CSI33 Data Structures
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1 Outline Department of Mathematics and Computer Science Bronx Community College November 21, 2018
2 Outline Outline 1 C++ Supplement 1.3: Balanced Binary Search Trees Balanced Binary Search Trees
3 Outline 1 C++ Supplement 1.3: Balanced Binary Search Trees Balanced Binary Search Trees
4 Improving The Worst-Case Performance for BSTs The Worst Case Scenario In the worst case, a binary search tree looks like a linked list, with all the links going the same way. The performance of the important methods (find, insert, delete) is Θ(n). 4
5 Improving The Worst-Case Performance for BSTs Goal: Keeping Any BST Balanced Ideally, to prevent a BST from becoming too unbalanced, it would be filled so that as many nodes as possible have left and right subtrees. This would be equivalent to being a complete binary tree. This is impractical, since it would take too long to rearrange the nodes for the tree to keep this shape every time a new node gets added or deleted.
6 Improving The Worst-Case Performance for BSTs A Workable Compromise We will only insist that, for a BST to be balanced, any node will have the property that the depths of its left and right subtrees will differ by one level at most. This can be efficiently enforced each time a node is inserted or deleted. The worst case height is about 1.44 log(n). The performance of the insert, delete, and find operations is Θ(log n).
7 Basic Facts The AVL Tree property An AVL tree is a binary search tree (so it has the Binary Search Property), which has the additional AVL Tree Property that for every node, the depths of its left and right subtrees will differ by at most one level.
8 Basic Facts The AVL Tree property An AVL tree is a binary search tree (so it has the Binary Search Property), which has the additional AVL Tree Property that for every node, the depths of its left and right subtrees will differ by at most one level. Inventors Such a tree is called an AVL Tree after its two co-inventors, G. M. Adelson-Velskii and E. M. Landis.
9 : Insertion Normal BST Insertion A value gets inserted into a BST by comparing its value with the current node (starting with the root). If the value is less, it changes the current node to the left subtree if it exists. If the value is greater, it changes the current node to the right subtree if it exists. If the value is equal, an error has occurred: value is already in the tree. The new node is made a leaf when the subtree on that side doesn t exist.
10 : Insertion AVL Insertion: Overview The height of each subtree is saved as a new attribute of every TreeNode object. Perform the insertion to the proper subtree (say, the left subtree). If the left subtree height is now 2 more than the right subtree, rebalance the tree at the current node. Similarly for the right subtree. Height of the current node = max(height left subtree, height right subtree)+1.
11 : Insertion AVL Insertion: Overview void AVLTree::insert(int value) root = insertrec( root, value);
12 : Insertion AVL Rebalancing TreeNode *AVLTree:: insertrec(treenode* t, int value) if (t == NULL) t = new TreeNode(value, NULL, NULL); else if (value < t-> item) t-> left = insertrec(t-> left, value); if (getheight(t-> left) - getheight(t-> right) == 2) //rebalance? // inserted into which subtree of left child? if (value < t-> left-> item) t = leftsinglerotate(t); // left subtree else t = rightleftrotate(t); // right subtree
13 : Insertion AVL Rebalancing: Double Rotation TreeNode *AVLTree:: rightleftrotate(treenode *t) t-> left = rightsinglerotate(t-> left); t = leftsinglerotate(t); return t; 2 3 t
14 : Insertion AVL Right Subtree Insertion: Rebalancing at node t TreeNode *AVLTree:: rightsinglerotate(treenode *t) TreeNode *grand = t; TreeNode * = t-> right; grand-> right = -> left; -> left = grand; t = ; // adjust heights of grand, return t; 2 t
15 : Insertion AVL Right Subtree Insertion: Rebalancing at node t TreeNode *AVLTree:: rightsinglerotate(treenode *t) TreeNode *grand = t; TreeNode * = t-> right; grand-> right = -> left; -> left = grand; t = ; // adjust heights of grand, return t; t, grand NONE
16 : Insertion AVL Right Subtree Insertion: Rebalancing at node t TreeNode *AVLTree:: rightsinglerotate(treenode *t) TreeNode *grand = t; TreeNode * = t-> right; grand-> right = -> left; -> left = grand; t = ; // adjust heights of grand, return t; t, grand NONE
17 : Insertion AVL Right Subtree Insertion: Rebalancing at node t TreeNode *AVLTree:: rightsinglerotate(treenode *t) TreeNode *grand = t; TreeNode * = t-> right; grand-> right = -> left; -> left = grand; t = ; // adjust heights of grand, return t; grand t, NONE
18 : Insertion AVL Right Subtree Insertion: Rebalancing at node t TreeNode *AVLTree:: rightsinglerotate(treenode *t) TreeNode *grand = t; TreeNode * = t-> right; grand-> right = -> left; -> left = grand; t = ; // adjust heights of grand, return t; t 3 grand
19 : Insertion AVL Left Subtree Insertion: Rebalancing at node t TreeNode *AVLTree:: leftsinglerotate(treenode *t) TreeNode *grand = t; TreeNode * = t-> left; grand-> left = -> right; -> right = grand; t = ; // adjust heights of grand, return t; t 4 6 8
20 : Insertion AVL Left Subtree Insertion: Rebalancing at node t TreeNode *AVLTree:: leftsinglerotate(treenode *t) TreeNode *grand = t; TreeNode * = t-> left; grand-> left = -> right; -> right = grand; t = ; // adjust heights of grand, return t; NONE t, grand 8
21 : Insertion AVL Left Subtree Insertion: Rebalancing at node t TreeNode *AVLTree:: leftsinglerotate(treenode *t) TreeNode *grand = t; TreeNode * = t-> left; grand-> left = -> right; -> right = grand; t = ; // adjust heights of grand, return t; NONE 6 t, grand 8
22 : Insertion AVL Left Subtree Insertion: Rebalancing at node t TreeNode *AVLTree:: leftsinglerotate(treenode *t) TreeNode *grand = t; TreeNode * = t-> left; grand-> left = -> right; -> right = grand; t = ; // adjust heights of grand, return t; t, NONE 6 grand 8
23 : Insertion AVL Left Subtree Insertion: Rebalancing at node t TreeNode *AVLTree:: leftsinglerotate(treenode *t) TreeNode *grand = t; TreeNode * = t-> left; grand-> left = -> right; -> right = grand; t = ; // adjust heights of grand, return t; t
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