computer notes - Data Structures - 21 by ecomputernotes com

Class No.21 Data Structures http://ecomputernotes.com 1

singleRightRotation ď&#x201A;´

TreeNode<int>* singleRightRotation(TreeNode<int>* k2) { if( k2 == NULL ) return NULL; k2 // k1 (first node in k2's left subtree) // will be the new root k1 TreeNode<int>* k1 = k2->getLeft(); // Y moves from k1's right to k2's left k2->setLeft( k1->getRight() ); Y k1->setRight(k2); X // reassign heights. First k2 int h = Max(height(k2->getLeft()), height(k2->getRight())); k2->setHeight( h+1 ); // k2 is now k1's right subtree h = Max( height(k1->getLeft()), k2->getHeight()); k1->setHeight( h+1 ); return k1; }

k1 k2 X

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singleRightRotation ď&#x201A;´

k1 k2 X

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singleRightRotation

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k1 k2 X

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singleRightRotation

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k1 k2 X

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singleRightRotation TreeNode<int>* singleRightRotation(TreeNode<int>* k2) { if( k2 == NULL ) return NULL; k2 // k1 (first node in k2's left subtree) // will be the new root k1 TreeNode<int>* k1 = k2->getLeft(); // Y moves from k1's right to k2's left k2->setLeft( k1->getRight() ); Y k1->setRight(k2); X // reassign heights. First k2 int h = Max(height(k2->getLeft()), height(k2->getRight())); k2->setHeight( h+1 ); // k2 is now k1's right subtree h = Max( height(k1->getLeft()), k2->getHeight()); k1->setHeight( h+1 );

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return k1; }

k1 k2 X

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return k1; }

k1 k2 X

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return k1; }

k1 k2 X

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height

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int height( TreeNode<int>* node ) { if( node != NULL ) return node->getHeight(); return -1; }

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height

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int height( TreeNode<int>* node ) { if( node != NULL ) return node->getHeight(); return -1; }

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singleLeftRotation ď&#x201A;´

TreeNode<int>* singleLeftRotation( TreeNode<int>* k1 ) { k1 if( k1 == NULL ) return NULL; // k2 is now the new root TreeNode<int>* k2 = k1->getRight(); X k1->setRight( k2->getLeft() ); // Y k2->setLeft( k1 ); // reassign heights. First k1 (demoted) int h = Max(height(k1->getLeft()), height(k1->getRight())); k1->setHeight( h+1 ); // k1 is now k2's left subtree h = Max( height(k2->getRight()), k1->getHeight()); k1 k2->setHeight( h+1 ); return k2; }

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Y Z

singleLeftRotation

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Y Z

singleLeftRotation

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Y Z

singleLeftRotation

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Y Z

singleLeftRotation

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Y Z

singleLeftRotation TreeNode<int>* singleLeftRotation( TreeNode<int>* k1 ) { k1 if( k1 == NULL ) return NULL; // k2 is now the new root TreeNode<int>* k2 = k1->getRight(); X k1->setRight( k2->getLeft() ); // Y k2->setLeft( k1 ); // reassign heights. First k1 (demoted) int h = Max(height(k1->getLeft()), height(k1->getRight())); k1->setHeight( h+1 ); // k1 is now k2's left subtree h = Max( height(k2->getRight()), k1->getHeight()); k1 k2->setHeight( h+1 ); return k2;

ď&#x201A;´ }

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Y Z

doubleRightLeftRotation ď&#x201A;´ TreeNode<int>* doubleRightLeftRotation(TreeNode<int>* k1) { if( k1 == NULL ) return NULL; // single right rotate with k3 (k1's right child) k1->setRight( singleRightRotation(k1->getRight())); // now single left rotate with k1 as the root return singleLeftRotation(k1); }

k1 k3

A k2

k3 D

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doubleRightLeftRotation

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TreeNode<int>* doubleRightLeftRotation(TreeNode<int>* k1) { if( k1 == NULL ) return NULL; // single right rotate with k3 (k1's right child) k1->setRight( singleRightRotation(k1->getRight())); // now single left rotate with k1 as the root return singleLeftRotation(k1); }

k1 k3

A k2

k3 D

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doubleRightLeftRotation

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k1 k1

A k3 B

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doubleRightLeftRotation ď&#x201A;´ TreeNode<int>* doubleLeftRightRotation(TreeNode<int>* k3) { if( k3 == NULL ) return NULL; // single left rotate with k1 (k3's left child) k3->setLeft( singleLeftRotation(k3->getLeft())); // now single right rotate with k3 as the root return singleRightRotation(k3); }

k2 D k2

D k1 C

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doubleRightLeftRotation

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TreeNode<int>* doubleLeftRightRotation(TreeNode<int>* k3) { if( k3 == NULL ) return NULL; // single left rotate with k1 (k3's left child) k3->setLeft( singleLeftRotation(k3->getLeft())); // now single right rotate with k3 as the root return singleRightRotation(k3); }

k2 D k2

D k1 C

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doubleRightLeftRotation

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D k1 C A

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Deletion in AVL Tree ď&#x201A;§ Delete is the inverse of insert: given a value X and an AVL tree T, delete the node containing X and rebalance the tree, if necessary. ď&#x201A;§ Turns out that deletion of a node is considerably more complex than insert

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Deletion in AVL Tree  Insertion in a height-balanced tree requires at most one single rotation or one double rotation.  We can use rotations to restore the balance when we do a deletion.  We may have to do a rotation at every level of the tree: log2N rotations in the worst case. http://ecomputernotes.com 24

Deletion in AVL Tree ď&#x201A;§ Here is a tree that causes this worse case number of rotations when we delete A. At every node in Nâ&#x20AC;&#x2122;s left subtree, the left subtree is one shorter than the right subtree. N F C A

I D

G E

K H

L M

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Deletion in AVL Tree ď&#x201A;§ Deleting A upsets balance at C. When rotate (D up, C down) to fix this N F C A

I D

G E

K H

L M

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Deletion in AVL Tree ď&#x201A;§ Deleting A upsets balance at C. When rotate (D up, C down) to fix this

N F C

I D

G E

K H

L M

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Deletion in AVL Tree  The whole of F’s left subtree gets shorter. We fix this by rotation about F-I: F down, I up. N F D C

I E

K H

L M

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Deletion in AVL Tree  The whole of F’s left subtree gets shorter. We fix this by rotation about F-I: F down, I up. N F D C

I E

K H

L M

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Deletion in AVL Tree ď&#x201A;§ This could cause imbalance at N. ď&#x201A;§ The rotations propagated to the root. N I F D C

K G

J H

L M

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Deletion in AVL Tree Procedure  Delete the node as in binary search tree (BST).  The node deleted will be either a leaf or have just one subtree.  Since this is an AVL tree, if the deleted node has one subtree, that subtree contains only one node (why?)  Traverse up the tree from the deleted node checking the balance of each node.

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Deletion in AVL Tree  There are 5 cases to consider.  Let us go through the cases graphically and determine what action to take.  We will not develop the C++ code for deleteNode in AVL tree. This will be left as an exercise.

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Deletion in AVL Tree Case 1a: the parent of the deleted node had a balance of 0 and the node was deleted in the parentâ&#x20AC;&#x2122;s left subtree.

Delete on this side

Action: change the balance of the parent node and stop. No further effect on balance of any higher node.

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Deletion in AVL Tree Here is why; the height of left tree does not change. 0

4 2 1

6 3

Deletion in AVL Tree Here is why; the height of left tree does not change. 0

4 2 1

6 3

4 2

6 3

remove(1)

Deletion in AVL Tree Case 1b: the parent of the deleted node had a balance of 0 and the node was deleted in the parentâ&#x20AC;&#x2122;s right subtree.

Delete on this side

Action: (same as 1a) change the balance of the parent node and stop. No further effect on balance of any higher node. 36

Deletion in AVL Tree Case 2a: the parent of the deleted node had a balance of 1 and the node was deleted in the parentâ&#x20AC;&#x2122;s left subtree.

Delete on this side

Action: change the balance of the parent node. May have caused imbalance in higher nodes so continue up the tree. 37