Download Chapter 18

Copyright © 2007 Ramez Elmasri and Shamkant B. Navathe
Slide 14- 1
Chapter Outline


Indexes as additional auxiliary access structure
Types of Single-level Ordered Indexes (ordered
files)






Primary Indexes
Clustering Indexes
Secondary Indexes
Multilevel Indexes
Dynamic Multilevel Indexes Using B-Trees
and B+-Trees
Indexes on Multiple Keys
Copyright © 2007 Ramez Elmasri and Shamkant B. Navathe
Indexes as Access Paths




A single-level index is an auxiliary file that makes
it more efficient to search for a record in the data
file.
The index is usually specified on one field of the
file (although it could be specified on several
fields)
One form of an index is a file of entries <field
value, pointer to record>, which is ordered by
field value
The index is called an access path on the field.
Copyright © 2007 Ramez Elmasri and Shamkant B. Navathe
Indexes as Access Paths (contd.)



The index file usually occupies considerably less disk
blocks than the data file because its entries are much
smaller
A binary search on the index yields a pointer to the file
record
Indexes can also be characterized as dense or sparse


A dense index has an index entry for every search key
value (and hence every record) in the data file.
A sparse (or nondense) index, on the other hand, has
index entries for only some of the search values
Copyright © 2007 Ramez Elmasri and Shamkant B. Navathe
Indexes Vs non-Index as Access
Paths (example)


Example: Given the following data file EMPLOYEE(NAME, SSN, ADDRESS,
JOB, SAL, ... )
Suppose that:


block size B=512 bytes
r=30000 records
Then, we get:



record size R=150 bytes
blocking factor Bfr= floor(B div R)= 512 div 150= 3 records/block
number of file blocks b= (r/Bfr)= (30000/3)= 10000 blocks
For an index on the SSN field, assume the field size VSSN=9 bytes, assume
the record pointer size PR=7 bytes. Then:





index entry size RI=(VSSN+ PR)=(9+7)=16 bytes
index blocking factor BfrI= B div RI= 512 div 16= 32 entries/block
number of index blocks b= (r/ BfrI)= (30000/32)= 938 blocks
binary search needs log2bI= log2938= 10 block accesses
This is compared to an average linear search cost of:


(b/2)= 30000/2= 15000 block accesses
If the file records are ordered, the binary search cost would be:

log2b= log230000= 15 block accesses
Copyright © 2007 Ramez Elmasri and Shamkant B. Navathe
Types of Single-Level Indexes

Primary Index







Defined on an ordered data file
The data file is ordered on a key field
Includes one index entry for each block in the data file; the
index entry has the key field value for the first record in the
block, which is called the block anchor
A similar scheme can use the last record in a block.
A primary index is a nondense (sparse) index, since it
includes an entry for each disk block of the data file and the
keys of its anchor record rather than for every search value.
Disadvantage - as any ordered file – insertion/deletion of
records.
move records to make space for the new record + change
some index entries
Copyright © 2007 Ramez Elmasri and Shamkant B. Navathe
Primary index on the ordering key field
Copyright © 2007 Ramez Elmasri and Shamkant B. Navathe
Types of Single-Level Indexes

Clustering Index





Defined on an ordered data file
The data file is ordered on a non-key field unlike primary
index
does not have distinct value for each record of the data file
Includes one index entry for each distinct value of the field;
the index entry points to the first data block that contains
records with that field value.
It is another example of nondense index where Insertion
and Deletion is relatively straightforward with a clustering
index.
Copyright © 2007 Ramez Elmasri and Shamkant B. Navathe
A Clustering Index Example

FIGURE 14.2
A clustering index on the
DEPTNUMBER ordering
non-key field of an
EMPLOYEE file.
Copyright © 2007 Ramez Elmasri and Shamkant B. Navathe
Another Clustering Index Example
Copyright © 2007 Ramez Elmasri and Shamkant B. Navathe
Types of Single-Level Indexes

Secondary Index



A secondary index provides a secondary means of
accessing a file for which some primary access already
exists.
The secondary index may be on a field which is a candidate
key and has a unique value in every record, or a non-key
with duplicate values.
The index is an ordered file with two fields.




The first field is of the same data type as some non-ordering
field of the data file that is an indexing field.
The second field is either a block pointer or a record pointer.
There can be many secondary indexes (and hence, indexing
fields) for the same file.
Includes one entry for each record in the data file; hence, it
is a dense index
Copyright © 2007 Ramez Elmasri and Shamkant B. Navathe
Example of a Dense Secondary Index
Copyright © 2007 Ramez Elmasri and Shamkant B. Navathe
An Example of a Secondary Index
Copyright © 2007 Ramez Elmasri and Shamkant B. Navathe
Secondary Index - Example






Example 2. Consider the data file has r = 30,000 fixed-length records of size R = 100 bytes
stored on a disk with block size B = 1024 bytes.
The file has b =3000 blocks. Suppose we want to search for a record with a specific value
for the secondary key—a non-ordering key field of the file that is V = 9 bytes long.
Without the secondary index, to do a linear search on the file would require b/2 = 3000/2 =
1500 block accesses on the average.
secondary index on nonordering key field of the file - block pointer P = 6 bytes, then each
index entry, Ri = V+P = (9 + 6) = 15 bytes. Blocking factor for the index, bfri = ⎣(B/Ri)⎦ =
⎣(1024/15)⎦ = 68 entries per block.
In a dense secondary index, the total number of index entries ri = the number of records in
the data file, which is 30,000. The #of blocks needed for the index, bi = ⎡(ri /bfri)⎤ =
⎡(3000/68)⎤ = 442 blocks.
A binary search on this secondary index needs ⎡(log2bi)⎤ = ⎡(log2442)⎤ = 9 block accesses.
To search for a record using the index, we need an additional block access to the data file
for a total of 9 + 1 = 10 block accesses—a vast improvement over the 1500 block accesses
needed on the average for a linear search, but slightly worse than the 7 block accesses
required for the primary index. This difference arose because the primary index was
nondense and hence shorter, with only 45 blocks in length.
Copyright © 2007 Ramez Elmasri and Shamkant B. Navathe
Slide 14- 14
Properties of Index Types
Copyright © 2007 Ramez Elmasri and Shamkant B. Navathe
Multi-Level Indexes

Because a single-level index is an ordered file, we can
create a primary index to the index itself;






In this case, the original index file is called the first-level
index and the index to the index is called the second-level
index.
We can repeat the process, creating a third, fourth, ..., top
level until all entries of the top level fit in one disk block
A multi-level index can be created for any type of firstlevel index (primary, secondary, clustering) as long as the
first-level index consists of more than one disk block
If first level has r1 entries and blocking factor bfri = f0
r2 = ceiling (r1/f0), r3 = ceiling(r2/f0)…
Total levels needed t = ceiling(logf0(r1)) or ⎡(logfo(r1))⎤
Copyright © 2007 Ramez Elmasri and Shamkant B. Navathe
A Two-level Primary Index
Copyright © 2007 Ramez Elmasri and Shamkant B. Navathe
Multi-Level Indexes

Such a multi-level index is a form of search tree



However, insertion and deletion of new index entries is a severe problem
because every level of the index is an ordered file.
Thus far - indexing uses an ordered index file, binary search (log2bi)
to locate pointers to a disk block/record, algorithm reduces the part of
the index file that we continue to search by a factor of 2.
multilevel index - reduce the search by bfri (blocking factor for the
index which is fan-out (fo) ), which is larger than 2. search space
reduced much faster. Instead of two halves at each step, record
search space is divided into n-ways (where n = fan-out), requires
approximately (logfobi) block accesses, which is a substantially
smaller. In most cases, the fan-out is much larger than 2.
Copyright © 2007 Ramez Elmasri and Shamkant B. Navathe
Slide 14- 18
Multi-Level Index - Example
Example 3. Suppose that the dense secondary index of Example 2 is converted into
a multilevel index. We calculated the index blocking factor bfri = 68 index entries
per block, which is also the fan-out fo for the multilevel index;
#of firstlevel blocks b1 = 442 blocks was also calculated.
#of second-level blocks, b2 = ⎡(b1/fo)⎤ = ⎡(442/68)⎤ = 7 blocks
#of third-level blocks, b3 = ⎡(b2/fo)⎤ = ⎡(7/68)⎤ = 1 block.
Hence, the third level is the top level of the index, and t = 3.
To access a record by searching the multilevel index, we must access one block at each
level plus one block from the data file, so we need t + 1 = 3 + 1 = 4 block accesses.
Compare this to Example 2, where 10 block accesses were
needed when a single-level index and binary search were used.
Copyright © 2007 Ramez Elmasri and Shamkant B. Navathe
Slide 14- 19
A Node in a Search Tree with Pointers to
Subtrees below It

A search tree of order p is a tree such that each node contains at most p − 1
search values and p pointers in the order <P1, K1, P2, K2, ..., Pq−1, Kq−1,
Pq>, where q ≤ p. Each Pi is a pointer to a child node (or a NULL pointer),
and each Ki is a search value
Copyright © 2007 Ramez Elmasri and Shamkant B. Navathe
Slide 14- 20
FIGURE 14.9
A search tree of order p = 3.
Copyright © 2007 Ramez Elmasri and Shamkant B. Navathe
Slide 14- 21
Dynamic Multilevel Indexes Using B-Trees
and B+-Trees

Most multi-level indexes use B-tree or B+-tree data
structures because of the insertion and deletion problem




This leaves space in each tree node (disk block) to allow for
new index entries
These data structures are variations of search trees that
allow efficient insertion and deletion of new search values.
In B-Tree and B+-Tree data structures, each node
corresponds to a disk block
Each node is kept between half-full and completely full
Copyright © 2007 Ramez Elmasri and Shamkant B. Navathe
Slide 14- 22
Dynamic Multilevel Indexes Using B-Trees
and B+-Trees (contd.)

An insertion into a node that is not full is quite
efficient




If a node is full the insertion causes a split into two
nodes
Splitting may propagate to other tree levels
A deletion is quite efficient if a node does not
become less than half full
If a deletion causes a node to become less than
half full, it must be merged with neighboring
nodes
Copyright © 2007 Ramez Elmasri and Shamkant B. Navathe
Slide 14- 23
Difference between B-tree and B+-tree



In a B-tree, pointers to data records exist at all
levels of the tree
In a B+-tree, all pointers to data records exists at
the leaf-level nodes
A B+-tree can have less levels (or higher capacity
of search values) than the corresponding B-tree
Copyright © 2007 Ramez Elmasri and Shamkant B. Navathe
Slide 14- 24
B-tree Structures
Copyright © 2007 Ramez Elmasri and Shamkant B. Navathe
Slide 14- 25
B-tree Example
Example 4. Suppose that the search field is a nonordering key field, and we construct
a B-tree on this field with p = 23 pointers. Assume that each node of the B-tree is 69
percent full. Each node, on the average, will have p * 0.69 = 23 * 0.69 or approximately
16 pointers and, hence, 15 search key field values. The average fan-out fo = 16.
We can start at the root and see how many values and pointers can exist, on the
average, at each subsequent level:
Root: 1 node 15 key entries 16 pointers
Level 1: 16 nodes 240 key entries 256 pointers
Level 2: 256 nodes 3840 key entries 4096 pointers
Level 3: 4096 nodes 61,440 key entries
At each level, we calculated the number of key entries by multiplying the total number
of pointers at the previous level by 15, the average number of entries in each
node. Hence, for the given block size, pointer size, and search key field size, a two level
B-tree holds 3840 + 240 + 15 = 4095 entries on the average; a three-level B-tree
holds 65,535 entries on the average.
Copyright © 2007 Ramez Elmasri and Shamkant B. Navathe
Slide 14- 26
The Nodes of a B+-tree

B+-tree


(a) Internal node of a B+-tree with q –1 search values.
(b) Leaf node of a B+-tree with q – 1 search values and q – 1 data pointers.
Copyright © 2007 Ramez Elmasri and Shamkant B. Navathe
Slide 14- 27
B+-tree Example
Example 5. To calculate the order p of a B+-tree, suppose that the search key
field is V = 9 bytes long, the block size is B = 512 bytes, a record pointer is Pr = 7
bytes, and a block pointer is P = 6 bytes.
An internal node of the B+-tree can have up to p tree pointers and p – 1 search
field values; these must fit into a single block. Hence,
we have:
(p * P) + ((p – 1) * V) ≤ B
(P * 6) + ((P − 1) * 9) ≤ 512
(15 * p) ≤ 521
We can choose p to be the largest value satisfying the above inequality, which
gives
p = 34. This is larger than the value of 23 for the B-tree
Copyright © 2007 Ramez Elmasri and Shamkant B. Navathe
Slide 14- 28
An Example of an Insertion in a B+-tree
Copyright © 2007 Ramez Elmasri and Shamkant B. Navathe
Slide 14- 29
An Example of a Deletion in a B+-tree
Copyright © 2007 Ramez Elmasri and Shamkant B. Navathe
Slide 14- 30
Summary

Types of Single-level Ordered Indexes






Primary Indexes
Clustering Indexes
Secondary Indexes
Multilevel Indexes
Dynamic Multilevel Indexes Using B-Trees
and B+-Trees
Indexes on Multiple Keys
Copyright © 2007 Ramez Elmasri and Shamkant B. Navathe
Slide 14- 31