Red Black Trees Two - Introduction to Algorithms - Lecture Slides, Slides of Computer Science

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Algorithms

Red-Black Trees

Review: Red-Black Trees

● Red-black trees :

■ Binary search trees augmented with node color

■ Operations designed to guarantee that the height

h = O(lg n )

● We described the properties of red-black trees

● We proved that these guarantee h = O(lg n )

● Next: describe operations on red-black trees

Review: Black-Height

● black-height: # black nodes on path to leaf

● What is the minimum black-height of a node

with height h?

● A: a height- h node has black-height ≥ h /

● Theorem: A red-black tree with n internal

nodes has height h ≤ 2 lg( n + 1)

■ Proved by (what else?) induction

Review: Proving Height Bound

● Thus at the root of the red-black tree:

n ≥ 2 bh( root )^ - 1

n ≥ 2 h /2^ - 1

lg( n+1 ) ≥ h /

h ≤ 2 lg( n + 1)

Thus h = O(lg n )

Red-Black Trees: An Example

● Color this tree: 7

5 9

1212

Red-black properties:

1. Every node is either red or black
2. Every leaf (NULL pointer) is black
3. If a node is red, both children are black
4. Every path from node to descendent leaf contains the same number of black nodes
5. The root is always black

● Insert 8

■ Where does it go?

Red-Black Trees:

The Problem With Insertion

1. Every node is either red or black
2. Every leaf (NULL pointer) is black
3. If a node is red, both children are black
4. Every path from node to descendent leaf contains the same number of black nodes
5. The root is always black

● Insert 8

■ Where does it go?

■ What color

should it be?

Red-Black Trees:

The Problem With Insertion

1. Every node is either red or black
2. Every leaf (NULL pointer) is black
3. If a node is red, both children are black
4. Every path from node to descendent leaf contains the same number of black nodes
5. The root is always black

Red-Black Trees:

The Problem With Insertion

● Insert 11

■ Where does it go?

1. Every node is either red or black
2. Every leaf (NULL pointer) is black
3. If a node is red, both children are black
4. Every path from node to descendent leaf contains the same number of black nodes
5. The root is always black

Red-Black Trees:

The Problem With Insertion

● Insert 11

■ Where does it go?

■ What color?

○ Can’t be red! (#3)

1. Every node is either red or black
2. Every leaf (NULL pointer) is black
3. If a node is red, both children are black
4. Every path from node to descendent leaf contains the same number of black nodes
5. The root is always black

Red-Black Trees:

The Problem With Insertion

● Insert 11

■ Where does it go?

■ What color?

○ Can’t be red! (#3) ○ Can’t be black! (#4)

1. Every node is either red or black
2. Every leaf (NULL pointer) is black
3. If a node is red, both children are black
4. Every path from node to descendent leaf contains the same number of black nodes
5. The root is always black

Red-Black Trees:

The Problem With Insertion

● Insert 10

■ Where does it go?

1. Every node is either red or black
2. Every leaf (NULL pointer) is black
3. If a node is red, both children are black
4. Every path from node to descendent leaf contains the same number of black nodes
5. The root is always black

Red-Black Trees:

The Problem With Insertion

● Insert 10

■ Where does it go?

■ What color?

1. Every node is either red or black
2. Every leaf (NULL pointer) is black
3. If a node is red, both children are black
4. Every path from node to descendent leaf contains the same number of black nodes
5. The root is always black

RB Trees: Rotation

● Our basic operation for changing tree structure

is called rotation :

● Does rotation preserve inorder key ordering?

● What would the code for rightRotate()

actually do?

y

x C

A B

x

A y

B C

rightRotate(y)

leftRotate(x)

rightRotate(y)

RB Trees: Rotation

● Answer: 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

● What is the running time?

y x C A B

x A y B C