DatabaseIM ISU1 Chapter 10 Functional Dependencies and Normalization for RDBs Fundamentals of...

IM ISU 1Database

Chapter 10

Functional Dependencies and

Normalization for RDBs

Fundamentals of Database Systems

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Informal Guidelines for Relation Schema Design

Four Informal Measures Semantics of the attributes Reducing the redundant values in tuples Reducing the null values in tuples Disallowing the possibility of generating spur

ious tuples

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Informal Guidelines for Relation Schema Design (cont.)

Semantics of the Relation Attributes Meaning of semantics: specifies how to inter

pret the attribute values stored in a tuple of the relation

In general, the easier it is to explain the semantics of the relation, the better the relation schema design will be

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ER-to-Relational Mapping (cont.)

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GUIDELINE 1» Design a relation schema so that it is easy to

explain its meaning» Do not combine attributes from multiple entity

types and relationship types into a single relation» Intuitively, if a relation schema corresponds to

one entity type or one relationship type, the meaning tends to be clear

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Poor design example:

EMP_DEPT mixes attributes of employees and departments

EMP_PROJ mixes attributes of employees and projects

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Redundant Information in Tuples and Update Anomalies One goal of schema design is to minimize th

e storage space of the base relations Example: compare Fig. 14.2 and 14.4

» Redundancy problem» Update anomalies problem

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Fig 14.2

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Fig 14.4

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Update Anomalies Insertion anomalies: two situations

» To insert a new employee tuple into EMP_DEPT, we must include either the attribute values for the department that the employee works for, or nulls

» It is difficult to insert a new department that has no employees as yet in the EMP_DEPT relation

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Deletion anomalies» If we delete from EMP_DEPT an employee tuple

that happens to represent the last employee working for a particular department, the information concerning that department is lost

Modification anomalies» If we change the value of one of the attributes of

a particular department, we must update the tuples of all employees in that department to avoid inconsistency

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GUIDELINE 2» Design the base relation schemas so that no

insertion, deletion, or modification anomalies are present in the relations

» If any anomalies are present, note them clearly and make sure that the programs that update the database will operate correctly

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Note» To improve the performance of certain queries, th

ese guidelines may sometimes have to be violated

» In general, it is advisable to use anomaly-free base relations and to specify views that include the JOINs for placing together the attributes frequently referenced in important queries

» Example: Specify EMP_DEPT as a view to speedup query

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Null Values in Tuples Problems with null values

» Waste space at the storage level » May lead to problems with understanding the me

aning of the attributes– The attribute does not apply to this tuple

– The attribute value for this tuple is unknown

– The value is known but absent

» How to account for them when aggregate operations such as COUNT or SUM are applied

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A NATURAL JOIN on EMP_PROJ1 and EMP_LOCS produces more tuples than those in EMP_PROJ

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GUIDELINE 4» Design relation schemas so that they can be JOIN

ed with equality conditions on attributes that are either primary keys or foreign keys in a way that guarantees that no spurious tuples are generated

» Do not have relations that contain matching attributes other than foreign key-primary key combinations

» If such relations are unavoidable, do not join them on such attributes

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Functional Dependencies

Definition Consider a relation schema R = {A1, A2, …, An}.

A functional dependency, denoted by X Y, for X, Y R, specifies a constraint on a relation state r of R such that for any two tuples t1 and t

2 in r,

if t1[X] = t2[X], we must have t1[Y] = t2[Y].

Note: if X is a candidate key of R, this implies that X Y for any subset of attributes Y of R

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Functional Dependencies (cont.)

Meaning The Y component of a tuple in r depend on,

or are determined by, the values of the X component

The values of the X component of a tuple uniquely (or functionally) determine the values of the Y component

We also say that there is a functional (FD) dependency from X to Y or that Y is functionally dependent on X

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Example: Relation schema EMP_PROJ 1. SSN ENAME

2. PNUMBER {PNAME, PLOCATION}

3. {SSN, PNUMBER} HOURS

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Notice A functional dependency is a property of the

relation schema (intension) R, not of a particular legal relation state (extension) r of R.

An FD cannot be inferred automatically from a given relation extension r but must be defined explicitly by someone who knows the semantics of the attributes of R.

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Inference Rules for FDs F: the set of functional dependencies specifie

d on a relation schema R Other dependencies can be inferred or deduce

d from the FDs in F The set of all such dependencies is called the

closure of F and is denoted by F+

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Functional Dependencies (cont.) Example

» Let F = {SSN {ENAME, BDATE, ADDRESS, DNUMBER}, DNUMBER {DNAME, DMGRSSN}}

» We can infer the following additional FDsSSN {DNAME, DMGRSSN}, SSN SSN, DNUMBER DNAME

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Functional Dependencies (cont.) Inference rules

» The rules that can be used to infer new dependencies from a given F

» Notation F X Y: X Y is inferred from the set F

» For simplicity, – {X, Y} Z is abbreviated to XY Z – {X, Y, Z} {U, V} is abbreviated to XYZ

UV

»There are six well defined rules

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» IR1 (reflexive rule): If X Y, then X Y » IR2 (augmentation rule): {X Y } XZ YZ » IR3 (transitive rule): {X Y, Y Z} X Z » IR4 (decomposition, or projective, rule):

{X YZ} X Y. » IR5 (union, or additive, rule):

{X Y, X Z} X YZ » IR6 (pseudotransitive rule):

{X Y, WY Z } WX Z

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IR1 through IR3 are known as Armstrong’s inference rules » It has been shown by Armstrong (1974) that

inference rules IR1 through IR3 are sound and complete

» In other words, the set of dependencies, which we called the closure of F, can be determined from F by using only inference rules IR1 through IR3

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Normalization

Introduction Normalization of data

» A process of analyzing the given relation schemas based on their FDs and primary keys to achieve the desirable properties of (1) minimizing redundancy(2) minimizing the insertion, deletion, and update

anomalies » Unsatisfactory relation schemas that do not meet t

he normal form tests are decomposed into smaller relation schemas

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Normalization (cont.) History

» Initially, Codd (1972a) proposed three normal forms based on FD, which he called first, second, and third normal form

» A stronger definition of 3NF—called Boyce-Codd normal form (BCNF)—was proposed later by Boyce and Codd

» Later, a fourth normal form (4NF) and a fifth normal form (5NF) were proposed, based on the concepts of multivalued dependencies and join dependencies

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Normalization (cont.)

Notice» Normal forms, when considered in isolation from

other factors, do not guarantee a good database design

– The lossless join or nonadditive join property, which guarantees that the spurious tuple generation problem

– The dependency preservation property, which ensures that each functional dependency is represented in some individual relations resulting after decomposition

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Normalization (cont.)» The database designers need not normalize to the

highest possible normal form. – Relations may be left in a lower normalization

status for performance reasons– The process of storing the join of higher norm

al form relations as a base relation—which is in a lower normal form—is known as denormalization

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Normalization (cont.) Related terminology

» Prime attribute – An attribute of relation schema R is called a

prime attribute of R if it is a member of some candidate key of R

» Nonprime attribute – An attribute is called nonprime if it is not a

prime attribute

»Example -- WORKS_ON relation– prime: both SSN and PNUMBER– nonprime: others

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Normal Forms

First Normal Form (1NF) Historically, it was defined to disallow multival

ued attributes, composite attributes, and their combinations

The domain of an attribute must include only atomic (simple, indivisible) values and that the value of any attribute in a tuple must be a single value from the domain of that attribute

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Normal Forms (cont.)

Example» Not in 1NF because DLOCATIONS is not an

atomic attribute

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Normal Forms (cont.) Three main techniques to achieve 1NF

1. Decomposes the non-1NF relation into two 1NF relations

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Normal Forms (cont.)2. Expand the key to distinguish each tuple

– Has the disadvantage of introducing redundancy

3. Divide the attribute into several atomic attributes– DLOCATIONS => DLOCATION1, DLOCATION2, an

d DLOCATION3 – The maximum number of values needs to be known – Has the disadvantage of introducing null values

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Normal Forms (cont.)» The first is superior because it does not suffer fro

m redundancy and it is completely general

The first normal form also disallows multivalued, composite attributes» These are called nested relations» For example:

EMP_PROJ(SSN, ENAME, {PROJS(PNUMBER, HOURS)})

– PROJS is a multivalued, composite attribute

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Normal Forms (cont.) Technique to normalize multivalued, compos

ite attributes into 1NF » Remove the nested relation attributes into a new r

elation » Propagate the primary key into new relation

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Normal Forms (cont.) Second Normal Form (2NF)

2NF is based on the concept of full functional dependency

X Y is a full functional dependency if removal of any attribute A from X invalidates the dependency

X Y is a partial dependency if some attribute A X can be removed and the dependency still holds

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Normal Forms (cont.) Example

» {SSN, PNUMBER} HOURS is a full dependency

» {SSN, PNUMBER} ENAME is partial

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Normal Forms (cont.) Testing for 2NF

» The test for 2NF involves testing for functional dependencies whose left-hand side attributes are part of the primary key

» If the primary key contains a single attribute, the test need not be applied at all

» A relation schema R is in 2NF if every nonprime attribute A in R is fully functionally dependent on the primary key of R

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Normal Forms (cont.)» Example

– EMP_PROJ is in 1NF but not in 2NF– The nonprime attribute ENAME violates 2NF

because FD2 is partial– The nonprime attributes PNAME and

PLOCATION also violates 2NF because FD3 is partial

Method for normalizing a non-2NF relation» Divide the relation into several relations in which

nonprime attributes are associated only with the part of the primary key on which they are fully functionally dependent

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Third Normal Form (3NF) 3NF is based on the concept of transitive

dependency X Y in a relation schema R is a transitive

dependency if there is a set of attributes Z that is neither a candidate key nor a subset of any key of R, and both X Z and Z Y hold

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Normal Forms (cont.) Example

» Both SSN DNUMBER and DNUMBER DMGRSSN hold

» DNUMBER is neither a key nor a subset of the key of EMP_DEPT

» SSN DMGRSSN is a transitive dependency

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Normal Forms (cont.) Testing for 3NF

» A relation schema R is in 3NF if it satisfies 2NF and no nonprime attribute of R is transitively dependent on the primary key

» Example: the EMP_DEPT relation

Method for normalizing a non-3NF relation» Decompose and set up a relation that includes the

nonkey attribute(s) that functionally determine(s) other nonkey attribute(s)

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Normal Forms (cont.)» Example

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General Normal Form Definitions

Preliminary The above definitions consider the primary

key only We have to consider more general definitions

that take into account relations with multiple candidate keys

General definition of prime attribute» An attribute that is part of any candidate key will

be considered as prime

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General Normal Form Definitions (cont.)

General Definition of 2NF 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

Example: relation schema LOTS» Two candidate keys: PROPERTY_ID# and

{COUNTY_NAME, LOT#}» FD1 and FD2 hold» Assume FD3 and FD4 also hold

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» TAX_RATE is partially dependent on the candidate key {COUNTY_NAME, LOT#}, due to FD3

» LOTS not in general 2NF

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» Normalization to general 2NF– Decompose it into the two relations LOTS1

and LOTS2

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General 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 Superkey of relation schema R

» A set of attributes S of R that contains a key of R

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Example» LOTS2 is in general 3NF» FD4 in LOTS1 violates 3NF

– AREA is not a superkey– PRICE is not a prime attribute in LOTS1

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» Normalization LOTS1 to general 3NF– Decompose it into the relation schemas

LOTS1A and LOTS1B

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Boyce-Codd Normal Form Definition

» A relation schema R is in Boyce-Codd Normal Form (BCNF) if whenever a nontrivial FD X A holds in R, then X is a superkey of R

» BCNF is stronger than 3NF– Every relation in BCNF is also in 3NF; however,

a relation in 3NF is not necessarily in BCNF

– The only difference is that condition (b) of 3NF

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General Normal Form Definitions (cont.) Example -- LOTS1A

» Suppose that – There are only two counties: Dekalb and Fulton – Lot sizes in Dekalb: restricted to 0.5, 0.6, ...,1.0 acres – Lot sizes in Fulton: restricted to 1.1, 1.2, ..., 2.0 acres

» There is an additional FD in relation LOTS1AFD5: AREA COUNTY_NAME

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» It is still is in 3NF because COUNTY_NAME is a prime attribute

» FD5 violates BCNF in LOTS1A because AREA is not a superkey of LOTS1A

» We can decompose LOTS1A into two BCNF relations LOTS1AX and LOTS1AY

– In LOTS1AY, there are only 16 possible AREA values

– This reduces the redundancy in LOTS1A tuples– But it loses the functional dependency FD2

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Summary» Most relation 3NF schemas are also in BCNF » Only if FD X A holds in R with X not being a s

uperkey and A being a prime attribute will R be in 3NF but not in BCNF

» General form

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General Normal Form Definitions (cont.) » Example -- TEACH

– FD1: {STUDENT, COURSE} INSTRUCTOR

– FD2: INSTRUCTOR COURSE

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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

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