CS639:DataManagementfor

DataScienceLecture4:RelationalAlgebra

TheodorosRekatsinas1

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Announcements• Assignment1

• HintsandGrading

Today’sLecture

1. TheRelationalModel&RelationalAlgebra

2. RelationalAlgebraPt.II

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1.TheRelationalModel&RelationalAlgebra

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Whatyouwilllearnaboutinthissection

1. TheRelationalModel

2. RelationalAlgebra:BasicOperators

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Levelsofabstraction

Motivation

TheRelationalmodelisprecise,implementable,andwecanoperateonit

(query/update,etc.)

Databasemapsinternallyintothisprocedurallanguage.

TheRelationalModel:Schemata

• RelationalSchema:

Students(sid: string, name: string, gpa: float)

AttributesString,float,int,etc.arethedomains oftheattributes

Relationname

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TheRelationalModel:Data

sid name gpa

001 Bob 3.2

002 Joe 2.8

003 Mary 3.8

004 Alice 3.5

Student

Anattribute (orcolumn)isatypeddataentrypresentineachtupleintherelation

Thenumberofattributesisthearity oftherelation

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TheRelationalModel:Data

sid name gpa

001 Bob 3.2

002 Joe 2.8

003 Mary 3.8

004 Alice 3.5

Student

Atuple orrow (orrecord)isasingleentryinthetablehavingtheattributesspecifiedbytheschema

Thenumberoftuplesisthecardinality oftherelation

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TheRelationalModel:DataStudent

Arelationalinstance isaset oftuplesallconformingtothesameschema

InpracticeDBMSsrelaxthesetrequirement,andusemultisets.

sid name gpa

001 Bob 3.2

002 Joe 2.8

003 Mary 3.8

004 Alice 3.5

• Arelationalschema describesthedatathatiscontainedinarelationalinstance

ToReiterate

LetR(f1:Dom1,…,fm:Domm)bearelationalschema then,aninstanceofRisasubsetofDom1 xDom2 x…xDomn

Inthisway,arelationalschema Risatotalfunctionfromattributenames totypes

• Arelationalschema describesthedatathatiscontainedinarelationalinstance

OneMoreTime

ArelationRofarity t isafunction:R:Dom1 x…xDomt à {0,1}

Then,theschemaissimplythesignatureofthefunction

I.e.returnswhetherornotatupleofmatchingtypesisamemberofit

Noteherethatordermatters,attributenamedoesn’t…We’ll(mostly)workwiththeothermodel(lastslide)in

whichattributenamematters,orderdoesn’t!

Arelationaldatabase

• Arelationaldatabaseschema isasetofrelationalschemata,oneforeachrelation

• Arelationaldatabaseinstance isasetofrelationalinstances,oneforeachrelation

Twoconventions:1. Wecallrelationaldatabaseinstancesassimplydatabases2. Weassumeallinstancesarevalid,i.e.,satisfythedomainconstraints

RemembertheCMS

• RelationDBSchema• Students(sid:string,name:string,gpa:float)• Courses(cid:string,cname:string,credits:int)• Enrolled(sid:string,cid:string,grade:string)

Sid Name Gpa101 Bob 3.2123 Mary 3.8

Students

cid cname credits564 564-2 4308 417 2

Coursessid cid Grade123 564 A

Enrolled

RelationInstances

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Notethattheschemasimposeeffectivedomain/typeconstraints,i.e.Gpacan’tbe“Apple”

2nd PartoftheModel:Querying

“FindnamesofallstudentswithGPA>3.5”

Wedon’ttellthesystem howorwhere togetthedata- justwhatwewant,i.e.,Queryingisdeclarative

SELECT S.nameFROM Students SWHERE S.gpa > 3.5;

Tomakethishappen,weneedtotranslatethedeclarativequeryintoaseriesofoperators…we’llseethisnext!

Virtuesofthemodel

• Physicalindependence(logicaltoo),Declarative

• Simple,elegantclean:Everythingisarelation

RelationalAlgebra

RDBMSArchitecture

HowdoesaSQLenginework?

SQLQuery

RelationalAlgebra(RA)

Plan

OptimizedRAPlan Execution

Declarativequery(fromuser)

Translatetorelationalalgebraexpression

Findlogicallyequivalent- butmoreefficient- RAexpression

Executeeachoperatoroftheoptimizedplan!

RDBMSArchitecture

HowdoesaSQLenginework?

SQLQuery

RelationalAlgebra(RA)

Plan

OptimizedRAPlan Execution

RelationalAlgebraallowsustotranslatedeclarative(SQL)queriesintopreciseandoptimizableexpressions!

• Fivebasicoperators:1. Selection: s2. Projection:P3. CartesianProduct:´4. Union:È5. Difference:-

• Derivedorauxiliaryoperators:• Intersection,complement• Joins(natural,equi-join,thetajoin,semi-join)• Renaming: r• Division

RelationalAlgebra(RA)

Keepinmind:RAoperatesonsets!

• RDBMSsusemultisets,howeverinrelationalalgebraformalismwewillconsidersets!

• Also:wewillconsiderthenamedperspective,whereeveryattributemusthaveauniquename• àattributeorderdoesnotmatter…

NowontothebasicRAoperators…

• Returnsalltupleswhichsatisfyacondition• Notation: sc(R)• Examples• sSalary >40000 (Employee)• sname =“Smith” (Employee)

• Theconditionccanbe=,,³,

1.Selection(𝜎)

SELECT *FROM StudentsWHERE gpa > 3.5;

SQL:

RA:𝜎"#$&'.)(𝑆𝑡𝑢𝑑𝑒𝑛𝑡𝑠)

Students(sid,sname,gpa)

sSalary >40000 (Employee)

SSN Name Salary1234545 John 2000005423341 Smith 6000004352342 Fred 500000

SSN Name Salary5423341 Smith 6000004352342 Fred 500000

Anotherexample:

• Eliminatescolumns,thenremovesduplicates• Notation:P A1,…,An (R)• Example:projectsocial-securitynumberandnames:• P SSN,Name (Employee)• Outputschema:Answer(SSN,Name)

2.Projection(Π)

SELECT DISTINCTsname,gpa

FROM Students;

SQL:

RA:Π45$67,"#$(𝑆𝑡𝑢𝑑𝑒𝑛𝑡𝑠)

Students(sid,sname,gpa)

P Name,Salary (Employee)

SSN Name Salary1234545 John 2000005423341 John 6000004352342 John 200000

Name SalaryJohn 200000John 600000

Anotherexample:

NotethatRAOperatorsareCompositional!

SELECT DISTINCTsname,gpa

FROM StudentsWHERE gpa > 3.5;

Students(sid,sname,gpa)

HowdowerepresentthisqueryinRA?

Π45$67,"#$(𝜎"#$&'.)(𝑆𝑡𝑢𝑑𝑒𝑛𝑡𝑠))

𝜎"#$&'.)(Π45$67,"#$(𝑆𝑡𝑢𝑑𝑒𝑛𝑡𝑠))

Aretheselogicallyequivalent?

• EachtupleinR1witheachtupleinR2• Notation:R1´ R2• Example:• Employee´ Dependents

• Rareinpractice;mainlyusedtoexpressjoins

3.Cross-Product(×)

SELECT *FROM Students, People;

SQL:

RA:𝑆𝑡𝑢𝑑𝑒𝑛𝑡𝑠×𝑃𝑒𝑜𝑝𝑙𝑒

Students(sid,sname,gpa)People(ssn,pname,address)

ssn pname address1234545 John 216 Rosse

5423341 Bob 217 Rosse

sid sname gpa001 John 3.4

002 Bob 1.3

𝑆𝑡𝑢𝑑𝑒𝑛𝑡𝑠×𝑃𝑒𝑜𝑝𝑙𝑒

×

ssn pname address sid sname gpa1234545 John 216 Rosse 001 John 3.4

5423341 Bob 217 Rosse 001 John 3.4

1234545 John 216 Rosse 002 Bob 1.3

5423341 Bob 216 Rosse 002 Bob 1.3

People StudentsAnotherexample:

• Changestheschema,nottheinstance• A‘special’operator- neitherbasicnorderived• Notation:r B1,…,Bn (R)

• Note:thisisshorthandfortheproperform(sincenames,notordermatters!):• r A1àB1,…,AnàBn (R)

Renaming(𝜌)

SELECTsid AS studId,sname AS name,gpa AS gradePtAvg

FROM Students;

SQL:

RA:𝜌4?@ABA,5$67,"C$A7D?EF"(𝑆𝑡𝑢𝑑𝑒𝑛𝑡𝑠)

Students(sid,sname,gpa)

Wecareaboutthisoperatorbecause weareworkinginanamedperspective

sid sname gpa001 John 3.4

002 Bob 1.3

𝜌4?@ABA,5$67,"C$A7D?EF"(𝑆𝑡𝑢𝑑𝑒𝑛𝑡𝑠)

Students

studId name gradePtAvg001 John 3.4

002 Bob 1.3

Students

Anotherexample:

• Notation:R1⋈R2

• JoinsR1 andR2 onequalityofallsharedattributes• IfR1 hasattributesetA,andR2 hasattributesetB,andtheyshareattributesA⋂B=C,canalsobewritten:R1⋈ 𝐶R2

• OurfirstexampleofaderivedRA operator:• Meaning:R1⋈ R2 =PAUB(sC=D(𝜌J→L(R1)´ R2))• Where:

• Therename𝜌J→L renamesthesharedattributesinoneoftherelations• TheselectionsC=Dchecksequalityofthesharedattributes• TheprojectionPAUBeliminatestheduplicatecommonattributes

NaturalJoin(⋈)

SELECT DISTINCTssid, S.name, gpa,ssn, address

FROM Students S,People P

WHERE S.name = P.name;

SQL:

RA:𝑆𝑡𝑢𝑑𝑒𝑛𝑡𝑠 ⋈ 𝑃𝑒𝑜𝑝𝑙𝑒

Students(sid,name,gpa)People(ssn,name,address)

ssn P.name address1234545 John 216 Rosse

5423341 Bob 217 Rosse

sid S.name gpa001 John 3.4

002 Bob 1.3

𝑆𝑡𝑢𝑑𝑒𝑛𝑡𝑠 ⋈ 𝑃𝑒𝑜𝑝𝑙𝑒

⋈

sid S.name gpa ssn address001 John 3.4 1234545 216 Rosse

002 Bob 1.3 5423341 216 Rosse

PeoplePStudentsSAnotherexample:

NaturalJoin

• GivenschemasR(A,B,C,D),S(A,C,E),whatistheschemaofR⋈S?

• GivenR(A,B,C),S(D,E),whatisR⋈S?

• GivenR(A,B),S(A,B),whatisR⋈S?

Example:ConvertingSQLQuery->RA

SELECT DISTINCTgpa,address

FROM Students S,People P

WHERE gpa > 3.5 ANDsname = pname;

Π"#$,$AAC744(𝜎"#$&'.)(𝑆 ⋈ 𝑃))

Students(sid,sname,gpa)People(ssn,sname,address)

LogicalEquivalenceofRAPlans

• GivenrelationsR(A,B)andS(B,C):

• Here,projection&selectioncommute:• 𝜎EM)(ΠE(𝑅)) = ΠE(𝜎EM)(𝑅))

• Whatabouthere?• 𝜎EM)(ΠP(𝑅))?= ΠP(𝜎EM)(𝑅))

1.Union(È) and2.Difference(–)

• R1È R2• Example:• ActiveEmployeesÈ RetiredEmployees

• R1– R2• Example:• AllEmployees -- RetiredEmployees

R1 R2

R1 R2

WhataboutIntersection(Ç) ?

• Itisaderivedoperator• R1Ç R2=R1– (R1– R2)• Alsoexpressedasajoin!• Example

• UnionizedEmployeesÇ RetiredEmployees

R1 R2

RAExpressionsCanGetComplex!

PersonPurchasePersonProduct

sname=fred sname=gizmo

P pidP ssn

seller-ssn=ssn

pid=pid

buyer-ssn=ssn

P name

RAhasLimitations!

• Cannotcompute“transitiveclosure”

• FindalldirectandindirectrelativesofFred• CannotexpressinRA!!!

• NeedtowriteCprogram,useagraphengine,ormodernSQL…

Name1 Name2 RelationshipFred Mary FatherMary Joe CousinMary Bill SpouseNancy Lou Sister