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CSE 480: Database Systems

Lecture 15: Relational Algebra

Reference:

Read Chapter 6.1-6.3.1 of the textbook

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What is Relational Algebra?

A formal way to express a query in relational model– A query consists of relational expressions describing the

sequence of operators applied to obtain the query result

Example :

LName, Sex ( DNo=4 OR Salary>100000 (EMPLOYEEEMPLOYEE) )

– and are the operators

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Why Study Relational Algebra?

SELECT A.Name, B.GradeFROM A, BWHERE A.Id = B.Id

Name, Grade (Id,Name(A B))

Query processing in DBMS

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Relational Algebra Operator

Output of a relational algebra expression is a relation

Types of operators in a relational algebra expression– Unary: op(Relation)

– Binary: Relation op Relation

Relational algebra operators– select, project, union, set difference, Cartesian product

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Select Operator ()

condition (R) (NOT THE SAME AS SELECT in SQL)

Resulting relation – contains only tuples in R that satisfy the given condition

– has the same attributes (columns) as the original relation

Example:

Student Student Status=‘Junior’(StudentStudent)

1123 John 123 Main Junior1234 Lee 123 Main Senior5556 Mary 7 Lake Dr Freshman9876 Bart 5 Pine St Junior

1123 John 123 Main Junior9876 Bart 5 Pine St Junior

Id Name Address Status Id Name Address Status

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Selection Condition

select_condition (relation)– Selection condition acts as a filter to the rows in relation

Selection condition is a Boolean expression– Simple selection condition:

<attribute> operator <constant> <attribute> operator <attribute>

where operator: <, , , >, =,

– <condition> AND <condition>– <condition> OR <condition>– NOT <condition>

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Selection Condition - Examples

Id>3000 OR Status=‘Freshman’ (Student) selectivity = 2/4

Id>3000 AND Id <3999 (Student) selectivity = 0/4

NOT(Status=‘Senior’) (Student) selectivity = 3/4

Status‘Senior’ (Student) selectivity = 3/4

Student

1123 John 123 Main Junior1234 Lee 123 Main Senior5556 Mary 7 Lake Dr Freshman9876 Bart 5 Pine St Junior

Id Name Address StatusSelectivity

Fraction of tuples selected by a selection condition

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Selection Condition – Incorrect Queries

Find professors who taught both CS315 and CS305– CrsCode=‘CS315’ AND CrsCode=‘CS305’ (TEACHING) X

Find courses taught by CS professors– PROFESSOR.Id=Teaching.ProfId AND PROFESSOR.DeptId=‘CS’ (TEACHING) X

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Properties of SELECT operator

Commutative:– <condition1>( < condition2> (R)) = <condition2> ( < condition1> (R))

Cascade of SELECT operations may be applied in any order:– <cond1>(<cond2> (<cond3> (R)) = <cond2> (<cond3> (<cond1> ( R)))

Cascade of SELECT operations may be replaced by a single selection with a conjunction of all the conditions:– <cond1>(< cond2> (<cond3>(R)) = <cond1> AND < cond2> AND < cond3>(R)))

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Project Operator

Produces a relation containing a subset of columns of its input relation

– attribute list(R)

Example:

Student

Name,Status(Student)

1123 John 123 Main Junior1234 Lee 123 Main Freshman5556 Mary 7 Lake Dr Senior9876 Bart 5 Pine St Junior

John JuniorLee FreshmanMary SeniorBart Junior

Id Name Address Status Name Status

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Project Operator

Resulting relation has no duplicates; therefore it can have fewer tuples than the input relation

Example:

StudentStudent Address(StudentStudent)

123 Main7 Lake Dr5 Pine St

Address

1123 John 123 Main Junior1234 Lee 123 Main Freshman5556 Mary 7 Lake Dr Senior9876 Bart 5 Pine St Junior

Id Name Address Status

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Properties of PROJECT Operator

Number of tuples in the result of projection <list>(R) is always less or equal to the number of tuples in R– If list of projected attributes includes a superkey, the resulting

relation has the same number of tuples as the input relation

PROJECT operator is not commutative– <list1> ( <list2> (R) ) <list2> ( <list1> (R) )

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Relational Algebra Expressions

Find the Ids and names of junior undergraduate students

Id, Name ( Status=’Junior’ (StudentStudent) )

1123 John9876 Bart

ResultResult

Id Name

StudentStudent

1123 John 123 Main Junior1234 Lee 123 Main Freshman5556 Mary 7 Lake Dr Senior9876 Bart 5 Pine St Junior

Id Name Address Status

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Relational Algebra Expressions

We may want to apply several relational algebra operations one after the other

– We can write the operations as a single relational algebra expression by nesting the operations, or

– We can apply one operation at a time and create intermediate result relations.

– In the latter case, we must give names to the relations that hold the intermediate results.

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Single versus Sequence of operations

Query: Find the first name, last name, and salary of all employees who work in department number 5

We can write a single relational algebra expression: – FNAME, LNAME, SALARY( DNO=5(EMPLOYEE))

OR We can explicitly show the sequence of operations, giving a name to each intermediate relation:– DEP5_EMPS DNO=5(EMPLOYEE)

– RESULT FNAME, LNAME, SALARY (DEP5_EMPS)

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RENAME operator

The RENAME operator is denoted by

We may rename the attributes of a relation or the relation name or both– S (B1, B2, …, Bn )(R)

Change the relation name from R to S, and Change column (attribute) names to B1, B2, …..Bn

– S(R) : Change the relation name only to S

– (B1, B2, …, Bn )(R) : Change the column (attribute) names only to B1, B2, …..Bn

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Example

Change relation name to Emp and attributes to First_name, Last_name, and Salary

Emp(First_name, Last_name, Salary ) ( FNAME,LNAME,SALARY (EMPLOYEE))

or

Emp(First_name, Last_name, Salary) FNAME,LNAME,SALARY (EMPLOYEE)

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Set Operators

A relation is a set of tuples; therefore set operations are applicable: – Intersection:

– Union: – Set difference (Minus):

Set operators are binary operators– Operator: Relation Relation Relation

Result of set operation is a relation that has the same schema as the combining relations

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Examples: Set Operations

A B

x1 x2

x3 x4

A B

x1 x2

x5 x6

X Y X Y X – Y

X Y

A B

x1 x2

A B

x1 x2

x3 x4

x5 x6

A B

x3 x4

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Example

STUDENT

INSTRUCTOR

STUDENT

INSTRUCTOR

STUDENT – INSTRUCTOR

INSTRUCTOR – STUDENT

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Union Compatible Relations

Set operations are limited to union compatible relationsunion compatible relations– Two relations are union compatible if:

they have the same number of attributes, and the domains of corresponding attributes are type compatible (i.e.

dom(Ai)=dom(Bi) for i=1, 2, ..., n).

The resulting relation has the same attribute names as the first operand relation

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Example

Tables: StudentStudent (SSN, Name, Address, Status) ProfessorProfessor (Id, Name, Office, Phone)are not union compatible.

But

Name (StudentStudent) and Name (ProfessorProfessor)

are union compatible

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Exercise

Relations:– EmployeeEmployee (SSN, Name, Address)

– ProfessorProfessor (SSN, Office, Phone)

– StudentStudent (SSN, Status)

Find the SSN of student employees SSN (EmployeeEmployee) SSN (StudentStudent)

Find the SSN of employees who are not professors SSN (EmployeeEmployee) – SSN (ProfessorProfessor)

Find the SSN of employees who are neither professors nor students

SSN (EmployeeEmployee) – ( SSN (StudentStudent) SSN (ProfessorProfessor))

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Cartesian Product

R S is expensive to compute:– Number of columns = degree(R) + degree(S)

– Number of rows = number of rows (R) × number of rows (S)

A B C D A B C D x1 x2 y1 y2 x1 x2 y1 y2 x3 x4 y3 y4 x1 x2 y3 y4 x3 x4 y1 y2 RR SS x3 x4 y3 y4 RR SS

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Example

BrokerBroker (BrokerId, BrokerName)

ClientClient (ClientId, ClientName)

List all the possible Broker-Client pairs– Broker Client

BrokerID BrokerName

1 Merrill Lynch

2 Morgan Stanley

3 Salomon Smith Barney

ClientId ClientName

A11 Bill Gates

B11 Steve JobsBrokerId BrokerName ClientId ClientName

1 Merrill Lynch A11 Bill Gates

1 Merrill Lynch B11 Steve Jobs

2 Morgan Stanley A11 Bill Gates

2 Morgan Stanley B11 Steve Jobs

3 Salomon Smith Barney A11 Bill Gates

3 Salomon Smith Barney B11 Steve Jobs

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More Complex Examples

Find the names of employees who are not supervisors

EmployeeNonsupAnswer

EmployeeEmployeeNonsup

.Employee.Nonsup,,

SSNSSNLnameMInitFname

SuperSSNSSN

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Example

Find the names of employees who earn more than their supervisors

EmpEmployeeAnswer

Employee)Sal,SSN1(Emp

SalSalary AND SSN1SuperSSN,

,

LnameFname

SalarySSN

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Summary (Relational Algebra Operators)

Unary operators– SELECT condition(R)

– PROJECT Attribute-List(R)

– RENAME S(A1,A2,…Ak)(R)

Set operators– R S

– R S

– R – S or S – R

Cartesian product– R S