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decomposed into R1(A, B) and R2(A, C, D, E) Now, neither of the dependencies in F contains only attributes from (A, C, D, E) so we might be misled into thinking R2 satis es BCNF In fact, there is a dependency AC D in F + (which can be inferred using the pseudotransitivity rule from the two dependencies in F ), which shows that R2 is not in BCNF Thus, we may need a dependency that is in F + , but is not in F , to show that a decomposed relation is not in BCNF An alternative BCNF test is sometimes easier than computing every dependency in F + To check if a relation Ri in a decomposition of R is in BCNF, we apply this test: For every subset of attributes in Ri , check that + (the attribute closure of under F ) either includes no attribute of Ri , or includes all attributes of Ri If the condition is violated by some set of attributes in Ri , consider the following functional dependency, which can be shown to be present in F + : ( + ) Ri The above dependency shows that Ri violates BCNF, and is a witness for the violation The BCNF decomposition algorithm, which we shall see in Section 762, makes use of the witness
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We are now able to state a general method to decompose a relation schema so as to satisfy BCNF Figure 713 shows an algorithm for this task If R is not in BCNF, we can decompose R into a collection of BCNF schemas R1 , R2 , , Rn by the algorithm The algorithm uses dependencies ( witnesses ) that demonstrate violation of BCNF to perform the decomposition The decomposition that the algorithm generates is not only in BCNF, but is also a lossless-join decomposition To see why our algorithm generates only lossless-join decompositions, we note that, when we replace a schema Ri with (Ri ) and ( , ), the dependency holds, and (Ri ) ( , ) = result := {R}; done := false; compute F + ; while (not done) do if (there is a schema Ri in result that is not in BCNF) then begin let be a nontrivial functional dependency that holds on Ri such that Ri is not in F + , and = ; result := (result Ri ) (Ri ) ( , ); end else done := true; Figure 713
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II Relational Databases
7 Relational Database Design
The McGraw Hill Companies, 2001
7
Relational-Database Design
We apply the BCNF decomposition algorithm to the Lending-schema schema that we used in Section 72 as an example of a poor database design: Lending-schema = (branch-name, branch-city, assets, customer-name, loan-number, amount) The set of functional dependencies that we require to hold on Lending-schema are branch-name assets branch-city loan-number amount branch-name A candidate key for this schema is {loan-number, customer-name} We can apply the algorithm of Figure 713 to the Lending-schema example as follows: The functional dependency branch-name assets branch-city holds on Lending-schema, but branch-name is not a superkey Thus, Lendingschema is not in BCNF We replace Lending-schema by Branch-schema = (branch-name, branch-city, assets) Loan-info-schema = (branch-name, customer-name, loan-number, amount) The only nontrivial functional dependencies that hold on Branch-schema include branch-name on the left side of the arrow Since branch-name is a key for Branch-schema, the relation Branch-schema is in BCNF The functional dependency loan-number amount branch-name holds on Loan-info-schema, but loan-number is not a key for Loan-info-schema We replace Loan-info-schema by Loan-schema = (loan-number, branch-name, amount) Borrower-schema = (customer-name, loan-number) Loan-schema and Borrower-schema are in BCNF Thus, the decomposition of Lending-schema results in the three relation schemas Branchschema, Loan-schema, and Borrower-schema, each of which is in BCNF These relation schemas are the same as those in Section 75, where we demonstrated that the resulting decomposition is both a lossless-join decomposition and a dependency-preserving decomposition The BCNF decomposition algorithm takes time exponential in the size of the initial schema, since the algorithm for checking if a relation in the decomposition satis es BCNF can take exponential time The bibliographical notes provide references to an
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