B-Trees, Part 2 Hash-Based Indexes

1
B-Trees, Part 2Hash-Based Indexes

• RG Chapter 10
• Lecture 10

2
Administrivia

• The new Homework 3 now available
• Due 1 week from Sunday
• Homework 4 available the week after
• Midterm exams available here

3
Review

• Last time discussed File Organization
• Unordered heap files
• Sorted Files
• Clustered Trees
• Unclustered Trees
• Unclustered Hash Tables
• Indexes
• B-Trees dynamic, good for changing data, range
  queries
• Hash tables fastest for equality queries,
  useless for range queries

4
Review (2)

• For any index, 3 alternatives for data entries
  k
• Data record with key value k
• ltk, rid of data record with search key value kgt
• ltk, list of rids of data records with search key
  kgt
• Choice orthogonal to the indexing technique

5
Today

• Indexes
• Composite Keys, Index-Only Plans
• B-Trees
• details of insertion and deletion
• Hash Indexes
• How to implement with changing data sets

6
Inserting a Data Entry into a 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.

7
Indexes with Composite Search Keys

• Composite Search Keys Search on a combination of
  fields.
• Equality query Every field value is equal to a
  constant value. E.g. wrt ltsal,agegt index
• age20 and sal 75
• Range query Some field value is not a constant.
  E.g.
• age 20 or age20 and sal gt 10
• Data entries in index sorted by search key to
  support range queries.
• Lexicographic order, or
• Spatial order.

Examples of composite key indexes using
lexicographic order.
11,80
11
12
12,10
name
age
sal
12,20
12
bob
10
12
13,75
13
cal
80
11
ltage, salgt
ltagegt
joe
12
20
sue
13
75
10,12
10
20
20,12
Data records sorted by name
75,13
75
80,11
80
ltsal, agegt
ltsalgt
Data entries in index sorted by ltsal,agegt
Data entries sorted by ltsalgt
8
Composite Search Keys

• To retrieve Emp records with age30 AND sal4000,
  an index on ltage,salgt would be better than an
  index on age or an index on sal.
• Choice of index key orthogonal to clustering etc.
• If condition is 20ltagelt30 AND 3000ltsallt5000
• Clustered tree index on ltage,salgt or ltsal,agegt is
  best.
• If condition is age30 AND 3000ltsallt5000
• Clustered ltage,salgt index much better than
  ltsal,agegt index!
• Composite indexes are larger, updated more often.

9
Index-Only Plans
SELECT D.mgr FROM Dept D, Emp E WHERE
D.dnoE.dno
ltE.dnogt
SELECT D.mgr, E.eid FROM Dept D, Emp E WHERE
D.dnoE.dno

• A number of queries can be answered without
  retrieving any tuples from one or more of the
  relations involved if a suitable index is
  available.

ltE.dno,E.eidgt
Tree index!
SELECT E.dno, COUNT() FROM Emp E GROUP BY
E.dno
ltE.dnogt
SELECT E.dno, MIN(E.sal) FROM Emp E GROUP BY
E.dno
ltE.dno,E.salgt
Tree index!
ltE. age,E.salgt or ltE.sal, E.agegt
SELECT AVG(E.sal) FROM Emp E WHERE E.age25
AND E.sal BETWEEN 3000 AND 5000
Tree!
10
Index-Only Plans (Contd.)

• Index-only plans are possible if the key is
  ltdno,agegt or we have a tree index with key
  ltage,dnogt
• Which is better?
• What if we consider the second query?

SELECT E.dno, COUNT () FROM Emp E WHERE
E.age30 GROUP BY E.dno
SELECT E.dno, COUNT () FROM Emp E WHERE
E.agegt30 GROUP BY E.dno
11
B-Trees Insertion and Deletion

• Insertion
• Find leaf where new record belongs
• If leaf is full, redistribute
• If siblings too full, split, copy middle key up
• If index too full, redistribute
• If index siblings full, split, push middle key up
• Deletion
• Find leaf where record exists, remove it
• If leaf is less than 50 empty, redistribute
• If siblings too empty, merge, remove key above
• If index node above too empty, redistribute
• If index siblings too empty, merge, move above
  key down

12
B-Trees For Homework 2

• Insertion
• Find leaf where new record belongs
• If leaf is full, redistribute
• If siblings too full, split, copy middle key up
• If index too full, redistribute
• If index siblings full, split, push middle key up
• Deletion
• Find leaf where record exists, remove it
• If leaf is less than 50 empty, redistribute
• If siblings too empty, merge, remove key above
• If index node above too empty, redistribute
• If index siblings too empty, merge, move above
  key down

This means that after deletion, nodes will often
be lt 50 full
13
B-Trees For Homework 2 (cont)

• Splits
• When splitting nodes, choose the middle key
• If there are even number of keys, choose ?middle?
• Your code must Handle Duplicate Keys
• We promise that there will never be more than 1
  page of duplicate values.
• Thus, when splitting, if the middle key is
  identical to the key to the left, you must find
  the closest splittable key to the middle.

14
B-Tree Demo
15
Hashing

• Static and dynamic hashing techniques exist
  trade-offs based on data change over time
• Static Hashing
• Good if data never changes
• Extendable Hashing
• Uses directory to handle changing data
• Linear Hashing
• Avoids directory, usually faster

16
Static Hashing

• primary pages fixed, allocated sequentially,
  never de-allocated overflow pages if needed.
• h(k) mod N bucket to which data entry with key
  k belongs. (N of buckets)

0
h(key) mod N
2
key
h
N-1
Primary bucket pages
Overflow pages
17
Static Hashing (Contd.)

• Buckets contain data entries.
• Hash fn works on search key field of record r.
  Must distribute values over range 0 ... N-1.
• h(key) (a key b) usually works well.
• a and b are constants lots known about how to
  tune h.
• Long overflow chains can develop and degrade
  performance.
• Extendible and Linear Hashing Dynamic techniques
  to fix this problem.

18
Extendible Hashing

• Situation Bucket (primary page) becomes full.
• Why not re-organize file by doubling of
  buckets?
• Reading and writing all pages is expensive!
• Idea Use directory of pointers to buckets,
• double of buckets by doubling the directory,
• splitting just the bucket that overflowed!
• Directory much smaller than file, doubling much
  cheaper.
• No overflow pages!
• Trick lies in how hash function is adjusted!

19
Extendible Hashing Details

• Need directory with pointer to each bucket
• Need hash function to incrementally double range
• can just use increasingly more LSBs of h(key)
• Must keep track of global depth
• how many times directory doubled so far
• Must keep track of local depth
• how many time each bucket has been split

20
Example
2
LOCAL DEPTH
Bucket A
16
4
12
32
GLOBAL DEPTH
2
2
Bucket B
13
00
1
21
5

• Directory is array of size 4.
• To find bucket for r, take last global depth
  bits of h(r) we denote r by h(r).
• If h(r) 5 binary 101, it is in bucket
  pointed to by 01.

01
2
10
Bucket C
10
11
2
DIRECTORY
Bucket D
15
7
19
DATA PAGES

• Insert If bucket is full, split it (allocate
  new page, re-distribute).

• If necessary, double the directory. (As we will
  see, splitting a
• bucket does not always require doubling we
  can tell by
• comparing global depth with local depth for
  the split bucket.)

21
Insert h(r)20 (Causes Doubling)
2
LOCAL DEPTH
3
LOCAL DEPTH
Bucket A
32
16
4
12
GLOBAL DEPTH
32
16
Bucket A
GLOBAL DEPTH
2
2
2
3
Bucket B
1
5
21
13
00
1
5
21
13
000
Bucket B
01
001
2
10
2
010
Bucket C
10
11
10
Bucket C
011
100
2
2
DIRECTORY
101
Bucket D
15
7
19
15
19
7
Bucket D
110
111
3
DIRECTORY
20
12
Bucket A2
4
(split image'
of Bucket A)
22
Now Insert h(r)9 (Causes Split Only)
3
3
LOCAL DEPTH
LOCAL DEPTH
32
32
16
16
Bucket A
Bucket A
GLOBAL DEPTH
GLOBAL DEPTH
3
3
2
3
1
9
1
5
21
13
000
000
Bucket B
Bucket B
001
001
2
2
010
010
10
10
Bucket C
Bucket C
011
011
100
100
2
2
101
101
15
15
19
7
19
7
Bucket D
Bucket D
110
110
111
111
3
3
DIRECTORY
Bucket A2
20
12
Bucket A2
20
12
4
4
(split image'
DIRECTORY
of Bucket A)
3
5
21
13
Bucket B2
23
Points to Note

• 20 binary 10100. Last 2 bits (00) tell us r
  belongs in A or A2. Last 3 bits needed to tell
  which.
• Global depth of directory Max of bits needed
  to tell which bucket an entry belongs to.
• Local depth of a bucket of bits used to
  determine if splitting bucket will also double
  directory
• When does bucket split cause directory doubling?
• If, before insert, local depth of bucket global
  depth.
• Insert causes local depth to become gt global
  depth
• directory is doubled by copying it over and
  fixing pointer to split image page.
• (Use of least significant bits enables efficient
  doubling via copying of directory!)

24
Directory Doubling

• Why use least significant bits in directory?
• Allows for doubling via copying!

2
2
3
16
4
12
32
16
4
12
32
000
2
2
2
001
13
1
21
5
010
13
00
1
21
5
011
01
2
2
100
10
10
10
11
101
110
2
2
111
15
7
19
15
7
19
25
Deletion

• Delete If removal of data entry makes bucket
  empty, can be merged with split image. If each
  directory element points to same bucket as its
  split image, can halve directory.

2
1
16
16
4
12
32
4
12
32
2
2
2
2
Delete 10
00
00
13
1
21
5
13
1
21
5
01
01
2
10
10
10
11
11
2
2
15
15
26
Deletion (cont)

• Delete If removal of data entry makes bucket
  empty, can be merged with split image. If each
  directory element points to same bucket as its
  split image, can halve directory.

1
2
16
4
12
32
00
1
16
4
12
32
01
Delete 15
1
10
13
1
21
5
2
2
11
00
13
1
21
5
01
10
1
1
11
16
4
12
32
0
1
2
1
15
13
1
21
5
27
Comments on Extendible Hashing

• If directory fits in memory, equality search
  answered with one disk access else two.
• 100MB file, 100 bytes/rec, 4K pages contains
  1,000,000 records (as data entries) and 25,000
  directory elements chances are high that
  directory will fit in memory.
• Directory grows in spurts, and, if the
  distribution of hash values is skewed, directory
  can grow large.
• Biggest problem
• Multiple entries with same hash value cause
  problems!
• If bucket already full of same hash value, will
  keep doubling forever! So must use overflow
  buckets if dups.

28
Linear Hashing

• This is another dynamic hashing scheme, an
  alternative to Extendible Hashing.
• LH handles the problem of long overflow chains
  without using a directory, and handles
  duplicates.
• Idea Use a family of hash functions h0, h1,
  h2, ...
• hi(key) h(key) mod(2iN) N initial buckets
• h is some hash function (range is not 0 to N-1)
• If N 2d0, for some d0, hi consists of applying
  h and looking at the last di bits, where di d0
  i.
• hi1 doubles the range of hi (similar to
  directory doubling)

29
Linear Hashing Example

• Lets start with N 4 Buckets
• Start at round 0, next 0, have 2round buckets
• Each time any bucket fills, split next bucket
• If (O hround(key) lt Next), use hround1(key)
  instead

Start
Add 9
Next
16
16
4
12
32
32
Add 20
9
13
1
21
5
13
1
21
5
Next
16
10
10
32
15
15
Nround
Nround
9
13
1
21
5
Next
4
12
10
15
Nround
20
4
12
30
Linear Hashing Example (cont)

• Overflow chains do exist, but eventually get
  split
• Instead of doubling, new buckets added
  one-at-a-time

Add 6
Add 17
16
16
32
32
9
13
1
21
5
Next
13
1
10
6
10
6
Next
Nround
Nround
15
15
4
12
20
4
12
20
21
5
9
31
Linear Hashing (Contd.)

• Directory avoided in LH by using overflow pages,
  and choosing bucket to split round-robin.
• Splitting proceeds in rounds. Round ends when
  all NR initial (for round R) buckets are split.
  Buckets 0 to Next-1 have been split Next to NR
  yet to be split.
• Current round number also called Level.
• Search To find bucket for data entry r, find
  hround(r)
• If hround(r) in range Next to NR , r belongs
  here.
• Else, r could be hround(r) or hround(r) NR
• must apply hround1(r) to find out.

32
Overview of LH File

• In the middle of a round.

Buckets split in this round
Bucket to be split
h
search key value
)
(
If
Level
Next
is in this range, must use
search key value
)
(
h
Level1
Buckets that existed at the
to decide if entry is in
beginning of this round
split image' bucket.
this is the range of
h
Level
split image' buckets
created (through splitting
of other buckets) in this round
33
Linear Hashing (Contd.)

• Insert Find bucket by applying hround /
  hround1
• If bucket to insert into is full
• Add overflow page and insert data entry.
• (Maybe) Split Next bucket and increment Next.
• Can choose any criterion to trigger split.
• Since buckets are split round-robin, long
  overflow chains dont develop!
• Doubling of directory in Extendible Hashing is
  similar switching of hash functions is implicit
  in how the of bits examined is increased.

34
Another Example of Linear Hashing

• On split, hLevel1 is used to re-distribute
  entries.

Round0, N4
Round0
PRIMARY
h
h
h
h
OVERFLOW
PRIMARY
0
1
PAGES
0
1
PAGES
PAGES
Next0
32
32
44
36
00
000
00
000
Next1
Data entry r
25
9
5
25
9
5
with h(r)5
01
001
01
001
30
30
14
18
10
14
18
10
Primary
10
10
010
010
bucket page
31
35
11
7
31
35
11
7
43
011
011
11
11
(This info is for illustration only!)
(The actual contents of the linear hashed file)
100
44
36
00
35
Example End of a Round
Round1
PRIMARY
OVERFLOW
h
h
PAGES
0
1
PAGES
Next0
Round0
00
000
32
PRIMARY
OVERFLOW
PAGES
h
PAGES
h
0
1
001
01
9
25
32
00
000
10
010
10
50
66
18
34
9
25
001
01
011
11
35
11
43
10
66
10
18
34
010
Next3
100
00
44
36
43
35
31
7
11
011
11
101
11
5
29
37
44
36
100
00
14
30
22
10
110
5
37
29
101
01
22
14
30
31
7
111
11
10
110
36
LH Described as a Variant of EH

• The two schemes are actually quite similar
• Begin with an EH index where directory has N
  elements.
• Use overflow pages, split buckets round-robin.
• First split is at bucket 0. (Imagine directory
  being doubled at this point.) But elements
  lt1,N1gt, lt2,N2gt, ... are the same. So, need
  only create directory element N, which differs
  from 0, now.
• When bucket 1 splits, create directory element
  N1, etc.
• So, directory can double gradually. Also, primary
  bucket pages are created in order. If they are
  allocated in sequence too (so that finding ith
  is easy), we actually dont need a directory!
  Voila, LH.

37
Summary

• Hash-based indexes best for equality searches,
  cannot support range searches.
• Static Hashing can lead to long overflow chains.
• Extendible Hashing avoids overflow pages by
  splitting a full bucket when a new data entry is
  to be added to it. (Duplicates may require
  overflow pages.)
• Directory to keep track of buckets, doubles
  periodically.
• Can get large with skewed data additional I/O if
  this does not fit in main memory.

38
Summary (Contd.)

• Linear Hashing avoids directory by splitting
  buckets round-robin, and using overflow pages.
• Overflow pages not likely to be long.
• Duplicates handled easily.
• Space utilization could be lower than Extendible
  Hashing, since splits not concentrated on dense
  data areas.
• Can tune criterion for triggering splits to
  trade-off slightly longer chains for better space
  utilization.
• For hash-based indexes, a skewed data
  distribution is one in which the hash values of
  data entries are not uniformly distributed!