Tree Rotations & Splay Trees

BST Structure

• BST's only perform well when balanced• But common cases lead

to unbalanced trees

Restoring Balance

• How could this unbalanced tree be balanced?

Restoring Balance

• How could this unbalanced tree be balanced?

Rotation

• Parent node become child, child becomes parent– Right rotation : Everyone moves right (clockwise)– Left rotation : Counter clockwise

Rotation Description

• Describe rotations by parent– Rotate 38 right

– Rotate 21 right

Child Issues

• Want to rotate 38 right (make 21 parent)– Where does 26 go?

Child Issues

• Right child of old child left child of old parent

Right Rotation

Left Rotation

Multi Step

• Want to balance this tree:

Multi Step

• Want to balance this tree:– Rotate root right is just as bad:

Multi Step

• Want to balance zig-zag pattern

Multi Step

• Want to balance zig-zag pattern– Rotate child of root left

Multi Step

• Want to balance zig-zag pattern– Rotate child of root left– Rotate root right

Deciding What to Rotate

• Figuring out where to rotate is tough– Need more info than available locally in BST• AVL, RB

Deciding What to Rotate

• Figuring out where to rotate is tough– Need more info than available locally in BST• AVL, RB

• Alternative:– Keep recent nodes at top of tree– Do a pretty good job balancing

Splay Tree

• Splaying : move node to root using rotates• Splay Tree : after any access, move node to

root– O(1) to access same element again– Faster to access nearby values– Amortized O(logn) average access

How it works:

• Insert 1-6 in order BST:http://www.cs.usfca.edu/~galles/visualization/BST.html

• Insert 1-6 in order Splay Tree:http://www.cs.usfca.edu/~galles/visualization/SplayTree.html

Splay Algorithm

• Rotating upwards to root does not always balance tree:

Find 1

• Insert 1-6, Find 1:

Find 1

• Insert 1-6, Find 1:

Zig-Zig and Zig-Zag

• Rotations done in 3 ways– Node is right below root: Zig (Rotate left or right)

Zig-Zig and Zig-Zag

• Rotations done in 3 ways– Path to node is left-left or right-right:

Zig-Zig (Rotate grandparent, then rotate parent)

Zig-Zig and Zig-Zag

• Rotations done in 3 ways– Path to node is left-right or right-left:

Zig-Zag (Rotate parent one way, then grand parent other way)

Sample:

• Splay 42

Sample:

• Splay 42– It is left-left child : Zig-Zig

Sample:

• Splay 42– It is left-left child : Zig-Zig• Rotate 52 right

Sample:

• Splay 42– It is left-left child : Zig-Zig• Rotate 52 right, rotate 47 right

Sample:

• Splay 42– Now it is a left-right child : Zig-Zag

Sample:

• Splay 42– Now it is a left-right child : Zig-Zag• 35 Rotates left

Sample:

• Splay 42– Now it is a left-right child : Zig-Zag• 35 Rotates left, 70 rotate right