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AVL Tree Smt Genap 2011-2012

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Outline AVL Tree ◦ Definition ◦ Properties ◦ Operations Smt Genap 2011-2012

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AVL Trees Unbalanced Binary Search Trees are bad. Worst case: operations take O(n). AVL (Adelson-Velskii & Landis) trees maintain balance. For each node in tree, height of left subtree and height of right subtree differ by a maximum of 1. Smt Genap 2011-2012 XX H H-1 H-2

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AVL Trees Smt Genap 2011-2012 10 5 3 20 2 13 10 5 3 20 1 43 5

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AVL Trees Smt Genap 2011-2012 12 816 410 26 14

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Insertion for AVL Tree After insert 1 Smt Genap 2011-2012 12 816 410 26 14 1

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Insertion for AVL Tree To ensure balance condition for AVL-tree, after insertion of a new node, we back up the path from the inserted node to root and check the balance condition for each node. If after insertion, the balance condition does not hold in a certain node, we do one of the following rotations: ◦ Single rotation ◦ Double rotation Smt Genap 2011-2012

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Insertions Causing Imbalance An insertion into the subtree: ◦ P (outside) - case 1 ◦ Q (inside) - case 2 An insertion into the subtree: ◦ Q (inside) - case 3 ◦ R (outside) - case 4 Smt Genap 2011-2012 R P Q k1k1 k2k2 Q k2k2 P k1k1 R H P =H Q =H R

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Single Rotation (case 1) Smt Genap 2011-2012 A k2k2 B k1k1 C CB A k1k1 k2k2 H A =H B +1 H B =H C

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Single Rotation (case 4) Smt Genap 2011-2012 C k1k1 B k2k2 A AB C k2k2 k1k1 H A =H B H C =H B +1

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Problem with Single Rotation Single rotation does not work for case 2 and 3 (inside case) Smt Genap 2011-2012 Q k2k2 P k1k1 R R P Q k1k1 k2k2 H Q =H P +1 H P =H R

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Double Rotation: Step Smt Genap 2011-2012 C k3k3 A k1k1 D B k2k2 C k3k3 A k1k1 D B k2k2 H A =H B =H C =H D

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Double Rotation: Step Smt Genap 2011-2012 C k3k3 A k1k1 D B k2k2

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Double Rotation Smt Genap 2011-2012 C k3k3 A k1k1 D B k2k2 C k3k3 A k1k1 D B k2k2 H A =H B =H C =H D

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Double Rotation Smt Genap 2011-2012 B k1k1 D k3k3 A C k2k2 B k1k1 D k3k3 A C k2k2 H A =H B =H C =H D

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Example Insert 3 into the AVL tree Smt Genap 2011-2012 3 11 8 20 4 16 27 8 8 11 4 20 3 16 27

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Example Insert 5 into the AVL tree Smt Genap 2011-2012 5 11 8 20 4 16 278 11 5 20 4 16 27 8

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AVL Trees: Exercise Insertion order: ◦ 10, 85, 15, 70, 20, 60, 30, 50, 65, 80, 90, 40, 5, 55 Smt Genap 2011-2012

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Remove Operation in AVL Tree Removing a node from an AVL Tree is the same as removing from a binary search tree. However, it may unbalance the tree. Similar to insertion, starting from the removed node we check all the nodes in the path up to the root for the first unbalance node. Use the appropriate single or double rotation to balance the tree. May need to continue searching for unbalanced nodes all the way to the root. Smt Genap 2011-2012

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Deletion X in AVL Trees Deletion: ◦ Case 1: if X is a leaf, delete X ◦ Case 2: if X has 1 child, use it to replace X ◦ Case 3: if X has 2 children, replace X with its inorder predecessor (and recursively delete it) Rebalancing Smt Genap 2011-2012

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Delete 55 (case 1) Smt Genap 2011-2012 60 2070 10406585 5153050 8090 55

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Delete 55 (case 1) Smt Genap 2011-2012 60 2070 10406585 5153050 8090 55

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Delete 50 (case 2) Smt Genap 2011-2012 60 2070 10406585 5153050 8090 55

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Delete 50 (case 2) Smt Genap 2011-2012 60 2070 10406585 51530 50 8090 55

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Delete 60 (case 3) Smt Genap 2011-2012 60 2070 10406585 5153050 8090 55 prev

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Delete 60 (case 3) Smt Genap 2011-2012 55 2070 10406585 5153050 8090

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Delete 55 (case 3) Smt Genap 2011-2012 55 2070 10406585 5153050 8090 prev

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Delete 55 (case 3) Smt Genap 2011-2012 50 2070 10406585 51530 8090

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Delete 50 (case 3) Smt Genap 2011-2012 50 2070 10406585 51530 8090 prev

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Delete 50 (case 3) Smt Genap 2011-2012 40 2070 10306585 515 8090

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Delete 40 (case 3) Smt Genap 2011-2012 40 2070 10306585 515 8090 prev

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Delete 40 : Rebalancing Smt Genap 2011-2012 30 2070 106585 515 8090 Case ?

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Delete 40: after rebalancing Smt Genap 2011-2012 30 7010 206585 5 15 8090 Single rotation is preferred!

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Minimum Element in AVL Tree An AVL Tree of height H has at least F H+3 -1 nodes, where F i is the i-th fibonacci number S 0 = 1 S 1 = 2 S H = S H-1 + S H-2 + 1 Smt Genap 2011-2012 S H-1 S H-2 H H-1 H-2

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AVL Tree: analysis (2) The depth of AVL Trees is at most logarithmic. So, all of the operations on AVL trees are also logarithmic. Smt Genap 2011-2012

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Summary Find element, insert element, and remove element operations all have complexity O(log n) for worst case Insert operation: top-down insertion and bottom up balancing Smt Genap 2011-2012

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Further Study Section 19.4 of Weiss book Smt Genap 2011-2012

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AVL Trees AVL (Adel`son-Vel`skii and Landis) tree = – A BST – With the property: For every node, the heights of the left and right subtrees differ at most.

AVL Trees AVL (Adel`son-Vel`skii and Landis) tree = – A BST – With the property: For every node, the heights of the left and right subtrees differ at most.

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