GrantWeddell - David R. Cheriton School of Computer...

Relational Algebra

Grant Weddell

Cheriton School of Computer ScienceUniversity of Waterloo

CS 348Introduction to Database Management

Winter 2017

CS 348 (Intro to DB Mgmt) Relational Algebra Winter 2017 1 / 15

Database Schema Used in Examples

RespEmpDeptNoProjNo

EmPTime

Project

EmEnDate

Emp_Act

EmpNoMajProj

MidInitLastName

Employee

WorkDeptHireDateSalary

FirstName

DeptNameMgrNoAdmrDept

DeptNo

Department

ProjNoActNoEmStDate

Relational Algebra

• the relational algebra consists of a set of operators• relational algebra is closed

• each operator takes as input zero or more relations• each operator defines a single output relation in terms of its input

relation(s)• relational operators can be composed to form expressions that

define new relations in terms of existing relations.

• Notation:R is a relation name; E is a relational algebra expression

Primary Relational Operators

• Relation Name: R

• Selection: �condition(E)• result schema is the same as E ’s• result instance includes the subset of the tuples of E that each

satisfies the condition• Projection: �attributes(E)

• result schema includes only the specified attributes• result instance could have as many tuples as E , except that

duplicates are eliminated

• Relation Name: R• Selection: �condition(E)

• result schema is the same as E ’s• result instance includes the subset of the tuples of E that each

satisfies the condition

• Projection: �attributes(E)• result schema includes only the specified attributes• result instance could have as many tuples as E , except that

• Relation Name: R• Selection: �condition(E)

• result schema is the same as E ’s• result instance includes the subset of the tuples of E that each

satisfies the condition• Projection: �attributes(E)

• result schema includes only the specified attributes• result instance could have as many tuples as E , except that

Primary Relational Operators (cont’d)

• Rename: �(R(F );E)• F is a list of terms of the form oldname ! newname• returns the result of E with columns renamed according to F .• remembers the result as R for future expressions

• Product: E1 � E2• result schema has all of the attributes of E1 and all of the attributes

of E2• result instance includes one tuple for every pair of tuples (one from

each expression result) in E1 and E2• sometimes called cross-product or Cartesian product• renaming is needed when E1 and E2 have common attributes

Primary Relational Operators (cont’d)

• Rename: �(R(F );E)• F is a list of terms of the form oldname ! newname• returns the result of E with columns renamed according to F .• remembers the result as R for future expressions

• Product: E1 � E2• result schema has all of the attributes of E1 and all of the attributes

of E2• result instance includes one tuple for every pair of tuples (one from

each expression result) in E1 and E2• sometimes called cross-product or Cartesian product• renaming is needed when E1 and E2 have common attributes

Cross Product Example

RAAA BBBa1 b1a2 b2a3 b3

SCCC DDDc1 d1c2 d2

R � SAAA BBB CCC DDDa1 b1 c1 d1a2 b2 c1 d1a3 b3 c1 d1a1 b1 c2 d2a2 b2 c2 d2a3 b3 c2 d2

Select,Project,Product Examples

• Note: Use Emp to mean the Employee relation, Proj the projectrelation

• Find the last names and hire dates of employees who make morethan $100000.

�LastName ;HireDate(�Salary>100000(Emp))

• For each project for which department E21 is responsible, find thename of the employee in charge of that project.

�ProjNo;LastName(�DeptNo=E21(�RespEmp=EmpNo(Emp � Proj )))

• Conditional join: E1 oncondition E2• equivalent to �condition (E1 � E2)• special case: equijoin

Proj on(RespEmp=EmpNo) Emp

• Natural join (E1 on E2)• The result of E1 on E2 can be formed by the following steps

1 form the cross-product of E1 and E2 (renaming duplicate attributes)2 eliminate from the cross product any tuples that do not have

matching values for all pairs of attributes common to schemas E1

and E2

3 project out duplicate attributes• if no attributes in common, this is just a product

• Conditional join: E1 oncondition E2• equivalent to �condition (E1 � E2)• special case: equijoin

Proj on(RespEmp=EmpNo) Emp

• Natural join (E1 on E2)• The result of E1 on E2 can be formed by the following steps

1 form the cross-product of E1 and E2 (renaming duplicate attributes)2 eliminate from the cross product any tuples that do not have

matching values for all pairs of attributes common to schemas E1

and E2

3 project out duplicate attributes• if no attributes in common, this is just a product

Example: Natural Join

• Consider the natural join of the Project and Department tables,which have attribute DeptNo in common

• the schema of the result will include attributes ProjName, DeptNo,RespEmp, MajProj, DeptName, MgrNo, and AdmrDept

• the resulting relation will include one tuple for each tuple in theProject relation (why?)

Set-Based Relational Operators

• Union (R [ S):• schemas of R and S must be “union compatible”• result includes all tuples that appear either in R or in S or in both

• Difference (R � S):• schemas of R and S must be “union compatible”• result includes all tuples that appear in R and that do not appear

in S• Intersection (R \ S):

• schemas of R and S must be “union compatible”• result includes all tuples that appear in both R and S

• Union Compatible:• Same number of fields.• ’Corresponding’ fields have the same type

• Intersection (R \ S):• schemas of R and S must be “union compatible”• result includes all tuples that appear in both R and S

in S• Intersection (R \ S):

• schemas of R and S must be “union compatible”• result includes all tuples that appear in both R and S

Relational Division

XA B Ca1 b1 c1a1 b1 c2a1 b2 c2a2 b1 c1a2 b1 c2a2 b2 c2a2 b3 c3a3 b1 c1

SB Cb1 c1b1 c2b2 c2

X =SAa1a2

Division is the Inverse of Product

RAa1a2

SB Cb1 c1b1 c2b2 c2

R � SA B Ca1 b1 c1a1 b1 c2a1 b2 c2a2 b1 c1a2 b1 c2a2 b2 c2

(R � S)=SAa1a2

Summary of Relational Operators

E ::= Rj �condition(E)

j �attributes(E)

j �(R(F );E)

j E1 � E2j E1 oncondition E2j E1 on E2j E1 [ E2j E1 \ E2j E1 � E2j E1 = E2

Algebraic Equivalences

• This:

�ProjNo;LastName(�DeptNo=E21(�RespEmp=EmpNo(E � P)))

• is equivalent to this:

�ProjNo;LastName(�DeptNo=E21(E onRespEmp=EmpNo P))

�ProjNo;LastName(E onRespEmp=EmpNo �DeptNo=E21(P))

�ProjNo;LastName( ( �LastName;EmpNo(E)) onRespEmp=EmpNo

( �ProjNo;RespEmp(�DeptNo=E21(P))))

• More on this topic later when we discuss database tuning. . .

• This:

Relational Completeness

Definition (Relationally Complete)A query language that is at least as expressive as relational algebra issaid to be relationally complete.

• The following languages are all relationally complete:• safe relational calculus• relational algebra• SQL

• SQL has additional expressive power because it captures duplicatetuples, unknown values, aggregation, ordering, . . .

GrantWeddell - David R. Cheriton School of Computer...

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