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TheRelationalModel

Lecture18

Today’sLecture

1. TheRelationalModel&RelationalAlgebra

2. RelationalAlgebraPt.II

2

Lecture18

1.TheRelationalModel&RelationalAlgebra

3

Lecture18>Section1

Whatyouwilllearnaboutinthissection

1. TheRelationalModel

2. RelationalAlgebra:BasicOperators

3. Execution

4. ACTIVITY:FromSQLtoRA&Back

4

Lecture18>Section1

Motivation

TheRelationalmodelisprecise,implementable,andwecanoperateonit

(query/update,etc.)

Databasemapsinternallyintothisprocedurallanguage.

Lecture18>Section1>TheRelationalModel

ALittleHistory

• RelationalmodelduetoEdgar“Ted”Codd,amathematicianatIBMin1970• ARelationalModelofDataforLargeSharedDataBanks". CommunicationsoftheACM 13 (6):377–387

• IBMdidn’twanttouserelationalmodel(takemoneyfromtheirInformationManagementSystem)

WonTuringaward1981


TheRelationalModel:Schemata

• RelationalSchema:


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

AttributesString,float,int,etc.arethedomains oftheattributes

Relationname

8

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


9

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


10

TheRelationalModel:DataStudent

Arelationalinstance isaset oftuplesallconformingtothesameschema

Recall: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

14


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

2nd PartoftheModel:Querying

“FindnamesofallstudentswithGPA>3.5”

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

Actually,Ishowedhowtodothistranslationforamuchricherlanguage!


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

Tomakethishappen,weneedtotranslatethedeclarativequeryintoaseriesofoperators…we’llseethisnext!

Virtuesofthemodel

• Physicalindependence(logicaltoo),Declarative

• Simple,elegantclean:Everythingisarelation

• Whydidittakemultipleyears?• Doubteditcouldbedoneefficiently.


RelationalAlgebra

Lecture18>Section1 >RelationalAlgebra

RDBMSArchitecture

HowdoesaSQLenginework?

SQLQuery

RelationalAlgebra(RA)

Plan

OptimizedRAPlan Execution

Declarativequery(fromuser)

Translatetorelationalalgebraexpresson

Findlogicallyequivalent- butmoreefficient- RAexpression

Executeeachoperatoroftheoptimizedplan!


RDBMSArchitecture


SQLQuery


Plan


RelationalAlgebraallowsustotranslatedeclarative(SQL)queriesintopreciseandoptimizable expressions!


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

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


We’lllookatthesefirst!

Andalsoatoneexampleofaderivedoperator(naturaljoin)andaspecialoperator(renaming)


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)



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;


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)


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• TheprojectionPAUBeliminatestheduplicate

commonattributes

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:ConvertingSFWQuery->RA

SELECT DISTINCTgpa,address

FROM Students S,People P

WHERE gpa > 3.5 ANDsname = pname;

HowdowerepresentthisqueryinRA?

Π"#$,$AAC744(𝜎"#$&'.)(𝑆 ⋈ 𝑃))


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

LogicalEquivalece ofRAPlans

• GivenrelationsR(A,B)andS(B,C):

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

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

We’lllookatthisinmoredepthlaterinthelecture…


RDBMSArchitecture


SQLQuery


Plan


WesawhowwecantransformdeclarativeSQLqueriesintoprecise,compositionalRAplans


RDBMSArchitecture


SQLQuery


Plan


We’lllookathowtothenoptimizetheseplanslaterinthisclass


RDBMSArchitecture

HowistheRA“plan”executed?

SQLQuery


Plan


Wealreadyknowhowtoexecuteallthebasicoperators!


RAPlanExecution

• NaturalJoin/Join:• Wesawhowtousememory&IOcostconsiderationstopickthecorrectalgorithmtoexecuteajoin with(BNLJ,SMJ,HJ…)!

• Selection:• Wesawhowtouseindexestoaidselection• Canalwaysfallbackonscan/binarysearchaswell

• Projection:• Themainoperationhereisfindingdistinctvaluesoftheprojecttuples;webrieflydiscussedhowtodothiswithe.g.hashingorsorting

Wealreadyknowhowtoexecuteallthebasicoperators!


Activity-16-1.ipynb

40

Lecture18>Section1>ACTIVITY

2.Adv.RelationalAlgebra

41

Lecture18>Section2

Whatyouwilllearnaboutinthissection

1. SetOperationsinRA

2. FancierRA

3. Extensions&Limitations

42

Lecture18>Section2

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

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


We’lllookatthese

Andalsoatsomeofthesederivedoperators

Lecture18>Section2

1.Union(È) and2.Difference(–)

• R1È R2• Example:• ActiveEmployeesÈ RetiredEmployees

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

R1 R2

R1 R2

Lecture18>Section2 >SetOperations

WhataboutIntersection(Ç) ?

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

• UnionizedEmployeesÇ RetiredEmployees

R1 R2

Lecture18>Section2 >SetOperations

FancierRA

Lecture18>Section2>FancierRA

ThetaJoin(⋈q)

• Ajointhatinvolvesapredicate• R1⋈q R2=s q (R1´ R2)• Hereq canbeanycondition

SELECT *FROM

Students,PeopleWHERE q;

SQL:

RA:𝑆𝑡𝑢𝑑𝑒𝑛𝑡𝑠 ⋈R 𝑃𝑒𝑜𝑝𝑙𝑒


Notethatnaturaljoinisathetajoin+aprojection.


Equi-join(⋈A=B)

• Athetajoinwhereq isanequality• R1⋈A=B R2=s A=B (R1´ R2)• Example:• Employee⋈SSN=SSN Dependents

SELECT *FROM

Students S,People P

WHERE sname = pname;

SQL:

RA:𝑆 ⋈45$67M#5$67 𝑃


Mostcommonjoininpractice!


Semijoin (⋉)

• R⋉ S=P A1,…,An (R⋈ S)• WhereA1,…,An aretheattributesinR• Example:• Employee⋉Dependents

SELECT DISTINCTsid,sname,gpa

FROM Students,People

WHEREsname = pname;

SQL:

RA:

𝑆𝑡𝑢𝑑𝑒𝑛𝑡𝑠 ⋉ 𝑃𝑒𝑜𝑝𝑙𝑒



SemijoinsinDistributedDatabases• Semijoins areoftenusedtocomputenaturaljoinsindistributeddatabases

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

Sendlessdatatoreducenetworkbandwidth!


RAExpressionsCanGetComplex!

PersonPurchasePersonProduct

sname=fred sname=gizmo

P pidP ssn

seller-ssn=ssn

pid=pid

buyer-ssn=ssn

P name


Multisets

Lecture18>Section2>Extensions&Limitations

RecallthatSQLusesMultisets

53

Tuple

(1,a)

(1,a)

(1, b)

(2,c)

(2,c)

(2,c)

(1,d)

(1,d)

Tuple 𝝀(𝑿)

(1,a) 2

(1,b) 1

(2,c) 3

(1, d) 2EquivalentRepresentationsofaMultiset

Multiset X

Multiset X

Note:Inasetallcountsare{0,1}.

𝝀 𝑿 =“CountoftupleinX”(Itemsnotlistedhaveimplicitcount0)


GeneralizingSetOperationstoMultisetOperations

54

Tuple 𝝀(𝑿)

(1,a) 2

(1,b) 0

(2,c) 3

(1, d) 0

Multiset X

Tuple 𝝀(𝒀)

(1,a) 5

(1,b) 1

(2,c) 2

(1, d) 2

Multiset Y

Tuple 𝝀(𝒁)

(1,a) 2

(1,b) 0

(2,c) 2

(1, d) 0

Multiset Z

∩ =

𝝀 𝒁 = 𝒎𝒊𝒏(𝝀 𝑿 , 𝝀 𝒀 )Forsets,thisisintersection


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Tuple 𝝀(𝑿)

(1,a) 2

(1,b) 0

(2,c) 3

(1, d) 0

Multiset X

Tuple 𝝀(𝒀)

(1,a) 5

(1,b) 1

(2,c) 2

(1, d) 2

Multiset Y

Tuple 𝝀(𝒁)

(1,a) 7

(1,b) 1

(2,c) 5

(1, d) 2

Multiset Z

∪ =

𝝀 𝒁 = 𝝀 𝑿 + 𝝀 𝒀Forsets,

thisisunion

GeneralizingSetOperationstoMultisetOperations


OperationsonMultisets

AllRAoperationsneedtobedefinedcarefullyonbags

• sC(R):preservethenumberofoccurrences

• PA(R):noduplicateelimination

• Cross-product,join:noduplicateelimination

Thisisimportant- relationalenginesworkonmultisets,notsets!


RAhasLimitations!

• Cannotcompute“transitiveclosure”

• FindalldirectandindirectrelativesofFred• CannotexpressinRA!!!

• NeedtowriteCprogram,useagraphengine,ormodernSQL…

Name1 Name2 RelationshipFred Mary FatherMary Joe CousinMary Bill SpouseNancy Lou Sister


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