# From Last Time: Remove (Delete) Operation

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1 CSE 32 Lecture : More on Search Trees Today s Topics: Lazy Operations Run Time Analysis of Binary Search Tree Operations Balanced Search Trees AVL Trees and Rotations Covered in Chapter of the text From Last Time: Remove (Delete) Operation Removing a node containing X:. Find the node containing X 2. Replace it with: If it has no children, with NULL If it has child, with that child If it has 2 children, with the node with the smallest value in its right subtree, (or largest value in left subtree) 3. Recursively remove node used in 2 and 3 Worst case: Recursion propagates all the way to a leaf node time is O(depth of tree) 5 2 2

2 Laziness in Data Structures A lazy operation is one that puts off work as much as possible in the hope that a future operation will make the current operation unnecessary Data Struct -ures 3 Lazy Deletion Idea: Mark node as deleted; no need to reorganize tree Skip marked nodes during Find or Insert Reorganize tree only when number of marked nodes exceeds a percentage of real nodes (e.g. 5%) Constant time penalty only due to marked nodes depth increases only by a constant amount if 5% are marked undeleted nodes (N nodes max N/2 marked) Modify Insert to make use of marked nodes whenever possible e.g. when deleted value is re-inserted Can also use lazy deletion for Lists

3 Run Time Analysis of BST operations All BST operations (except MakeEmpty) are O(d), where d is the depth of the accessed node in the tree MakeEmpty takes O(N) for a tree with N nodes frees all nodes We know: log N d N- for a binary tree with N nodes What is the best case tree? What is the worst case tree? Best Case Running Time of Insert/Remove/etc. =? Worst Case Running Time =? Average Case Running Time =? 5 The best, the worst, and the average For a binary tree with N nodes, depth d of any node satisfies: log N d N- So, best case running time of BST operations is O(log N) Worst case running time is O(N) Average case running time = O(average value of d) = O(log N) Can prove that average depth over all nodes = O(log N) if all insertion sequences equally likely. See Chap. in textbook for proof

4 Can we do better? Worst case running time of BST operations is O(N) E.g. What happens when you Insert elements in ascending (or descending) order? Insert 2,,, 8,, 2 into an empty BST Problem: Lack of balance Tree becomes highly asymmetric Idea: Can we restore balance by re-arranging tree according to depths of left and right subtrees? Goal: Get depth down from O(N) to O(log N) Idea #: Achieving the perfect balance First try at balancing trees: Perfect balance Re-arrange to get a complete tree after every operation Recall: A tree is complete if there are no holes when scanning from top to bottom, left to right Problem: Too expensive to re-arrange E.g. Insert 2 in the example shown Need a looser constraint 5 8 Insert 2 & make tree 5 complete 2 8 8

5 Idea #2: Leave it to the professionals Many efficient algorithms exist for balancing trees in order to achieve faster running times for the BST operations Adelson-Velskii and Landis (AVL) trees (2) Splay trees and other self-adjusting trees (8) B-trees and other multiway search trees (2) AVL Trees AVL trees are height-balanced binary search trees Balance factor of a node = height(left subtree) - height(right subtree) 3 2 Heights An AVL tree can only have balance factors of,, or at every node For every node, heights of left and right subtree differ by no more than Height of an empty subtree = Balance factors -(-) = 2 Implementation: Store current heights in each node (-)- = - 8

6 Which of these are AVL trees? AVL Trees: Examples and Non-Examples Balance factors - Not AVL -(-) = 2 (-)- = - 8 AVL - 5 AVL AVL Not AVL 2

7 The good news about AVL Trees Can prove: Height of an AVL tree of N nodes is always O(log N) How? Can show: Height h. log(n+2)-.328 Prove using recurrence relation for minimum number of nodes S(h) in an AVL tree of height h: S(h) = S(h-) + S(h-2) + Use Fibonacci numbers to get bound on S(h) bound on height h Height = See textbook for details O(log N) The really good news about AVL Trees Can prove: Height of an AVL tree of N nodes is always O(log N) All operations (e.g. Find, Remove using lazy deletion, etc.) on an AVL tree are O(log N) except Insert Insert 3 Why is Insert different? 5 -

8 The bad news about AVL Trees 5 - Insert No longer an AVL tree (i.e. not balanced anymore) 5 Restoring Balance in (the life of) an AVL Tree Problem: Insert may cause balance factor to become 2 or 2 for some node on the path from insertion point to root node Idea: After Inserting the new node,. Back up to root updating heights along the access path 2. If Balance Factor = 2 or 2, adjust tree by rotation around deepest such node

9 Rotating to restore Balance: A Simple Example Insert Rotate AVL Not AVL AVL Next Class: Rotating and Splaying for Fun and Profit To Do: Finish Reading Chapter 8

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The Union-Find Problem Kruskal s algorithm for finding an MST presented us with a problem in data-structure design. As we looked at each edge, cheapest first, we had to determine whether its two endpoints