j - ptr->element);
         }
         else
            break;
}

In practice the test j - ptr->element <= n - m can be omitted and the algorithm becomes :

void SKIP(char *x, int m, char *y, int n) {
   int i, j;
   List ptr, z[ASIZE];

   /* Preprocessing */
   memset(z, NULL, ASIZE*sizeof(List));
   for (i = 0; i < m; ++i) {
      ptr = (List)malloc(sizeof(struct _cell));
      if (ptr == NULL)
         error("SKIP");
      ptr->element = i;
      ptr->next = z[x[i]];
      z[x[i]] = ptr;
   }

   /* Searching */
   for (j = m - 1; j < n; j += m)
      for (ptr = z[y[j]]; ptr != NULL; ptr = ptr->next)
         if (memcmp(x, y + j - ptr->element, m) == 0)
            OUTPUT(j - ptr->element);
}

The example

Preprocessing phase

Skip search Z table
Z table used by Skip Search algorithm.