Database System Concepts, 6th Ed.
©Silberschatz, Korth and Sudarshan
See www.db-book.com for conditions on re-use
Chapter 11: Indexing and Hashing
©Silberschatz, Korth and Sudarshan11.2Database System Concepts - 6 th Edition
Chapter 11: Indexing and Hashing
Basic Concepts
Ordered Indices
B+-Tree Index Files
B-Tree Index Files
Static Hashing
Dynamic Hashing
Comparison of Ordered Indexing and Hashing
Index Definition in SQL
Multiple-Key Access
©Silberschatz, Korth and Sudarshan11.3Database System Concepts - 6 th Edition
Basic Concepts
Indexing mechanisms used to speed up access to desired data.
E.g., author catalog in library
Search Key - attribute to set of attributes used to look up records in a
file.
An index file consists of records (called index entries) of the form
Index files are typically much smaller than the original file
Two basic kinds of indices:
Ordered indices: search keys are stored in sorted order
Hash indices: search keys are distributed uniformly across
“buckets” using a “hash function”.
search-key pointer
©Silberschatz, Korth and Sudarshan11.4Database System Concepts - 6 th Edition
Index Evaluation Metrics
Access types supported efficiently. E.g.,
records with a specified value in the attribute
or records with an attribute value falling in a specified range of
values.
Access time
Insertion time
Deletion time
Space overhead
©Silberschatz, Korth and Sudarshan11.5Database System Concepts - 6 th Edition
Ordered Indices
In an ordered index, index entries are stored sorted on the search key
value. E.g., author catalog in library.
Primary index: in a sequentially ordered file, the index whose search
key specifies the sequential order of the file.
Also called clustering index
The search key of a primary index is usually but not necessarily the
primary key.
Secondary index: an index whose search key specifies an order
different from the sequential order of the file. Also called
non-clustering index.
Index-sequential file: ordered sequential file with a primary index.
©Silberschatz, Korth and Sudarshan11.6Database System Concepts - 6 th Edition
Dense Index Files
Dense index — Index record appears for every search-key
value in the file.
E.g. index on ID attribute of instructor relation
©Silberschatz, Korth and Sudarshan11.7Database System Concepts - 6 th Edition
Dense Index Files (Cont.)
Dense index on dept_name, with instructor file sorted on
dept_name
©Silberschatz, Korth and Sudarshan11.8Database System Concepts - 6 th Edition
Sparse Index Files
Sparse Index: contains index records for only some search-key
values.
Applicable when records are sequentially ordered on search-key
To locate a record with search-key value K we:
Find index record with largest search-key value < K
Search file sequentially starting at the record to which the index
record points
©Silberschatz, Korth and Sudarshan11.9Database System Concepts - 6 th Edition
Sparse Index Files (Cont.)
Compared to dense indices:
Less space and less maintenance overhead for insertions and
deletions.
Generally slower than dense index for locating records.
Good tradeoff: sparse index with an index entry for every block in file,
corresponding to least search-key value in the block.
©Silberschatz, Korth and Sudarshan11.10Database System Concepts - 6 th Edition
Secondary Indices Example
Index record points to a bucket that contains pointers to all the
actual records with that particular search-key value.
Secondary indices have to be dense
Secondary index on salary field of instructor
©Silberschatz, Korth and Sudarshan11.11Database System Concepts - 6 th Edition
Primary and Secondary Indices
Indices offer substantial benefits when searching for records.
BUT: Updating indices imposes overhead on database
modification --when a file is modified, every index on the file
must be updated,
Sequential scan using primary index is efficient, but a
sequential scan using a secondary index is expensive
Each record access may fetch a new block from disk
Block fetch requires about 5 to 10 milliseconds, versus
about 100 nanoseconds for memory access
©Silberschatz, Korth and Sudarshan11.12Database System Concepts - 6 th Edition
Multilevel Index
If primary index does not fit in memory, access becomes
expensive.
Solution: treat primary index kept on disk as a sequential file
and construct a sparse index on it.
outer index – a sparse index of primary index
inner index – the primary index file
If even outer index is too large to fit in main memory, yet
another level of index can be created, and so on.
Indices at all levels must be updated on insertion or deletion
from the file.
©Silberschatz, Korth and Sudarshan11.13Database System Concepts - 6 th Edition
Multilevel Index (Cont.)
©Silberschatz, Korth and Sudarshan11.14Database System Concepts - 6 th Edition
Index Update: Deletion
Single-level index entry deletion:
Dense indices – deletion of search-key is similar to file record
deletion.
Sparse indices –
 if an entry for the search key exists in the index, it is
deleted by replacing the entry in the index with the next
search-key value in the file (in search-key order).
 If the next search-key value already has an index entry, the
entry is deleted instead of being replaced.
If deleted record was the
only record in the file with its
particular search-key value,
the search-key is deleted
from the index also.
©Silberschatz, Korth and Sudarshan11.15Database System Concepts - 6 th Edition
Index Update: Insertion
Single-level index insertion:
Perform a lookup using the search-key value appearing in
the record to be inserted.
Dense indices – if the search-key value does not appear in
the index, insert it.
Sparse indices – if index stores an entry for each block of
the file, no change needs to be made to the index unless a
new block is created.
 If a new block is created, the first search-key value
appearing in the new block is inserted into the index.
Multilevel insertion and deletion: algorithms are simple
extensions of the single-level algorithms
©Silberschatz, Korth and Sudarshan11.16Database System Concepts - 6 th Edition
Secondary Indices
Frequently, one wants to find all the records whose values in
a certain field (which is not the search-key of the primary
index) satisfy some condition.
Example 1: In the instructor relation stored sequentially by
ID, we may want to find all instructors in a particular
department
Example 2: as above, but where we want to find all
instructors with a specified salary or with salary in a
specified range of values
We can have a secondary index with an index record for
each search-key value
©Silberschatz, Korth and Sudarshan11.17Database System Concepts - 6 th Edition
B+-Tree Index Files
Disadvantage of indexed-sequential files
performance degrades as file grows, since many overflow
blocks get created.
Periodic reorganization of entire file is required.
Advantage of B+-tree index files:
automatically reorganizes itself with small, local, changes,
in the face of insertions and deletions.
Reorganization of entire file is not required to maintain
performance.
(Minor) disadvantage of B+-trees:
extra insertion and deletion overhead, space overhead.
Advantages of B+-trees outweigh disadvantages
B+-trees are used extensively
B+-tree indices are an alternative to indexed-sequential files.
©Silberschatz, Korth and Sudarshan11.18Database System Concepts - 6 th Edition
Example of B+-Tree
©Silberschatz, Korth and Sudarshan11.19Database System Concepts - 6 th Edition
B+-Tree Index Files (Cont.)
All paths from root to leaf are of the same length
Each node that is not a root or a leaf has between n/2 and
n children.
A leaf node has between (n–1)/2 and n–1 values
Special cases:
If the root is not a leaf, it has at least 2 children.
If the root is a leaf (that is, there are no other nodes in
the tree), it can have between 0 and (n–1) values.
A B+-tree is a rooted tree satisfying the following properties:
©Silberschatz, Korth and Sudarshan11.20Database System Concepts - 6 th Edition
B+-Tree Node Structure
Typical node
Ki are the search-key values
Pi are pointers to children (for non-leaf nodes) or pointers to
records or buckets of records (for leaf nodes).
The search-keys in a node are ordered
K1 < K2 < K3 < . . . < Kn–1
(Initially assume no duplicate keys, address duplicates later)
©Silberschatz, Korth and Sudarshan11.21Database System Concepts - 6 th Edition
Leaf Nodes in B+-Trees
For i = 1, 2, . . ., n–1, pointer Pi points to a file record with
search-key value Ki,
If Li, Lj are leaf nodes and i < j, Li’s search-key values are less
than or equal to Lj’s search-key values
Pn points to next leaf node in search-key order
Properties of a leaf node:
©Silberschatz, Korth and Sudarshan11.22Database System Concepts - 6 th Edition
Non-Leaf Nodes in B+-Trees
Non leaf nodes form a multi-level sparse index on the leaf
nodes. For a non-leaf node with m pointers:
All the search-keys in the subtree to which P1 points are
less than K1
For 2 i n – 1, all the search-keys in the subtree to which
Pi points have values greater than or equal to Ki–1 and less
than Ki
All the search-keys in the subtree to which Pn points have
values greater than or equal to Kn–1
©Silberschatz, Korth and Sudarshan11.23Database System Concepts - 6 th Edition
Example of B+-tree
Leaf nodes must have between 3 and 5 values
((n–1)/2 and n –1, with n = 6).
Non-leaf nodes other than root must have between 3
and 6 children ((n/2 and n with n =6).
Root must have at least 2 children.
B+-tree for instructor file (n = 6)
©Silberschatz, Korth and Sudarshan11.24Database System Concepts - 6 th Edition
Observations about B+-trees
Since the inter-node connections are done by pointers,
“logically” close blocks need not be “physically” close.
The non-leaf levels of the B+-tree form a hierarchy of sparse
indices.
The B+-tree contains a relatively small number of levels
 Level below root has at least 2* n/2 values
 Next level has at least 2* n/2 * n/2 values
 .. etc.
If there are K search-key values in the file, the tree height is
no more than  logn/2(K)
thus searches can be conducted efficiently.
Insertions and deletions to the main file can be handled
efficiently, as the index can be restructured in logarithmic time
(as we shall see).
©Silberschatz, Korth and Sudarshan11.25Database System Concepts - 6 th Edition
Queries on B+-Trees
Find record with search-key value V.
1. C=root
2. While C is not a leaf node {
1. Let i be least value s.t. V Ki.
2. If no such exists, set C = last non-null pointer in C
3. Else { if (V= Ki ) Set C = Pi +1 else set C = Pi}
}
3. Let i be least value s.t. Ki = V
4. If there is such a value i, follow pointer Pi to the desired record.
5. Else no record with search-key value k exists.
©Silberschatz, Korth and Sudarshan11.26Database System Concepts - 6 th Edition
Handling Duplicates
With duplicate search keys
In both leaf and internal nodes,
 we cannot guarantee that K1 < K2 < K3 < . . . < Kn–1
 but can guarantee K1 K2 K3 . . . Kn–1
Search-keys in the subtree to which Pi points
 are Ki,, but not necessarily < Ki,
 To see why, suppose same search key value V is present
in two leaf node Li and Li+1. Then in parent node Ki must
be equal to V
©Silberschatz, Korth and Sudarshan11.27Database System Concepts - 6 th Edition
Handling Duplicates
We modify find procedure as follows
traverse Pi even if V = Ki
As soon as we reach a leaf node C check if C has
only search key values less than V
if so set C = right sibling of C before checking
whether C contains V
Procedure printAll
uses modified find procedure to find first
occurrence of V
Traverse through consecutive leaves to find all
occurrences of V
** Errata note: modified find procedure missing in first printing of 6th edition
©Silberschatz, Korth and Sudarshan11.28Database System Concepts - 6 th Edition
Queries on B+-Trees (Cont.)
If there are K search-key values in the file, the height of the tree is no
more than logn/2(K).
A node is generally the same size as a disk block, typically 4
kilobytes
and n is typically around 100 (40 bytes per index entry).
With 1 million search key values and n = 100
at most log50(1,000,000) = 4 nodes are accessed in a lookup.
Contrast this with a balanced binary tree with 1 million search key
values — around 20 nodes are accessed in a lookup
above difference is significant since every node access may need
a disk I/O, costing around 20 milliseconds
©Silberschatz, Korth and Sudarshan11.29Database System Concepts - 6 th Edition
Updates on B+-Trees: Insertion
1. Find the leaf node in which the search-key value would appear
2. If the search-key value is already present in the leaf node
1. Add record to the file
2. If necessary add a pointer to the bucket.
3. If the search-key value is not present, then
1. add the record to the main file (and create a bucket if
necessary)
2. If there is room in the leaf node, insert (key-value, pointer)
pair in the leaf node
3. Otherwise, split the node (along with the new (key-value,
pointer) entry) as discussed in the next slide.
©Silberschatz, Korth and Sudarshan11.30Database System Concepts - 6 th Edition
Updates on B+-Trees: Insertion (Cont.)
Splitting a leaf node:
take the n (search-key value, pointer) pairs (including the one
being inserted) in sorted order. Place the first n/2 in the original
node, and the rest in a new node.
let the new node be p, and let k be the least key value in p. Insert
(k,p) in the parent of the node being split.
If the parent is full, split it and propagate the split further up.
Splitting of nodes proceeds upwards till a node that is not full is found.
In the worst case the root node may be split increasing the height
of the tree by 1.
Result of splitting node containing Brandt, Califieri and Crick on inserting Adams
Next step: insert entry with (Califieri,pointer-to-new-node) into parent
©Silberschatz, Korth and Sudarshan11.31Database System Concepts - 6 th Edition
B+-Tree Insertion
B+-Tree before and after insertion of “Adams”
©Silberschatz, Korth and Sudarshan11.32Database System Concepts - 6 th Edition
B+-Tree Insertion
B+-Tree before and after insertion of “Lamport”
©Silberschatz, Korth and Sudarshan11.33Database System Concepts - 6 th Edition
Splitting a non-leaf node: when inserting (k,p) into an already full
internal node N
Copy N to an in-memory area M with space for n+1 pointers and n
keys
Insert (k,p) into M
Copy P1,K1, …, K n/2-1,P n/2 from M back into node N
Copy Pn/2+1,K n/2+1,…,Kn,Pn+1 from M into newly allocated node
N’
Insert (K n/2,N’) into parent N
Read pseudocode in book!
Crick
Insertion in B+-Trees (Cont.)
Adams Brandt Califieri Crick Adams Brandt
Califieri
©Silberschatz, Korth and Sudarshan11.34Database System Concepts - 6 th Edition
Examples of B+-Tree Deletion
Deleting “Srinivasan” causes merging of under-full leaves
Before and after deleting “Srinivasan”
©Silberschatz, Korth and Sudarshan11.35Database System Concepts - 6 th Edition
Examples of B+-Tree Deletion (Cont.)
Deletion of “Singh” and “Wu” from result of previous example
Leaf containing Singh and Wu became underfull, and borrowed a value
Kim from its left sibling
Search-key value in the parent changes as a result
©Silberschatz, Korth and Sudarshan11.36Database System Concepts - 6 th Edition
Example of B+-tree Deletion (Cont.)
Before and after deletion of “Gold” from earlier example
Node with Gold and Katz became underfull, and was merged with its sibling
Parent node becomes underfull, and is merged with its sibling
Value separating two nodes (at the parent) is pulled down when merging
Root node then has only one child, and is deleted
©Silberschatz, Korth and Sudarshan11.37Database System Concepts - 6 th Edition
Updates on B+-Trees: Deletion
Find the record to be deleted, and remove it from the main file and
from the bucket (if present)
Remove (search-key value, pointer) from the leaf node if there is no
bucket or if the bucket has become empty
If the node has too few entries due to the removal, and the entries in
the node and a sibling fit into a single node, then merge siblings:
Insert all the search-key values in the two nodes into a single node
(the one on the left), and delete the other node.
Delete the pair (Ki–1, Pi), where Pi is the pointer to the deleted
node, from its parent, recursively using the above procedure.
©Silberschatz, Korth and Sudarshan11.38Database System Concepts - 6 th Edition
Updates on B+-Trees: Deletion
Otherwise, if the node has too few entries due to the removal, but the
entries in the node and a sibling do not fit into a single node, then
redistribute pointers:
Redistribute the pointers between the node and a sibling such that
both have more than the minimum number of entries.
Update the corresponding search-key value in the parent of the
node.
The node deletions may cascade upwards till a node which has n/2
or more pointers is found.
If the root node has only one pointer after deletion, it is deleted and
the sole child becomes the root.
©Silberschatz, Korth and Sudarshan11.39Database System Concepts - 6 th Edition
Non-Unique Search Keys
Alternatives to scheme described earlier
Buckets on separate block (bad idea)
List of tuple pointers with each key
 Extra code to handle long lists
 Deletion of a tuple can be expensive if there are many
duplicates on search key (why?)
 Low space overhead, no extra cost for queries
Make search key unique by adding a record-identifier
 Extra storage overhead for keys
 Simpler code for insertion/deletion
 Widely used
©Silberschatz, Korth and Sudarshan11.40Database System Concepts - 6 th Edition
B+-Tree File Organization
Index file degradation problem is solved by using B+-Tree indices.
Data file degradation problem is solved by using B+-Tree File
Organization.
The leaf nodes in a B+-tree file organization store records, instead of
pointers.
Leaf nodes are still required to be half full
Since records are larger than pointers, the maximum number of
records that can be stored in a leaf node is less than the number of
pointers in a nonleaf node.
Insertion and deletion are handled in the same way as insertion and
deletion of entries in a B+-tree index.
©Silberschatz, Korth and Sudarshan11.41Database System Concepts - 6 th Edition
B+-Tree File Organization (Cont.)
Good space utilization important since records use more space than
pointers.
To improve space utilization, involve more sibling nodes in redistribution
during splits and merges
Involving 2 siblings in redistribution (to avoid split / merge where
possible) results in each node having at least entries
Example of B+-tree File Organization
 3/2n
©Silberschatz, Korth and Sudarshan11.42Database System Concepts - 6 th Edition
Other Issues in Indexing
Record relocation and secondary indices
If a record moves, all secondary indices that store record pointers
have to be updated
Node splits in B+-tree file organizations become very expensive
Solution: use primary-index search key instead of record pointer in
secondary index
 Extra traversal of primary index to locate record
– Higher cost for queries, but node splits are cheap
 Add record-id if primary-index search key is non-unique
©Silberschatz, Korth and Sudarshan11.43Database System Concepts - 6 th Edition
Indexing Strings
Variable length strings as keys
Variable fanout
Use space utilization as criterion for splitting, not number of
pointers
Prefix compression
Key values at internal nodes can be prefixes of full key
 Keep enough characters to distinguish entries in the
subtrees separated by the key value
– E.g. “Silas” and “Silberschatz” can be separated by
“Silb”
Keys in leaf node can be compressed by sharing common
prefixes
©Silberschatz, Korth and Sudarshan11.44Database System Concepts - 6 th Edition
Bulk Loading and Bottom-Up Build
Inserting entries one-at-a-time into a B+-tree requires 1 IO per entry
assuming leaf level does not fit in memory
can be very inefficient for loading a large number of entries at a time
(bulk loading)
Efficient alternative 1:
sort entries first (using efficient external-memory sort algorithms
discussed later in Section 12.4)
insert in sorted order
 insertion will go to existing page (or cause a split)
 much improved IO performance, but most leaf nodes half full
Efficient alternative 2: Bottom-up B+-tree construction
As before sort entries
And then create tree layer-by-layer, starting with leaf level
 details as an exercise
Implemented as part of bulk-load utility by most database systems
©Silberschatz, Korth and Sudarshan11.45Database System Concepts - 6 th Edition
B-Tree Index Files
Similar to B+-tree, but B-tree allows search-key values to
appear only once; eliminates redundant storage of search
keys.
Search keys in nonleaf nodes appear nowhere else in the Btree;
an additional pointer field for each search key in a
nonleaf node must be included.
Generalized B-tree leaf node
Nonleaf node – pointers Bi are the bucket or file record
pointers.
©Silberschatz, Korth and Sudarshan11.46Database System Concepts - 6 th Edition
B-Tree Index File Example
B-tree (above) and B+-tree (below) on same data
©Silberschatz, Korth and Sudarshan11.47Database System Concepts - 6 th Edition
B-Tree Index Files (Cont.)
Advantages of B-Tree indices:
May use less tree nodes than a corresponding B+-Tree.
Sometimes possible to find search-key value before reaching leaf
node.
Disadvantages of B-Tree indices:
Only small fraction of all search-key values are found early
Non-leaf nodes are larger, so fan-out is reduced. Thus, B-Trees
typically have greater depth than corresponding B+-Tree
Insertion and deletion more complicated than in B+-Trees
Implementation is harder than B+-Trees.
Typically, advantages of B-Trees do not out weigh disadvantages.
©Silberschatz, Korth and Sudarshan11.48Database System Concepts - 6 th Edition
Multiple-Key Access
Use multiple indices for certain types of queries.
Example:
select ID
from instructor
where dept_name = “Finance” and salary = 80000
Possible strategies for processing query using indices on
single attributes:
1. Use index on dept_name to find instructors with
department name Finance; test salary = 80000
2. Use index on salary to find instructors with a salary of
$80000; test dept_name = “Finance”.
3. Use dept_name index to find pointers to all records
pertaining to the “Finance” department. Similarly use index
on salary. Take intersection of both sets of pointers
obtained.
©Silberschatz, Korth and Sudarshan11.49Database System Concepts - 6 th Edition
Indices on Multiple Keys
Composite search keys are search keys containing more
than one attribute
E.g. (dept_name, salary)
Lexicographic ordering: (a1, a2) < (b1, b2) if either
a1 < b1, or
a1=b1 and a2 < b2
©Silberschatz, Korth and Sudarshan11.50Database System Concepts - 6 th Edition
Indices on Multiple Attributes
With the where clause
where dept_name = “Finance” and salary = 80000
the index on (dept_name, salary) can be used to fetch only records
that satisfy both conditions.
Using separate indices in less efficient — we may fetch many
records (or pointers) that satisfy only one of the conditions.
Can also efficiently handle
where dept_name = “Finance” and salary < 80000
But cannot efficiently handle
where dept_name < “Finance” and balance = 80000
May fetch many records that satisfy the first but not the second
condition
Suppose we have an index on combined search-key
(dept_name, salary).
©Silberschatz, Korth and Sudarshan11.51Database System Concepts - 6 th Edition
Other Features
Covering indices
Add extra attributes to index so (some) queries can avoid fetching
the actual records
 Particularly useful for secondary indices
– Why?
Can store extra attributes only at leaf
Database System Concepts, 6th Ed.
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See www.db-book.com for conditions on re-use
Hashing
©Silberschatz, Korth and Sudarshan11.53Database System Concepts - 6 th Edition
Static Hashing
A bucket is a unit of storage containing one or more records (a
bucket is typically a disk block).
In a hash file organization we obtain the bucket of a record directly
from its search-key value using a hash function.
Hash function h is a function from the set of all search-key values K
to the set of all bucket addresses B.
Hash function is used to locate records for access, insertion as well
as deletion.
Records with different search-key values may be mapped to the
same bucket; thus entire bucket has to be searched sequentially to
locate a record.
©Silberschatz, Korth and Sudarshan11.54Database System Concepts - 6 th Edition
Example of Hash File Organization
There are 10 buckets,
The binary representation of the ith character is assumed to be the
integer i.
The hash function returns the sum of the binary representations of
the characters modulo 10
E.g. h(Music) = 1 h(History) = 2
h(Physics) = 3 h(Elec. Eng.) = 3
Hash file organization of instructor file, using dept_name as key
(See figure in next slide.)
©Silberschatz, Korth and Sudarshan11.55Database System Concepts - 6 th Edition
Example of Hash File Organization
Hash file organization of instructor file, using dept_name as key
(see previous slide for details).
©Silberschatz, Korth and Sudarshan11.56Database System Concepts - 6 th Edition
Hash Functions
Worst hash function maps all search-key values to the same bucket;
this makes access time proportional to the number of search-key
values in the file.
An ideal hash function is uniform, i.e., each bucket is assigned the
same number of search-key values from the set of all possible values.
Ideal hash function is random, so each bucket will have the same
number of records assigned to it irrespective of the actual distribution of
search-key values in the file.
Typical hash functions perform computation on the internal binary
representation of the search-key.
For example, for a string search-key, the binary representations of
all the characters in the string could be added and the sum modulo
the number of buckets could be returned. .
©Silberschatz, Korth and Sudarshan11.57Database System Concepts - 6 th Edition
Handling of Bucket Overflows
Bucket overflow can occur because of
Insufficient buckets
Skew in distribution of records. This can occur due to two
reasons:
 multiple records have same search-key value
 chosen hash function produces non-uniform distribution of key
values
Although the probability of bucket overflow can be reduced, it cannot
be eliminated; it is handled by using overflow buckets.
©Silberschatz, Korth and Sudarshan11.58Database System Concepts - 6 th Edition
Handling of Bucket Overflows (Cont.)
Overflow chaining – the overflow buckets of a given bucket are
chained together in a linked list.
Above scheme is called closed addressing.
An alternative, called open addressing, which does not use
overflow buckets, is not suitable for database applications.
©Silberschatz, Korth and Sudarshan11.59Database System Concepts - 6 th Edition
Hash Indices
Hashing can be used not only for file organization, but also for indexstructure
creation.
A hash index organizes the search keys, with their associated record
pointers, into a hash file structure.
Strictly speaking, hash indices are always secondary indices
if the file itself is organized using hashing, a separate primary
hash index on it using the same search-key is unnecessary.
However, we use the term hash index to refer to both secondary
index structures and hash organized files.
©Silberschatz, Korth and Sudarshan11.60Database System Concepts - 6 th Edition
Example of Hash Index
hash index on instructor, on attribute ID
©Silberschatz, Korth and Sudarshan11.61Database System Concepts - 6 th Edition
Deficiencies of Static Hashing
In static hashing, function h maps search-key values to a fixed set of B
of bucket addresses. Databases grow or shrink with time.
If initial number of buckets is too small, and file grows, performance
will degrade due to too much overflows.
If space is allocated for anticipated growth, a significant amount of
space will be wasted initially (and buckets will be underfull).
If database shrinks, again space will be wasted.
One solution: periodic re-organization of the file with a new hash
function
Expensive, disrupts normal operations
Better solution: allow the number of buckets to be modified dynamically.
©Silberschatz, Korth and Sudarshan11.62Database System Concepts - 6 th Edition
Dynamic Hashing
Good for database that grows and shrinks in size
Allows the hash function to be modified dynamically
Extendable hashing – one form of dynamic hashing
Hash function generates values over a large range — typically b-bit
integers, with b = 32.
At any time use only a prefix of the hash function to index into a
table of bucket addresses.
Let the length of the prefix be i bits, 0 i 32.
 Bucket address table size = 2i. Initially i = 0
 Value of i grows and shrinks as the size of the database grows
and shrinks.
Multiple entries in the bucket address table may point to a bucket
(why?)
Thus, actual number of buckets is < 2i
 The number of buckets also changes dynamically due to
coalescing and splitting of buckets.
©Silberschatz, Korth and Sudarshan11.63Database System Concepts - 6 th Edition
General Extendable Hash Structure
In this structure, i2 = i3 = i, whereas i1 = i – 1 (see next
slide for details)
©Silberschatz, Korth and Sudarshan11.64Database System Concepts - 6 th Edition
Use of Extendable Hash Structure
Each bucket j stores a value ij
All the entries that point to the same bucket have the same values on
the first ij bits.
To locate the bucket containing search-key Kj:
1. Compute h(Kj) = X
2. Use the first i high order bits of X as a displacement into bucket
address table, and follow the pointer to appropriate bucket
To insert a record with search-key value Kj
follow same procedure as look-up and locate the bucket, say j.
If there is room in the bucket j insert record in the bucket.
Else the bucket must be split and insertion re-attempted (next slide.)
 Overflow buckets used instead in some cases (will see shortly)
©Silberschatz, Korth and Sudarshan11.65Database System Concepts - 6 th Edition
Insertion in Extendable Hash Structure (Cont)
If i > ij (more than one pointer to bucket j)
allocate a new bucket z, and set ij = iz = (ij + 1)
Update the second half of the bucket address table entries originally
pointing to j, to point to z
remove each record in bucket j and reinsert (in j or z)
recompute new bucket for Kj and insert record in the bucket (further
splitting is required if the bucket is still full)
If i = ij (only one pointer to bucket j)
If i reaches some limit b, or too many splits have happened in this
insertion, create an overflow bucket
Else
 increment i and double the size of the bucket address table.
 replace each entry in the table by two entries that point to the
same bucket.
 recompute new bucket address table entry for Kj
Now i > ij so use the first case above.
To split a bucket j when inserting record with search-key value Kj:
©Silberschatz, Korth and Sudarshan11.66Database System Concepts - 6 th Edition
Deletion in Extendable Hash Structure
To delete a key value,
locate it in its bucket and remove it.
The bucket itself can be removed if it becomes empty (with
appropriate updates to the bucket address table).
Coalescing of buckets can be done (can coalesce only with a
“buddy” bucket having same value of ij and same ij –1 prefix, if it is
present)
Decreasing bucket address table size is also possible
 Note: decreasing bucket address table size is an expensive
operation and should be done only if number of buckets becomes
much smaller than the size of the table
©Silberschatz, Korth and Sudarshan11.67Database System Concepts - 6 th Edition
Use of Extendable Hash Structure: Example
©Silberschatz, Korth and Sudarshan11.68Database System Concepts - 6 th Edition
Example (Cont.)
Initial Hash structure; bucket size = 2
©Silberschatz, Korth and Sudarshan11.69Database System Concepts - 6 th Edition
Example (Cont.)
Hash structure after insertion of “Mozart”, “Srinivasan”,
and “Wu” records
©Silberschatz, Korth and Sudarshan11.70Database System Concepts - 6 th Edition
Example (Cont.)
Hash structure after insertion of Einstein record
©Silberschatz, Korth and Sudarshan11.71Database System Concepts - 6 th Edition
Example (Cont.)
Hash structure after insertion of Gold and El Said records
©Silberschatz, Korth and Sudarshan11.72Database System Concepts - 6 th Edition
Example (Cont.)
Hash structure after insertion of Katz record
©Silberschatz, Korth and Sudarshan11.73Database System Concepts - 6 th Edition
Example (Cont.)
And after insertion of
eleven records
©Silberschatz, Korth and Sudarshan11.74Database System Concepts - 6 th Edition
Example (Cont.)
And after insertion of
Kim record in previous
hash structure
©Silberschatz, Korth and Sudarshan11.75Database System Concepts - 6 th Edition
Extendable Hashing vs. Other Schemes
Benefits of extendable hashing:
Hash performance does not degrade with growth of file
Minimal space overhead
Disadvantages of extendable hashing
Extra level of indirection to find desired record
Bucket address table may itself become very big (larger than
memory)
 Cannot allocate very large contiguous areas on disk either
 Solution: B+-tree structure to locate desired record in bucket
address table
Changing size of bucket address table is an expensive operation
Linear hashing is an alternative mechanism
Allows incremental growth of its directory (equivalent to bucket
address table)
At the cost of more bucket overflows
©Silberschatz, Korth and Sudarshan11.76Database System Concepts - 6 th Edition
Comparison of Ordered Indexing and Hashing
Cost of periodic re-organization
Relative frequency of insertions and deletions
Is it desirable to optimize average access time at the expense of
worst-case access time?
Expected type of queries:
Hashing is generally better at retrieving records having a
specified value of the key.
If range queries are common, ordered indices are to be
preferred
In practice:
PostgreSQL supports hash indices, but discourages use due to
poor performance
Oracle supports static hash organization, but not hash indices
SQLServer supports only B+-trees
©Silberschatz, Korth and Sudarshan11.77Database System Concepts - 6 th Edition
Bitmap Indices
Bitmap indices are a special type of index designed for efficient
querying on multiple keys
Records in a relation are assumed to be numbered sequentially
from, say, 0
Given a number n it must be easy to retrieve record n
 Particularly easy if records are of fixed size
Applicable on attributes that take on a relatively small number
of distinct values
E.g. gender, country, state, …
E.g. income-level (income broken up into a small number of
levels such as 0-9999, 10000-19999, 20000-50000, 50000-
infinity)
A bitmap is simply an array of bits
©Silberschatz, Korth and Sudarshan11.78Database System Concepts - 6 th Edition
Bitmap Indices (Cont.)
In its simplest form a bitmap index on an attribute has a bitmap for
each value of the attribute
Bitmap has as many bits as records
In a bitmap for value v, the bit for a record is 1 if the record has the
value v for the attribute, and is 0 otherwise
©Silberschatz, Korth and Sudarshan11.79Database System Concepts - 6 th Edition
Bitmap Indices (Cont.)
Bitmap indices are useful for queries on multiple attributes
not particularly useful for single attribute queries
Queries are answered using bitmap operations
Intersection (and)
Union (or)
Complementation (not)
Each operation takes two bitmaps of the same size and applies the
operation on corresponding bits to get the result bitmap
E.g. 100110 AND 110011 = 100010
100110 OR 110011 = 110111
NOT 100110 = 011001
Males with income level L1: 10010 AND 10100 = 10000
 Can then retrieve required tuples.
 Counting number of matching tuples is even faster
©Silberschatz, Korth and Sudarshan11.80Database System Concepts - 6 th Edition
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
Deletion needs to be handled properly
Existence bitmap to note if there is a valid record at a record location
Needed for complementation
 not(A=v): (NOT bitmap-A-v) AND ExistenceBitmap
Should keep bitmaps for all values, even null value
To correctly handle SQL null semantics for NOT(A=v):
 intersect above result with (NOT bitmap-A-Null)
©Silberschatz, Korth and Sudarshan11.81Database System Concepts - 6 th Edition
Efficient Implementation of Bitmap Operations
Bitmaps are packed into words; a single word and (a basic CPU
instruction) computes and of 32 or 64 bits at once
E.g. 1-million-bit maps can be and-ed with just 31,250 instruction
Counting number of 1s can be done fast by a trick:
Use each byte to index into a precomputed array of 256 elements
each storing the count of 1s in the binary representation
 Can use pairs of bytes to speed up further at a higher memory
cost
Add up the retrieved counts
Bitmaps can be used instead of Tuple-ID lists at leaf levels of
B+-trees, for values that have a large number of matching records
Worthwhile if > 1/64 of the records have that value, assuming a
tuple-id is 64 bits
Above technique merges benefits of bitmap and B+-tree indices
©Silberschatz, Korth and Sudarshan11.82Database System Concepts - 6 th Edition
Index Definition in SQL
Create an index
create index <index-name> on <relation-name>
(<attribute-list>)
E.g.: create index b-index on branch(branch_name)
Use create unique index to indirectly specify and enforce the
condition that the search key is a candidate key is a candidate key.
Not really required if SQL unique integrity constraint is supported
To drop an index
drop index <index-name>
Most database systems allow specification of type of index, and
clustering.
Database System Concepts, 6th Ed.
©Silberschatz, Korth and Sudarshan
See www.db-book.com for conditions on re-use
End of Chapter
©Silberschatz, Korth and Sudarshan11.84Database System Concepts - 6 th Edition
Figure 11.01
©Silberschatz, Korth and Sudarshan11.85Database System Concepts - 6 th Edition
Figure 11.15
©Silberschatz, Korth and Sudarshan11.86Database System Concepts - 6 th Edition
Partitioned Hashing
Hash values are split into segments that depend on each
attribute of the search-key.
(A1, A2, . . . , An) for n attribute search-key
Example: n = 2, for customer, search-key being
(customer-street, customer-city)
search-key value hash value
(Main, Harrison) 101 111
(Main, Brooklyn) 101 001
(Park, Palo Alto) 010 010
(Spring, Brooklyn) 001 001
(Alma, Palo Alto) 110 010
To answer equality query on single attribute, need to look up
multiple buckets. Similar in effect to grid files.
©Silberschatz, Korth and Sudarshan11.87Database System Concepts - 6 th Edition
Grid Files
Structure used to speed the processing of general multiple searchkey
queries involving one or more comparison operators.
The grid file has a single grid array and one linear scale for each
search-key attribute. The grid array has number of dimensions
equal to number of search-key attributes.
Multiple cells of grid array can point to same bucket
To find the bucket for a search-key value, locate the row and column
of its cell using the linear scales and follow pointer
©Silberschatz, Korth and Sudarshan11.88Database System Concepts - 6 th Edition
Example Grid File for account
©Silberschatz, Korth and Sudarshan11.89Database System Concepts - 6 th Edition
Queries on a Grid File
A grid file on two attributes A and B can handle queries of all following
forms with reasonable efficiency
(a1 A a2)
(b1 B b2)
(a1 A a2  b1 B b2),.
E.g., to answer (a1 A a2  b1 B b2), use linear scales to find
corresponding candidate grid array cells, and look up all the buckets
pointed to from those cells.
©Silberschatz, Korth and Sudarshan11.90Database System Concepts - 6 th Edition
Grid Files (Cont.)
During insertion, if a bucket becomes full, new bucket can be created
if more than one cell points to it.
Idea similar to extendable hashing, but on multiple dimensions
If only one cell points to it, either an overflow bucket must be
created or the grid size must be increased
Linear scales must be chosen to uniformly distribute records across
cells.
Otherwise there will be too many overflow buckets.
Periodic re-organization to increase grid size will help.
But reorganization can be very expensive.
Space overhead of grid array can be high.
R-trees (Chapter 23) are an alternative