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Multivalued Dependencies

Multivalued Dependencies. by Asmerom Tekeste. Normal Forms. BCNF. 5NF. 2NF. 4NF. 3NF. 1NF. Functional dependencies. Multivalued dependencies. Join dependencies. Recall Normal Forms. Example :. R (A, B, C, D, E). FD: { B, C, D, E}. A. Normal Form of R (A,B,C,D,E).

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Multivalued Dependencies

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  1. Multivalued Dependencies by Asmerom Tekeste

  2. Normal Forms BCNF 5NF 2NF 4NF 3NF 1NF Functional dependencies Multivalued dependencies Join dependencies

  3. Recall Normal Forms Example: R (A, B, C, D, E) FD: { B, C, D, E} A

  4. Normal Form of R (A,B,C,D,E) 1NF: no multivalues 2NF: no FDs where a subset of the key to the relation is on the left 3NF: no non-trivial FDs, either the determinant is a superkey or the RHS of the FD is a member of some key BCNF: the determinant of any non-trivial FD is a superkey for the relation

  5. Normal Form (cont) Stars

  6. Normal Form (cont) • R T x U x V(R is similar to the Cartesian product of relations T, U, and V) ~ ~ Relation U Relation V Relation T - The relation Stars is in BCNF

  7. Problems with relation Stars The relation Stars doesn’t seem to be sufficiently normalized Reasons: • No reason to associate address with one movie and not • another • When we repeat address and movie facts in all combinations, • there is obvious redundancy • - Relation R contains unnecessary duplication of data

  8. Introducing MVD Definitions: • A MVD: A1A2…An B1B2…Bn for a Relation R is • “non- trival” if • 1. none of the Bs are among the As • 2. not all of the attributes of R are among the As and Bs • A MVD is “trivial” if • A1A2…An  B1B2…Bmwhere B1B2…Bm is a subset of A1A2…Anor (A1A2…AnU B1B2…Bn ) contains all attributes of R • A relation cannot be decomposed any further (under 4NF • rules) if it has a trivial MVD

  9. More on MVD Definitions(cont): A1A2…AnB1B2…Bm holds for relation R if: For all tuples t, u in R If t[A1A2...An] = u[A1A2...An], then there exists a v in R such that: (1) v[A1A2...An] = t[A1A2...An] = u[A1A2...An] (2) v[B1B2…Bm] = t[B1B2…Bm] (3) v[C1C2…Ck] = u[C1C2…Ck], where C1C2…Ck is all attributes in R except (A1A2...An  B1B2…Bm)

  10. More on MVD • Intuitively, A1A2…An  B1B2…Bm says that the relationship • between A1A2…An and B1B2…Bm is independent of the relationship • between A1A2…An and R -{B1B2…Bm} • - MVD's uncover situations where independent facts related to a certain object are being squished together in one relation • Functional dependenciesrule out certain tuples from being in a • relation • - if A  B, then we can’t have two tuples with the same A values but different B • - a.k.a. equality-generating dependencies • Multivalued dependenciesrequire that other tuples of a certain form be present in the relation • - a.k.a. tuple-generating dependencies

  11. MVD Example: is A  C

  12. Theories of MVD REPLICATION FD is an MVD If A1A2…An  B1B2…Bm then A1A2…An  B1B2…Bm holds COMPLEMENTATION If A1A2…An  B1B2…Bm then A1A2…An  C1C2…Ck where C1C2…Ck is all attributes in R except (A1A2…An  B1B2…Bm) AUGMENTATION If XY and WZ then WX YZ

  13. Theories of MVD (cont) TRANSITIVITY If XY and YZ then X  (Z-Y) REFLEXIVITY ifAis a set of attributes andA B, then ABholds. REPLICATION if A  Bholds, then AB.

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