Chapter10in normalization for Data base management system .ppt

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Copyright © 2007 Ramez Elmasri and Shamkant B. Navathe

Chapter 10
Functional Dependencies and
Normalization for Relational
Databases

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Chapter Outline
 1 Informal Design Guidelines for Relational Databases
 1.1Semantics 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 FD
 2.2 Inference Rules for FDs
 2.3 Equivalence of Sets of FDs
 2.4 Minimal Sets of FDs

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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 Multiple Keys)
 5 BCNF (Boyce-Codd Normal Form)

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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?

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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 Normal Form)
 - BCNF (Boyce-Codd Normal Form)
 Additional types of dependencies, further normal forms,
relational design algorithms by synthesis are discussed in
Chapter 11

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1.1 Semantics of the Relation Attributes
 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.

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Figure 10.1 A simplified COMPANY
relational database schema

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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

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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.

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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.

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EXAMPLE OF AN 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.

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Figure 10.3 Two relation schemas
suffering from update anomalies

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Figure 10.4 Example States for
EMP_DEPT and EMP_PROJ

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Guideline to 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.

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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

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1.4 Spurious Tuples
 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.

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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 11).

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2.1 Functional Dependencies (1)
 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

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Functional Dependencies (2)
 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

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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

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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])

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2.2 Inference Rules for FDs (1)
 Given a set of FDs F, we can infer additional FDs that
hold whenever the FDs in F hold
 Armstrong's inference rules:
 IR1. (Reflexive) If Y subset-of X, then X -> Y
 IR2. (Augmentation) If X -> Y, then XZ -> YZ

(Notation: XZ stands for X U Z)
 IR3. (Transitive) If X -> Y and Y -> Z, then X -> Z
 IR1, IR2, IR3 form a sound and complete set of
inference rules
 These are rules hold and all other rules that hold can be
deduced from these

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Inference Rules for FDs (2)
 Some additional inference rules that are useful:
 Decomposition: If X -> YZ, then X -> Y and X ->
Z
 Union: If X -> Y and X -> Z, then X -> YZ
 Psuedotransitivity: If X -> Y and WY -> Z, then
WX -> Z
 The last three inference rules, as well as any
other inference rules, can be deduced from IR1,
IR2, and IR3 (completeness property)

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Inference Rules for FDs (3)
 Closure of a set F of FDs is the set F+
of all FDs
that can be inferred from F
 Closure of a set of attributes X with respect to F
is the set X+
of all attributes that are functionally
determined by X
 X+
can be calculated by repeatedly applying IR1,
IR2, IR3 using the FDs in F

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2.3 Equivalence of Sets of FDs
 Two sets of FDs F and G are equivalent if:
 Every FD in F can be inferred from G, and
 Every FD in G can be inferred from F
 Hence, F and G are equivalent if F+
=G+
 Definition (Covers):
 F covers G if every FD in G can be inferred from F

(i.e., if G+
subset-of F+
)
 F and G are equivalent if F covers G and G covers F
 There is an algorithm for checking equivalence of sets of
FDs

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2.4 Minimal Sets of FDs (1)
 A set of FDs is minimal if it satisfies the
following conditions:
1. Every dependency in F has a single attribute for
its RHS.
2. We cannot remove any dependency from F and
have a set of dependencies that is equivalent to
F.
3. We cannot replace any dependency X -> A in F
with a dependency Y -> A, where Y proper-
subset-of X ( Y subset-of X) and still have a set
of dependencies that is equivalent to F.

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Minimal Sets of FDs (2)
 Every set of FDs has an equivalent minimal set
 There can be several equivalent minimal sets
 There is no simple algorithm for computing a
minimal set of FDs that is equivalent to a set F of
FDs
 To synthesize a set of relations, we assume that
we start with a set of dependencies that is a
minimal set
 E.g., see algorithms 11.2 and 11.4

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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

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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

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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 (Chapter 11)
 Additional properties may be needed to ensure a
good relational design (lossless join, dependency
preservation; Chapter 11)

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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, BCNF or 4NF)
 Denormalization:
 The process of storing the join of higher normal form
relations as a base relation—which is in a lower normal form

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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.

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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.

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3.2 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 relation

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Figure 10.8 Normalization into 1NF

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Figure 10.9 Normalization nested
relations into 1NF

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3.3 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

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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

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Figure 10.10 Normalizing into 2NF and
3NF

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Figure 10.11 Normalization into 2NF and
3NF

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3.4 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

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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.

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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

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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
 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

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General Normal Form Definitions (2)
 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
 NOTE: Boyce-Codd normal form disallows
condition (b) above

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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
 The goal is to have each relation in BCNF (or 3NF)

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Figure 10.12 Boyce-Codd normal form

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Figure 10.13 a relation TEACH that is in
3NF but not in BCNF

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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
10.12 (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 11.3)

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Achieving the BCNF by Decomposition (2)
 Three possible decompositions for relation TEACH
 {student, instructor} and {student, course}
 {course, instructor } and {course, student}
 {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 in section
11.1.4 under Property LJ1. Verify that the third decomposition above
meets the property.

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Chapter Outline
 Informal Design Guidelines for Relational
Databases
 Functional Dependencies (FDs)
 Definition, Inference Rules, Equivalence of Sets of
FDs, Minimal Sets of FDs
 Normal Forms Based on Primary Keys
 General Normal Form Definitions (For Multiple
Keys)
 BCNF (Boyce-Codd Normal Form)

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