Lecture+19 20+Red+Black+Trees

8/3/2019 Lecture+19 20+Red+Black+Trees

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Lecture 15

Umar Manzoor


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Chapter 13, Red Black Trees,

Introduction to Algorithms ByCormen et al


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Red-Black Trees (Intro)

BSTs perform dynamic set operations such asSEARCH, INSERT, DELETE etc in O(h) time.

If the tree height is large, performance may be nobetter than a linked list, for large n.

RBTs are balanced in order to guarantee O(lg n)worst case time for set dynamic operations


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Red-Black Trees (Intro) A RBT is a binary search tree where each

node is colored red or black

No path from root to leaf is twice as long asany other. (approximately balanced)

A node contains: color, key, left, right, p

A NIL is a leaf, all other nodes are internal

nodes.


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Red-Black Trees (Properties)

A binary search tree is a red-black tree if:

1. Every node is either red or black

2. The root is black3. Every leaf (NIL) is black

4. A rednodes children are black

5. For each node, all paths from a node todescendant leaves contain the same number ofblack nodes


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Red-Black Trees (Intro)

We use sentinel nil[ T ]to aid us in dealingwith boundary conditions in pseudo code.

nil[ T ]object with color field black, otherfields are set to arbitrary values.


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Black-Height

Black Height of a node: bh(x)Excluding the node x itself, the number of black

nodes on any path down to a leaf.

Black Height of tree = Black Height of root

node


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Rotations

The operations TREE-INSERT and TREE-DELETE modify the tree, they may violate

the red-black properties.

To restore these, we

Change color of nodes

Change pointer structure (rotations)

Two kinds, left rotation and right rotation


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Rotations

Left-Rotation on a node x:Right child yis not nil[ T ]

Left rotation pivots around the link from x to y

i.e., it makes ythe new root of the subtree, withxas ys left child and ys left child asxs rightchild.


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Inserting Nodes

A node is inserted into the tree just as inBST.

When a node is inserted, it is colored redinitially.

A procedure is called at the end of theinsertion to correct the tree structure


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Inserting Nodes

Suppose we are inserting a node zin the tree. The following red-black properties can be

violated:

2. The root is black (when the first red node is inserted)4. A red node has black children (when the parent of

inserted node is red)

A violation of property 4 leads to three cases: Case 1: zs uncle yis red

Case 2: zs uncle yis black and zis a right child

Case 3: zs uncle yis black and zis a left child


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Inserting a node z

Case 1: z,p[z]and zs uncle yall are red.

Since p[p[z]]is black, color it red, color p[z] and yblack.

Check for the three cases from p[p[z]], i.e., now zisthe node p[p[z]]


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Inserting a node z

Case 2: z,p[z]are red,zs uncle yis black, zisright child

Use a left rotation, we get to case 3.

Case 3: z,p[z]are red,zs uncle yis black, zis

left childChange color of p[p[z]]to red, p[z]to black and

perform a right rotation on p[p[z]]


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Example

Input : { 31, 22, 19, 11, 7, 5, 18, 34, 44, 59,80, 85, 60, 32}

31 31

22


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Lecture+19 20+Red+Black+Trees