Midterm

Still 20 left…

Average so far: 26.8

Percentage under 25: ≈ 40

Percentage under 22.5: ≈ 25

1 bonus to pass 75% of the class

Highest: 46.5

It is always possible to scale the grades and to ask dumb questions.Surely, you would have an easier

time that way.

If you remember, you’re paying to get an education. That means being good at a given topic, not just a meaningless piece of paper.

Using bonus means having more work. But it gives you a chance to understand what you’re supposed to know.

Balanced trees: AVL

Rotations

Examples

How to use

Unbalanced trees

Balanced trees: AVL

This tree is unbalanced: its height h could be made smaller.

This has an impact on performances, since operations are in O(h).

lookup(67) → 4 steps

Now, let’s imagine a world of prefect happiness in which trees

are balanced.

Unbalanced trees

Balanced trees: AVL

lookup(67) → 4 steps

Now, let’s imagine a world of prefect happiness in which trees

are balanced.

Before balancing

After balancing

lookup(67) → 3 steps

Unbalanced trees

Balanced trees: AVL

Since it’s a desirable feature, can we keep a tree balanced at all time?

…and how do we do it?

AVL Tree

Balanced trees: AVL

A brand-new idea: Adelson-Velskii and Landis in 1962.

What’s the idea? Rotations.

In an ArrayList you had shiftLeft and shiftRight to maintain some properties: to maintain the balance, we have rotateRight and rotateLeft.

A

B S1

S2 S3 S1S3

S2

B

A

rotateRight

rotateLeft

RotateRight: Code

Balanced trees: AVL

A

B S1

S2 S3

public void rotateRight(Node n){

Node tmp = n.getLeftChild();

n.setLeftChild(tmp.getRightChild());

tmp.setRightChild(n);

}

rotateRight(A):

n

tmp = B;

tmp

A.setLeftChild(S3);

B.setRightChild(A);

RotateRight: How to remember

Balanced trees: AVL

A

B S1

S2 S3

I rotate A and B clockwise (right).

A

B

Then I inverse the order of the subtrees.

S2

S3 S1

This is one way to remember. Practice and find your own way.

RotateRight: Example

Balanced trees: AVL

10

136

8

7

3

1 5

This tree is not balanced: the left part is

very unbalanced.

Let’s do a rotate right!

RotateRight: Example

Balanced trees: AVL

6B

3

1 5

S2

8

7

S3

10

13

A

S1

Balanced trees: AVL

RotateLeft: Codepublic void rotateLeft(Node n){

Node tmp = n.getRightChild();

n.setRightChild(tmp.getLeftChild());

tmp.setLeftChild(n);

}A

S1S3

B

S2tmp

n

RotateLeft: Example

Balanced trees: AVL

3

71

5 9

4 6 8

B

A

S1S3

S2

RotateLeft: Example

Balanced trees: AVL

3 B

7 A

9

8S1

5

4 6S3

1S2

How to use

Balanced trees: AVL

An AVL keeps the following property:

« For all node that is not a leaf, the heights of its children can differ by at most 1. »

Information is inserted exactly as in a Binary Search Tree, but we check when the heights differ by more than 1 and then we perform rotations.

There are different situations, and each require a particular rotation.

How to use

Balanced trees: AVL

There are different situations, and each require a particular rotation.

Case L.(left)

How to use

Balanced trees: AVL

There are different situations, and each require a particular rotation.

Case R.(right)

Balanced trees: AVL

Single rotation

Double rotation

How to use

Balanced trees: AVL

If the tree is unbalanced on the right side{If the right subtree is unbalanced on the left side

Double rotationElse

Single left rotation}else if the tree is unbalanced on the left side{

If the left subtree is unbalanced on the right sideDouble rotation

ElseSingle right rotation

}

To correct the balance, find where it’s unbalanced and rotate.

Note that this is really just the idea and it needs more for implementation.

Balanced trees: AVL

Some more examples

from http://faculty.ksu.edu.sa/mhussain/CSC212/Lecture%20-%20AVL%20%20Tree.pdf

Balanced trees: AVL

Some more examples

from http://faculty.ksu.edu.sa/mhussain/CSC212/Lecture%20-%20AVL%20%20Tree.pdf

Balanced trees: AVL

Some more examplesShow the AVL resulting from the insertion of 12, 3, 2, 5, 4, 7, 9.