Balanced Tree (AVL Tree & Red-Black Tree)

Balanced TreeAVL Tree

&RED-BLACK

Tree

BySamrin Ahmed Riya

ID : 011142021Sanzida Akter

ID : 011142032

Balanced Tree Node based binary search tree Automatically balance it’s height in the face of arbitrary item

insertions and deletions

Fig : Balanced Tree

AVL Tree

AVL Tree

A special kind of binary search tree Self balancing tree Height of right sub tree ˞ ˞ height of left sub tree ≤ 1

Features :

Georgy Adelson-Velsky and Evgenii Landis' tree Named after the inventors (1962)

Examples

AVL Tree or not

YESEach left sub-tree has height 1 greater than each right sub-tree

NOLeft sub-tree has height 3, but right sub-tree has height 1

Operations

Insertion

Deletion

Traversal

Searching

Rotation for Balancing

It is performed as in binary search trees. For insertions, one rotation is sufficient. Sometimes it needs two rotations.

Insertion

Insertion(0,0)1

Insert 1Elements :1 2 3 6 15 -2 -5 -8

Insertion

(0,0)2

(0,1)1Insert 2Elements :

1 2 3 6 15 -2 -5 -8

Insertion

Rotation needed

(0,0)3

(0,1)2

(0,2)1Insert 3Elements :

1 2 3 6 15 -2 -5 -8

1 (0,0)3

(1,1)2Insert 3

(0,0)

InsertionElements :1 2 3 6 15 -2 -5 -8

1 (0,1)3

(1,2)2Insert 6

(0,0)

(0,0)6

InsertionElements :1 2 3 6 15 -2 -5 -8

1 (0,2)3

(1,3)2Insert 15

(0,0)

(0,1)6

(0,0)15

Rotation needed

InsertionElements :1 2 3 6 15 -2 -5 -8

1 (1,1)6

(1,2)2Insert 15

(0,0)

(0,0)153 (0,0)

InsertionElements :1 2 3 6 15 -2 -5 -8

1 (1,1)6

(2,2)2Insert -2

(1,0)

(0,0)153 (0,0)-2 (1,0)

InsertionElements :1 2 3 6 15 -2 -5 -8

1 (1,1)6

(3,2)2Insert -5

(2,0)

(0,0)153 (0,0)-2 (1,0)

-5 (0,0)

Rotation needed

InsertionElements :1 2 3 6 15 -2 -5 -8

-2 (1,1)6

(2,2)2Insert -5

(1,1)

(0,0)1 3(0,0)

-5 (0,0)

(0,0)

(0,0)15

InsertionElements :1 2 3 6 15 -2 -5 -8

-2 (1,1)6

(3,2)2Insert -8

(2,1)

(0,0)1 3(0,0)

-5 (1,0)

(0,0)

(0,0)15

-8 (0,0)

InsertionElements :1 2 3 6 15 -2 -5 -8

Deletion Deletion can make the tree unbalanced One rotation is needed for rebalancing Sometimes it needs two rotations

Insertion & Deletion (Algorithms)

LeftRotationAVL (x: BinTree) { x := ( x. rightChild .key ,

(x.key , x. leftChild , x. rightChild . leftChild ) , x. rightChild . rightChild );

}

RightRotation (x: BinTree) { x := ( x. leftChild .key ,

x. leftChild . leftChild , (x.key , x. leftChild . rightChild , x. rightChild ) );

}

Insertion & Deletion (Algorithms) RightLeftRotation (x: BinTree) {

x := ( x. rightChild . leftChild .key , ( x.key , x. leftChild , x. rightChild . leftChild . leftChild ) , ( x. rightChild .key , x. rightChild . leftChild . rightChild , x. rightChild . rightChild ) ); }

LeftRightRotation(x: BinTree) { x := ( x. leftChild . rightChild .key ,

( x. leftChild .key , x. leftChild . leftChild , x. leftChild . rightChild . leftChild ) , ( x.key , x. leftChild . rightChild . rightChild , x. rightChild ) ); }

Traversal Maintains preorder, inorder and postorder traversal Depends on the height of the tree

Traversal (Algorithms)Preorder Traversal

void preorder(node *t) { if (t != NULL) {

printf(“%d ”, t->element); preorder(t->leftChild); preorder(t->rightChild);

} }

Traversal (Algorithms)Inorder Traversal

void inorder(node *t) { if (t != NULL) {

inorder(t->leftChild); printf(“%d ”, t->element); inorder(t->rightChild);

} }

Traversal (Algorithms)Postorder Traversal

void postorder(node *t) { if (t != NULL) {

postorder(t->leftChild); /* L */ postorder(t->rightChild); /* R */ printf(“%d ”, t->element); /* V */

} }

Searching Similar to normal unbalanced binary search tree. Successful searches are limited by the height of the tree. Unsuccessful searching time is very close to the height of the

tree.

AVL Tree

Applications of AVL Tree Used in many search applications where data is constantly

entering/leaving. To security concerns and to parallel code. Creating new types of data structures.

Red-Black Tree

• A balancing binary search tree.

• A data structure requires an extra one bit color field in each node which is red or black.

• Leonidas J. Guibas and Robert Sedgewick derived the red-black tree from the symmetric binary B-tree.

Introduction

Example of Red-Black Tree

• The root and leaves (NIL’s) are black. • A RED parent never has a RED child.• in other words: there are never two successive RED nodes in a path

• Every path from the root to an empty subtree contains the same number of BLACK nodes• called the black height

• We can use black height to measure the balance of a red-black tree.

Properties of Red-Black Tree

Average

Space O(n)

Search O(log2 n)

Traversal O(n)

Insertion O(log2 n)

Deletion O(log2 n)

Red-black tree Operations

• Basic operation for changing tree structure is called rotation:

Red-Black Trees: Rotation

x

y

y

x

• x keeps its left child• y keeps its right child• x’s right child becomes y’s left child• x’s and y’s parents change

A B

C A

B C

Red-Black Trees: Rotation

Rotation Example• Rotate left about 9:

12

5 9

7

8

11

Rotation Example• Rotate left about 9:

5 12

7

9

118

LEFT-ROTATE(T, x) y ← x->right x->right← y->left y->left->p ← x y->p ← x->p

if x->p = Null then T->root ← y

else if x = x->p->left then x->p->left ← y else x->p->right ← y

y->left ← xx->p ← y

RIGHT-ROTATE(T, x) y ← x->left x->left← y->right y->right->p ← x y->p ← x->p

if x->p = Null then T->root ← y else if x = x->p->right then x->p->right ← y else x->p->left ← y

y->right ← xx->p ← y

Runtime : O(1) for Both.

Rotation Algorithm

Red-Black Trees: Insertion

• Insertion: the basic idea• Insert x into tree, color x red• Only r-b property 3 might be violated (if p[x] red)• If so, move violation up tree until a place is found

where it can be fixed• Total time will be O(log n)

Red-Black Insertion: Case 1

B

x

● Case 1: “uncle” is red● In figures below, all ’s are equal-black-height

subtrees

C

A D

C

A D

y

new x

Same action whether x is a left or a right child

B

x case 1

Red-Black Insertion: Case 2

B

x

● Case 2:■ “Uncle” is black■ Node x is a right child

● Transform to case 3 via a left-rotation

CA

CBy

A

x

case 2

y

Transform case 2 into case 3 (x is left child) with a left rotationThis preserves property 4: all downward paths contain same number of black nodes

Red-Black Insertion: Case 3● Case 3:

■ “Uncle” is black■ Node x is a left child

● Change colors; rotate right

BAx

case 3CB

A

x

y C

Perform some color changes and do a right rotationAgain, preserves property 4: all downward paths contain same number of black nodes

Red-Black Insert: Cases 4-6• Cases 1-3 hold if x’s parent is a left child

• If x’s parent is a right child, cases 4-6 are symmetric (swap left for right)

Insertion Example

Insert 6547

7132

93

Insertion Example

Insert 6547

7132

65 93

Insert 6547

7132

65 93

Insert 82

Insertion Example

82

Insert 65 47

7132

65 93

Insert 82

Insertion Example

82

Insert 6547

7132

65 93

Insert 82

65

71

93

change nodes’ colors

Insertion Example

9365

71

82

Insert 65

47

32

Insert 82

Insert 87

87

Insertion Example

9365

71

82

Insert 65

47

32

Insert 82

Insert 87

87

Insertion Example

9365

71

87

Insert 65

47

32

Insert 82

Insert 87

82

Insertion Example

9365

87

Insert 65

47

32

Insert 82

Insert 87

82

71

87

93

change nodes’ colors

Insertion Example

87

93

65

Insert 65

47

32Insert 82Insert 87

82

71

Insertion Example

TreeNode<T> rbInsert(TreeNode<T> root,TreeNode<T> x)// returns a new root{ root=bstInsert(root,x); // a modification of BST insertItem x.setColor(red); while (x != root and x.getParent().getColor() == red) { if (x.getParent() == x.getParent().getParent().getLeft()) { //parent is left child y = x.getParent().getParent().getRight() //uncle of x if (y.getColor() == red) {// uncle is red x.getParent().setColor(black); y.setColor(black); x.getParent().getParent().setColor(red); x = x.getParent().getParent(); } else { // uncle is black if (x == x.getParent().getRight()) { x = x.getParent(); root = left_rotate(root,x); } x.getParent().setColor(black); x.getParent().getParent().setColor(red); root = right_rotate(root,x.getParent().getParent()); }} } else // ... symmetric to if } // end while root.setColor(black); return root;}

Insertion Algorithm

Red-Black Tree Deletion• If n has no children, we only have to remove n from the tree.• If n has a single child, we remove n and connect its parent to its child.• If n has two children, we need to :• Find the smallest node that is larger than n, call it m.• Remove m from the tree and Replace the value of n with m.• Then restores the red-black tree properties.

Red-Black Tree Deletion AlgorithmTreeNode<T> rbDelete(TreeNode<T> root,TreeNode<T> z)//return new root, z contains item to be deleted{ TreeNode<T> x,y; // find node y, which is going to be removed if (z.getLeft() == null || z.getRight() == null) y = z; else { y = successor(z); // or predecessor z.setItem(y.getItem); // move data from y to z } // find child x of y if (y.getRight() != null) x = y.getRight(); else x = y.getLeft(); // Note x might be null; create a pretend node if (x == null) { x = new TreeNode<T>(null); x.setColor(black); }

Red-black tree Searching:• Searching a node from a red-black tree doesn’t require more than the

use of the BST procedure, which takes O(log n) time.

Red-Black Trees efficiency All operations work in time O(height) hence, all operations work in time O(log n)! – much

more efficient than linked list or arrays implementation of sorted list!

Red-Black Tree Application• Completely Fair Scheduler in Linux Kernel.

• Computational Geometry Data structures.

• To keep track of the virtual memory segments for a process - the start address of the range serves as the key.

• Red–black trees are also particularly valuable in functional programming.

ComparisonFor small data :• Insert: RB tree will be faster because on average it uses less rotation.• Lookup: AVL tree is faster, because it has less depth.• Delete: RB tree is faster for it’s runtime.

For large data :• Insert: AVL tree is faster, because it maintains O(log n) which is better than RB

tree.• Lookup: AVL tree is faster. (same as in small data case)• Delete: AVL tree is faster on average, but in worst case RB tree is faster.

Thank You