Skip List

Skip list is a data structure that implements dictinaries. It assumes that random numbers can be generated easily in any computer systems and uses randomization in its structure.

The expected worst case performance of a skip list is similar to that of an AVL tree.

A skip list S for dictionary D consists a series of lists {S₀, S₁, ..., S_h} where S₀ contains every item of D plus the two special items with keys -∞ and +∞. For 1 <= i <= (h-1), list S_i contains a randomly generated subset (usually half) of the items in list S_i-1 plus the two special items. List S_h contains only the two special items.

Traversal operations in skip list:

• after(p): goes to the position following p on the same level;
• before(p): goes to the position preceding p on the same level;
• below(p): goes to the position below p on the same tower;
• above(p): goes to the position above p on the same tower;

Main operations:

• Search(k)

  Position * SkipSearch(k)
  {
      initialize p to be the top-most and left position of the skip list    
      (i.e., p is placed on the left node of Sh)
      while (p->below != NULL) {
          p = p->below;
          while (p->after->key <= k) {
              p = p->after;
          }
      }
      return p;
  }
  

• Lookup (k):

  Data SkipLookup(k)
  {
      p = SkipSearch(k);
      if (p->key == k)
          return p->data
      else
          Exception 404 "No such item"
  }
  

• Insert(k, obj)

  void SkipInsert(k, obj)
  {
      p = SkipSearch(k);
      if (p->key == k)
          Exception "Duplicate Key"
      else {
          th = 0;
          q = insertAfterAbove(p, NULL, (k, obj));
          while (th < h && random() < 0.5) {
              while (p->above == NULL)
                  p = p->before;
              p = p->above;
              q = insertAfterAbove(p, q, (k, NULL));
              th++;
          }
          if (th == h) {
              add a top list; // with only the two special elements
              h++;
          }
      }
  }
  

  The probability that the height of the newly inserted tower rises to h is 2^-h.

• Remove(k)

  Data & SkipRemove(k)
  {
      p = SkipSearch(k);
      if (p->key != k)
          Exception 404 "No such item
      else {
          Data *temp = p->obj
          Position *q;
          // delete the whole tower
          while (p != NULL) {
              p->before->after = p->after;
              p->after->before = p->before;
              q = p;
              p = p->above;
              delete q;
          }
          return (*temp);
      }
  }