btree, data structures

DATA STRUCTURESB-TREE

Jibrael Jos : Sep 2009

Avoid Taking Printout : Use RTF Outline in case needed 2

IntroductionMultiway TreesB TreeApplicationStructureAlgo : Insert / Delete

Agenda

Please Do Not Take Printout : Use RTF Outline in case needed 3

Data Structures

AVL Trees Red Black B-tree Hashing / Indexing Techniques Graphs


Path Has to be enjoyed

Walking Walking in Rain !! Certification

Effort ~ Satisfaction


Research

Shoulders of Giants

Research on an area to reach a level of expertise

Mindmap and Research Path


B Tree

Critic

Maths

Summattion

Series

Variations

B*, B+

Application

Industry


Methodology

One Book to Another One Link to Another


Binary Search Tree

What happens if data is loaded in a binary search tree in this order

23, 32, 45, 11, 43 , 41

1,2,3,4,5,6,7,8

What is AVL tree


Multiway Trees

< K1>= K2

K1

K2

>= K1

<K2

m-way trees

Reduce the depth of the tree to O(logmn)with m-way trees

m children, m-1 keys per node m = 10 : 106 keys in 6 levels vs 20 for a

binary tree but ........

K1 K2 K3

K1

K2

K3

K1

K2

K3

K1

K2

K3

K1

K2

K3

m-way trees

But you have to search through the m keys in each node!

Reduces your gain from having fewer levels!

m-way trees50

100

150

35

45

110

120

60

70

125

135

85

95

90

75

175

Anand B

B-trees

All leaves are on the same level All nodes except for the root and the leaves

have at least m/2 children at most m children

Each node is at least

half full of keys

BTREE

74

78

85

9711

14 125

135

21

102


Disk

1 track = 5000 Chars1 Cylinder = 20 tracks1 disk unit = 200 cylinders

Time Taken

Seek Time Latency Time Transmission Time

Overcoming Latency Time ??

72.5 + o.o5n millisec to read n chars


3 level


Multiway Tree

M – ary tree

3 levels :

Cylinder , Track , Record : Index Seq (RDBMS)

Tables with less change


BTree

If level is 3, m =199 then what is N

How many split per insertion ?


Multiway Trees : Application NDPL , Delhi: Electricity Billing

3 lakh consumers Table indexed as BTREE

UCO Bank, Jaipur One DD takes 10 minutes to print Saviour : BTREE

B-trees - Insertion

Insertion B-tree property : block is at least half-full

of keys Insertion into block with m keys

block overflows split block promote one key split parent if necessary if root is split, tree becomes one level

deeper

Insert Node

74

78

85

9711

14 125

135

21

102

63

After Insert 63

11

14 125

135

63

74

21

78

102

85

97

Insert Node

74

78

85

9711

14 125

135

21

102

99

After Insert 99

11

14 125

135

74

78

21

85

102

97

99

Split Node

74

78

85

97

74

78

85

97

4

node

0

63


Structure of Btree

node firstPtr numEntries Entries[1.. M-1] End

Entry key rightPtr End Entry

Split Node : Final

78

63

74

3

node

0

85

97

2

rightPtr

43

2

median

entry

toNdx

fromNdx

Split Node : Final

85

74

78

3

node

4

97

99

2

rightPtr

43

1

median

entry

toNdx

fromNdx

Traversal

42

45

63

7411

14 85 95

21

78


DeleteDelete Walk ThroughReflowBorrow LeftBorrow RightCombineDelete Mid

Agenda


Delete : For 78

42

1

16

21

2 57

78

2

45

52

2 63

74

2 85

97

2

Btree Delete Delete() Delete() Delete Mid() Reflow() Reflow() If shorter delete root


Btree Delete If (root null) print (“Attempt to delete from null tree”) Else shorter = delete (root, target) if Shorter delete root Return root

42

1

16

21

2 57

78

2

45

52

2 63

74

2 85

97

2

Target = 78

B


Delete(root , deleteKey) If (root null) data does not exist Else entryNdx= searchNode(root, deleteKey) if found entry to be deleted if leaf node underflow=deleteEntry() else underflow=deleteMid (left) if underflow underflow=reflow()

42

1

16

21

2 57

78

2

45

52

2 63

74

2 85

97

2

Target = 78

B

D


Delete Else Part Else if deleteKey less than first entry subtree=firstPtr else subtree=rightPtr underflow= delete (subtree,deleteKey) if underflow underflow= reflow() Return underflow

42

1

16

21

2 57

78

2

45

52

2 63

74

2 85

97

2

Target = 78

B

D


Delete(root , deleteKey) If (root null) data does not exist Else entryNdx= searchNode(root, deleteKey) if found entry to be deleted if leaf node underflow=deleteEntry() else underflow=deleteMid (root,entryIndx,left) if underflow underflow=reflow(root,entryIndx)

42

1

16

21

2 57

78

2

45

52

2 63

74

2 85

97

2

Target = 78

B

D

D

DM



42

1

16

21

2 57

74

2

45

52

2 63

1 85

97

2

74 replaces 78

B

D

D



42

1

16

21

2

45

52

2

After Reflow

57

1

63

74

85

97

4

B

D

D


Delete Else Part Else if deleteKey less than first entry subtree=firstPtr else subtree=rightPtr underflow= delete (subtree,deleteKey) if underflow underflow= reflow(root,entryIndx) Return underflow

Before Reflow

42

1

16

21

2

45

52

2

57

1

63

74

85

97

4

B

D


Delete Else Part Else if deleteKey less than first entry subtree=firstPtr else subtree=rightPtr underflow= delete (subtree,deleteKey) if underflow underflow= reflow(root,entryIndx) Return underflow

After Reflow

0

45

52

2 63

74

85

97

4

16

21

42

57

4

B

D


BTREE Delete If (root null) print (“Attempt to delete from null tree”) Else shorter = delete (root, target) if Shorter delete root Return root

0

45

52

2 63

74

85

97

4

16

21

42

57

4

B


BTREE Delete If (root null) print (“Attempt to delete from null tree”) Else shorter = delete (root, target) if Shorter delete root Return root

45

52

2 63

74

85

97

4

16

21

42

57

4

B


Delete : For 78

42

1

16

21

2 57

78

2

45

52

2 63

74

2 85

97

2

Btree Delete Delete() Delete() Delete Mid() Reflow() Reflow() If shorter delete root


Delete : Reflow

1: Try to borrow right.

2: If 1 failed try to borrow from left

3: Cannot Borrow (1,2 failed) Combine


Delete Reflow

Underflow=false If RT->no > min Entries BorrowRight (root,entryNdx,LT,RT) Else If LT->no > min Entries BorrowLeft (root,entryNdx,LT,RT) Else combine (root,entryNdx,LT,RT) if root->no < min entries underflow=True Return underflow


Borrow Left

8 78

2

85

145

63

74

3

Node >= 74 < 78

Node >= 78 < 85


Combine

65

71

2

63

1

21

57

78

3

42

45

2

59

61

2


Combine

65

71

2

63

1

21

57

78

3

59

61

2

42

45

57

3


Combine

65

71

2

21

57

78

3

59

61

2

42 45

57 63

4


Combine

65

71

2

21

78

2

59

61

2

42 45

57 63

4


Delete Mid

If leaf exchange data and delete leaf

entry Else traverse right to locate

predecessor deleteMid(right) if underflow reflow


Delete Mid

42

1

16

21

2 57

78

2

45

52

2 63

74

2 85

97

2

Case 1: To Delete 78 we replace with 74


Delete Mid

42

1

16

21

2 57

78

2

45

52

2 63

74

2 85

97

2

75

76

2Case 2:To Delete 78 we replace with 76

Hence recursive call of Delete Mid to locate predecessor


order

Order Min Max3 2 34 2 45 3 56 3 6… … …

m m/2 m


Get the Order Right Keys are 4 Subtrees Max is 5 = Order is 5 Minimum = 3 (which is subtrees) Min Keys is 2

45

52

2 63

74

85

97

4

16

21

42

57

4


2-3 Tree

Order 3 ….. So how many keys in a node

This rule is valid for non root leaf

Root can have 0, 2, 3 subtrees


2 -3 Tree

42

1

16

2 57

78

2

45

52

2 63

2 85

97

2


2-3-4 Tree

Order 4 ….. So how many keys in a node

This rule is valid for non root leaf

Root can have 0, 2, 3 subtrees


Structure of B + tree

Non leaf node firstPtr numEntries Entries[1.. M-1] End

Entry key rightPtr End Entry

Leaf node firstPtr numEntries Entries[1.. M-1] Next Leaf Node End


B + Tree

42

1

57

78

2

45

52

2 63

74

2 85

97

2

Implies there are more nodes


B * Tree

Space Usage

BTREE nodes can be 50% Empty (1/2)

So rule modified to two third (2/3)

Also when node overflows instead of being split immed distributed with siblings

And even when split happens all siblings are equally distributed (pg 462)

B+-trees

B+ trees All the keys in the nodes are dummies Only the keys in the leaves point to “real”

data Linking the leaves

Ability to scan the collection in orderwithout passing through the higher nodes


Reference My Course Furzon

Chapter 10 Volume 3 Knuth : 5.4.9 (Disks ) 6.2.4 (Multiway)

Action Item Do research on BTREE , AVL , Red

Black

btree, data structures

Education

use rtf outline

case needed9

case needed48213node

case needed2agenda

case needed3

case needed7

case needed8

case needed47