Chapter 11: Indexing and
Hashing

Indexing
Basic Concepts
 Ordered Indices
 B+-Tree Index Files


Hashing
Static
 Dynamic Hashing

1
Hashing


Static hashing
Dynamic hashing
2
Hashing
key  h(key)
<key>
.
.
.
Buckets
(typically 1
disk block)
3
Example hash function



Key = ‘x1 x2 … xn’ n byte character string
Have b buckets
h: add x1 + x2 + ….. xn

compute sum modulo b
4
 This may not be best function …
Good hash  Expected number of
function:
keys/bucket is the
same for all buckets
5
Within a bucket:


Do we keep keys sorted?
Yes, if CPU time critical
& Inserts/Deletes not too frequent
6
Next: example to illustrate
inserts, overflows, deletes
h(K)
7
EXAMPLE 2 records/bucket
INSERT:
h(a) = 1
h(b) = 2
h(c) = 1
h(d) = 0
0
d
1
a
c
2
b
e
3
h(e) = 1
8
EXAMPLE: deletion
Delete:
e
f
c
0
a
1
b
c d
e
2
3
f
g
d
maybe move
“g” up
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Rule of thumb:



Try to keep space utilization
between 50% and 80%
Utilization = # keys used
total # keys that fit
If < 50%, wasting space
If > 80%, overflows significant
depends on how good hash
function is & on # keys/bucket
10
How do we cope with growth?


Overflows and reorganizations
Dynamic hashing


Extensible
Linear
11
Extensible hashing: two ideas
(a) Use i of b bits output by hash function
b
h(K) 
00110101
use i  grows over time….
12
(b) Use directory
h(K)[i ]
.
.
.
to bucket
.
.
.
13
Example: h(k) is 4 bits; 2 keys/bucket
i= 1
1
0001
00
01
1 2
1001
1010 1100
Insert 1010
i= 2
1 2
1100
10
11
New directory
14
Example continued
i= 2
00
01
10
11
Insert:
0111
2
0000
0001
1 2
0001 0111
0111
2
1001
1010
2
1100
0000
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Example continued
0000 2
i= 2
0001
00
0111 2
01
10
11
Insert:
1001
1001 3
1001
1010 1001 2 3
1010
1100 2
i=3
000
001
010
011
100
101
110
111
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Extensible hashing: deletion


No merging of blocks
Merge blocks
and cut directory if possible
(Reverse insert procedure)
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Deletion example:

Run thru insert example in reverse!
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Summary
Extensible hashing
+ Can handle growing files
- with less wasted space
- with no full reorganizations
- Indirection
(Not bad if directory in memory)
-
Directory doubles in size
(Now it fits, now it does not)
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Advanced indexing


Multiple attributes
Bitmap indexing
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Multiple-Key Access


Use multiple indices for certain types of
queries.
Example:
select account-number
from account
where branch-name = “Perryridge” and balance =
1000

Possible strategies?
21
Indices on Multiple Attributes

where branch-name = “PP” and balance =
1000 Suppose we have an index on combined search-key
PP,1500
PP,1560
CC,200
PP,800
PP,1500
PP,800
PP,1000
PP,1300
CC,200
PP,300
CC,200
DD,200
DD,300
AB,200
AC,200
AA,2000
AA,2300
AA,2500
AB,200
BB,1000
(branch-name, balance).
22
Suppose we have an index on combined search-key
(branch-name, balance).

where branch-name = “PP” and balance <
1000
search pp,1000
PP,1500
PP,1560
CC,200
PP,800
PP,1500
PP,800
PP,1000
PP,1300
CC,200
PP,300
CC,200
DD,200
DD,300
AB,200
AC,200
AA,2000
AA,2300
AA,2500
AB,200
BB,1000
search pp,0
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PP,1500
PP,1560
PP,800
PP,1000
PP,1300
CC,200
PP,300
CC,200
DD,200
DD,300
CC,200
PP,800
PP,1500
BB,1000
AB,200

AB,200
AC,200
AA,2000
AA,2300
AA,2500
Suppose we have an index on combined search-key
(branch-name, balance).
NO!
where branch-name < “PP” and balance =
1000?
24
Bitmap Indices

An index designed for multiple valued search
keys
25
Bitmap Indices (Cont.)
The income-level value of record 3 is L1
Bitmap(size = table size)
Unique values
of gender
Unique values
of income-level
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Bitmap Indices (Cont.)

Some properties of bitmap indices



Number of bitmaps for each attribute?
Size of each bitmap?
When is the bitmap matrix sparse and what attributes are good
for bitmap indices?
27
Bitmap Indices (Cont.)

Bitmap indices generally very small compared
with relation size




E.g. if record is 100 bytes, space for a single bitmap
is 1/800 of space used by relation.
If number of distinct attribute values is 8, bitmap is
only 1% of relation size
What about insertion?
Deletion?
28
Bitmap Indices Queries
Sample query: Males with income level L1
10010 AND 10100 = 10000
even faster!
What about the number of males with income level L1?
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Bitmap Indices Queries

Queries are answered using bitmap operations



Intersection (and)
Union (or)
Complementation (not)
30