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Database System Concepts, 5th Ed.

Silberschatz, Korth and Sudarshan

See www.db-book.com for conditions on re-use

Chapter 7: Relational Database DesignChapter 7: Relational Database Design

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Silberschatz, Korth and Sudarshan7.2Database System Concepts - 5th Edition, July 28, 2005.

First Normal FormFirst Normal Form

Domain is atomicif its elements are considered to be indivisible units Examples of non-atomic domains:

Set of names, composite attributes

Identification numbers like CS101 that can be broken up into parts

A relational schema R is in first normal form if the domains of all attributes of R

are atomic

Non-atomic values complicate storage and encourage redundant (repeated)

storage of data

Example: Set of accounts stored with each customer, and set of owners

stored with each account

We assume all relations are in first normal form (and revisit this in Chapter9)

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First Normal Form (Contd)First Normal Form (Contd)

Example

SupplierPart (S#, P#, status, city, Qty.)

S#,p# Qty

S# city

city status

S#status

Insertion anomalies

Cant insert information about supplier until supplier supplies certain part

Deletion anomalies

If we delete first tuple of supplier we not only delete shipment information but alsoinformation about supplier

Updation anomalies

The city value for a particular supplier appears many times and hence can lead toinconsistency while updation

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Functional DependenciesFunctional Dependencies

Constraints on the set of legal relations.

Require that the value for a certain set of attributes determines uniquely the

value for another set of attributes.

Let R be a relation schema

E R andF R

The functional dependency

E p F

holds on R if and only if for any legal relations r(R), whenever any two tuples t1and t2of r agree on the attributes E, they also agree on the attributesF. That is,

t1[E] = t2[E] t1[F ] = t2[F]

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Functional Dependencies (Cont.)Functional Dependencies (Cont.)

Example1:

Consider r(A,B ) with the following instance of r.

On this instance, A p B does NOThold, but B p A does hold.

Example2:

Consider r(A,B,C,D ) with the following instance of r.

On this instance, A p C hold, but Cp A does NOThold.

1 41 5

3 7

A B C D

a1 b1 c1 d1

a1 b2 c1 d2

a2 b2 c2 d2

a2 b3 c2 d3a3 b3 c2 d4

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Functional Dependencies (Cont.)Functional Dependencies (Cont.)

Functional Dependency can be Trivial

E p F is trivial if F E

Non trivial

E p F is non trivial if F E

A functional dependency is a generalization of the notion of a key.

K is a superkey for relation schema R if and only if Kp R

K is a candidate key for R if and only if

Kp R, and

for no E K, E p R

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Closure of a Set of Functional DependenciesClosure of a Set of Functional Dependencies

Given a set F of functional dependencies, there are certain other functionaldependencies that are logically implied by F.

For example: If A p B and B p C, then we can infer thatA p C

The set ofallfunctional dependencies logically implied by F is the closure of F.

We denote the closure of F byF+.

F+ is a superset of F.

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Closure of a Set of Functional DependenciesClosure of a Set of Functional Dependencies

We can find all of F+ by applying ArmstrongsAxioms:

if F E, then E p F (reflexivity)

ifE p F, then K E p K F (augmentation)

ifE p F, and F p K, then E p K (transitivity)

IfE p F holds andE p K holds, then E p F K holds (union)

IfE p F K holds, then E p F holds andE p K holds (decomposition)

IfE p F holds andK F p H holds, then E K p H holds (pseudotransitivity)

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ExampleExample

R = (A, B, C, G, H, I)

F = {Ap BAp C

CGp H

CGp I

B p H}

some members of F+

Ap H

by transitivity from Ap B and B p H

AGp I

by augmentingAp C with G, to getAGp CG

and then transitivity with CGp I CGp HI

by augmenting CGp I to infer CGp CGI,

and augmenting of CGp H to infer CGIp HI,

and then transitivity

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Procedure for Computing FProcedure for Computing F++

To compute the closure of a set of functional dependencies F:

F+ = F

repeat

for each functional dependency f in F+

apply reflexivity and augmentation rules on f

add the resulting functional dependencies to F+for each pair of functional dependencies f1and f2in F

+

if f1 and f2can be combined using transitivity

then add the resulting functional dependency to F+

until F+ does not change any further

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Second Normal FormSecond Normal Form

Arelation R is in second normal form if and only if it is in 1

st

NF and every nonkey attribute is irreducibly dependant on primary key

So we decompose the relation of 1NF as

(S# ,P#, Qty)

(S#, City, Status)

Insertion anomalies Cant insert information about status of a city until supplier resides in

that city

Deletion anomalies

If we delete first tuple of supplier we not only delete Supplier

information but also information about status of a city Updation anomalies

The city value appears many times and hence can lead to

inconsistency while updation

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Example (Convert it in 2NF)Example (Convert it in 2NF)

ProjectNo. ProjectName EmployeeNo. EmployeeName Ratecategory Rate

1203 Madagascar

travel site

11 Jessica

Brookes

A 90

1203 Madagascar travel site 12 AndyEvans B 80

1203 Madagascat

travel site

16 Max Fat C 70

1506 Online

estate

agency

11 Jessica

Brookes

A 90

1506 Online

estate

agency

17 Alex

Branton

B 70

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We look for partial dependencies

We look for fields that depend on only part of the

key and not the entire key.

Field Project No Employee No

Project Name 3

Employee 3

Rate Category 3

Rate 3

We remove partial dependencies

The fields listed are only dependent on part of

the key so we remove them from the table.

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We create new tables

Clearly we cant take the data out and leave it out

of our database.We put it into a new tableconsisting of the field that has the partial

dependency and the field it is dependent on.

Looking at our example we will need to create

two new tables:

Dependent On Partially

Dependent

Project No Project

Name

Dependent On Partially

Dependent

Employee No Employee

Name

Ratecategory

Rate

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EmployeeNo EmployeeName RateCategory

11 Jessica

Brookes

A

12 Andy

Evans

B

16 Max Fat C

17 Alex

Branton

A

Rate

Category

Rate

A 90

B 80

C 70

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Closure ofAttribute SetsClosure ofAttribute Sets

Given a set of attributes E define the closure ofE underF (denoted byE+

) asthe set of attributes that are functionally determined byE under F

Algorithm to compute E+, the closure ofE under F

result := E;

while (changes to result) do

for each F p K i nFdo

begin

ifF result then result := result K

end

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Example ofAttribute Set ClosureExample ofAttribute Set Closure

R = (A

, B, C, G, H, I) F = {Ap B

Ap CCGp HCGp IB p H}

(AG)+

1. result = AG

2. result = ABCG ( Ap C andAp B)

3. result = ABCGH (CG p H and CG AGBC)

4. result = ABCGHI (CG p I and CG AGBCH)

IsA

G a candidate key?1. Is AG a super key?

1. Does AGp R? == Is (AG)+ R

2. Is any subset ofAG a superkey?

1. Does A p R? == Is (A)+ R

2. Does Gp R? == Is (G)+

R

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Uses ofAttribute ClosureUses ofAttribute Closure

There are several uses of the attribute closure algorithm: Testing for superkey:

To test ifE is a superkey, we compute E+, and check ifE+ contains all

attributes of R.

Testing functional dependencies

To check if a functional dependencyE p F holds (or, in other words, is inF+), just check if F E+.

That is, we compute E+ by using attribute closure, and then check if it

containsF.

Is a simple and cheap test, and very useful

Computing closure of F For each K R, we find the closure K+, and for each S K+, we output a

functional dependencyK p S.

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Extraneous AttributesExtraneous Attributes

Consider a set F of functional dependencies and the functional dependencyEp F in F.

AttributeA is extraneous in E ifA E

and F logically implies (F {E p F}) {(E A) p F}.

AttributeA is extraneous inF ifA F

and the set of functional dependencies

(F { E pF}) {E p(F A)} logically implies F.

Example: Given F = {A p C, AB p C }

B is extraneous in AB p C because {A p C, AB p C} logically implies A

p C (I.e. the result of dropping B from AB p C).

Example: Given F = {A p C, AB p CD}

C is extraneous in AB p CD since AB p C can be inferred even after

deleting C

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Canonical CoverCanonical Cover

Acanonical coverfor F is a set of dependencies Fcsuch that

F logically implies all dependencies in Fc, and

Fclogically implies all dependencies in F, and

No functional dependency in Fccontains an extraneous attribute, and

Each left side of functional dependency in Fc is unique.

To compute a canonical cover for F:repeat

Use the union rule to replace any dependencies in FE1 pF1 andE1 pF2with E1 pF1 F2

Find a functional dependencyE pF with anextraneous attribute either in E or inF

If an extraneous attribute is found, delete it from E pF

until F does not change

Note: Union rule may become applicable after some extraneous attributes have beendeleted, so it has to be re-applied

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Computing a Canonical CoverComputing a Canonical Cover

R = (A, B, C)

F = {Ap BC

B p C

Ap B

AB p C}

CombineAp BC andAp B into Ap BC

Set is now {Ap BC, B p C, AB p C}

A is extraneous in AB p C

Check if the result of deleting A from AB p C is implied by the other dependencies

Yes: in fact, B p C is already present!

Set is now {Ap BC, B p C}

C is extraneous in A p BC

Check ifAp C is logically implied byAp B and the other dependencies

Yes: using transitivity on Ap B and B p C.

Can use attribute closure ofA in more complex cases

The canonical cover is: Ap B

B p C

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LosslessLossless--join Decompositionjoin Decomposition

A decomposition of R into R1

and R2

is lossless join if and only if at

least one of the following dependencies is in F+:

R1 R2p R1

R1 R2p R2

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Dependency PreservationDependency Preservation

Let Fibe the set of dependencies F+ that include only attributes in R

i.

A decomposition is dependency preserving, if

(F1 F2 Fn)+ = F+

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ExampleExample

R = (A

, B, C )F = {A p B

B p C}

Key = {A}

Decomposition R1 = (A, B), R2= (B, C)

Lossless-join decomposition

Dependency preserving

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A Lossy DecompositionA Lossy Decomposition

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BCNFBCNF Boyce/codd Normal FormBoyce/codd Normal Form

Insertion anomalies

Cant insert information about teacher teaches a subject until a student

is enrolled for that subject

Deletion anomalies

If we delete first tuple of a subject opted by a student we not only

delete Student information but also information about teacher teaching

that subject.

Updation anomalies

The subject value for a teacher appears many times and hence can

lead to inconsistency while updation

So we decompose the relation as

(Student, Teacher)

(Teacher, Subject)

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Fourth Normal FormFourth Normal Form

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The above relation is in BCNF

Still the above relation has lot of redundancy and hence can lead to update anomalies

The reason is the presence of multi valued dependencies

Multi Valued Dependency

Given a relation R with subset of attributes A,B and C, then we say that B is multi

dependant on A written as AB, if and only if the set of B values matching a given

AC value only depends on A value and is independent on C Value. Course teacher

Each course has set of teachers which is not dependent on text

Course text

Each course has set of text which is not dependent on teacher

Fourth Normal Form A relation R is in 4NF if there exist a non trivial multivalued dependencyAB, then

all attributes of R are functionally dependent on A.

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Fifth Normal FormFifth Normal Form

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