1. Instructor: Mohamed Eltabakh meltabakh@cs.wpi.edu
Normalization- 3NF
Part III
Instructor: Mohamed Eltabakh

2. Announcements Homework 2 is due NOW !!!
Homework 3 will be out today (Nov. 15) and due on Nov. 22, 8:00AM
Midterm on Nov. 22
Until Normalization (Normalization is included)
Next lecture is mostly revision + short quiz

3. Third Normal Form: Motivation
There are some situations where
BCNF is not dependency preserving
Solution: Define a weaker normal form, called Third Normal Form (3NF)
Allows some redundancy (we will see examples later)
But all FDs can be checked on individual relations without computing a join
There is always a lossless-join, dependency-preserving decomposition into 3NF

4. Normal Form : 3NF Relation R is in 3NF if, for every FD in F+ α  β,
where α ⊆ R and β ⊆ R, at least one of the following holds:
α → β is trivial (i.e.,β⊆α)
α is a superkey for R
Each attribute in β-α is part of a candidate key (prime attribute)

5. Testing for 3NF
Use attribute closure to check for each dependency α → β, if α is a superkey
If α is not a superkey, we have to verify if each attribute in (β- α) is contained in a candidate key of R

6. 3NF: Example Is relation Lot in 3NF ?
Lot (ID, county, lotNum, area, price, taxRate)
Candidate key: <county, lotNum>
FDs:
county  taxRate
area  price
Is relation Lot in 3NF ? NO
Decomposition based on county  taxRate
Lot (ID, county, lotNum, area, price)
County (county, taxRate)
Are relations Lot and County in 3NF ?
Lot is not

7. 3NF: Example (Cont’d) Is every relation in 3NF ?
Lot (propNo, county, lotNum, area, price)
County (county, taxRate)
Candidate key for Lot: <county, lotNum>
FDs:
county  taxRate
area  price
Decomposition based on area  price
Lot (propNo, county, lotNum, area)
County (county, taxRate)
Area (area, price)
Is every relation in 3NF ?
YES

8. Main Idea of the 3NF Decomposition
Use the decomposition algorithm as in BCNF
But to ensure dependency preservation
If α  β is not preserved, then create relation (α, β) where α is the key
To ensure the result of decomposition is dependency-preserving and lossless
Use the canonical cover in the decomposition 7

9. Canonical Cover of FDs Given set of FDs (F) with functional closure F+
Canonical Cover (Minimal Cover) = G
Is the smallest set of FDs that produce the same F+
There are no extra attributes in the L.H.S or R.H.S of and dependency in G
Given set of FDs (F) with functional closure F+
Canonical cover of F is the minimal subset of FDs (G), where
G+ = F+
Every FD in the canonical cover is needed, otherwise some dependencies are lost 8

10. Example : Canonical Cover
Given F:
A  B, ABCD  E, EF  GH, ACDF  EG
Then the canonical cover G:
A  B, ACD  E, EF  GH 8

11. Computing the Canonical Cover
Given a set of functional dependencies F, how to compute the canonical cover G
Use the next algorithm for this step

12. Finding Extraneous Attributes

13. Example : Canonical Cover (Lets Check L.H.S)
Given F = {A  B, ABCD  E, EF  G, EF  H, ACDF  EG}
Union Step: {A  B, ABCD  E, EF  GH, ACDF  EG}
Test ABCD  E
Check A:
{BCD}+ = {BCD}  A cannot be deleted
Check B:
{ACD}+ = {A B C D E}  Then B can be deleted
Now the set is: {A  B, ACD  E, EF  GH, ACDF  EG}
Test ACD  E
Check C:
{AD}+ = {ABD}  C cannot be deleted
Check D:
{AC}+ = {ABC}  D cannot be deleted 8

14. Example: Canonical Cover (Lets Check L.H.S-Cont’d)
Now the set is: {A  B, ACD  E, EF  GH, ACDF  EG}
Test EF  GH
Check E:
{F}+ = {F}  E cannot be deleted
Check F:
{E}+ = {E}  F cannot be deleted
Test ACDF  EG
None of the H.L.S can be deleted 8

15. Example: Canonical Cover (Lets Check R.H.S)
Now the set is: {A  B, ACD  E, EF  GH, ACDF  EG}
Test EF  GH
Check G:
{EF}+ = {E F H}  G cannot be deleted
Check H:
{EF}+ = {E F G}  H cannot be deleted
Test ACDF  EG
Check E:
{ACDF}+ = {A B C D F E G}  E can be deleted
Now the set is: {A  B, ACD  E, EF  GH, ACDF  G} 8

16. Example: Canonical Cover (Lets Check R.H.S-Cont’d)
Now the set is: {A  B, ACD  E, EF  GH, ACDF  G}
Test ACDF  G
Check G:
{ACDF}+ = {A B C D F E G}  G can be deleted
Now the set is: {A  B, ACD  E, EF  GH}
The canonical cover is:
{A  B, ACD  E, EF  GH} 8

17. Use of Canonical Cover
Used in the decomposition of relations to be in 3NF
The resulting decomposition is lossless and dependency preserving

18. Summary of Normalization
Normalization forms
First Normal Form (1NF)
BCNF
Third Normal Form (2NF)
Fourth Normal Form (4NF) – Not covered
Used to ensure the database design is in a good form
Decomposing the relation according to functional dependencies