Skiplists

A skiplist is a probabilistic data structure that allows for fast search, insertion, and deletion operations, similar to balanced trees like red-black trees or AVL trees, but with a simpler implementation. It is essentially a series of sorted linked lists, where each list is a subsequence of the list below it.

The lowest level list contains all elements in sorted order. Each subsequent level contains a subset of the elements from the level below. Elements are promoted to higher levels probabilistically, typically with a coin flip. This means that an element might appear in level 0, level 1, level 2, and so on, up to some maximum level.

Searching for an element begins at the highest level. The search traverses forward in the current level until the next element is greater than the target element or the end of the list is reached. At that point, the search drops down to the next lower level and continues the process. This "staircase" traversal allows skiplists to achieve an average time complexity of O(log n) for search, insertion, and deletion, where n is the number of elements in the skiplist.

Insertion involves finding the appropriate position for the new element in the lowest level and then probabilistically promoting it to higher levels. Deletion involves removing the element from all levels it appears in. The probabilistic nature of skiplists means that while worst-case performance can be O(n), this is highly unlikely with a reasonable probability distribution.

Skiplists are often favored for their simplicity in implementation compared to balanced trees, while still offering comparable performance characteristics. They are used in various applications where efficient dynamic ordered sets or maps are required.