AVL Tree
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AVL Tree 27 th Mar 2007
description
AVL Tree. AVL Trees. Pencarian node pada unbalanced Binary Search Trees adalah tidak efisien . Worst case: operations take O(n). AVL ( Adelson-Velskii & Landis) trees adalah BST yang diseimbangkan - PowerPoint PPT Presentation
Transcript of AVL Tree
AVL Tree
27th Mar 2007
AVL Trees Pencarian node pada unbalanced Binary Search
Trees adalah tidak efisien. Worst case: operations take O(n).
AVL (Adelson-Velskii & Landis) trees adalah BST yang diseimbangkan
Untuk setiap node pada tree, beda tinggi antara left subtree dan right subtree maximum 1 saja.
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X X
H
H-1H-2
AVL Trees
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12
8 16
4 10
2 6
14
Untuk setiap node pada tree, beda tinggi antara left subtree dan right subtree maximum 1 saja.
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AVL property violated here
AVL Trees
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10
5
3
20
2
1 3
10
5
3
20
1
43
5
Insertion for AVL TreeAfter insert 1
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12
8 16
4 10
2 6
14
1
Insertion for AVL Tree
Untuk memastikan AVL-tree dalam kondisi seimbang, setelah suatu node di-insert-kan, perlu dicek keseimbangan untuk tiap node
Jika setelah insertion ada node yang tidak seimbang, maka dilakukan operasi berikut :Single rotation
Double rotation
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Insertions Causing Imbalance
Kemungkinan insertion yang menyebabkan ketidakseimbangan Insertion pada P (outside) case1 Insertion pada Q (inside) case2
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RP
Q
k1
k2
Q
k2
P
k1
R
AVL TreeHP=HQ=HR
Kemungkinan insertion yang menyebabkan ketidakseimbangan Insertion pada Q (inside) case 3 Insertion pada R (outside) case 4
Single Rotation (case 1)
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A
k2
B
k1
CCBA
k1
k2
insertion pada A menyebabkan ketidakseimbangan
Single Rotation (case 4)
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C
k1
B
k2
A
A B C
k2
k1
insertion pada C menyebabkan ketidakseimbangan
Problem with Single Rotation Single rotation does not work for case 2 and 3
(inside case)
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Q
k2
P
k1
RR
P
Q
k1
k2
Insertion pada Q kemudian single rotation, tetap tidak seimbang
Double Rotation: Stepsama dengan dua kali single rotation
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C
k3
A
k1
D
B
k2
C
k3
A
k1
D
B
k2
Double Rotation: Step
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C
k3
A
k1
DB
k2
Double Rotation
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C
k3
A
k1
D
B
k2
C
k3
A
k1
DB
k2
Double Rotation
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B
k1
D
k3
A
C
k2
B
k1
D
k3
A C
k2
ExampleInsert 3 into the AVL tree
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3
11
8 20
4 16 27
8
8
11
4 20
3 16 27
ExampleInsert 5 into the AVL tree
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5
11
8 20
4 16 27 8
11
5 20
4 16 27
8
AVL Trees: Exercise
Insertion order:10, 85, 15, 70, 20, 60, 30, 50, 65, 80, 90, 40,
5, 55
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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.
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Deletion X in AVL Trees
Deletion:Case 1: if X is a leaf, delete XCase 2: if X has 1 child, use it to replace XCase 3: if X has 2 children, replace X with its
inorder predecessor (and recursively delete it)
Rebalancing
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Delete 55 (case 1)
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60
20 70
10 40 65 85
5 15 30 50 80 90
55
Delete 55 (case 1)
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60
20 70
10 40 65 85
5 15 30 50 80 90
55
Delete 50 (case 2)
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60
20 70
10 40 65 85
5 15 30 50 80 90
55
Delete 50 (case 2)
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60
20 70
10 40 65 85
5 15 30 50 80 90
55
Delete 60 (case 3)
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60
20 70
10 40 65 85
5 15 30 50 80 90
55
prev
Delete 60 (case 3)
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55
20 70
10 40 65 85
5 15 30 50 80 90
Delete 55 (case 3)
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55
20 70
10 40 65 85
5 15 30 50 80 90
prev
Delete 55 (case 3)
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50
20 70
10 40 65 85
5 15 30 80 90
Delete 50 (case 3)
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50
20 70
10 40 65 85
5 15 30 80 90
prev
Delete 50 (case 3)
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40
20 70
10 30 65 85
5 15 80 90
Delete 40 (case 3)
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40
20 70
10 30 65 85
5 15 80 90
prev
Delete 40 : Rebalancing
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30
20 70
10 65 85
5 15 80 90
Case ?
Delete 40: after rebalancing
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30
7010
20 65 855
15 80 90
Single rotation is preferred!
Minimum Element in AVL Tree An AVL Tree of height H has at least FH+3-1 nodes,
where Fi is the i-th fibonacci number S0 = 1 S1 = 2 SH = SH-1 + SH-2 + 1
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SH-1
SH-2
H
H-1H-2
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
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