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FUNCTIONAL DEPENDENCIES. 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) –

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definition
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.

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

example scp relation
Example (SCP Relation)

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

example scp relation contd
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.

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

exercise
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#>

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

extended definition over basic one
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.

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

trivial non trivial dependencies
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.

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

closure of a set of dependencies
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.

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

algorithm
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;

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

example
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

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

solution
Solution
  • We initialize the result CLOSURE [Z, S] to <A, B>
  • 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>.

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

solution12
Solution

3. 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.

  • On the third iteration (For FD B E), we add E to the closure, which now has the value <A, B, C, E>
  • On the fourth iteration, (for FD CD  EF), remains unchanged.

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

solution13
Solution
  • Inner loop times, on the first iteration no change, second, it expands to <A,B, C, E, F> third & fourth, no change.
  • Again inner loop four times, no change, and so the whole process terminates.

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

armstrong rules
Armstrong rules
  • Reflexivity: if B is a subset of A, then A  B.
  • Augmentation: if A  B then AC  BC
  • Transitivity: it A  B and B  C then A  C.
  • Self – determination: A  A.
  • Decomposition: If A  BC, then AB, AC.
  • Union: it A  B and A  C, then A  BC
  • Composition: if A  B, C  D then AC  BD.
  • If A  B and C  D, then All (C – B)  BD

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

armstrong rules contd
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+.

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

example16
Example
  • A  BC,
  • B  C
  • A  B
  • AB  C
  • AC  D

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

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

solution17
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  D

We observe that the FD A  B occurs twice. So one occurrence will be eliminated.

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

solution18
Solution
  • 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.

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

solution19
Solution

3. Next, we observe that the FD AB  C can be eliminated, because again we have

A  C

By augmentation AB  CB

By 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  C

A  D

This set is irreducible.

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

example20
Example
  • A  BC
  • B  E
  • CD  EF

Show that FD AD  F for R.

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

solution21
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

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

learning objectives
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

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

normalization24
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.

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

normalization25
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.

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

normalization26
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

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

data redundancy
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.

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

data redundancy28
Data Redundancy

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

data redundancy29
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.

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

update anomalies
Update Anomalies
  • Relations that contain redundant information may potentially suffer from update anomalies.
  • Types of update anomalies include:
    • Insertion,
    • Deletion,
    • Modification.

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

functional dependency
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.

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

functional dependency32
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.

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

example functional dependency
Example - Functional Dependency

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

functional dependency34
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.

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

functional dependency35
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.

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

the process of normalization
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.

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

relationship between normal forms
Relationship Between Normal Forms

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

unnormalized form unf
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.

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

first normal form 1nf
First Normal Form (1NF)
  • A relation in which intersection of each row and column contains one and only one value.

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

unf to 1nf
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).

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

unf to 1nf41
UNF to 1NF
  • All key attributes defined
  • No repeating groups in table
  • All attributes dependent on

primary key

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

second normal form 2nf
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)

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

1nf to 2nf
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.

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

2nf conversion results
2NF Conversion Results

Figure 4.5

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

third normal form 3nf
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.

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

2nf to 3nf
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.

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

3nf conversion results
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)

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

general definitions of 2nf and 3nf
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 anycandidate 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.

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

boyce codd normal form bcnf
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.

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

boyce codd normal form bcnf50
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.

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

boyce codd normal form bcnf51
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).

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

3nf table not in bcnf
3NF Table Not in BCNF

Figure 4.7

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

decomposition of table structure to meet bcnf
Decomposition of Table Structure to Meet BCNF

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

bcnf conversion results
BCNF Conversion Results

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

review of normalization unf to bcnf
Review of Normalization (UNF to BCNF)

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

review of normalization unf to bcnf56
Review of Normalization (UNF to BCNF)

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

review of normalization unf to bcnf57
Review of Normalization (UNF to BCNF)

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

review of normalization unf to bcnf58
Review of Normalization (UNF to BCNF)

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

fourth normal form 4nf
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.

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

fourth normal form 4nf mvd
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.

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

fourth normal form 4nf61
Fourth Normal Form (4NF)
  • MVD between attributes A, B, and C in a relation using the following notation:

A ¾¾ØØ B

A ¾¾ØØ C

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

fourth normal form 4nf62
Fourth Normal Form (4NF)
  • MVD can be further defined as being trivial or nontrivial.
  • MVD A ¾¾ØØ 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.

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

fourth normal form 4nf63
Fourth Normal Form (4NF)
  • Defined as a relation that is in BCNF and contains no nontrivial MVDs.

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

4nf example
4NF - Example

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

3nf table not in bcnf65
3NF Table Not in BCNF

Figure 4.7

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

decomposition of table structure to meet bcnf66
Decomposition of Table Structure to Meet BCNF

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

decomposition into bcnf
Decomposition into BCNF

Figure 4.9

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

4nf conversion results
4NF Conversion Results

Set of Tables in 4NF

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

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

denormalization
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

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

unnormalized table defects
Unnormalized Table Defects
  • Data updates less efficient
  • Indexing more cumbersome
  • No simple strategies for creating views

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

summary
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

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