Hash-Based Indexes

1
Hash-Based Indexes

• Chapter 11

2
Introduction Hash-based Indexes

• Best for equality selections.
• Cannot support range searches.
• Static and dynamic hashing techniques exist
• Trade-offs similar to ISAM vs. B trees.

3
Static Hashing

• h(k) mod N bucket to which data entry with key
  k belongs. (N of buckets)

0
h(key) mod N
1
key
h
N-1
Primary bucket pages
Overflow pages
4
Static Hashing h(k) mod N

• primary pages fixed (N of buckets)
• allocated sequentially
• never de-allocated
• overflow pages if needed.

0
h(key) mod N
2
key
h
N-1
Primary bucket pages
Overflow pages
5
Static Hashing

• h(k) mod N bucket to which data entry with key
  k belongs with N of buckets
• Hash function works on search key of record r.
• h() must distribute values over range 0 ...
  N-1.
• For example, h(key) (a key b)
• a and b are constants
• lots known about how to tune h.

6
Static Hashing Cons

• Primary pages fixed space ? static structure.
• Fixed bucket is the problem
• Rehashing can be done ? Not good for search.
• In practice, instead use overflow chains.
• Long overflow chains can develop and degrade
  performance.
• Solution Employ dynamic techniques to fix this
  problem
• Extendible hashing, or
• Linear Hashing

7
Extendible Hashing

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

8
Example
LOCAL DEPTH
2
Bucket A

• Directory array4
• To find bucket for r, take last global depth
  bits of function h(r)

16
4
12
32
GLOBAL DEPTH
2
2
Bucket B
00
13
1
21
5
01
2
10
Bucket C
10
11
2
DIRECTORY
Bucket D
15
7
19
DATA PAGES
9
Example
LOCAL DEPTH
2
Bucket A

• If h(r) 5 101,
• it is in bucket pointed to by 01.
• If h(r) 4 100,
• it is in bucket pointed to by 00.

16
4
12
32
GLOBAL DEPTH
2
2
Bucket B
00
13
1
21
5
01
2
10
Bucket C
10
11
2
DIRECTORY
Bucket D
15
7
19
DATA PAGES
10
Insertion
2
LOCAL DEPTH
Bucket A
16
4
12
32
GLOBAL DEPTH
2
2
Bucket B
13
00
1
21
5

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

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

• Splitting may
• double the directory, or
• simply link in a new page.
• To tell what to do
• Compare global depth with local depth for split
  bucket.

11
Insert h(r) 6 binary 110
12
Insert h(r) 6 binary 110
13
Insert h(r)20 binary 10100
14
Insert h(r)20
Split Bucket A into two buckets A1 and A2.
3
20
Bucket A2
4
12
3
Bucket A1
32
16
15
Insert h(r)20
Bucket A
LOCAL DEPTH
2
16
4
12
32
GLOBAL DEPTH
2
2
Bucket B
13
00
1
21
5
01
2
10
Bucket C
10
11
Bucket D
2
DIRECTORY
15
7
19
DATA PAGES
16
Insert h(r)20
2
LOCAL DEPTH
3
LOCAL DEPTH
Bucket A
16
32
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
2
3
Bucket A2
20
4
12
DIRECTORY
20
12
Bucket A2
4
(split image'
of Bucket A)
(split image'
of Bucket A)
17
Points to Note

• 20 binary 10100.
• Last 2 bits (00) tell us r belongs in A (or not
  A).
• Last 3 bits needed to tell A1 or A2
• Using bits
• 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 an entry belongs to this bucket.

18
More Points to Note

• Bits
• 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 an entry belongs to this bucket.
• When does bucket split cause directory doubling?
• Before insert, local depth of bucket global
  depth.
• Insert causes local depth to become lt global
  depth
• Directory is doubled by copying it over and
  fixing pointer to split image page.

19
Extendible Hashing Delete

• 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.

20
Comments on Extendible Hashing

• If directory fits in memory,
  then equality search answered with one disk
  access else two.
• 100MB file, 100 bytes/rec, 4K pages contain
  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.
• If the distribution of hash values is skewed,
  directory can grow large.

21
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 full bucket when new data to be added
• Directory to keep track of buckets, doubles
  periodically