BCNF Decomposition algorithm not working

Question

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.

Answer 1

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.