Lecture 6: Design Theory · FDs for Relational Schema Design •High-level idea: why do we care about FDs? 1. Start with some relational schema 2. Find out its functional dependencies

Lecture6:DesignTheory

Lectures6

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Lecture6

Lecture6:DesignTheoryI

Lecture6

Today’sLecture

1. Normalforms&functionaldependencies• ACTIVITY:FindingFDs

2. Findingfunctionaldependencies

3. Closures,superkeys &keys• ACTIVITY:Thekeyorakey?

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Lecture6

1.Normalforms&functionaldependencies

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Lecture6>Section1

Whatyouwilllearnaboutinthissection

1. Overviewofdesigntheory&normalforms

2. Dataanomalies&constraints

3. Functionaldependencies

4. ACTIVITY:FindingFDs

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Lecture6>Section1

DesignTheory

• Designtheoryisabouthowtorepresentyourdatatoavoidanomalies.

• Itisamostlymechanicalprocess• Toolscancarryoutroutineportions

• Wehaveanotebookimplementingallalgorithms!• We’llplaywithitintheactivities!

Lecture6>Section1>Overview

NormalForms• 1st NormalForm(1NF) =Alltablesareflat

• 2nd NormalForm =disused

• Boyce-Codd NormalForm(BCNF)

• 3rd NormalForm(3NF)

• 4thand5th NormalForms =seetextbooks

DBdesignsbasedonfunctionaldependencies,intendedtopreventdataanomalies

Ourfocusforthislecture+thenexttwoones


1st NormalForm(1NF)

Student CoursesMary {CS564,CS368}Joe {CS564,CS552}… …

Violates1NF.

1NFConstraint:Typesmustbeatomic!

Student CoursesMary CS564Mary CS368Joe CS564Joe CS552

In1st NF


DataAnomalies&Constraints

Lecture6>Section1>Dataanomalies&constraints

ConstraintsPrevent(some)AnomaliesintheData

Student Course RoomMary CS564 B01Joe CS564 B01Sam CS564 B01.. .. ..

Ifeverycourseisinonlyoneroom,containsredundantinformation!

Apoorlydesigneddatabasecausesanomalies:



Student Course RoomMary CS564 B01Joe CS564 C12Sam CS564 B01.. .. ..

Ifweupdatetheroomnumberforonetuple,wegetinconsistentdata=anupdate anomaly




Student Course Room.. .. ..

Ifeveryonedropstheclass,welosewhatroomtheclassisin!=adelete anomaly





Similarly,wecan’treservearoomwithoutstudents=aninsertanomaly


… CS368 C12



Student CourseMary CS564Joe CS564Sam CS564.. ..

Course RoomCS564 B01CS368 C12

Today:developtheorytounderstandwhythisdesignmaybebetterand howtofindthisdecomposition…

Isthisformbetter?

• Redundancy?• Updateanomaly?• Deleteanomaly?• Insertanomaly?


FunctionalDependencies

Lecture6>Section1>Functionaldependencies

FunctionalDependency

A->Bmeansthat“whenevertwotuplesagreeonAthentheyagreeonB.”

Def:LetA,Bbesets ofattributesWewriteAà BorsayAfunctionallydeterminesBif,foranytuplest1 andt2:

t1[A]=t2[A]impliest1[B]=t2[B]

andwecallAà Bafunctionaldependency


APictureOfFDs

A1 … Am B1 … Bn

Defn (again):GivenattributesetsA={A1,…,Am} andB={B1,…Bn}inR,


A1 … Am B1 … Bn

APictureOfFDs

ti

tj

Defn (again):GivenattributesetsA={A1,…,Am} andB={B1,…Bn}inR,

Thefunctionaldependency Aà BonRholdsifforanyti,tj inR:


APictureOfFDsDefn (again):GivenattributesetsA={A1,…,Am} andB={B1,…Bn}inR,


ti[A1]=tj[A1]ANDti[A2]=tj[A2]AND…ANDti[Am]=tj[Am]

A1 … Am B1 … Bn

ti

tj

Ift1,t2agreehere..


APictureOfFDsDefn (again):GivenattributesetsA={A1,…,Am} andB={B1,…Bn}inR,


if ti[A1]=tj[A1]ANDti[A2]=tj[A2]AND…ANDti[Am]=tj[Am]

then ti[B1]=tj[B1]ANDti[B2]=tj[B2]AND…ANDti[Bn]=tj[Bn]

A1 … Am B1 … Bn

ti

tj

Ift1,t2agreehere.. …theyalsoagreehere!


FDsforRelationalSchemaDesign

• High-levelidea:whydowecareaboutFDs?

1. Startwithsomerelationalschema

2. Modelitsfunctionaldependencies(FDs)

3. Usethesetodesignabetterschema1. Onewhichminimizesthepossibilityofanomalies


FunctionalDependenciesasConstraints


Note:TheFD{Course}->{Room}holdsonthisinstance

Afunctionaldependencyisaformofconstraint

• Holds onsomeinstancesnotothers.

• Partoftheschema,helpsdefineavalidinstance.

Lecture6>Section1

Recall:aninstance ofaschemaisamultiset oftuplesconformingtothatschema,i.e.atable

FunctionalDependenciesasConstraints


However,cannotprovethattheFD{Course}->{Room}ispartoftheschema

Notethat:• YoucancheckifanFDis

violated byexaminingasingleinstance;

• However,youcannotprovethatanFDispartoftheschemabyexaminingasingleinstance.• Thiswouldrequirechecking

everyvalidinstance

Lecture6>Section1

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MoreExamplesAnFDisaconstraintwhichholds,ordoesnothold onaninstance:

EmpID Name Phone PositionE0045 Smith 1234 ClerkE3542 Mike 9876 SalesrepE1111 Smith 9876 SalesrepE9999 Mary 1234 Lawyer


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{Position}à {Phone}

EmpID Name Phone PositionE0045 Smith 1234 ClerkE3542 Mike 9876 ¬ SalesrepE1111 Smith 9876 ¬ SalesrepE9999 Mary 1234 Lawyer

MoreExamples


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EmpID Name Phone PositionE0045 Smith 1234 ® ClerkE3542 Mike 9876 SalesrepE1111 Smith 9876 SalesrepE9999 Mary 1234 ® Lawyer

butnot {Phone}à {Position}

MoreExamples


ACTIVITY

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Lecture6>Section1>ACTIVITY

A B C D E

1 2 4 3 63 2 5 1 81 4 4 5 71 2 4 3 63 2 5 1 8

Findatleastthree FDswhichareviolatedonthisinstance:

{ } à { }{ } à { }{ } à { }

2.Findingfunctionaldependencies

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Lecture6>Section2


1. “Good”vs.“Bad”FDs:Intuition

2. FindingFDs

3. Closures

4. ACTIVITY:Computetheclosures

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Lecture6>Section2

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“Good”vs.“Bad”FDs

Wecanstarttodevelopanotionofgoodvs.bad FDs:


Intuitively:

EmpID ->Name,Phone,Positionis“goodFD”• Minimalredundancy,

lesspossibilityofanomalies

Lecture6>Section2>Goodvs.BadFDs

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Wecanstarttodevelopanotionofgoodvs.bad FDs:


Intuitively:

EmpID ->Name,Phone,Positionis“goodFD”

ButPosition->Phoneisa“badFD”• Redundancy!

Possibilityofdataanomalies



GivenasetofFDs(fromuser)ourgoalisto:1. FindallFDs,and2. Eliminatethe“BadOnes".

Returningtoouroriginalexample…canyouseehowthe“badFD”{Course}->{Room}couldleadtoan:• UpdateAnomaly• InsertAnomaly• DeleteAnomaly• …



FDsforRelationalSchemaDesign

• High-levelidea:whydowecareaboutFDs?

1. Startwithsomerelationalschema

2. Findoutitsfunctionaldependencies(FDs)

3. Usethesetodesignabetterschema1. Onewhichminimizespossibilityofanomalies

Lecture6>Section2>FindingFDs

Thispartcanbetricky!

FindingFunctionalDependencies

• Therecanbeaverylargenumber ofFDs…• Howtofindthemallefficiently?

• Wecan’tnecessarilyshowthatanyFDwillholdonallinstances…• Howtodothis?

Wewillstartwiththisproblem:GivenasetofFDs,F,whatotherFDsmusthold?


Equivalenttoasking:GivenasetofFDs,F={f1,…fn},doesanFDghold?

Inferenceproblem:Howdowedecide?




1.{Name}à {Color}2.{Category}à {Department}3.{Color,Category}à {Price}

Name Color Category Dep PriceGizmo Green Gadget Toys 49Widget Black Gadget Toys 59Gizmo Green Whatsit Garden 99

Which/howmanyotherFDsdo?!?

ProvidedFDs:Products

GiventheprovidedFDs,wecanseethat{Name,Category}à {Price}mustalsoholdonanyinstance…

Example:


Equivalenttoasking:GivenasetofFDs,F={f1,…fn},doesanFDghold?

Inferenceproblem:Howdowedecide?

Answer:ThreesimplerulescalledArmstrong’sRules.

1. Split/Combine,2. Reduction,and3. Transitivity…ideasbypicture



1.Split/Combine

A1 … Am B1 … Bn

A1,…,Am à B1,…,Bn


1.Split/Combine

A1 … Am B1 … Bn


…isequivalenttothefollowingn FDs…

A1,…,Am à Bi fori=1,…,n


1.Split/Combine

A1 … Am B1 … Bn


…isequivalentto…

Andvice-versa,A1,…,Am à Bi fori=1,…,n


2.Reduction/TrivialA1 … Am

A1,…,Am à Aj foranyj=1,…,m


3.TransitiveClosure

A1 … Am B1 … Bn C1 … Ck

A1,…,Am à B1,…,Bn andB1,…,Bn à C1,…,Ck


3.TransitiveClosure

A1 … Am B1 … Bn C1 … Ck

A1,…,Am à B1,…,Bn andB1,…,Bn à C1,…,Ck

impliesA1,…,Am à C1,…,Ck



1.{Name}à {Color}2.{Category}à {Department}3.{Color,Category}à {Price}

Name Color Category Dep PriceGizmo Green Gadget Toys 49Widget Black Gadget Toys 59Gizmo Green Whatsit Garden 99

Which/howmanyotherFDshold?

ProvidedFDs:Products

Example:



1.{Name}à {Color}2.{Category}à {Dept.}3.{Color,Category}à{Price}

Which/howmanyotherFDshold?

ProvidedFDs:InferredFDs:

Example:

Inferred FD Ruleused

4. {Name,Category}->{Name} ?5.{Name,Category}->{Color} ?6.{Name,Category} ->{Category} ?7.{Name,Category ->{Color,Category} ?8.{Name,Category}->{Price} ?



1.{Name}à {Color}2.{Category}à {Dept.}3.{Color,Category}à{Price}

Canwefindanalgorithmicwaytodothis?

ProvidedFDs:InferredFDs:

Example:

Inferred FD Ruleused

4. {Name,Category}->{Name} Trivial5.{Name,Category}->{Color} Transitive (4->1)6.{Name,Category} ->{Category} Trivial7.{Name,Category ->{Color,Category} Split/combine (5+6)8.{Name,Category}->{Price} Transitive(7-> 3)


Closures

Lecture6>Section2 >Closures

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ClosureofasetofAttributes

Given asetofattributesA1,…,An andasetofFDsF:Thentheclosure,{A1,…,An}+ isthesetofattributesB s.t. {A1,…,An}à B

{name} à {color}{category} à {department}{color, category} à {price}

Example: F=

ExampleClosures:

{name}+ = {name, color}{name, category}+ ={name, category, color, dept, price}{color}+ = {color}


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ClosureAlgorithm

StartwithX={A1,…,An}andsetofFDsF.

Repeatuntil Xdoesn’tchange;do:

if {B1,…,Bn}à CisentailedbyF

and {B1,…,Bn}⊆ X

then addCtoX.

Return XasX+


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ClosureAlgorithmStartwithX={A1,…,An},FDsF.Repeatuntil Xdoesn’tchange;do:if {B1,…,Bn}à CisinFand {B1,

…,Bn}⊆ X:then addCtoX.

Return XasX+

{name} à {color}

{category} à {dept}

{color, category} à{price}

F=

{name, category}+ ={name, category}


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Return XasX+

{name} à {color}



F=


{name, category}+ ={name, category, color}


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Return XasX+

{name} à {color}



F=



{name, category}+ ={name, category, color, dept}


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Return XasX+

F=


{name, category}+ ={name, category, color, dept, price}


{name, category}+ ={name, category, color, dept}{name} à {color}




Example

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Compute{A,B}+ ={A,B,}

Compute{A,F}+ ={A,F,}

R(A,B,C,D,E,F) {A,B} à {C}{A,D} à {E}{B} à {D}{A,F} à {B}


Example

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Compute{A,B}+ ={A,B,C,D}

Compute{A,F}+ ={A,F,B}



Example

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Compute{A,B}+ ={A,B,C,D,E}

Compute{A,F}+ ={A,B,C,D,E,F}



3.Closures,Superkeys &Keys

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Lecture6>Section3


1. ClosuresPt.II

2. Superkeys &Keys

3. ACTIVITY:Thekeyorakey?

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Lecture6>Section3

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WhyDoWeNeed theClosure?

• WithclosurewecanfindallFD’seasily

• TocheckifX® A

1. ComputeX+

2. CheckifAÎ X+

NoteherethatX isaset ofattributes,butA isasingleattribute.

RecalltheSplit/combine rule:Xà A1,…,Xà AnimpliesXà {A1,…,An}

Lecture6>Section3>ClosuresPt.II

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UsingClosuretoInferALLFDs{A,B} à C{A,D} à B{B} à D

Example:GivenF=Step1:ComputeX+,foreverysetofattributesX:

{A}+ = {A}{B}+ = {B,D}{C}+ = {C}{D}+ = {D}{A,B}+ = {A,B,C,D}{A,C}+ = {A,C}{A,D}+ = {A,B,C,D}{A,B,C}+ = {A,B,D}+ = {A,C,D}+ = {A,B,C,D} {B,C,D}+ = {B,C,D}{A,B,C,D}+ = {A,B,C,D}

Noneedtocomputethese-why?


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Example:GivenF=Step1:ComputeX+,foreverysetofattributesX:

{A}+ = {A}, {B}+ = {B,D}, {C}+ = {C}, {D}+ = {D}, {A,B}+ = {A,B,C,D}, {A,C}+ = {A,C}, {A,D}+ = {A,B,C,D}, {A,B,C}+ = {A,B,D}+ = {A,C,D}+ = {A,B,C,D}, {B,C,D}+ = {B,C,D}, {A,B,C,D}+ = {A,B,C,D}

Step2:EnumerateallFDsXà Y,s.t. YÍ X+ andXÇ Y=Æ:

{A,B} à {C,D}, {A,D} à {B,C},{A,B,C} à {D}, {A,B,D} à {C},{A,C,D} à {B}


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Example:GivenF=




“YisintheclosureofX”


Step1:ComputeX+,foreverysetofattributesX:

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Example:GivenF=




TheFDXà Yisnon-trivial


Step1:ComputeX+,foreverysetofattributesX:

Superkeys andKeys

Lecture6>Section3>Superkeys &Keys

KeysandSuperkeys

Asuperkey isasetofattributesA1,…,An s.t.foranyother attributeB inR,wehave {A1,…,An}à B

Akey isaminimal superkey

I.e.allattributesarefunctionallydeterminedbyasuperkey

Meaningthatnosubsetofakeyisalsoasuperkey


FindingKeysandSuperkeys

• ForeachsetofattributesX

1. ComputeX+

2. IfX+=setofallattributesthenXisasuperkey

3. IfXisminimal,thenitisakey


ExampleofFindingKeysProduct(name, price, category, color)

{name, category} à price{category} à color

Whatisakey?


ExampleofKeysProduct(name, price, category, color)

{name, category} à price{category} à color

{name, category}+ = {name, price, category, color}= thesetofallattributes⟹ thisisasuperkey⟹ thisisakey,sinceneithername norcategoryaloneisasuperkey


Activity-6.ipynb

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Lecture6>Section3 >ACTIVITY

Documents

Lecture 6: Design Theory · FDs for Relational Schema Design •High-level idea: why do we care about FDs? 1. Start with some relational schema 2. Find out its functional dependencies