Database managment System Relational Algebra

CSEC 321
Lecture 2
Relational Algebra

Relational Database: Definitions
• Relational database: a set of relations.
• Relation: made up of 2 parts:
– Schema : specifies name of relation, plus name
and type of each column.
• E.g. Students(sid: string, name: string, login: string, age:
integer, gpa: real)
– Instance : a table, with rows and columns.
• #rows = cardinality
• #fields = degree / arity
• Can think of a relation as a set of rows or
tuples.
– i.e., all rows are distinct

Ex: Instance of Students Relation
sid name login age gpa
53666 Jones jones@cs 18 3.4
53688 Smith smith@eecs 18 3.2
53650 Smith smith@math 19 3.8
• Cardinality = 3, arity = 5 , all rows distinct
• Do all values in each column of a relation instance
have to be distinct?

Relational Query Languages
• Query languages: Allow manipulation and retrieval of
data from a database.
• Relational model supports simple, powerful QLs:
– Strong formal foundation based on logic.
– Allows for much optimization.
• Query Languages != programming languages!
– QLs not intended to be used for complex calculations.
– QLs support easy, efficient access to large data sets.

Preliminaries
• A query is applied to relation instances, and the
result of a query is also a relation instance.

Relational Algebra: 5 Basic Operations
• Selection ( s ) Selects a subset of rows from relation
(horizontal).
• Projection ( p ) Retains only wanted columns from relation
(vertical).
• Cross-product (  ) Allows us to combine two relations.
• Set-difference ( — ) Tuples in r1, but not in r2.
• Union (  ) Tuples in r1 or in r2.
Since each operation returns a relation, operations can be
composed! (Algebra is “closed”.)

sid sname rating age
22 dustin 7 45.0
31 lubber 8 55.5
58 rusty 10 35.0
28 yuppy 9 35.0
31 lubber 8 55.5
44 guppy 5 35.0
58 rusty 10 35.0
sid bid day
22 101 10/10/96
58 103 11/12/96
R1
S1
S2
bid bname color
101 Interlake blue
102 Interlake red
103 Clipper green
104 Marine red
Boats
Example Instances

Projection (p)
page S( )2• Examples: ;
• Retains only attributes that are in the “projection list”.
• Schema of result:
– exactly the fields in the projection list, with the same names that they
had in the input relation.
• Projection operator has to eliminate duplicates (How do they
arise? Why remove them?)
– Note: real systems typically don’t do duplicate elimination unless the
user explicitly asks for it. (Why not?)
psname rating
S
,
( )2

Projection (p)
age
35.0
55.5
28 yuppy 9 35.0
31 lubber 8 55.5
44 guppy 5 35.0
58 rusty 10 35.0
S2
sname rating
yuppy 9
lubber 8
guppy 5
rusty 10
)2(
,
S
ratingsname
p
page S( )2

Selection (s)
srating
S
8
2( )
sname rating
yuppy 9
rusty 10
p ssname rating rating
S
,
( ( ))
8
2
• Selects rows that satisfy selection condition.
• Result is a relation.
Schema of result is same as that of the input relation.
• Do we need to do duplicate elimination?
28 yuppy 9 35.0
31 lubber 8 55.5
44 guppy 5 35.0
58 rusty 10 35.0

Union and Set-Difference
• Both of these operations take two input relations,
which must be union-compatible:
– Same number of fields.
– `Corresponding’ fields have the same type.
• For which, if any, is duplicate elimination required?

Union
22 dustin 7 45.0
31 lubber 8 55.5
58 rusty 10 35.0
44 guppy 5 35.0
28 yuppy 9 35.0
22 dustin 7 45.0
31 lubber 8 55.5
58 rusty 10 35.0
28 yuppy 9 35.0
31 lubber 8 55.5
44 guppy 5 35.0
58 rusty 10 35.0
S1
S2
S S1 2

Set Difference
22 dustin 7 45.0
31 lubber 8 55.5
58 rusty 10 35.0
28 yuppy 9 35.0
31 lubber 8 55.5
44 guppy 5 35.0
58 rusty 10 35.0
S1
S2
22 dustin 7 45.0
S2 – S1
28 yuppy 9 35.0
44 guppy 5 35.0
S S1 2

Cross-Product
• S1  R1: Each row of S1 paired with each row of R1.
• Q: How many rows in the result?
• Result schema has one field per field of S1 and R1,
with field names `inherited’ if possible.
– May have a naming conflict: Both S1 and R1 have a field
with the same name.
– In this case, can use the renaming operator:
 ( ( , ), )C sid sid S R1 1 5 2 1 1  

Cross Product Example
(sid) sname rating age (sid) bid day
22 dustin 7 45.0 22 101 10/10/96
22 dustin 7 45.0 58 103 11/12/96
31 lubber 8 55.5 22 101 10/10/96
31 lubber 8 55.5 58 103 11/12/96
58 rusty 10 35.0 22 101 10/10/96
58 rusty 10 35.0 58 103 11/12/96
22 dustin 7 45.0
31 lubber 8 55.5
58 rusty 10 35.0
sid bid day
22 101 10/10/96
58 103 11/12/96
R1
S1
S1 x R1 =

Compound Operator: Intersection
• In addition to the 5 basic operators, there are several
additional “Compound Operators”
– These add no computational power to the language, but are
useful shorthands.
– Can be expressed solely with the basic ops.
• Intersection takes two input relations, which must be
union-compatible.
• Q: How to express it using basic operators?
R  S = R  (R  S)

Natural Join Example
22 dustin 7 45.0
31 lubber 8 55.5
58 rusty 10 35.0
sid bid day
22 101 10/10/96
58 103 11/12/96
R1
S1
S1 R1 =
sid sname rating age bid day
22 dustin 7 45.0 101 10/10/96
58 rusty 10 35.0 103 11/12/96

Other Types of Joins
• Condition Join (or “theta-join”):
• Result schema same as that of cross-product.
• May have fewer tuples than cross-product.
• Equi-Join: Special case: condition c contains only
conjunction of equalities.
R c S c R S  s ( )
(sid) sname rating age (sid) bid day
22 dustin 7 45.0 58 103 11/12/96
31 lubber 8 55.5 58 103 11/12/96
11
.1.1
RS
sidRsidS 


Examples
22 dustin 7 45.0
31 lubber 8 55.5
58 rusty 10 35.0
bid bname color
101 Interlake Blue
102 Interlake Red
103 Clipper Green
104 Marine Red
sid bid day
22 101 10/10/96
58 103 11/12/96
Reserves
Sailors
Boats

Find names of sailors who’ve reserved boat #103
• Solution 1: p ssname bid
serves Sailors(( Re ) )
103

• Solution 2: p ssname bid
serves Sailors( (Re ))
103


Find names of sailors who’ve reserved a red boat
• Information about boat color only available in
Boats; so need an extra join:
p ssname color red
Boats serves Sailors((
' '
) Re )

 
 A more efficient solution:
p p p ssname sid bid color red
Boats s Sailors( ((
' '
) Re ) )

 
 A query optimizer can find this given the first solution!

Find sailors who’ve reserved a red boat or a green
boat
• Can identify all red or green boats, then find
sailors who’ve reserved one of these boats:
 s( ,(
' ' ' '
))Tempboats
color red color green
Boats
  
p sname Tempboats serves Sailors( Re ) 

Find sailors who’ve reserved a red and a green boat
• Cut-and-paste previous slide?

 (Tempboats,(s
color'red'color'green'
Boats))
p sname Tempboats serves Sailors( Re ) 

Find sailors who’ve reserved a red and a green boat
• Previous approach won’t work! Must identify
sailors who’ve reserved red boats, sailors who’ve
reserved green boats, then find the intersection
(note that sid is a key for Sailors):
 p s( , ((
' '
) Re ))Tempred
sid color red
Boats serves


p sname Tempred Tempgreen Sailors(( ) ) 
 p s( , ((
' '
) Re ))Tempgreen
sid color green
Boats serves



Summary
• Relational Algebra: a small set of operators
mapping relations to relations
– Operational, in the sense that you specify the
explicit order of operations
– A closed set of operators! Can mix and match.
• Basic ops include: s, p, , , —
• Important compound ops: ,

SQL - A language for Relational DBs
• SQL (a.k.a. “Sequel”), standard
language
• Data Definition Language (DDL)
– create, modify, delete relations
– specify constraints
– administer users, security, etc.
• Data Manipulation Language (DML)
– Specify queries to find tuples that satisfy
criteria
– add, modify, remove tuples

SQL Overview
• CREATE TABLE <name> ( <field> <domain>, … )
• INSERT INTO <name> (<field names>)
VALUES (<field values>)
• DELETE FROM <name>
WHERE <condition>
• UPDATE <name>
SET <field name> = <value>
WHERE <condition>
• SELECT <fields>
FROM <name>
WHERE <condition>

Creating Relations in SQL
• Creates the Students relation.
– Note: the type (domain) of each field is
specified, and enforced by the DBMS
whenever tuples are added or modified.
CREATE TABLE Students
(sid CHAR(20),
name CHAR(20),
login CHAR(10),
age INTEGER,
gpa FLOAT)

Table Creation (continued)
• Another example: the Enrolled table
holds information about courses
students take.
CREATE TABLE Enrolled
(sid CHAR(20),
cid CHAR(20),
grade CHAR(2))

Adding and Deleting Tuples
• Can insert a single tuple using:
INSERT INTO Students (sid, name, login, age, gpa)
VALUES (‘53688’, ‘Smith’, ‘smith@ee’, 18, 3.2)
• Can delete all tuples satisfying some condition
(e.g., name = Smith):
DELETE
FROM Students S
WHERE S.name = ‘Smith’
Powerful variants of these commands are available;
more later!

Keys
• Keys are a way to associate tuples in
different relations
• Keys are one form of integrity constraint
(IC)
sid cid grade
53666 Carnatic101 C
53666 Reggae203 B
53650 Topology112 A
53666 History105 B
Enrolled Students
PRIMARY KeyFOREIGN Key

Primary Keys
• A set of fields is a superkey if:
– No two distinct tuples can have same values in all key fields
• A set of fields is a key for a relation if :
– It is a superkey
– No subset of the fields is a superkey
• what if >1 key for a relation?
– One of the keys is chosen (by DBA) to be the primary key.
Other keys are called candidate keys.
• E.g.
– sid is a key for Students.
– What about name?
– The set {sid, gpa} is a superkey.

Primary and Candidate Keys in SQL
• Possibly many candidate keys (specified using
UNIQUE), one of which is chosen as the primary key.
• Keys must be used carefully!
• “For a given student and course, there is a single grade.”
“Students can take only one course, and no two students
in a course receive the same grade.”
(sid CHAR(20)
cid CHAR(20),
grade CHAR(2),
PRIMARY KEY (sid,cid))
(sid CHAR(20)
cid CHAR(20),
grade CHAR(2),
PRIMARY KEY (sid),
UNIQUE (cid, grade))
vs.

Foreign Keys, Referential Integrity
• Foreign key: Set of fields in one relation
that is used to `refer’ to a tuple in another
relation.
– Must correspond to the primary key of the other
relation.
– Like a `logical pointer’.
• If all foreign key constraints are enforced,
referential integrity is achieved (i.e., no
dangling references.)

Foreign Keys in SQL
• E.g. Only students listed in the Students relation
should be allowed to enroll for courses.
– sid is a foreign key referring to Students:
(sid CHAR(20),cid CHAR(20),grade CHAR(2),
PRIMARY KEY (sid,cid),
FOREIGN KEY (sid) REFERENCES Students )
sid cid grade
53666 Carnatic101 C
53666 Reggae203 B
53650 Topology112 A
53666 History105 B
Enrolled
Students
11111 English102 A

Enforcing Referential Integrity
• Consider Students and Enrolled; sid in Enrolled is a
foreign key that references Students.
• What should be done if an Enrolled tuple with a non-
existent student id is inserted? (Reject it!)
• What should be done if a Students tuple is deleted?
– Also delete all Enrolled tuples that refer to it?
– Disallow deletion of a Students tuple that is referred to?
– Set sid in Enrolled tuples that refer to it to a default sid?
– (In SQL, also: Set sid in Enrolled tuples that refer to it to a
special value null, denoting `unknown’ or `inapplicable’.)
• Similar issues arise if primary key of Students tuple is
updated.

Integrity Constraints (ICs)
• IC: condition that must be true for any
instance of the database; e.g., domain
constraints.
– ICs are specified when schema is defined.
– ICs are checked when relations are modified.
• A legal instance of a relation is one that
satisfies all specified ICs.
– DBMS should not allow illegal instances.
• If the DBMS checks ICs, stored data is more
faithful to real-world meaning.
– Avoids data entry errors, too!

Where do ICs Come From?
• ICs are based upon the semantics of the real-world
that is being described in the database relations.
• We can check a database instance to see if an IC is
violated, but we can NEVER infer that an IC is true by
looking at an instance.
– An IC is a statement about all possible instances!
– From example, we know name is not a key, but the
assertion that sid is a key is given to us.
• Key and foreign key ICs are the most common; more
general ICs supported too.
• In the real world, sometimes the constraint should
hold but doesn’t --> data cleaning!

Relational Query Languages
• A major strength of the relational model:
supports simple, powerful querying of data.
• Queries can be written intuitively, and the
DBMS is responsible for efficient evaluation.
– The key: precise semantics for relational queries.
– Allows the optimizer to extensively re-order
operations, and still ensure that the answer does
not change.

The SQL Query Language
• The most widely used relational query
language.
– Current std is SQL:2003; SQL92 is a basic subset
• To find all 18 year old students, we can write:
SELECT *
FROM Students S
WHERE S.age=18
• To find just names and logins, replace the first line:
SELECT S.name, S.login
sid name age gpa
53666 Jones 18 3.4
53688 Smith 18 3.2
53650 Smith
login
jones@cs
smith@ee
smith@math 19 3.8

Querying Multiple Relations
• What does the following query compute?
SELECT S.name, E.cid
FROM Students S, Enrolled E
WHERE S.sid=E.sid AND E.grade='A'
sid cid grade
53831 Carnatic101 C
53831 Reggae203 B
53650 Topology112 A
53666 History105 B
Given the following instance of
Enrolled
S.name E.cid
Smith Topology112
we get:

Semantics of a Query
• A conceptual evaluation method for the previous
query:
1. do FROM clause: compute cross-product of Students and
Enrolled
2. do WHERE clause: Check conditions, discard tuples that fail
3. do SELECT clause: Delete unwanted fields
• Remember, this is conceptual. Actual evaluation will
be much more efficient, but must produce the same
answers.

Cross-product of Students and Enrolled Instances
S.sid S.name S.login S.age S.gpa E.sid E.cid E.grade
53666 Jones jones@cs 18 3.4 53831 Carnatic101 C
53666 Jones jones@cs 18 3.4 53832 Reggae203 B
53666 Jones jones@cs 18 3.4 53650 Topology112 A
53666 Jones jones@cs 18 3.4 53666 History105 B
53688 Smith smith@ee 18 3.2 53831 Carnatic101 C
53688 Smith smith@ee 18 3.2 53831 Reggae203 B
53688 Smith smith@ee 18 3.2 53650 Topology112 A
53688 Smith smith@ee 18 3.2 53666 History105 B
53650 Smith smith@math 19 3.8 53831 Carnatic101 C
53650 Smith smith@math 19 3.8 53831 Reggae203 B
53650 Smith smith@math 19 3.8 53650 Topology112 A
53650 Smith smith@math 19 3.8 53666 History105 B

Relational Model: Summary
• A tabular representation of data.
• Simple and intuitive, currently the most widely used
– Object-relational support in most products
– XML support added in SQL:2003, most systems
• Integrity constraints can be specified by the DBA,
based on application semantics. DBMS checks for
violations.
– Two important ICs: primary and foreign keys
– In addition, we always have domain constraints.
• Powerful query languages exist.
– SQL is the standard commercial one
• DDL - Data Definition Language
• DML - Data Manipulation Language

Databases for Programmers
• Programmers think about objects
(structs)
– Nested and interleaved
• Often want to “persist” these things
• Options
– encode opaquely and store
– translate to a structured form
• relational DB, XML file
– pros and cons?

YUCK!!
• How do I “relationalize” my objects?
• Have to write a converter for each
class?
• Think about when to save things into
the DB?
• Good news:
– Can all be automated
– With varying amounts of trouble

Object-Relational Mappings
• Roughly:
– Class ~ Entity Set
– Instance ~ Entity
– Data member ~ Attribute
– Reference ~ Foreign Key

Details, details
• We have to map this down to tables
• Which table holds which class of object?
• What about relationships?
• Solution #1: Declarative Configuration
– Write a description file (often in XML)
• E.g. Enterprise Java Beans (EJBs)
• Solution #2: Convention
– Agree to use some conventions
• E.g. Rails

Ruby on Rails
• Ruby: an OO scripting language
– and a pretty nice one, too
• Rails: a framework for web apps
– “convention over configuration”
• great for standard web-app stuff!
– allows overriding as needed
• Very ER-like

Rails and ER
• Models
– Employees
– Departments
lot
name
Employees
ssn
Works_In
since
dname
budgetdid
Departments

Some Rails “Models”
app/models/state.rb
class State < ActiveRecord::Base
has_many :cities
end
app/models/city.rb
class City < ActiveRecord::Base
belongs_to :state
end

Further Reading
• Chapter 18 (through 18.3) in Agile Web
Development with Rails

Database managment System Relational Algebra

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