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Silberschatz Korth Sudarshan: Database System Concepts, Fourth Edition
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II Relational Databases
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7 Relational Database Design
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Some relation schemas, even though they are in BCNF, do not seem to be suf ciently normalized, in the sense that they still suffer from the problem of repetition of information Consider again our banking example Assume that, in an alternative design for the bank database schema, we have the schema BC-schema = (loan-number, customer-name, customer-street, customer-city) The astute reader will recognize this schema as a non-BCNF schema because of the functional dependency customer-name customer-street customer-city that we asserted earlier, and because customer-name is not a key for BC-schema However, assume that our bank is attracting wealthy customers who have several addresses (say, a winter home and a summer home) Then, we no longer wish to enforce the functional dependency customer-name customer-street customer-city If we remove this functional dependency, we nd BC-schema to be in BCNF with respect to our modi ed set of functional dependencies Yet, even though BC-schema is now in BCNF, we still have the problem of repetition of information that we had earlier To deal with this problem, we must de ne a new form of constraint, called a multivalued dependency As we did for functional dependencies, we shall use multivalued dependencies to de ne a normal form for relation schemas This normal form, called fourth normal form (4NF), is more restrictive than BCNF We shall see that every 4NF schema is also in BCNF, but there are BCNF schemas that are not in 4NF
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781 Multivalued Dependencies
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Functional dependencies rule out certain tuples from being in a relation If A B, then we cannot have two tuples with the same A value but different B values Multivalued dependencies, on the other hand, do not rule out the existence of certain tuples Instead, they require that other tuples of a certain form be present in the relation For this reason, functional dependencies sometimes are referred to as equalitygenerating dependencies, and multivalued dependencies are referred to as tuplegenerating dependencies Let R be a relation schema and let R and R The multivalued dependency holds on R if, in any legal relation r(R), for all pairs of tuples t1 and t2 in r such that t1 [ ] = t2 [ ], there exist tuples t3 and t4 in r such that t1 [ ] = t2 [ ] = t3 [ ] = t4 [ ] t3 [ ] = t1 [ ] t3 [R ] = t2 [R ] t4 [ ] = t2 [ ] t4 [R ] = t1 [R ]
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Silberschatz Korth Sudarshan: Database System Concepts, Fourth Edition
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II Relational Databases
7 Relational Database Design
The McGraw Hill Companies, 2001
7
Relational-Database Design
t1 t2 t3 t4
a1 ai a1 ai a1 ai a1 ai
ai + 1 aj bi + 1 bj ai + 1 aj bi + 1 bj
R aj + 1 an bj + 1 bn bj + 1 bn aj + 1 an
Tabular representation of
This de nition is less complicated than it appears to be Figure 716 gives a tabular picture of t1 , t2 , t3 , and t4 Intuitively, the multivalued dependency says that the relationship between and is independent of the relationship between and R If the multivalued dependency is satis ed by all relations on schema R, then is a trivial multivalued dependency on schema R Thus, is trivial if or = R To illustrate the difference between functional and multivalued dependencies, we consider the BC-schema again, and the relation bc (BC-schema) of Figure 717 We must repeat the loan number once for each address a customer has, and we must repeat the address for each loan a customer has This repetition is unnecessary, since the relationship between a customer and his address is independent of the relationship between that customer and a loan If a customer (say, Smith) has a loan (say, loan number L-23), we want that loan to be associated with all Smith s addresses Thus, the relation of Figure 718 is illegal To make this relation legal, we need to add the tuples (L-23, Smith, Main, Manchester) and (L-27, Smith, North, Rye) to the bc relation of Figure 718 Comparing the preceding example with our de nition of multivalued dependency, we see that we want the multivalued dependency customer-name customer-street customer-city to hold (The multivalued dependency customer-name loan-number will do as well We shall soon see that they are equivalent) As with functional dependencies, we shall use multivalued dependencies in two ways: 1 To test relations to determine whether they are legal under a given set of functional and multivalued dependencies 2 To specify constraints on the set of legal relations; we shall thus concern ourselves with only those relations that satisfy a given set of functional and multivalued dependencies loan-number L-23 L-23 L-93 Figure 717 customer-name Smith Smith Curry customer-street North Main Lake customer-city Rye Manchester Horseneck
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