Skip lists are fascinating!

Skip lists are a fascinating data structure: very simple, and yet have the same asymptotic efficiency as much more complicated AVL trees and red-black trees. 

skip list can be directly used to implement some operations that are not efficient on a typical sorted set:

• Find the element in the set that is closest to some given value, in O(log N) time.
• Find the k-th largest element in the set, in O(log N) time. Requires a simple  augmentation of the the skip list with partial counts.
• Count the number of elements in the set whose values fall into a given range, in O(log N) time. Also requires a simple augmentation of the skip list.

Instead of ensuring that the level-2 list skips every other node, a skip list is designed in a way that the level-2 list skips one node on average. In some places, it may skip two nodes, and in other places, it may not skip any nodes. But overall, the structure of a skip list is very similar to the structure of a sorted multi-level list.
It allows simple O(log N) insertions and deletions.

• Insertion: decide how many lists will this node be a part of. With a probability of 1/2, make the node a part of the lowest-level list only. With 1/4 probability, the node will be a part of the lowest two lists. With 1/8 probability, the node will be a part of three lists. And so forth. Insert the node at the appropriate position in the lists that it is a part of.

public void Insert(int value)
    {
        // Determine the level of the new node. Generate a random number R. The number of
        // 1-bits before we encounter the first 0-bit is the level of the node. Since R is
        // 32-bit, the level can be at most 32.
        int level = 0;
        for (int R = _rand.Next(); (R & 1) == 1; R >>= 1)
        {
            level++;
            if (level == _levels) { _levels++; break; }
        }

        // Insert this node into the skip list
        Node newNode = new Node(value, level + 1);
        Node cur = _head;
        for (int i = _levels - 1; i >= 0; i--)
        {
            for (; cur.Next[i] != null; cur = cur.Next[i])
            {
                if (cur.Next[i].Value > value) break;
            }

            if (i <= level) { newNode.Next[i] = cur.Next[i]; cur.Next[i] = newNode; }
        }
    }

Deletion: remove the node from all sorted lists that it is a part of.

   public bool Remove(int value)
    {
        Node cur = _head;

        bool found = false;
        for (int i = _levels - 1; i >= 0; i--)
        {
            for (; cur.Next[i] != null; cur = cur.Next[i])
            {
                if (cur.Next[i].Value == value)
                {
                    found = true;
                    cur.Next[i] = cur.Next[i].Next[i];
                    break;
                }

                if (cur.Next[i].Value > value) break;
            }
        }

        return found;
    }

   public bool Contains(int value)
    {
        Node cur = _head;
        for (int i = _levels - 1; i >= 0; i--)
        {
            for (; cur.Next[i] != null; cur = cur.Next[i])
            {
                if (cur.Next[i].Value > value) break;
                if (cur.Next[i].Value == value) return true;
            }
        }
        return false;
    }

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