Tree Balancing: AVL Trees

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Tree Balancing AVL Trees

• Dr. Yingwu Zhu

2
Recall in BST

• The insertion order of items determine the shape
  of BST
• Balanced search T(n)O(logN)
• Unbalanced T(n) O(n)
• Key issue
• A need to keep a BST balanced!
• Introduce AVL trees, by Russian mathematican

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AVL Tree Definition

• First, a BST
• Second, height-balance property balance factor
  of each node is 0, 1, or -1
• Question what is balance factor?

BF Height of the left subtree height of the
right subtree Height of levels in a
subtree/tree
4
Determine balance factor
A
B
D
G
E
C
F
H
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ADT AVL Trees

• Data structure to implement

Balance factor
Data
Left
Right
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ADT AVL Trees

• Basic operations
• Constructor, search, travesal, empty
• Insert keep balanced!
• Delete keep balanced!
• See P842 for class template
• Similar to BST

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Example
RI
Insert DE, what happens? Need rebalancing?
PA
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Basic Rebalancing Rotation

• Single rotation
• Right rotation the inserted item is on the left
  subtree of left child of the nearest ancestor
  with BF of 2
• Left rotation the inserted item is on the right
  subtree of right child of the nearest ancestor
  with BF of -2
• Double rotation
• Left-right rotation the inserted item is on the
  right subtree of left child of the nearest
  ancestor with BF of 2
• Right-left rotation the inserted item is on the
  left subtree of right child of the nearest
  ancestor with BF of -2

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How to perform rotations

• Rotations are carried out by resetting links
• Two steps
• Determine which rotation
• Perform the rotation

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Right Rotation

• Key identify the nearest ancestor of inserted
  item with BF 2
• A the nearest ancestor. B left child
• Step1 reset the link from parent of A to B
• Step2 set the left link of A equal to the right
  link of B
• Step3 set the right link of B to A
• Examples

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A
B
L(A)
L(B)
R(B)
h
12
Exercise 1
10
4
How about insert
8
20
6
9
13
Left Rotation

• Step1 reset the link from parent of A to B
• Step2 set the right link of A to the left link
  of B
• Step3 set the left link of B to A
• Examples

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Example
A
B
L(A)
h
L(B)
R(B)
15
Exercise 2
7
18
What if insert
5
15
10
20
16
Exercise 3
12
1
How about insert
8
16
4
10
14
2
6
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Basic Rebalancing Rotation

• Double rotation
• Left-right rotation the inserted item is on the
  right subtree of left child of the nearest
  ancestor with BF of 2
• Right-left rotation the inserted item is on the
  left subtree of right child of the nearest
  ancestor with BF of -2

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Double Rotations

• Left-right rotation
• Right-left rotation
• How to perform?
• 1. Rotate child and grandchild nodes of the
  ancestor
• 2. Rotate the ancestor and its new child node

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Example
A
C
What if insert
B
20
Example
A
B
R(A)
h1
C
L(B)
h1
R(C)
L(C)
h
21
Exercise 4
12
7
How about insert
8
16
4
10
14
2
6
22
Exercise 5
4
2
6
15
What if insert
1
3
5
7
16
23
Exercise 5
4
2
6
14
What if insert
1
3
5
15
16
7
24
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Summary on rebalancing

• The key is you need to identify the nearest
  ancestor of inserted item
• Determine the rotation by definition
• Perform the rotation