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CS 225
Data Structures
- Feb. 27 – AVL
L Tre rees
Wad ade Fag agen-Ulm lmschneid ider
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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 - - 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
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A B C D
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A B C D
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A B C D
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A B C D
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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:
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AVL Trees
Three issues for consideration:
- Rotations
- Maintaining Height
- Detecting Imbalance
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AVL Tree Rotations
Four templates for rotations:
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t
t1 t2 t3 t4
Finding the Rotation
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t
t1 t2 t3 t4
Finding the Rotation
t
t1 t2 t3 t4
If an insertion occurred in subtrees t3 or t4 and a subtree was detected at t:
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t
tL tR
Finding the Rotation
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t
tL tR
Finding the Rotation
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t
tL tA tB
Finding the Rotation
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t
t1 t2 t3 t4
Theorem: If an insertion occurred in subtrees t3 or t4 and a subtree was detected at t, then a __________ rotation about t restores the balance of the tree. We gauge this by noting the balance factor of t->right is ______.
Finding the Rotation
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Example:
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t
t1 t2 t3 t4
Finding the Rotation
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t
t1 t2 t3 t4
Finding the Rotation
t
t1 t2 t3 t4
Theorem: If an insertion occurred in subtrees t2 or t3 and a subtree was detected at t:
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t
t1 t2 t3 t4
Theorem: If an insertion occurred in subtrees t2 or t3 and a subtree was detected at t, then a __________ rotation about t restores the balance of the tree. We gauge this by noting the balance factor of t->right is ______.
Finding the Rotation
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In Insertion into an AVL Tree
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struct TreeNode { T key; unsigned height; TreeNode *left; TreeNode *right; }; 1 2 3 4 5 6
_insert(6.5)
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Insert (pseudo code): 1: Insert at proper place 2: Check for imbalance 3: Rotate, if necessary 4: Update height
In Insertion into an AVL Tree
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struct TreeNode { T key; unsigned height; TreeNode *left; TreeNode *right; }; 1 2 3 4 5 6
_insert(6.5)
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template <typename K, typename V> void AVL<K, D>::_insert(const K & key, const V & data, TreeNode *& cur) { if (cur == NULL) { cur = new TreeNode(key, data); } else if (key < cur->key) { _insert( key, data, cur->left ); } else if (key > cur->key) { _insert( key, data, cur->right );} _ensureBalance(cur); } 151 152 153 157 160 166 167
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template <typename K, typename V> void AVL<K, D>::_ensureBalance(TreeNode *& cur) { // Calculate the balance factor: int balance = height(cur->right) - height(cur->left); // Check if the node is current not in balance: if ( balance == -2 ) { int l_balance = height(cur->left->right) - height(cur->left->left); if ( l_balance == -1 ) { ____________________________; } else { ____________________________; } } else if ( balance == 2 ) { int r_balance = height(cur->right->right) - height(cur->right->left); if( r_balance == 1 ) { _____________________________; } else { _____________________________; } } _updateHeight(cur); }; 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136
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AVL Tree Analysis
We know: insert, remove and find runs in: __________. We will argue that: h = _________.
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AVL Tree Analysis
Definition of big-O: …or, with pictures: