A4B33ALG 2011 / 06
AVL tree -- G.M. Adelson-Velskij & E.M. Landis, 1962
AVL tree is a BST with additional properties
which keep it acceptably balanced.
2
1
0
-1
8
-1
13
0
-1
34
-1
1
1
-1
22
51
Operations
Find, Insert, Delete
also apply to AVL tree.
0
40
0
-1
-1
45
68
-1
76
0
-1
92
-1
-1
AVL rule:
There are two integers associated
with each node:
Depth of the left and depth of
the right subtree of the node.
Note: Depth of an empty tree is -1.
The difference of the heights of
the left and the right subtree
may be only -1 or 0 or 1
in each node of the tree.
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A4B33ALG 2011 / 06
Inserting a node may disbalance the AVL tree
The subtrees height difference
may be only -1, 0, 1 !!
Insert 30
3
2
0
-1
8
-1
13
34
1
-1
-1
Changed depths
0
30
1
1
-1
22
51
0
40
0
-1
-1
45
68
-1
76
0
-1
92
-1
-1
-1
Left subtree of node 51 is too deep,
the tree is no more an AVL tree.
3
A4B33ALG 2011 / 06
Rotation R yields a balanced tree
51
3
Insert 30
2
0
-1
8
13
-1
34
1
-1
1
-1
22
-1
0
40
0
30
1
0
-1
-1
45
68
76
0
-1
-1
92
-1
-1
-1
34
2
2
0 13 1
The tree balanced
by single R rotation
-1
8
-1
-1
1
22
-1
0
30
-1
-1
40
51
0
-1
1
0
45
-1
-1
68
76
-1
0
-1
92
-1
4
A4B33ALG 2011 / 06
Rotate to obtain a balanced tree
51
3
2
Insert 30
0
-1
8
-1
13
34
1
-1
1
-1
22
-1
0
40
0
30
1
0
-1
-1
45
68
76
0
-1
-1
92
-1
-1
Direction of rotation
-1
34
2
0 13 1
Rotation R in node 51
Unaffected
subtrees
-1
2
8
-1
-1
1
22
-1
0
30
-1
-1
40
51
0
-1
1
0
45
-1
-1
68
76
-1
0
-1
92
-1
5
A4B33ALG 2011 / 06
Rotation R in general, before and after
A
x+2
x+1
Before
B
C
x
D
-1
E
Z
-1
x+1
After
x
B
x+1
D
x
E
-1
Z
A
x
C
-1
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A4B33ALG 2011 / 06
Rotation L in general, before and after
x
Before
A
x+2
C
x
B
x+1
D
E
Rotation L is a mirror image of
rotation R, there is no other
difference between the two.
Z
-1
x+1
After
x
C
A
B
-1
x+1
D
x
E
-1
Z
-1
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A4B33ALG 2011 / 06
AVL tree
2
1
0
-1
8
-1
13
34
-1
1
1
0
-1
51
36
0
40
0
-1 -1
45
-1
68
-1
76
0
-1
92
-1
-1
Demonstration AVL tree for rotation LR
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A4B33ALG 2011 / 06
Inserting a node may disbalance the AVL tree
The subtrees height difference
may be only -1, 0, 1 !!
Insert 38
3
1
0
-1
8
-1
13
34
-1
36
-1
Left subtree of node 51
is too deep, the tree
is no more an AVL tree.
1
2
1
-1
51
0
40
0
-1
0 -1
38
45
-1
-1
68
-1
76
0
-1
92
-1
Changed depths
-1
Rotation R would not help, the right
subtree of node 34 would become
relatively too deep compared
to the new right subtree of the root.
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A4B33ALG 2011 / 06
Rotation LR yields a balanced tree
51
3
Insert 38
1
0
-1
8
13
34
-1
2
1
-1
-1
36
-1
0
40
0
-1
0 -1
38
-1
45
68
76
0
-1
-1
92
-1
-1
-1
40
Balanced tree
after double
rotation LR.
2
1
0
-1
1
8
-1
13
-1
34
2
0
1
-1
36
-1
-1
0
38
-1
-1
45
51
-1
1
0
-1
68
76
-1
0
-1
92
-1
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A4B33ALG 2011 / 06
Rotate to obtain a balanced tree
51
3
Insert 38
1
0
-1
8
13
34
1
-1
-1
36
-1
Rotation LR
in nodes 34 and 51
0
-1
8
-1
13
R
2
L
-1
40
0
-1
0 -1
38
-1
34
0
-1
45
68
-1
0
-1
-1
92
-1
Direction
of rotations
40
Unaffected
subtrees
2
0
1
-1
76
-1
2
1
1
36
-1
-1
0
38
-1
-1
45
51
-1
1
0
-1
68
76
-1
0
-1
92
-1
11
A4B33ALG 2011 / 06
Rotation LR in general, before and after
X+2
x+1
Before
B
C
E
x
0
F
-1
G
Z
-1
x+1
x
x
x+1
D
After
A
B
E
x+1
x
x
F
D
-1
Z
G
A
x
C
-1
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A4B33ALG 2011 / 06
Rotation RL in general, before and after
x
A
X+2
C
x+1
B
x+1
Before
0
E
G
Rotation RL is a mirror image of
rotation LR, there is no other
difference between the two.
After
x
C
A
F
-1
x+1
E
D
x
Z
-1
x+1
x
x
G
F
-1
Z
B
x
D
-1
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A4B33ALG 2011 / 06
Rules for aplying rotations L, R, LR, RL in Insert operation
Travel from the inserted node up to the root
and update subtree depths in each node along the path.
If a node is disbalanced and you came to it along two consecutive edges
* in the up and right direction
perform rotation R in this node,
* in the up and left direction
perform rotation L in this node,
* first in the in the up and left and then in the up and right direction
perform rotation LR in this node,
* first in the in the up and right and then in the up and left direction
perform rotation RL in this node,
After one rotation in the Insert operation the AVL tree is balanced.
After one rotation in the Delete operation the AVL tree might still
not be balanced, all nodes on the path to the root have to be checked.
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A4B33ALG 2011 / 06
Delete in AVL tree
Demonstration AVL tree for rotation after deletion
1
-1
16
51
2
0
1
Delete 16
-1
28
-1
0
-1
55
70
84
0
-1
-1
93
-1
-1
1. Remove node using the same method as in BST.
2. Travel from the place of deletion up to the root.
Update subtree heights in each node, and if necessary
apply the corresponding rotation.
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A4B33ALG 2011 / 06
Delete in AVL tree
1
Delete 16
-1
16
51
2
0
1
-1
28
-1
-1
0
-1
28
70
0
51
55
84
0
-1
-1
93
-1
-1
2
-1
1
0
Changed depths
-1
55
70
-1
84
0
-1
93
-1
-1
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A4B33ALG 2011 / 06
Delete in AVL tree
0
-1
28
51
L
2
-1
1
0
Changed depths
-1
55
70
84
0
R
-1
-1
93
-1
-1
In the disbalanced node (51), check the root (84) of the subtree opposite
to the one which you came from.
If the heights of subtrees of that root are equal apply a single rotation R or L.
If the hight of the more distant subtree of that root is bigger than the height
of the less distant one apply a single rotation R or L.
In the remaining case apply a double rotation RL or LR.
In this example, apply RL.
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A4B33ALG 2011 / 06
Delete in AVL tree
0
-1
28
51
L
2
-1
1
Delete 16
0
70
84
0
R
-1
-1
93
-1
Changed depths
-1
0
After rotation RL
in nodes 84 and 51
0
-1
28
-1
51
0
-1
55
70
-1
0
-1
55
-1
84
0
-1
93
-1
18
A4B33ALG 2011 / 06
Possibility of multiple rotations in operation Delete.
Example. The AVL tree is
originally balanced.
1
3
2
4
Balanced.
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A4B33ALG 2011 / 06
Implementation ot the AVL tree operations
...
// homework...
Asymptotic complexities of Find, Insert, Delete in BST and AVL
BST with n nodes
AVL tree with n nodes
Operation
Balanced
Maybe not
balanced
Balanced
Find
(log(n))
(n)
(log(n))
Insert
(log(n))
(n)
(log(n))
Delete
(log(n))
(n)
(log(n))
20
A4B33ALG 2011 / 06
B-tree -- Rudolf Bayer, Edward M. McCreight, 1972
All lengths of paths from the root to the leaves are equal.
B-tree is perfectly balanced.
Keys in the nodes are kept sorted.
Fixed k > 1 dictates the same size of all nodes.
Each node except for the root contains at least k and at most 2k keys
and if it is not a leaf it has at least k+1 and at most 2k+1 children.
The root can contain any number of keys from 1 to 2k.
If it is not simultaneously a leaf it has at least 2 and at most 2k+1
children.
X Y
keys < X
X < keys < Y
Y < keys
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A4B33ALG 2011 / 06
B-tree -- Find
Find 17
18
8 14
1 2 4 5
10 12
15 16 17
26 41
19 20 22 25
27 36
42 45 60
Search in the node is sequential (or binary or other...).
If the node is not a leaf and the key is not in the node
then the search continues in the appropriate child node.
If the node is a leaf and the key is not in the node
then the key is not in the tree.
22
A4B33ALG 2011 / 06
B-tree -- Insert
B-tree
8 17 26
2 4
10 12 14 16
Insert 5
19 22 25
36 41 42 45
8 17 26 41
2 4 5
10 12 14 16
19 22 25
36 41 42 45
Insert 20
8 17 26
2 4 5
10 12 14 16
19 20 22 25
36 41 42 45
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A4B33ALG 2011 / 06
B-tree -- Insert
Insert 27
2 4 5
8 17 26
10 12 14 16
19 20 22 25
36 41 42 45
27
27 36 41 42 45
Sort outside the tree.
Select median,
create new node,
move to it the values
bigger than the median.
41
27 36
42 45
27
8 17 26 41
Try to insert the median
into the parent node.
19 20 22 25
27 36
42 45
Success.
24
A4B33ALG 2011 / 06
B-tree -- Insert
Insert 15
2 4 5
8 17 26 41
10 12 14 16
19 20 22 25
27 36
42 45
15
10 12 14 15 16
Sort outside the tree.
Select median,
create new node,
move to it the values
bigger than the median.
14
10 12
15 16
?
Try to insert the median
into the parent node.
14
8 17 26 41
Success?
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A4B33ALG 2011 / 06
B-tree -- Insert
K 15 inserted into a leaf...
Key
...
8 17 26 41
... key 14 goes to parent node
14
2 4 5
10 12
15 16
19 20 22 25
27 36
42 45
The parent node is full – repeat the process analogously.
Sort values
Select median, create new node,
move to it the values bigger
than the median together with
the corresponding references.
Cannot propagate the median into
the parent (there is no parent),
create a new root and store the
median there.
8 14 17 26 41
17
8 14
26 41
17
8 14
26 41
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A4B33ALG 2011 / 06
B-tree -- Insert
Recapitulation - insert 15
8 17 26 41
2 4 5
10 12 14 16
19 20 22 25
27 36
42 45
Insert 15
17
8 14
2 4 5
10 12
15 16
Unchanged nodes
26 41
19 20 22 25
27 36
42 45
Each lavel acquired one new node, a new root was created too,
the tree grows upwards and remains perfectly balanced.
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A4B33ALG 2011 / 06
B-tree -- Delete
Delete in a sufficiently
full leaf.
Delete 4
2 4 5
17
8 14
10 12
26 41
15 16
19 20 22 25
27 36
42 45 60
27 36
42 45 60
17
8 14
2 5
10 12
15 16
26 41
19 20 22 25
28
A4B33ALG 2011 / 06
B-tree -- Delete
The deleted key is substituted
by the smallest bigger key,
like in a BST.
Delete in an internal node
17
Delete 17
8 14
2 5
10 12
26 41
19 20 22 25
15 16
27 36
42 45 60
The smallest bigger key is always in the leaf in a B-tree.
If the leaf is sufficiently full the delete operation is complete.
19
8 14
2 5
10 12
15 16
26 41
20 22 25
27 36
42 45 60
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A4B33ALG 2011 / 06
B-tree -- Delete
Delete in an
insufficiently full leaf.
Delete 27
2 5
19
8 14
10 12
The neighbour leaf
is sufficiently full.
26 41
15 16
20 22 25
Merge the keys of the two leaves
with the dividing key in the parent
into one sorted list.
27 36
42 45 60
26 41
42 45 60
36
36 41 42 45 60
Insert the median of the sorted list
into the parent and distribute
the remainig keys into
the left and right children of the median.
26 42
36 41
45 60
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A4B33ALG 2011 / 06
B-tree -- Delete
Recapitulation - delete 27
19
8 14
2 5
10 12
26 41
15 16
20 22 25
27 correctly deleted
27 36
42 45 60
Unchanged nodes
19
8 14
2 5
10 12
15 16
26 42
41
20 22 25
27 41
36
36
42 60
45
45 60
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A4B33ALG 2011 / 06
B-tree -- Delete
Delete in an
insufficiently full node.
Delete 12
2 5
19
8 14
10 12
None of the neighbours
is sufficiently full.
15 16
26 42
41
20 22 25
27 41
36
36
42 60
45
45 60
8 14
10 12
Merge the keys of the node and of one
of the neighbours and the median
in the parent into one sorted list.
Move all these keys to the original node,
delete the neighbour, remove the original
median and associated reference
from the parent.
15 16
8 14
10 14 15 16
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A4B33ALG 2011 / 06
B-strom -- Delete
Deleted 12
19
8
2 5
The parent violates
B-tree rules.
10 14 15 16
26 42
41
20 22 25
27 41
36
36
42 60
45
45 60
If the parent of the deleted node is not sufficiently full
apply the same deleting strategy to the parent and continue the process
towards the root until the rules of B-tree are satisfied.
19
8 19
26
41 26 42
8
26 42
41
8 19 26 42
26 41
33
A4B33ALG 2011 / 06
B-tree -- Delete
Recapitulation - delete 12
19
8 14
2 5
10 12
26 42
41
15 16
20 22 25
27 41
36
36
42 60
45
45 60
Key 12 was deleted and the tree was reconstructed accordingly.
Unchanged nodes
8 19
26
41 26 42
2 5
10 14 15 16
20 22 25
27 41
36
36
42 60
45
45 60
34