AVL Trees Section 4.4 39 AVL Tree A balanced tree Ensures log - - PowerPoint PPT Presentation

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AVL Trees Section 4.4 39 AVL Tree A balanced tree Ensures log - - PowerPoint PPT Presentation

Dec 30, 2022 273 likes β€’416 views

AVL Trees Section 4.4 39 AVL Tree A balanced tree Ensures log running time for search, insert, and delete A simple and relaxed definition for balance log log : Too restrictive 40 Balanced Tree Height(null) =

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SLIDE 1

AVL Trees

Section 4.4

39

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SLIDE 2

AVL Tree

A balanced tree Ensures 𝑃 log π‘œ running time for search, insert, and delete A simple and relaxed definition for balance log π‘œ ≀ β„Ž ≀ log π‘œ : Too restrictive

40

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SLIDE 3

Balanced Tree

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4 3 5 2 6 1 7

Height(root->left) = Height(root->right)

Too weak Height(null) = -1

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SLIDE 4

Balanced Tree

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4 2 6 1 7

Height(node->left) = Height(node->right)

Could be impossible to satisfy

3 5

Height(null) = -1

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SLIDE 5

AVL Balance Condition

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5 3 6 2 7 1 4

πΌπ‘“π‘—π‘•β„Žπ‘’ π‘œπ‘π‘’π‘“ β†’ π‘šπ‘“π‘”π‘’ βˆ’ πΌπ‘“π‘—π‘•β„Žπ‘’ π‘œπ‘π‘’π‘“ β†’ π‘ π‘—π‘•β„Žπ‘’ ≀ 1

Height(null) = -1

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SLIDE 6

AVL Example

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5 2 7 1 6 3 4

Is this an AVL Tree? Yes

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SLIDE 7

AVL Example

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5 4 6 2 7 1 3

Is this an AVL Tree? No

Violating node

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SLIDE 8

AVL Example

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Is this an AVL Tree?

Yes

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SLIDE 9

Balancing an AVL Tree

For simplicity, we assume that we keep the height of each subtree at its root An imbalance can occur as a result of an insertion or deletion To balance an AVL tree, we carry out a rotation operation

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SLIDE 10

Insertion

Call BST.insert Update the height as you climb up to the root After each height update, check for an AVL tree violation and fix using rotation

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SLIDE 11

Violation after Insertion

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Violating Node Case 1 Case 2 Case 3 Case 4 + New Node Cases 1 and 4 are symmetric Cases 2 and 3 are symmetric

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SLIDE 12

Case 1 – Single Rotation

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k2 k1 Violating Node X Y Z Status upon insertion in X k2 is in violation k2 k1 X Y Z Rotation Is this a BST? Is this an AVL Tree? Yes Yes

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SLIDE 13

Case 2 – Single Rotation?

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k2 k1 Violating Node X Y Z Status upon insertion in Y k2 is in violation k2 k1 X Y Z Single Rotation Is this a BST? Is this an AVL Tree? Yes No

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SLIDE 14

Case 2 – Double Rotation

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k3 k1 Violating Node A B D k2 C k2 k1 A B C k3 D