Transcript Indexing Structure for File

Chapter 18
Indexing
Structures for
Files
Copyright © 2011 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Indexes as Access Paths
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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 © 2011 Ramez Elmasri and Shamkant Navathe
Indexes as Access Paths (cont.)
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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
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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 © 2011 Ramez Elmasri and Shamkant Navathe
Indexes as Access Paths (cont.)
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Example: Given the following data file EMPLOYEE(NAME, SSN,
ADDRESS, JOB, SAL, ... )
Suppose that:
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record size R=100 bytes
block size B=1024bytes r=30000
records
Then, we get:
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blocking factor Bfr= B div R= 1024 / 150= 10 records/block
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number of file blocks b= (r/Bfr)= (30000/10)= 3000 blocks
For an index on the SSN field, assume the field size VSSN=9 bytes,
assume the record pointer size PR=6 bytes. Then:
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index entry size RI=(VSSN+ PR)=(9+6)=15 bytes
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index blocking factor BfrI= B div RI= 1024 / 15= 68 entries/block
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number of index blocks b= (r/ BfrI)= (30000/68)= 45 blocks
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binary search needs log2bI= log245= 6 block accesses
 We need one more block access to the data file. Total we need 7
block accesses
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This is compared to an average linear search cost of:
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(b/2)= 30000/2= 15000 block accesses
If the file records are ordered, the binary search cost would be:
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log2b= log23000= 12 block accesses
Copyright © 2011 Ramez Elmasri and Shamkant Navathe
Types of Single-Level Indexes
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Primary Index
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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.
A major problem with a primary index—as with any ordered
file—is insertion and deletion of records.
Copyright © 2011 Ramez Elmasri and Shamkant Navathe
Primary Index
on the Ordering
Key Field
Copyright © 2011 Ramez Elmasri and Shamkant Navathe
Types of Single-Level Indexes
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Clustering Index
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Defined on an ordered data file
The data file is ordered on a non-key field unlike primary index,
which requires that the ordering field of the data file have a distinct
value for each record.
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.
A clustering index is another example of a nondense index
because it has an entry for every distinct value of the indexing
field
Record insertion and deletion still cause problems because the
data records are physically ordered. To alleviate the problem of
insertion, it is common to reserve a whole block (or a cluster of
contiguous blocks) for each value of the clustering field; all
records with that value are placed in the block (or block cluster).
This makes insertion and deletion relatively straightforward.
Copyright © 2011 Ramez Elmasri and Shamkant Navathe
A Clustering
Index
Example
Copyright © 2011 Ramez Elmasri and Shamkant Navathe
Another
Clustering
Index
Example
Copyright © 2011 Ramez Elmasri and Shamkant Navathe
Types of Single-Level Indexes
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Secondary Index
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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.
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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 © 2011 Ramez Elmasri and Shamkant Navathe
Example of
a Dense
Secondary
Index
Copyright © 2011 Ramez Elmasri and Shamkant Navathe
Example of
a Secondary
Index
Copyright © 2011 Ramez Elmasri and Shamkant Navathe
Properties of Index Types
Copyright © 2011 Ramez Elmasri and Shamkant Navathe
Multi-Level Indexes
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Because a single-level index is an ordered file, we can create a
primary index to the index itself;
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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 first-level
index (primary, secondary, clustering) as long as the first-level
index consists of more than one disk block
The value bfri is called the fan-out of the multilevel index, and
usually is referred by the symbol fo.
 Each level reduces the number of entries at the previous level
by a factor of fo.
 a multilevel index with r1 first-level entries will have
approximately t levels, where t = ⎡(logfo(r1))⎤.
Copyright © 2011 Ramez Elmasri and Shamkant Navathe
A Two-Level
Primary Index
Copyright © 2011 Ramez Elmasri and Shamkant Navathe
Multi-Level Indexes
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Such a multi-level index is a form of search tree
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However, insertion and deletion of new index
entries is a severe problem because every level of
the index is an ordered file.
Copyright © 2011 Ramez Elmasri and Shamkant Navathe
A Node in a Search Tree with Pointers to
Subtrees Below It
Copyright © 2011 Ramez Elmasri and Shamkant Navathe
Copyright © 2011 Ramez Elmasri and Shamkant Navathe
Dynamic Multilevel Indexes Using BTrees and B+-Trees
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Most multi-level indexes use B-tree or B+-tree data
structures because of the insertion and deletion problem
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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 © 2011 Ramez Elmasri and Shamkant Navathe
Dynamic Multilevel Indexes Using BTrees and B+-Trees (cont.)
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An insertion into a node that is not full is quite
efficient
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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 © 2011 Ramez Elmasri and Shamkant Navathe
Difference between B-tree and B+-tree
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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 © 2011 Ramez Elmasri and Shamkant Navathe
B-Trees.
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The B-tree is always balanced and that the space wasted by deletion, if any, never becomes
excessive.
A B-tree of order p, when used as an access structure on a key field to search for records in
a data file, can be defined as follows:
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1 . Each internal node in the B-tree is of the form
<P1, <K1, Pr1>, P2, <K2, Pr2>, ..., <Kq–1, Prq–1>, Pq>
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where q ≤ p. Each Pi is a tree pointer—a pointer to another node in the B-tree. Each Pri is a
data pointer—a pointer to the record whose search key field value is equal to Ki (or to the data
file block containing that record).
2. Within each node, K1 < K2 < ... < Kq−1.
3. For all search key field values X in the subtree pointed at by Pi (the ith sub-tree) we have:
Ki–1 < X < Ki for 1 < i < q; X < Ki for i = 1; and Ki–1 < X for i = q.
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4. Each node has at most p tree pointers.
5. Each node, except the root and leaf nodes, has at least ⎡(p/2)⎤ tree pointers. The root node
has at least two tree pointers unless it is the only node in the tree.
6. A node with q tree pointers, q ≤ p, has q – 1 search key field values (and hence has q – 1
data pointers).
7. All leaf nodes are at the same level. Leaf nodes have the same structure as internal nodes
except that all of their tree pointers Pi are N U LL.
Copyright © 2011 Ramez Elmasri and Shamkant Navathe
B-tree Structures
Copyright © 2011 Ramez Elmasri and Shamkant Navathe
B+-Trees
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The structure of the internal nodes of a B+-tree of order p is as follows:
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1 . Each internal node is of the form
<P1, K1, P2, K2, ..., Pq – 1, Kq –1, Pq>
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where q ≤ p and each Pi is a tree pointer.
2. Within each internal node, K1 < K2 < ... < Kq−1.
3. For all search field values X in the subtree pointed at by Pi, we have Ki−1 < X≤ Ki for 1 < i < q;
X ≤ Ki for i = 1; and Ki−1 < X for i = q
4. Each internal node has at most p tree pointers.
5. Each internal node, except the root, has at least ⎡(p/2)⎤ tree pointers. The root node has at
least two tree pointers if it is an internal node.
6. An internal node with q pointers, q ≤ p, has q − 1 search field values.
The structure of the leaf nodes of a B+-tree of order p is as follows:
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1 . Each leaf node is of the form
<<K1, Pr1>, <K2, Pr2>, ..., <Kq–1, Prq–1>, Pnext>
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where q ≤ p, each Pri is a data pointer, and Pnext points to the next leaf node of the B+-tree.
2. Within each leaf node, K1 ≤ K2 ... , Kq−1, q ≤ p.
3. Each Pri is a data pointer that points to the record whose search field value is Ki or to a file
block containing the record (or to a block of record pointers that point to records whose search
field value is Ki if the search field is not a key).
4. Each leaf node has at least ⎡(p/2)⎤ values.
5. All leaf nodes are at the same level.
Copyright © 2011 Ramez Elmasri and Shamkant Navathe
The Nodes of a B+-tree
Copyright © 2011 Ramez Elmasri and Shamkant Navathe
Example B+ Tree
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Search begins at root, and key comparisons
direct it to a leaf.
Search for 5*, 15*, all data entries >= 24* ...
Root
13
2*
3*
5*
13*
14* 16* 17*
17
24
19* 20* 24*
30
25* 27* 30*
33* 34* 38* 39*
* Based on the search for 15*, we know it is not in the tree!
Copyright © 2011 Ramez Elmasri and Shamkant Navathe
Inserting a Data Entry into a B+ Tree
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Find correct leaf L.
Put data entry onto L.
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If L has enough space, done!
Else, must split L (into L and a new node L2)
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This can happen recursively
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Redistribute entries evenly, copy up middle key.
Insert index entry pointing to L2 into parent of L.
To split index node, redistribute entries evenly, but
push up middle key. (Contrast with leaf splits.)
Splits “grow” tree; root split increases height.
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Tree growth: gets wider or one level taller at top.
Copyright © 2011 Ramez Elmasri and Shamkant Navathe
Example of
an Insertion
in a B+-tree
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Deleting a Data Entry from a B+ Tree
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Start at root, find leaf L where entry belongs.
Remove the entry.
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If L is at least half-full, done!
If L has only d-1 entries,
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Try to re-distribute, borrowing from sibling (adjacent node
with same parent as L).
If re-distribution fails, merge L and sibling.
If merge occurred, must delete entry (pointing to L or
sibling) from parent of L.
Merge could propagate to root, decreasing height.
Copyright © 2011 Ramez Elmasri and Shamkant Navathe
Example of
a Deletion in
a B+-tree
Copyright © 2011 Ramez Elmasri and Shamkant Navathe
Bulk Loading of a B+ Tree
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If we have a large collection of records, and we
want to create a B+ tree on some field, doing so
by repeatedly inserting records is very slow.
Bulk Loading can be done much more efficiently.
Initialization: Sort all data entries, insert pointer
to first (leaf) page in a new (root) page.
Root
3* 4*
Sorted pages of data entries; not yet in B+ tree
6* 9*
10* 11*
12* 13* 20* 22* 23* 31* 35* 36*
Copyright © 2011 Ramez Elmasri and Shamkant Navathe
38* 41* 44*
Bulk Loading (Contd.)
Root
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Index entries for leaf
pages always
entered into rightmost index page just
above leaf level.
3*
When this fills up, it
splits. (Split may go
up right-most path to
the root.)
Much faster than
repeated inserts,
especially when one
considers locking!
10
20
Data entry pages
6
4*
3* 4*
12
6* 9*
23
not yet in B+ tree
10* 11* 12* 13* 20*22* 23* 31* 35* 36* 38*41* 44*
Root
20
10
6
6* 9*
35
12
Data entry pages
not yet in B+ tree
35
23
38
10* 11* 12* 13* 20*22* 23* 31* 35* 36* 38*41* 44*
Copyright © 2011 Ramez Elmasri and Shamkant Navathe
Summary of Bulk Loading
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Option 1: multiple inserts.
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Slow.
Does not give sequential storage of leaves.
Option 2: Bulk Loading
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Has advantages for concurrency control.
Fewer I/Os during build.
Leaves will be stored sequentially (and linked, of
course).
Can control “fill factor” on pages.
Copyright © 2011 Ramez Elmasri and Shamkant Navathe