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AVL Search Trees

AVL Search Trees. What is an AVL Tree? AVL Tree Implementation. Why AVL Trees? Rotations. What is an AVL Tree?. An AVL tree is a binary search tree with a height balance property: For each node v, the heights of the subtrees of v differ by at most 1.

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AVL Search Trees

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  1. AVL Search Trees • What is an AVL Tree? • AVL Tree Implementation. • Why AVL Trees? • Rotations.

  2. What is an AVL Tree? • An AVL tree is a binary search tree with a height balance property: • For each node v, the heights of the subtrees of v differ by at most 1. • A subtree of an AVL tree is also an AVL tree. • For each node of an AVL tree: Balance factor = height(right subtree) - height(left subtree) • An AVL node can have a balance factor of -1, 0, or +1. -1 -1 7 7 -2 1 -1 1 3 10 3 10 0 1 13 1 0 1 4 13 1 0 Not an AVL Tree AVL Tree 2 0 2 0

  3. MyComparable AbstractObject Container AbstractContainer Tree AbstractTree BinaryTree BinarySearchTree SearchableContainer SearchTree AVL Trees Implementation • The class hierarchy of AVL trees is: AVLTree

  4. AVL Trees Implementation (Contd.) • public class AVLTree extends BinarySearchTree • { 3 protected int height; 4 public AVLTree() 5 { 6 height = -1; • } 8 • public int getHeight(){ return height } ; 10 • protected void adjustHeight() • { 13 if(isEmpty()) 14 height = -1; 15 else 16 height = 1 + Math.max(left.getHeight() , right.getHeight()); 17 } 18 19 protected int getBalanceFactor() 20 { 21 if( isEmpty()) 22 return 0; 23 else 24 return right.getHeight() - left.getHeight(); 25 } 26 // . . . 27 }

  5. Why AVL Trees? • Insertion or deletion in an ordinary Binary Search Tree can cause large imbalances. • In the worst case searching an imbalanced Binary Search Tree is O(n). • An AVL tree is rebalanced after each insertion or deletion. • The height-balance property ensures that the height of an AVL tree with n nodes is O(log n). • Searching, insertion, and deletion are all O(log n).

  6. What is a Rotation? • A rotation is a process of switching children and parents among two or three adjacent nodes to restore balance to a tree. • An insertion or deletion may cause an imbalance in an AVL tree. • The deepest node, which is an ancestor of a deleted or an inserted node, and whose balance factor has changed to -2 or +2 requires rotation to rebalance the tree. -1 -2 50 50 Insert 35 -1 -2 45 45 78 78 0 0 0 -1 40 40 0 Deepest unbalanced node 35

  7. What is a Rotation? (contd.) -2 -1 50 50 • There are two kinds of single rotation: Right Rotation. Left Rotation. • A double right-left rotation is a right rotation followed by a left rotation. • A double left-right rotation is a left rotation followed by a right rotation. rotation -2 0 45 40 78 78 0 0 -1 40 0 0 35 45 0 35

  8. y x x y T3 T1 T2 T2 T1 T3 Single Right Rotation • Right rotation: • The left child x of a node y becomes y's parent. • y becomes the right child of x. • The right child T2 of x, if any, becomes the left child of y. deepest unbalanced node a right rotation of x about y

  9. x y y x T1 T3 T2 T2 T3 T1 Single Left Rotation • Left rotation: • The right child y of a node x becomes x's parent. • x becomes the left child of y. • The left child T2 of y, if any, becomes the right child of x. deepest unbalanced node a left rotation of y about x

  10. Single Right Rotation Implementation • protected void rightRotate() • { 3 if( isEmpty()) 4 throw new InvalidOperationException(); 5 BinaryTree temp = right; 6 right = left; 7 left = right.left; 8 right.left = right.right; 9 right.right = temp; 10 Object tmpObj = key; 11 key = right.key; 12 right.key = tempObj; 13 getRightAVL().adjustHeight(); 14 adjustHeight(); 15 }

  11. Double Right-Left Rotation x deepest unbalanced node x y z right rotation of y about z T1 T1 y z T2 T4 T2 T3 T3 T4 y left rotation of Y about X x z T1 T2 T3 T4

  12. x w T4 v T3 T2 T1 Double Left-Right Rotation x deepest unbalanced node v left rotation of w about v T4 w T1 T2 T3 w right rotation of W about X v x T1 T2 T3 T4

  13. Double Rotation implementation 1 protected void rotateRightLeft() 2 { 3 if( isEmpty()) 4 throw new InvalidOperationException(); 5 getRightAVL().rotateRight(); 6 rotateLeft(); 7 } 1 protected void rotateLeftRight() 2 { 3 if( isEmpty()) 4 throw new InvalidOperationException(); 5 getLeftAVL().rotateLeft(); 6 rotateRight(); 7 }

  14. x y y x T1 T3 T2 T2 T3 T1 BST ordering property after a rotation • A rotation does not affect the ordering property of a BST (Binary Search Tree). a right rotation of x about y BST ordering property requirement: BST ordering property requirement: T1 < x < y T1 < x < y x < T2 < y Similar x < T2 < y x < y < T3 x < y < T3 • Similarly for a left rotation.

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