Chapter 13 Normalization. 2 Chapter 13 - Objectives u Purpose of normalization. u Problems associated with redundant data. u Identification of various

Chapter 13

Normalization

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Chapter 13 - Objectives

Purpose of normalization.

Problems associated with redundant data.

Identification of various types of update anomalies such as insertion, deletion, and modification anomalies.

How to recognize appropriateness or quality of the design of relations.

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Chapter 13 - Objectives

How functional dependencies can be used to group attributes into relations that are in a known normal form.

How to undertake process of normalization.

How to identify most commonly used normal forms, namely 1NF, 2NF, 3NF, and Boyce–Codd normal form (BCNF).

How to identify fourth (4NF) and fifth (5NF) normal forms.

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Recommended Reading

Chapter 13 of the text book Module 9 of the Study Book

– Available from the Course Website

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Normalization

Main objective in developing a logical data model for relational database systems is to create an accurate representation of the data, its relationships, and constraints.

To achieve this objective, must identify a suitable set of relations.

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From Conceptual Model to Logical Model

1. Analyse requirements 2. Draw up ERM diagram

– See chapters 11 & 123. Map ERM diagram to Relational Model

– See separate slides + Chapter 154. Normalize relations

– See chapter 135. Optimize relations

– ‘de-’normalize to gain performance– Not covered in this course

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Normalization

Four most commonly used normal forms are first (1NF), second (2NF) and third (3NF) normal forms, and Boyce–Codd normal form (BCNF).

Based on functional dependencies among the attributes of a relation.

A relation can be normalized to a specific form to prevent possible occurrence of update anomalies.

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Data Redundancy

Major aim of relational database design is to group attributes into relations to minimize data redundancy and reduce file storage space required by base relations.

Problems associated with data redundancy are illustrated by comparing the following Staff and Branch relations with the StaffBranch relation.

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Data Redundancy

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Data Redundancy

StaffBranch relation has redundant data: details of a branch are repeated for every member of staff.

In contrast, branch information appears only once for each branch in Branch relation and only branchNo is repeated in Staff relation, to represent where each member of staff works.

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Update Anomalies

Relations that contain redundant information may potentially suffer from update anomalies.

Types of update anomalies include:– Insertion

» E.g. Adding branch without staff

– Deletion» E.g. Deleting Mary Howe

– Modification» E.g. Changing address of B003

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Lossless-join and Dependency Preservation Properties

Two important properties of decomposition:

- Lossless-join property enables us to find any instance of original relation from corresponding instances in the smaller relations.

- Dependency preservation property enables us to enforce a constraint on original relation by enforcing some constraint on each of the smaller relations.

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Section 1

Functional Dependencies

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Functional Dependency (FD)

Main concept associated with normalization Formally related to concept of keys Functional Dependency

– Describes relationship between attributes in a relation.

– If A and B are attributes of relation R, B is functionally dependent on A (denoted

A B), if each value of A in R is associated with exactly one value of B in R.

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Functional Dependency

Property of the meaning (or semantics) of the attributes in a relation.

Diagrammatic representation:

Determinant of a functional dependency refers to attribute or group of attributes on left-hand side of the arrow.

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Example - Functional Dependency

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Functional Dependency

Main characteristics of functional dependencies used in normalization:

– have a 1:1 relationship between attribute(s) on left and right-hand side of a dependency;

– hold for all time;– are nontrivial

» Impossible not to be satisfied

» RHS is a subset of LHS

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Closure of FDs

Complete set of functional dependencies for a given relation can be very large.

Important to find an approach that can reduce set to a manageable size.

Need to identify set of functional dependencies (X) for a relation that is smaller than complete set of functional dependencies (Y) for that relation and has property that every functional dependency in Y is implied by functional dependencies in X.

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Closure of FDs

Set of all functional dependencies implied by a given set of functional dependencies X called closure of X (written X+).

Set of inference rules, called Armstrong’s axioms, specifies how new functional dependencies can be inferred from given ones.

These rules are both sound and complete Computing closure of a set of FDs is expensive

– Size of closure may be exponential in the number of attributes!

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Armstrong’s Axioms

Let A, B, and C be subsets of the attributes of relation R. Armstrong’s axioms are as follows:

1. Reflexivity

If B is a subset of A, then A B

2. Augmentation

If A B, then AC C

3. Transitivity

If A B and B C, then A CFour other rules mentioned in Study Book can be derived from

Amstrong’s Axioms.

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Attribute Closure

Typically want to check if given FD X Y is in F+

– To check if X is a candidate key for relation R– (Also need to check if X is minimal)

X+ is called closure of attribs X under F– Defined as the set of attribs functionally determined by X

under F– X Y is in F+ iff Y X+

Example– Is AG a candidate key for R(A,B,C,G,H,I) under set of FDs

F = {A B, A C, CG H, CG I, B H} ?– Compute AG+ using algorithm in StudyBook– Check to see if AG is minimal (compute A+ and G+)

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Minimal Covers

Minimal Sets of FDs– Set of all FDs for a relation can be large– Reduce number of FDs by eliminating redundancies

– Textbook gives different definition than StudyBook» Both amount to the same concept

Unfortunately there can be several minimal covers

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Minimal Covers

Textbook definition:– A set of FDs Y is covered by a set of FDs X if every

FD in Y is in X+

– X is minimal set of FDs if:» Every FD in X has a single attribute on its RHS

» A B cannot be replaced by C B with C A without losing equivalence

» Not possible to remove a FD in X without losing equivalence

– A minimal cover for X is a minimal set of FDs Xmin that is equivalent to X

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Summary

Relation R yields set of FDs F– Possible to compute closure F+

» Includes all FDs derivable from F, needed for BCNF

– Possible to compute attribute closure of X under F» Useful to identify candidate keys

– Possible to compute minimal cover FM

» Excludes redundancies, needed for 3NF

Relationship between F, F+, and FM:

– F F+ and F FM+, but not FM F

Examples on next 7 slides are from Study Book

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Examples – (a) functional dependencies

staffNo sName, position, salary, branchNo, bAddress {PK} branchNo bAddress bAddress branchNo branchNo, position salary bAddress, position salary

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Three methods to identify FDs for a relation:– Create adequate test data

» populate the relation with several tuples (rows), inserting all possible data combinations, while identifying which combinations should not be valid

– Reason about the relation itself » look for rules (also called predicates)

– Translate any given assumptions about the data

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Diagrammatic notation:

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Examples – (b) closure of set of FDs

Given R(A,B,C,G,H,I) and F = {A B, A C, CG H, CG I, B H}

Compute F+ using Amstrong’s Axioms:

– A H (since A B and B H) (Transitivity)

– CG CG H and CG I(Union)

– AG AG CG and CG I(Aug+trans)

– … Can be very large Needed for BCNF definition

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Examples – (c) attribute closure

Given R(A,B,C,G,H,I) and F = {A B, A C, CG H, CG I, B H}

Is AG a candidate key for R? Calculate AG+: (ref. algorithm in StudyBook)

– Result := AG

– Result := AGB (A B and A result(AG) )

– Result := AGBC (A C and A AGB )

– Result := AGBCH (B H and B result(AGBC) )

– Result := AGBCHI (CG I and CG result(AGBCH) ) Thus, AG is at least a super key; if minimal, also a candidate key Calculate A+ and G+ separately in the same manner

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Examples – (d) minimal set of FDs

Given R(A,B,C) and F = {A BC, B C, A B, AB C} Compute FM (ref. algorithm in StudyBook)

Merge all FDs with the same LHS in F: A BC (A B is merged)

AB C contains extra attrib A in LHS since B CF – {AB C} {B C} = {A BC, B C}

C is extra attrib in RHS of A BC since B CF – {A BC} {A B} = {A B, B C}

1. Thus, FM = {A B, B C}

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Examples – (e) finding candidate keys (1/2)

Given R(A,B,C,D,E) and F = {A B, BC E, ED A} Step 1: find super keys

– A) check whether determinants are super keys» A+ = AB BC+ = BCE ED+ = ABED (No)

– B) create super keys using F» (R – A+) A = ACDE is a super key for R

» (R – BC+) BC = ABCD is a super key for R

» (R – ED+) ED = CDE is a super key for R

Step 2: find candidate keys

– For each super key, try minimalizing

– Compute attrib. closure of all subsets, test equality with R

– Example: super key ACDE, take all subsets

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Examples – (e) finding candidate keys (2/2)

Super key ACDE, check subsets:» AC+ = ABCE not R

» AD+ = ABD not R

» AE+ = ABE not R

» CD+ = CD not R

» CE+ = CE not R

» DE+ = ABDE not R

» ACD+ = ABCDE candidate key

» ACE+ = ABCE not R

» CDE+ = ABCDE candidate key

» ADE+ = ABDE not R

Need to do the same for the 2 other super keys All candidate keys for R: ACD, CDE, and BCD

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Section 2

Normalization

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The Process of Normalization

Formal technique for analyzing (decomposing) a relation based on its primary key and functional dependencies between its attributes.

Often executed as a series of steps. Each step corresponds to a specific normal form, which has known properties.

As normalization proceeds, relations become progressively more restricted (stronger) in format and also less vulnerable to update anomalies.

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Relationship Between Normal Forms

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Three methods, increasing difficulty

– Text Book approach» Up to 3NF, based on primary key only

» BCNF based on all candidate keys for a relation

» Will be assessed

– Text Book General Definitions for 2NF, 3NF» Based on all candidate keys for a relation

» Catches some extra update anomalies

» May be assessed

– Study Book extra information» Based on all candidate keys for a relation

» Formal algorithms using F+ and FM

» Will not be assessed

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Unnormalized Form (UNF)

A table that contains one or more repeating groups.– Multiple values for a group of attribs per tuple,

given a primary key

To create an unnormalized table: – transform data from information source

(e.g. form) into table format with columns and rows.

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First Normal Form (1NF)

A relation in which the intersection of each row and column contains one and only one value.– Cfr. ‘multi-valued’ attributes in E-ER Model

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UNF to 1NF

Nominate an attribute or group of attributes to act as the key for the unnormalized table.

Identify repeating group(s) in unnormalized table which repeats for the key attribute(s).

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UNF to 1NF

Remove repeating group by:– entering appropriate data into the empty

columns of rows containing repeating data (‘flattening’ the table).

Or by– placing repeating data along with copy of the

original key attribute(s) into a separate relation Both approaches induce 1NF, but update anomalies

are still very likely

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UNF to 1NF: example First approach

CR76 John Kay PG4 6 Lawrence CO40 T. Murphy

PG16 5 Novar Dr CO93 T. Shaw

CR56 A. Stewart PG4 6 Lawrence CO40 T. Murphy

PG36 2 Manor D CO93 T. Shaw


CR76 John Kay PG16 5 Novar Dr CO93 T. Shaw


CR56 A. Stewart PG36 2 Manor D CO93 T. Shaw

UNF

1NF

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UNF to 1NF: example Second approach

CR76 John Kay

CR56 A. Stewart





1NF

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Second Normal Form (2NF)

Based on concept of full functional dependency:– A and B are (groups of) attributes of a relation, – B is fully dependent on A if B is functionally

dependent on A but not on any proper subset of A.

2NF - A relation that is in 1NF and every non-primary-key attribute is fully functionally dependent on the primary key.

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1NF to 2NF

Identify primary key for the 1NF relation.

Identify functional dependencies in the relation.

If partial dependencies exist on the primary key remove them by placing them in a new relation along with copy of their determinant.

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1NF to 2NF: example





1NF

Fd1 {PK}

Fd2 Partial dependency on PK

Fd3 Partial dependency on PK

2NF

Client(clientNo, cName)Prop(propNo, pAdd, pOwnNo, pOwnName)PropRent(clientNo, propNo)

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Third Normal Form (3NF)

Based on concept of transitive dependency:– A, B and C are (groups of) attributes of a relation such that

if A B and B C,

– then C is transitively dependent on A through B. (Provided that A is not functionally dependent on B or C).

3NF - A relation that is in 1NF and 2NF and in which no non-primary-key attribute is transitively dependent on the primary key.

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2NF to 3NF

Identify the primary key in the 2NF relation.

Identify functional dependencies in the relation.

If transitive dependencies exist on the primary key remove them by placing them in a new relation along with copy of their determinant.

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2NF to 3NF: example

PG4 6 Lawrence CO40 T. Murphy

PG16 5 Novar Dr CO93 T. Shaw

PG4 6 Lawrence CO40 T. Murphy

PG36 2 Manor D CO93 T. Shaw

2NF

Fd1 {PK}

Fd2 transitive dependency on PK

3NF

Client(clientNo, cName)Prop(propNo, pAdd, pOwnNo)Owner(pOwnNo, pOwnName)PropRent(clientNo, propNo)

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General Definitions of 2NF and 3NF

Previous definitions are only an approximation

Second normal form (2NF)– A relation that is in 1NF and every non-primary-

key attribute is fully functionally dependent on any candidate key.

Third normal form (3NF)– A relation that is in 1NF and 2NF and in which no

non-primary-key attribute is transitively dependent on any candidate key.

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Boyce–Codd Normal Form (BCNF)

Based on functional dependencies that take into account all candidate keys in a relation, however BCNF also has additional constraints compared with general definition of 3NF.

BCNF - A relation is in BCNF if and only if every determinant is a candidate key.

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Boyce–Codd normal form (BCNF)

Difference between 3NF and BCNF is that for a functional dependency A B, 3NF allows this dependency in a relation if B is a primary-key attribute and A is not a candidate key.

Whereas, BCNF insists that for this dependency to remain in a relation, A must be a candidate key.

Every relation in BCNF is also in 3NF. However, relation in 3NF may not be in BCNF.

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Boyce–Codd normal form (BCNF)

Violation of BCNF is quite rare.

Potential to violate BCNF may occur in a relation that:– contains two (or more) composite candidate

keys;– the candidate keys overlap (ie. have at least

one attribute in common).

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Example BCNF Violation

clientNo interviewDate interviewTime staffNo roomNo

CR76 13 May 02 10.30 SG5 G101

CR56 13 May 02 12.00 SG5 G101

CR74 13 May 02 12.00 SG37 G102

CR56 1 July 02 10.30 SG5 G103

fd1 PK

fd2 CK

fd3 CK

fd4

Relation is in ‘simple 3NF’ (not in ‘general 3NF’)But not in BCNF since LHS of fd4 is not a candidate key

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Problems with BCNF

3NF properties:– Always possible to decompose to 3NF

» Where decomposition is lossless, and» Dependencies are preserved

BCNF properties:– When decomposing to BCNF, dependencies may be

lost (e.g. fd3 in previous slide)

Stay at 3NF or go to BCNF?– 3NF: more redundancy– BCNF: loss of constraint

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Review of Normalization (UNF to BCNF)

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UNF

1NF

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1NF

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Resulting relational schema:– Property(propertyNo, pAddress)– Staff(staffNo, sName)– Inspection(propertyNo, iDate, iTime, comments,

staffNo)– StaffCar(staffNo, iDate, carReg)

Loss of fd5, since fd4 causes redundancy PK, FK mechanism can be used to compute lossless join

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Fourth Normal Form (4NF)

Although BCNF removes anomalies due to functional dependencies, another type of dependency called a multi-valued dependency (MVD) can also cause data redundancy.

Possible existence of MVDs in a relation is due to 1NF and can result in data redundancy.

branchNo sName oName

B03 {Beech, Ford} {Farrel, Murphy}

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Additional Normal Forms

4NF and 5NF These are optional w.r.t. the process of normalization

Assessment:– Concepts may be assessed– Normalization to 4NF and/or 5NF will not be

assessed

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Fourth Normal Form (4NF) - MVD

MVD

– Dependency between attributes (for example, A, B, and C) in a relation,

– such that for each value of A there is a set of values for B and a set of values for C.

– However, set of values for B and C are independent of each other.

branchNo sName oName

B03 Beech Farrel

B03 Beech Murphy

B03 Ford Farrel

B03 Ford Murphy

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MVD between attributes A, B, and C in a relation using the following notation:

A B

A C

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MVD can be further defined as being trivial or nontrivial.

– MVD A B in relation R is defined as being trivial if (a) B is a subset of A or (b) A B = R.

– MVD is defined as being nontrivial if neither (a) nor (b) are satisfied.

– Trivial MVD does not specify a constraint on a relation, while a nontrivial MVD does specify a constraint.

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Defined as a relation that is in BCNF and contains no nontrivial MVDs.

Note that we must allow non-trivial MVDs that are in fact FDs

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4NF - Example

BCNF

4NF

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Fifth Normal Form (5NF)

A relation decomposed into two relations must have lossless-join property, which ensures that no spurious tuples are generated when relations are reunited through a natural join.

However, there are requirements to decompose a relation into more than two relations.

Although rare, these cases are managed by join dependency and fifth normal form (5NF).

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Fifth Normal Form (5NF) A relation that has no (non-trivial) join dependency. Example of a JD

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5NF - Example

PropItemSup(A,B,C) satisfies JD (R1(A,B), R2(B,C), R3(A,C))

Any join between two relations produces spurious tuplesJoin between the three produces legal state of PropItemSup

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Chapter 13 Normalization. 2 Chapter 13 - Objectives u Purpose of normalization. u Problems associated with redundant data. u Identification of various