lecture 13 avl trees king fahd university of petroleum & minerals college of computer science...

Lecture 13

AVL Trees

King Fahd University of Petroleum & MineralsCollege of Computer Science & Engineering

Information & Computer Science Department

AVL Search Trees

What is an AVL Tree? AVL Tree Implementation. Why AVL Trees? Rotations. Insertion into an AVL tree Deletion from an AVL tree

2

What is an AVL Tree?

An AVL tree is a binary search tree with a height balance property: • For each node v, the heights of the subtrees of v differ by at most 1.

A subtree of an AVL tree is also an AVL tree. For each node of an AVL tree:

Balance factor = height(right subtree) - height(left subtree)

An AVL node can have a balance factor of -1, 0, or 1. Determine whether the trees below are AVL trees or not.

3

1

2

4

10

13

7

3

1

2

10

13

7

3

AVL Trees Implementation

public class AVLTree extends BinarySearchTree{ protected int height; public AVLTree(){ height = -1;}

public int getHeight(){ return height } ;

protected void adjustHeight(){ if(isEmpty()) height = -1; else height = 1 + Math.max(left.getHeight() , right.getHeight()); }

protected int getBalanceFactor(){ if( isEmpty()) return 0; else return right.getHeight() - left.getHeight(); } // . . .}

4

Why AVL Trees?

Insertion or deletion in an ordinary Binary Search Tree can cause large imbalances.

In the worst case searching an imbalanced Binary Search Tree is O( )

An AVL tree is rebalanced after each insertion or deletion.• The height-balance property ensures that the height of

an AVL tree with n nodes is O( ).• Searching, insertion, and deletion are all O( ).

5

n

log n

log n

What is a Rotation?

A rotation is a process of switching children and parents among two or three adjacent nodes to restore balance to a tree.

An insertion or deletion may cause an imbalance in an AVL tree.

The deepest node, which is an ancestor of a deleted or an inserted node, and whose balance factor has changed to -2 or +2 requires rotation to rebalance the tree.

45

40

78

50 -1

0

-1 0 45

40

78

50-2

-1

-2 0

350

Insert 35

Deepest unbalanced node

6

What is a Rotation? (contd.)

There are two kinds of single rotation:

Right Rotation. Left Rotation.

A double right-left rotation is a right rotation followed by a left rotation.

A double left-right rotation is a left rotation followed by a right rotation.

45

40

78

50-2

-1

-2 0

350

45

40 78

50-1

0

0

0

350

rotation

7

Single Right Rotation

Single right rotation:• The left child x of a node y becomes y's parent.• y becomes the right child of x.

• The right child T2 of x, if any, becomes the left child of y.

x

T1 T2

y

T3

deepest unbalanced node a right rotation of x about y

y

T3T2

x

T1

Note: The pivot of the rotation is the deepest unbalanced node

8

Single Left Rotation

Single left rotation:• The right child y of a node x becomes x's parent.• x becomes the left child of y.

• The left child T2 of y, if any, becomes the right child of x.

y

T3T2

x

T1

deepest unbalanced node a left rotation of y about x

x

T1 T2

y

T3

Note: The pivot of the rotation is the deepest unbalanced node

9

Single Right Rotation Implementation

protected void rightRotate(){ if( isEmpty()) throw new InvalidOperationException(); BinaryTree temp = right; right = left; left = right.left; right.left = right.right; right.right = temp; Object tmpObj = key; key = right.key; right.key = tempObj; getRightAVL().adjustHeight(); adjustHeight();}

10

Single Right Rotation Implementation (example)

11

Single Right Rotation Implementation (example) contd

12


13


14


15


16


17


18


19


20

Double Right-Left Rotation

z

T4T3

y

T2

x

T1y

T3T2

z

T4

x

T1

z

T4T3

x

T1

y

T2

right rotation of y about z

deepest unbalanced node

left rotation of Y about XNote: First pivot is the right child of the deepest unbalanced node; second pivot is the deepest unbalanced node

21

Double Left-Right Rotation

w

T2 T3

v

T1

x

T4v

T1 T2

w

T3

x

T4

x

T4T3

v

T1

w

T2

left rotation of w about v

deepest unbalanced node

left rotation of W about XNote: First pivot is the left child of the deepest unbalanced node; second pivot is the deepest unbalanced node

22

Double Rotation implementation

1 protected void rotateRightLeft()2 {3 if( isEmpty())4 throw new InvalidOperationException();5 getRightAVL().rotateRight();6 rotateLeft();7 }

1 protected void rotateLeftRight()2 {3 if( isEmpty())4 throw new InvalidOperationException();5 getLeftAVL().rotateLeft();6 rotateRight();7 }

23

BST ordering property after a rotation

A rotation does not affect the ordering property of a BST (Binary Search Tree).

y

T3T2

x

T1

x

T1 T2

y

T3

a right rotation of x about y

BST ordering property requirement: BST ordering property requirement:

T1 < x < y T1 < x < y

x < T2 < y Similar x < T2 < y

x < y < T3 x < y < T3

• Similarly for a left rotation.

24

Insertion

Insert using a BST insertion algorithm. Rebalance the tree if an imbalance occurs. An imbalance occurs if a node's balance factor changes from -1

to -2 or from+1 to +2. Rebalancing is done at the deepest or lowest unbalanced

ancestor of the inserted node.

There are three insertion cases:1. Insertion that does not cause an imbalance.

2. Same side (left-left or right-right) insertion that causes an imbalance. Requires a single rotation to rebalance.

3. Opposite side (left-right or right-left) insertion that causes an imbalance. Requires a double rotation to rebalance.

25

Insertion: case 1

Example: An insertion that does not cause an imbalance.

Insert 14

26

Insertion: case 2

Case 2a: The lowest node (with a balance factor of -2) had a taller left-subtree and the insertion was on the left-subtree of its left child.

Requires single right rotation to rebalance.

Insert 3

right rotation, with node 10 as pivot

-2

-1

27

Insertion: case 2 (contd)

Case 2b: The lowest node (with a balance factor of +2) had a taller right-subtree and the insertion was on the right-subtree of its right child.

Requires single left rotation to rebalance.

Insert 45left rotation, with node 30 as the pivot

+2

+1

28

Insertion: case 3

Case 3a: The lowest node (with a balance factor of -2) had a taller left-subtree and the insertion was on the right-subtree of its left child.

Requires a double left-right rotation to rebalance.

Insert 7

left rotation, with node 5 as the pivot

right rotation, with node 10 as the pivot

-2

+1

29

Insertion: case 3 (contd)

Case 3b: The lowest node (with a balance factor of +2) had a taller right-subtree and the insertion was on the left-subtree of its right child.

Requires a double right-left rotation to rebalance.

Insert 15


left rotation, with node 9 as the pivot

+2

-1

30

AVL Rotation Summary

-1

-2

+1

-2

-1 +1

+2+2

Single right rotation

Double left-right rotation

Single left rotation

Double right-left rotation

31

Insertion Implementation

The insert method of the AVLTree class is:

Recall that the insert method of the BinarySearchTree class is:

public void insert(Comparable comparable){ if(isEmpty()) attachKey(comparable); else { Comparable key = (Comparable) getKey(); if(comparable.compareTo(key)==0) throw new IllegalArgumentException("duplicate key"); else if (comparable.compareTo(key)<0) getLeftBST().insert(comparable); else getRightBST().insert(comparable); }}

public void insert(Comparable comparable){ super.insert(comparable); balance();}

32

Insertion Implementation (contd)

The AVLTree class overrides the attachKey method of the BinarySearchTree class:

public void attachKey(Object obj) { if(!isEmpty()) throw new InvalidOperationException(); else { key = obj; left = new AVLTree(); right = new AVLTree(); height = 0; } }

33

Insertion Implementation (contd)

protected void balance(){ adjustHeight(); int balanceFactor = getBalanceFactor(); if(balanceFactor == -2){ if(getLeftAVL().getBalanceFactor() < 0) rotateRight(); else rotateLeftRight(); } else if(balanceFactor == 2){ if(getRightAVL().getBalanceFactor() > 0) rotateLeft(); else rotateRightLeft(); }}

34

Deletion

Delete by a BST deletion by copying algorithm. Rebalance the tree if an imbalance occurs. There are three deletion cases:

1. Deletion that does not cause an imbalance.2. Deletion that requires a single rotation to rebalance.3. Deletion that requires two or more rotations to rebalance.

Deletion case 1 example:

Delete 14

35

Deletion: case 2 examples

Delete 40right rotation, with node 35 as the pivot

36

Deletion: case 2 examples (contd)

Delete 32left rotation, with node 44 as the pivot

37

Deletion: case 3 examples

Delete 40

0



38

lecture 13 avl trees king fahd university of petroleum & minerals college of computer science...

Documents