Splay tree
| Splay tree | |||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Type | tree | ||||||||||||||||||||
| Invented | 1985 | ||||||||||||||||||||
| Invented by | Daniel Dominic Sleator and Robert Endre Tarjan | ||||||||||||||||||||
| Time complexity in big O notation | |||||||||||||||||||||
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| Algorithm | Average | Worst Case | |
|---|---|---|---|
| Space | O(n) | O(n) | |
| Search | O(log n) | amortized O(log n) | |
| Insert | O(log n) | amortized O(log n) | |
| Delete | O(log n) | amortized O(log n) |
A splay tree is a self-adjusting binary search tree with the additional property that recently accessed elements are quick to access again. It performs basic operations such as insertion, look-up and removal in O(log n) amortized time. For many sequences of non-random operations, splay trees perform better than other search trees, even when the specific pattern of the sequence is unknown. The splay tree was invented by Daniel Sleator and Robert Tarjan in 1985.
All normal operations on a binary search tree are combined with one basic operation, called splaying. Splaying the tree for a certain element rearranges the tree so that the element is placed at the root of the tree. One way to do this is to first perform a standard binary tree search for the element in question, and then use tree rotations in a specific fashion to bring the element to the top. Alternatively, a top-down algorithm can combine the search and the tree reorganization into a single phase.
Good performance for a splay tree depends on the fact that it is self-optimizing, in that frequently accessed nodes will move nearer to the root where they can be accessed more quickly. The worst-case height—though unlikely—is O(n), with the average being O(log n). Having frequently used nodes near the root is an advantage for many practical applications (also see Locality of reference), and is particularly useful for implementing caches and garbage collection algorithms.
Advantages include:
The most significant disadvantage of splay trees is that the height of a splay tree can be linear. For example, this will be the case after accessing all n elements in non-decreasing order. Since the height of a tree corresponds to the worst-case access time, this means that the actual cost of an operation can be high. However the amortized access cost of this worst case is logarithmic, O(log n). Also, the expected access cost can be reduced to O(log n) by using a randomized variant.
The representation of splay trees can change even when they are accessed in a 'read-only' manner (i.e. by find operations). This complicates the use of such splay trees in a multi-threaded environment. Specifically, extra management is needed if multiple threads are allowed to perform find operations concurrently. This also makes them unsuitable for general use in purely functional programming, although even there they can be used in limited ways to implement priority queues.
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