13.1 Introduction

2. 13.1 Introduction • Multi-valued dependency (MVD)  4NF • Join dependency (JD)  5NF Advanced Database System

3. 13.2 Multi-valued Dependencies and Fourth Normal Form (1/5) • Example (See Fig. 13.1) Assumptions: 1. For a given course c, there can be any number m of corresponding teachers and any number n of corresponding texts (m>0, n>0). 2. Teachers and texts are quite independent of one another. 3. A given teacher or a given text can be associated with any number of courses. Advanced Database System

4. 13.2 Multi-valued Dependencies and Fourth Normal Form (2/5) • Eliminating the relation-valued attributes (See Fig. 13.2) CTX satisfies the constraint If tuples (c, t1, x1), (c, t2, x2) both appear then tuples (c, t1, x2), (c, t2, x1) both appear also. Advanced Database System

5. 13.2 Multi-valued Dependencies and Fourth Normal Form (3/5) • CTX involves a good deal of redundancy  update anomalies In fact, CTX is in BCNF, since it is “all key.” • Multi-valued dependencies (MVDs) are a generalization of functional dependencies. Two MVDs in CTX: Course →> Teacher Course →> Text • Multi-valued dependence: Let R be a relvar, and let A, B, and C be subsets of the attributes of R. Then B is multi-dependent on A, A→>B, if and only if, in every legal value of R, the set of B values matching a given AC value pair depends only on the A value and is independent of the C value. Advanced Database System

6. 13.2 Multi-valued Dependencies and Fourth Normal Form (4/5) • MVDs always go together in pairs. e.g., Course →> Teacher|Text • The two projections CT and CX do not involve any such MVDs. (See Fig. 13.3) Advanced Database System

7. 13.2 Multi-valued Dependencies and Fourth Normal Form (5/5) • Theorem: Let R{A, B, C} be a relvar, where A, B, and C are sets of attributes. Then R is equal to the join of its projections on {A, B} and {A, C} if and only if R satisfies the MVDs A→>B|C. • Fourth normal form: Relvar R is in 4NF if and only if, whenever there exist subsets A and B of the attributes of R such that the nontrivial MVD A→>B is satisfied, then all attributes of R are also functionally dependent on A. • In other words, the only nontrivial dependencies in R are of the form K→X. • If we start with a relvar involving two or more independent RVAs, it is better to separate the RVAs first. Advanced Database System

8. 13.3 Join Dependencies and Fifth Normal Form (1/5) • A relvar is “n-decomposable” if it can be nonloss-decomposed into n projections but not into m, where 1 < m and m < n. (See Fig. 13.4) Advanced Database System

9. 13.3 Join Dependencies and Fifth Normal Form (2/5) • The constraint (Constraint 3D): if the pair (s1, p1) appears in SP and the pair (p1, j1) appears in PJ and the pair (j1, s1) appears in JS then the triple (s1, p1, j1) appears in SPJ  if (s1, p1, j2), (s2, p1, j1), (s1, p2, j1) appear in SPJ then (s1, p1, j1) appears in SPJ also • A relvar will be n-decomposable for some n > 2 if and only if it satisfies some such (n-way) cyclic constraint. Advanced Database System

10. 13.3 Join Dependencies and Fifth Normal Form (3/5) • Join dependency: Let R be a relvar, and let A, B, ..., Z be subsets of the attributes of R. Then R satisfies the JD ＊{A, B, ..., Z} if and only if every legal value of R is equal to the join of its projections on A, B, ..., Z. • Relvar SPJ suffers from a number of update anomalies, anomalies that disappear when it is 3-decomposed. (See Fig. 13.5) Advanced Database System

11. 13.3 Join Dependencies and Fifth Normal Form (4/5) • An MVD is just a special case of a JD, or that JDs are a generalization of MVDs. • JDs are the most general form of dependency possible. • Fifth normal form: A relvar R is in 5NF, also called projection-join normal form (PJ/NF) if and only if every nontrivial join dependency that is satisfied by R is implied by the candidate keys of R. • Relvar SPJ is in 4NF, but not in 5NF. Relvar SPJ can be 3-decomposable and 3-decomposability is not implied by the fact that the combination {S#, P#, J#} is a candidate key. Advanced Database System

12. 13.3 Join Dependencies and Fifth Normal Form (5/5) • Example: ＊{ {S#, Sname, Status}, {S#, City} } This JD is implied by the fact that {S#} is a candidate key. • Example: ＊{ {S#, Sname}, {S#, Status}, {Sname, City} } This JD is implied by the fact that {S#} and {Sname} are both candidate keys. • Given a relvar R, we can tell if R is in 5NF as long as we know all candidate keys and all JDs in R. However, discovering all of those JDs might itself be a nontrivial exercise. • 5NF is the ultimate normal form with respect to projection and join. e.g., the supplier relvar S Advanced Database System

13. 13.4 The Normalization Procedure Summarized (1/3) • Given some 1NF relvar R and some set of FDs, MVDs, and JDs that apply to R, we systematically reduce R to a collection of “smaller” relvars that are equivalent to R. • The overall process: 1. 1NF  2NF: to eliminate FDs that are not irreducible 2. 2NF  3NF: to eliminate transitive FDs 3. 3NF  BCNF: to eliminate remaining FDs in which the determinant is not a candidate key. 4. BCNF  4NF: to eliminate MVDs that are not also FDs 5. 4NF  5NF: to eliminate JDs that are not implied by the candidate key. Advanced Database System

14. 13.4 The Normalization Procedure Summarized (2/3) • Several points: 1. Done in a nonloss way, and preferably in a dependency-preserving way as well. 2. There is a very attractive parallelism among the definitions of BCNF, 4NF, and 5NF. 3. The overall objectives: a. To eliminate certain kinds of redundancy b. To avoid certain update anomalies c. To produce a design that is “good” representation of the real world d. To simplify the enforcement of certain integrity constraints Advanced Database System

15. 13.4 The Normalization Procedure Summarized (3/3) 4. The normalization guidelines are only guidelines, and occasionally there might be good reasons for not normalizing “all the way.” 5. The notions of dependency and further normalization are semantic in nature. 6. The ideas of normalization are not a panacea. a. JDs, MVDs and FDs are not the only kinds of constraints that can arise in practice. b. The decomposition might not be unique. c. The BCNF and dependency preservation objectives can be in conflict. d. Not all redundancies can be eliminated in the normalization procedure. Advanced Database System

16. 13.5 A Note on Denormalization • It is often claimed that “denormalization” is necessary to achieve good performance. • Full normalization  lots of logically separate relvars  lots of physically separate stored files  lots of I/O • More specifically, the objective is to reduce the number of joins that need to be done at run time by doing some of those joins ahead of time, as part of the database design. • Example (See Fig. 13.6) Advanced Database System

17. 13.5 A Note on Denormalization (Cont.) • Some problems: a. Once we start denormalizing, it is not clear where we should stop. b. There can be retrieval problems too. e.g., Summarize P Per P{Color} Add Avg (Weight) As Avwt Summarize PSQ {P#, Color, Weight} Per PSQ{Color} Add Avg (Weight) As Avwt c. When we say that denormalization is good for performance, what we really mean is that it is good for the performance of specific applications Advanced Database System

18. 13.6 Orthogonal Design (A Digression) (See Fig. 13.7) • The Principle of Orthogonal Design (initial version): Within a given database, no two distinct base relvars should have overlapping meanings. Advanced Database System

19. 13.6 Orthogonal Design (A Digression) (Cont.) (See Fig. 13.8) • The Principle of Orthogonal Design (final version): Let A and B be distinct base relvars. Then there must not exist nonloss decompositions of A and B into A1, A2, …, Am and B1, B2, …, Bn such that some projection Ai in the set A1, A2, …, Am and some projection Bj in the set B1, B2, …, Bn have overlapping meanings. Advanced Database System

20. 13.7 Other Normal Forms • Dependency theory • Domain-key normal form • “Restriction-union” normal form (3,3)NF  4NF • Sixth normal form Advanced Database System

21. The End. Advanced Database System