Database Management Systems, R. Ramakrishnan and J. Gehrke1 Schema Refinement and Normal Forms...

Database Management Systems, R. Ramakrishnan and J. Gehrke 1

Schema Refinement and Normal Forms

Chapter 19


The Evils of Redundancy Redundancy is at the root of several problems

associated with relational schemas:– redundant storage, insert/delete/update anomalies

Integrity constraints, in particular functional dependencies, can be used to identify schemas with such problems and to suggest refinements.

Main refinement technique: decomposition (replacing ABCD with, say, AB and BCD, or ACD and ABD).

Decomposition should be used judiciously:– Is there a reason to decompose a relation?– What problems (if any) does the decomposition cause?


Example in page 607:

Hourly_Emps(ssn, name, lot, rating, hourly_wages, hours_worked)

suppose hourly_wages attribute is determined by the rating attribute.For a given rating value, there is only one permissible hourly_wagesvalue.

This integrity constraint (IC) is an example of functional dependency. See problems in next page


Figure 19.1

Problems due to R W :– Redundant storage– 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 N L R W H

123-22-3666 Attishoo 48 8 10 40

231-31-5368 Smiley 22 8 10 30

131-24-3650 Smethurst 35 5 7 30

434-26-3751 Guldu 35 5 7 32

612-67-4134 Madayan 35 8 10 40


Can null values help?

Null values cannot help eliminate redundant storage or update anomalies

The insertion anomaly in the previous page may be solved by using null values, null values cannot address all insertion anomalies: we cannot record the hourly wages for a rating unless there is an employee with that rating (primary key field cannot store null)

Null values can not solve deletion anomalies either


Use of Decompositions

Decomposing Hourly_Emps into two relations:

page 609

Are all problems solved?


Problems related to decomposition

Queries over the original relation may require us to join the decomposed relations. If such queries are common, the performance penalty of decomposing the relation may not be acceptable.

In this case we may choose to live with some of the problems of redundancy and not decompose the relation.

It is important to be aware of the potential problems caused by such residual redundancy in the design and to take steps to avoid them (e.g., by adding some checks to application code).

A good DB designer should have a firm grasp of normal forms and what problems they do (or do not) alleviate, the technique of decomposition, and potential problems with decompositions.


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 X Y Y


Examples motivating schema refinement

ER design CAN generate some schemas with redundancy problems, because it is a complex, subjective process, and certain constraints are NOT expressible in terms of ER diagrams.– Constraints on an entity set– Constraints on a relationship set– Identifying attributes of entities– Identifying entity sets


Constraints on Entity Set

Consider relation obtained from Hourly_Emps:– Hourly_Emps (ssn, name, lot, rating, hrly_wages, hrs_worke

d) Notation: We will denote this relation schema by listin

g 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 usin

g the relation name. (e.g., Hourly_Emps for SNLRWH) Some FDs on Hourly_Emps:

– ssn is the key: S SNLRWH – rating determines hrly_wages: R W


Example (Contd.)

Problems due to R W :– Redundant storage– 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 N L R W H

123-22-3666 Attishoo 48 8 10 40

231-31-5368 Smiley 22 8 10 30

131-24-3650 Smethurst 35 5 7 30

434-26-3751 Guldu 35 5 7 32

612-67-4134 Madayan 35 8 10 40

This FD (RW) can NOTbe expressed in terms of theER model. Only FDs that determine all attributes of a relation (i.e., key constraints)can be expressed in the ERmodel. Therefore we can NOTdetect it when we consideredHourly_Emps as an entity setduring ER modeling.


Example (Contd.)S N L R W H

123-22-3666 Attishoo 48 8 10 40

231-31-5368 Smiley 22 8 10 30

131-24-3650 Smethurst 35 5 7 30

434-26-3751 Guldu 35 5 7 32

612-67-4134 Madayan 35 8 10 40

S N L R H

123-22-3666 Attishoo 48 8 40

231-31-5368 Smiley 22 8 30

131-24-3650 Smethurst 35 5 30

434-26-3751 Guldu 35 5 32

612-67-4134 Madayan 35 8 40

R W

8 10

5 7

Hourly_Emps2

Wages


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

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)

lot

dname

budgetdid

sincename

Works_In DepartmentsEmployees

ssn

lot

dname

budget

did

sincename

Works_In DepartmentsEmployees

ssn

Before:

After:


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 whenev

er 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 X Y – 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 (Contd.) Couple of additional rules (that follow from AA):

– Union: If X Y and X Z, then X YZ (XX XY and XYYZ; XXYZ)

– Decomposition: If X YZ, then X Y and X Z (YZY and YZ Z) Example: Contracts(cid,sid,jid,did,pid,qty,value), and:

– C is the key: C CSJDPQV– A project purchases a given part using a single contract: JP C– A 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 expen

sive. (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.

– Check if Y is in Does F = {A B, B C, C D E } imply A E?

– i.e, is A E in the closure ? Equivalently, is E in ?

X

X

F

AF


Apply algorithm in Figure 19.4 to compute attribute closure for A+

closure = A;repeat until there is no change: { if there is an FD U V in F s.t. U closure, then set closure = closure V }

U V closure = A FD: A Bclosure = AB FD: B Cclosure = ABC

A+ = {A,B,C} and E A+ , hence A E F+


Another Example ― 習題 19.2 第一小題

Given a relation R with 5 attributes ABCDE and the following FDs: A B, BC E, and ED A. We want to find all keys for R. Any key K for R must satisfy K ABCDE. There are 5 + 10 + 10 + 5 + 1 = 31 different possibilities.

Let’s first try K = A; we need to show A ABCDEclosure = A;

U V closure = A FD: A Bclosure = AB

A+ = {A,B} and ABCDE A+ , hence A ABCDE F+. In other words, A is not a key for R.



Now you try K = BCD; you need to show BCD ABCDE



Show BCD ABCDE

closure = BCD; U V

closure = BCD FD: BC Eclosure = BCDE FD: ED AClosure=ABCDE

BCD+ = {A,B,C,D,E} and ABCDE BCD+ , hence BCD ABCDE F+ . In other words, BCD is a key for R.



You try K = ACD; you need to show ACD ABCDE

You try K = CDE; you need to show CDE ABCDE



Hence, there are three keys for R: CDE, ACD, BCD


Normal Forms

Returning to the issue of schema refinement, the first question to ask is whether any refinement is needed!

If a relation is in a certain normal form (BCNF, 3NF etc.), it is known that certain kinds of problems are avoided/minimized. This can be used to help us decide whether decomposing the relation will help.

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 valu

e, and if so, they’ll all have the same B value!


Normal Forms

Normal forms based on FDs: – First normal form (1NF) – second normal form (2NF) – third normal form (3NF) – Boyce-Codd normal form (BCNF)

1NF: every field contains only atomic values, that is, no lists or sets.


Boyce-Codd Normal Form (BCNF)

Reln R with FDs F is in BCNF if, for every X A (single attribute) in F– 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 (see Figure 19.5).

Given a Reln R with a set F of FDs, to show whether R is in BCNF, it is sufficient to check whether the left side of each FD in F is a superkey


Another Example ― 習題 19.2 第三小題

Given a relation R with 5 attributes ABCDE and the following FDs: A B, BC E, and ED A. Is R in BCNF?

We have shown that there are 3 keys for R: CDE, ACD, BCD.

R is not in BCNF because none of A, BC, and ED contain a key.


Boyce-Codd Normal Form (BCNF)

BCNF ensures that no redundancy can be detected using FD information alone (because key is unique).

It is thus the most desirable normal form (from the point of view of redundancy) if we take into account only FD information.


Third Normal Form (3NF) Reln R with FDs F is in 3NF if, for every X A in F

– A X (called a trivial FD), or– X contains a key for R (i.e., X is a superkey), or– A is part of some key for R.

Minimality of a key is crucial in third condition above! If R is in BCNF, obviously in 3NF. If R is in 3NF, some redundancy is possible. It is a com

promise, used when BCNF not achievable (e.g., no ``good’’ decomp, or performance considerations).– Lossless-join, dependency-preserving decomposition of R in

to a collection of 3NF relations always possible.


Another Example ― 習題 19.2 第二小題

Given a relation R with 5 attributes ABCDE and the following FDs: A B, BC E, and ED A. Is R in 3NF?

We have shown that there are 3 keys for R: CDE, ACD, BCD.

R is in 3NF because the attribute on the right hand side of each FD, i.e., B, E, and A, are all parts of keys.


Suppose a dependency X A causes a violation of 3NF (2 cases to consider) X is a proper subset of some key K (see Fig. 19.7): th

is is called partial dependency. In this case, we store (X,A) pairs redundantly. As an example, consider the Reserves relation with

attribute SBDC where S is sid, B is bid, D is day, and C denotes the credit card to which the reservation is charged. The only key is SBD and we have S C (sid determines credit card). In this case, we store the credit card number for a sailor as many times as there are reservations for that sailor.


Suppose a dependency X A causes a violation of 3NF (the second case) X is not a proper subset of any key (see Fig. 19.8): su

ch a dependency is called a transitive dependency. As an example, consider the Hourly_Emps relation

with attributes SNLRWH. The only key is S, but there is a FD R W, which gives rise to the chain S R W.

The consequence is that we cannot record the fact that employee S has rating R without knowing the hourly wage for that rating.

This condition leads to insertion, deletion, and update anomalies.


Redundancy is possible with 3NF

Let us revisit the Reserves relation with attributes SBDC and the FD S C, which states a sailor uses a unique credit card to pay for reservations.

S is not a key, and C is not part of a key (the only key is SBD). Hence, this relation is not in 3NF; (S,C) pairs stored redundantly.


Redundancy is possible with 3NF

If we also know that credit cards uniquely identify the owner, we have the FD C S, which means that CBD is also a key for Reserves.

Therefore, the dependency S C does not violate 3NF, and Reserves is in 3NF.

Nonetheless, in all tuples containing the same S value, the same (S,C) pair is redundantly recorded.


Second Normal Form (2NF)

2NF are relations in which partial dependencies are not allowed.

Thus, if a relation is in 3NF (which precludes both partial and transitive dependencies), it is also in 2NF.


習題 19.1 第 3 小題Given a set of FDs for the relation schema R(A,B,C,D) withPrimary key AB under which R is in 1NF but not in 2NF.



Answer: A C or A D or B C or B D.



Answer: D C or C D or AC D or AD C or BC D or BD C


3NF but not BCNF

Given a set of FDs for the relation schema R(A,B,C,D) withPrimary key AB under which R is in 3NF but not in BCNF.


3NF but not BCNF

Given a set of FDs for the relation schema R(A,B,C,D) withprimary key AB under which R is in 3NF but not in BCNF.

Answer: C A or C B or D A or D B or CD A or CD B or CDA B or CDB A


BCNF

Given a set of FDs for the relation schema R(A,B,C,D) withPrimary key AB under which R is in BCNF.


BCNF

Given a set of FDs for the relation schema R(A,B,C,D) withPrimary key AB under which R is in BCNF.

Answer: AB ABCD


習題 19.1 第 5 小題Consider the relation schema R(A,B,C), which has the FD B C. If A is a candidate key for R, is it possible for R to be in BCNF? If so, under what conditions? If not, explain why not.


習題 19.1 第 5 小題Consider the relation schema R(A,B,C), which has the FD B C. If A is a candidate key for R, is it possible for R to be in BCNF? If so, under what conditions? If not, explain why not.

Answer: If B is also a key, then R is in BCNF.


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.


Example Decomposition

Decompositions should be used only when needed.– SNLRWH has FDs S SNLRWH and R W– Second FD causes violation of 3NF; W values repeatedly a

ssociated with R values. Easiest way to fix this is to create a relation RW to store these associations, and to remove W from the main schema:

i.e., we decompose SNLRWH into SNLRH and RW


Example (Contd.)S N L R W H

123-22-3666 Attishoo 48 8 10 40

231-31-5368 Smiley 22 8 10 30

131-24-3650 Smethurst 35 5 7 30

434-26-3751 Guldu 35 5 7 32

612-67-4134 Madayan 35 8 10 40

S N L R H

123-22-3666 Attishoo 48 8 40

231-31-5368 Smiley 22 8 30

131-24-3650 Smethurst 35 5 30

434-26-3751 Guldu 35 5 32

612-67-4134 Madayan 35 8 40

R W

8 10

5 7

Hourly_Emps2

Wages


Example Decomposition

The information to be stored consists of SNLRWH tuples. If we just store the projections of these tuples onto SNLRH and RW, are there any potential problems that we should be aware of?


Problems with Decompositions

There are three potential problems to consider: Some queries become more expensive.

e.g., How much did employee Guldu earn? (salary = W*H)

Given instances of the decomposed relations, we may not be able to reconstruct the corresponding instance of the original relation!

Fortunately, not in the SNLRWH example. Checking some dependencies may require joining the inst

ances of the decomposed relations. Fortunately, not in the SNLRWH example.

Tradeoff: Must consider these issues vs. redundancy.


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 wi

th redundancy be lossless! (Avoids Problem (2).)

X Y X Y


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.

A B C1 2 34 5 67 2 81 2 87 2 3

A B C1 2 34 5 67 2 8

A B1 24 57 2

B C2 35 62 8

LossyDecomp.


An example of lossless decomposition

We decompose Hourly_Emps with attributes SNLRWH into SNLRH and RW. R is common to both decomposed relations, and RW holds, this decomposition is lossless.


習題 19.8 第 2 小題的 (b)Is the following decomposition of R = ABCDEG, with the FD set F = {AB C, AC B, AD E, B D, BC A, E G}, lossless-join?

{ABC, ACDE, ADG}

首先 ABC join ACDE = ABCDE. 共同 attributes 是AC. 這個 decomposition 是否為 lossless, 我們必須檢查 AC B or AC DE 是否 F+ ?答案是肯定 , 因為 AC B F.接下來 ABCDE join ADG = ABCDEG. 共同 attributes 是AD. 這個 decomposition 是否為 lossless, 我們必須檢查 AD BCE or AD G 是否 F+ ?答案是肯定 , 因為 AD E 與 E G F, 故 AD G F

+.故此 decomposition 是 lossless-join


習題 19.8 第 2 小題的 (a)Is the following decomposition of R = ABCDEG, with the FD set F = {AB C, AC B, AD E, B D, BC A, E G}, lossless-join?

{AB, BC, ABDE, EG}

Answer: this decomposition is not lossless-join.


Dependency Preserving Decomposition

Consider CSJDPQV, C is key, JP C and SD P.– BCNF decomposition: CSJDQV and SDP (lossless)– Problem: Checking JP C requires a join!

Dependency preserving decomposition (Intuitive):– If R is decomposed into X, Y and Z, and we enforce the FD

s that hold on X, on Y and on Z, then all FDs that were given to hold on R must also hold. (Avoids Problem (3).)

Projection of set of FDs F: If R is decomposed into X, ... projection of F onto X (denoted FX ) is the set of FDs U V in F+ (closure of F ) such that U, V are in X.


Dependency Preserving Decompositions (Contd.)

Decomposition of R into X and Y is dependency preserving if (FX FY ) + = 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? (next page) Is C A

preserved?????



F = {AB, B C, C A} F+ = {A C, B A, C B, AB, B C, C A} F{A,B} = {A B, B A}

F{B,C} = {B C, C B}

(F{A,B} F{B,C} )+ = {A B, B A, B C, C B}+ = {C A, A C, A B, B A, B C, C B} = F+



Dependency preserving does not imply lossless join:– ABC, A B, decomposed into AB and BC.

And vice-versa! (Example?)


習題 19.8 第 2 小題的 (b)Is the following decomposition of R = ABCDEG, with the FD set F = {AB C, AC B, AD E, B D, BC A, E G}, is dependency-preserving?

{ABC, ACDE, ADG}


習題 19.8 第 2 小題的 (b)Is the following decomposition of R = ABCDEG, with the FD set F = {AB C, AC B, AD E, B D, BC A, E G}, is dependency-preserving?

{ABC, ACDE, ADG}

F = {AB C, AC B, AD E, B D, BC A, E G}F+ = {AC D, AD G, AB C, AC B, AD E, B D, B

C A, E G}F{ABC} = {AB C, AC B, BC A}F{ACDE} = {AC D, AD E}F{ADG} = {AD G}(F{ABC} F{ACDE} F{ADG} )+ = {AB C, AC B, BC A, AC D, AD E, AD G} F+ ( 少了 B D, E G)故此 decomposition 非 dependency-preserving.


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.

– e.g., CSJDPQV, key C, JP C, SD P, J S– To deal with SD P, decompose into SDP, CSJDQV.– To deal with J S, decompose CSJDQV into JS and CJDQV

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!


BCNF and Dependency Preservation

In general, there may not be a dependency preserving decomposition into BCNF.– e.g., CSZ, CS Z, Z C (CS is key)– 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 dep

endency preserving decomposition. JPC tuples stored only for checking FD! (Redundancy!) Hence, redundancy can still occur across relations, even though

there is no redundancy within a relation.


Decomposition into 3NF

Obviously, 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 ad

dition 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.


Minimal Cover for a Set of FDs e.g., AB, ABCD E, EF G, EF H, ACDF EG has the f

ollowing minimal cover:– A B, ACD E, EF G and EF H

Steps:1. Put FDs in a standard form: AB, ABCD E, EF G, EF H, ACD

F E, ACDF G 2. Minimize the left side of each FD: ABCD E can be minimized to be

come ACD E (use algorithm in Fig. 15.6)3. Delete redundant FDs: ACDF G can be deleted because we have A

B, ACD E, EF G (again use algorithm in Fig. 15.6); Similarly ACDF E can be deleted as well


Minimal Cover for a Set of FDs

AB, ABCD E, EF G, EF H, ACDF EG

AB, ABCD E, EF G, EF H, ACDF E, ACDF G

AB, ACD E, EF G, EF H, ACDF E, ACDF G

AB, ACD E, EF G, EF H

Step 1

Step 2

Step 3


Algorithm in Figure 15.6

XY F+ is equivalent to Y X+

closure = X;do until no change { if there is U V in F and U closure then closure = closure V}


Minimal Cover for a Set of FDs Step 2: ABCD E (?) F+ is equivalent to E (?) {ABCD}+

closure UV U closure {ABCD} ACD E {ACD} {ABCD} {ABCDE} E {ABCDE} {ABCD}+

Hence, ABCD E F+


Dependency-preserving decomposition into 3NF

Let R be a relation with a set F of FDs that is a minimal cover, and let R1, R2, …, Rn be a lossless-join decomposition of R. For 1 i n, suppose that each Ri is in 3NF and let Fi denote the projection of F onto the attributes of Ri. Do the following:– Identify the set N of dependencies in F that are not preser

ved, that is, not included in the closure of the union of Fis – For each FD XA in N, create a relation schema XA and ad

d it to the decomposition of R.


Summary of Schema Refinement

If a relation is in BCNF, it is free of redundancies that can be detected using FDs. Thus, trying to ensure that all relations are in BCNF is a good heuristic.

If a relation is not in BCNF, we can try to decompose it into a collection of BCNF relations.– Must consider whether all FDs are preserved. If a lossles

s-join, dependency preserving decomposition into BCNF is not possible (or unsuitable, given typical queries), should consider decomposition into 3NF.

– Decompositions should be carried out and/or re-examined while keeping performance requirements in mind.

Database Management Systems, R. Ramakrishnan and J. Gehrke1 Schema Refinement and Normal Forms...

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