Indexing Techniques. Advanced DatabasesIndexing Techniques2 The Problem What can we introduce to make search more efficient? –Indices! What is an index?

Indexing Techniques

Advanced Databases Indexing Techniques 2

The Problem

• What can we introduce to make search more efficient?– Indices!

• What is an index?

… …


Definitions

• Index: an auxiliary data structure to speed up record retrieval• Search key: the field/s of a table which is/are indexed• Storage: index files that contain index records

– Each entry storing• Actual data record • or, search key value k and record ID <k,rid> • or, search key value k and list of records IDs <k,rid list>

• Types: ordered and unordered (hash) indices

Page i Page i+1

Paul

Anna

Tim


Types of Ordered Indices (1/3)

• Assuming ordered data files• Depending on which field is indexed

– Primary index: search key is ordering key field• Pointer for each page

– Secondary index: search key is non ordering field

Paul00112233Anna00112234Matt00112235Tim00112236

Carol00112237Rob00112238

00112233001122350011223600112238

AnnaCarolPaulTim

primary

secondary



• Depending on the density of index records– Dense index: an index record for each distinct search key value, ie

every record

– Sparse index: index records for only some search key values• search key value for first record in page• pointer to page

Paul00112233Anna00112234

Matt00112235

Tim00112236Carol00112237

Rob00112238

00112233001122350011223600112238

sparse

001122330011223400112235001122360011223700112238

dense



• Ordering field is nonkey (may have duplicates)– Clustered index

– Unclustered index

Paul00112233

Anna00112234

Matt00112235

Tim00112236

Carol00112237

Rob00112238Paul01112233

Tim01112236Tim02112236

AnnaCarolMattPaulRobTim

001122330011223400112235001122360011223700112238011122330111223602112236

clustered

unclustered


Indices Exercise

• 215 records• 128 bytes/record• 210 bytes/page• ordered file equality search on ordering field, unspanned

organization– without an index

– with a primary index• on field of size 12 bytes• assume pointer 4 bytes long


Multi-level Indices (1/2)

• If access using first-level index is still expensive• Build a sparse index on the first-level index

– Multi-level Index

• Fan-out: index blocking factor

Paul00112233Anna00112234

Matt00112235

Tim00112236Carol00112237

Rob00112238

0011223300112234

00112235

001122360011223700112238

00112233

00112235

00112236

first-level index

second-level index


Multi-level Indices (2/2)

• 26 index records/page (fan-out)• 215 index records• 1st-level

– 29 pages

• 2nd-level– 29 index records

– 23 pages

• 3rd-level– 23 index records

– 1 page

• 1 <= 215 / (26)t

• t = ceil(log26 215 ) = 3

• t = ceil(logfo#index-records)


Dynamic multi-level indices

• So far assumed indices are physically ordered files– expensive insertions and deletions

• Dynamic multi-level indices– B trees

– B+ trees


Tree-structured Indices

• For each node: K1 < K2 < … Kq-1

• For each value X in subtree pointed to by Pi

– Ki-1< X < Ki, 1<i<q

– X < Ki, i=1

– Ki-1< X, i=q

P1 K1 … Ki-1 Pi Ki … Kq-1 Pq

X X X


B tree

• Problems: empty nodes, unbalanced trees– solution: B trees

…

…

… …

…

…

…

… …


B tree: Definition

• Each node: <P1,<K1, Pr1>, P2,…,<Kq-1, Prq-1>, Pq>• Pi tree pointer, Ki search value, Pri data pointer • For each node: K1 < K2 < … Kq-1

• For each value X in subtree pointed to by Pi – Ki-1< X < Ki, 1<i<q– X < Ki, i=1– Ki-1< X, i=q

• Each node at most q pointers– B tree is order q

• Each node at least ceil(q/2) tree pointers– except from root

• Internal node with p pointers has p-1 values• All leaves at the same level

– balanced tree


B tree: Example

5 8

ø 1 ø 3 ø ø 6 ø 7 ø ø 9 ø 12 ø

tree pointer

data pointer

ø null pointer


B+ tree

• Most implementations of B tree are B+ tree• Data pointers only in leaves

– more entries in internal nodes than regular B trees

– less internal nodes

– less levels

– faster access


B+ tree: Definition

• Internal nodes: <P1,K1, P2,…, Pq-1, Kq-1, Pq>

• Leaf nodes: <<K1, Pr1>, <K2, Pr2>,…,<Kp-1, Prp-1>, Pnext>

• Pri points a data records or block of pointers of such records

• leaf order

120 150 180

150 156 179 180 200

100 101 110 120 130


100 101 110 120 130 150 156 179 180 200

3 5 11 30 35

120 150 18030

100

B+ tree: Search

• At each level, find smallest Ki larger than search key

• Follow associated pointer Pi


B+ tree: Insert

• Nodes may overflow or underflow• Ignoring overflow or underflow• Inserting data record with with search key value k

– find leaf node

– if k found• add record to file, create indirect block if there isn’t one• add record pointer to indirect block

– if k not found• add data record to file• insert record pointer in leaf node (all search keys in order)


B+ tree: Delete

• Ignoring overflow or underflow• Find leaf node with search key value k• Find data record pointer, delete record• delete index record

– and indirect block, if any, if empty


B+ tree: Simple Insert

• Insert 42

100 101 110 120 130 150 156 179 180 200

3 5 11 30 35

120 150 18030

100k < 100

42


B+ tree: Leaf Overflow (1/2)

• Insert 9

100 101 110 120 130 150 156 179 180 200

3 5 11 30 35 42

120 150 18030

100k < 100


B+ tree: Leaf Overflow (2/2)

• first ceil(n/2) in existing node, rest in new leaf node• n=3+1=4

100 101 110 120 130 150 156 179 180 200

120 150 1809 30

100k < 100

3 5 30 35 429 11


9 30

k < 100

3 5 30 35 429 11

B+ tree: Internal Node Overflow (1/3)

• Insert 210, insert 205

100 101 110 120 130 150 156 179 180 200 210

120 150 180

100



• Leaf Split

9 30

k < 100

3 5 30 35 429 11

100 101 110 120 130 150 156 179 180 200

120 150 180

100

205 210



9 30

k < 100

3 5 30 35 429 11

100 101 110 120 130 150 156 179 180 200

120

100 150

205 210

180 205


B+ tree: New Root (1/2)

• Insert 210, insert 205

100 101 110 120 130 150 156 179 180 200

120 150 180

205 210


B+ tree: New Root (2/2)

180 205

100 101 110 120 130 150 156 179 180 200

120

205 210

150


Index Insert Exercise

• Insert 8, 7, 41

9 30

3 5 30 35 429 11


B+ tree: Delete

• Simple delete case• Underflow case:

– redistribute records

– coalesce with siblings

– update parents


B+ tree: Simple Delete (1/2)

• Delete 110

180 205

100 101 110 120 130 150 156 179 180 200

120

205 210 215

150


B+ tree: Simple Delete (2/2)

• Leaf Updated

180 205

100 101 120 130 150 156 179 180 200

120

205 210 215

150


B+ tree: Delete Redistribution (1/2)

• Delete 180

180 205

100 101 120 130 150 156 179 180 200

120

205 210 215

150


B+ tree: Delete Redistribution (2/2)

• Redistribute entries– left or right sibling

179 205

100 101 120 130 150 156 179 200

120

205 210

150


B+ tree: Delete Coalesce (1/4)

• Delete 101

179 205

100 101 120 130 150 156 179 200

120

205 210 215

150



• Leaf updated• No redistribution

– sibling coalesce

179 205

100 120 130 150 156 179 200

120

205 210 215

150



• Leaf updated• No redistribution

– sibling coalesce

179 205

100 120 130 150 156 179 200

205 210 215

150



• Redistribution

205

100 120 130 150 156 179 200

150

205 210 215

179

Hashing Techniques


Static Hashing (1/2)

• Store records in buckets with overflow chains• Allocate a fixed number of buckets M• Problems:

– small M• long overflow chains, slow search-delete-insert

null

h

null


Static Hashing (2/2)

• Problems:– large M

• wasted space, slow scan null

h

null

null


Dynamic Hashing

• Splitting and coalescing buckets as the database grows-shrinks• One scheme: Extendible Hashing• Hash function generates large values, eg 32 bits

– use i bits, change i as database size changes

• If overflow, double the number of buckets– use i+1 bits of the hash function

– but, expensive: read all pages M and distribute records in 2*M pages

• solution: use a directory and double the size of the directory– only split bucket that overflowed


Extendible Hashing (1/4)

h(18) = 10010

2

01

00

11

10

16 20

2

1

2

2

Directory

Buckets

3 7

2

A

B

C

D

18



h(4) = 00100

2

01

00

11

10

16 20

2

1

2

2

3 7

2

A

B

C

D

18



2

01

00

11

10

16

3

1

2

2

3 7

2

A

B

C

D

18

20 4

3A1



3

001

000

011

010

16

3

1

2

2

3 7

2

A

B

C

D

18

20 4

3A1

101

100

111

110

• Global Depth• Local Depth• If bucket full:

– split bucket

– increment LD

• If GD=LD– increment GD

– double directory


Extendible Hashing: Delete

• If deletion make bucket empty– merge with split image

• If directory pointers point to same bucket as split image– directory halved


Extendible Hashing: Summary

• Avoids overflow pages• Directory can get large• Key search requires just 2 page reads• Space utilization fluctuates

– 59-90% for uniformly distributed records


Extendible Hashing: Exercise

• Initially GD = LD = 1• M = 2 buckets• Hash function: h(k) = k mod 2i

• inserts: 14, 18, 22, 3, 9• deletes 9, 22, 3

1

01

00

12 8

1

5

1

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Indexing Techniques. Advanced DatabasesIndexing Techniques2 The Problem What can we introduce to make search more efficient? –Indices! What is an index?