- EN6 Chapter 15 / EN7 Chapter 14: Basics of Functional Dependencies and
Normalization for Relational Databases

- EN6 Chapter 16 / EN7 Chapter 15: Relational Database Design Algorithms and Further Dependencies

Our first design point will be to make sure that relation and entity attributes have clear semantics. That is, we can easily explain attributes, and they are all related. E&N spells this out as follows:

E&N
Guideline 1: Design a relation schema so that it is easy to explain its
meaning. Do not combine attributes from multiple entity types and
relationship types into a single relation.

As examples, consider

EMP_DEPT

Ename

Ssn

Bdate

Address

Dnumber

Dname

Dmgr_ssn

This mixes employee information with department information.

Another example is

EMP_PROJ

Ssn

Pnumber

Hours

Ename

Pname

Plocation

Both these later records can lead to

- insertion anomalies: when
adding an employee, we must assign them to a department or else use
NULLs. When adding a new department with no employees, we have to use
NULLs for the employee Ssn, which is supposed to be the primary key!

- deletion anomalies: if we
delete the last EMP_DEP record from a department, we have lost the
department!

- update anomalies: what if we update some EMP_DEPs with the new Dmgr_ssn, but not others?

E&N
Guideline 2: design your database so there are no insertion, deletion,
or update anomalies. If this is not possible, document any anomalies
clearly so that the update software can take them into account.

The third guideline is about reducing NULLs, at least for frequently used data. Consider the example of a disjoint set of inherited entity types, implemented as a single fat entity table. Secretaries, Engineers, and Managers each have their own attributes; every record has NULLs for two out of these three.

E&N
Guideline 3: NULLs should be used only for exceptional conditions. If
there are many NULLs in a column, consider a separate table.

The fourth guideline is about joins that give back spurious tuples. Consider

EMP_LOCS

Ename

Plocation

EMP_PROJ1

Ssn

Pnumber

Hours

Pname

Plocation

If we join these two tables on field Plocation, we do not get what we want! (We would if we made EMP_LOCS have Ssn instead of Ename, and then joined the two on the Ssn column.) We can create each of these as a view:

create view emp_locs as select e.lname,
p.plocation from employee e, works_on w, project p

where e.ssn=w.essn and w.pno=p.pnumber;

create view emp_proj1 as select e.ssn, p.pnumber, w.hours, p.pname, p.plocation

from employee e, works_on w, project p where e.ssn=w.essn and w.pno = p.pnumber;

where e.ssn=w.essn and w.pno=p.pnumber;

create view emp_proj1 as select e.ssn, p.pnumber, w.hours, p.pname, p.plocation

from employee e, works_on w, project p where e.ssn=w.essn and w.pno = p.pnumber;

Now let us join these on plocation:

select * from emp_locs el, emp_proj1
ep where el.plocation = ep.plocation;

Oops. What is wrong?

E&N
Guideline 4: Design relational schemas so that they can be joined with
equality conditions on attributes that are appropriatedly related
⟨primary key, foreign key⟩ pairs in a way that guarantees that no
spurious tuples are generated. Avoid relations that contain matching
attributes that are not ⟨foreign key, primary key⟩ combinations.

Problem: what do we have to do to

For example, if X is a set including the key attributes, then X⟶{all attributes}.

Like key constraints, FD constraints are not based on specific sets of records. For example, in the US, we have {zipcode}⟶{city}, but we no longer have {zipcode}⟶{areacode}.

In the earlier EMP_DEPT we had FDs

Ssn ⟶ Ename, Bdate, Address, Dnumber

Dnumber ⟶ Dname, Dmgr_ssn

In EMP_PROJ, we had FDs

Ssn ⟶ Ename

Pnumber ⟶ Pname, Plocation

{Ssn, Pnumber} ⟶ Hours

Sometimes FDs are a problem, and we might think that just discreetly removing them would be the best solution. But they often represent important business rules; we can't really do that either. At the very least, if we don't "normalize away" a dependency we run the risk that data can be entered so as to make the dependency

A superkey (or key superset) of a relation schema is a set of attributes S so that no two tuples of the relationship can have the same values on S. A key is thus a minimal superkey: it is a superkey with no extraneous attributes that can be removed. For example, {Ssn, Dno} is a superkey for EMPLOYEE, but Dno doesn't matter (and in fact contains no information relating to any other employee attributes); the key is {Ssn}.

Note that, as with FDs, superkeys are related to the sematics of the relationships, not to particular data in the tables.

Relations can have multiple keys, in which case each is called a candidate key. For example, in table DEPARTMENT, both {dnumber} and {dname} are candidate keys. For arbitrary performance-related reasons we designated one of these the primary key; other candidate keys are known as secondary keys. (Most RDBMSs create an

A prime attribute is an attribute (ie column name) that belongs to some candidate key. A nonprime attribute is not part of any key. For DEPARTMENT, the prime attributes are dname and dnumber; the nonprime are mgr_ssn and mgr_start.

A dependency X⟶A is full if the dependency fails for every proper subset X' of X; the dependency is partial if not, ie if there is a proper subset X' of X such that X'⟶A. If A is the set of

Alternative ways for dealing with the multivalued location attribute would be making ⟨dnumber, location⟩ the primary key, or supplying a fixed number of location columns loc1, loc2, loc3, loc4. For the latter approach, we must know in advance how many locations we will need; this method also introduces NULL values.

Note that if a relation has a single-attribute primary key, as does EMP_DEPT, then 2NF is automatic. (Actually, the general definition of 2NF requires this for every candidate key; a relation with a single-attribute primary key but with some multiple-attribute other key would still have to be checked for 2NF.)

We say that X⟶Y is a full functional dependency if for every proper subset X' of X, X' does not functionally determine Y. Thus, 2NF means that for every nonprime attribute A, the dependency K⟶A is full: no nonprime attribute depends on less than the full key.

In the earlier EMP_PROJ relationship, the primary key K is {Ssn, Pnumber}. 2NF fails because {Ssn}⟶Ename, and {Pnumber}⟶Pname, {Pnumber}⟶Plocation.

original: ⟨K1, K2, A2⟩ -- A1 removed

new: ⟨K1, A1⟩ -- both sides of K1⟶A1

To put a table in 2NF, use factoring. If two new tables have a common full dependency (key) on some subset K' of K, combine them. For EMP_PROJ, the result of factoring is:

⟨Ssn, Pnumber, Hours⟩

⟨Ssn, Ename⟩

⟨Pnumber, Pname, Plocation⟩

Note that Hours is the only attribute with a

The table EMP_DEPT is in 2NF, as is any table with a single key attribute.

Note that we might have a table ⟨K1, K2, K3, A1, A2, A3⟩, with dependencies like these:

{K1,K2,K3}⟶A1 is full

{K1,K2}⟶A2 is full (neither K1 nor K2 alone determines A2)

{K2,K3}⟶A2 is full

{K1,K3}⟶A3 is full

{K2,K3}⟶A3 is full

The decomposition could be this, if we factor on {K1,K2}⟶A2 and then {K1,K3}⟶A3:

⟨K1, K2, K3, A1⟩

⟨K1, K2, A2⟩

⟨K1, K3, A3⟩

or it could be this, if we first factor on {K2,K3}⟶A2 and then on {K2,K3}⟶A3.

⟨K1, K2, K3, A1⟩

⟨K2, K3, A2⟩

⟨K2, K3, A3⟩

The latter two can be coalesced into ⟨K2, K3, A2, A3⟩. However, there is no way of continuing the first factorization approach and the second to find some common final factorization.

Remember, dependency constraints can be arbitrary! Dependency constraints are often best thought of as "externally imposed rules"; they come out of the user-input-and-requirements phase of the DB process. Trying to pretend that there is not a dependency constraint is sometimes a bad idea.

Consider the LOTS example of Fig 15.12.

Attributes for LOTS are

- property_ID
- county
- lot_num
- area
- price
- tax_rate

FD3: county ⟶ tax_rate

FD4: area ⟶ price

(For farmland, FD4 is not completely unreasonable, at least if price refers to the price for tax purposes. In Illinois, the formula is tax_price = area × factor_determined_by_soil_type).

2NF fails because of the dependency county ⟶ tax_rate; FD4 does not violate 2NF. E&N suggest the decomposition into LOTS1(property_ID, county, lot_num, area, price) and LOTS2(county, tax_rate).

We can algorithmically use a 2NF-violating FD to define a decomposition into new tables. If X⟶A is the FD, we remove A from table R, and construct a new table with attributes those of X plus A. E&N did this above for FD3: county ⟶ tax_rate.

Before going further, perhaps two points should be made about decomposing too far. The first is that all the functional dependencies should still appear in the set of decomposed tables; the second is that reassembling the decomposed tables with the "obvious" join should give us back the original table, that is, the join should be

A lossless join means no

In Fig 15.5 there was a proposed decomposition into EMP_LOCS(ename, plocation) and EMP_PROJ1(ssn,pnumber,hours,pname,plocation). The join was not lossless.

2NF: If K represents the set of attributes making up the primary key, every nonprime attribute A (that is an attribute not a member of any key) is functionally dependent on K (ie K⟶A), but that this fails for any proper subset of K (no proper subset of K functionally determines A).

3NF: 2NF + there is no dependency X⟶A for nonprime attribute A and for an attribute set X that does not contain a key (ie X is not a superkey).

Recall that a prime attribute is an attribute that belongs to some key. A nonprime attribute is not part of any key. The reason for the nonprime-attribute restriction on 3NF is that if we do have a dependency X⟶A, the general method for improving the situation is to "factor out" A into another table, as below. This is fine if A is a nonprime attribute, but if A is a prime attribute then we just demolished a key for the original table! Thus, dependencies X⟶A involving a nonprime A are easy to fix; those involving a prime A are harder.

If X is a proper subset of a key, then we've ruled out X⟶A for nonprime A in the 2NF step. If X is a superkey, then X⟶A is automatic for all A. The remaining case is where X may contain some (but not all) key attributes, and also some nonkey attributes. An example might be a relation with attributes K1, K2, A, and B, where K1,K2 is the key. If we have a dependency K1,A⟶B, then this violates 3NF. A dependency A⟶B would also violate 3NF.

Note again that if A were a prime attribute, then it would be part of a key, and factoring it out might break that key!

One question that comes up when we factor (for

The harder question is making sure R1⋈R2 does not contain added records. If ⟨k1,k2,a,b⟩ is in R1⋈R2, we know that it came from ⟨k1,k2,a⟩ in R1 and ⟨k1,a,b⟩ in R2. Each of these partial records came from the decomposition, so there must be b' so ⟨k1,k2,a,b'⟩ is in R, and there must be k2' so ⟨k1,k2',a,b⟩ is in R, but in the most general case we need not have b=b' or k2=k2'. Here we use the key constraint, though: if ⟨k1,k2,a,b⟩ is in R, and ⟨k1,k2,a,b'⟩ is in R, and k1,k2 is the key, then b=b'. Alternatively we could have used the dependency K1,A⟶B: if this dependency holds, then it implies that if R contains ⟨k1,k2,a,b⟩ and ⟨k1,k2,a,b'⟩, then b=b'.

This worked for either of two reasons: R1 contained the original key, and R2's new key was the lefthand side of a functional dependency that held in R.

In general, if we factor a relation R=⟨A,B,C⟩ into R1=⟨A,B⟩ and R2=⟨A,C⟩, by projection, then the join R1⋈R2 on column A

123456789 |
1 |
32.5 |

453453453 |
1 |
20 |

we see that the factored tables would contain ⟨123456789,1⟩ and ⟨1,20⟩, and so the join would contain ⟨123456789,1,20⟩ violating the key constraint.

The relationship EMP_DEPT of EN fig 15.11 is not 3NF, because of the dependency dnumber ⟶ dname (or dnumber ⟶ dmgr_ssn).

Can we factor this out?

The LOTS1 relation above (EN fig 15.12) is not 3NF, because of Area ⟶ Price. So we factor on Area ⟶ Price, dividing into LOTS1A(property_ID, county,lot_num,area) and LOTS1B(area,price). Another approach would be to drop price entirely, if it is in fact proportional to area, and simply treat it as a computed attribute.

4343

An aid to dealing with this sort of situation is to notice that in effect we have a three-stage dependency: K1⟶B⟶C. These are often best addressed by starting with the downstream (B⟶C) dependency.

As for 3NF, we can use factoring to put a set of tables into BCNF. However, there is now a serious problem: by factoring out a prime attribute A, we can destroy an existing key constraint! To say this is undesirable is putting it mildly.

The canonical "formal" example of a relation in 3NF but not BCNF is ⟨A, B, C⟩ where we also have C⟶B. Factoring as above leads to ⟨A, C⟩ and ⟨C, B⟩. We have lost the key A,B! However, this isn't quite all it appears, because from C⟶B we can conclude A,C⟶B, and thus that A,C is also a key, and might be a better choice of key than A,B.

LOTS1A from above was 3NF and BCNF. But now let us suppose that DeKalb county lots have sizes <= 1.0 acres, while Fulton county lots have sizes >1.0 acres; this means we now have an additional dependency FD5: area⟶county. This violates BCNF, but not 3NF as county is a prime attribute. (The second author Shamkant Navathe is at Georgia Institute of Technology so we can assume DeKalb County is the one in Georgia, not the one in Illinois, and hence is pronounced "deKAB".)

If we fix LOTS1A as in Fig 15.13, dividing into LOTS1AX(property_ID,area,lot_num) and LOTS1AY(area,county), then we lose the functional dependency FD2: (county,lot_num)⟶property_ID.

Where has it gone? This was more than just a random FD; it was a candidate key for LOTS1A.

All databases enforce primary-key constraints. One could use a CHECK statement to enforce the lost FD2 statement, but this is often a lost cause.

CHECK (not exists (select ay.county,
ax.lot_num, ax.property_ID, ax2.property_ID

from LOTS1AX ax, LOTS1AX ax2, LOTS1AY ay

where ax.area = ay.area and ax2.area = ay.area // join condition

and ax.lot_num = ax2.lot_num

and ax.property_ID <> ax2.property_ID))

from LOTS1AX ax, LOTS1AX ax2, LOTS1AY ay

where ax.area = ay.area and ax2.area = ay.area // join condition

and ax.lot_num = ax2.lot_num

and ax.property_ID <> ax2.property_ID))

We might be better off ignoring FD5 here, and just allowing for the possibility that area does not determine county, or determines it only "by accident".

Generally, it is good practice to normalize to 3NF, but it is often not possible to achieve BCNF. Sometimes, in fact, 3NF is too inefficient, and we re-consolidate for the sake of efficiency two tables factored apart in the 3NF process.

EMP_DEPENDENTS

ename |
depname |

Smith |
John |

Smith |
Anna |

EMP_PROJECTS

ename |
projname |

Smith |
projX |

Smith |
projY |

Joining gives

ename |
depname |
projname |

Smith |
John |
X |

Smith |
John |
Y |

Smith |
Anna |
X |

Smith |
Anna |
Y |

Fourth normal form attempts to recognize this in reverse, and undo it. The point is that we have a table ⟨X,Y,Z⟩ (where X, Y, or Z may be a set of attributes), and it turns out to be possible to decompose it into ⟨X,Y⟩ and ⟨X,Z⟩ so the join is lossless. Furthermore, neither Y nor Z depend on X, as was the case with our 3NF/BCNF decompositions.

Specifically, for the "cross product phenomenon" above to occur, we need to know that if t1 = ⟨x,y1,z1⟩ and t2 = ⟨x,y2,z2⟩ are in ⟨X,Y,Z⟩, then so are t3 = ⟨x,y1,z2⟩ and t4 = ⟨x,y2,z1⟩. (Note that this is the same condition as in E&N, 15.6.1, p 533, but stated differently.)

If this is the case, then X is said to multidetermine Y (and Z). More to the point, it means that if we decompose into ⟨X,Y⟩ and ⟨X,Z⟩ then the join will be lossless.

Are you really supposed even to look for things like this? Probably not.

16.4 and the problems of NULLs

"There is no fully satisfactory relational
design theory as yet that includes NULL values"

[When joining], "particular care must be
devoted to watching for potential NULL values in foreign keys"

Unless you have a clear reason for doing otherwise, don't let foreign-key columns be NULL. Of course, we did just that in the EMPLOYEE table: we let dno be null to allow for employees not yet assigned to a department. An alternative would be to create a fictional department "unassigned", with department number 0. However, we then have to assign Department 0 a manager, and it has to be a real manager in the EMPLOYEE table.

As an example, suppose we have the following tables, with key fields underlined. (Unlike the company database we are used to, projects here each have their own managers.)

Now suppose management works mostly with the following view:

create view **assign** as

select e.ssn, p.pnumber, e.name, e.job_title, w.hours, p.pname, p.mgr_ssn, s.name

from employee e, project p, works_on w, employee s

where e.ssn = works_on.essn and works_on.pid = project.pnumber and p.mgr_ssn = s.ssn

That is a four-table join! Perhaps having a table select e.ssn, p.pnumber, e.name, e.job_title, w.hours, p.pname, p.mgr_ssn, s.name

from employee e, project p, works_on w, employee s

where e.ssn = works_on.essn and works_on.pid = project.pnumber and p.mgr_ssn = s.ssn

Similarly, our earlier

To retrieve a particular customer's invoice, we have to search three tables. Suppose we adopt instead a "bigger" table for invoice:

part2num, part2desc,part2quan, part3num, part3desc,part3quan, part4num, part4desc,part4quan

That is,

This means that most queries about the database can be displayed by examining only a single table. If this is a web interface, there might be a button users could click to "see full order", but the odds are it would seldom be clicked, especially if users discovered that response was slow.

The important thing for the software developers is to make sure that the

select i.invoicenum, i.cust_id, i.cust_name,
c.cust_name, i.cust_phone, c.cust_phone

from invoice2 i join customer c on i.cust_id = c.cust_id

where i.cust_name <> c.cust_name or i.cust_phone <> c.cust_phone

from invoice2 i join customer c on i.cust_id = c.cust_id

where i.cust_name <> c.cust_name or i.cust_phone <> c.cust_phone

We might even keep both

The value invoice2.cust_phone might be regarded as redundant, but it also might be regarded as the phone

Denormalization can be regarded as a form of

Denormalization works best if all the original tables involved are too large to be stored in memory; the cost of a join with a table small enough to fit in memory is quite small. For this reason, the