+ data structures b-tree jibrael jos : sep 2009. + agenda introduction multiway trees b tree...

+

Data StructuresB-treeJibrael Jos : Sep 2009

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+

Agenda

IntroductionMultiway TreesB TreeApplicationStructureAlgo : Insert / Delete

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

3+B Tree

Critic

Maths

Summattion

Series

Variations B*, B+

Application Industry


4+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


5+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

+B-trees

All leaves are on the same level

All nodes except for the root and the leaveshave at least m/2 children at most m children

Anand B

Each node is at least

half full of keys

+ BTREE

74

78

85

9711

14 125

135

21

102


11+Multiway Tree

M – ary tree

3 levels :

Cylinder , Track , Record : Index Seq (RDBMS)

Tables with less change


12+BTree

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

How many split per insertion ?


13+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

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20+Structure of Btree

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

Entry key rightPtrEnd 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

Avoid Taking Printout : Use RTF Outline in case needed 24

+

Agenda

DeleteDelete Walk ThroughReflowBorrow LeftBorrow RightCombineDelete Mid


25+ Delete : For 78Btree Delete Delete() Delete() Delete Mid() Reflow() Reflow() If shorter delete root

42

1

16

21

2 57

78

2

45

52

2 63

74

2 85

97

2


26+Btree DeleteIf (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


27+Delete(root , deleteKey)If (root null) data does not existElse 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


28+Delete Else PartElse 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


29+Delete(root , deleteKey)If (root null) data does not existElse 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


32+Delete Else PartElse 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


33+Delete Else PartElse 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


34+BTREE DeleteIf (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


35+BTREE DeleteIf (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


38+ Delete : For 78Btree Delete Delete() Delete() Delete Mid() Reflow() Reflow() If shorter delete root

42

1

16

21

2 57

78

2

45

52

2 63

74

2 85

97

2


39+Delete : Reflow

1: Try to borrow right.

2: If 1 failed try to borrow from left

3: Cannot Borrow (1,2 failed) Combine


40+Delete Reflow

Underflow=falseIf 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=TrueReturn underflow


41+ Borrow Left

8 78

2

85

145

63

74

3

Node >= 74 < 78

Node >= 78 < 85


42+ Combine

65

71

2

63

1

21

57

78

3

42

45

2

59

61

2


43+ Combine

65

71

2

63

1

21

57

78

3

59

61

2

42

45

57

3


44+ Combine

65

71

2

21

57

78

3

59

61

2

42 45

57 63

4


45+ Combine

65

71

2

21

78

2

59

61

2

42 45

57 63

4


46+Delete Mid

If leaf exchange data and delete leaf entryElse traverse right to locate predecessor deleteMid(right) if underflow reflow


47+ 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


48+ 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


49+order

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

m m/2 m


50+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


51+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


52+ 2 -3 Tree

42

1

16

2 57

78

2

45

52

2 63

2 85

97

2


53+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

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54+Structure of B + tree

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

Entry key rightPtrEnd Entry

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


55+ B + Tree

42

1

57

78

2

45

52

2 63

74

2 85

97

2

Implies there are more nodes


56+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

+ data structures b-tree jibrael jos : sep 2009. + agenda introduction multiway trees b tree...

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