1

CS 728 Advanced Database Systems

Chapter 17Database File Indexing Techniques, B-

Trees, and B+-Trees

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Indexes as Access Paths

A single-level index is an auxiliary file that makes it more efficient to search for a record in the data file.

The index is usually specified on one field of the file (although it could be specified on several fields)

One form of an index is a file of entries <field value, pointer to record>, which is ordered by field value

The index is called an access path on the field.

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Indexes as Access Paths (cont.)

The index file usually occupies considerably less disk blocks than the data file because its entries are much smaller

A binary search on the index yields a pointer to the file record

Indexes can also be characterized as dense or sparse dense index

has index entry for every record in the file. sparse (nondense) index

has index entries for only some of the search-key values.

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Sparse Vs. Dense Indices

Sparse primaryindex sorted

on Id

Dense secondaryindex sorted

on Name

Ordered file sorted on Id

Id Name Dept

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Sparse Vs. Dense Indices

Ashby, 25, 3000

Smith, 44, 3000

Ashby

Cass

Smith

22

25

30

40

44

44

50Sparse primaryindex on Name

Ordered file on NameDense secondary

index on Age

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Bristow, 30, 2007

Basu, 33, 4003

Cass, 50, 5004

Tracy, 44, 5004

Daniels, 22, 6003

Jones, 40, 6003

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Sparse Indices

Sparse index contains index records for only some search-key values. Some keys in the data file will not have an entry

in the index file

Applicable when records are sequentially ordered on search-key (ordered files)

Normally keeps only one key per data block

Less space (can keep more of index in memory)

Less maintenance overhead for insertions and deletions.

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Ordered File

2010

4030

6050

8070

10090

Sparse/Primary Index

10305070

90110130150

170190210230

Sparse Indices

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Example Given the following data file EMPLOYEE(NAME, SSN, ADDRESS,

SAL) Suppose that:

record size R=150 bytes block size B=512 bytes r=30000 records blocking factor Bfr= B/R=512/150=3 records/block number of file blocks b=(r/Bfr)=(30000/3)=10000 blocks

For an index on the SSN field, assume the field size VSSN=9 bytes, assume the record pointer size PR=7 bytes. Then: index entry size RI=(VSSN+ PR)=(9+7)=16 bytes index blocking factor BfrI= B/RI= 512/16= 32 entries/block number of index blocks b= (r/BfrI)=(30000/32)=938 blocks binary search needs log2bI= log2938= 10 block accesses This is compared to an average linear search cost of:

(b/2)= 30000/2= 15000 block accesses If the file records are ordered, the binary search cost would be:

log2b= log230000= 15 block accesses

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Types of Single-Level Indexes

primary index: is specified on the ordering key field of an ordered file,

where every record has a unique value for that field. The index has the same ordering as the one of the file.

clustering index: is specified on the ordering field of an ordered file. The index has the same ordering as the one of the file. An ordered file can have at most one primary index or one

clustering index, but not both.

secondary index: is specified on any nonordering field of the file. The index has different ordering than the one of the file. A file can have several secondary indices in addition to its

primary/clustering index.

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Primary Indices

Defined on an ordered data file The data file is ordered on a key field

Includes one index entry for each block in the data file; the index entry has the key field value for the first record in the block, which is called the block anchor

A similar scheme can use the last record in a block. Finding a record is efficient – do a binary search A primary index is a nondense (sparse) index, since

it includes an entry for each disk block of the data file and the keys of its anchor record rather than for every search value.

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Primary Index on the Ordering Key Field

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Clustering Indices

Defined on an ordered data file

The data file is ordered on a non-key field unlike primary index, which requires that the ordering field of the data file have a distinct value for each record.

Includes one index entry for each distinct value of the clustering field (rather than for every record). Sparse index (nondense)

The index entry points to the first data block that contains records with that field value.

A file can have at most one primary index or one clustering index, but not both.

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Clustering Indices

A clustering index on the DEPNo ordering nonkey field of an EMPLOYEE file.

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Clustering Indices

Clustering index with a separate block cluster for each group of records that share the same value for the clustering field.

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Secondary Indices

Secondary index: is specified on any nonordering field of the file.

Non-ordering field can be a key (unique) or a non-key (duplicates)

The index has different ordering than the one of the file.

A file can have several secondary indices in addition to its primary index.

there is one index entry for each data record

index record points either to the block in which the record is stored, or to the record itself

Hence, such an index is dense

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Secondary Indices

A secondary index usually needs more storage space and longer search time than does a primary index. It has larger number of entries.

Sequential scan using primary index is efficient, but a sequential scan using a secondary index is expensive each record access may fetch a new block from disk

The index is an ordered file with two fields. The first field is of the same data type as some non-

ordering field of the data file that is an indexing field. The second field is either a block pointer or a record

pointer.

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Example of a Dense Secondary Index

A dense secondary index (with block pointers) on a nonordering KEY field.

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Example of a Secondary Index

A dense secondary index (with record pointers) on a non-ordering non-key field.

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Index Types and Indexing Fields

Data file ordered by indexing field

Data file not ordered by

indexing field

Indexing field is key Primary Index Secondary index(Key)

Indexing field is nonkey Clustering Index Secondary index(NonKey)

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Multi-Level Indexes

Because a single-level index is an ordered file, we can create a primary index to the index itself; In this case, the original index file is called the first-

level index and the index to the index is called the second-level index.

We can repeat the process, creating a third, fourth, ..., top level until all entries of the top level fit in one disk block

A multi-level index can be created for any type of first-level index (primary, secondary, clustering) as long as the first-level index consists of more than one disk block

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A Two-Level

Primary Index

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Multi-Level Indexes

Such a multi-level index is a form of search tree However, insertion and deletion of new index

entries is a severe problem because every level of the index is an ordered file.

A Node in a search tree with pointers to subtrees below it.

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Dynamic Multilevel Indexes Using B-Trees and B+-Trees

Most multi-level indexes use B-tree or B+-tree data structures because of the insertion and deletion problem This leaves space in each tree node (disk block) to

allow for new index entries

These data structures are variations of search trees that allow efficient insertion and deletion of new search values.

In B-Tree and B+-Tree data structures, each node corresponds to a disk block

Each node is kept between half-full and completely full

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Dynamic Multilevel Indexes Using B-Trees and B+-Trees (cont.)

An insertion into a node that is not full is quite efficient If a node is full the insertion causes a split into

two nodes

Splitting may propagate to other tree levels

A deletion is quite efficient if a node does not become less than half full

If a deletion causes a node to become less than half full, it must be merged with neighboring nodes

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Difference between B-tree and B+-tree

In a B-tree, pointers to data records exist at all levels of the tree

In a B+-tree, all pointers to data records exists at the leaf-level nodes

A B+-tree can have less levels (or higher capacity of search values) than the corresponding B-tree

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B-tree Structures

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B+-Tree Index

A B+-tree, of order f (fan-out --- maximum node capacity), is a rooted tree satisfying the following: All paths from root to leaf are of the same length

(balanced tree)

Each non-leaf node (except the root) has between f/2 and up to f tree pointers (f-1 key values).

A leaf node has between f/2 and f-1 data pointers (plus a pointer for sibling node).

If the root is not a leaf, it has at least 2 children.

If the root is a leaf (that is, there are no other nodes in the tree), it can have between 0 and f-1 values.

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The Nodes of a B+-tree

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B+-Tree Non-leaf Node Structure

Ki are the search-key values, K1 K2 K3 … Kf-1

all keys in the subtree to which P1 points are K1. all keys in the subtree to which Pf points are Kf-1. for 2 i f-1, all keys in the subtree to which Pi points

have values Ki-1 and Ki.

Pi are pointers to children nodes (tree nodes).

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B+-Tree Leaf Node Structure

for i = 1, 2, …, f-1, pointer Pri is a data pointer, that either points to a file record with search-key value Ki, or block of record pointers that point to records having

search-key value Ki. (if search-key is not a key) Pnext points to next leaf node in search-key order.

Within each leaf node, K1 K2 K3 … Kf-1

If Li, Lj are leaf nodes and i j, then Li’s search-key values Lj’s search-key values

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Sample Leaf Node

From non-leaf node

to next leafin sequence

57

81

95

To r

eco

rd

wit

h k

ey 5

7

To r

eco

rd

wit

h k

ey 8

1

To r

eco

rd

wit

h k

ey 9

5

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Sample Non-Leaf Node

to keys to keys to keys to keys 57 57 k 81 81 k 95 95

57

81

95

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Example of a B+-Tree

Root f=4

35

110

130

179

11

3 5 11

30

35

100

101

110

120

130

150

156

179

180

200

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Number of pointers/keys for B+-Tree

Full node min. node

Non-leaf

Leaf

f=4

12

01

50

18

0

30

3 5 11

30

35

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Observations about B+-Trees

In a B+-tree, data pointers are stored only at the leaf nodes of the tree hence, the structure of leaf nodes differs from

the structure of internal nodes.

The leaf nodes have an entry for every value of the search field, along with a data pointer to the record.

Some search field values from the leaf nodes are repeated in the internal nodes.

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B+-Trees: Search

Let a be a search key value and T the pointer to the root of the tree that has f pointer.

Search(a, T) If T is non-leaf node:

for the first i that satisfy a Ki, 1 i f-1 call Search(a, Pi), else call Search(a, Pf).

Else //T is a leaf node if no value in T equals a, report not found. else if Ki in T equals a, follow pointer Pri to

read the record/block.

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B+-Trees: Search

In processing a query, a path is traversed in the tree from the root to some leaf node.

If there are n search-key values in the file, the path is no longer than log f/2(n) (worst

case).

With 1 million search key values and f = 100, at most log50(1000000) = 4 nodes are accessed in a lookup.

Contrast this with a balanced binary tree with 1 million search key values -- around 20 nodes are accessed in a lookup.

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B+-Trees: Insertion

Find the leaf node in which the search-key value would appear

If the search-key value is found in the leaf node, add the record to main file and if necessary add to the block a pointer to the record

If the search-key value is not there, add the record to the main file and then: If there is room in the leaf node, insert (key-

value, pointer) pair in the leaf node Otherwise, split the node along with the new

(key-value, pointer) entry

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B+-Trees: Insertion

Splitting a node: take the f (search-key value, pointer) pairs

(including the one being inserted) in sorted order. place the first (f+1)/2 in the original node x, and

the rest in a new node y. let k be the largest key value in x. insert (k, y) in the parent node in their correct

sequence. If the parent is full

the entries in the parent node up to Pj, where j = (f+1)/2 are kept, while the jth search value is moved to the parent, no replicated.

A new internal node will hold the entries from Pj+1 to the end of the entries in the node.

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B+-Trees: Insertion

The splitting of nodes proceeds upwards till a node that is not full is found.

In the worst case the root node may be split increasing the height of the tree by 1.

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Insertion – Example 3

Insert key = 31

f=4

3 5 11

30

32

11

32

31

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Insertion – Example 3

Insert key = 7f=4

3 5 11

30

31

11

31

3 5

7

5

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Insertion – Example 3

Insert key = 160f=4

10

0

120

140

179

150

156

179

180

200

156

17

9

160

179

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Insertion – Example 3

New root, insert 45f=4

3 12

25

1 2 3 10

12

20

25

30

32

40

40

45

32

25new root

46

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B+-Trees: Deletion

Find the record to be deleted, and remove it from the main file and from the bucket (if present).

Remove (search-key value, pointer) from the leaf node.

If the node has too few entries due to the removal, and the entries in the node and a sibling fit into a single node, then Insert all the search-key values in the two nodes

into a single node (the one on the left), and delete the other node.

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B+-Trees: Deletion

Delete the pair (Ki-1, Pi), where Pi is the pointer to the deleted node, from its parent, recursively using the above procedure.

Otherwise, if the node has too few entries due to the removal, and the entries in the node and a sibling DO NOT fit into a single node, then Redistribute the pointers between the node and a

sibling such that both have more than the minimum number of entries.

Update the corresponding search-key value in the parent of the node.

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B+-Trees: Deletion

The node deletions may cascade upwards till a node which has f/2 or more pointers is found.

If the root node has only one pointer after deletion, it is deleted and the sole child becomes the root.

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Merge with Sibling

Delete 45

10

40

50

20

40

45

50

f=4

50

51

10

35

50

15

30

35

40

5035

30

f=4

Redistribute Keys

Delete 40

52

40

45

30

37

25

26

20

22

10

141 3

3 14

26

3730

22

22

new root

f=4

Non-leaf Merging

Delete 37

30

30

53

Example of a Deletion in a

B+-tree

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Extra Reading

Read Examples 1 to 7.