Database Schema Refinement and Normal Forms Explained
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LECTURE 5: Schema Refinement
and Normal Forms
SOME OF THESE SLIDES ARE BASED ON YOUR
TEXT BOOK
Anomalies
Insertion anomalies
Cannot record filmType without starName
Deletion anomalies
If we delete the last star, we also lose the movie info.
Modification (update) anomalies
DECOMPOSITION
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Is there a possibility of an update, deletion, and
insertiona anomalies in the table below? Explain with
examples
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Set valued attributes: example
Consider a database of customer transactions
You have customers represented by cid, and transactions
represented by tid, where each transaction has multiple
items, represented by itemid, together with name, quantity
and the cost of the item
Design a database to keep track of the transactions
of customers (set valued, vs flat design)
Example queries:
what is the total sales in dollars per item
What is the total sales per customer
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Schema Refinement
functional dependencies
, can be used to identify
schemas with problems and to suggest refinements.
Decomposition
is used for schema refinement.
Example FD
Database of beer drinkers
Example FD
title year
length
title year
filmType
title year
studioName
title year
length filmType studioName
Functional Dependencies (FDs)
A
functional dependency
X Y
holds over relation R if, for
every allowable instance
r
of R:
t1 r, t2 r,
(
t1
) = (
t2
) implies (
t1
) = (
t2
)
i.e., given two tuples in
r
, if the X values agree, then the Y values must
also agree. (X and Y are
sets
of attributes.)
t1
t2
Functional Dependencies (FDs)
Does the following relation instance satisfy X->Y ?
Functional Dependencies (FDs)
A
functional dependency
X Y
holds over relation R if, for
every allowable instance
r
of R:
t1 r, t2 r,
(
t1
) = (
t2
) implies (
t1
) = (
t2
)
i.e., given two tuples in
r
, if the X values agree, then the Y values must
also agree. (X and Y are
sets
of attributes.)
An FD is a statement about
all
allowable relations.
Must be identified based on semantics of application.
Given some allowable instance
r1
of R, we can check if it violates some
FD
f
, but we cannot tell if
f
holds over R!
K is a candidate key for R means that K R
However, K R does not require K to be
minimal
!
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Functional Dependencies (FDs)
Does the following relation instance satisfy X->Y ?
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Functional Dependencies (FDs)
If X is a candidate key, then X -> Y Z !
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Functional Dependencies (FDs)
If Y Z -> X can we say YZ is a candidate key?
Example: Constraints on Entity Set
Consider relation obtained from Hourly_Emps:
Hourly_Emps (
ssn
, name, lot, rating, hrly_wages
,
hrs_worked
)
Notation
:
We will denote this relation schema by listing the attributes:
SNLRWH
This is really the
set
of attributes {S,N,L,R,W,H}.
Sometimes, we will refer to all attributes of a relation by using the relation
name. (e.g., Hourly_Emps for SNLRWH)
Hourly_Emps
Example: Constraints on Entity Set
Some FDs on Hourly_Emps:
ssn
is the key:
S SNLRWH
rating
determines
hrly_wages
:
R W
Did you notice anything wrong with the following instance ?
Example: Constraints on Entity Set
Some FDs on Hourly_Emps:
ssn
is the key:
S SNLRWH
rating
determines
hrly_wages
:
R W
Salary should be the same for a given rating!
Example
Problems due to R W :
Update anomaly
:
Can we change W in just the 1st tuple of
SNLRWH?
Insertion anomaly
:
What if we want to insert an employee and don’t
know the hourly wage for his rating?
Deletion anomaly
:
If we delete all employees with rating 5, we lose
the information about the wage for rating 5!
Hourly_Emps2
Wages
Hourly_Emps
Refining an ER Diagram
1st diagram translated:
Workers(S,N,L,D,S) Departments(D,M,B)
Lots associated with workers.
Suppose all workers in a dept are assigned the same lot: D L
Refining an ER Diagram
Suppose all workers in a dept are assigned the same lot: D L
Redundancy; fixed by:
Workers2(S,N,D,S) Dept_Lots(D,L)
Can fine-tune this:
Workers2(S,N,D,S) Departments(D,M,B,L)
Reasoning About FDs
Given some FDs, we can usually infer additional FDs:
ssn did
,
did lot
implies
ssn lot
An FD
f
is
implied by
a set of FDs
F
if
f
holds whenever all FDs in
F
hold.
=
closure of F
is the set of all FDs that are implied by
F
.
Armstrong’s Axioms (X, Y, Z are
sets of
attributes):
Reflexivity
:
If X Y, then Y X (
a trivial FD
)
Augmentation
:
If X Y, then XZ YZ for any Z
Transitivity
:
If X Y and Y Z, then X Z
These are
sound
and
complete
inference rules for FDs!
Reasoning About FDs
For example, in the above schema
S N -> S is a trivial FD
since {S,N} is a superset of {S}Reasoning About FDs
For example, in the above schema
If S N -> R W, then S N
L
-> R W
L
(by augmentation)
Reasoning About FDs (Contd.)
Couple of additional rules (that follow from AA):
Union
:
If X -> Y and X -> Z, then X -> YZ
Proof:
From X -> Y, we have XX -> XY (by augmentation)
Note that XX is X, therefore X -> XY
From X -> Z, we have XY -> YZ (by augmentation)
From X -> XY and XY -> YZ, we have X -> YZ (by transitiviy)
Reasoning About FDs (Contd.)
Couple of additional rules (that follow from AA):
Decomposition
:
If X -> YZ, then X -> Y and X -> Z
Try to prove it at home/dorm/IC/Vitamin/DD/Bus!
Reasoning About FDs (Contd.)
Example:
Contracts(
cid,sid,jid,did,pid,qty,value
),
and:
C is the key:
C CSJDPQV
Project purchases each part using single contract:
JP C
Dept purchases at most one part from a supplier:
SD P
JP C, C CSJDPQV imply JP CSJDPQV
SD P implies SDJ JP
SDJ JP, JP CSJDPQV imply SDJ CSJDPQV
Reasoning About FDs (Contd.)
Computing the closure of a set of FDs ( ) can be expensive.
(Size of closure is exponential in # attrs!)
Example:
A database with 3 attributes (A,B,C)
F = {A->B, B->C}
Find the closure of F denoted by F
+
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Example
Suppose we are given a relation scheme R = (A,B,C,G,H,I), and
the set of functional dependencies
, F provided below
:
A → B
A → C
CG → H
CG → I
B → H
Is
A → H logically implied
by F?
Is
AG → I
logically implied by F?
Reasoning About FDs (Contd.)
Computing the closure of a set of FDs ( ) can be expensive.
(Size of closure is exponential in # attrs!)
Typically, we just want to check if a given FD
X Y
is in the
closure of a set of FDs
F
. An efficient check:
Compute
attribute closure
of X (denoted ) wrt
F:
Set of all attributes A such that X A is in
There is a linear time algorithm to compute this.
For each FD Y -> Z in F, if is a superset of Y then add Y to
Reasoning About FDs (Contd.)
Does F = {A B, B C, C D E } imply A E?
i.e,
is A E in the closure ? Equivalently, is E in ?
Lets compute A
+
Initialize A
+
to {A} : A
+
= {A}
From A -> B, we can add B to
A
+
:
A
+
= {A, B}
From B -> C, we can add C to A
+
:
A
+
= {A, B, C}
We can not add any more attributes, and A
+
does not contain E
therefore A -> E does not hold.
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DB Design Guidelines
Design a relation schema with a clearly defined
semantics
Design the relation schemas so that there is not
insertion, deletion, or modification anomalies. If
there may be anomalies, state them clearly
Avoid attributes which may frequently have null
values as much as possible.
Make sure that relations can be combined by key-
foreign key links
Normal Forms
Normal forms are standards for a good DB schema (introduced
by Codd in 1972)
If a relation is in a certain normal form (such as
BCNF, 3NF
etc.), it is known that certain kinds of problems are
avoided/minimized.
Normal forms help us decide if decomposing a relation helps.
Normal Forms
First Normal Form:
No set valued attributes (only atomic values)
Normal Forms (Contd.)
Role of FDs in detecting redundancy:
Consider a relation R with 3 attributes, ABC.
No FDs hold:
There is no redundancy here.
Given A -> B:
Several tuples could have the same A value, and if so,
they’ll all have the same B value!
Normal Forms (Contd.)
Second Normal Form
: Every non-prime attribute should be
fully functionally dependent on every key (i.e., candidate
keys).
In other words: “
No non-prime attribute in the table is
functionally dependent on a proper subset of any candidate
key
”
Prime attribute
: any attribute that is part of a key
Non-prime attributes
: rest of the attributes
Ex: If AB is a key, and C is a non-prime attribute, then if A->C holds then
A
partially determines C (there is a partial functional dependency to a key)
Third Normal Form (3NF)
Reln R with FDs
F
is in
3NF
if, for all X A in
A X (called a
trivial
FD), or
X contains a key for R, or
A is part of some key for R.
If R is in 3NF, some redundancy is possible.
What Does 3NF Achieve?
If 3NF violated by X -> A, one of the following holds:
X is a subset of some key K
We store (X, A) pairs redundantly.
X is not a proper subset of any key.
There is a chain of FDs K -> X -> A, which means that we
cannot associate an X value with a K value unless we also associate
an A value with an X value.
But:
even if reln is in 3NF, these problems could arise.
e.g., Reserves SBDC, S C, C S is in 3NF, but for
each reservation of sailor S, same (S, C) pair is stored.
There is a stricter normal form (BCNF).
Boyce-Codd Normal Form (BCNF)
Reln R with FDs
F
is in
BCNF
if, for all X A in
A X (called a
trivial
FD), or
X contains a key for R. (i.e., X is a superkey)
In other words, R is in BCNF if the only non-trivial
FDs that hold over R are key constraints.
No dependency in R that can be predicted using FDs
alone.
If we are shown two tuples that agree upon
the X value, we cannot infer the A value in
one tuple from the A value in the other.
If example relation is in BCNF, the 2 tuples
must be identical (since X is a key).
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Normal Forms Contd.
Person(SSN, Name, Address, Hobby)
F = {SSN Hobby -> Name Address, SSN ->Name Address}Is the above relation in 2
nd
normal form ?
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Normal Forms Contd.
Ex: R = ABCD,
F
={AB->CD, AC->BD}
What are the (candidate) keys for R?
Is R in 3NF?
Is R in BCNF?
Is there a redundancy in the above instance
With respect to
F
?
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Example
An employee can be assigned to at most one project, but many
employees participate in a project
EMP_PROJ(ENAME,
SSN
, ADDRESS, PNUMBER, PNAME, PMGRSSN)
PMGRSSN is the SSN of the manager of the project
Is this a good design?
Decomposition of a Relation Scheme
Suppose that relation R contains attributes
A1 ... An.
A
decomposition
of R consists of replacing R by two or more
relations such that:
Each new relation scheme contains a subset of the attributes of R (and
no attributes that do not appear in R), and
Every attribute of R appears as an attribute of one of the new relations.
Intuitively, decomposing R means we will store instances of the
relation schemes produced by the decomposition, instead of
instances of R.
E.g., Can decompose
SNLRWH
into
SNLRH
and
RW
.
Decomposition of a Relation Scheme
We can decompose
SNLRWH
into
SNL
and
RWH
.
Example Decomposition
SNLRWH has FDs S -> SNLRWH and R -> W
Is this in 3NF?
R->W violates 3NF (W values repeatedly associated with R values)
In order to fix the problem, we need to create a relation RW to store the
R->W associations, and to remove W from the main schema:
i.e., we decompose SNLRWH into SNLRH and RW
Problems with Decompositions
There are potential problems to consider:
Some queries become more expensive.
e.g., How much did Ali earn ? (salary = W*H)
Problems with Decompositions
Given instances of the decomposed relations, we may not be
able to reconstruct the corresponding instance of the original
relation!
Lossless Join Decompositions
Decomposition of R into X and Y is
lossless-join
w.r.t.
a set of FDs F if, for every instance
r
that satisfies F:
(
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.
Definition extended to decomposition into 3 or more
relations in a straightforward way.
It is essential that all decompositions used to deal
with redundancy be lossless!
(Avoids Problem (2).)
More on Lossless Join
The decomposition of R into
X and Y is
lossless-join wrt F
if and only if
the closure of F
contains:
X Y X, or
X Y Y
In particular, the
decomposition of R into
UV and R - V is lossless-join
if U V holds over R.
Person(SSN, Name, Address, Hobby)
F = {SSN Hobby -> Name Address, SSN ->Name Address}Person
Person1
Hobby
Person(SSN, Name, Address, Hobby)
F = {SSN Hobby -> Name Address, SSN ->Name Address}9/10/2024
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exercise
F1 = {AB->C, BC->AD, D->E, CF->B}
Does AB ->D hold wrt F1?
Does D->A hold wrt F1?
Is AB a key wrt F1?
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Projecting FDs
Suppose that we have R(A,B,C,D) with
F = {A->B, B->C, C->D}
We would like to project the FDs to R1(A,C,D)
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Projecting FDs
Need to compute the closure of
F = {A->B, B->C, C->D}
A
simple exponential algorithm is:
1.
For each set of attributes
X
, compute
X
+
.
2.
Add
X
->
A
for all
A
in
X
+
-
X
.
3.
Finally, use only FD’s involving projected attributes.
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Projecting FDs
Suppose that we have R(A,B,C,D) with
F = {A->B, B->C, C->D}
We would like to project the FDs to R1(A,C,D)
Using the
algorithm
in the previous slide
:
{A}+
= {A,B,C,D}, therefore A->C, and A->D hold in F1
{C}+
= {C,D}, therefore only C->D is in F1
{D}+
= {D}
No need to check the supersets of A since {A}+
includes all
the attributes
For {C,D}+
= {C,D} we have only {A->C, A->D, C->D} inthe projected FDs
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Problems with Decompositions (Contd.)
Checking some dependencies may require joining the instances
of the decomposed relations.
Dependency Preserving Decomposition
Consider CSJDPQV, C is key, JP -> C and SD -> P.
BCNF decomposition: CSJDQV and SDP
Problem: Checking JP -> C requires a join!
Dependency preserving decomposition
:
A dependency X->Y that appear in F should either appear in one of the
sub relations or should be inferred from the dependencies in one of the
subrelations.
Projection of set of FDs F
:
If R is decomposed into X, ...
projection of F onto X (denoted
F
X
) is the set of FDs
U -> V in F
+
(
closure of F
)
such that
U, V are in X
.
Ex: R=
ABC, F={ A -> B, B -> C, C -> A}
F
+
includes FDs, {A->B, B->C, C->A, B->A, A->C, C->B }
F
AB
= {A->B, B->A}, FAC
={C->A, A->C}Dependency Preserving Decompositions (Contd.)
Decomposition of R into X and Y is
dependency
preserving
if
(F
X
union 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
+
, not F, in this definition:
ABC, A -> B, B -> C, C -> A, decomposed into AB and BC.
Is this dependency preserving? Is C ->A preserved?????
F
+
includes FDs, {A->B, B->C, C->A, B->A, A->C, C->B }
F
AB
= {A->B, B->A}, FBC
={B->C, C->B},
F
AB
U
F
BC
= {A->B, B->A, B->C, C->B}
Does the closure of F
AB
U
F
BC
imply C->A ?
Dependency Preserving Decompositions (Contd.)
Dependency preserving does not imply lossless join:
Ex: ABC, A -> B, decomposed into AB and BC, is lossy.
And vice-versa! (Example?)
Decomposition into BCNF
Consider relation R with FDs F.
If X -> Y violates BCNF,
decompose R into R - 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.
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!
Example: Decomposition into BCNF
R= ABCDEFGH with FDs
ABH->C
A->DE
BGH->F
F->ADH
BH->GE
Is R in BCNF?
Which FD violates the BCNF ?
ABH -> C ?
No, since ABH is a superkey
A->DE violates BCNF
Since attribute closure of A is ADE and therefore A is not a superkey
Decompose R=ABCDEFGH into R1=ADE and R2=ABCFGH
Example: Decomposition into BCNF
R= ABCDEFG with FDs
ABH->C
A->DE
BGH->F
F->ADH
BH->GE
R1=ADE F1= {A->DE}
R2=ABCFGH F2= {ABH->C, BGH->F, F->AH, BH->G}
Note that the new FDs are obtained by projecting the original FDs on the
attributes in the new relations.
For example BH->GE is decomposed into {BH->G, BH->E} and BH->E is notincluded in F1 or F2, BH->G is included into R2.
Is the decomposition of R into R1 and R2 dependency preserving?
R1 is in BCNF, but we need to apply the algorithm on R2 since it is not in BCNF
BCNF and Dependency Preservation
In general,
there may not be a dependency preserving
decomposition into BCNF
.
e.g., CSZ, CS -> Z, Z -> C
Can’t decompose while preserving 1st FD; not in BCNF.
Similarly, decomposition of CSJDQV into SDP, JS and CJDQV is
not dependency preserving (w.r.t. the FDs JP -> C, SD -
> P and J -> S).
However, it is a lossless join decomposition.
In this case, adding JPC to the collection of relations gives us a
dependency preserving decomposition.
JPC tuples stored only for checking FD!
(
Redundancy!
)
Decomposition into 3NF
The algorithm for lossless join decomp into BCNF can be used
to obtain a lossless join decomp into 3NF (typically, can stop
earlier).
To ensure dependency preservation, one idea:
If X -> Y is not preserved, add relation XY.
Problem is that XY may violate 3NF! e.g., consider the addition of CJP
to `preserve’ JP -> C. What if we also have J -> C ?
Refinement:
Instead of the given set of FDs F, use a
minimal
cover for F
.
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
Obtaining the Minimal Cover
Algorithm Steps:
1.
Put the FDs in standard form (single attribute on the right hand side)
2.
Minimize the left hand side of each FD
3.
Delete the redundant FDs
The steps should be performed in the
above order
(think why)
Is there a
unique
minimal cover for a given set of FDs?
Obtaining the Minimal Cover
Example:
F = {ABCD->E, E->D, A->B, AC->D}
Notice that the right hand sides have a single attr (if not we had to
decompose the right hand sides first)
Can we remove
B
from the left hand side of A
B
CD->E?
Check if ACD->E is implied by F
In order to do this, find the attribute closure ACD wrt F
If
B
is in the attribute closure, then ACD->E is implied by F, and therefore
we can replace ABCD->E with ACD->E (note that given ACD->E, we
have ABCD->E)
Can we remove
D
from AC
D
->E
Check if AC->E is implied by F’ (obtained by replacing ABCD->E in F with
ACD->E)
F’’ = {AC->E, E->D, A->B, AC->D}
Can we drop any FDs in F’’ ?
Could we drop any FDs in F before minimizing the left hand sides?
Dependency Preserving
Decomposition into 3
rd
NF
Let R be the relation to be decomposed into 3
rd
NF
and F be the FDs that is a minimal cover
Algorithm Steps
Perform lossless-join decomposition of R into R1,R2,..Rn
Project the FDs in F into F1,F2,…,Fn (that correspond to R1,R2,…,Rn)
Identify the set of FDs that are not preserved (I.e., that are not in the
closure of the union of F1,F2,…,Fn)
For each FD X->A that is not preserved, create a relation schema XA
and it to the decomposition.
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69
Example
Consider the relation R = ABCDE with FDs:
A->B
BC->E
ED->A
Lets first find all the keys
Lets check if R is in 3NF
Lets check if R is in BCNF
Explore the concept of schema refinement and normal forms in database management. Understand issues like insertion, deletion, and update anomalies in database tables. Learn about set-valued attributes and their implications in designing databases for tracking customer transactions.
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Presentation Transcript
LECTURE 5: Schema Refinement and Normal Forms SOME OF THESE SLIDES ARE BASED ON YOUR TEXT BOOK 9/10/2024 1
title year length filmType studioName starName Star Wars 1977 124 color Fox Carrie Fisher Star Wars 1977 124 color Fox Mark Haill Star Wars 1977 124 color Fox Harrison Ford Mighty Ducks 1991 104 color Disney Emilo Estevez Wayne s World 1992 95 color Paramount Dana Carvey Wayne s World 1992 95 color Paramount Mike Meyers Anomalies Insertion anomalies Cannot record filmType without starName Deletion anomalies If we delete the last star, we also lose the movie info. Modification (update) anomalies
DECOMPOSITION title year length filmType studioName starName Star Wars 1977 124 color Fox Carrie Fisher Star Wars 1977 124 color Fox Mark Haill Star Wars 1977 124 color Fox Harrison Ford Mighty Ducks 1991 104 color Disney Emilo Estevez Wayne s World 1992 95 color Paramount Dana Carvey Wayne s World 1992 95 color Paramount Mike Meyers title year starName Star Wars 1977 Carrie Fisher title year length filmType studioName Star Wars 1977 Mark Haill Star Wars 1977 124 color Fox Star Wars 1977 Harrison Ford Mighty Ducks 1991 104 color Disney Mighty Ducks 1991 Emilo Estevez Wayne s World 1992 95 color Paramount Wayne s World 1992 Dana Carvey Wayne s World 1992 Mike Meyers
Is there a possibility of an update, deletion, and insertiona anomalies in the table below? Explain with examples StudentNum CourseNum Student Name Address Course S21 9201 Jones Edinburgh Accounts 1 S21 9267 Jones Edinburgh physics S24 9267 Smith Glasgow physics S30 9201 Richards Manchester Accounts 1 S30 9322 Richards Manchester Maths 9/10/2024 4
Set valued attributes: example Consider a database of customer transactions You have customers represented by cid, and transactions represented by tid, where each transaction has multiple items, represented by itemid, together with name, quantity and the cost of the item Design a database to keep track of the transactions of customers (set valued, vs flat design) Example queries: what is the total sales in dollars per item What is the total sales per customer 9/10/2024 5
Schema Refinement functional dependencies, can be used to identify schemas with problems and to suggest refinements. Decomposition is used for schema refinement. 9/10/2024 6
Example FD Database of beer drinkers BEER LIKED FAV BEER NAME ADDR MANUF John Doe NY, Soho Bud Light BW Leffe John Doe NY, Soho Leffe LF Leffe Elisa Day DC, Dupont Gusto Weiss Gusto Chimay Blue Elisa Day DC, Dupont Chimay Blue Chimay Chimay Blue
Example FD title year length title year filmType title year studioName title year length filmType studioName TITLE YEAR LENGTH FILMTYPE studioName starName Star Wars 1977 124 color Fox Carrie Fisher Star Wars 1977 124 color Fox Mark Hamill Star Wars 1977 124 color Fox Harrison Ford Mighty Ducks 1991 104 color Disney Emilio Estevez Wayne s World 1992 95 color Paramount Dana Carvey Wayne s World 1992 95 color Paramount Mike Meyers
Functional Dependencies (FDs) A functional dependency X Y holds over relation R if, for every allowable instance r of R: t1 r, t2 r, (t1) = (t2) implies (t1) = (t2) i.e., given two tuples in r, if the X values agree, then the Y values must also agree. (X and Y are sets of attributes.) X Y Y X X 1 2 1 2 Y a b a b Z p q r p t1 t2
Functional Dependencies (FDs) Does the following relation instance satisfy X->Y ? X 1 2 1 2 Y a b a c Z p q r p
Functional Dependencies (FDs) A functional dependency X Y holds over relation R if, for every allowable instance r of R: t1 r, t2 r, (t1) = (t2) implies (t1) = (t2) i.e., given two tuples in r, if the X values agree, then the Y values must also agree. (X and Y are sets of attributes.) An FD is a statement about all allowable relations. Must be identified based on semantics of application. Given some allowable instance r1 of R, we can check if it violates some FD f, but we cannot tell if f holds over R! K is a candidate key for R means that K R However, K R does not require K to be minimal! X Y Y X
Functional Dependencies (FDs) Does the following relation instance satisfy X->Y ? X 1 2 1 3 Y a b a b Z p q r p 9/10/2024 12
Functional Dependencies (FDs) If X is a candidate key, then X -> Y Z ! X 1 2 1 3 Y a b a b Z p q r p 9/10/2024 13
Functional Dependencies (FDs) If Y Z -> X can we say YZ is a candidate key? X 1 2 1 3 Y a b a b Z p q r p 9/10/2024 14
Example: Constraints on Entity Set Consider relation obtained from Hourly_Emps: Hourly_Emps (ssn, name, lot, rating, hrly_wages, hrs_worked) Notation: We will denote this relation schema by listing the attributes: SNLRWH This is really the set of attributes {S,N,L,R,W,H}. Sometimes, we will refer to all attributes of a relation by using the relation name. (e.g., Hourly_Emps for SNLRWH) S N L R W H Hourly_Emps
Example: Constraints on Entity Set Some FDs on Hourly_Emps: ssn is the key: S SNLRWH rating determines hrly_wages: R W Did you notice anything wrong with the following instance ? S N L R 1 2 3 2 W 100 200 250 300 H
Example: Constraints on Entity Set Some FDs on Hourly_Emps: ssn is the key: S SNLRWH rating determines hrly_wages: R W Salary should be the same for a given rating! S N L R 1 2 3 2 W 100 200 250 200 H
S 123-22-3666 Attishoo 231-31-5368 Smiley 131-24-3650 Smethurst 35 5 7 434-26-3751 Guldu 612-67-4134 Madayan N L 48 8 10 40 22 8 10 30 R W H Example 30 32 35 5 7 35 8 10 40 Problems due to R W : Update anomaly: Can we change W in just the 1st tuple of SNLRWH? Insertion anomaly: What if we want to insert an employee and don t know the hourly wage for his rating? Deletion anomaly: If we delete all employees with rating 5, we lose the information about the wage for rating 5!
S 123-22-3666 Attishoo 231-31-5368 Smiley 131-24-3650 Smethurst 35 5 7 434-26-3751 Guldu 612-67-4134 Madayan N L 48 8 10 40 22 8 10 30 R W H Hourly_Emps 30 32 35 5 7 35 8 10 40 Hourly_Emps2 Wages S 123-22-3666 Attishoo 231-31-5368 Smiley 131-24-3650 Smethurst 35 5 30 434-26-3751 Guldu 612-67-4134 Madayan N L 48 8 40 22 8 30 R H R W 8 10 5 7 35 5 32 35 8 40
Refining an ER Diagram 1st diagram translated: Workers(S,N,L,D,S) Departments(D,M,B) Lots associated with workers. Suppose all workers in a dept are assigned the same lot: D L since name dname ssn lot did budget Works_In Employees Departments
Refining an ER Diagram Suppose all workers in a dept are assigned the same lot: D L Redundancy; fixed by: Workers2(S,N,D,S) Dept_Lots(D,L) Can fine-tune this: Workers2(S,N,D,S) Departments(D,M,B,L) budget since name dname ssn did lot Works_In Employees Departments
Reasoning About FDs Given some FDs, we can usually infer additional FDs: ssn did, did lot implies ssn lot An FD f is implied by a set of FDs F if f holds whenever all FDs in F hold. = closure of F is the set of all FDs that are implied by F. Armstrong s Axioms (X, Y, Z are sets of attributes): Reflexivity: If X Y, then Y X (a trivial FD) Augmentation: If X Y, then XZ YZ for any Z Transitivity: If X Y and Y Z, then X Z These are sound and complete inference rules for FDs! F+
Reasoning About FDs S N L R W H For example, in the above schema S N -> S is a trivial FD since {S,N} is a superset of {S}
Reasoning About FDs S N L R W H For example, in the above schema If S N -> R W, then S N L -> R W L (by augmentation)
Reasoning About FDs (Contd.) Couple of additional rules (that follow from AA): Union: If X -> Y and X -> Z, then X -> YZ Proof: From X -> Y, we have XX -> XY (by augmentation) Note that XX is X, therefore X -> XY From X -> Z, we have XY -> YZ (by augmentation) From X -> XY and XY -> YZ, we have X -> YZ (by transitiviy)
Reasoning About FDs (Contd.) Couple of additional rules (that follow from AA): Decomposition: If X -> YZ, then X -> Y and X -> Z Try to prove it at home/dorm/IC/Vitamin/DD/Bus!
Reasoning About FDs (Contd.) Example: Contracts(cid,sid,jid,did,pid,qty,value), and: C is the key: C CSJDPQV Project purchases each part using single contract: JP C Dept purchases at most one part from a supplier: SD P JP C, C CSJDPQV imply JP CSJDPQV SD P implies SDJ JP SDJ JP, JP CSJDPQV imply SDJ CSJDPQV
Reasoning About FDs (Contd.) F+ Computing the closure of a set of FDs ( ) can be expensive. (Size of closure is exponential in # attrs!) Example: A database with 3 attributes (A,B,C) F = {A->B, B->C} Find the closure of F denoted by F+
Example Suppose we are given a relation scheme R = (A,B,C,G,H,I), and the set of functional dependencies, F provided below: A B A C CG H CG I B H Is A H logically implied by F? Is AG I logically implied by F? 9/10/2024 29
Reasoning About FDs (Contd.) F+ Computing the closure of a set of FDs ( ) can be expensive. (Size of closure is exponential in # attrs!) Typically, we just want to check if a given FD X Y is in the closure of a set of FDs F. An efficient check: Compute attribute closure of X (denoted ) wrt F: Set of all attributes A such that X A is in There is a linear time algorithm to compute this. For each FD Y -> Z in F, if is a superset of Y then add Y to X+ F+ X+ X+
Reasoning About FDs (Contd.) Does F = {A B, B C, C D E } imply A E? i.e, is A E in the closure ? Equivalently, is E in ? F+ A+ Lets compute A+ Initialize A+ to {A} : A+ = {A} From A -> B, we can add B to A+ : A+ = {A, B} From B -> C, we can add C to A+ : A+ = {A, B, C} We can not add any more attributes, and A+ does not contain E therefore A -> E does not hold.
DB Design Guidelines Design a relation schema with a clearly defined semantics Design the relation schemas so that there is not insertion, deletion, or modification anomalies. If there may be anomalies, state them clearly Avoid attributes which may frequently have null values as much as possible. Make sure that relations can be combined by key- foreign key links 9/10/2024 32
Normal Forms Normal forms are standards for a good DB schema (introduced by Codd in 1972) If a relation is in a certain normal form (such as BCNF, 3NF etc.), it is known that certain kinds of problems are avoided/minimized. Normal forms help us decide if decomposing a relation helps.
Normal Forms First Normal Form: No set valued attributes (only atomic values) sid 1 name ali phones {5332344568, 2165533561} 2 3 4 veli ayse fatma
Normal Forms (Contd.) Role of FDs in detecting redundancy: Consider a relation R with 3 attributes, ABC. No FDs hold: There is no redundancy here. Given A -> B: Several tuples could have the same A value, and if so, they ll all have the same B value!
Normal Forms (Contd.) Second Normal Form : Every non-prime attribute should be fully functionally dependent on every key (i.e., candidate keys). In other words: No non-prime attribute in the table is functionally dependent on a proper subset of any candidate key Prime attribute: any attribute that is part of a key Non-prime attributes: rest of the attributes Ex: If AB is a key, and C is a non-prime attribute, then if A->C holds then A partially determines C (there is a partial functional dependency to a key)
Third Normal Form (3NF) F+ Reln R with FDs F is in 3NF if, for all X A in A X (called a trivial FD), or X contains a key for R, or A is part of some key for R. If R is in 3NF, some redundancy is possible.
What Does 3NF Achieve? If 3NF violated by X -> A, one of the following holds: X is a subset of some key K We store (X, A) pairs redundantly. X is not a proper subset of any key. There is a chain of FDs K -> X -> A, which means that we cannot associate an X value with a K value unless we also associate an A value with an X value. But: even if reln is in 3NF, these problems could arise. e.g., Reserves SBDC, S C, C S is in 3NF, but for each reservation of sailor S, same (S, C) pair is stored. There is a stricter normal form (BCNF).
Boyce-Codd Normal Form (BCNF) F+ Reln R with FDs F is in BCNF if, for all X A in A X (called a trivial FD), or X contains a key for R. (i.e., X is a superkey) In other words, R is in BCNF if the only non-trivial FDs that hold over R are key constraints. No dependency in R that can be predicted using FDs alone. If we are shown two tuples that agree upon the X value, we cannot infer the A value in one tuple from the A value in the other. If example relation is in BCNF, the 2 tuples must be identical (since X is a key). X Y x x A y1 a y2 ?
Normal Forms Contd. Person(SSN, Name, Address, Hobby) F = {SSN Hobby -> Name Address, SSN ->Name Address} SSN Name Address Hobby 111111 Celalettin Sabanci D. Stamps 111111 Celalettin Sabanci D. Coins 555555 Elif Mutlukent Skating 555555 Elif Mutlukent Surfing 666666 Sercan Esentepe Math Is the above relation in 2nd normal form ? 9/10/2024 40
Normal Forms Contd. Ex: R = ABCD, F={AB->CD, AC->BD} What are the (candidate) keys for R? Is R in 3NF? Is R in BCNF? A 1 2 B 1 1 C 3 3 D 4 4 Is there a redundancy in the above instance With respect to F ? 9/10/2024 41
Example An employee can be assigned to at most one project, but many employees participate in a project EMP_PROJ(ENAME, SSN, ADDRESS, PNUMBER, PNAME, PMGRSSN) PMGRSSN is the SSN of the manager of the project Is this a good design? 9/10/2024 42
Decomposition of a Relation Scheme Suppose that relation R contains attributes A1 ... An. A decomposition of R consists of replacing R by two or more relations such that: Each new relation scheme contains a subset of the attributes of R (and no attributes that do not appear in R), and Every attribute of R appears as an attribute of one of the new relations. Intuitively, decomposing R means we will store instances of the relation schemes produced by the decomposition, instead of instances of R. E.g., Can decompose SNLRWH into SNLRH and RW.
Decomposition of a Relation Scheme We can decompose SNLRWH into SNL and RWH. S N L R W H S N L R W H
Example Decomposition SNLRWH has FDs S -> SNLRWH and R -> W Is this in 3NF? R->W violates 3NF (W values repeatedly associated with R values) In order to fix the problem, we need to create a relation RW to store the R->W associations, and to remove W from the main schema: i.e., we decompose SNLRWH into SNLRH and RW S N L R H R W
Problems with Decompositions There are potential problems to consider: Some queries become more expensive. e.g., How much did Ali earn ? (salary = W*H) S N L R H R W
Problems with Decompositions Given instances of the decomposed relations, we may not be able to reconstruct the corresponding instance of the original relation!
Lossless Join Decompositions Decomposition of R into X and Y is lossless-join w.r.t. a set of FDs F if, for every instance r that satisfies F: (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. Definition extended to decomposition into 3 or more relations in a straightforward way. It is essential that all decompositions used to deal with redundancy be lossless! (Avoids Problem (2).) X Y X Y
A B 1 4 7 More on Lossless Join 2 5 2 A B C 1 2 4 5 7 2 The decomposition of R into X and Y is lossless-join wrt F if and only if the closure of F contains: X Y X, or X Y Y In particular, the decomposition of R into UV and R - V is lossless-join if U V holds over R. 3 6 8 B 2 5 2 C 3 6 8 A B C 1 2 4 5 7 2 1 2 7 2 3 6 8 8 3
Person(SSN, Name, Address, Hobby) F = {SSN Hobby -> Name Address, SSN ->Name Address} Person SSN Name Address Hobby 111111 Celalettin Sabanci D. Stamps 111111 Celalettin Sabanci D. Coins 555555 Elif Mutlukent Skating 555555 Elif Mutlukent Surfing 666666 Sercan Esentepe Math Person1 Hobby SSN Name Address SSN Hobby 111111 Celalettin Sabanci D. 111111 Stamps 555555 Elif Mutlukent 111111 Coins 666666 Sercan Esentepe 555555 Skating 555555 Surfing 666666 Math
