Splay Trees

CSE 331Section 2James Daly

Reminder

• Homework 2 is out• Due Thursday in class

• Project 2 is out• Covers tree sets• Due next Friday at midnight

Review: Binary Tree

• Every node has at most 2 children• Left and right child

• Variation: n-ary trees have at most n children

Review: Binary Search Tree

• For every node• Left descendents smaller (l ≤ k)• Right descendents bigger (r ≥ k)

k

<k >k

Review: AVL Trees

• Named for Adelson-Velskii & Landis• Self-balancing binary search tree• Goal:

• Keep BST in balance• Ability to check / track balance

Review: AVL Trees

• For every node, the height of both subtrees differs by at most 1.

RotateRight(&t)

// Rotates t down to the right// Brings up left childleft ← t.lefttemp = left.rightleft.right ← tt.left ← tempt ← left

Left-Left / Right-Right Case

D

Ba6

c5

e5

B

De5

c5

a6

Left-Right / Right-Left Case

F

Ba5

e4

g5

Dc5

D

Ba5

c5

Fe

4g5

Ba5

c5

e4

D

Fg5

Motivation

• Frequently we care more about how long it takes to do a string of operations than any one• O(n) isn’t too bad – if we do it only once• Want O(m log n) for m searches

• Recently used items are more likely to be called again• Basis for caches

Splay Tree

• Not kept rigorously balanced• When a node is accessed, rotate it to the

top• Next search for the same item will be very

quick• No need for extra information

• AVL Tree: Node height• Red-Black Tree: Node color

Wrong way

k1

k2

k3

k4

k5

A

B C

D

E

F

Wrong way

k2

k3

k4

k5

k1

A B

C

D

E

F

Wrong way

k2

k3

k4

k5

k1

A B C D

E

F

Wrong way

k2

k3

k4

k5

k1

A B

C D

E

F

Wrong way

k2

k3

k4

k5k1

A B

C D

E

F

Problem

• K3 was pushed down almost as far as K2 came up.

• Easy to show that you could keep selecting bad nodes• O(n2) time total.

• Need a smarter way to do this

Splaying

• Find X• If root, done• If parent(X) = root, rotate up• Otherwise, X has both parent and grandparent

• Two cases: Zig-Zag and Zig-Zig

Zig-Zag

X

P

G

A

B C

D

X

P G

A CB D

Zig-Zig

X

P

G

A B

C

D

X

P

G

A

B

C D

Splaying

• Tends to reduce the height of the tree• Many items will be half as deep as before• Some items may be at most 2 deeper than

before

Right way

k1

k2

k3

k4

k5

A

B C

D

E

F

Right way

k1

k2

k3

k4

k5

A CB D

E

F

Right way

k1

k2

k3

k4

k5A

C

B

D E F

Insertion

• Insert X as normal for a BST• Splay X to the top

Deletion

• Remove X as normal for a BST• Splay its parent to the top