CS 225 Data Structures Feb. 27 AVL L Tre rees Wad ade Fag - PowerPoint PPT Presentation

CS 225 Data Structures Feb. 27 – AVL L Tre rees Wad ade Fag agen-Ulm lmschneid ider

Left ft Rotation 38 13 51 10 25 40 84 89 66 95

38 13 51 10 25 84 A 89 B C D

38 84 13 51 51 89 10 25 84 A A C B D 89 B C D

84 38 51 89 84 13 89 51 10 25 A C B D 40 95 66

38 13 51 10 25 37

38 13 51 10 25 37

BST Rotation Summary ry - Four kinds of rotations (L, R, LR, RL) - All rotations are local (subtrees are not impacted) - All rotations are constant time: O(1) - BST property maintained GOAL : We call these trees:

AVL Trees Three issues for consideration: - Rotations - Maintaining Height - Detecting Imbalance

AVL Tree Rotations Four templates for rotations:

Finding the Rotation t t 1 t 2 t 3 t 4

If an insertion occurred in Finding the Rotation subtrees t 3 or t 4 and a subtree was detected at t: t t t 1 t 1 t 2 t 2 t 3 t 4 t 3 t 4

Finding the Rotation t t L t R

Finding the Rotation t t L t R

Finding the Rotation t t L t A t B

Finding the Rotation Theorem: t If an insertion occurred in subtrees t 3 or t 4 and a subtree was detected at t , then a __________ rotation about t restores the balance of the t 1 tree. t 2 We gauge this by noting the balance t 3 t 4 factor of t->right is ______.

Example: 38 13 51 10 25 40 84 89 66

Finding the Rotation t t 1 t 4 t 2 t 3

Theorem: Finding the Rotation If an insertion occurred in subtrees t 2 or t 3 and a subtree t was detected at t : t t 1 t 1 t 4 t 4 t 2 t 3 t 2 t 3

Finding the Rotation Theorem: t If an insertion occurred in subtrees t 2 or t 3 and a subtree was detected at t , then a __________ rotation about t restores the balance of the t 1 tree. t 4 We gauge this by noting the balance factor of t->right is ______. t 2 t 3

_insert(6.5) Insertion into an AVL Tree In 8 5 10 3 6 9 12 1 struct TreeNode { 7 4 11 1 2 T key; 3 unsigned height; 4 TreeNode *left; 2 5 TreeNode *right; 6 };

_insert(6.5) In Insertion into an AVL Tree Insert (pseudo code): 1: Insert at proper place 2: Check for imbalance 3: Rotate, if necessary 8 4: Update height 5 10 3 6 9 12 1 struct TreeNode { 7 4 11 1 2 T key; 3 unsigned height; 4 TreeNode *left; 2 5 TreeNode *right; 6 };

151 template <typename K, typename V> 152 void AVL<K, D>::_insert(const K & key, const V & data, TreeNode *& cur) { 153 if (cur == NULL) { cur = new TreeNode(key, data); } 157 else if (key < cur->key) { _insert( key, data, cur->left ); } 160 else if (key > cur->key) { _insert( key, data, cur->right );} 166 _ensureBalance(cur); 167 }

119 template <typename K, typename V> 120 void AVL<K, D>::_ensureBalance(TreeNode *& cur) { 121 // Calculate the balance factor: 122 int balance = height(cur->right) - height(cur->left); 123 124 // Check if the node is current not in balance: 125 if ( balance == -2 ) { 126 int l_balance = height(cur->left->right) - height(cur->left->left); 127 if ( l_balance == -1 ) { ____________________________; } 128 else { ____________________________; } 129 } else if ( balance == 2 ) { 130 int r_balance = height(cur->right->right) - height(cur->right->left); 131 if( r_balance == 1 ) { _____________________________; } 132 else { _____________________________; } 133 } 134 135 _updateHeight(cur); 136 };

8 5 10 3 6 9 12 4 7 11 1 2

AVL Tree Analysis We know: insert, remove and find runs in: __________. We will argue that: h = _________.

AVL Tree Analysis Definition of big-O: …or, with pictures: