1d index structures

1
1-d index structures

• B-tree
• Logarithmic complexity for equality and range
  searches
• Hashing
• Best for equality searches
• Static hashing
• Long overflow chains are possible
• Dynamic hashing
• Linear hashing
• Extendible hashing

2
Static hashing

• Hash function and buckets
• Primary and overflow pages
• Single disk I/O for search
• More for data resident in overflow pages
• Heuristic pages 80 full initially

0
h(key) mod N
1
key
h
N-1
Primary bucket pages
Overflow pages
3
Dynamic hashing Extendible hashing

• Directory
• Address of buckets
• Global depth
• Local depth at buckets to decide if doubling of
  directory is needed
• Mapping from directory to buckets
• Many-to-one for buckets with local depth lt global
  depth
• One-to-one for buckets with local depth global
  depth

4
Example

• Directory is array of size 4.
• Bucket for record r has entry with index
  global depth least significant bits of h(r)
• If h(r) 5 binary 101, it is in bucket
  pointed to by 01.
• If h(r) 7 binary 111, it is in bucket
  pointed to by 11.

2
LOCAL DEPTH
Bucket A
16
4
12
32
GLOBAL DEPTH
2
1
Bucket B
00
13
1
7
5
01
10
2
Bucket C
10
11
DIRECTORY
5
Handling Inserts

• Find bucket where record belongs.
• If theres room, put it there.
• Else, if bucket is full, split it
• increment local depth of original page
• allocate new page with new local depth
• re-distribute records from original page.
• add entry for the new page to the directory

6
Example Insert 21, then 19, 15

• 21 10101
• 19 10011
• 15 01111

LOCAL DEPTH
Bucket A
GLOBAL DEPTH
2
2
1
Bucket B
00
13
1
7
5
21
01
2
10
Bucket C
10
11
DIRECTORY
15
19
7
DATA PAGES
7
Insert h(r)20 (Causes Doubling)
LOCAL DEPTH
Bucket A
GLOBAL DEPTH
2
2
Bucket B
1
5
21
13
00
01
2
10
Bucket C
10
11
2
Bucket D
15
7
19
of Bucket A)
8
Dynamic hashing Linear hashing

• Directory not needed
• Buckets during a round
• Set A buckets split during current round
• Set B buckets not split during current round
• Set C new buckets created during current round
• Next pointer to the beginning of set B next
  bucket to be split
• Not necessarily the overflowing bucket
• Heuristics for splitting and advancement of Next.

9
Main idea

• Use a family of hash functions h0, h1, h2, ...
• hi(key) h(key) mod(2iN)
• N initial buckets
• h is some hash function
• hi1 doubles the range of hi (similar to
  directory doubling)

10

• Algorithm proceeds in rounds. Current round
  number is Level.
• There are NLevel ( N 2Level) buckets at the
  beginning of a round
• Buckets 0 to Next-1 have been split Next to
  NLevel 1 have not been split yet this round.
• Round ends when all initial buckets have been
  split (i.e. Next NLevel).
• To start next round, increment level and reset
  Next.

11
LH Search Algorithm

• To find bucket for data entry r, find hLevel(r)
• If hLevel(r) gt Next (i.e., hLevel(r) is a bucket
  that hasnt been involved in a split this round)
  then r belongs in that bucket.
• Else, r could belong to bucket hLevel(r) or
  bucket hLevel(r) NLevel . Apply hLevel1(r) to
  find out.

12
Example Search 44 (11100), 9
(01001)
Level0, Next0, N4
h
h
0
1
00
000
001
01
10
010
011
11
PRIMARY
(This info is for illustration only!)
PAGES
13
Insert 43
Level0, Next 1, N4
14
Insert operation

• Find appropriate bucket
• If bucket is full
• Add overflow page and insert data entry.
• Split Next bucket and increment Next.
• Note This is likely NOT the bucket being
  inserted to!!!
• to split a bucket, create a new bucket and use
  hLevel1 to re-distribute entries.
• Since buckets are split round-robin, long
  overflow chains dont develop.

15
Example End of a Round
Level1, Next 0
Insert 37, 29, 22, 66, 34, 50
Level0, Next 3
PRIMARY
OVERFLOW
PAGES
h
PAGES
h
0
1
32
00
000
9
25
001
01
10
66
10
18
34
010
Next3
43
35
31
7
11
011
11
44
36
100
00
5
37
29
101
01
22
14
30
10
110
16
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.
• 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.

17
Summary (Contd.)

• Linear Hashing avoids directory by splitting
  buckets round-robin, and using overflow pages.
• Overflow pages not likely to be long.
• 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.

18
References

• Database textbook, e.g.,
• Database management systems, Ramakrishnan and
  Gehrke, Mcgraw Hill, 2nd edition.