COSC 2007 Data Structures II

Chapter 12Advanced Implementation of

Tables II

2

Topics

AVL trees Definition Operations

Insertion Deletion Traversal Search

3

AVL (Adel'son-Vel'skii & Landis) Trees

AVL Tree Balanced BST: Adelson_Velskii and Landii Height is guaranteed to be O(log2 n) After each insertion & deletion, the tree is checked to

see if it is still an AVL tree or not Tree is rebalanced by rearranging its nodes by some

rotations around some pivot Advantages:

Permit insertion, deletion, and retrieval to be done in a dynamic application

Worst-case performance of O(log n)

4

AVL Trees

Terminology & Definitions Height-balanced p-tree:

For each node in the tree, the difference in height of its two subtrees is at most p

Height-balanced 1-tree: For each node in the tree, the difference in height of

its two subtrees is at most 1. AVL-tree is a height-balanced 1-tree

60

7040

30 50

5645

5

AVL Trees

Terminology & Definitions Balance Difference (BD)

Height of the node's right subtree - Height of the node's left subtree

Pivot node (first bad node) Deepest node on a search path, after inserting the new node

that has a BD value +2, -2

Search Path: Path from the root to the point which a node is to be inserted or

found

60

7040

30 50

5645

6

AVL Trees: Example BST PropertyAt every node X, values inleft subtree are smaller thanthe value in X, and values in right subtree are largerthan the value in X. 6

2

4

3

1

8

9

7

AVL Trees: Example

BST PropertyAt every node X, values inleft subtree are smaller thanthe value in X, and values in right subtree are largerthan the value in X.

AVL Balance PropertyAt every node X, the heightof the left subtree differs from the height of the right subtree by at most 1.6

2

4

3

1

8

9

8

AVL Trees: Example AVL Balance PropertyAt every node X, the heightof the left subtree differs from the height of the right subtree by at most 1.

height(X) = max(height(left(X)), height(right(X))) + 1

6

2

4

3

1

8

9

9

ADT AVL Tree

Operations: Same as BST Only the type of the tree will be AVL-Tree

instead of a BST Insert Delete Retrieve Traverse

6

2

4

3

1

8

9

10

ADT AVL Tree

Insertion: Search path:

Path from the root to the inserted node Idea:

Attach the new node as in a BST. Start from the new node Find the deepest node on the search path that has a BD value =

+2 or –2 after insertion of the new node. Call it the Pivot Node. Adjust the tree so that the values of balance are correct by rotating

and it is still an AVL tree. We are done, and do not need to continue up the tree

Adding a node to the short subtree change the BAL value from the pivot down to the node

Adding a node to the tall subtree of the pivot node requires adjustment to the tree's structure

6

2

4

3

1

8

9

11

ADT AVL Tree Insert operation:

Case 1: Every node on the search path is balanced, having

BD=0. No possibility that the tree will be non-AVL after insert

Just adjust the tree by assigning new BD values to each node on the search path

8040

600

00 8040

60

20

Insert 20

0-1

0

-1

700

8040

60

20

Insert 70

-1-1

0

0

12

ADT AVL Tree

Insertion: Case 2:

New node is added to the short subtree of the pivot node

No rotation is required Adjusting tree by changing value of BD for each

node on the search path starting with the pivot node

700

8040

60

20

1--1

0

0

70

0

8040

60

20

Insert 50

1-0

0

0

50

0

70

0

8040

60

20

Insert 90

00

0

0

20 90

00

13

ADT AVL Tree

Insertion: Case 3:

New node is added to the tall subtree of the pivot node

Adding the new node unbalances the tree (not an AVL tree)

Rotation is required

14

ADT AVL Tree Insertion:

Case 3-a: Single Rotation

10020 50

10

8040

60

30

Insert 5

0

10020 50

10

8040

60

30

5

-1

-1

0

-2

Pivot100

20

50

10

80

40

60

3050 0

-1

-1

0

0

00

0

+1

Rebalance

15

ADT AVL Tree

Insert operation: Case 3-b: Double rotation

The new node is added to the tall subtree of the pivot node Adding the new node unbalances the tree (not an AVL tree) Example: Insert 35

100

30

50

20

80

40

60

35100

-1

-1

0

0

00

0

+1

Double Rotation

10020 50

10

8040

60

30

35 0

0

+1

+1

-2 (Pivot)

Insert 35

16

ADT AVL Tree

General Rebalancing Idea: Single Rotation

Longer path is on the outside of the tree Some rotations decrease tree height

0

Single rotation20

40

-1

-2 (Pivot)

hh+1

h

Tree height = h+3

20

400

0

hh+1

h

Tree height = h+2

17

ADT AVL Tree

General Rebalancing Idea: Single Rotation

Some rotations keep the tree height

Single rotation

0

20

40

-1

-2 (Pivot)

h+1

h

Tree height = h+3

h+1

20

400

0

h+1h

Tree height = h+3

h+1

18

ADT AVL Tree

General Rebalancing Idea: Double Rotation

Longer path is on the inside of the tree 3 nodes are involved

Double rotation20

40

+1

-2 (Pivot)

h

+1

h

Tree height = h+4

h

30

h

+1

-1200 0

h

+1

40

h

Tree height = h+3

h

30

h

+1

-1

19

Node Deletion in AVL Trees

There are 4 cases: Case 1:

No node in the tree contains the data to be deleted No deletion occurs

Case 2: The node to be deleted has no children.

Deletion occurs. Check for BAL of all nodes should be performed. If rebalancing is required, perform it

Case 3: The node to be deleted has one child

Connect this child to the previous parent of the deleted node and then check for balancing

Case 4: The node to be deleted has two children

Search for the rightmost (or the leftmost) node in the left (in the right) subtree of the node removed and replace the removed node with this node. Then check for balancing

20

Efficiency of AVL Trees

Changes required by insertion into or deletion from an AVL tree are limited to a path between the root and the leaf Time complexity?

21

AVL Trees

Traversal Time complexity?

Search