Tree Balancing: AVL Trees

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Presentation transcript:

Tree Balancing: AVL Trees Dr. Yingwu Zhu

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 mathematician

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

Determine balance factor G F H

ADT: AVL Trees Data structure to implement Balance factor Data Left Right

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

Example RI Insert “DE”, what happens? Need rebalancing? PA

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

How to perform rotations Rotations are carried out by resetting links Two steps: Determine which rotation Perform the rotation

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

A B L(A) L(B) R(B) h new

Exercise #1 10 4 How about insert 8 20 6 9

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

Example A B L(A) h L(B) R(B) new

Exercise #2 7 18 What if insert 5 15 10 20

Exercise #3 12 1 How about insert 8 16 4 10 14 2 6

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

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

Example A C What if insert B

Example A B R(A) h+1 C L(B) h+1 L(C) R(C) h new

Exercise #4 12 7 How about insert 8 16 4 10 14 2 6

Exercise #5 4 2 6 15 What if insert 1 3 5 7 16

Exercise #6 4 2 6 14 What if insert 1 3 5 15 7 16

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