Lecture 9 : Balanced Search Trees

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Lecture 9 Balanced Search Trees

• Bong-Soo Sohn
• Assistant Professor
• School of Computer Science and Engineering
• Chung-Ang University

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Balanced Binary Search Tree

• In Binary Search Tree
• Average and maximum search times will be
  minimized when BST is maintained as a complete
  tree at all times. O (lg N)
• If not balanced, the search time degrades to O(N)

Idea Keep BST as balanced as possible
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AVL tree

• "height-balanced binary tree
• the height of the left and right subtrees of
  every node differ by at most one

5
3
6
2
9
2
5
4
7
1
4
12
7
3
3
14
8
11
10
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Non-AVL tree example
6
4
9
3
5
7
8
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Balance factor

• BF (height of right subtree - height of left
  subtree)
• So, BF -1, 0 or 1 for an AVL tree.

0
-1
1
0
0
-1
0
-1
-2
0
1
0
0
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AVL tree rebalancing

• When the AVL property is lost we can rebalance
  the tree via one of four rotations
• Single right rotation
• Single left rotation
• Double left rotation
• Double right rotation

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Single Left Rotation (SLR)

• when A is unbalanced to the left
• and B is left-heavy

A
B
SLR at A
B
T3
T1
A
T1
T2
T2
T3
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Single Right Rotation (SRR)

• when A is unbalanced to the right
• and B is right-heavy

A
B
SRR at A
T1
B
A
T3
T2
T3
T1
T2
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Double Left Rotation (DLR)

• When C is unbalanced to left
• And A is right heavy

SRR at C
SLR at A
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Double Right Rotation (DRR)

• When C is unbalanced to right
• And A is left heavy

SRR at C
SRR at A
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Insertion in AVL tree

• An AVL tree may become out of balance in two
  basic situations
• After inserting a node in the right subtree of
  the right child
• After inserting a node in the left subtree of the
  right child

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insertion

• Insertion of a node in the right subtree of the
  right child
• Involves SLR at node P

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insertion

• Insertion of a node in the left subtree of the
  right child
• Involves DRR

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insertion

• In each case the tree is rebalanced locally
• insertion requires one single or one double
  rotation
• O(constant) for rebalancing
• Cost of search for insert is still O(lg N)
• Experiments have shown that 53 of insertions do
  not bring the tree out of balance

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Deletion in AVL tree

• Not covered here
• Requires O(lg N) rotations in worst case

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example

• Given the following AVL tree, insert the node
  with value 9

6
3
12
7
13
4
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