btree, data structures
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DATA STRUCTURESB-TREE
Jibrael Jos : Sep 2009
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IntroductionMultiway TreesB TreeApplicationStructureAlgo : Insert / Delete
Agenda
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Data Structures
AVL Trees Red Black B-tree Hashing / Indexing Techniques Graphs
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Path Has to be enjoyed
Walking Walking in Rain !! Certification
Effort ~ Satisfaction
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Research
Shoulders of Giants
Research on an area to reach a level of expertise
Mindmap and Research Path
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B Tree
Critic
Maths
Summattion
Series
Variations
B*, B+
Application
Industry
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Methodology
One Book to Another One Link to Another
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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
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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
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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
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3 level
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Multiway Tree
M – ary tree
3 levels :
Cylinder , Track , Record : Index Seq (RDBMS)
Tables with less change
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BTree
If level is 3, m =199 then what is N
How many split per insertion ?
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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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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
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DeleteDelete Walk ThroughReflowBorrow LeftBorrow RightCombineDelete Mid
Agenda
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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
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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
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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
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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
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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
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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
74
2
45
52
2 63
1 85
97
2
74 replaces 78
B
D
D
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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
45
52
2
After Reflow
57
1
63
74
85
97
4
B
D
D
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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
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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
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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
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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
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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
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Delete : Reflow
1: Try to borrow right.
2: If 1 failed try to borrow from left
3: Cannot Borrow (1,2 failed) Combine
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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
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Borrow Left
8 78
2
85
145
63
74
3
Node >= 74 < 78
Node >= 78 < 85
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Combine
65
71
2
63
1
21
57
78
3
42
45
2
59
61
2
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Combine
65
71
2
63
1
21
57
78
3
59
61
2
42
45
57
3
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Combine
65
71
2
21
57
78
3
59
61
2
42 45
57 63
4
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Combine
65
71
2
21
78
2
59
61
2
42 45
57 63
4
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Delete Mid
If leaf exchange data and delete leaf
entry Else traverse right to locate
predecessor deleteMid(right) if underflow reflow
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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
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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
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order
Order Min Max3 2 34 2 45 3 56 3 6… … …
m m/2 m
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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
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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
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2 -3 Tree
42
1
16
2 57
78
2
45
52
2 63
2 85
97
2
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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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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
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B + Tree
42
1
57
78
2
45
52
2 63
74
2 85
97
2
Implies there are more nodes
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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
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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
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