2. 2
Decomposing relations
• In previous lecture, we saw that we could
‘decompose’ the bad relation schema
Data(sid,sname,address,cid,cname,grade)
to a ‘better’ set of relation schema
Student(sid,sname,address)
Course(cid,cname)
Enrolled(sid,cid,grade)
3. 3
Are all decompositions
good?
• Consider our motivating example:
Data(sid,sname,address,cid,cname,grade)
• Alternatively we could decompose into
R1(sid,sname,address)
R2(cid,cname,grade)
• But this decomposition loses information about
the relationship between students and courses
4. 4
Decomposition
• A decomposition of a relation R=R(A1:1, …,
An:n) is a collection of relations {R1, …, Rk} and a
set of queries
)
,
,
( 1
0 k
R
R
Q
R K
=
)
(R
Q
R i
i =
if
then
}
,
,
,
{ 1
0 k
Q
Q
Q K
such that
This is Tim’s somewhat
non-standard definition….
5. 5
Special Case: Lossless-
join decomposition
• {R1,…,Rk} is a lossless-join
decomposition of R with respect
to an FD set F, if for every relation
instance r of R that satisfies F,
R1
(r) V … V Rk
(r) = r
(this means project on the attributes of the relation’s schema)
7. 7
Lossless-join: Example
sid sname addres
s
cid cname grade
124 Julia USA 206 Database A++
204 Kim Essex 202 Semantics C
124 Julia USA 201 S/Eng I A+
206 Tim London 206 Database B-
124 Julia USA 202 Semantics B+
What happens if we decompose on
(sid,sname,address) and (cid,cname,grade)?
8. 8
Dependency preservation
• Intuition: If R is decomposed into R1, R2 and
R3, say, and we enforce the FDs that hold
individually on R1, on R2 and on R3, then all
FDs that were given to hold on R must also
hold
• Reason: Otherwise checking updates for
violation of FDs may require computing joins
9. 9
Dependency preservation
• The projection of an FD set F onto a set
of attributes Z, written Fz is defined
{XY | XYF+
and XYZ}
• A decomposition ={R1,…,Rk} is
dependency preserving if
F+
=(FR1
… FRk
)+
GOAL OF SCHEMA REFINEMENT: REDUCE REDUNDANCY
WHILE PRESERVING DEPENDENCIES IN A LOSSLESS-JOIN
MANNER.
10. 10
Dependency preservation:
example
• Take R=R(city, street&no, zipcode) with FDs:
– city,street&no zipcode
– zipcode city
• Decompose to
– R1(street&no,zipcode)
– R2(city,zipcode)
• Claim: This is a lossless-join decomposition
• Is it dependency preserving?
11. 11
Boyce-Codd normal form
“Represent Every Fact Only ONCE”
• A relation R with FDs F is said to be in
Boyce-Codd normal form (BCNF) if for
all XA in F+
then
– Either AX (‘trivial dependency’), or
– X is a superkey for R
• Intuition: A relation R is in BCNF if the left
side of every non-trivial FD contains a key
12. 12
BCNF: Example
• Consider R=R(city, street&no, zipcode) with
FDs:
– city,street&no zipcode
– zipcode city
• This is not in BCNF, because zipcode is not
a superkey for R
– We potentially duplicate information relating
zipcodes and cities
13. 13
BCNF: Example
BankerSchema(brname,cname,bname)
• With FDs
– bname brname
– brname,cname bname
• Not in BCNF (Why?)
• We might decompose to
– BBSchema(bname,brname)
– CBrSchema(cname,bname)
• This is in BCNF
• BUT this is not dependency-preserving
14. 14
Third normal form
• A relation R with FDs F is said to be in third normal
form (3NF) if for all XA in F+
then
– Either AX (‘trivial dependency’), or
– X is a superkey for R, or
– A is a member of some candidate key for R
• Notice that 3NF is strictly weaker than BCNF
• (A prime attribute is one which appears in a
candidate key)
• It is always possible to find a dependency-
preserving lossless-join decomposition
that is in 3NF.
15. 15
3NF: Example
• Recall R=R(city, street&no, zipcode) with
FDs:
– city,street&no zipcode
– zipcode city
• We saw earlier that this is not in BCNF
• However this is in 3NF, because city is a
member of a candidate key ({city,street&no})
16. 16
Prehistory: First normal
form
• First normal form (1NF) is now considered
part of the formal definition of the relational
model
• It states that the domain of all attributes must
be atomic (indivisible), and that the value of
any attribute in a tuple must be a single
value from the domain
• NOTE: Modern databases have moved away
from this restriction
17. 17
Prehistory: Second
normal form
• A partial functional dependency XY is
an FD where for some attribute AX, (X-
{A})Y
• A relation schema R is in second normal
form (2NF) if every non-prime attribute A
in R is not partially dependent on any key
of R
19. 19
Not the end of problems…
• ONLY TRIVIAL FDs!! (see Date)
• Is in BCNF!
• Obvious insertion anomalies…
Course Teacher Book
Databases gmb Date
Databases gmb Elmasri
Databases jkmm Date
Databases jkmm Elmasri
OSF gmb Silberschatz
OSF tlh Slberschatz
20. 20
Decomposition
• Even though its in BCNF, we’d prefer to
decompose it to the schema
– Teaches(Course,Teacher)
– Books(Course,Title)
• We need to extend our underlying theory
to capture this form of redundancy
21. 21
Further normal forms
• We can generalise the notion of FD to a
‘multi-valued dependency’, and define two
further normal forms (4NF and 5NF)
• These are detailed in the textbooks
• In practise, BCNF (preferably) and 3NF (at
the very least) are good enough
22. 22
Design goals: Summary
• Our goal for relational database design is
– BCNF
– Lossless-join decomposition
– Dependency preservation
• If we can’t achieve this, we accept
– Lack of dependency preservation, or
– 3NF
23. 23
Summary
You should now understand:
• Decomposition of relations
• Lossless-join decompositions
• Dependency preserving decompositions
• BCNF and 3NF
• 2NF and 1NF
Next lecture: More algebra, more SQL
