relational algebra...relational algebra ´ unary operations ´selection: σ ´project: π ´rename:...

Relational AlgebraCS 4750 – Database Systems

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Why Relational Algebra?

´ Relational Algebraisthemathematical foundation forhowwedefinerelationships andgetdata inandoutofthesystem

´ Doeseveryquerylanguage (e.g.SQL)followRelational Algebra?No!

´ Someexamples…´EasytodoinRA

´VeryhardinSQL

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Relational Algebra

´ UnaryOperations

´Selection:σ´Project:π´Rename:ρ

Anoperatortakesone(unary)ortwo(binary)relations asinputandreturnsanewsingle relationasoutput

´ BinaryOperations

´Union:U

´Setdifference:−´Cartesian product: ×

´ Sixfundamental operations:

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Relational Algebra´ Basedonthesixfundamentaloperationswecandefine…´ Additionalrelationaloperations(addnoadditional‘power’)

´ Setintersection:∩´ Naturaljoin: ⋈

´ Division:÷´ Assignment:←

´ Extendedoperations(withadditionalexpressiveness)´ AggregatefunctionG

´ Outerjoin

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Selection (σ) – Sigma

´ σp(r), wherep =selection predicate,r=relation(table)

´ Comparisons intheselection predicate:=, <, ≤, >, ≥, ≠´Comparisonsarenotonlybetweenanattributeandavalue,butmayalsobeacomparisonbetweentwoattributes

´ Combining severalpredicates intoalargerpredicatecanbedonebyusingtheconnectives: AND,OR,andNOT

´ Selects specific tuples (rows)fromarelation (table)

´ Limits therowsbased onwhatwe’relookingfor

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Selection (σ)

´σcity = “Zurich” AND postcode > 8010 (customer)´ “Findalltuples inthecustomerrelationthatareinthecityofZurichand

haveapostcode greaterthan8010.”

´ Result?

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CD store Relational Database example © B.Signer

Selection (σ)

´ !!!Attention !!!

´ Theselection operation inrelational algebrahasadifferent meaningthattheSELECT statement used inSQL´SELECTinSQLcorrespondstoaprojection inrelationalalgebra

´Wewilltalkaboutprojectionnext

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Selection (SQL) == Projection (RA)8

´ Example: ThetableE(forEMPLOYEE)

Projection (π) – Pi

´ πA1, A2, …, Am (r), whereAi =attribute

´ Returnspecificattributes (columns)fromarelation

´ Identical tuplescollapse intoasingle tuple(duplicates removed)intheresulting relation

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Projection (π) – Pi

Identical tuplescollapse intoasingle tuple(duplicates removed)intheresulting relation

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Rename (ρ) – Rho

´ ρx(E), renames theresultofexpression Etox

´ ρx(A1, A2, …, An)(E), renamestheresultofexpression Etoxandrenames theattributes toA1, A2, …, An

´ρperson(name, location) (πname, city (customer))

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Rename the result of expression E to x; rename the resultant attributes to A1, A2, …, An

ρfriend_contact(ID, primary_email, alternative_email)(contact)

ID email1mi1y [email protected]

mi1y [email protected]

[email protected]

[email protected]

contact

= ID primary_emailmi1y [email protected]


[email protected]

[email protected]

friend_contact

Rename (ρ) – Another Example

Union (U)

´ r U s, whererandsaretworelations´ Unify(combine)tuples fromtworelations

´ Twothingsmust apply´ randsmusthavethesamedegree(samenumberofattributes)

´Thecorrespondingattributedomainsmusthaveacompatibletype

´ “Or”:“FindthenamesofSailorswhorentedaGreenor Redboat”

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Intersect (∩)

´ r ∩ s, whererandsaretworelations´ Unify(combine)tuples thatarepresent inbothrelations

´ SamerulesforUnionmustapply

´ Union(U)is“or”

´ Intersect(∩)is“and”

´ “And”:“Findthenames ofpeople living inVirginiaandwhoownahome”

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Set Difference (−)

´ r − s, whererandsaretworelations´ Findthetuplesthatareinonerelationbutarenotinanother

´ Twothingsmust apply´ randsmusthavethesamedegree(samenumberofattributes)

´Thecorrespondingattributedomainsmusthaveacompatibletype

´ πname (supplier) − πname (customer)´ “Findthenamesofsupplierswhoarenotcustomers”

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Compatible type

CS1_studentscomputingID name

mi1y Mickey

mi2e Minnie

do3d Donald

da4y Daisy

do5d Donald

Find tuples that are in one relation but are not in another

Same number of attributes

computingID name

go9y Goofy

mi2e Minnie

do3d Donald

pl4o Pluto

do5d Donald

pe6e Pete

CS1_students – CS2_students

computingID name

mi1y Mickey

da4y Daisy

Attribute names match

− =

CS2_students

Set Difference (−) – Another Example

Cartesian (Cross) Product (×)

´ r × s, whererandsaretworelations´ Combines informationfromanytwotables

´ Schemaoftheresult isthecombinedschemas ofthetworelations´Relation1has3attributes

´Relation2has4attributes

´TheresultingrelationaftercombiningthetwotablesusingtheCartesianproductwillhave7attributes

´Theattributesofthe1st relationfollowedbytheattributesofthe2nd

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Cartesian Product (×)

´ Question: Customerwhoate>1itemfromthemenu

´ Crossatablewithitself, canrenameoneofthetables

´ MatchwhereC1 =C2 andD1 ≠ D2

´ So,customerAnnate>1dish

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Customer Dish

Ann Chicken

Bob Fish

Ann Veal

EatLog

C1 D1 C2 D2

Ann Ch Ann Ch

Ann Ch Bob Fi

Ann Ch Ann Ve

Bob Fi Ann Ch

Bob Fi Bob Fi

Bob Fi Ann Ve

Ann Ve Ann Ch

Ann Ve Bob Fi

Ann Ve Ann Ve

EatLogxEatLog

à

à

Cartesian Product (×) Example19

A B Ca b cd a fc b d

D E Fx y zp q r

A B C D E Fa b c x y za b c p q rd a f x y zd a f p q rc b d x y zc b d p q r

R

S

R × SNote: RxS is not the same as SxR!

ID email


mi2e [email protected]


ID email




ID email




Combinetworelations(mergecolumns)Cartesian(Cross)Product(×)– AnotherExampleUsuallynotmeaningfulwhenitisperformedalone.

×

C1× C2

=

ID email




(Let’scall itC1)contact

(Let’scall itC2)

ID1 email1










ID2 email2




Meaningfulwhenitisfollowedbyotheroperations.Findallstudentswhohavemorethanoneemail(2steps:crossproduct,thenselect tuples)

σID1=ID2 ^ email1≠ email2 (C1× C2)

ID1 email1



ID2 email2



Natural Join (⋈) (“Inner Join”)

´ r ⋈ s = πR US (σr.A1=s.A1 ^ r.A2=s.A2 ^ … ^ r.An=s.An (r x s))´ whereR U S = {A1, A2, …, An}´ Takethecrossproductoftwotables, select therowsyoucareabout

´ Binaryoperation

´ Combine certainSELECTIONSandaCARTESIANPRODUCTintooneoperation´FormsaCartesianproductofitstwoarguments

´Performsaselectionforcingequalityonthoseattributesthatappearinbothrelationschemas

´Removesduplicates

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Natural Join (⋈) (“Inner Join”)

´r ⋈ s = πR US (σr.A1=s.A1 ^ r.A2=s.A2 ^ … ^ r.An=s.An (r x s))

´ Natural joinisanassociative operation

´ Usedquiteoften!

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It performs a PROJECTION on the unionof all the attributes of both relations

It forms a SELECTION on tuples with equal, similarly named attributes

That is, keep all tuples of the cartesian product r x s that have the same value for the shared attributes or r(R) and s(S)

Natural Join (⋈) Example23

A B Ca b cd b cb b fc a d

B C Db c db c ea d b

A B C Da b c da b c ed b c dd b c ec a d bR

S

R ⋈ S

TheNaturalJoineliminatesreplicatedattributes(e.g.columns“B”and“C”showuponlyonceintheR ⋈ S table.)

Assignment (←)

´ v← E´ Theexpression (E)ontheRHSoftheassignment operatorisassigned

temporarily totherelation variable(v)ontheLHS

Temp1 ={someexpression}

Temp2 ={someotherexpression}

Result =Temp1 UTemp2

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Division (÷)

´ r ÷ s´ Wheresisasubset ofr

´ “Forall”queries

´ “AB÷ B” Find“A”forall“B”,where“A”and“B”areattributes

´ (AllA’spairedwiththeirB’s)÷ (Allpossible B’s)

“AB”set ÷ “B”set

Result: ListofA’s

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Division (÷)

´ Exampler ÷ s

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Aggregate Function (G)

´G1, G2, …, Gm G F1(A1), F2(A2), …, Fn(An) (R)´ Where

´Gn =attributesonwhichtogroup,

´Fi =aggregatefunctionsonanattribute (An),

´R=relation

´ Theaggregatefunctions canbe:´Min,max,sum,count,average

´ Countofnumberofsailors: G count(sid) (Sailors)´ Result isarelation,singlecolumn: Sidwithresult value

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´ “Numberofeachofthecolorsoftheboats”

´ Countofnumberofboats,groupedbycolor:

colorG count(bid) (Boats)

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color count(bid)Red 2Blue 1

Yellow 1Green 1


´ Justaddcommastoincludemultiple groupingcolums

´ E.g.UsingBankDatabase

´ branch-nameG sum(amount), max(amount) (loan)

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branch-name sum(amount) max(amount)Downtown 2500 1500Perryridge 2800 1500

RelationalAlgebra:SUMMARYBinaryoperations

Union rUs -- “or”

Intersection r∩ s-- “and”

Setdifference r− s

AdditionalExpressiveness

AssignmentAggregatefunction

Takeonerelation, returnanewrelationUnaryoperations

Selection σp(r)

Project πA1, A2, …, Am (r)

Rename ρx(A1, A2, …, An)(E)

Findtuplesthatsatisfyagivencondition

Slicearelation,returnanewrelationwithcertainattributes

RenametheresultofexpressionEtox;renameresultantattributestoA1,A2,…

Combinetuplesfrom2relations[require:samenumberofattributes;compatibledomains][result:sameattributes]

Combinetuplesfrom2relations[require:samenumberofattributes;compatibledomains][result:sameattributes]

Findtuplesthatareinonerelationbutarenotinanother[require:samenumberofattributes;compatibledomains][result:sameattributes]

Cartesianproduct r × s

Naturaljoin r⋈ s

Division r÷ s

Taketworelation,returnanewrelation

Combine2relations,allcombination[result:combinedattributes]

Selecttuplesthatsatisfythematchingconditions fromcombinedrelations[result:combinedattributes]

SimilartoAB÷ BFind“A”forall“B”[require:thereexistsB’sattributes inA][result:Aschema]

Additional Material(Not on exam ~ but may come in useful sometimes!)

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Outer Join

´ Leftouter join:=⋈ (mostcommon)– discussed next

´ Rightouterjoin:⋈=´ Fullouter join:=⋈= (mostuncommon)

´ Extensions ofthenaturaljoinoperation

´ Computes thenaturaljoinandthenaddsthetuples fromonerelationthatdonotmatchtheotherrelation

´ Padsthetupleswithnullvalues

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Outer Join

´ r1 =⋈ r2 Example – Leftouterjoin

´ Takes alltuples inthe left relation (r1)thatdidnotmatch withanytupleintheright relation(r2),padsthetupleswithnullvaluesforallotherattributes fromtheright relation(r2),andaddsthemtotheresultofthenaturaljoin

´ Want: List of all customers with their ordersYou want to know if you have any customers who still haven’t placed any orders yet. So you need ALL of the rows from the left relation r1 (ALL customer names) but you might not need all of the rows from relation r2

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Name C_id …… … …

Name Order# Items …… … … …

Customers Table (r1) Orders Table (r2)

Outer Join´ Giventhefollowingexample “CDStore”Database

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Outer Join38


Outer Join

´ Leftouter join

´ customer =⋈ order´ Result: (nullscomefromtherelation onthe⋈ side- order)

´ (Inorder table,customers“53”and“2”madepurchases)

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relational algebra...relational algebra ´ unary operations ´selection: σ ´project: π ´rename:...

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