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Heurstic 1 for finding keys

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• Heursstic 1: Find all keys - increase/decrease until key
  • Problem:

    Find all keys in the following relations:

    R(A, B, C, D, E, F) ℉ = { A → BC BD → EF F → A }

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  • Heuristic
    • Pretend that the LHS (left hand side) and RHS of → are nutrients (vitamins :-)).

      The items on the LHS of → are nutrients that you consume The items on the RHS of → are nutrients that your body acquires

      Obviously: the following facts are true

      A → A B → B C → C D → D E → E F → F

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      The set of functional dependencies ℉ are the added benefits :

      ℉ = { A → BC BD → EF F → A }

      • Consuming A will also acquires BC
      • Consuming BD will also acquires EF
      • Consuming F will also acquires A

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      Problem statement:

      acquire every nutrient (A, B, C, D, E, F) using the minimum number of intakes

    How do you go set up your minimal menu ?

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    Step 1: compute closure of all functional dependencies in ℉:

    A+ = A, B, C BD+ = A, B, C, D, E, F F+ = F, A, B, C

    Step 2:

    • If deficient , then add missing attributes ("nutrients") to the LHS until the closure is equal to the relation (complete set) or
    • If sufficient , then remove extraneous attributes ("nutrients") from the LHS until the set is minimal

    Example:

    A+ = A, B, C Deficient : Try adding any combinations of: D, E or F Add 1 attribute: AD+ = ABCDEF key ! AE+ = ABCE AF+ = ABCF Add 2 attributes, but do not include D: AEF+ = ABCEF Add 3 attributes, but do not include D: Not possible - done. BD+ = A, B, C, D, E, F Sufficient : Try deleting any combinations of: B or D Delete 1 attribute: B+ = B D+ = D Delete 2 attributes: Empty set - done. Conclusion: BD is minimal (is a key) F+ = F, A, B, C Deficient : Try adding any combinations of: D or E Add 1 attribute: DF+ = ABCDEF key ! EF+ = ABCEF Add 2 attributes, but do not include F: Not possible - done.

    Done !!!

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  • The keys of the relation R are:

    AD, BD, DF