1. Multivalued Dependency Tamer Abuelata

2. Introduction Goal in Databases: Goal in Databases: BCNF (Boyce Codd Normal Form) BCNF (Boyce Codd Normal Form) Losslessness Losslessness Dependency preservation Dependency preservation

3. Remember… Boyce Codd Normal Form (BCNF) eliminates all redundancy that can be discovered based on functional dependencies.

4. Issue Some relation schemas, even though they are in BCNF, do not seem to be sufficiently normalized. Some relation schemas, even though they are in BCNF, do not seem to be sufficiently normalized. They still contain repetitions They still contain repetitions

5. Case study Consider the bank database schema: cust_loan = (loan_number, cust_id, cust_name, cust_street, cust_city) This is BCNF because of the functional dependency: cust_id -> cust_name, cust_street cust_city And because cust_id is not a key for cust_loan

6. Case Study But what if some customers have several addresses? But what if some customers have several addresses? We no longer wish to enforce the func. dependency: cust_id ->cust_street cust_city But we still want to enforce cust_id -> cust_name

7. Case Study Following BCNF decomposition algorithm Following BCNF decomposition algorithm we get: R1 = (cust_id, cust_name) R2 = (loan_number, cust_id, cust_street, cust_city) (both in BCNF)

8. Case Study The issue Despite R2 in BCNF, there is redundancy. We repeat the address of each residence for each loan that the customer has.

9. Case Study We can therefore decompose further into: loan_cust_id = (loan_number, cust_id) cust_residence = (cust_id, cust_street, cust_city) But there is no constraint that lead us to do that. To deal with this, we need a few form of constraint: 4NF.

10. 4NF We can use multivalued dependencies to define the fourth normal form

11. 4NF A relation schema R is in fourth normal form with respect to a set D of functional and multivalued dependencies if, for all multivlued dependencies in D+ of the form A -->-> B at least one of the following holds: A -->-> B is a trivial multivalued dependency A -->-> B is a trivial multivalued dependency A is a superkey for schema R A is a superkey for schema R

12. Multivalued Dependency Requires that other tuples of a certain form be present in the relation. Requires that other tuples of a certain form be present in the relation. Also referred to as: Also referred to as: tuple-generating dependency

13. Example R relation schema, A and B follow the multivalued dependency: R relation schema, A and B follow the multivalued dependency: A -->-> B The relationship between A and B is independent of the relation between A and R – B The relationship between A and B is independent of the relation between A and R – B If A -->-> B is satisfied by all relations on R then If A -->-> B is satisfied by all relations on R then A -->-> B is a trivial multivalued dependency A -->-> B is a trivial multivalued dependency

14. Example Let’s reconsider Let’s reconsider R2 = (loan_number, cust_id, cust_street, cust_city) loan_numbercust_idcust_streetcust_city L-2399-123NorthRye L-2399-123MainManchester L-9315-106LakeHorseneck

15. Example loan_numbercust_idcust_streetcust_city L-2399-123NorthRye L-2399-123MainManchester L-9315-106LakeHorseneck We must repeat the loan number once for each address a customer has and we must repeat the address for each loan a customer has.

16. Example loan_numbercust_idcust_streetcust_city L-2399-123NorthRye L-2399-123MainManchester L-9315-106LakeHorseneck This repetition is unnecessary since the relationship between a customer and his address is independent of the relationship between that customer and a loan.

17. Example Therefore this relation is illegal Therefore this relation is illegal loan_numbercust_idcust_streetcust_city L-2399-123NorthRye L-2799-123MainManchester

18. Example loan_numbercust_idcust_streetcust_city L-2399-123NorthRye L-2799-123MainManchester To make it legal we should add tuples To make it legal we should add tuples (L23, 99-123, Main, Manchester) and (L27, 99-123, North, Rye)

19. Example loan_numbercust_idcust_streetcust_city L-2399-123NorthRye L-2799-123MainManchester L-2399-123MainManchester L-2799-123NorthRye Updated table (legal) Updated table (legal)

20. Example loan_numbercust_idcust_streetcust_city L-2399-123NorthRye L-2799-123MainManchester L-2399-123MainManchester L-2799-123NorthRye We want Cust_id -->-> cust_street cust_city We want Cust_id -->-> cust_street cust_city to hold

21. Conclusion We can use multivalued dependencies To test relations to determine whether they are legal under a given set of functional and multivalued dependencies To test relations to determine whether they are legal under a given set of functional and multivalued dependencies To specify constraints on the set of legal relations To specify constraints on the set of legal relations