I am trying to make sense of an example of 3NF decomposition using the 4-step algorithm mentioned by Ullman here, but I'm not understanding what my lecturer is doing with the last step (or, worse, I'm not understanding the algorithm itself).
I realize this is a bit of a newbie question, but I did all the googling but couldn't find anything illuminating and I've been sitting here scratching my head for a few hours now.
the algorithm in question is:
- Find a canonical cover F_c for F
- For each FD X->Y in F_c create a relation with schema XY
- Eliminate a relation if its schema is a subset of another
- If none of the schemas created so far contains a key or R add a relation schema containing a key of R.
Now, the example my lecturer gave us starts with
R<A,B,C,D,E,F> CE->A, C->D, A->B, D->BE, B->F, AD->CF
Fc is shown to be
C->A, C->D, A->B, D->B, D->E, B->F, AD->C
and the keys are C and AD.
All's well until now.
Now we apply step 2 of the algorithm.
R_i: A,B | D,B,E | F,B | C,D,A | A,D,C ---------+----------+----------+-----------+----------- F_i A->B | D->B, | B->F | C->D | AD->C | D->E | | C->A |
Now this is were it gets ugly. The lecturer remarks that the fifth one, R_5 is a subset (a trivial subset, say I) of the fourth and proceeds to... "merge" them, leaving us with
R_i: A,B | D,B,E | F,B | C,D,A ---------+----------+----------+----------- F_i A->B | D->B, | B->F | C->D | D->E | | C->A | | | AD->C - - - - - notice this!
Notice how AD->C popped up in F_4.
The question is simply: why?
Why didn't he remove the fifth column altogether?
Is he applying step 3 of the algorithm? But it only instructs to eliminate the offending schemas, it doesn't say anything about adding dependencies to other schemas.
Is he applying step 4 of the algorithm? But it says that the schemas must contain a key of R, not all of them (which would then make some sense).
Is this simply a trivial mistake? Is the final result correct anyway?