17-Relational Design Algorithms-07-06-2023.pdf

CHAPTER 14
Basics of Functional Dependencies
and Normalization for Relational
Databases
Slide 14- 2

Chapter Outline
 1 Informal Design Guidelines for Relational Databases
 1.1 Semantics of the Relation Attributes
 1.2 Redundant Information in Tuples and Update Anomalies
 1.3 Null Values in Tuples
 1.4 Spurious Tuples
 2 Functional Dependencies (FDs)
 2.1 Definition of Functional Dependency
Slide 14- 3

Chapter Outline
 3 Normal Forms Based on Primary Keys
 3.1 Normalization of Relations
 3.2 Practical Use of Normal Forms
 3.3 Definitions of Keys and Attributes Participating in Keys
 3.4 First Normal Form
 3.5 Second Normal Form
 3.6 Third Normal Form
 4 General Normal Form Definitions for 2NF and 3NF (For
Multiple Candidate Keys)
 5 BCNF (Boyce-Codd Normal Form)
Slide 14- 4

Chapter Outline
 6 Multivalued Dependency and Fourth Normal Form
 7 Join Dependencies and Fifth Normal Form
Slide 14- 5

1. Informal Design Guidelines for
Relational Databases (1)
 What is relational database design?
 The grouping of attributes to form "good" relation
schemas
 Two levels of relation schemas
 The logical "user view" level
 The storage "base relation" level
 Design is concerned mainly with base relations
 What are the criteria for "good" base relations?
Slide 14- 6

Informal Design Guidelines for Relational
Databases (2)
 We first discuss informal guidelines for good relational
design
 Then we discuss formal concepts of functional
dependencies and normal forms
 - 1NF (First Normal Form)
 - 2NF (Second Normal Form)
 - 3NF (Third Noferferferfewrmal Form)
 - BCNF (Boyce-Codd Normal Form)
 Additional types of dependencies, further normal forms,
relational design algorithms by synthesis are discussed in
Chapter 15
Slide 14- 7

1.1 Semantics of the Relational
Attributes must be clear
 GUIDELINE 1: Informally, each tuple in a relation should
represent one entity or relationship instance. (Applies to
individual relations and their attributes).
 Attributes of different entities (EMPLOYEEs,
DEPARTMENTs, PROJECTs) should not be mixed in the
same relation
 Only foreign keys should be used to refer to other entities
 Entity and relationship attributes should be kept apart as
much as possible.
 Bottom Line: Design a schema that can be explained
easily relation by relation. The semantics of attributes
should be easy to interpret.
Slide 14- 8

Figure 14.1 A simplified COMPANY
relational database schema
Slide 14- 9
Figure 14.1 A
simplified COMPANY
relational database
schema.

1.2 Redundant Information in Tuples and
Update Anomalies
 Information is stored redundantly
 Wastes storage
 Causes problems with update anomalies
 Insertion anomalies
 Deletion anomalies
 Modification anomalies
Slide 14- 10

EXAMPLE OF AN UPDATE ANOMALY
 Consider the relation:
 EMP_PROJ(Emp#, Proj#, Ename, Pname,
No_hours)
 Update Anomaly:
 Changing the name of project number P1 from
“Billing” to “Customer-Accounting” may cause this
update to be made for all 100 employees working
on project P1.
Slide 14- 11

EXAMPLE OF AN INSERT ANOMALY
No_hours)
 Insert Anomaly:
 Cannot insert a project unless an employee is
assigned to it.
 Conversely
 Cannot insert an employee unless an he/she is
assigned to a project.
Slide 14- 12

EXAMPLE OF A DELETE ANOMALY
No_hours)
 Delete Anomaly:
 When a project is deleted, it will result in deleting
all the employees who work on that project.
 Alternately, if an employee is the sole employee
on a project, deleting that employee would result in
deleting the corresponding project.
Slide 14- 13

Figure 14.3 Two relation schemas
suffering from update anomalies
Slide 14- 14
Figure 14.3
Two relation schemas
suffering from update
anomalies. (a)
EMP_DEPT and (b)
EMP_PROJ.

Figure 14.4 Sample states for
EMP_DEPT and EMP_PROJ
Slide 14- 15
Figure 14.4
Sample states for EMP_DEPT
and EMP_PROJ resulting from
applying NATURAL JOIN to the
relations in Figure 14.2. These
may be stored as base
relations for performance
reasons.

Guideline for Redundant Information in
Tuples and Update Anomalies
 GUIDELINE 2:
 Design a schema that does not suffer from the
insertion, deletion and update anomalies.
 If there are any anomalies present, then note them
so that applications can be made to take them into
account.
Slide 14- 16

1.3 Null Values in Tuples
 GUIDELINE 3:
 Relations should be designed such that their
tuples will have as few NULL values as possible
 Attributes that are NULL frequently could be
placed in separate relations (with the primary key)
 Reasons for nulls:
 Attribute not applicable or invalid
 Attribute value unknown (may exist)
 Value known to exist, but unavailable
Slide 14- 17

1.4 Generation of Spurious Tuples – avoid
at any cost
 Bad designs for a relational database may result
in erroneous results for certain JOIN operations
 The "lossless join" property is used to guarantee
meaningful results for join operations
 GUIDELINE 4:
 The relations should be designed to satisfy the
lossless join condition.
 No spurious tuples should be generated by doing
a natural-join of any relations.
Slide 14- 18

Spurious Tuples (2)
 There are two important properties of decompositions:
a) Non-additive or losslessness of the corresponding join
b) Preservation of the functional dependencies.
 Note that:
 Property (a) is extremely important and cannot be
sacrificed.
 Property (b) is less stringent and may be sacrificed. (See
Chapter 15).
Slide 14- 19

2. Functional Dependencies
 Functional dependencies (FDs)
 Are used to specify formal measures of the
"goodness" of relational designs
 And keys are used to define normal forms for
relations
 Are constraints that are derived from the meaning
and interrelationships of the data attributes
 A set of attributes X functionally determines a set
of attributes Y if the value of X determines a
unique value for Y
Slide 14- 20

2.1 Defining Functional Dependencies
 X  Y holds if whenever two tuples have the same value
for X, they must have the same value for Y
 For any two tuples t1 and t2 in any relation instance r(R): If
t1[X]=t2[X], then t1[Y]=t2[Y]
 X  Y in R specifies a constraint on all relation instances
r(R)
 Written as X  Y; can be displayed graphically on a
relation schema as in Figures. ( denoted by the arrow: ).
 FDs are derived from the real-world constraints on the
attributes
Slide 14- 21

Examples of FD constraints (1)
 Social security number determines employee
name
 SSN  ENAME
 Project number determines project name and
location
 PNUMBER  {PNAME, PLOCATION}
 Employee ssn and project number determines
the hours per week that the employee works on
the project
 {SSN, PNUMBER}  HOURS
Slide 14- 22

Examples of FD constraints (2)
 An FD is a property of the attributes in the
schema R
 The constraint must hold on every relation
instance r(R)
 If K is a key of R, then K functionally determines
all attributes in R
 (since we never have two distinct tuples with
t1[K]=t2[K])
Slide 14- 23

Defining FDs from instances
 Note that in order to define the FDs, we need to
understand the meaning of the attributes involved
and the relationship between them.
 An FD is a property of the attributes in the
schema R
 Given the instance (population) of a relation, all
we can conclude is that an FD may exist between
certain attributes.
 What we can definitely conclude is – that certain
FDs do not exist because there are tuples that
show a violation of those dependencies.
Slide 14- 24

Figure 14.7 Ruling Out FDs
Slide 14- 25
Note that given the state of the TEACH relation, we can
say that the FD: Text → Course may exist. However, the
FDs Teacher → Course, Teacher → Text and
Couse → Text are ruled out.

Figure 14.8 What FDs may exist?
Slide 14- 26
 A relation R(A, B, C, D) with its extension.
 Which FDs may exist in this relation?

3 Normal Forms Based on Primary Keys
 3.1 Normalization of Relations
 3.2 Practical Use of Normal Forms
 3.3 Definitions of Keys and Attributes
Participating in Keys
 3.4 First Normal Form
 3.5 Second Normal Form
 3.6 Third Normal Form
Slide 14- 27

3.1 Normalization of Relations (1)
 Normalization:
 The process of decomposing unsatisfactory "bad"
relations by breaking up their attributes into
smaller relations
 Normal form:
 Condition using keys and FDs of a relation to
certify whether a relation schema is in a particular
normal form
Slide 14- 28

Normalization of Relations (2)
 2NF, 3NF, BCNF
 based on keys and FDs of a relation schema
 4NF
 based on keys, multi-valued dependencies :
MVDs;
 5NF
 based on keys, join dependencies : JDs
 Additional properties may be needed to ensure a
good relational design (lossless join, dependency
preservation; see Chapter 15)
Slide 14- 29

3.2 Practical Use of Normal Forms
 Normalization is carried out in practice so that the
resulting designs are of high quality and meet the
desirable properties
 The practical utility of these normal forms becomes
questionable when the constraints on which they are
based are hard to understand or to detect
 The database designers need not normalize to the
highest possible normal form
 (usually up to 3NF and BCNF. 4NF rarely used in practice.)
 Denormalization:
 The process of storing the join of higher normal form
relations as a base relation—which is in a lower normal
form
Slide 14- 30

3.3 Definitions of Keys and Attributes
Participating in Keys (1)
 A superkey of a relation schema R = {A1, A2, ....,
An} is a set of attributes S subset-of R with the
property that no two tuples t1 and t2 in any legal
relation state r of R will have t1[S] = t2[S]
 A key K is a superkey with the additional
property that removal of any attribute from K will
cause K not to be a superkey any more.
Slide 14- 31

Definitions of Keys and Attributes
Participating in Keys (2)
 If a relation schema has more than one key, each
is called a candidate key.
 One of the candidate keys is arbitrarily designated
to be the primary key, and the others are called
secondary keys.
 A Prime attribute must be a member of some
candidate key
 A Nonprime attribute is not a prime attribute—
that is, it is not a member of any candidate key.
Slide 14- 32

3.4 First Normal Form
 Disallows
 composite attributes
 multivalued attributes
 nested relations; attributes whose values for an
individual tuple are non-atomic
 Considered to be part of the definition of a
relation
 Most RDBMSs allow only those relations to be
defined that are in First Normal Form
Slide 14- 33

Figure 14.9 Normalization into 1NF
Slide 14- 34
Figure 14.9
Normalization into 1NF. (a)
A relation schema that is not
in 1NF. (b) Sample state of
relation DEPARTMENT. (c)
1NF version of the same
relation with redundancy.

Figure 14.10 Normalizing nested relations
into 1NF
Slide 14- 35
Figure 14.10
Normalizing nested relations into 1NF. (a) Schema of the EMP_PROJ relation with a
nested relation attribute PROJS. (b) Sample extension of the EMP_PROJ relation
showing nested relations within each tuple. (c) Decomposition of EMP_PROJ into
relations EMP_PROJ1 and EMP_PROJ2 by propagating the primary key.

3.5 Second Normal Form (1)
 Uses the concepts of FDs, primary key
 Definitions
 Prime attribute: An attribute that is member of the primary
key K
 Full functional dependency: a FD Y -> Z where removal
of any attribute from Y means the FD does not hold any
more
 Examples:
 {SSN, PNUMBER} -> HOURS is a full FD since neither SSN
-> HOURS nor PNUMBER -> HOURS hold
 {SSN, PNUMBER} -> ENAME is not a full FD (it is called a
partial dependency ) since SSN -> ENAME also holds
Slide 14- 36

Second Normal Form (2)
 A relation schema R is in second normal form
(2NF) if every non-prime attribute A in R is fully
functionally dependent on the primary key
 R can be decomposed into 2NF relations via the
process of 2NF normalization or “second
normalization”
Slide 14- 37

Figure 14.11 Normalizing into 2NF and
3NF
Slide 14- 38
Figure 14.11
Normalizing into 2NF and 3NF.
(a) Normalizing EMP_PROJ into
2NF relations. (b) Normalizing
EMP_DEPT into 3NF relations.

Figure 14.12 Normalization into 2NF and
3NF
Slide 14- 39
Figure 14.12
Normalization into 2NF
and 3NF. (a) The LOTS
relation with its
functional dependencies
FD1 through FD4.
(b) Decomposing into
the 2NF relations LOTS1
and LOTS2. (c)
Decomposing LOTS1
into the 3NF relations
LOTS1A and LOTS1B.
(d) Progressive
normalization of LOTS
into a 3NF design.

3.6 Third Normal Form (1)
 Definition:
 Transitive functional dependency: a FD X -> Z
that can be derived from two FDs X -> Y and Y ->
Z
 Examples:
 SSN -> DMGRSSN is a transitive FD
 Since SSN -> DNUMBER and DNUMBER ->
DMGRSSN hold
 SSN -> ENAME is non-transitive
 Since there is no set of attributes X where SSN -> X
and X -> ENAME
Slide 14- 40

Third Normal Form (2)
 A relation schema R is in third normal form (3NF) if it is
in 2NF and no non-prime attribute A in R is transitively
dependent on the primary key
 R can be decomposed into 3NF relations via the process
of 3NF normalization
 NOTE:
 In X -> Y and Y -> Z, with X as the primary key, we consider
this a problem only if Y is not a candidate key.
 When Y is a candidate key, there is no problem with the
transitive dependency .
 E.g., Consider EMP (SSN, Emp#, Salary ).
 Here, SSN -> Emp# -> Salary and Emp# is a candidate key.
Slide 14- 41

Normal Forms Defined Informally
 1st normal form
 All attributes depend on the key
 2nd normal form
 All attributes depend on the whole key
 3rd normal form
 All attributes depend on nothing but the key
Slide 14- 42

4. General Normal Form Definitions (For
Multiple Keys) (1)
 The above definitions consider the primary key
only
 The following more general definitions take into
account relations with multiple candidate keys
 Any attribute involved in a candidate key is a
prime attribute
 All other attributes are called non-prime
attributes.
Slide 14- 43

4.1 General Definition of 2NF (For
Multiple Candidate Keys)
 A relation schema R is in second normal form
(2NF) if every non-prime attribute A in R is fully
functionally dependent on every key of R
 In Figure 14.12 the FD
County_name → Tax_rate violates 2NF.
So second normalization converts LOTS into
LOTS1 (Property_id#, County_name, Lot#, Area, Price)
LOTS2 ( County_name, Tax_rate)
Slide 14- 44

4.2 General Definition of Third Normal
Form
 Definition:
 Superkey of relation schema R - a set of attributes
S of R that contains a key of R
 A relation schema R is in third normal form (3NF)
if whenever a FD X → A holds in R, then either:
 (a) X is a superkey of R, or
 (b) A is a prime attribute of R
 LOTS1 relation violates 3NF because
Area → Price ; and Area is not a superkey in
LOTS1. (see Figure 14.12).
Slide 14- 45

4.3 Interpreting the General Definition of
Third Normal Form
 Consider the 2 conditions in the Definition of 3NF:
A relation schema R is in third normal form (3NF) if
whenever a FD X → A holds in R, then either:
 (a) X is a superkey of R, or
 (b) A is a prime attribute of R
 Condition (a) catches two types of violations :
- one where a prime attribute functionally determines
a non-prime attribute. This catches 2NF violations due to
non-full functional dependencies.
-second, where a non-prime attribute functionally
determines a non-prime attribute. This catches 3NF
violations due to a transitive dependency.
Slide 14- 46

4.3 Interpreting the General Definition of
Third Normal Form (2)
 ALTERNATIVE DEFINITION of 3NF: We can restate the definition
as:
A relation schema R is in third normal form (3NF) if
every non-prime attribute in R meets both of these
conditions:
 It is fully functionally dependent on every key of R
 It is non-transitively dependent on every key of R
Note that stated this way, a relation in 3NF also meets
the requirements for 2NF.
 The condition (b) from the last slide takes care of the
dependencies that “slip through” (are allowable to) 3NF
but are “caught by” BCNF which we discuss next.
Slide 14- 47

5. BCNF (Boyce-Codd Normal Form)
 A relation schema R is in Boyce-Codd Normal Form
(BCNF) if whenever an FD X → A holds in R, then X is a
superkey of R
 Each normal form is strictly stronger than the previous
one
 Every 2NF relation is in 1NF
 Every 3NF relation is in 2NF
 Every BCNF relation is in 3NF
 There exist relations that are in 3NF but not in BCNF
 Hence BCNF is considered a stronger form of 3NF
 The goal is to have each relation in BCNF (or 3NF)
Slide 14- 48

Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 14- 49
Figure 14.13 Boyce-Codd normal form
Figure 14.13
Boyce-Codd normal form. (a) BCNF normalization of
LOTS1A with the functional dependency FD2 being lost in
the decomposition. (b) A schematic relation with FDs; it is
in 3NF, but not in BCNF due to the f.d. C → B.

Figure 14.14 A relation TEACH that is in
3NF but not in BCNF
Slide 14- 50
Figure 14.14
A relation TEACH that is in 3NF
but not BCNF.

Achieving the BCNF by Decomposition (1)
 Two FDs exist in the relation TEACH:
 fd1: { student, course} -> instructor
 fd2: instructor -> course
 {student, course} is a candidate key for this relation and
that the dependencies shown follow the pattern in Figure
14.13 (b).
 So this relation is in 3NF but not in BCNF
 A relation NOT in BCNF should be decomposed so as to
meet this property, while possibly forgoing the
preservation of all functional dependencies in the
decomposed relations.
 (See Algorithm 15.3)
Slide 14- 51

Achieving the BCNF by Decomposition (2)
 Three possible decompositions for relation TEACH
 D1: {student, instructor} and {student, course}
 D2: {course, instructor } and {course, student}
 D3: {instructor, course } and {instructor, student} 
 All three decompositions will lose fd1.
 We have to settle for sacrificing the functional dependency
preservation. But we cannot sacrifice the non-additivity property
after decomposition.
 Out of the above three, only the 3rd decomposition will not generate
spurious tuples after join.(and hence has the non-additivity property).
 A test to determine whether a binary decomposition (decomposition
into two relations) is non-additive (lossless) is discussed under
Property NJB on the next slide. We then show how the third
decomposition above meets the property.
Slide 14- 52

Test for checking non-additivity of Binary
Relational Decompositions
 Testing Binary Decompositions for Lossless
Join (Non-additive Join) Property
 Binary Decomposition: Decomposition of a
relation R into two relations.
 PROPERTY NJB (non-additive join test for
binary decompositions): A decomposition D =
{R1, R2} of R has the lossless join property with
respect to a set of functional dependencies F on R
if and only if either
 The f.d. ((R1 ∩ R2)  (R1- R2)) is in F+, or
 The f.d. ((R1 ∩ R2)  (R2 - R1)) is in F+.

Test for checking non-additivity of Binary
Relational Decompositions
If you apply the NJB test to the 3
decompositions of the TEACH relation:
 D1 gives Student  Instructor or Student 
Course, none of which is true.
 D2 gives Course  Instructor or Course 
Student, none of which is true.
 However, in D3 we get Instructor  Course or
Instructor  Student.
Since Instructor  Course is indeed true, the NJB
property is satisfied and D3 is determined as a non-
additive (good) decomposition.

Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 14-55
General Procedure for achieving BCNF
when a relation fails BCNF
Here we make use the algorithm from Chapter
15 (Algorithm 15.5):
 Let R be the relation not in BCNF, let X be a subset-of R,
and let X  A be the FD that causes a violation of BCNF.
Then R may be decomposed into two relations:
 (i) R –A and (ii) X υ A.
 If either R –A or X υ A. is not in BCNF, repeat the
process.
Note that the f.d. that violated BCNF in TEACH was Instructor Course.
Hence its BCNF decomposition would be :
(TEACH – COURSE) and (Instructor υ Course), which gives
the relations: (Instructor, Student) and (Instructor, Course) that we
obtained before in decomposition D3.

5. Multivalued Dependencies and Fourth
Normal Form (1)
Definition:
 A multivalued dependency (MVD) X —>> Y specified on relation
schema R, where X and Y are both subsets of R, specifies the
following constraint on any relation state r of R: If two tuples t1 and
t2 exist in r such that t1[X] = t2[X], then two tuples t3 and t4 should
also exist in r with the following properties, where we use Z to
denote (R 2 (X υ Y)):
 t3[X] = t4[X] = t1[X] = t2[X].
 t3[Y] = t1[Y] and t4[Y] = t2[Y].
 t3[Z] = t2[Z] and t4[Z] = t1[Z].
 An MVD X —>> Y in R is called a trivial MVD if (a) Y is a subset of
X, or (b) X υ Y = R.

Multivalued Dependencies and Fourth Normal
Form (3)
Definition:
 A relation schema R is in 4NF with respect to a set of
dependencies F (that includes functional dependencies
and multivalued dependencies) if, for every nontrivial
multivalued dependency X —>> Y in F+, X is a superkey
for R.
 Note: F+ is the (complete) set of all dependencies
(functional or multivalued) that will hold in every relation
state r of R that satisfies F. It is also called the closure of
F.

Figure 14.15 Fourth and fifth normal
forms.
Slide 14- 58
Figure 14.15
Fourth and fifth normal forms. (a) The EMP relation with two MVDs: Ename –>> Pname and Ename –>>
Dname. (b) Decomposing the EMP relation into two 4NF relations EMP_PROJECTS and EMP_DEPENDENTS.
(c) The relation SUPPLY with no MVDs is in 4NF but not in 5NF if it has the JD(R1, R2, R3). (d)
Decomposing the relation SUPPLY into the 5NF relations R1, R2, R3.

6. Join Dependencies and Fifth Normal Form
(1)
Definition:
 A join dependency (JD), denoted by JD(R1, R2, ..., Rn),
specified on relation schema R, specifies a constraint
on the states r of R.
 The constraint states that every legal state r of R should
have a non-additive join decomposition into R1, R2, ..., Rn;
that is, for every such r we have
 * (R1(r), R2(r), ..., Rn(r)) = r
Note: an MVD is a special case of a JD where n = 2.
 A join dependency JD(R1, R2, ..., Rn), specified on
relation schema R, is a trivial JD if one of the relation
schemas Ri in JD(R1, R2, ..., Rn) is equal to R.

Join Dependencies and Fifth Normal Form (2)
Definition:
 A relation schema R is in fifth normal form
(5NF) (or Project-Join Normal Form (PJNF))
with respect to a set F of functional, multivalued,
and join dependencies if,
 for every nontrivial join dependency JD(R1, R2, ...,
Rn) in F+ (that is, implied by F),
 every Ri is a superkey of R.
 Discovering join dependencies in practical databases
with hundreds of relations is next to impossible.
Therefore, 5NF is rarely used in practice.

Chapter Summary
 Informal Design Guidelines for Relational
Databases
 Functional Dependencies (FDs)
 Normal Forms (1NF, 2NF, 3NF)Based on Primary
Keys
 General Normal Form Definitions of 2NF and 3NF
(For Multiple Keys)
 BCNF (Boyce-Codd Normal Form)
 Fourth and Fifth Normal Forms
Slide 14- 61

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