Normalization
DB Tuning
CS186 Final Review Session
Plan
• Functional Dependencies, Rules of Inference
• Candidate Keys
• Normal forms (BCNF/3NF)
• Decomposition
– BCNF
– Lossless
– Dependency preserving
– 3NF + Minimal cover
Functional Dependencies
• A functional dependency
X
Y holds over relation schema R if, for every allowable instance r of R: t1
r, t2
r, implies p
X
( t1 ) =
Y
( t1 ) =
X
( t2 )
Y
( t2 )
(where t1 and t2 are tuples;X and Y are sets of attributes)
• In other words:
X
Y means
Given any two tuples in r , if the X values are the same, then the Y values must also be the same. (but not vice versa!!)
• Can read “ ” as “determines”
Rules of Inference
• Armstrong’s Axioms (X, Y, Z are sets of attributes):
–
Reflexivity : If X
Y, then X
Y
– Augmentation : If X
Y, then XZ
YZ for any Z
–
Transitivity : If X
Y and Y
Z, then X
Z
• Some additional rules (that follow from AA):
– Union : If X
Y and X
Z, then X
YZ
–
Decomposition : If X
YZ, then X
Y and X
Z
Plan
• Functional Dependencies, Rules of Inference
•
Candidate Keys
• Normal forms (BCNF/3NF)
• Decomposition
– BCNF
– Lossless
– Dependency preserving
– 3NF + Minimal cover
Candidate Keys
• R = {A, B, C, D, E}
• F = { B
CD, D
E, B
A, E
C, AD
B }
• Is B
E in F + ?
• Is AD a key for R? AD + = AD
B + = B AD + = ABD and B is a key, so Yes!
B + = BCD
• Is AD a candidate key for R?
B
B
+
+
= BCDA
A + = A, D+ = DEC
… A,D not keys, so Yes!
= BCDAE … Yes! and B is a key for R too!
• Is D a key for R?
… No! AD is a key, so ADE is a superkey, but not a candidate key
D + = D
D + = DE
D + = DEC
… Nope!
Plan
• Functional Dependencies, Rules of Inference
• Candidate Keys
• Normal forms (BCNF/3NF)
• Decomposition
– BCNF
– Lossless
– Dependency preserving
– 3NF + Minimal cover
Boyce-Codd Normal Form (BCNF)
• Reln R with FDs
F is in BCNF if, for all X
A in F +
– A
X (called a trivial FD), or
– X is a superkey for R.
• In other words: “R is in BCNF if the only nontrivial FDs over R are key constraints
.”
Third Normal Form (3NF)
• Reln R with FDs
F is in 3NF if, for all X
A in F +
A
X (called a trivial FD), or
X is a superkey of R, or
A is part of some candidate key (not superkey!) for R.
(sometimes stated as “A is prime
”
)
• If R is in BCNF, obviously in 3NF.
–
Lossless-join, dependency-preserving decomposition of R into a collection of 3NF relations always possible.
Plan
• Functional Dependencies, Rules of Inference
• Candidate Keys
• Normal forms (BCNF/3NF)
•
Decomposition
– BCNF
– Lossless
– Dependency preserving
– 3NF + Minimal cover
Lossless Decomposition
•
The decomposition of R into X and Y is lossless with respect to F if and only if the closure of F contains:
X
Y
X, or
X
Y
Y i.e. the common attributes form a superkey for one side or the other
•
Useful result : If W
Z holds over R and W
Z is empty, then decomposition of R into R-Z and WZ is loss-less.
Dependency Preserving Decompositions
• Decomposition of R into X and Y is dependency preserving if (F
X
F
Y
) + = F +
– i.e., if we consider only dependencies in the closure F + that can be checked in X without considering Y, and in Y without considering
X, these imply all dependencies in F + .
• Important to consider F + in this definition:
– ABC, A
B, B
C, C
A, decomposed into AB and BC.
• F+ also contains B
A, A
C, C
B …
•
F AB contains A
B and B
A ; F BC contains B
C and
C
B
•
So, (F AB
F BC)
+ contains C
A
BCNF Decomposition
• For each FD in
F+ that violates BCNF, X
A
• Decompose R into (R - A) and XA
• If either (R - A) or XA is not in BCNF, decompose recursively
• Guaranteed to be lossless but not dependency preserving
Minimal Cover for a Set of FDs
• Minimal cover G for a set of FDs F:
– Closure of F = closure of G.
– Right hand side of each FD in G is a single attribute.
– If we modify G by deleting an FD or by deleting attributes from an FD in G, the closure changes.
• Intuitively, every FD in G is needed, and `` as small as possible
’’ in order to get the same closure as F.
• e.g., A
B, ABCD
E, EF
GH, ACDF
EG has the following minimal cover:
– A
B, ACD
E, EF
G and EF
H
– Do we need ACDF
EG? It can be derived from ACD
E and
EF
G. Same for ACDF
E, ACDF
G
3NF Decomposition
• Compute minimal cover
• For each FD X
A in minimal cover that is not preserved
– Add relation XA
• Guaranteed to be lossless AND dependency preserving
Tuning the Schema
Contracts (Cid, Sid, Jid, Did, Pid, Qty, Val)
Depts (Did, Budget, Report)
Suppliers (Sid, Address)
Parts (Pid, Cost)
Projects (Jid, Mgr)
• We will concentrate on Contracts , denoted as CSJDPQV .
The following ICs are given to hold:
is the primary key .
– C and JP are candidate keys
– 3NF normal form
BCNF Decomposition
• Use SD
P, we get SDP and CSJDQV
• Lossless but not dependency-preserving (JP
C)
• Three options
– Leave it in 3NF without decomposition
– Add JPC as an extra table (redundancy across relations)
– Create an assertion to enforce JP
C
• Acceptable when updates are infrequent
Check Assertion (JP
C)
PartInfo: SDP
ContractInfo: CSJDQV
CREATE ASSERTION checkDep
CHECK (NOT EXISTS
(SELECT *
Lossless join on SD
FROM PartInfo PI, ContractInfo CI
Group By JP
WHERE PI.supplierid=CI.supplierid
AND PI.deptid = CI.deptid
)
GROUP BY CI.projectid, PI.partid
HAVING COUNT (cid) > 1 Count C
)
Questions?
Don’t forget..
• Relational algebra, calculus
• Query optimization
– Nested loop, index nested, block nested, hash..
– Computing costs
– Enumerating plans
Good luck!
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