DS5: Splay Trees

San Skulrattanakulchai

September 24, 2018

Splay Trees

A splay tree is an efficient form of binary search tree. It achieves \(O(\log n)\) time per binary search tree operation in an amortized sense instead of the worst-case sense.
Unlike red-black trees, splay trees keep no balance information.
Instead, it uses the following simple heuristic: each time an item is accessed, move it to the root.
Hence splay trees are self-adjusting. However, we have to implement this heuristic correctly in order to achieve the desired efficiency.
The correct way to move the accessed item to the root is called splaying.

Splaying

Here is the code for a splay operation at node x.

splay(x) {  
    while x is not the root do {  
        if x.p is the root then {  
            rotate x.p so x becomes the root  
        } else if x and x.p are both left children,
        or both right children then {  
            rearrange zig-zig path to opposite zig-zig path
        } else {  
            rearrange zig-zag path to sibling path   
        }
    }
}

Splay tree algorithms

search(k) — do a search like in the binary search tree. Let x be the last node accessed. Then either x.key = k, or k is not in the tree but k’s parent would be x if k were in the tree. Splay x.
insert(k) — use the bst algorithm to insert k, say in the new node x. Splay x.
delete(z) — execute the BST algorithm. It sets y to z or its successor, and replaces y by child x. Splay y.p.

Running Time

The time for each splay tree operation is proportional to the time for the splay.
Splay Tree Lemma. Starting with any bst of \(n\) nodes, any sequence of \(m\ge n\) splays takes time \(O(m\log n)\).
Splay Tree Theorem. Starting from an empty tree, any sequence of \(m\) search, insert, and delete operations takes total time \(O(m\log n)\). Here \(n\) is the greatest number of nodes ever in the tree.
Although some operations can take \(\Theta(n)\) time, on the average (over any sequence of operations) each operation takes \(O(\log n)\) time.
The proof of the Splay Tree Theorem is beyond the scope of this course.
Several conjectures concerning splay tree properties remain open. (See the wiki page for splay tree.)
An example is
- Dynamic Optimality Conjecture. Starting with any bst, on any sequence of node accesses, the time for splaying is at most a constant more than any other bst algorithm.