AVL Trees CSC 172 SPRING 2002 LECTURE 17. A PROBLEM WITH BSTs Common operations on balanced BST are O(log(n)) Alas, when the tree goes out of balance,

AVL Trees

CSC 172

SPRING 2002

LECTURE 17

A PROBLEM WITH BSTs

Common operations on balanced BST are O(log(n))

Alas, when the tree goes out of balance, performance degrades (worst case : chain O(n))

There are several data structures that modify the BST to maintain balance.

AVL Trees

The first balanced binary search tree

Named after discoverers Adelson-Velskii and Landis.

DEFINITION:An AVL tree is a binary search tree with the additional

balance property that, for any node in the tree, the height of the left and right subtrees can differ by at most 1. (Height of an empty subtree is –1).

12

8 16

2 6

4 10 14

AN AVL TREE

What if we insert “7”?-1 -1 -1 -1

-1 -1 -1 -1

-1

0 0

001

12What is the height?

3

12

8 16

2 6

4 10 14

NOT AN AVL TREE

-1 -1

-1 -1

-1 -1 -1 -10 1

002

13

4

-1

7

-1 0

Out of balance

AVL THEOREMAn AVL tree of n element has height O(log n)

In fact, An AVL tree of height H has at least FH+3 –1 nodes, where Fi is the ith Fibonacci number

Let SH be the size of smallest AVL tree of height H.

Clearly S0=1 and S1=2

SH must have subtrees of height H-1 and H-2These subtrees must have fewest nodes for height

So, SH = SH-1 + SH-2 + 1Minimum number AVL trees are Fibonacci trees

Smallest AVL Tree of height H

SH-

1

H-1SH-

2

H-2

H

SH = FH+3 – 1

FH+3 – 1 >= (3/2)H // do this as exercise

Hint: (3/2)H-1+(3/2)H=(3/2)H-2(3/2 + 1) > (3/2)H-2(9/4)

SH >= (3/2)H

Log2(SH) >= H Log2(3/2)

H <= 1.75 Log2(SH)

Adds can unbalance a BST

Only nodes on the path from the root to the insertion point can have their balances altered

If we restore the unbalance node, we balance the tree

12

8 16

2 6

4 10 14

AN AVL TREE

What if we insert “15”?

-1 -1 -1 -1

-1 -1 -1 -1

-1

0 0

001

12

3

If we insert 7We unbalance the whole tree

12

8 16

2 6

4 10 14

AN AVL TREE

What if we insert “15”?

-1 -1 -1 -1

-1 -1 -1

-1

0 0

101

22

3

If we insert 7We unbalance the whole tree

15-1 -1

0

4 cases

1. Insertion in left sub-tree of left child

2. Insertion in right sub-tree of left child

3. Insertion in left sub-tree of right child

4. Insertion in right sub-tree of right child

1 & 4 are symmetric

2 & 3 are symmetric

Cases 1 & 4

A B

C

k1

k2Insertion extends Tree ‘A’

HH-2

H-2 H-2

Cases 1 & 4

A B

C

k1

k2Insertion extends Tree ‘A’

H-2

H-1 H-2

“Rotation” fixesbalance

Cases 1 & 4

AB C

k2

k1Insertion extends Tree ‘A’

H-2H-1 H-2

“Rotation” fixesbalance

H

Cases 2 & 3

A B

C

k1

k2Insertion extends Tree ‘B’

HH-2

H-2 H-2

Cases 2 & 3

A B

C

k1

k2Insertion extends Tree ‘B’

H-2

H-2 H-1

Cases 2 & 3

AB C

k2

k1Insertion extends Tree ‘B’

H-2H-2 H-1

“Rotation” Does not fixbalance

Double Rotation

If x is out of balance for cases 2&3

1. Rotate between X’s child and grandchild

2. Rotate between X and it’s new child

Cases 2 & 3

A B

C

k1

k2Insertion extends Tree ‘B’

HH-2

H-2 H-2

Cases 2 & 3

A C

D

k1

k2Insertion extends Tree ‘B’ or ‘C’

HH-2

H-2 H-3

k3

B

Cases 2 & 3

A C

D

k1

k2Insertion extends Tree ‘B’ or ‘C’

H-2

H-2 H-2

k3

B

Rotate grandchildWith child

Cases 2 & 3

AC

D

k3

k2Insertion extends Tree ‘B’ or ‘C’

H-2

H-2

H-2

k1

B

Rotate grandchildWith child

Rotate X with new child

Cases 2 & 3

A C D

k2

k3Insertion extends Tree ‘B’ or ‘C’

H-2

H-2

H-2

k1

B

Rotate grandchildWith child

Rotate X with new child

Implementation

Insert

Fixup

Rotate

Need to keep track of balance

Rotationprivate void rotateLeft(Entry p) {

Entry r = p.right;p.right = r.left;if (r.left != null) r.left.parent = p;r.parent = p.parent;if (p.parent == null) {root = r; r.parent = null;}else if (p.parent.left == p) p.parent.left = r;else p.parent.right = r;r.left = p;p.parent = r;

}

element left right parentroot

50

80

element left right parentroot

50

80

90

r

Entry r = p.right;

p

element left right parentroot

50

80

90

r

Entry r = p.right;p.right = r.left;

p

element left right parentroot

50

80

90

r

Entry r = p.right;p.right = r.left;if (r.left != null) r.left.parent = p;if (p.parent == null)root = r; r.parent = p.parent;

p

element left right parentroot

50

80

90

r

Entry r = p.right;p.right = r.left;if (r.left != null) r.left.parent = p;r.parent = p.parent;if (p.parent == null) root = r;else if …else …r.left = p;

p

element left right parentroot

50

80

90

r

Entry r = p.right;p.right = r.left;if (r.left != null) r.left.parent = p;r.parent = p.parent;if (p.parent == null){ root = r; ..}else if …else …r.left = p;p.parent = r;

p

Node (Entry) classprivate static class Entry {

Object element;char balanceFactor = ‘=‘; // new nodes are balanced// we could set this to R or L indicating child with > heightEntry left = null, right = null, parent;Entry (Object element, Entry parent) {

this.element = element;this.parent = parent;

}}

50

R

20

L

10

=

80

R

70

=

100

=

92

=

50

=

public boolen add(Object o){

if (root == null) {

root = new Entry(o,null);

size++;

return true;

}// empty tree

else {

Entry temp = root,

ancestor = null; // we keep track of nearest unbalanced ancestor

int comp;

while (true) {

comp = ((Comparable)o).compareTo(temp.element);

if (comp == 0) return false;

if (comp < 0) {

if (temp.balanceFactor != ‘=‘) ancestor = temp;

if (temp.left != null) temp = temp.left;

else {

temp.left = new Entry(o,temp);

fixAfterInsertion(ancestor,temp.left);

size++;

}

}// comp < 0

else { // comp > 0

if (temp.balanceFactor != ‘=‘) ancestor = temp;

if (temp.right != null) temp = temp.right;

else {

temp.rig = new Entry(o,temp);

fixAfterInsertion(ancestor,temp.right);

size++;

}

}// comp < 0

}// while

}// root not null

}//method add

Adjusting paths50

=

25

=

15

=

70

=

60

=

30

=

55

=

90

=

Adjusting paths50

R

25

=

15

=

70

L

60

L

30

=

55

=

90

=

protected void adjustPath(Entry to, Entry inserted) {

Object o = inserted.element;

Entry temp = inserted.parent;

while (temp != to) {

if (((Comparable)o).compareTo(temp.element)<0)

temp.balanceFactor = ‘L’;

else

temp.balanceFactor = ‘R’;

temp= temp.parent

}// while

} //adjust path

protected void fixAfterInsertion(Entry ancestor, Entry inserted) {

Object o = inserted element;

if (ancestor == null) {

if (((Comparable)o).compareTo(root.element)<0)

root.balanceFactor = ‘L’;

else

root.balanceFactor = ‘R’;

adjustPath(root,inserted);

} // Case 1: all ancestor of inserted element have ‘=‘ balanceFactor

if ((ancestor.balanceFactor == ‘L’ &&

((Comparable)o).compareTo(ancestor.element)>0)||

(ancestor.balanceFactor == ‘R’ &&

((Comparable)o).compareTo(ancestor.element)<0)){

ancestor.balanceFactor = ‘=’;

adjustPath(ancestor,inserted);

} // Case 2: insertion causes ancestor’s balanceFactor to ‘=‘

if ((ancestor.balanceFactor == ‘R’ &&((Comparable)o).compareTo(ancestor.right.element)>0){

ancestor.balanceFactor = ‘=’;rotateLeft(ancestor);adjustPath(ancestor.parent,inserted);

} // Case 3: ancestor’s balance factor = ‘R’ // and o > ancestor’s right child

if ((ancestor.balanceFactor == ‘L’ &&((Comparable)o).compareTo(ancestor.left.element)<0){

ancestor.balanceFactor = ‘=’;rotateRight(ancestor);adjustPath(ancestor.parent,inserted);

} // Case 4: ancestor’s balance factor = ‘L’ // and o < ancestor’s right child

if (ancestor.balanceFactor == ‘L’ &&((Comparable)o).compareTo(ancestor.left.element)>0){

rotateLeft(ancestor.left);rotateRight(ancestor);adjustLeftRight(ancestor,inserted);

} // Case 5: ancestor’s balanceFactor = ‘L’ // and o > ancestor’s left child

else{rotateRight(ancestor.right);rotateLeft(ancestor);adjustRightLeft(ancestor,inserted);

} // Case 6: ancestor’s balanceFactor = ‘R’ // and o < ancestor’s right child

}// fixAfterInsertion

protected void adjustLeftRight(Entry ancestor, Entry inserted) {

Object o = inserted.element;

if (ancestor.parent == inserted) ancestor.balanceFactor = ‘=‘;

else if (((Comparable)o).compareTo(ancestor.parent.element)<0){

ancestor.balanceFactor = ‘R’;

adjustPath(ancestor.parent.left,inserted);

}// o < ancestor’s parent

else {

ancestor.balanceFactor = ‘=’;

ancestor.parent.left.balanceFactor = ‘L’;

adjustPath(ancestor,inserted);

}// while

} //adjustLeftRight

AdjustLeftRight Case 1

50

L

30

=

40

=

50

=

40

=

30

=

AdjustLeftRight Case250

L

10

=5

=

15

=35

=

40

=30

=

45

=

20

=

90

=

70

=

100

=Rotate Left around 20Right around 50

AdjustLeftRight Case240

=

10

=5

=

15

=

30

R35

=

20

=

50

R

45

=

90

=70

=

100

=

AdjustLeftRight Case350

L

10

=5

=

15

=42

=

40

=30

=

45

=

20

=

90

=

70

=

100

=Rotate Left around 20Right around 50

AdjustLeftRight Case340

=

10

=5

=

15

=

30

=42

=

20

L

50

=

45

L

90

=70

=

100

=

protected void adjustRightLeft(Entry ancestor, Entry inserted) {

Object o = inserted.element;

if (ancestor.parent == inserted) ancestor.balanceFactor = ‘=‘;

else if (((Comparable)o).compareTo(ancestor.parent.element)>0){

ancestor.balanceFactor = ‘L’;

adjustPath(ancestor.parent.right,inserted);

}// o < ancestor’s parent

else {

ancestor.balanceFactor = ‘=’;

ancestor.parent.right.balanceFactor = ‘R’;

adjustPath(ancestor,inserted);

}// while

} //adjustRightLeft