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Today’s class

Today’s class. 3NF Decomposition Multi-valued Dependencies (MVD) 4NF. Example 4 . Let R=R(A,B,C) with FDs F={ B C, ACB} Keys: {A,C}, {A,B} BC violates BCNF condition R1={B,C} with BC ({B} is the key) R2={B,A} What about ACB BCNF decomposition is not Dependency Preserving.

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Today’s class

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  1. Today’s class 3NF Decomposition Multi-valued Dependencies (MVD) 4NF

  2. Example 4 • Let R=R(A,B,C) with FDs F={ BC, ACB} • Keys: {A,C}, {A,B} • BC violates BCNF condition • R1={B,C} with BC ({B} is the key) • R2={B,A} • What about ACB • BCNF decomposition is not Dependency Preserving

  3. 3NF • Less stringent than BCNF • Dependency preserving at the cost of a little redundancy • A relation R is in 3NF iff every non-trivial FD X  Y is such that X is either a superkey or a member of key • Example 4 satisfies 3NF condition • Since ‘B’ is a part of a key ({A,B})

  4. Example 4 (Re-visited) • Let R=R(A,B,C) with FDs F={ BC,ACB} • Keys: {A,C}, {A,B} • BC, AC B satisfy 3NF condition • R is in 3NF • R may contain redundancy due to BC

  5. Note • 3NF decomposition is similar to 4NF decomposition • A relation is in 4NF => it is in 3NF

  6. Multi-valued Dependency – EX1 Consider the following relations Name  Street City

  7. Multi-valued Dependency – EX2 Consider the following relation CTB with NO FDs and the key is CTB. But with the constraint that the recommended texts for a course are independent of the instructor. CTB is in BCNF but contains redundancy

  8. MVD • The MVD X  Y is said to hold over R if, each X value is associated with a set of Y values and this set is ‘independent’ of the values in the other attributes • The MVD X  Y hold over r=r(R) implies if t1, t2 R and t1.X=t2.X, then there must be some t3r such that t3.XY=t1.XY and t3.Z=t2.Z • In previous example: C  T • FD restricts tuples where as MVD adds tuples • If (a, b1, c1), (a, b2, c2)  r=R(X,Y,Z) then (a, b1, c2) and (a, b2, c1) must be in r

  9. Theory of MVDs • Replication • X  Y => X  Y • Converse is not true • MVD complementation • If X  Y, then X  R-X-Y • In the given example • C  T => C  B • MVD transitivity • If X  Y and Y  Z, then X  Z-Y

  10. Note • Splitting rule does not hold for MVDs • Ex: name  street city • Keys are determined using FD’s only • AB  C then AB  AC (i.e. adding attributes from left is okay) • A MVD X  Y is non-trivial • None of Y is in X • R  X  Y

  11. 4NF • R is said to be in 4NF if for every non-trivial MVD X  Y that holds on R, X is a superkey • Refer Example 2 • CTB is not in 4NF as CT is a nontrivial MVD and C is not a key (given CTB is the key) • 4NF decomposition • R1 (C,T) is in 4NF • R2 (C,B) is in 4NF • 4NF forbids non-trivial MVD’s unless they are actually FD’s allowed by BCNF

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