CMSC424: Database Design

1. CMSC424: Database Design Lecture 9 CMSC424, Spring 2005

2. Relational Database Design • Find a “good” collection of relation schemas; otherwise • Repetition of Information. • Leads to anomalies • Inability to represent certain information. • Use integrity constraints to refine the schema • Main refinement technique: Decomposition • E.g. break ABCD into AB and BCD • Must be careful with decomposition CMSC424, Spring 2005

3. Decomposition • All attributes of an original schema (R) must appear in the decomposition (R1, R2): • R = R1  R2 • Lossless-join decomposition.For all possible legal relations r on schema R • r = R1 (r) R2 (r) • How do you define legal ? CMSC424, Spring 2005

4. Examples of Join Decompositions CMSC424, Spring 2005

5. Goal — Devise a Theory for the Following • Decide whether a particular relation R is in “good” form. • In the case that a relation R is not in “good” form, decompose it into a set of relations {R1, R2, ..., Rn} such that • each relation is in good form • the decomposition is a lossless-join decomposition • Our theory is based on: • functional dependencies • multivalued dependencies CMSC424, Spring 2005

6. Normal Forms • 1st, 2nd, 3rd, 4th Normal Forms • Boyce-Codd Normal Form (BCNF) CMSC424, Spring 2005

7. First Normal Form • Atomic Domains • Sometimes a function of how domain is used, rather than intrinsic properties of the domain • A set of values as an attribute not acceptable • Non-atomic values complicate storage and encourage redundant (repeated) storage of data • We will assume 1st normal form compliance CMSC424, Spring 2005

8. Rest of the normal forms… • Need: • functional dependencies • multi-valued dependencies • What are they ? • Constraints on the set of legal relations CMSC424, Spring 2005

9. Functional Dependencies • Let R be a relation schema •   R and   R • The functional dependency •  holds onR if and only if for any legal relations r (R): t1[] = t2 []  t1[ ] = t2 [ ] • Example: Consider r(A,B) with the following instance of r. • On this instance, AB does NOT hold, but BA does hold. • Beware: • Difference between holding in one instance, and holding in all legal relations • 4 • 1 5 • 3 7 CMSC424, Spring 2005

10. Functional Dependencies (Cont.) • Generalization of the notion of keys • K is a superkey for relation schema R if and only if K R • K is a candidate key for R if and only if • K R, and • for no   K,  R • Functional dependencies allow us to express constraints that cannot be expressed using superkeys. CMSC424, Spring 2005

11. Functional Dependencies • If a relation r is legal under a set F of functional dependencies, we say that rsatisfiesF. • We say that Fholds onR if all legal relations on R satisfy the set of functional dependencies F. • What’s the difference between r and R above ? CMSC424, Spring 2005

12. Functional Dependencies (Cont.) • A functional dependency is trivial if it is satisfied by all instances of a relation • E.g. • customer-name, loan-number customer-name • customer-name customer-name • In general,   is trivial if    CMSC424, Spring 2005

13. Closure of a Set of Functional Dependencies • Given a set F set of functional dependencies, there are certain other functional dependencies that are logically implied by F. • E.g. If A  B and B  C, then we can infer that A  C • F+ (closure of F) • Set of all functional dependencies logically implied by F • Armstrong’s Axioms: • if   , then   (reflexivity) • if  , then    (augmentation) • if  , and   , then   (transitivity) • These rules are • sound • complete CMSC424, Spring 2005

14. Example • R = (A, B, C, G, H, I)F = { A BA CCG HCG IB H} • some members of F+ • A H • AG I • CG HI • if   , then   (reflexivity) • if  , then    (augmentation) • if  , and   , then   (transitivity) CMSC424, Spring 2005

15. To compute the closure of a set of functional dependencies F: F+ = Frepeatfor each functional dependency f in F+ apply reflexivity and augmentation rules on fadd the resulting functional dependencies to F+for each pair of functional dependencies f1and f2 in F+iff1 and f2 can be combined using transitivitythen add the resulting functional dependency to F+until F+ does not change any further Procedure for Computing F+ CMSC424, Spring 2005

16. Closure of Functional Dependencies (Cont.) • We can further simplify manual computation of F+ by using the following additional rules. • If   holds and  holds, then    holds (union) • If    holds, then   holds and  holds (decomposition) • If   holds and   holds, then   holds (pseudotransitivity) • The above rules can be inferred from Armstrong’s axioms. CMSC424, Spring 2005

17. Closure of Attribute Sets • For an attriute set , + =Closureof underF • is the set of attributes that are functionally determined by  under F:is in F+   + • Algorithm to compute + : result := ;while (changes to result) do for each in F do begin if  result then result := result end CMSC424, Spring 2005

18. Example of Attribute Set Closure • R = (A, B, C, G, H, I) • F = {A BA C CG HCG IB H} • (AG)+ • 1. result = AG • 2. result = ABCG (A C and A  B) • 3. result = ABCGH (CG H and CG  AGBC) • 4. result = ABCGHI (CG I and CG  AGBCH) • Is AG a candidate key? Is AG a super key? • Does AG R? == Is (AG)+  R Is any subset of AG a superkey? • Does AR? == Is (A)+  R • Does GR? == Is (G)+  R CMSC424, Spring 2005

19. There are several uses of the attribute closure algorithm: Testing for superkey: To test if  is a superkey, we compute +, and check if +contains all attributes of R. Testing functional dependencies To check if a functional dependency    holds (or, in other words, is in F+), just check if   +. That is, we compute +by using attribute closure, and then check if it contains . Is a simple and cheap test, and very useful Computing closure of F For each   R, we find the closure +, and for each S  +, we output a functional dependency   S. Uses of Attribute Closure CMSC424, Spring 2005

20. Canonical Cover • Sets of functional dependencies may have redundant dependencies that can be inferred from the others • Eg: A  C is redundant in: {A  B, B  C, A  C} • Parts of a functional dependency may be redundant • E.g. on RHS: {A  B, B  C, A  CD} can be simplified to {A  B, B  C, A  D} • E.g. on LHS: {A  B, B  C, AC  D} can be simplified to {A  B, B  C, A  D} CMSC424, Spring 2005

21. Extraneous Attributes • Consider a set F of functional dependencies and the functional dependency   in F. • Attribute A is extraneous in  if A   and F logically implies (F – {})  {( – A) }. • Attribute A is extraneous in  if A  and the set of functional dependencies (F – {})  {(– A)} logically implies F. • Note: implication in the opposite direction is trivial in each of the cases above, since a “stronger” functional dependency always implies a weaker one CMSC424, Spring 2005

22. Testing if an Attribute is Extraneous • Consider a set F of functional dependencies and the functional dependency   in F. • To test if attribute A   is extraneousin compute ({} – A)+ using the dependencies in F check that ({} – A)+ contains A; if it does, A is extraneous • To test if attribute A  is extraneous in  compute + using only the dependencies in F’ = (F – {})  {(– A)}, check that + contains A; if it does, A is extraneous CMSC424, Spring 2005

23. Canonical Cover • A canonical coverfor F is a set of dependencies Fc such that • F logically implies all dependencies in Fc, and • Fclogically implies all dependencies in F, and • No functional dependency in Fccontains an extraneous attribute, and • Each left side of functional dependency in Fcis unique. • To compute a canonical cover for F:repeatUse the union rule to replace any dependencies in F11 and 12 with 112 Find a functional dependency  with an extraneous attribute either in  or in  If an extraneous attribute is found, delete it from until F does not change • Note: Union rule may become applicable after some extraneous attributes have been deleted, so it has to be re-applied CMSC424, Spring 2005

24. Example of Computing a Canonical Cover • R = (A, B, C)F = {A BC B C A BABC} • Combine A BC and A B into A BC • Set is now {A BC, B C, ABC} • A is extraneous in ABC • Check if the result of deleting A from ABC is implied by the other dependencies • Yes: in fact, BC is already present! • Set is now {A BC, B C} • C is extraneous in ABC • Check if A C is logically implied by A B and the other dependencies • Yes: using transitivity on A B and B  C. • Can use attribute closure of A in more complex cases • The canonical cover is: A B B C CMSC424, Spring 2005

25. Normalization Using Functional Dependencies • When we decompose a relation schema R with a set of functional dependencies F into R1, R2,.., Rn we want • Lossless-join decomposition • No redundancy • Dependency preservation CMSC424, Spring 2005

26. Example • R = (A, B, C)F = {A B, B C) • Can be decomposed in two different ways • R1 = (A, B), R2 = (B, C) • Lossless-join decomposition: • R1  R2 = {B} and B BC • Dependency preserving • R1 = (A, B), R2 = (A, C) • Lossless-join decomposition: • R1  R2 = {A} and A  AB • Not dependency preserving (cannot check B C without computing R1 R2) CMSC424, Spring 2005