1. Chapter 11: Indexing and Hashing
Basic Concepts
Ordered Indices
B+-Tree Index Files
Hashing
Static
Dynamic Hashing
More: bitmap indexing

2. Hashing
Static hashing
Dynamic hashing

3. Hashing
key  h(key)
<key>
Buckets
(typically 1
disk block) .

4. Example hash function Key = ‘x1 x2 … xn’ n byte character string
Have b buckets
h: add x1 + x2 + ….. xn
compute sum modulo b

5.  This may not be best function …
Good hash  Expected number of
function: keys/bucket is the
same for all buckets

6. Within a bucket: Do we keep keys sorted? Yes, if CPU time critical
& Inserts/Deletes not too frequent

7. Next: example to illustrate inserts, overflows, deletes
h(K)

8. EXAMPLE 2 records/bucket
INSERT:
h(a) = 1
h(b) = 2
h(c) = 1
h(d) = 0 1 2 3 d a c b e
h(e) = 1

9. Delete: e f c EXAMPLE: deletion a b d c d e f g 1 2 3 maybe move1 2 3 a d b d c c e
maybe move
“g” up f g

10. 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

11. How do we cope with growth?
Overflows and reorganizations
Dynamic hashing
Extensible
Linear

12. Extensible hashing: two ideas
(a) Use i of b bits output by hash function b
h(K) 
use i  grows over time….

13. (b) Use directory
h(K)[i ] to bucket . .

14. Example: h(k) is 4 bits; 2 keys/bucket
New directory 2 00 01 10 11
i = 1
i =
0001 1 1
1001 1
1100
1010
1100
Insert 1010

15. Example continued 2
0000
0111
0001
i = 2 00 01 10 11 1
0001
0111 2
1001
1010
Insert:
0111
0000 2
1100

16. Example continued i = 3 2 0000 0001 i = 2 2 0111 1001 1010 2 1001 1010
110
111 3
i =
0000 2
i =
0001 2 00 01 10 11
0111 2
1001
1010 2
1001
1010
Insert:
1001 2
1100

17. Extensible hashing: deletion
No merging of blocks
Merge blocks and cut directory if possible
(Reverse insert procedure)

18. Deletion example:
Run thru insert example in reverse!

19. 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) -

20. Advanced indexing
Multiple attributes
Bitmap indexing

21. 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?

22. Indices on Multiple Attributes
where branch-name = “PP” and balance = 1000
Suppose we have an index on combined search-key
(branch-name, balance).
BB,1000
CC,200
PP,800
PP,1500
AB,200
AA,2000
AA,2300
AA,2500
CC,200
DD,200
DD,300
PP,1000
PP,800
PP,1300
PP,1500
PP,1560
AB,200
AC,200
CC,200
PP,300

23. where branch-name = “PP” and balance < 1000
Suppose we have an index on combined search-key
(branch-name, balance).
where branch-name = “PP” and balance < 1000
search pp,0
BB,1000
CC,200
PP,800
PP,1500
AB,200
AA,2000
AA,2300
AA,2500
CC,200
DD,200
DD,300
PP,1000
PP,800
PP,1300
PP,1500
PP,1560
AB,200
AC,200
CC,200
PP,300
search pp,1000

24. where branch-name < “PP” and balance = 1000?
Suppose we have an index on combined search-key
(branch-name, balance).
NO!
where branch-name < “PP” and balance = 1000?
BB,1000
CC,200
PP,800
PP,1500
AB,200
AA,2000
AA,2300
AA,2500
CC,200
DD,200
DD,300
PP,1000
PP,800
PP,1300
PP,1500
PP,1560
AB,200
AC,200
CC,200
PP,300

25. Bitmap Indices
An index designed for multiple valued search keys

26. 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

27. 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?

28. 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?

29. Bitmap Indices Queries
Sample query: Males with income level L1
10010 AND = 10000
even faster!
What about the number of males with income level L1?

30. Bitmap Indices Queries
Queries are answered using bitmap operations
Intersection (and)
Union (or)
Complementation (not)