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Instructor: Marina Gavrilova
If you don’t find it in the index, look very carefully through
the entire catalogue.
-- Sears, Roebuck, and Co., Consumer’s Guide , 1897
Outline of Presentation
1.
2.
3.
4.
5.
Data on external storage – review
Data Structure for file organization
•
Heaps and Sorted
•
Indexes
•
B and B+ tree Indexes
•
Hash based indexes
•
Index classification
Comparing file organization
•
Assumptions
•
Cost of operations
Summary
Review Questions
Goal
In this lecture we will study different type of data
structures for file organization and discuss
difference between them based on operations and
cost model analysis.
World Largest Databases
 10. World Data Centre for Climate
This database is controlled and maintained by the German Climate Computing Centre as
well as the Max Planck Institute for Meteorology. This database could be examined to
find the patterns that led to the severe changes in the climatic conditions.
9. National Energy Research Scientific Computing Center
The National Energy Research Scientific Computing Center is the second largest
database of the world. It is controlled by the Lawrence Berkeley National Laboratory in
the United States of America. It holds research information related to atomic energy,
high energy physics, theories related to various topics, etc.





8. This is similar to Sprint and is the oldest company that deals with telecommunications. It holds
over 310 terabytes of information and almost 2 trillion rows- making the call records extremely
extensive. In addition to that, one can also find old records. Therefore, if your grandfather ever made
a call using AT&T, the company will probably still have the records 
7. Google
Google has never made the true size of their database public. However, the type and amount of
information found on the website is overwhelming. According to statistics, over 90 million searches
are carried out every day. Google has been called the king of internet databases.
6. Sprint
The telecommunication company has over 50 million subscribers. In the past, it offered long distance
packages as well. The records are highly detailed and holds at least 3 trillion rows of database, over
350 call records on a daily basis and 70,000 insertions every second. Sprint is quite notorious and
infamous and expands quite rapidly.
5. ChoicePoint is basically a phone book which contains information about the population residing in the
United States of America. It holds criminal histories as well as driving records. It has been said that the
database would reach the moon and back at least 75 times. ChoicePoint has helped a number of
authorities solve difficult and complicated cases in the past.
4. YouTube
YouTube has been in operation for two years and holds a massive library of videos. A loyal user base
follows the website and according to records-over a 100 million clips are watched on a daily basis. The
size of this database seems to double every 5 months. Therefore, it goes without saying that the overall
statistics are staggering.
3. Amazon
This website holds over 250,000 textbooks and users can comment and interact with other users which
makes Amazon the largest community on the web. Amazon has 55 million customers and above 40
terabytes of data.
 2. CIA database collects information on everything ranging from places to things to
people. Even though the accurate size of this database is unknown, it holds both private
and public information. 100 articles are added to the library every month and includes
population statistics, maps as swell as military capabilities.
 1. Library of Congress
 Even after the onset of the digital age, the Library of Congress is still among the largest
databases of the world. It holds over 125 million items which consist of colonial
newspapers, cook books and government proceedings. The library expands every day.
http://www.worldsbiggests.com/2010/02/top-10-largestdatabases-in-world.html
Data Structures for File Organizations
Many data representations
exist, each ideal for some
situations, and not so good in
others:
2.1 Indexes: Data structures to
organize records via trees or
hashing.


Like sorted files, they speed up
searches for a subset of records,
based on values in certain
(“search key”) fields
Updates are much faster than
in sorted files.
K-d Trees
Data Structures for File Organizations
2.2 Sorted Files: Best if records must be retrieved in some
order, or only a `range’ of records is needed.
2.3 Heap (random order) files: Suitable when typical
access is a file scan retrieving all records.
2.1 Indexes
 An index on a file speeds up selections on the
search key fields for the index.


Any subset of the fields of a relation can be the
search key for an index on the relation.
Search key is not the same as unique key (minimal
set of fields that uniquely identify a record in a
relation).
 An index contains a collection of data entries, and
supports efficient retrieval of all data entries k*
with a given key value k.
 Given data entry k*, we can find record with key k in
at most one disk I/O.
B-Trees
 A B-Tree is a tree in which each node may have
multiple children and multiple keys.
 It is specially designed to allow efficient searching
for keys.
 Like a binary search tree each key has the property
that all keys to the left are lower and all keys to the
right are greater.
B-Trees
B-Tree
 From node 10 in the tree all keys to the left are less than 10
and all keys to the right are greater than 10 and less than 20.
 The key in a given node represents an upper or lower bound
on the sets of keys below it in the tree.
B-Trees
 A tree may also have nodes with several ordered keys.
For example, if each node can have three keys, then it
will also have four references (pointers to children).
Node of a B-Tree
 In this node (:20:40:60:) the reference to the left of 20
refers to nodes with keys less than 20, the reference
between 20 & 40 refers to nodes with keys from 21 to 39,
the reference between keys 40 & 60 to nodes with keys
between 41 and 59, and finally the reference to the right
of 60 refers to nodes with keys with values greater than
61.
B-Trees
 Organizational basis of the B-Tree
For m references there must be (m-1) keys in a given node.
 Typically a B-tree is specified in terms of the maximum number
of successors that a given node may have.
 This is also equivalent to the number of references

that may occupy a single node, also called the order of
the tree.
 However, sometimes order is defined as the number of keys
(but not in this course).
B-Trees
 Constraints

For an order m B-tree no node has more than m subtrees.
 Every node except the root and the leaves must have at least
m/2 subtrees.
 A leaf node must have at least m/2 -1 keys.
 The root has 0 or >= 2 subtrees.
 Terminal or leaf nodes are all at the same depth.
 Within a node, the keys are in ascending order
B-Trees
 Construction of a B-Tree
The B-tree is built differently than a binary search
tree.

 The binary search tree is constructed starting at the
root and working toward the leaves.
 A B-tree is constructed from the leaves and as it
grows the tree is pushed upward.
B-Trees
 Construction of a B-Tree

Suppose, the tree of order 4 and each node can hold
a maximum of 3 keys.
 The keys are always kept in ascending order within
a node.
 Because the tree is of order 4, every node except the
root and leaves must have at least 2 subtrees
(or one key which has a pointer to a node containing
keys which are less than the key in the parent node and
a pointer to a node containing key(s) which are greater
than the key in the parent node).
 This essentially defines a minimum number of keys
which must exist within any given node.
B-Trees
 Construction of a B-Tree
(continued)
 If random data are used for the insertions into the
B-tree, it generally will be within a level of minimum
height.
However, as the data become ordered the B-tree
degenerates.
 The worst case is for data which is sorted in which
case an order 4 B-tree becomes an order 2 tree or a
binary search tree.
This obviously results in much wasted space and a
substantial loss of search efficiency.
B-Tree insertion example
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B-Tree insertion
Steps for Insertion
 If after inserting the node into the appropriate sorted
order, no inner node is over its key capacity, the process is
finished.
 If some node has more than the maximum amount of
child nodes then it is split into two nodes, each with the
minimum amount of child nodes. This process continues
action recursively in the parent node.
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B-Trees
 Deletions from B-Trees

Deletions also must be done from the leaves.
Simple Deletion: Remove some key from the leaf
and there are still enough keys in the leaf so that there
are (m/2-1) keys in total.
The removal of keys from the leaves can occur under two
circumstances:
- when the key actually exists in the leaf of the tree,
and
- when the key exists in an internal leaf and must be
moved to a leaf by determining which leaf position
contains the key closest to the one to be removed.
B-Tree deletion
 Locate the in-order successor of the key to remove and
replace it with the key
 If the leaf node is in legal state (min capacity not violated)
then finished.
 If some inner node is in an illegal state then:
 Redistribute Its siblings node (a child of the same parent
node) can transfer one of its keys to the current node.
 Concatenate Its siblings does not have an extra key to
share. In that case both these nodes are merged into a
single node (together with a key from a parent) and
pointers updated accordingly. The process continues until
the parent node remains in a legal state or until the root
node is reached.
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B-Trees
 Efficiency of B-Trees
Height:

Same as the height of a binary tree.

In binary tree, the height of a binary tree is related
to the number of nodes through log2.

Here, the height of a B-Tree is related through log m
where m is the order of the tree:
height = logm n + 1
Summary
 A B-Tree is a tree in which each node may have
multiple children and multiple keys.
 It is specially designed to allow efficient searching
for keys and is much more compact than BST tree.
 B-tree insertion involves splitting the node
 Insertion is easier than deletion operation
B and B+ Tree Indexes
Non-leaf
Pages
Leaf
Pages
(Sorted by search key)
Leaf pages contain data entries, and are chained (prev & next)
 Non-leaf pages have index entries; only used to direct searches:

index entry
P0
K 1
P1
K 2
P 2
K m Pm
Example B+ Tree
Note how data entries
in leaf level are sorted
Root
17
Entries <= 17
5
2*
3*
Entries > 17
27
13
5*
7* 8*
14* 16*
22* 24*
30
27* 29*
33* 34* 38* 39*
 Find 28*? 29*? All > 15* and < 30*
 Insert/delete: Find data entry in leaf, then change
it. Need to adjust parent sometimes.
 And change sometimes propagates to the root!
Hash-Based Indexes
Approaches to Search
1. Sequential and list methods
(lists, tables, arrays).
2. Direct access by key value (hashing)
3. Tree indexing methods.
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Definition
Hashing is the process of
mapping a key value to a
position in a table.
A hash
function maps key values to positions.
A hash
table is an array that holds the records.
Searching in a hash table can be done in O(1) regardless of the
hash table size.
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Applications of Hashing
 Compilers use hash tables to keep track of declared variables
 A hash table can be used for on-line spelling checkers — if
misspelling detection (rather than correction) is important, an
entire dictionary can be hashed and words checked in constant
time
 Game playing programs use hash tables to store seen positions,
thereby saving computation time if the position is encountered
again


Hash functions can be used to quickly check for inequality — if
two elements hash to different values they must be different
Storing sparse data
30
Hash-Based Indexes
 Index is a collection of buckets.
 Bucket = primary page plus zero or more overflow
pages.
 Buckets contain data entries.
 Hashing function h: h(r) = bucket in which (data
entry for) record r belongs. h looks at the search key
fields of r.
 No need for “index entries” in this scheme.
Alternatives for Data Entry k* in Index
 In a data entry k* we can store:
 Data record with key value k, or
 <k, rid of data record with search key value k>, or
 <k, list of rids of data records with search key k>
 Choice of alternative for data entries depends on the
indexing technique used to locate data entries with a
given key value k.
 Examples of indexing techniques: B+ trees, hash-
based structures
 Typically, index contains auxiliary information that
directs searches to the desired data entries
Alternatives for Data Entries (Contd.)
 Alternative 1: Data record with key value k
 If this is used, index structure is a file organization for
data records.
 At most one index on a given collection of data
records can use Alternative 1. (Otherwise, large data
records are duplicated, leading to redundant storage
and potential inconsistency.)
 If data records are very large, # of pages containing
data entries is high. Implies size of auxiliary
information in the index is also large, typically.
Alternatives for Data Entries
 Alternative 2
<k, rid of data record with search key value k>:

Data entries typically much smaller than data records.
So, better than Alternative 1 with large data records,
especially if search keys are small.

Alternative 3 <k, list of rids of data records with search
key k>
Alternative 3 more compact than Alternative 2, but leads
to variable sized data entries even if search keys are of
fixed length.

Index Classification
 Primary vs. secondary: If search key contains primary
key, then called primary index.

Unique index: Search key contains a candidate key.
 Clustered vs. un-clustered: If order of data records is
the same as, or `close to’, order of data entries, then
called clustered index.


A file can be clustered on at most one search key.
Cost of retrieving data records through index varies
greatly based on whether index is clustered or not!
Clustered vs. Unclustered Index
 Suppose that Alternative (2) is used for data entries, and
that the data records are stored in a Heap file.


To build clustered index, first sort the Heap file (with some
free space on each page for future inserts).
Overflow pages may be needed for inserts. (Thus, order of
data recs is `close to’, but not identical to, the sort order.)
CLUSTERED
Index entries
direct search for
data entries
Data entries
UNCLUSTERED
Data entries
(Index File)
(Data file)
Data Records
Data Records
Clustering Definition

Clustering is the unsupervised
classification of patterns
(observations, data items or
feature vectors) into groups
(clusters).
– A.K. Jain, M. N. Murty,
P. J. Flynn, Data Clustering:
A Review
Clustering a collection of
points
37
Clustering Properties

Linear increase in processing time with increase in size of dataset
(Scalability).

Ability to detect clusters of different shapes and densities.

Minimal input parameter.

Robust with regard to noise.

Insensitive to data input order.

Extensible to higher dimensions.
Osmar R. Zaΐane, Andrew Foss, Chi-Hoon Lee, Weinan Wang, “On Data Clustering Analysis: Scalability,
Constraints and Validation”, Advances in Knowledge Discovery and Data Mining, Springer-Verlag, 2002.
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Clusters
t7.10k dataset (9 visible clusters, n = 10,000)
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A
B
C
E
A
K-Means
k=9
B
CURE
k = 9, α = 0.3 and 10
representative points
per cluster
C
ROCK
θ = 0.975 and k = 1000
D
CHAMELEON
K-NN = 10, MinSize =
2.5%, k = 9
E
DBSCAN
є = 5.9, MinPts = 4
F
DBSCAN
є = 5.5, MinPts = 4
G
WaveCluster
Resolution = 5, Г = 1.5
H
WaveCluster
Resolution = 5, Г =
1.999397
F
G
Clustering results on t7.10k dataset
D
H
Osmar R. Zaΐane, Andrew Foss, Chi-Hoon Lee, Weinan
Wang, “On Data Clustering Analysis: Scalability,
Constraints and Validation”
40
Example: Indexing & Searching One of the
World's Largest Compound Database

http://accelrys.com/resource-center/case-studies/pharmacopeia-database.html
The Data Challenge
Pharmacopeia's corporate compound collection contains over seven million molecules. These are typically
small drug-like molecules: organic compounds, of which 99% fall within a range of molecular weights range
from 250 to 750. A typical biotechnology company has a database of only hundreds of thousands of
compounds, while major pharmaceuticals may have collections approaching the same order of magnitude as
Pharmacopeia's.
A Single, Integrated, Open System
Pharmacopeia has tested the DS Accord Chemistry Cartridge for use in data mining and subsequent analysis
to assess novel libraries of compounds that are planned for synthesis. Creating such a library can take over a
man-year. To enable such library construction and analysis, Accord users will need to routinely conduct
sophisticated and varied searches on the multi-million compound database. Pharmacopeia's developers took
just a couple of hours to build a prototype of a customized client user interface using Oracle forms and
PL/SQL. This enabled substructure searching within their internal compound collection.
Cost Model for Our Analysis
We ignore CPU costs, for simplicity:





B: The number of data pages
R: Number of records per page
D: (Average) time to read or write disk page
Measuring number of page I/O’s ignores gains of prefetching a sequence of pages; thus, even I/O cost is
only approximated.
Average-case analysis; based on several simplistic
assumptions.
* Good enough to show the overall trends!
Comparing File Organizations





Heap files (random order; insert at eof)
Sorted files, sorted on <age, sal>
Clustered B+ tree file, Alternative (1), search key
<age, sal>
Heap file with unclustered B + tree index on search
key <age, sal>
Heap file with unclustered hash index on search key
<age, sal>
Operations to Compare





Scan: Fetch all records from disk
Equality search
Range selection
Insert a record
Delete a record
Assumptions in Our Analysis
 Heap Files:
 Equality selection on key; exactly one match.
 Sorted Files:
 Files compacted after deletions.
 Indexes:
 Alt (2), (3): data entry size = 10% size of record
 Hash: No overflow buckets.


80% page occupancy => File size = 1.25 data size
Tree: 67% occupancy (this is typical).

Implies file size = 1.5 data size
Assumptions (contd.)
 Scans:
 Leaf levels of a tree-index are chained.
 Index data-entries plus actual file scanned for
unclustered indexes.
 Range searches:
 We use tree indexes to restrict the set of data records
fetched, but ignore hash indexes.
Cost of Operations
(a) Scan
(b)
Equality
(c ) Range
(d) Insert
(e) Delete
(1) Heap
(2) Sorted
(3) Clustered
(4) Unclustered
Tree index
(5) Unclustered
Hash index
B: The number of data pages
R: Number of records per page
D: (Average) time to read or
write disk page
* Several assumptions underlie these (rough) estimates!
Cost of Operations
(a) Scan
(b) Equality
(c ) Range
(d) Insert (e) Delete
(1) Heap
BD
0.5BD
BD
2D
(2) Sorted
BD
Dlog 2B
D(log 2 B +
# pgs with
match recs)
(3)
1.5BD
Dlog F 1.5B D(log F 1.5B
Clustered
+ # pgs w.
match recs)
(4) Unclust. BD(R+0.15)
D(1 +
D(log F 0.15B
Tree index
log F 0.15B) + # pgs w.
match recs)
(5) Unclust. BD(R+0.125) 2D
BD
Hash index
Search
+ BD
Search
+D
Search
+BD
Search
+D
Search
+D
Search
+ 2D
Search
+ 2D
Search
+ 2D
Search
+ 2D
* Several assumptions underlie these (rough) estimates!
Choice of Indexes
 What indexes should we create?
 Which relations should have indexes? What field(s)
should be the search key? Should we build several
indexes?
 For each index, what kind of an index should it be?
 Clustered? Hash/tree?
Choice of Indexes (Contd.)
 One approach: Consider the most important queries in turn.
Consider the best plan using the current indexes, and see if a
better plan is possible with an additional index. If so, create
it.
 Obviously, this implies that we must understand how a DBMS
evaluates queries and creates query evaluation plans!
 For now, we discuss simple 1-table queries.
 Before creating an index, must also consider the impact on
updates in the workload!

Trade-off: Indexes can make queries go faster, updates slower.
Require disk space, too.
Index Selection Guidelines
 Attributes in WHERE clause are candidates for index keys.
 Exact match condition suggests hash index.
 Range query suggests tree index.

Clustering is especially useful for range queries; can also help on equality
queries if there are many duplicates.
 Multi-attribute search keys should be considered when a
WHERE clause contains several conditions.
 Order of attributes is important for range queries.
 Such indexes can sometimes enable index-only strategies for important
queries.

For index-only strategies, clustering is not important!
 Try to choose indexes that benefit as many queries as possible.
Since only one index can be clustered per relation, choose it
based on important queries that would benefit the most from
clustering.
Summary
 Many alternative file organizations exist, each
appropriate in some situation.
 If selection queries are frequent, sorting the file or
building an index is important.


Hash-based indexes only good for equality search.
Sorted files and tree-based indexes best for range search;
also good for equality search. (Files rarely kept sorted in
practice; B+ tree index is better.)
 Index is a collection of data entries plus a way to
quickly find entries with given key values.
Summary (Contd.)
 Data entries can be actual data records, <key, rid>
pairs, or <key, rid-list> pairs.

Choice orthogonal to indexing technique used to
locate data entries with a given key value.
 Can have several indexes on a given file of data
records, each with a different search key.
 Indexes can be classified as clustered vs.
unclustered, primary vs. secondary. Differences
have important consequences for
utility/performance.
Summary (Contd.)
 Understanding the nature of the workload for the
application, and the performance goals, is essential to
developing a good design.

What are the important queries and updates? What
attributes/relations are involved?
 Indexes must be chosen to speed up important queries
(and perhaps some updates!).




Index maintenance overhead on updates to key fields.
Choose indexes that can help many queries, if possible.
Build indexes to support index-only strategies.
Clustering is an important decision; only one index on a
given relation can be clustered!
Review Questions
1. What is an external storage and name at least three of them.
2. How does a DBMS organize files of data records on disk to minimize I/O costs?
3. How can we compare different file data structures.
4. What is hash based indexes ?
5. What are different ways of index classification.
6. How can we apply cost model analysis in file data structure comparison?
7. What is an index, and why is it used?
8. Explain with example B+ tree indexes.
9. Why is I/O cost so important for database operations?
10. What are important properties of indexes?