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Functional Dependency and Normalization

• Informal design guidelines for relation schemas.

• Functional dependencies.

• Normal forms.

• Normalization.

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Informal Design Guidelines

• Semantics of relations and attributes.

• Guideline 1: Design a relation schema so that it is easy to explain its meaning. (Fig. 14.1, 14.2)

Do not combine attributes from multiple entity types and relationship types into single relation. (Fig. 14.3)

• Reducing redundant values in tuples saves storage space and avoid update anomalies. (Fig. 14.4)

- Insertion anomalies. - Deletion anomalies. - Modification anomalies.

• Guideline 2: Design the base relation schemas so that no insertion, deletion, or modification anomalies occur.

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• Insert Anomalies

• Inserting a dept with no employee info – null values need to assign, which will create problems

• Inconsistency problem with insertion of new tuple

Deletion Anomalies

– If we delete last employee, dept info is deleted.

-Modification anomalies – if we change manager of department 5, we must update all the tuples

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Figure 14.1 Simplified version of the COMPANY relational database schema.

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Figure 14.2 Example relations for the schema of Figure 14.1

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Figure 14.3 Two relation schemas and their functional dependencies. Both suffer from update anomalies. (a) The EMP_DEPT relation schema.

(b) The EMP_PROJ relation schema.

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Figure 14.4 Example relations for the schemas in Figure 14.3 that result from applying NATURAL JOIN to the relations in Figure 14.2.

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Figure 14.5 Alternative (bad) representation of the EMP_PROJ relation. (a) Representing EMP_PROJ of Figure 14.3(b) by two relation schemas: EMP_LOCS and EMP_PROJ1. (b) Result of projecting the populated relation EMP_PROJ of Figure

14.4 on the attributes of EMP_LOCS and EMP_PROJ1.

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Figure 14.5 (continued)

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Figure 14.6 Result of applying the NATURAL JOIN operation to the tuples above dotted lines in EMP_PROJ1 and EMP_LOCS, with generated spurious tuples marked

by an asterisk.

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Informal Design Guidelines• Reducing the null values in tuples. e.g., if 10% of employees have

offices, it is better to have a separate relation, EMP_OFFICE, rather than an attribute OFFICE_NUMBER in EMPLOYEE.

• Guideline 3: Avoid placing attributes in a base relation whose values are mostly null.

• Disallowing spurious tuples. - Spurious tuples: tuples that are not in the original relation but generated by natural join of decomposed subrelations. - Example: decompose EMP_PROJ into EMP_LOCS and EMP_PROJ1. (Fig. 14.5) - natural join of EMP_LOCS and EMP_PROJ1 results in spurious tuples. (Fig. 14.6)• Guideline 4: Design relation schemas so that they can be naturally

JOINed on primary keys or foreign keys in a away that guarantees no spurious tuples are generated.

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Functional Dependencies

• A functional dependency, denoted by X Y, between two sets of attributes X and Y (X and Y are subsets of R) specifies a constraint on the possible tuples that can form a relation instance r of R: for any two tuples t1 and t2 in r such that t1[X]= t2[X], we must have t1[Y]= t2[Y].

• If X Y, we say X functionally determines Y or Y is functionally dependent on X.

• We abbreviate functional dependency by FD. X is called the left-hand side of the FD. Y is called the right-hand side of the FD.

• A functional dependency is a property of the meaning or semantics of the attributes, I.e., a property of the relation schema. They must hold on all relation states (extensions) of R. Relation extensions r(R) that satisfy the FD are called legal extensions.

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Figure 14.7 The teach relation state with an apparent functional dependency text COURSE. However, COURSE

TEXT is ruled out.

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Functional Dependencies (Cont.)

• Examples. 1. SSN ENAME 2. PNUMBER {PNAME, PLOCATION} 3. {SSN, PNUMBER} HOURS 4. Others?

• Diagrammatic notation for displaying FDs. (Fig. 14.3)

• FD is property of the relation schema R, not of a particular relation state/instance r(R).

• FDs cannot be inferred from a given relation extension r, but must be defined explicitly by someone who knows the semantics of the attributes of R. (Fig. 14.7)

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Figure 14.3 Two relation schemas and their functional dependencies. Both suffer from update anomalies. (a) The EMP_DEPT relation schema.

(b) The EMP_PROJ relation schema.

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Functional Dependencies (Cont)

• From the FDs: F = {SSN { ENAME, BDATE, ADDRESS, DNUMBER}, DNUMBER {DNAME, DMGRSSN}} we can infer the following FDs: SSN {ENAME, DMGRSSN}, SSN SSN, DNUMBER DNAME

• A FD X Y is inferred from a set of dependencies F specified on R if X Y holds in every relation state r that is a legal extension of R.

• F |= X Y denotes X Y is inferred from F.• The closure of F, denoted by F+, is the set of all FDs that can be

inferred from F.

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Functional Dependencies (Cont.)

• Inference rules for FDs.

• Abbreviated notation: XYZ UV for {X, Y, Z} {U, V}

• Reflective: If Y X, then X Y

• Augmentation: {X Y} |= XZ YZ

• Transitive: {X Y, Y Z} |= X Z

• Decomposition (projective): {X YZ} |= X Y

• Union (additive): {X Y, X Z} |= X YZ

• Pseudotransitive: {X Y, WY Z} |= WX Z

• The first three rules are sound and complete, called Armstrong's inference rules.

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Functional Dependencies (Cont.)

• Closure of X under F, denoted by X+, is the set of all attributes that are functionally determined by X under F.

• Algorithms for determining X+

X+ := X;

repeat

oldX+ := X+;

for each FD Y Z in F do

if Y X+ then X+ :=X+ Z;

until oldX+ = X+;

• Example:

F = {SSN ENAME, PNUMBER {PNAME, PLOCATION},

{SSN, PNUMBER} HOURS}

{SSN}+ = {SSN, ENAME}

{PNUMBER}+ = ?

{SSN, PNUMBER}+ = ?

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Functional Dependencies (Cont.)

• Equivalence of sets of FDs.

• E is covered by F if every FD in E is also in F+, i.e., every FD in E can be inferred from F.

• E and F are equivalent if E+ = F+, i.e, E covers F and F covers E.

• F is minimal if - every dependency in F has a single attribute for its right hand side; - we cannot remove any FD from F and still have a set of FDs equivalent to F; - we cannot replace any FD X A in F with a FD Y A where Y X and still have a set of FDs equivalent to F.

• Minimal set: a standard or canonical form with no redundancies.

• A minimal cover of F is a minimal set of dependencies, Fmin, that is equivalent to F.

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Functional Dependencies (Cont.)

• Compute a minimal cover

Algorithm 14.2 Find a minimal cover G for F.

1. G := f;

2. Replace each FD X A1, A2,…, AK in G by the k FDs X A1, X A2,

X AK;

3. for each FD X A in G

for each attribute B X

if (X – B)+ with-respect-to G contains A

then replace X A with X – {B} A in G;

4. For each FD X A in G

if X+ with-respect-to G-{X A} contains A

then remove X A from G;

• There is at least one minimal cover for any F, maybe several.

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Normal Forms• Superkey, candidate key or key, primary key.

• A FD X Y is a full functional dependency if removal of any attribute from X means that the dependency does not hold any more; otherwise, it is a partial functional dependency.

• An attribute is prime if it is a member of any key (Primary or candidate).

• A relation R is in first normal form if domains of attributes include only atomic values. (Fig. 14.8, 14.9)

• A relation R is in second normal form if every non-prime attribute A in R is not partially dependent on any key of R.

• Alternatively, R is in 2NF if every non-prime attribute A in R is fully dependent on every key of R.

• Examples. (Fig. 14.10 a, b)

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Figure 14.8 Normalization into 1NF. (a) Relational schema that is not in 1NF. (b) Example relation instance. (c) 1NF relation with redundancy.

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Figure 14.9 Normalizing nested relations into 1NF. (a) Schema of the EMP_PROJ relation with a “nested relation” PROJS. (b) Example extension of the EMP_PROJ

relation showing nested relations within each tuple.

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Figure 14.9 (continued) (c) Decomposing EMP_PROJ into 1NF relations EMP_PROJ1 and

EMP_PROJ2 by propagating the primary key.

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Figure 14.10 The normalization process. (a) Normalizing EMP_PROJ into 2NF relations. (b) Normalizing EMP_DEPT into 3NF relations.

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Normal Forms

• A relation R is in third normal form if for every FD X A that holds on R, either

- X is a superkey of R, or - A is a prime attribute of R.(Alternative Def . - No transitive dependencies – If there is a set of attributes Z that

is neither a candidate key nor a subset of any key (primary or candidate) of R , X Z and Z Y holds.

SSN DMGRSSN is transitive as SSN Dnumber DMGRSSN (Emp-dept) and dnumber is neither a key nor a subset of key.

• Example. (Fig. 14.10 c)

• A relation R is in Boyce-Codd normal form if for every FD X A that holds on R, X is a superkey of R.

• Example. (Fig. 14.12)

• Increasing Order of restrictiveness: 1NF, 2NF, 3NF, BCNF. For example, if a relation schema R is in BCNF, it is in 3NF.

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Figure 14.11 Normalization to 2NF and 3NF. (a) The lots relation schema and its functional dependencies FD1 through FD4. (b) Decomposing lots into the 2NF

relations LOTS1 and LOTS2.

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Figure 14.11 (continued) (c) Decomposing LOTS1 into the 3NF relations LOTS1A and LOTS1B.

(d) Summary of normalization of lots.

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Figure 14.12 Boyce-Codd normal form. (a) BCNF normalization with the dependency of FD2 being “lost” in the decomposition. (b) A relation R in 3NF but

not in BCNF.

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Figure 14.13 A relation TEACH that is in 3NF but not in BCNF.

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Normalization

• Database design revisited. Top-down approach – conceptual design. A more purist way – decomposition.

• Normalization: a process in which unsatisfactory relational schemas are decomposed into smaller relation schemas that possess desirable properties.

• Starting with a single universal relation schema R = A1, A2,…. An that includes all the attributes of the database.

• Decompose R into a set of relation schemas D ={R1, R2,… Rm} using the FDs specified by the database designers. D is called a decomposition of R.

• Guidelines for normalization: normal forms, attribute preservation, dependency preservation, lossless join.

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Normalization (Cont.)

• Attribute preservation. No attributes are lost. m

U Ri = R i=1

• Dependency preservation.

(F(R1) F(R2) ……. F(Rm) )+ = F+

where F(R1) is the set of FDs, X Y , in F+ such that

X Y Ri.

• A decomposition D={R1, R2,…., Rm} of R has the lossless join property with respect to the set of dependencies F on R if, for every relation state r of R that satisfies F,

*(<R1>(r),…, <Rm>(r)) = r

where <Ri> are the attributes in Ri.

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Normalization (Cont.)

• Decomposition into 3NF relation schemas Algorithm 15.1 Dependency-preserving and lossless

decomposition into 3NF relation schemas. 1. Find a minimal cover G for F (Algorithm 14.2) 2. For each left-hand side X of a FD in G

create a relation schema {X A1 A2 … Ak} in D where X A1, X A2,…., X Ak are the only dependencies in G with X as left-hand side;

3. Place any remaining (unplaced) attributes in a single relation schema;

4. If none of the relation schemas contains a key of R, create one more relation schema that contains attributes that form a key for R.

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Normalization (Cont.)

• Determine a key

Algorithm 15.4a Find a key K for R.

1. K := R;

2. For each attribute A in K

if (K – {A})+ with-respect-to F contains A then remove A from K;

• Example. (Fig. 14.11)

• It is not always possible to find a decomposition that preserves dependencies and in BCNF. (Fig. 14.12)

• The lossless join decomposition is based on the assumption that no null values are allowed for the join attributes.