Normalization

FUNCTIONAL DEPENDENCIES

Deepak Gour, Faculty – DBMS, School of Engineering, SPSU

Definition • Let R be the relation, and let x and y be the

arbitrary subset of the set of attributes of R. Then we say that Y is functionally dependent on x – in symbol.

X Y

(Read x functionally determines y) –

If and only if each x value in R has associated with it precisely one y value in R

In other words

Whenever two tuples of R agree on their x value, they also agree on their Y value.


Example (SCP Relation)

S # City P # QTY

S1 London P1 100

S1 London P2 100

S2 Paris P1 200

S2 Paris P2 200

S3 Delhi P2 300

S4 Kolkata P2 400

S4 Kolkata P2 400

S4 Kolkata P5 400


Example (SCP Relation) (contd..)One FD : - ( { S#} {City})

• Because every tuple of that relation with a given S# value also has the same city value.

• The left and right hand side of an FD are sometimes called determinant and the dependents respectively.


Exercise Check whether following relation satisfy FD

as not

• < S#, P# > <QTY>

• <S#, P#> <City>

• < S#, P#> <City, QTY>

• <S#, P#> <S#>

• <S#, P#> <S#, P#, QTY, City>

• <OTY> <S#>


Extended definition over basic one • Let R be the relation variable, and let x and y be

arbitrary subset of the set of attributes of R. Then we says that Y is functionally dependent on x – in symbol.

X Y(Read x functionally determines y)

• If and only if, in every possible legal value of R, each x value has associated with it precisely one y value

Or in other words• In every possible legal value of R, whenever two

tuple agree on their X values, they also agree on their Y value.


TRIVIAL & NON-TRIVIAL DEPENDENCIES

• One-way to reduce the size of the set of FD we need to deal with is to eliminate the trivial dependencies.

• An FD is trivial if and only if the right hand side is a subset of the left hand side.

e.g. <S#, P#> <S#>. (Trivial)

• Nontrivial dependencies are the one, which are not trivial.


CLOSURE of a set of dependencies

• The set of all FDs that are implied by a given set S of FDs is called the closure of S, denoted by S+

• So we need an algorithm which compute S+ from S.


Algorithm CLOSURE (Z, S): = Z;

DO “ forever”

For each FD X -> Y in S

Do;

if X < CLOSURE [Z, S] /* <= “subset of” */

then CLOSURE [Z,S] : = CLOSURE [Z, Z] U Y;

end;

If CLOSURE [Z, S] did not change on this iteration.

Then leave loop; /* Computation complete */

End;


Example Suppose use are given R with attributes

A, B, C, D, E, F, and FDs

• A BC

• E CF

• B E

• CD EF

Then compute the closure (A, B)+ of the set of attributes under this set of FD’s


Solution1. We initialize the result CLOSURE [Z, S]

to <A, B>

2. We now go round the inner loop four times, once for each for the given FDs. An the first iteration (For FD A BC), we find that the left hand side is indeed a subset of CLOSURE (Z, S) as computed so for, so we add attributes (B and C) to the result. CLOSURE [Z, S] is now the set <A, B, C>.


Solution3. On the second iteration (for FD E CF>.

we find that the left hand side is not a subset of the result as computed so for, which than remain unchanged.

4. On the third iteration (For FD B E), we add E to the closure, which now has the value <A, B, C, E>

5. On the fourth iteration, (for FD CD EF), remains unchanged.


Solution

6. Inner loop times, on the first iteration no change, second, it expands to <A,B, C, E, F> third & fourth, no change.

7. Again inner loop four times, no change, and so the whole process terminates.


Armstrong rules 1. Reflexivity: if B is a subset of A, then A B.

2. Augmentation: if A B then AC BC

3. Transitivity: it A B and B C then A C.

4. Self – determination: A A.

5. Decomposition: If A BC, then AB, AC.

6. Union: it A B and A C, then A BC

7. Composition: if A B, C D then AC BD.

8. If A B and C D, then All (C – B) BD


Armstrong rules (contd..)• Now we define a set of FD to be

irreducible as minimal; if and only if it satisfies the following two properties.

(1) The right hand side of every FD in S involve just one attribute (i.e., it is a singleton set)

(2) The left hand side of every FD in S is irreducible in turn meaning that no attribute can be discarded from the determinant without changing the CLUSURE S+.


Example

• A BC,

• B C

• A B

• AB C

• AC D

Compute an irreducible set of FD that is equivalent to this given set.


Solution(1) The step is to rewrite the FD such that

each has a singleton right hand side.• A B• A C• B C• A B• AB C• AC DWe observe that the FD A B occurs twice.

So one occurrence will be eliminated.


Solution2. Next, attributed C can be eliminated from

the left hand side of the FD AC D

• Because we have A C,

• By augmentation A AC

• And we are given AC D,

• So A D by transitivity;

Thus C on the left hand side is redundant.


Solution3. Next, we observe that the FD AB C can be

eliminated, because again we haveA C

By augmentation AB CBBy decomposition AB C

4. Finally, the FD A C is implied by the FD A B and B C, so it can be eliminated.Now we have A B

B CA D

This set is irreducible.


Example

• A BC

• B E

• CD EF

Show that FD AD F for R.


Solution

1. A BC (given)

2. A C (1, decomposition)

3. AD CD (2, augmentation)

4. CD EF (given)

5. AD EF (3 & 4, transitivity)

6. AD F (5, decomposition

Normalization


Learning Objectives• Definition of normalization and its purpose

in database design• Types of normal forms 1NF, 2NF, 3NF,

BCNF, and 4NF • Transformation from lower normal forms to

higher normal forms• Design concurrent use of normalization and

E-R modeling are to produce a good database design

• Usefulness of denormalization to generate information efficiently


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.


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.


Normalization

• Normalization is the process for assigning attributes to entities– Reduces data redundancies– Helps eliminate data anomalies– Produces controlled redundancies to link tables

• Normalization stages– 1NF - First normal form– 2NF - Second normal form– 3NF - Third normal form– 4NF - Fourth normal form


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.


Data Redundancy


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.


Update Anomalies

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

• Types of update anomalies include:– Insertion,– Deletion,– Modification.


Functional Dependency• Main concept associated with normalization.

• 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.


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.


Example - Functional Dependency


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.


Functional Dependency • 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.


The Process of Normalization

• Formal technique for analyzing 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.


Relationship Between Normal Forms


Unnormalized Form (UNF)

• A table that contains one or more repeating groups.

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

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


First Normal Form (1NF)

• A relation in which intersection of each row and column contains one and only one value.


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).


UNF to 1NF

• All key attributes defined

• No repeating groups in table

• All attributes dependent on

primary key


Second Normal Form (2NF)

• Based on concept of full functional dependency:– A and B are 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 (no partial dependency)


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.


2NF Conversion ResultsFigure 4.5


Third Normal Form (3NF)

• Based on concept of transitive dependency:– A, B and C are 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.


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.


3NF Conversion Results

• Prevent referential integrity violation by adding a JOB_CODE

PROJECT (PROJ_NUM, PROJ_NAME)ASSIGN (PROJ_NUM, EMP_NUM, HOURS)EMPLOYEE (EMP_NUM, EMP_NAME, JOB_CLASS)JOB (JOB_CODE, JOB_DESCRIPTION, CHG_HOUR)


General Definitions of 2NF and 3NF

• 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.


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.


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.


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 (i.e. have at least

one attribute in common).


3NF Table Not in BCNF

Figure 4.7


Decomposition of Table Structure to Meet BCNF


BCNF Conversion Results


Review of Normalization (UNF to BCNF)








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.


Fourth Normal Form (4NF) - 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.


Fourth Normal Form (4NF)• MVD between attributes A, B, and C in a

relation using the following notation:A B

A C


Fourth Normal Form (4NF)• 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.


Fourth Normal Form (4NF)• Defined as a relation that is in BCNF and

contains no nontrivial MVDs.


4NF - Example


3NF Table Not in BCNF

Figure 4.7


Decomposition of Table Structure to Meet BCNF


Decomposition into BCNF

Figure 4.9


4NF Conversion Results

Multivalued Dependencies (an employee can work for many services and on many projects

Set of Tables in 4NF


Denormalization• Normalization is one of many database design

goals • Normalized table requirements

– Additional processing– Loss of system speed

• Normalization purity is difficult to sustain due to conflict in:– Design efficiency– Information requirements– Processing


Unnormalized Table Defects• Data updates less efficient

• Indexing more cumbersome

• No simple strategies for creating views


Summary• We will use normalization in database

design to create a set of relations in 3FN normal form:– Each entity has a unique primary key, and each

attribute depends upon the primary key– No partial dependency– No transitive dependency