CS660: RedBlack, B-Trees

CS660 Combinatorial Algorithms
Fall Semester, 1996
RedBlack, B-Trees

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San Diego State University -- This page last updated Oct 3, 1996

Contents of RedBlack, B-Trees

RedBlack, B-Trees Slide # 1

References

Introduction to Algorithms, Chapters 14, 19

If your web browser can run Java applets try the web site:
http://langevin.usc.edu/. It demonstrates additions and deletions to/from Red-Black trees and other types of trees.

RedBlack, B-Trees Slide # 2

Red-Black Trees

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

1. Every node is either red or black

2. Every leaf (nil) is black

3. If a node is red, then both its children are black

4. Every simple path from a node to a descendant leaf contains the same number of black nodes

Black-height of a node x, bh(x), is the number of black nodes on any path from x to a leaf, not counting x

RedBlack, B-Trees Slide # 3

Lemma

A red-black tree with n internal nodes has height at most 2lg(n+1)

proof

Show that subtree starting at x contains at least 2^bh(x)-1 internal nodes. By induction on height of x:
if x is a leaf then bh(x) = 0, 2^bh(x)-1 Assume x has height h, x's children have height h -1 x's children black-height is either bh(x) or bh(x) -1 By induction x's children subtree has 2^bh(x)-1-1 internal nodes So subtree starting at x contains 2^bh(x)-1-1 + 2^bh(x)-1-1 + 1 = 2^bh(x)-1 internal nodes

let h = height of the tree rooted at x

bh(x) >= h/2

So n >= 2^h/2-1 <=> n + 1 >= 2^h/2 <=> lg(n+1) >= h/2

h <= 2lg(n+1)

RedBlack, B-Trees Slide # 4

Inserting in Red-Black Tree

Color the node Red
Insert as in a regular BST
If have parent is red

Case 1
x is node of interest, x's uncle is Red

Decrease x's black height by one

RedBlack, B-Trees Slide # 5

Case 2
x's uncle is Black, x is a Right child

Transform to case 3

Case 3
x's uncle is Black, x is a Left child

Terminal case, tree is Red-Black tree

Insertion takes O(lg(n)) time
Requires at most two rotations

RedBlack, B-Trees Slide # 6

Deleting in a Red-Black Tree

Find node to delete
Delete node as in a regular BST
Node to be deleted will have at most one child

If we delete a Red node tree still is a Red-Black tree
Assume we delete a black node

Let x be the child of deleted node
If x is red, color it black and stop

If x is black mark it double black and apply the following:

Case 1
x's sibling is red

x stays at same black height
Transforms to case 2b then terminates

RedBlack, B-Trees Slide # 7

Case 2a
x's sibling is black
x's parent is black

Decreases x black height by one
Case 2b
x's sibling is black
x's parent is red

Terminal case, tree is Red-Black tree

RedBlack, B-Trees Slide # 8

Case 3
x's sibling is black
x's parent is either
x's sibling's left child is red
x's sibling's right child is black

x stays at same black height Transforms to case 4

RedBlack, B-Trees Slide # 9

Case 4
x's sibling is black
x's parent is either
x's sibling's left child is either
x's sibling's right child is red

Terminal case, tree is Red-Black tree

Delete time is O(lg(n))
At most three rotations are done

RedBlack, B-Trees Slide # 10

B-Trees, (a,b)-Trees

(a,b)-Trees
Let a and b be integers with a >= 2 and 2a-1 <= b. A tree T is an (a,b)-tree if

a) All leaves of T have the same depth

b) All internal nodes v of T satisfy c(v) <= b

c) All internal nodes v of T except the root satisfy c(v) >= a

d) The root of T satisfies c(v) >= 2

c(v) = number of children of node v

RedBlack, B-Trees Slide # 11

B-Trees of degree t
A tree T is a B-Trees of degree t if

a) All leaves of T have the same depth

b) All nodes of T except the root have at least t-1 keys

c) All nodes of T except the root have at most 2t-1 keys

d) The root of T at least one key

e) A node with n keys has n+1 children

c(v) = number of children of node v

RedBlack, B-Trees Slide # 12

Theorem. If n >= 1, then for any n-key B-tree T of height h and degree t >= 2 then

proof.

take log of both sides.

RedBlack, B-Trees Slide # 13

Theorem. The worst case search time on a n-key B-tree T of degree t is O(lg(n)).

A node in T has t-1 <= K <= 2t-1 keys in sorted order.

Worst case:
K = t-1 for all nodes
searching for X not in the tree


Given a node, W, in T, how much work does it take to find the subtree of W that would contain X?

Using binary search it takes
= = comparisons

Since the height of the tree is in worst case the total amount of work is:

RedBlack, B-Trees Slide # 14

Insertion in a B-Tree

Inserting X into B-tree T of degree t
A full node is one that contains 2t-1 keys

1. Find the leaf that should contain X

2. If the path from the root to the leaf contains a full node split the node when you first search it. Example t = 2, Insert 25

Full Node is split, Then insert 25 into subtree b

3. Insert X into the proper leaf

RedBlack, B-Trees Slide # 15

Example t = 2, Insert 25

RedBlack, B-Trees Slide # 16

Deletion in a B-Tree

Deleting X from B-tree T of degree t
A minimal node is one that contains t-1 keys and is not the root

In the search path from the root to node containing X, if you come across a minimal node add a key to it.

Case 3. Searching node W that does not contain X. Let c be the child of W that would contain X.

Case 3a. if c has t-1 keys and a sibling has t or more keys, steal a key from the sibling
Example t = 2, Delete 250

RedBlack, B-Trees Slide # 17

Case 3b. if c has t-1 keys and all siblings have t-1 keys, merge c with a sibling
Example 1. t = 2, Delete 250

Example 2. t = 2, Delete 250

RedBlack, B-Trees Slide # 18

Case 2. Internal node W contains X.

Case 2a. If the child y of W that precedes X in W has at least t keys, steal predecessor of W
Example 1. t = 2, Delete 50