1. B-Trees, Part 2 Hash-Based Indexes
R&G Chapter 10
Lecture 10
The slides for this text are organized into chapters. This lecture covers Chapter 10.
Chapter 1: Introduction to Database Systems
Chapter 2: The Entity-Relationship Model
Chapter 3: The Relational Model
Chapter 4 (Part A): Relational Algebra
Chapter 4 (Part B): Relational Calculus
Chapter 5: SQL: Queries, Programming, Triggers
Chapter 6: Query-by-Example (QBE)
Chapter 7: Storing Data: Disks and Files
Chapter 8: File Organizations and Indexing
Chapter 9: Tree-Structured Indexing
Chapter 10: Hash-Based Indexing
Chapter 11: External Sorting
Chapter 12 (Part A): Evaluation of Relational Operators
Chapter 12 (Part B): Evaluation of Relational Operators: Other Techniques
Chapter 13: Introduction to Query Optimization
Chapter 14: A Typical Relational Optimizer
Chapter 15: Schema Refinement and Normal Forms
Chapter 16 (Part A): Physical Database Design
Chapter 16 (Part B): Database Tuning
Chapter 17: Security
Chapter 18: Transaction Management Overview
Chapter 19: Concurrency Control
Chapter 20: Crash Recovery
Chapter 21: Parallel and Distributed Databases
Chapter 22: Internet Databases
Chapter 23: Decision Support
Chapter 24: Data Mining
Chapter 25: Object-Database Systems
Chapter 26: Spatial Data Management
Chapter 27: Deductive Databases
Chapter 28: Additional Topics 1

2. Administrivia The new Homework 3 now available Due 1 week from Sunday
Homework 4 available the week after
Midterm exams available here

3. Review Last time discussed File Organization Unordered heap files
Sorted Files
Clustered Trees
Unclustered Trees
Unclustered Hash Tables
Indexes
B-Trees – dynamic, good for changing data, range queries
Hash tables – fastest for equality queries, useless for range queries

4. Review (2) For any index, 3 alternatives for data entries k*:
Data record with key value k
<k, rid of data record with search key value k>
<k, list of rids of data records with search key k>
Choice orthogonal to the indexing technique 2

5. Today: Indexes Composite Keys, Index-Only Plans B-Trees
details of insertion and deletion
Hash Indexes
How to implement with changing data sets

6. Inserting a Data Entry into a B+ Tree
Find correct leaf L.
Put data entry onto L.
If L has enough space, done!
Else, must split L (into L and a new node L2)
Redistribute entries evenly, copy up middle key.
Insert index entry pointing to L2 into parent of L.
This can happen recursively
To split index node, redistribute entries evenly, but push up middle key. (Contrast with leaf splits.)
Splits “grow” tree; root split increases height.
Tree growth: gets wider or one level taller at top. 6

7. Indexes with Composite Search Keys
Composite Search Keys: Search on a combination of fields.
Equality query: Every field value is equal to a constant value. E.g. wrt <sal,age> index:
age=20 and sal =75
Range query: Some field value is not a constant. E.g.:
age =20; or age=20 and sal > 10
Data entries in index sorted by search key to support range queries.
Lexicographic order, or
Spatial order.
Examples of composite key
indexes using lexicographic order.
11,80 11
12,10 12
name
age
sal
12,20 12
13,75
bob 12 10 13
<age, sal>
cal 11 80
<age>
joe 12 20
10,12
sue 13 75 10
20,12
Data records
sorted by name 20
75,13 75
80,11 80
<sal, age>
<sal>
Data entries in index
sorted by <sal,age>
Data entries
sorted by <sal> 13

8. Composite Search Keys
To retrieve Emp records with age=30 AND sal=4000, an index on <age,sal> would be better than an index on age or an index on sal.
Choice of index key orthogonal to clustering etc.
If condition is: 20<age<30 AND 3000<sal<5000:
Clustered tree index on <age,sal> or <sal,age> is best.
If condition is: age=30 AND 3000<sal<5000:
Clustered <age,sal> index much better than <sal,age> index!
Composite indexes are larger, updated more often. 20

9. Index-Only Plans <E.dno>
SELECT D.mgr
FROM Dept D, Emp E
WHERE D.dno=E.dno
<E.dno>
SELECT D.mgr, E.eid
FROM Dept D, Emp E
WHERE D.dno=E.dno
A number of queries can be answered without retrieving any tuples from one or more of the relations involved if a suitable index is available.
<E.dno,E.eid>
Tree index!
SELECT E.dno, COUNT(*)
FROM Emp E
GROUP BY E.dno
<E.dno>
SELECT E.dno, MIN(E.sal)
FROM Emp E
GROUP BY E.dno
<E.dno,E.sal>
Tree index!
<E. age,E.sal> or
<E.sal, E.age>
SELECT AVG(E.sal)
FROM Emp E
WHERE E.age=25 AND
E.sal BETWEEN 3000 AND 5000
Tree! 21

10. Index-Only Plans (Contd.)
Index-only plans are possible if the key is <dno,age> or we have a tree index with key <age,dno>
Which is better?
What if we consider the second query?
SELECT E.dno, COUNT (*)
FROM Emp E
WHERE E.age=30
GROUP BY E.dno
SELECT E.dno, COUNT (*)
FROM Emp E
WHERE E.age>30
GROUP BY E.dno

11. B-Trees: Insertion and Deletion
Find leaf where new record belongs
If leaf is full, redistribute*
If siblings too full, split, copy middle key up
If index too full, redistribute*
If index siblings full, split, push middle key up
Deletion
Find leaf where record exists, remove it
If leaf is less than 50% empty, redistribute*
If siblings too empty, merge, remove key above
If index node above too empty, redistribute*
If index siblings too empty, merge, move above key down

12. B-Trees: For Homework 2
Insertion
Find leaf where new record belongs
If leaf is full, redistribute*
If siblings too full, split, copy middle key up
If index too full, redistribute*
If index siblings full, split, push middle key up
Deletion
Find leaf where record exists, remove it
If leaf is less than 50% empty, redistribute*
If siblings too empty, merge, remove key above
If index node above too empty, redistribute*
If index siblings too empty, merge, move above key down
This means that after deletion, nodes will often be < 50% full

13. B-Trees: For Homework 2 (cont)
Splits
When splitting nodes, choose the middle key
If there are even number of keys, choose middle
Your code must Handle Duplicate Keys
We promise that there will never be more than 1 page of duplicate values.
Thus, when splitting, if the middle key is identical to the key to the left, you must find the closest splittable key to the middle.

14. B-Tree Demo

15. Hashing
Static and dynamic hashing techniques exist; trade-offs based on data change over time
Static Hashing
Good if data never changes
Extendable Hashing
Uses directory to handle changing data
Linear Hashing
Avoids directory, usually faster

16. 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)
h(key) mod N 2
key h
N-1
Primary bucket pages
Overflow pages 3

17. Static Hashing (Contd.)
Buckets contain data entries.
Hash fn works on search key field of record r. Must distribute values over range N-1.
h(key) = (a * key + b) usually works well.
a and b are constants; lots known about how to tune h.
Long overflow chains can develop and degrade performance.
Extendible and Linear Hashing: Dynamic techniques to fix this problem. 4

18. Extendible Hashing Situation: Bucket (primary page) becomes full.
Why not re-organize file by doubling # of buckets?
Reading and writing all pages is expensive!
Idea: Use directory of pointers to buckets,
double # of buckets by doubling the directory,
splitting just the bucket that overflowed!
Directory much smaller than file, doubling much cheaper.
No overflow pages!
Trick lies in how hash function is adjusted! 5

19. Extendible Hashing Details
Need directory with pointer to each bucket
Need hash function to incrementally double range
can just use increasingly more LSBs of h(key)
Must keep track of global “depth”
how many times directory doubled so far
Must keep track of local “depth”
how many time each bucket has been split

20. Example If h(r) = 5 = binary 101, it is in bucket pointed to by 01.
LOCAL DEPTH 2
Bucket A 4*
12*
32*
16*
Example
GLOBAL DEPTH 2 2
Bucket B 00 1* 5*
21*
13*
Directory is array of size 4.
To find bucket for r, take last `global depth’ # bits of h(r); we denote r by h(r).
If h(r) = 5 = binary 101, it is in bucket pointed to by 01. 01 10 2
Bucket C 11
10* 2
DIRECTORY
Bucket D
15* 7*
19*
DATA PAGES
Insert: If bucket is full, split it (allocate new page, re-distribute).
If necessary, double the directory. (As we will see, splitting a
bucket does not always require doubling; we can tell by
comparing global depth with local depth for the split bucket.) 6

21. Insert h(r)=20 (Causes Doubling)
LOCAL DEPTH 3
LOCAL DEPTH
Bucket A 4*
12*
32*
16*
GLOBAL DEPTH
32*
16*
Bucket A
GLOBAL DEPTH 2 2 3 2
Bucket B 00 1* 5*
21*
13*
000 1* 5*
21*
13*
Bucket B 01
001 10 2 2
010
Bucket C 11
10*
011
10*
Bucket C
100 2
DIRECTORY 2
101
Bucket D
15* 7*
19*
15* 7*
19*
110
Bucket D
111 3
DIRECTORY 4*
12*
20*
Bucket A2
(`split image'
of Bucket A) 7

22. Now Insert h(r)=9 (Causes Split Only)3 3
LOCAL DEPTH
LOCAL DEPTH
32*
16*
Bucket A
32*
16*
Bucket A
GLOBAL DEPTH
GLOBAL DEPTH 3 2 3 3
000 1* 5*
21*
13*
Bucket B
000 1* 9*
Bucket B
001
001 2 2
010
010
011
10*
Bucket C
011
10*
Bucket C
100
100
101 2
101 2
15* 7*
19*
Bucket D
15* 7*
19*
110
110
Bucket D
111
111 3 3
DIRECTORY 4*
12*
20*
Bucket A2 4*
12*
20*
Bucket A2
(`split image'
DIRECTORY
of Bucket A) 3 5*
21*
13*
Bucket B2 7

23. Points to Note
20 = binary 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 splitting bucket will also double directory
When does bucket split cause directory doubling?
If, before insert, local depth of bucket = global depth.
Insert causes local depth to become > global depth;
directory is doubled by copying it over and `fixing’ pointer to split image page.
(Use of least significant bits enables efficient doubling via copying of directory!) 8

24. Directory Doubling Why use least significant bits in directory?
Allows for doubling via copying! 2 2
16* 3 4*
12*
32* 4*
12*
32*
16*
000 2 2
001 2 00 1* 5*
21*
13*
010 1* 5*
21*
13* 01
011 10 2
100 2
10* 11
10*
101
110 2 2
111
15* 7*
19*
15* 7*
19*

25. Deletion
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. 2 1 4*
12*
32*
16* 4*
12*
32*
16* 2 2
Delete 10 2 2 00 1* 5*
21*
13* 00 1* 5*
21*
13* 01 01 10 2 10 11
10* 11 2 2
15*
15*

26. Deletion (cont)
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. 1 2 4*
12*
32*
16* 1 00 4*
12*
32*
16*
Delete 15 01 1 10 1* 5*
21*
13* 2 2 11 00 1* 5*
21*
13* 01 10 1 1 11 4*
12*
32*
16* 2 1 1
15* 1* 5*
21*
13*

27. 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 contains 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, and, if the distribution of hash values is skewed, directory can grow large.
Biggest problem:
Multiple entries with same hash value cause problems!
If bucket already full of same hash value, will keep doubling forever! So must use overflow buckets if dups. 9

28. Linear Hashing
This is another dynamic hashing scheme, an alternative to Extendible Hashing.
LH handles the problem of long overflow chains without using a directory, and handles duplicates.
Idea: Use a family of hash functions h0, h1, h2, ...
hi(key) = h(key) mod(2iN); N = initial # buckets
h is some hash function (range is not 0 to N-1)
If N = 2d0, for some d0, hi consists of applying h and looking at the last di bits, where di = d0 + i.
hi+1 doubles the range of hi (similar to directory doubling) 10

29. Linear Hashing Example
Let’s start with N = 4 Buckets
Start at “round” 0, “next” 0, have 2round buckets
Each time any bucket fills, split “next” bucket
If (O ≤ hround(key) < Next), use hround+1(key) instead
Start
Add 9
Next 4*
12*
32*
16*
32*
16*
Add 20 1* 5*
21*
13*
Next 1* 5*
21*
13* 9*
10*
10*
32*
16*
Nround
15*
Nround
15*
Next 1* 5*
21*
13* 9* 4*
12*
10*
Nround
15* 4*
12*
20*

30. Linear Hashing Example (cont)
Overflow chains do exist, but eventually get split
Instead of doubling, new buckets added one-at-a-time
Add 6
Add 17
32*
16*
32*
16*
Next 1* 5*
21*
13* 9* 1*
13*
10* 6*
Next
10* 6*
Nround
15*
Nround
15* 4*
12*
20* 4*
12*
20* 5*
21* 9*

31. Linear Hashing (Contd.)
Directory avoided in LH by using overflow pages, and choosing bucket to split round-robin.
Splitting proceeds in `rounds’. Round ends when all NR initial (for round R) buckets are split. Buckets 0 to Next-1 have been split; Next to NR yet to be split.
Current round number also called Level.
Search: To find bucket for data entry r, find hround(r):
If hround(r) in range `Next to NR’ , r belongs here.
Else, r could be hround(r) or hround(r) + NR;
must apply hround+1(r) to find out. 11

32. Overview of LH File In the middle of a round. h Level
Buckets split in this round:
Bucket to be split If h (
search key value )
Level
Next
is in this range, must use h
Level+1 (
search key value )
Buckets that existed at the
to decide if entry is in
beginning of this round:
`split image' bucket.
this is the range of h
Level
`split image' buckets:
created (through splitting
of other buckets) in this round 11

33. Linear Hashing (Contd.)
Insert: Find bucket by applying hround / hround+1:
If bucket to insert into is full:
Add overflow page and insert data entry.
(Maybe) Split Next bucket and increment Next.
Can choose any criterion to `trigger’ split.
Since buckets are split round-robin, long overflow chains don’t develop!
Doubling of directory in Extendible Hashing is similar; switching of hash functions is implicit in how the # of bits examined is increased. 12

34. Another Example of Linear Hashing
On split, hLevel+1 is used to re-distribute entries.
Round=0, N=4
Round=0 h h
PRIMARY h h
PRIMARY
OVERFLOW 1
Next=0
PAGES 1
PAGES
PAGES
32*
44*
36*
32*
000 00
000 00
Next=1
Data entry r 9*
25* 5*
with h(r)=5 9*
25* 5*
001 01
001 01
14*
18*
10*
30*
14*
18*
10*
30*
010 10
Primary
010 10
bucket page
31*
35* 7*
11*
31*
35* 7*
11*
43*
011 11
011 11
(This info
is for illustration
only!)
(The actual contents
of the linear hashed
file)
100 00
44*
36* 13

35. Example: End of a Round Round=1 PRIMARY OVERFLOW h 1 h PAGES PAGES
PAGES
PAGES
Next=0
Round=0
000 00
32*
PRIMARY
OVERFLOW h
PAGES 1 h
PAGES
001 01 9*
25*
000 00
32*
010 10
66*
18*
10*
34*
50*
001 01 9*
25*
011 11
43*
35*
11*
010 10
66*
18*
10*
34*
Next=3
100 00
44*
36*
011 11
31*
35* 7*
11*
43*
101 11 5*
37*
29*
100 00
44*
36*
101 01 5*
37*
29*
110 10
14*
30*
22*
14*
30*
22*
31* 7*
110 10
111 11 14

36. LH Described as a Variant of EH
The two schemes are actually quite similar:
Begin with an EH index where directory has N elements.
Use overflow pages, split buckets round-robin.
First split is at bucket 0. (Imagine directory being doubled at this point.) But elements <1,N+1>, <2,N+2>, ... are the same. So, need only create directory element N, which differs from 0, now.
When bucket 1 splits, create directory element N+1, etc.
So, directory can double gradually. Also, primary bucket pages are created in order. If they are allocated in sequence too (so that finding i’th is easy), we actually don’t need a directory! Voila, LH. 15

37. 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. (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. 16

38. Summary (Contd.)
Linear Hashing avoids directory by splitting buckets round-robin, and using overflow pages.
Overflow pages not likely to be long.
Duplicates handled easily.
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.
For hash-based indexes, a skewed data distribution is one in which the hash values of data entries are not uniformly distributed! 17