CS 44321 CS4432: Database Systems II Basic indexing

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CS4432: Database Systems II

Basic indexing

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Indexing : helps to retrieve data quicker for certain queries

value= 1,000,000

Select * FROM Emp WHERE salary = 1,000,000;Select * FROM Emp WHERE salary = 1,000,000;

Chapter 13

value

record

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Topics

• Sequential Index Files (chap 13.1)• Secondary Indexes (chap 13.2)

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Sequential File

2010

4030

6050

8070

10090

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Sequential File

2010

4030

6050

8070

10090

Dense Index

10203040

50607080

90100110120

Every record

is in index.

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Sequential File

2010

4030

6050

8070

10090

Sparse Index

10305070

90110130150

170190210230

Only first record

per block in index.

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Sequential File

2010

4030

6050

8070

10090

Sparse 2nd level

10305070

90110130150

170190210230

1090

170250

330410490570

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Note : DATA FILE or INDEX are “ordered files”.

Question:How would we lay them out on disk ?

- contiguous layout on disk ? - block-chained layout on disk ?

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Questions:

• Do we want to build a dense 2nd-level index for a dense index?

• Can we even do this ?

Sequential File2010

4030

6050

8070

10090

2nd level?1030507090

110130150170190210230

1090

170250330410490570

1st level?

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Notes on pointers:

(1)Block pointer (used in sparse index) can be smaller than record pointer (used in dense index)

BP

RP

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K1

K3

K4

K2

R1

R2

R3

R4

say:1024 Bper block

• if we want K3 block:• get it at offset (3-1)*1024 = 2048 bytes

Note : If file is contiguous, then we can omit pointers

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Sparse vs. Dense Tradeoff

• Sparse: Less index space per record can keep more of index in

memory (Later: sparse better for insertions)

• Dense: Can tell if any record exists without accessing file

(Later: dense needed for secondary indexes)

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Terms

• Index sequential file• Search key ( primary key)• Primary index (on sequencing field)• Secondary index• Dense index (contains all search

key values)• Sparse index• Multi-level index

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Next:

• Duplicate keys

• Deletion/Insertion

• Secondary indexes

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Duplicate keys

1010

2010

3020

3030

4540

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1010

2010

3020

3030

4540

1010

2010

3020

3030

4540

10101020

20303030

10101020

20303030

Dense index ! Point to each value !

Duplicate keys

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1010

2010

3020

3030

4540

Dense index. Point to each distinct value!

10203040

Duplicate keys

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1010

2010

3020

3030

4540

10102030

Sparse index: point to start of block !

Duplicate keys

care

ful if lookin

gfo

r 2

0 o

r 3

0!

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1010

2010

3020

3030

4540

10203030

Sparse index, another way ?

Duplicate keys

– place first new key from block

shouldthis be40?

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Duplicate values, primary index

• Index may point to first instance ofeach value only

File Index

Summary

aaa

b

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Next:

• Duplicate keys



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Deletion from sparse index

2010

4030

6050

8070

10305070

90110130150

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2010

4030

6050

8070

10305070

90110130150

– delete record 40

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2010

4030

6050

8070

10305070

90110130150


4040

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2010

4030

6050

8070

10305070

90110130150

– delete records 30 & 40

5070

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Deletion from dense index

2010

4030

6050

8070

10203040

50607080

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Deletion from dense index

2010

4030

6050

8070

10203040

50607080


4040

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Insertion, sparse index case

2010

30

5040

60

10304060

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2010

30

5040

60

10304060

– insert record 34

34

• our lucky day! we have free space where we need it!

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2010

30

5040

60

10304060


15

2030

20

• Immediate reorganization• Other variations?

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• Just Illustrated: -Immediate reorganization

• Now Variation:– insert new block (chained file)

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2010

30

5040

60

10304060


25

overflow blocks(reorganize later...)

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Insertion, dense index case

• Similar

• Often more expensive . . .

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Next:

• Duplicate keys



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Secondary indexesSequencefield

5030

7020

4080

10100

6090

Can I make a

secondary

index sparse ?

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5030

7020

4080

10100

6090

• Sparse index

302080

100

90...

does not make sense!

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5030

7020

4080

10100

6090

• Must be dense index !10203040

506070...

105090...

sparsehighlevel

allowed?

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With secondary indexes:

• Lowest level is dense• Other levels are sparse

Also: Pointers are record pointers

(not block pointers; not computed)

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Duplicate values & secondary indexes

1020

4020

4010

4010

4030

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1020

4020

4010

4010

4030

10101020

20304040

4040...

one option...

Problem:excess overhead!

• disk space• search time

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1020

4020

4010

4010

4030

10

another option...

4030

20Problem:variable sizerecords inindex!

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1020

4020

4010

4010

4030

10203040

5060...

Another idea :Chain records with same key !

Problems:• Need to add fields to data records for each index• Need to follow chain to know records

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Summary : Conventional Indexes

– Basic Ideas: sparse, dense, multi-level…

– Duplicate Keys– Deletion/Insertion– Secondary indexes

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Multi-level Index StructuresSequencefield

5030

7020

4080

10100

6090

firstlevel

(dense,if non-

sequential)

10203040

506070...

105090...

highLevel

(alwayssparse)

1

2

5

43

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Sequential indexes : pros/cons ?

Advantage:- Simple- Index is sequential file

good for scans - Search efficient for static data

Disadvantage:

- Inserts expensive, and/or- Lose sequentiality & balance

- Then search time unpredictable

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Example Sequential Index

continuous

free space

102030

405060

708090

39313536

323834

33

overflow area(not sequential)

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Another type of index

• Give up “sequentiality” of index• Predictable performance under

updates• Achieve always balance of “tree” • Automate restructuring under

updates

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Root

B+Tree Example n=3

100

120

150

180

30

3 5 11

30

35

100

101

110

120

130

150

156

179

180

200

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Sample non-leaf

to keys to keys to keys to keys

< 57 57 k<81 81k<95 95

57

81

95

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Sample leaf node:

From non-leaf node

to next leafin

sequence5

7

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 8

5

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In textbook’s notationn=3

Leaf:

Non-leaf:

30

35

30

30 35

30

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Size of nodes: n+1 pointersn keys

(fixed)

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Don’t want nodes to be too empty

• Use at least

Non-leaf: (n+1)/2pointers

Leaf: (n+1)/2 pointers to data

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Full nodemin. node

Non-leaf

Leaf

n=3

12

01

50

18

0

30

3 5 11

30

35

counts

even if

null

Non-leaf: (n+1)/2 pointers

Leaf: (n+1)/2 pointers to data

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B+tree rules tree of order n

(1) All leaves at same lowest level(balanced tree)

(2) Pointers in leaves point to records except for “sequence pointer”

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Root

B+Tree Example : Searches

100

120

150

180

30

3 5 11

30

35

100

101

110

120

130

150

156

179

180

200

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Insert into B+tree

(a) simple case– space available in leaf

(b) leaf overflow(c) non-leaf overflow(d) new root

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(a) Insert key = 32 n=33 5 11

30

31

30

100

32

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(a) Insert key = 7 n=3

3 5 11

30

31

30

100

3 5

7

7

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(c) Insert key = 160 n=3

10

0

120

150

180

150

156

179

180

200

160

18

0

160

179

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(d) New root, insert 45 n=3

10

20

30

1 2 3 10

12

20

25

30

32

40

40

45

40

30new root

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Recap: Insert Data into B+ Tree

• Find correct leaf L. • Put data entry onto L.

– If L has enough space, done!– Else, must split L (into L and a new node L2)

• Redistribute entries evenly, copy up middle key.• Insert index entry pointing to L2 into parent of L.

• This can happen recursively– To split index node, redistribute entries evenly, but

push up middle key. (Contrast with leaf splits.)

• Splits “grow” tree; root split increases height. – Tree growth: gets wider or one level taller at top.

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(a) Simple case (b) Coalesce with neighbor (sibling)

(c) Re-distribute keys(d) Cases (b) or (c) at non-leaf

Deletion from B+tree

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(a) Delete key = 11 n=33 5 11

30

31

30

100

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(b) Coalesce with sibling– Delete 50

10

40

100

10

20

30

40

50

n=4

40

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(c) Redistribute keys– Delete 50

10

40

100

10

20

30

35

40

50

n=4

35

35

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40

45

30

37

25

26

20

22

10

141 3

10

20

30

40

(d) Coalese and Non-leaf coalese– Delete 37

n=4

40

30

25

25

new root

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B+tree deletions in practice

– Often, coalescing is not implemented– Too hard and not worth it!

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Delete Data from B+ Tree

• Start at root, find leaf L where entry belongs.• Remove the entry.

– If L is at least half-full, done! – If L has only d-1 entries,

• Try to re-distribute, borrowing from sibling (adjacent node with same parent as L).

• If re-distribution fails, merge L and sibling.

• If merge occurred, must delete entry (pointing to L or sibling) from parent of L.

• Merge could propagate to root, decreasing height.

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• Concurrency control harder in B-Trees• B-tree consumes more space• DBA does not know when to reorganize• DBA does not know how full to load pages of new index• Buffering

– B-tree: has fixed buffer requirements– Static index: must read several overflow blocks to be efficient (large & variable size

buffers needed)

Comparison: B-trees vs. static indexed sequential file

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• Speaking of buffering… Is LRU a good policy for B+tree

buffers?Of course not!

Should try to keep root in memory at all times

(and perhaps some nodes from second level)

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ComparisonB-tree vs. indexed seq.

file• Less space, so

lookup faster• Inserts managed

by overflow area• Requires

temporary restructuring

• Unpredictable performance

• Consumes more space, so lookup slower

•Each insert/delete potentially restructures

•Build-in restructuring

• Predictable performance

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Interesting problem:

For B+tree, how large should n be?

…

n is number of keys / node

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assumptions: n children per node and N records in database

(1) Time to read B-Tree node from disk is (tseek + tread*n) msec.(2) Once in main memory, use binary search to locate key, (a + b log_2 n) msec(3) Need to search (read) log_n (N) tree nodes

(4) t-search = (tseek + tread*n + (a + b*log_2(n)) * log n (N)

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Can get: f(n) = time to find a record

f(n)

nopt n

FIND nopt by f’(n) = 0

What happens to nopt as:•Disk gets faster? CPU get faster? …

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Bulk Loading of B+ Tree

• For large collection of records, create B+ tree.• Method 1: Repeatedly insert records slow.• Method 2: Bulk Loading more efficient.

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Bulk Loading of B+ Tree

• Initialization: – Sort all data entries – Insert pointer to first (leaf) page in new (root) page.

3* 4* 6* 9* 10* 11* 12* 13* 20* 22* 23* 31* 35* 36* 38* 41* 44*

Sorted pages of data entries; not yet in B+ treeRoot

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Bulk Loading (Contd.)

• Index entries for leaf pages always entered into right-most index page

• When this fills up, it splits.

Split may go up right-most path to root.

3* 4* 6* 9* 10*11* 12*13* 20*22* 23* 31* 35*36* 38*41* 44*

Root

Data entry pages

not yet in B+ tree3523126

10 20

3* 4* 6* 9* 10* 11* 12*13* 20*22* 23* 31* 35*36* 38*41* 44*

6

Root

10

12 23

20

35

38

not yet in B+ treeData entry pages

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Summary of Bulk Loading

• Method 1: multiple inserts.– Slow.– Does not give sequential storage of leaves.

• Method 2: Bulk Loading – Has advantages for concurrency control.– Fewer I/Os during build.– Leaves will be stored sequentially (and

linked) – Can control “fill factor” on pages.

Documents

CS 44321 CS4432: Database Systems II Basic indexing