B Trees and its Variants

B-Trees & its Variants

Advanced Data Structure Spring 2007

Zareen Alamgir

Motivation

Yet another Tree!

Why do we need another Tree-Structure ?

Data Retrieval from External Storage In database programs, the data is too large to fit in

memory, therefore, it is stored on secondary storage (disks or tapes).

Disk access is very expensive, the disk I/O operation takes milliseconds while CPU processes data on the order of nanoseconds, one million times faster.

When dealing with external storage the disk accesses dominate the running time.

Balanced Binary Search Trees Balanced binary search trees (AVL & Red-Black) have

good performance if the entire data can fit in the main memory.

These trees are not optimized for external storage and require many disk accesses, thus give poor performance.

Reduce Disk Accesses Data is transfer to and from the disk in block. (typically block

are of 512, 2048, 4096 or 8192 bytes)

We can reduce disk accesses by Storing multiple records in a block on the disk. Reducing the height of the tree by increasing the

number of subtrees of a node.

To achieve above goals we use Multiway (m-way) search tree, which is a generalization of BST, binary search tree.

12 19 32 39 73 84

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34 37 90 9475 7969 7130 31 45 513 5 15 17 21 23

Multiway(m-way) Search Trees

Multiway(m-way) Search Trees In an m-way tree all the nodes have degree ≤ m. Each node has the following structure:

Ki are the keys, where 1 ≤ i ≤ q-1, q<=m

Pi are pointers to subtrees, where 1 ≤ i ≤ q, q<=m

The keys in each node are in ascending order K1 ≤ K2 ≤ ... ≤ Ki

The key Ki is larger than keys in subtree pointed by Pi and smaller than keys in subtree pointed by Pi+1 .

The subtrees are the m-way trees.

P1 K1

keys<K1

P2 Kq-1 PqKi-1 Pi Ki

Ki-1<keys<Ki

Kq-1<keys

Multiway(m-way) Search Trees M-way tree is a generalization of BST, its working, benefits

& issues are same. Problems

The tree is not balanced. Leaf nodes are on different

levels. Bad space usage, tree can

become skew.

Benefits Fast information

retrieval. Fast update.

M-way tree

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25 62

90 9469 7130 313 5 15 21

17 19

B-Trees

B-Trees B-Tree is a balanced m-way tree that is tuned to

minimize disk accesses.

The node size of B-Tree is usually equal to the disk block size and the number of keys in a node depends on Key size Disk block size Data organization (keys or entire data records are store in nodes)

Access paths from root to leaf nodes are small.

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25 62

34 37 90 9475 7969 7130 31 45 513 5 15 17 21 23

B-Tree: Definition A B-tree of order m has following properties

The root has at least two subtrees unless it is a leaf. Each non-root and each non-leaf node holds q-1 keys

and q pointers to subtrees, where . Each leaf node holds q-1 keys where . All leaves are on the same level.

It is clear that B-tree is always at least half full, has fewer levels and is perfectly balanced.

mqm 2/ mqm 2/

FrP1 K1

keys<K1

P2 Kq-1 PqKi-1 Pi Ki

Ki-1<keys<Ki

Kq-1<keys

B-Tree B-Tree can have a field KeyTally, KT, to indicate the number of keys

currently stored in the node.

B-Tree node usually contains key and data pointer pair. The data pointer points to the data record which is not stored in the node, with this scheme we can pack more keys & pointers in a B-Tree node.

KT P1

keys<K1

Kq-1 PqKi-1 Pi Ki

Ki-1<keys<Ki

Kq-1<keys

K1 D1 Ki-1 Di-1 Ki Di Kq-1 Dq-1

Data pointer Data pointer

Height of B-Tree Maximum height of the B-Tree with n keys is important as it

bound the number of disk accesses.

The height of the tree is maximum when each node has minimum number of the subtree pointers, . 2/mq

Height of B-Tree The height of B-tree is maximum if all nodes have minimum

number of keys.

1 key in the root + 2(q-1) keys on the second level +……+ 2qh-2(q-1) keys in the leaves (level h).

12

1log

21

:as given is hheight of Tree- Bin keys ofnumber theThus,

21

1

1)1(21

nprogressio geometric offormula theApplying

2)1(1

1)-(q2q1)-2q(q1)-2(q 1

1

1

1

2

0

2-h

nh

qn

q

q

qq

qq

q

h

h

h

ih

i

Height of B-Tree The height of B-tree is minimum if all nodes are full, thus we have

m-1 keys in the root + m(m-1) keys on the second level +……+ mh-1(m-1) keys in the leaf nodes

12

1log)1(log

)1(log

1

:as given is hheight of Tree- Bin keys ofnumber theThus,

1

1

1)1(

nprogressio geometric offormula theApplying

)1()1(

1)-(m1)-(m1)-m(m 1)-(m1

0

1

0

1-h2

nhn

nh

mn

m

m

mm

mmmm

mm

qm

m

h

h

h

h

i

iih

i

Height of B-Tree If number of nodes in B-tree equal 2,000,000 (2 million)

and m=200 then maximum height of B-tree is 3, where as the binary tree would be of height 20.

Note: Order m is chosen so that B-tree node size is nearly equal to the disk block size.

Search in a B-Tree Search in a B-tree is similar to the search in BST

except that in B-tree we make a multiway branching decision instead of binary branching in BST.

Search key 71

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B-Tree Insert Operation Insertion in B-tree is more complicated than in BST.

In BST, the keys are added in top down fashion resulting in a unbalanced tree.

B-tree is built bottom up, the keys are added in the leaf node, if the leaf node is full another node is created, keys are evenly distributed and middle key is promoted to the parent. If parent is full, the process is repeated.

B-tree can also be built in top down fashion using pre-splitting technique.

Basic IdeaFind position for the key

in the appropriate leaf node

Is node full ?

Split node: •Create a new node • Move half of the keys from the full node to the new node•Promote the median key (before split) to the parent.Split guarantees that each node has keys.

If parent is full

12/ m

Cases in B-Tree Insert Operation In B-tree insertion we have the following cases:

Case 1: The leaf node has room for the new key. Case 2: The leaf in which key is to be placed is full.

This case can lead to the increase in tree height.

Now we explain these cases in detail.

B-Tree Insert Operation Case 1: The leaf node has room for the new key.

10 25

14 19 20 23 32 38

Insert 3

3

85

Insert 3 in order

Find appropriate leaf node for key 3

B-Tree Insert Operation Case 2: The leaf in which key is to be placed is full.

10

3 5 8 14 19 20 23 32 38

Insert 16

No room for key 16 in leaf node

14 20 2319

Insert key 19 in parent node in order

2516

Find appropriate leaf node for key 16

16

19

Move median key 19 up andSplit node: create a new node and move keys to the new node.

B-Tree Insert Operation Case 2: The leaf in which key is to be placed is full and

this lead to the increase in tree height.45 55 67 81

14 19 20 23 32 389 12 47 51 59 75

13 27 33 38

32 38 47 51 59 75 32 38 47 51 59 75 32 38 47 51 59 75 32 38 47 51 59 75

48 52 57 61 72 77 86 92

55

B-Tree Insert Operation Case 2: The height of the tree increases.

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55

13 27

3

2

3

8

4

7

5

1

5

9

7

5

3

2

3

8

4

7

5

1

5

9

7

5

3

2

3

8

4

7

5

1

5

9

7

5

3

2

3

8

4

7

5

1

5

9

7

5

48 52 57 61 72 77 86 92

Insert 16

16

No room for key 16,Move median key 19 up & Split node

Insert 19 in parent node in order

No room for 19 in parent,Split parent node

14 16 20 2319

33 38

Insert 27 in parent in order

67 8145

55

2719

No room for 27 in parent, Split node

B-Tree Delete Operation Deletion is analogous to insertion, but a little

more complicated.

Two major cases Deletion from leaf node Deletion from nonleaf node

Apply delete by copying technique used in BST to reduce this case to deleting a key from leaf.

In delete by copying, the key to be deleted is replaced by the largest key in the right subtree or smallest in left subtree (which is always a leaf).

B-Tree Delete Operation Leaf node deletion cases:

After deletion node is at least half full. After deletion underflow occurs

Redistribute: if number of keys in siblings > . Merge nodes if number of keys in siblings < . Merging leads to decrease in tree height.

12

m

12

m

B-Tree Delete Operation After deletion node is at least half full. (inverse of insertion

case 1)

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3 5 8 14 19 32 38 40

Search key 3

Key found, delete key 3.Move others keys in the node to eliminate the gap.

45

B-Tree Delete Operation Underflow occurs, evenly redistribute the keys if left or right

sibling has keys .

Delete 14

10

5 8 14 19 32 38

Underflow occurs, evenly redistribute keysin the underflow node, in its sibling and the separator key.

40

25

12/ m

Search key 14

45

B-Tree Delete Operation Underflow occurs and the keys in the left & right sibling are

. Merge the underflow node and a sibling.

12/ m

Delete 25

10

5 8 19 25 38 40

Underflow occurs, merge nodes.

32

Move separator key down.

Move the keys to underflow node and discard the sibling.

B-Tree Delete Operation Underflow occurs, height decreases after merging.

8

3 5 21 27 47 66

Delete 21

79 85

73 75 78 81 83 88 90 92

70

Underflow occurs, merge nodes by moving separator key and the keys in sibling node to the underflow node.

32

Underflow occurs, merge nodes

Issues in B-tree In B-tree, accessing data in sequential order is not

efficient. In-order traversal of the B-tree requires many disk accesses.

In B-tree, data pointers are stored in each node, thus resulting in less subtree pointers per node and more tree levels.

KT free spaceP1

keys<K1

Kq-1 PqKi-1 Pi Ki

Ki-1<keys<Ki

Kq-1<keys

K1 D1 Ki-1 Di-1 Ki Di Kq-1 Dq-1

Data pointer Data pointer

B+-Trees

B+ -Tree- Variant of B-Tree Resolves the issues in B-tree.

In B+ -tree Pointers to data is stored only in leaf nodes. Internal nodes contain only keys and subtree pointers

Can accommodate more keys in internal nodes. Less disk accesses due to fewer levels in the tree.

B+ -tree provides faster sequential access of data.

B+-Tree Structure B+-tree consist of two parts

Index Set Provides indexes for fast access of data. Consist of internal nodes that store only key & tree pointers.

Sequence Set Consist of leaf nodes that contain data pointers. Provide efficient sequential access of data(using doubly linked list).

Index Set

Sequence SetSequential Search

B+-Tree: Index Node Structure The basic structure of the B+-tree index node of

order m is same as that of B-tree node of order m.

The root has at least two subtrees unless it is a leaf. Each non-root index node holds q-1 keys and q subtree

pointers, where .

Only difference is that index node do not contain data pointers.

FrP1 K1

keys<K1

P2 Kq-1 PqKi-1 Pi Ki

Ki-1<keys<Ki

Kq-1<keys

mqm 2/

B+-Tree: Sequence Node Structure The structure of the B+-tree sequence node is as

follows:

K1 D1 Ki-1 Di-1 Ki Di Kq-1 Dq-1

Data pointer

Pointer to next leaf node in tree

Data pointer Data pointer Data pointer

Search in a B+-Tree The search in B+-tree works similar to B-tree but it

always ends at the leaf node because Pointer to the data is stored in the leaf node. Existence of the key in index set does not guarantee that the

particular record is present in the tree. A key can occur multiple time in the index set. (this does not

create problem because the key in index set node act only as a separator).

B+-tree of order m=5

Note 62 is not present in the sequence set

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30 62

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B+-Tree Insert Case: The leaf node has room for the key to be inserted.

14 32

14 19 20 23 32 38

Insert 3

3

85

Find appropriate leaf node for key 3, and insert in order.

B+-Tree Insert Operation Case 2: The leaf in which key is to be placed is full.

10

3 5 8 14 19 20 23 32 38

Insert 16

No room for key 16,Split node: create a new node and move keys to the new node.

14

2516

Find appropriate leaf node for key 16

16 20 2319 19

Modify Sequence Set next node links

Insert a copy of the first key of the new node in the parent node in order.

19

2/m

B+ -Tree Insert Operation Case: Only root node exists, and it is full.

10 17

Insert 18 Find appropriate position for key 18

Create a new index set node and make it a root node.

No room for key 18,Split node: create a new sequence set node and move keys to the new node.

2/m18

Insert the first key of the new sequence set node in the new root.

18

18

18

20 23

B+-Tree Delete Operation B+-tree deletion follows same rules as that of B-

tree deletion but the separator in index set node is not removed when a key is deleted.

Deletion cases: After deletion node is at least half full. After deletion underflow occurs

Redistribute: if number of keys in siblings > . Merge nodes if number of keys in siblings < . Merging lead to decrease in tree height.

12

m

12

m

B+-Tree Delete Case: After deletion node is at least half full.

14 32

3 5 8 14 19 32 38 40

Search key 14

Key found, delete key 14.Move others keys in the node to eliminate the gap.

Delete 14

21

Note: key 14 in the parent node is still a valid separator key. It is not modified.

B+-Tree Delete Operation Underflow occurs, evenly redistribute the keys if left or right

sibling has keys .

Delete 14

14

5 8 14 19 32 38

Underflow occurs, evenly redistribute keysin the underflow node and in the sibling node.

Note: unlike B-tree, the separator key in the parent node is not included. Why?

40

32

12/ m

Search key 14 & delete it. Separator key in the parent isno longer valid

Insert a copy of the first key of the sibling node in parent in order.

38

B+-Tree Delete Operation Underflow occurs and the keys in the left & right sibling are

. Merge the underflow node and a sibling.

12/ m

14

5 8 19 32 38 40

Underflow occurs, merge nodes.

Move keys in sibling to underflow node and discard the sibling node.

38

Delete 32 Search key 32 & delete it.

45 55 67 81

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Efficient Sequential Access in B+-Tree

Sequence Set Start

For efficient sequential access, start from the beginning of the sequence set and traverse the sequence set using the next pointers in sequence set nodes

Comparison B-Tree & B+-TreeB-Tree Data pointers stored in all

nodes No redundant keys Search can end at any

node Slow sequential access Higher trees

B+ -Tree Data pointers stored in leaf

nodes (sequence set)

Redundant keys exist Search ends always at

leaves Efficient sequential access Flatter trees (no data pointers

in index set nodes)

B* -Trees

B* -Tree -- Variant of B-Tree Each node of a B-tree represent a block of secondary memory,

therefore, accessing a node is expensive operation. Thus, the fewer nodes that are created, the better.

In B*-tree is a variant of B-tree introduced by Donald Knuth and named by Douglas Comer.

In a B*-tree, all nodes except the root are required to be at least two-thirds full, not just half full as in B-tree.

The number of keys in all non-root nodes in a B*-tree of order m is k, where .

Average utilization of B*-tree is 81%.

13

12

mkm

B* -Tree Insert Operation In B*-tree, the frequency of node splitting is decreased by

delaying a split.

A split in a B*-tree is delayed by attempting to redistribute the keys between node and its sibling when node overflows.

In B*-tree split operation two nodes are split into three instead of one into two as in B-tree.

All three nodes participating in the split are guaranteed to be two-thirds full after split.

B*-Tree Insert Operation Overflow occurs, evenly redistribute the keys between node

and its sibling.

Overflow occurs, evenly redistribute keysin the overflow node, in its sibling including the separator key in parent and new key.

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Insert 2222

35 42 47 51 53

32

2924

B*-Tree Insert Operation Overflow occurs, sibling is full, split node.

Overflow occurs, sibling is full, split node.

10 12 15 16 21 24 47

Insert 72

72

51 53 55 57 5935 42

32

25 29

B* -Tree Delete Operation B*-tree deletion follows same rules as that of B-tree deletion.

Deletion cases: After deletion node is at least two third full. After deletion underflow occurs

Redistribute: if number of keys in siblings > . Merge nodes if number of keys in siblings < .

Examples of deletion are omitted as they are similar to B-tree deletion.

3

12 m

3

12 m

Questions ?

References Data Structure and Algorithms in C++, Adam Drozek.

Introduction to Algorithms, T.H.Cormen, C.E.Leiserson, R.L.Rivest, and C.Stein.

Fundamentals of Database Systems, Elmasri Navathe.

Fundamentals of Data Structures in C++, E.Horowitz, S.Sahni and D.Mehta

The Ubiquitous B-Tree, DOUGLAS COMER, ACM Computing Surveys (CSUR), Volume 11 ,Issue 2(June 1979).

The End