# 程序代写代做代考 database js algorithm Schema Refinement and Normal Forms

Schema Refinement and Normal Forms

Normal Forms. BCNF and 3NF

Decompositions

CS430/630
Lecture 17

Slides based on “Database Management Systems” 3rd ed, Ramakrishnan and Gehrke

Decomposition of a Relation Schema

 A decomposition of R replaces it by two or more relations

 Each new relation schema contains a subset of the attributes of R

 Every attribute of R appears in one of the new relations

 E.g., SNLRWH decomposed into SNLRH and RW

 Decompositions should be used only when needed

 Cost of join will be incurred at query time

 Problems may arise with (improper) decompositions

 Reconstruction of initial relation may not be possible

 Dependencies cannot be checked on smaller tables

Lossless Join Decompositions

 Decomposition of R into X and Y is lossless-join if:

 (r) (r) = r

 It is always true that r (r) (r)

 In general, the other direction does not hold!

 If it does, the decomposition is lossless-join.

 It is essential that all decompositions used to deal with

redundancy be lossless!

X

Y

 
X

 
Y

Incorrect Decomposition

A B C

1 2 3

4 5 6

7 2 8

1 2 8

7 2 3

A B C

1 2 3

4 5 6

7 2 8

A B

1 2

4 5

7 2

B C

2 3

5 6

2 8

Natural

Join

Condition for Lossless-join

 The decomposition of R into X and Y is lossless-join wrt

F if and only if the closure of F contains:

 X Y X, or

 X Y Y

 In particular, the decomposition of R into UV and R – V is

lossless-join if U V holds over R.

Dependency Preserving Decomposition

 Consider CSJDPQV, C is key, JP C and SD P.

 Consider decomposition: CSJDQV and SDP

 Problem: Checking JP C requires a join!

 Dependency preserving decomposition (Intuitive):

 If R is decomposed into X and Y, and we enforce the FDs that hold on

X, Y then all FDs that were given to hold on R must also hold

 Projection of set of FDs F: If R is decomposed into X, …

projection of F onto X (denoted FX ) is the set of FDs U V

in F+ (closure of F ) such that U, V are in X.

 

Dependency Preserving Decompositions

 Decomposition of R into X and Y is dependency preserving if

(FX U FY )
+ = F +

 Dependencies that can be checked in X without considering Y, and in

Y without considering X, together represent all dependencies in F +

 Dependency preserving does not imply lossless join:

 ABC, A B, decomposed into AB and BC. 

Normal Forms

 If a relation is in a certain normal form (BCNF, 3NF etc.), it is

known that certain kinds of problems are avoided/minimized.

 Role of FDs in detecting redundancy:

 Consider a relation R with attributes AB

 No FDs hold: There is no redundancy

 Given A B:

 Several tuples could have the same A value

 If so, they’ll all have the same B value!

Boyce-Codd Normal Form (BCNF)

 Relation R with FDs F is in BCNF if, for all X A in

 A X (called a trivial FD), or

 X contains a key for R

 The only non-trivial FDs allowed are key constraints

 BCNF guarantees no anomalies occur

F

Decomposition into BCNF

 Consider relation R with FDs F. If X Y violates BCNF,

decompose R into R – Y and XY.

 Repeated application of this idea will give us a collection of relations

that are in BCNF; lossless join decomposition, and guaranteed to

terminate.

 e.g., CSJDPQV, key C, JP C, SD P, J S

 To deal with SD P, decompose into SDP, CSJDQV.

 To deal with J S, decompose CSJDQV into JS and CJDQV

  

Decomposition into BCNF

 In general, several dependencies may cause violation of BCNF.

The order in which we “deal with” them could lead to very

different sets of relations!

CSJDPQV

SDP CSJDQV

SD P 

JS CJDQV

J S 

BCNF and Dependency Preservation

 In general, there may not be a dependency preserving

decomposition into BCNF

 e.g., ABC, AB C, C A

 Can’t decompose while preserving first FD; not in BCNF

 

Third Normal Form (3NF)

 Relation R with FDs F is in 3NF if, for all X A in

 A X (called a trivial FD), or

 X contains a key for R, or

 A is part of some key for R (A here is a single attribute)

 Minimality of a key is crucial in third condition above!

 If R is in BCNF, it is also in 3NF.

 If R is in 3NF, some redundancy is possible

 compromise used when BCNF not achievable

 e.g., no “good’’ decomposition, or performance considerations

 Lossless-join, dependency-preserving decomposition of R into a

collection of 3NF relations always possible.

F

Decomposition into 3NF

 Lossless join decomposition algorithm also applies to 3NF

 To ensure dependency preservation, one idea:

 If X Y is not preserved, add relation XY

 Refinement: Instead of the given set of FDs F, use a minimal

cover for F

 Example: CSJDPQV, JP C, SD P, J S

 Choose SD P, result is SDP and CSJDQV

 Choose J S, result is JS and CJDQV, all 3NF



Summary of Schema Refinement

 BCNF: relation is free of FD redundancies

 Having only BCNF relations is desirable

 If relation is not in BCNF, it can be decomposed to BCNF

 Lossless join property guaranteed

 But some FD may be lost

 3NF is a relaxation of BCNF

 Guarantees both lossless join and FD preservation

 Decompositions may lead to performance loss

 performance requirements must be considered when using

decomposition

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