1. FUNCTIONAL DEPENDENCIES
4/21/2017
FUNCTIONAL DEPENDENCIES

2. Deepak Gour, Faculty – DBMS, School of Engineering, SPSU
Definition
4/21/2017
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

3. Example (SCP Relation)
4/21/2017
Example (SCP Relation)
S #
City
P #
QTY S1
London P1
100 P2 S2
Paris
200 S3
Delhi
300 S4
Kolkata
400 P5
Deepak Gour, Faculty – DBMS, School of Engineering, SPSU

4. Example (SCP Relation) (contd..)
4/21/2017
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

5. Deepak Gour, Faculty – DBMS, School of Engineering, SPSU
Exercise
4/21/2017
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

6. Extended definition over basic one
4/21/2017
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

7. TRIVIAL & NON-TRIVIAL DEPENDENCIES
4/21/2017
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

8. CLOSURE of a set of dependencies
4/21/2017
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

9. Deepak Gour, Faculty – DBMS, School of Engineering, SPSU
Algorithm
4/21/2017
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

10. Deepak Gour, Faculty – DBMS, School of Engineering, SPSU
Example
4/21/2017
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

11. Deepak Gour, Faculty – DBMS, School of Engineering, SPSU
Solution
4/21/2017
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

12. Deepak Gour, Faculty – DBMS, School of Engineering, SPSU
Solution
4/21/2017
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

13. Deepak Gour, Faculty – DBMS, School of Engineering, SPSU
Solution
4/21/2017
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

14. Deepak Gour, Faculty – DBMS, School of Engineering, SPSU
Armstrong rules
4/21/2017
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

15. Armstrong rules (contd..)
4/21/2017
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

16. Deepak Gour, Faculty – DBMS, School of Engineering, SPSU
Example
4/21/2017
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

17. Deepak Gour, Faculty – DBMS, School of Engineering, SPSU
Solution
4/21/2017
(1) The step is to rewrite the FD such that each has a singleton right hand side.
A  B
A  C
B  C
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

18. Deepak Gour, Faculty – DBMS, School of Engineering, SPSU
Solution
4/21/2017
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

19. Deepak Gour, Faculty – DBMS, School of Engineering, SPSU
Solution
4/21/2017
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

20. Deepak Gour, Faculty – DBMS, School of Engineering, SPSU
Example
4/21/2017
A  BC
B  E
CD  EF
Show that FD AD  F for R.
Deepak Gour, Faculty – DBMS, School of Engineering, SPSU

21. Deepak Gour, Faculty – DBMS, School of Engineering, SPSU
Solution
4/21/2017
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

22. Normalization

23. Deepak Gour, Faculty – DBMS, School of Engineering, SPSU
Learning Objectives
4/21/2017
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

24. Deepak Gour, Faculty – DBMS, School of Engineering, SPSU
4/21/2017
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

25. Deepak Gour, Faculty – DBMS, School of Engineering, SPSU
Normalization
4/21/2017
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

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

27. Deepak Gour, Faculty – DBMS, School of Engineering, SPSU
4/21/2017
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

28. Deepak Gour, Faculty – DBMS, School of Engineering, SPSU
Data Redundancy
4/21/2017
Deepak Gour, Faculty – DBMS, School of Engineering, SPSU

29. Deepak Gour, Faculty – DBMS, School of Engineering, SPSU
Data Redundancy
4/21/2017
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

30. Deepak Gour, Faculty – DBMS, School of Engineering, SPSU
Update Anomalies
4/21/2017
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

31. 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

32. Functional Dependency
4/21/2017
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

33. Example - Functional Dependency
4/21/2017
Example - Functional Dependency
Deepak Gour, Faculty – DBMS, School of Engineering, SPSU

34. 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

35. 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

36. The Process of Normalization
4/21/2017
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

37. Relationship Between Normal Forms
4/21/2017
Relationship Between Normal Forms
Deepak Gour, Faculty – DBMS, School of Engineering, SPSU

38. Unnormalized Form (UNF)
4/21/2017
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

39. Deepak Gour, Faculty – DBMS, School of Engineering, SPSU
First Normal Form (1NF)
4/21/2017
A relation in which intersection of each row and column contains one and only one value.
Deepak Gour, Faculty – DBMS, School of Engineering, SPSU

40. Deepak Gour, Faculty – DBMS, School of Engineering, SPSU
4/21/2017
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

41. Deepak Gour, Faculty – DBMS, School of Engineering, SPSU
4/21/2017
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

42. Second Normal Form (2NF)
4/21/2017
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

43. Deepak Gour, Faculty – DBMS, School of Engineering, SPSU
4/21/2017
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

44. Deepak Gour, Faculty – DBMS, School of Engineering, SPSU
2NF Conversion Results
Figure 4.5
Deepak Gour, Faculty – DBMS, School of Engineering, SPSU

45. Deepak Gour, Faculty – DBMS, School of Engineering, SPSU
4/21/2017
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

46. Deepak Gour, Faculty – DBMS, School of Engineering, SPSU
4/21/2017
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

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

48. 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.
Deepak Gour, Faculty – DBMS, School of Engineering, SPSU

49. 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

50. 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

51. 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

52. Deepak Gour, Faculty – DBMS, School of Engineering, SPSU
3NF Table Not in BCNF
Figure 4.7
Deepak Gour, Faculty – DBMS, School of Engineering, SPSU

53. Decomposition of Table Structure to Meet BCNF
Deepak Gour, Faculty – DBMS, School of Engineering, SPSU

54. BCNF Conversion Results
Deepak Gour, Faculty – DBMS, School of Engineering, SPSU

55. Review of Normalization (UNF to BCNF)
Deepak Gour, Faculty – DBMS, School of Engineering, SPSU

56. Review of Normalization (UNF to BCNF)
Deepak Gour, Faculty – DBMS, School of Engineering, SPSU

57. Review of Normalization (UNF to BCNF)
Deepak Gour, Faculty – DBMS, School of Engineering, SPSU

58. Review of Normalization (UNF to BCNF)
Deepak Gour, Faculty – DBMS, School of Engineering, SPSU

59. 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

60. 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

61. 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

62. 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

63. 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

64. Deepak Gour, Faculty – DBMS, School of Engineering, SPSU
4NF - Example
Deepak Gour, Faculty – DBMS, School of Engineering, SPSU

65. Deepak Gour, Faculty – DBMS, School of Engineering, SPSU
3NF Table Not in BCNF
Figure 4.7
Deepak Gour, Faculty – DBMS, School of Engineering, SPSU

66. Decomposition of Table Structure to Meet BCNF
Deepak Gour, Faculty – DBMS, School of Engineering, SPSU

67. Decomposition into BCNF
Figure 4.9
Deepak Gour, Faculty – DBMS, School of Engineering, SPSU

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

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

70. Unnormalized Table Defects
Data updates less efficient
Indexing more cumbersome
No simple strategies for creating views
Deepak Gour, Faculty – DBMS, School of Engineering, SPSU

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