avl tree smt genap 2011-2012. outline avl tree ◦ definition ◦ properties ◦ operations smt...
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AVL TreeAVL Tree
Smt Genap 2011-2012
OutlineOutlineAVL Tree
◦Definition◦Properties◦Operations
Smt Genap 2011-2012
AVL TreesAVL TreesUnbalanced 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
X X
H
H-1H-2
AVL TreesAVL Trees
Smt Genap 2011-2012
10
5
3
20
2
1 3
10
5
3
20
1
43
5
AVL TreesAVL Trees
Smt Genap 2011-2012
12
8 16
4 10
2 6
14
Insertion for AVL TreeInsertion for AVL TreeAfter insert 1
Smt Genap 2011-2012
12
8 16
4 10
2 6
14
1
Insertion for AVL TreeInsertion for AVL TreeTo 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
Insertions Causing Insertions Causing ImbalanceImbalance
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
RP
Q
k1
k2
Q
k2
P
k1
R
HP=HQ=HR
Single Rotation (case 1)Single Rotation (case 1)
Smt Genap 2011-2012
A
k2
B
k1
CCBA
k1
k2
HA=HB+1HB=HC
Single Rotation (case 4)Single Rotation (case 4)
Smt Genap 2011-2012
C
k1
B
k2
AA B C
k2
k1
HA=HB
HC=HB+1
Problem with Single Problem with Single RotationRotationSingle rotation does not work for case
2 and 3 (inside case)
Smt Genap 2011-2012
Q
k2
P
k1
RR
P
Q
k1
k2
HQ=HP+1HP=HR
Double Rotation: StepDouble Rotation: Step
Smt Genap 2011-2012
C
k3
A
k1
D
B
k2
C
k3
A
k1
D
B
k2
HA=HB=HC=HD
Double Rotation: StepDouble Rotation: Step
Smt Genap 2011-2012
C
k3
A
k1
DB
k2
Double RotationDouble Rotation
Smt Genap 2011-2012
C
k3
A
k1
D
B
k2
C
k3
A
k1
DB
k2
HA=HB=HC=HD
Double RotationDouble Rotation
Smt Genap 2011-2012
B
k1
D
k3
A
C
k2
B
k1
D
k3
A C
k2
HA=HB=HC=HD
ExampleExampleInsert 3 into the AVL tree
Smt Genap 2011-2012
3
11
8 20
4 16 27
8
8
11
4 20
3 16 27
ExampleExampleInsert 5 into the AVL tree
Smt Genap 2011-2012
5
11
8 20
4 16 27 8
11
5 20
4 16 27
8
AVL Trees: ExerciseAVL Trees: ExerciseInsertion order:
◦10, 85, 15, 70, 20, 60, 30, 50, 65, 80, 90, 40, 5, 55
Smt Genap 2011-2012
Remove Operation in AVL Remove Operation in AVL TreeTreeRemoving 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
Deletion X in AVL TreesDeletion X in AVL TreesDeletion:
◦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
Delete 55 (case 1)Delete 55 (case 1)
Smt Genap 2011-2012
60
20 70
10 40 65 85
5 15 30 50 80 90
55
Delete 55 (case 1)Delete 55 (case 1)
Smt Genap 2011-2012
60
20 70
10 40 65 85
5 15 30 50 80 90
55
Delete 50 (case 2)Delete 50 (case 2)
Smt Genap 2011-2012
60
20 70
10 40 65 85
5 15 30 50 80 90
55
Delete 50 (case 2)Delete 50 (case 2)
Smt Genap 2011-2012
60
20 70
10 40 65 85
5 15 30 50 80 90
55
Delete 60 (case 3)Delete 60 (case 3)
Smt Genap 2011-2012
60
20 70
10 40 65 85
5 15 30 50 80 90
55
prev
Delete 60 (case 3)Delete 60 (case 3)
Smt Genap 2011-2012
55
20 70
10 40 65 85
5 15 30 50 80 90
Delete 55 (case 3)Delete 55 (case 3)
Smt Genap 2011-2012
55
20 70
10 40 65 85
5 15 30 50 80 90
prev
Delete 55 (case 3)Delete 55 (case 3)
Smt Genap 2011-2012
50
20 70
10 40 65 85
5 15 30 80 90
Delete 50 (case 3)Delete 50 (case 3)
Smt Genap 2011-2012
50
20 70
10 40 65 85
5 15 30 80 90
prev
Delete 50 (case 3)Delete 50 (case 3)
Smt Genap 2011-2012
40
20 70
10 30 65 85
5 15 80 90
Delete 40 (case 3)Delete 40 (case 3)
Smt Genap 2011-2012
40
20 70
10 30 65 85
5 15 80 90
prev
Delete 40 : RebalancingDelete 40 : Rebalancing
Smt Genap 2011-2012
30
20 70
10 65 85
5 15 80 90
Case ?
Delete 40: after Delete 40: after rebalancingrebalancing
Smt Genap 2011-2012
30
7010
20 65 855
15 80 90
Single rotation is preferred!
Minimum Element in AVL Minimum Element in AVL TreeTreeAn AVL Tree of height H has at least FH+3-1
nodes, where Fi is the i-th fibonacci numberS0 = 1S1 = 2SH = SH-1 + SH-2 + 1
Smt Genap 2011-2012
SH-1SH-2
H
H-1H-2
AVL Tree: analysis (2)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
SummarySummaryFind 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
Further StudyFurther StudySection 19.4 of Weiss book
Smt Genap 2011-2012
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