RED-BLACK TREE

INTRODUCTION 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.

EXAMPLE OF RED BLACK TREE

PROPERTIES 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.

RED BLACK TREE OPERATIONS

Average

Space O(n)Search

O(log2 n)Traversal

*O(n)Insertion

O(log2 n)

Deletion O(log2 n)

RED BLACK TREES: ROTATION Basic operation for changing tree structure is

called rotation:

RB TREES: ROTATION

x

y

y

x

A lot of pointer manipulation 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 AB C

ROTATION ALGORITHMLEFT-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 EXAMPLE Rotate left about 9:

12

5 9

7

8

11

ROTATION EXAMPLE Rotate left about 9:

5 12

7

9

118

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)

Insertion AlgorithmTreeNode<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;}

RB INSERT: 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

RB INSERT: 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

RB INSERT: 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

RB 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

INSERTION EXAMPLE

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

RB 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 RETRIEVAL: Retrieving a node from a red-black tree

doesn’t require more than the use of the BST procedure, which takes O(log n) time.

RB TREES EFFICIENCY All operations work in time O(height) and we have proved that heigh is O(log n)

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.

Red-black trees make less structural changes to balance themselves .

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

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

COMPARISON BETWEEN AVL AND RB TREE For small data:

insert: RB tree & avl tree has constant number of max rotation but RB tree will be faster because on average RB tree use less rotation.

lookup: AVL tree is faster, because AVL tree has less depth.

delete: RB tree has constant number of max rotation but AVL tree can have O(log N) times of rotation as worst. and on average RB tree also has less number of rotation thus RB tree is faster.

COMPARISON BETWEEN AVL AND RB TREE (CONTINUED) for large data:

insert: AVL tree is faster. because you need to lookup for a particular node before insertion. as you have more data the time difference on looking up the particular node grows proportional to O(log N). but AVL tree & RB tree still only need constant number of rotation at the worst case. Thus the bottle neck will become the time you lookup for that particular node.

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. because you also need to lookup for a very deep node to swap before removal (similar to the reason of insertion). on average both trees has constant number of rotation. but RB tree has a constant upper bound for rotation.

Thank You