I have the following problem: R(ABCDEFG) and F ={ AB->CD, C->EF, G->A, G->F, CE ->F}. Clearly, B & G should be part of the key as they are not part of the dependent set. Further, BG+ = ABCDEFG, hence candidate key. Clearly, AB->CD violates BCNF. But when I follow the algorithm, I do not end up in any answer. Probably doing something wrong. Can anyone show me how to apply the algorithm correctly to reach the decomposition?
Thanks in advance.
First, you should calculate a canonical cover of the dependencies. In this case it is:
{ A B → C
A B → D
C → E
C → F
G → A
G → F } >
Since the (unique) candidate key is B G, no functional dependency satisfies the BNCF. Then, starting from any functional dependency X → Y that violates the BCNF, we calculate the closure of X, X+, and replace the original relation R<T,F> with two relations, R1<X+> and R2<T - X+ + X>. So, chosing in this case the dependency A B → C, we replace the original relation with:
R1 < (A B C D E F) ,
{ A B → C
A B → D
C → E
C → F} >
and:
R2 < (A B G) ,
{ G → A } >
Then we check each decomposed relation to see if it satisfies the BCNF, otherwise we apply recursively the algorithm.
In this case, for instance, in R1 the key is A B, so C -> E violates the BCNF, and we replace R1 with:
R3 < (C E F) ,
{ C → E
C → F } >
and:
R4 < (A B C D) ,
{ A B → C
A B → D } >
two relations that satisfy the BCNF.
Finally, since R2 too does not satisfy the BCNF (beacuse the key is B G), we decompose R2 in:
R5 < (A G) ,
{ G → A } >
and:
R6 < (B G) ,
{ } >
that are in BCNF.
So the final decomposition is constituted by the relations: R3, R4, R5, and R6. We can also note that the dependency G → F on the original relation is lost in the decomposition.
