Relational algebra

Department of Information Technology 1Data base Technologies (ITB4201)
Introduction to Relational Algebra
Dr. C.V. Suresh Babu
Professor
Department of IT
Hindustan Institute of Science & Technology

Discussion Topics
• DBMS Architecture
• Relational Algebra
• Union and Differences
• Selectin
• Projection
• Cartesian Product
• Renaming
• Join
• Limitations of Relational Algebra
• Quiz

DBMS Architecture
How does a SQL engine work ?
• SQL query  relational
algebra plan
• Relational algebra plan 
Optimized plan
• Execute each operator of the
plan
Query Optimization
and Execution
Relational Operators
Files and Access Methods
Buffer Management
Disk Space Management
DB
PracticeTheory
Relational Algebra
Relational Model
Relational Calculus

Review – Why do we need Query Languages anyway?
• Two key advantages
– Less work for user asking query
– More opportunities for optimization
• Relational Algebra
– Theoretical foundation for SQL
– Higher level than programming language
• but still must specify steps to get desired result
• Relational Calculus
– Formal foundation for Query-by-Example
– A first-order logic description of desired result
– Only specify desired result, not how to get it

Relational Algebra
• Formalism for creating new relations from existing ones
• Its place in the big picture:
Declartive
query
language
Algebra Implementation
SQL,
relational calculus
Relational algebra
Relational bag algebra

Relational Algebra
• Five operators:
– Union: 
– Difference: -
– Selection: s
– Projection: P
– Cartesian Product: 
• Derived or auxiliary operators:
– Intersection, complement
– Joins (natural,equi-join, theta join, semi-join)
– Renaming: r

1. Union and 2. Difference
• R1  R2
• Example:
– ActiveEmployees  RetiredEmployees
• R1 – R2
• Example:
– AllEmployees -- RetiredEmployees

What about Intersection ?
• It is a derived operator
• R1  R2 = R1 – (R1 – R2)
• Also expressed as a join
• Example
– UnionizedEmployees  RetiredEmployees

3. Selection
• Returns all tuples which satisfy a condition
• Notation: sc(R)
• Examples
– sSalary > 40000 (Employee)
– sname = “Smith” (Employee)
• The condition c can be =, <, , >, , <>

sSalary > 40000 (Employee)
SSN Name Salary
1234545 John 200000
5423341 Smith 600000
4352342 Fred 500000
SSN Name Salary
5423341 Smith 600000
4352342 Fred 500000

4. Projection
• Eliminates columns, then removes duplicates
• Notation: P A1,…,An (R)
• Example: project social-security number and names:
– P SSN, Name (Employee)
– Output schema: Answer(SSN, Name)

P Name,Salary (Employee)
SSN Name Salary
1234545 John 200000
5423341 John 600000
4352342 John 200000
Name Salary
John 20000
John 60000

5. Cartesian Product
• Each tuple in R1 with each tuple in R2
• Notation: R1  R2
• Example:
– Employee  Dependents
• Very rare in practice; mainly used to express joins

Cartesian Product Example
Employee
Name SSN
John 999999999
Tony 777777777
Dependents
EmployeeSSN Dname
999999999 Emily
777777777 Joe
Employee x Dependents
Name SSN EmployeeSSN Dname
John 999999999 999999999 Emily
John 999999999 777777777 Joe
Tony 777777777 999999999 Emily
Tony 777777777 777777777 Joe

Renaming
• Changes the schema, not the instance
• Notation: r B1,…,Bn (R)
• Example:
– rLastName, SocSocNo (Employee)
– Output schema:
Answer(LastName, SocSocNo)

Renaming Example
Employee
Name SSN
John 999999999
Tony 777777777
LastName SocSocNo
John 999999999
Tony 777777777
rLastName, SocSocNo (Employee)

Natural Join
• Notation: R1 || R2
• Meaning: R1 || R2 = PA(sC(R1  R2))
• Where:
– The selection sC checks equality of all common attributes
– The projection eliminates the duplicate common attributes

Natural Join Example
Employee
Name SSN
John 999999999
Tony 777777777
Dependents
SSN Dname
999999999 Emily
777777777 Joe
Name SSN Dname
John 999999999 Emily
Tony 777777777 Joe
Employee Dependents =
PName, SSN, Dname(s SSN=SSN2(Employee x rSSN2, Dname(Dependents))

Natural Join
• R= S=
• R || S=
A B
X Y
X Z
Y Z
Z V
B C
Z U
V W
Z V
A B C
X Z U
X Z V
Y Z U
Y Z V
Z V W

Natural Join
• Given the schemas R(A, B, C, D), S(A, C, E), what is the
schema of R || S ?
• Given R(A, B, C), S(D, E), what is R || S ?
• Given R(A, B), S(A, B), what is R || S ?

Theta Join
• A join that involves a predicate
• R1 || q R2 = s q (R1  R2)
• Here q can be any condition

Eq-join
• A theta join where q is an equality
• R1 || A=B R2 = s A=B (R1  R2)
• Example:
– Employee || SSN=SSN Dependents
• Most useful join in practice

Semijoin
• R | S = P A1,…,An (R || S)
• Where A1, …, An are the attributes in R
• Example:
– Employee | Dependents

Semijoins in Distributed Databases
• Semijoins are used in distributed databases
SSN Name
. . . . . .
SSN Dname Age
. . . . . .
Employee
Dependents
network
Employee | ssn=ssn (s age>71 (Dependents))
T = P SSN s age>71 (Dependents)
R = Employee |T
Answer = R ||Dependents

Complex RA Expressions
Person Purchase Person Product
sname=fred sname=gizmo
P pidP ssn
seller-ssn=ssn
pid=pid
buyer-ssn=ssn
P name

Operations on Bags
A bag = a set with repeated elements
All operations need to be defined carefully on bags
• {a,b,b,c}{a,b,b,b,e,f,f}={a,a,b,b,b,b,b,c,e,f,f}
• {a,b,b,b,c,c} – {b,c,c,c,d} = {a,b,b,d}
• sC(R): preserve the number of occurrences
• PA(R): no duplicate elimination
• Cartesian product, join: no duplicate elimination
Important ! Relational Engines work on bags, not sets !
Reading assignment: 5.3 – 5.4

Finally: RA has Limitations !
• Cannot compute “transitive closure”
• Find all direct and indirect relatives of Fred
• Cannot express in RA !!! Need to write program
Name1 Name2 Relationship
Fred Mary Father
Mary Joe Cousin
Mary Bill Spouse
Nancy Lou Sister

Test Yourself
1. Which of the following is not a valid binary operation in the Relational Algebra?
a) Project
b) Union
c) Set difference
d) Cartesian product
2. The operation of a relation x, produces y, such that y contains only selected attributes of x. Such a operation is:?
a) Projection
b) Intersection
c) Union
d) Difference
3. Which of the following is not a valid unary operation in the relational algebra?
a) select
b) min
c) project
d) rename
4. Which clause from the following corresponds to the project operation of the relational algebra?
a) from
b) select
c) where
d) none of these
5. The intersect operation:
a) automatically eliminates duplicates
b) automatically eliminates duplicates. If we provide all clause with intersect
c) never eliminates duplicates
d) none of these

Answers
1. Which of the following is not a valid binary operation in the Relational Algebra?
a) Project
b) Union
c) Set difference
d) Cartesian product
2. The operation of a relation x, produces y, such that y contains only selected attributes of x. Such a operation is:?
a) Projection
b) Intersection
c) Union
d) Difference
3. Which of the following is not a valid unary operation in the relational algebra?
a) select
b) min
c) project
d) rename
4. Which clause from the following corresponds to the project operation of the relational algebra?
a) from
b) select
c) where
d) none of these
5. The intersect operation:
a) automatically eliminates duplicates
b) automatically eliminates duplicates. If we provide all clause with intersect
c) never eliminates duplicates
d) none of these

Relational algebra

Recommended

Recommended

More Related Content

What's hot

What's hot (20)

Similar to Relational algebra

Similar to Relational algebra (20)

More from Dr. C.V. Suresh Babu

More from Dr. C.V. Suresh Babu (20)

Recently uploaded

Recently uploaded (20)

Relational algebra