HashBased Indexes

1
Hash-Based Indexes

• Chapter 11

2
Introduction

• Hash-based indexes are 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

• 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
4
Static Hashing

• Buckets contain data entries (alternative 1 to 3
  possible).
• Hash function works on search key field of record
  r.
• h(k) mod N bucket to which data entry with key
  k belongs with N of buckets
• h() must distribute values over range 0 ... N-1.
• h(key) (a key b) works well.
• a and b are constants
• lots known about how to tune h.

5
Static Hashing Cons

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

6
Extendible Hashing

• Situation 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
• splitting just the bucket that overflowed!
• Discussion
• Directory much smaller than file, so doubling
  it is much cheaper. Only one page of data
  entries is split.
• No overflow pages ever.
• Trick lies in how hash function is adjusted!

7
Example
LOCAL DEPTH
2
Bucket A

• Directory is array of size 4.
• To find bucket for r, take last global depth
  bits of function h(r)
• Examples
• If h(r) 5 binary 101, it is in bucket
  pointed to by 01.
• If h(r) 4 binary 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
8
ExampleInsertion
2
LOCAL DEPTH
Bucket A
16
4
12
32
GLOBAL DEPTH
2
2
Bucket B
13
00
1
21
5

• To find bucket for r, take last global depth
  bits of h(r)
• If h(r) 5 binary 101, r 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).

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

9
Insert h(r) 6 binary 110
10
Insert h(r)20
11
Insert h(r)20
12
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)
13
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 an entry belongs to this bucket.

14
More Points to Note

• 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 gt global
  depth
• directory is doubled by copying it over and
  fixing pointer to split image page.
• Why not first several bits?
• Use of least significant bits enables efficient
  doubling via copying of directory!

15
Directory Doubling

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

6 110
6 110
3
3
000
000
001
100
2
2
010
010
00
00
1
1
011
110
6
01
10
0
0
100
001
6
6
10
01
1
1
101
101
6
11
11
6
6
110
011
111
111
vs.
Least Significant
Most Significant
16
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.

17
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 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.

18
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.
• Large Number of 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.