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:
XYX, or
XYY
In particular, the decomposition of R into UV and R – V is
lossless-join ifU Vholds over R.
Dependency Preserving Decomposition
Consider CSJDPQV,C is key,JP CandSD P.
Consider decomposition: CSJDQV and SDP
Problem:CheckingJPCrequires 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
(FXU 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 XAin
AX (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 XY violates BCNF,
decompose R intoR – 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,JPC,SD P, JS
To deal with SDP, decompose intoSDP, 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
SDP
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 XAin
AX (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:
IfX Yis not preserved,add relation XY
Refinement:Instead of the given set of FDs F, use a minimal
cover for F
Example: CSJDPQV, JPC, SDP, J S
Choose SD P, result is SDP and CSJDQV
Choose J S, result is JS and CJDQV, all 3NF
Add CJP relation
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
Reviews
There are no reviews yet.