Objectives � Memahami variant dari Binary Search Tree yang balanced Struktur Data & Algoritme ( Data Structures & Algorithms ) AVL Tree Denny ( denny@cs.ui.ac.id ) Suryana Setiawan ( setiawan@cs.ui.ac.id ) Fakultas I lm u Kom puter Universitas I ndonesia Sem ester Genap - 2 0 0 4 / 2 0 0 5 Version 2 .0 - I nternal Use Only SDA/ TOPIC/ V2.0/ 2 Outline Motivation � AVL Tree � Unbalanced Binary Search Tree would make all operations take O(n) operation - worst case. � Definition � Property � operations SDA/ TOPIC/ V2.0/ 3 SDA/ TOPIC/ V2.0/ 4 1
AVL Trees AVL Trees � Untuk setiap node dalam tree, ketinggian subtree di 10 anak kiri dan subtree di anak kanan hanya berbeda 10 maksimum 1. 5 5 20 5 20 X X 43 3 3 H H-2 1 H-1 2 1 3 SDA/ TOPIC/ V2.0/ 5 SDA/ TOPIC/ V2.0/ 6 AVL Trees Insertion for AVL Tree � After insert 1 12 12 8 16 8 16 4 10 14 4 10 14 2 6 2 6 1 SDA/ TOPIC/ V2.0/ 7 SDA/ TOPIC/ V2.0/ 8 2
Insertion for AVL Tree Unbalanced Condition � To ensure balance condition for AVL-tree, after H P = H Q = H R k 2 k 1 insertion of a new node, we back up the path from the inserted node to root and check the balance k 1 k 2 condition for each node. � If after insertion, the balance condition does not hold in a certain node, we do one of the following rotations: R P P Q Q R � Single rotation � Double rotation � An insertion into the subtree: � An insertion into the subtree: � P (outside) - case 1 � Q (inside) - case 3 � Q (inside) - case 2 � R (outside) - case 4 SDA/ TOPIC/ V2.0/ 9 SDA/ TOPIC/ V2.0/ 10 Single Rotation (case 1) Single Rotation (case 4) H A = H B + 1 H A = H B H B = H C H C = H B + 1 k 2 k 1 k 2 k 1 k 1 k 2 k 1 k 2 A C C A A B B B B C C A SDA/ TOPIC/ V2.0/ 11 SDA/ TOPIC/ V2.0/ 12 3
Problem with Single Rotation Double Rotation: Step � Single rotation does not work for case 2 and 3 ( inside case ) k 3 k 3 k 2 k 1 k 1 k 2 k 1 k 2 k 2 k 1 R P D D P R Q Q A C B C B A H Q = H P + 1 H P = H R H A = H B = H C = H D SDA/ TOPIC/ V2.0/ 13 SDA/ TOPIC/ V2.0/ 14 Double Rotation: Step Double Rotation k 3 k 2 k 2 k 1 k 1 k 3 k 1 k 3 k 2 D B C A D A B C A D B C H A = H B = H C = H D SDA/ TOPIC/ V2.0/ 15 SDA/ TOPIC/ V2.0/ 16 4
Double Rotation Example � Insert 3 into the AVL tree k 2 k 1 11 11 k 3 k 1 k 3 k 2 8 8 20 4 20 A A B C D 4 16 27 3 16 D 8 27 B C 3 H A = H B = H C = H D SDA/ TOPIC/ V2.0/ 17 SDA/ TOPIC/ V2.0/ 18 Example AVL Trees: Exercise � Insertion order: � Insert 5 into the AVL tree � 10, 85, 15, 70, 20, 60, 30, 50, 65, 80, 90, 40, 5, 55 11 11 8 8 20 5 20 4 16 27 4 16 8 27 5 SDA/ TOPIC/ V2.0/ 19 SDA/ TOPIC/ V2.0/ 20 5
Remove Operation in AVL Tree Deletion X in AVL Trees � Removing from AVL Tree is same as removing from a � Deletion: binary search tree procedure with this difference that � Case 1: jika X adalah leaf, delete X it may unbalance the tree. � Case 2: jika X punya 1 child, X digantikan oleh child � Similar to insertion, starting from the removed node tsb. we check all the nodes in the path up to the root for � Case 3: jika X punya 2 child, ganti X secara rekursif the first unbalance node. dengan predecessor-nya secara inorder � We use the appropriate single or double rotation to � Rebalancing balance the tree. � We may need to continue searching for unbalanced node all the way to the root. SDA/ TOPIC/ V2.0/ 21 SDA/ TOPIC/ V2.0/ 22 Delete 55 (case 1) Delete 55 (case 1) 60 60 20 70 20 70 10 40 65 85 10 40 65 85 80 90 80 90 5 15 30 50 5 15 30 50 55 55 SDA/ TOPIC/ V2.0/ 23 SDA/ TOPIC/ V2.0/ 24 6
Delete 50 (case 2) Delete 50 (case 2) 60 60 20 70 20 70 10 40 65 85 10 40 65 85 50 80 90 80 90 5 15 30 50 5 15 30 55 55 SDA/ TOPIC/ V2.0/ 25 SDA/ TOPIC/ V2.0/ 26 Delete 60 (case 3) Delete 60 (case 3) 60 55 20 70 20 70 10 40 65 85 10 40 65 85 prev 80 90 80 90 5 15 30 50 5 15 30 50 55 SDA/ TOPIC/ V2.0/ 27 SDA/ TOPIC/ V2.0/ 28 7
Delete 55 (case 3) Delete 55 (case 3) 55 50 20 70 20 70 prev 10 40 65 85 10 40 65 85 80 90 80 90 5 15 30 50 5 15 30 SDA/ TOPIC/ V2.0/ 29 SDA/ TOPIC/ V2.0/ 30 Delete 50 (case 3) Delete 50 (case 3) 50 40 prev 20 70 20 70 10 40 65 85 10 30 65 85 80 90 80 90 5 15 30 5 15 SDA/ TOPIC/ V2.0/ 31 SDA/ TOPIC/ V2.0/ 32 8
Delete 40 (case 3) Delete 40 : Rebalancing 40 30 20 70 20 70 prev Case ? 10 30 65 85 10 65 85 80 90 80 90 5 15 5 15 SDA/ TOPIC/ V2.0/ 33 SDA/ TOPIC/ V2.0/ 34 Delete 40: after rebalancing 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 30 � S 1 = 2 10 70 � S H = S H-1 + S H-2 + 1 20 65 85 5 80 90 15 Single rotation is preferred! H H-2 S H-2 H-1 S H-1 SDA/ TOPIC/ V2.0/ 35 SDA/ TOPIC/ V2.0/ 36 9
AVL Tree: analysis (1) AVL Tree: analysis (2) � The depth of AVL Trees at is at most logarithmic. ≈ i φ / 5 F i � So, all of the operations on AVL trees are also logarithmic. = + ≈ φ (1 5 ) / 2 1 . 618 � The worst-case height is at most 44 percent more than the minimum possible for binary trees. ( ) AVL Tree of height H has at least roughly + H 3 φ / 5 < + − 1 . 44 log( 2 ) 1 . 328 H N SDA/ TOPIC/ V2.0/ 37 SDA/ TOPIC/ V2.0/ 38 Summary Further Study � Find element, insert element, and remove element � http://telaga.cs.ui.ac.id/WebKuliah/IKI101 operations all have complexity O(log n) for worst 00/resources/animation/data- case structure/avl/avltree.html (mirror from � Insert operation: top-down insertion and bottom up http://www.geocities.com/SiliconValley/ balancing Heights/8471/avltree.html ) � Chapter 18 SDA/ TOPIC/ V2.0/ 39 SDA/ TOPIC/ V2.0/ 40 10
What’s Next � B Trees SDA/ TOPIC/ V2.0/ 41 11
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