Splay Trees

Two great things that go great together

Nathan Tenney

WSU Tri-Cities

Splay Trees

• In the discussion of lists, we covered the idea of self-organizing lists.
• A similar concept exists in Binary Search Trees as well.
• As an item is accessed, perform a series of rotations to place the requested node at the root of the tree.
• At worst, operations on a splay tree take $O\left(M \log N\right)$

Splay Trees

• Purpose of a splay tree
  • Improve access to the most recently sought after item
    • Depending on observations, this may include item insertion
• Simple Solution:
  • Perform single rotations until the node is at the root
    • Improves access to the requested node.
    • May make searches for recently accessed items worse

Splay Trees

• Single rotations don’t restructure the tree in a beneficial way
• Instead, consider a series of double rotations to get the node to the root
• Three main cases to consider
  • Zig-Zag - Node is a right child of a left child, or a left child of a right child
  • Zig-Zig - Node is a right child of a right child, or a left child of a left child
  • Zig - Parent is the root

Zig-Zag

• Rotate node around its parent
• Rotate node around its new parent

Zig-Zig

• Start at a level above the node.
• Rotate parent around grandparent
• Rotate node around parent

Zig

• Simply rotate node around parent
• Only done as a last step, and only when node is at an odd depth to begin with

Splay Tree Operations

• Operations on a splay tree are done much the same as on other trees
  • Insertion and deletion occur at the leaf level
  • After each operation, rotations must occur (rotate inserted node to root)
  • How is this accomplished?
    • Insertion
    • Deletion
    • Failed Search