1. Lecture 07: E/R Diagrams and Functional Dependencies
Wednesday, October 12, 2005

2. Outline Finish E/R diagrams (Chapter 2) The relational data model: 3.1
And E/R diagrams to relations (3.2, 3.3)
The relational data model: 3.1
Functional dependencies: 3.4

3. Relational Schema Design
Person
buys
Product
name
price
ssn
Conceptual Model:
Relational Model:
plus FD’s
Normalization:
Eliminates anomalies

4. Data Anomalies When a database is poorly designed we get anomalies:
Redundancy: data is repeated
Updated anomalies: need to change in several places
Delete anomalies: may lose data when we don’t want

5. Relational Schema Design
Recall set attributes (persons with several phones):
Name
SSN
PhoneNumber
City
Fred
Seattle
Joe
Westfield
One person may have multiple phones, but lives in only one city
Anomalies:
Redundancy = repeat data
Update anomalies = Fred moves to “Bellevue”
Deletion anomalies = Joe deletes his phone number: what is his city ?

6. Relation Decomposition
Break the relation into two:
Name
SSN
PhoneNumber
City
Fred
Seattle
Joe
Westfield
Name
SSN
City
Fred
Seattle
Joe
Westfield
SSN
PhoneNumber
Anomalies have gone:
No more repeated data
Easy to move Fred to “Bellevue” (how ?)
Easy to delete all Joe’s phone number (how ?)

7. Relational Schema Design (or Logical Design)
Main idea:
Start with some relational schema
Find out its functional dependencies
Use them to design a better relational schema

8. Functional Dependencies
A form of constraint
hence, part of the schema
Finding them is part of the database design
Also used in normalizing the relations

9. Functional Dependencies
Definition:
If two tuples agree on the attributes
A1, A2, …, An
then they must also agree on the attributes
B1, B2, …, Bm
Formally:
A1, A2, …, An  B1, B2, …, Bm

10. When Does an FD Hold
Definition: A1, ..., Am  B1, ..., Bn holds in R if:
t, t’  R, (t.A1=t’.A1  ...  t.Am=t’.Am  t.B1=t’.B1  ...  t.Bn=t’.Bn ) R A1
... Am B1 Bm t
if t, t’ agree here
then t, t’ agree here t’

11. Examples EmpID Name Phone Position E0045 Smith 1234 Clerk E3542 Mike
An FD holds, or does not hold on an instance:
EmpID
Name
Phone
Position
E0045
Smith
1234
Clerk
E3542
Mike
9876
Salesrep
E1111
E9999
Mary
Lawyer
EmpID  Name, Phone, Position
Position  Phone
but not Phone  Position

12. Example EmpID Name Phone Position E0045 Smith 1234 Clerk E3542 Mike
Salesrep
E1111
E9999
Mary
Lawyer
Position  Phone

13. Example EmpID Name Phone Position E0045 Smith 1234  Clerk E3542 Mike
Clerk
E3542
Mike
9876
Salesrep
E1111
E9999
Mary
Lawyer
but not Phone  Position

14. Typical Examples of FDs
Product: name  price, manufacturer
Person: ssn  name, age
zip  city
city, state  zip

15. Example FD’s are constraints: On some instances they hold name  color
On others they don’t
name  color
category  department
color, category  price
name
category
color
department
price
Gizmo
Gadget
Green
Toys 49
Tweaker 99
No: color,category  price doesn’t hold
Does this instance satisfy all the FDs ?

16. Example name  color category  department color, category  price
Gizmo
Gadget
Green
Toys 49
Tweaker
Black 99
Stationary
Office-supp. 59
yes
What about this one ?

17. An Interesting Observation
name  color
category  department
color, category  price
If all these FDs are true:
Then this FD also holds:
name, category  price
Why ??

18. Goal: Find ALL Functional Dependencies
Anomalies occur when certain “bad” FDs hold
We know some of the FDs
Need to find all FDs, then look for the bad ones

19. Inference Rules for FD’s
A1, A2, …, An  B1, B2, …, Bm
Splitting rule
and
Combing rule
Is equivalent to A1
... Am B1 Bm
A1, A2, …, An  B1
A1, A2, …, An  B2
A1, A2, …, An  Bm

20. Inference Rules for FD’s (continued)
Trivial Rule
A1, A2, …, An  Ai
where i = 1, 2, ..., n A1 … Am
Why ?

21. Inference Rules for FD’s (continued)
Transitive Closure Rule If
A1, A2, …, An  B1, B2, …, Bm
and
B1, B2, …, Bm  C1, C2, …, Cp
then
A1, A2, …, An  C1, C2, …, Cp
Why ?

22. A1 … Am B1 Bm C1
... Cp

23. Example (continued) Which Rule did we apply ? Inferred FD
1. name  color
2. category  department
3. color, category  price
Start from the following FDs:
Infer the following FDs:
Inferred FD
Which Rule did we apply ?
4. name, category  name
5. name, category  color
6. name, category  category
7. name, category  color, category
8. name, category  price

24. Example (continued) 1. name  color 2. category  department
3. color, category  price
Answers:
Inferred FD
Which Rule did we apply ?
4. name, category  name
Trivial rule
5. name, category  color
Transitivity on 4, 1
6. name, category  category
7. name, category  color, category
Split/combine on 5, 6
8. name, category  price
Transitivity on 3, 7
THIS IS TOO HARD ! Let’s see an easier way.

25. Inference Rules for FDs
The three simple rules are all we need to derive all possible FDs
Called “Armstrong Rules”
However, they are clumsy to use in practice
Better: use “closure” of a set of attributes (next)

26. Closure of a set of Attributes
Given a set of attributes A1, …, An
The closure, {A1, …, An}+ , is the set of attributes B s.t. A1, …, An  B
Example:
name  color
category  department
color, category  price
Closures: name+ = {name, color} {name, category}+ = {name, category, color, department, price} color+ = {color}

27. Closure Algorithm Start with X={A1, …, An}.
Repeat until X doesn’t change do:
if B1, …, Bn  C is a FD and
B1, …, Bn are all in X
then add C to X.
Example:
name  color
category  department
color, category  price
{name, category}+ = { }
name, category, color, department, price
Hence:
name, category  color, department, price

28. Example In class: A, B  C R(A,B,C,D,E,F) A, D  E B  D A, F  B
Compute {A,B}+ X = {A, B, }
Compute {A, F}+ X = {A, F, }

29. Why Do We Need Closure With closure we can find all FD’s easily
To check if X ® A
Compute X+
Check if A Î X+

30. Using Closure to Infer ALL FDs
Example:
A, B  C A, D  B
B  D
Step 1: Compute X+, for every X:
A+ = A, B+ = BD, C+ = C, D+ = D
AB+ = ABCD, AC+ = AC, AD+ = ABCD
ABC+ = ABD+ = ACD+ = ABCD (no need to compute– why ?)
BCD+ = BCD, ABCD+ = ABCD
Step 2: Enumerate all FD’s X  Y, s.t. Y  X+ and XY = :
AB  CD, ADBC, ABC  D, ABD  C, ACD  B

31. Another Example Enrollment(student, major, course, room, time)
course  time
What else can we infer ? [in class, or at home]

32. Back to Conceptual Design
Now we know how to find more FDs, it’s easy
Search for “bad” FDs
If there are such, then decompose the table into two tables, repeat for the subtables.
When done, the database schema is normalized
Unfortunately, there are several normal forms…

33. Normal Forms First Normal Form = all attributes are atomic
Second Normal Form (2NF) = old and obsolete
Third Normal Form (3NF) = will discuss
Boyce Codd Normal Form (BCNF) = will discuss
Others...