Databases 2

1. Databases 2 Storage optimization: functionaldependencies, normalforms, datawarehouses

2. Functional Dependencies • X -> A is an assertion about a relation R that whenever two tuples of R agree on all the attributes of X, then they must also agree on the attribute A. • Say “X -> A holds in R.” • Notice convention: …,X, Y, Z represent sets of attributes; A, B, C,… represent single attributes. • Convention: no set formers in sets of attributes, just ABC, rather than {A,B,C }.

3. Example • Drinkers(name, addr, beersLiked, manf, favBeer). • Reasonable FD’s to assert: • name -> addr (the name determine the address) • name -> favBeer (the name determine the favourite beer) • beersLiked -> manf (every beer has only one manufacturer)

4. Example name -> addr BeersLiked -> manf name -> FavBeer

5. FD’s With Multiple Attributes • No need for FD’s with more than one attribute on right. • But sometimes convenient to combine FD’s as a shorthand. • Example: name -> addr and name -> favBeer become name -> addr favBeer • More than one attribute on left may be essential. • Example: bar beer -> price

6. Keys of Relations • K is a key for relation R if: • Set K functionally determines all attributes of R • For no proper subset of K is (1) true. • If K satisfies (1), but perhaps not (2), then K is a superkey. • Consequence: there are no two tuples having the same value in every attribute of the key. • Note E/R keys have no requirement for minimality, as in (2) for relational keys.

7. Example • Consider relation Drinkers(name, addr, beersLiked, manf, favBeer). • {name, beersLiked} is a superkey because together these attributes determine all the other attributes. • name -> addr favBeer • beersLiked -> manf

8. Example Every pair is different => rest of the attributes are determined

9. Example, Cont. • {name, beersLiked} is a key because neither {name} nor {beersLiked} is a superkey. • name doesn’t -> manf; beersLiked doesn’t -> addr. • In this example, there are no other keys, but lots of superkeys. • Any superset of {name, beersLiked}.

10. Example name doesn’t -> manf BeersLiked doesn’t -> addr

11. E/R and Relational Keys • Keys in E/R are properties of entities • Keys in relations are properties of tuples. • Usually, one tuple corresponds to one entity, so the ideas are the same. • But --- in poor relational designs, one entity can become several tuples, so E/R keys and Relational keys are different.

12. Example name, Beersliked relational key Beers Drinkers In E/R name is a key for Drinkers, and beersLiked is a keyfor Beers

13. Where Do Keys Come From? • We could simply assert a key K. Then the only FD’s are K -> A for all atributes A, and K turns out to be the only key obtainable from the FD’s. • We could assert FD’s and deduce the keys by systematic exploration. • E/R gives us FD’s from entity-set keys and many-one relationships.

14. Armstrong’s axioms • Let X,Y,Z  R (i.e. X,Y,Z are attribute sets of R) Armstrong’s axioms: • Reflexivity: if Y  X then X -> Y • Augmentation: if X -> Y then for every Z: XZ -> YZ • Transitivity: if X -> Y and Y -> Z then X -> Z

15. FD’s From “Physics” • While most FD’s come from E/R keyness and many-one relationships, some are really physical laws. • Example: “no two courses can meet in the same room at the same time” tells us: hour room -> course.

16. Inferring FD’s: Motivation • In order to design relation schemas well, we often need to tell what FD’s hold in a relation. • We are given FD’s X1 -> A1, X2 -> A2,…, Xn -> An , and we want to know whether an FD Y -> B must hold in any relation that satisfies the given FD’s. • Example: If A -> B and B -> C hold, surely A -> C holds, even if we don’t say so.

17. Inference Test • To test if Y -> B, start assuming two tuples agree in all attributes of Y. • Use the given FD’s to infer that these tuples must also agree in certain other attributes. • If B is eventually found to be one of these attributes, then Y -> B is true; otherwise, the two tuples, with any forced equalities form a two-tuple relation that proves Y -> B does not follow from the given FD’s.

18. Closure Test • An easier way to test is to compute the closure of Y, denoted Y +. • Basis: Y+ = Y. • Induction: Look for an FD’s left side X that is a subset of the current Y+. If the FD is X -> A, add A to Y+.

19. X A new Y+ Y+

20. Finding All Implied FD’s • Motivation: “normalization,” the process where we break a relation schema into two or more schemas. • Example: ABCD with FD’s AB ->C, C ->D, and D ->A. • Decompose into ABC, AD. What FD’s hold in ABC ? • Not only AB ->C, but also C ->A !

21. Basic Idea • To know what FD’s hold in a projection, we start with given FD’s and find all FD’s that follow from given ones. • Then, restrict to those FD’s that involve only attributes of the projected schema.

22. Simple, Exponential Algorithm • For each set of attributes X, compute X+. • Add X ->A for all A in X + - X. • However, drop XY ->A whenever we discover X ->A. • Because XY ->A follows from X ->A. • Finally, use only FD’s involving projected attributes.

23. A Few Tricks • Never need to compute the closure of the empty set or of the set of all attributes: • ∅ += ∅ • R + =R • If we find X + = all attributes, don’t bother computing the closure of any supersets of X: • X + = R and X  Y => Y + = R

24. Example • ABC with FD’s A ->B and B ->C. Project onto AC. • A +=ABC ; yields A ->B, A ->C. • We do not need to compute AB + or AC +. • B +=BC ; yields B ->C. • C +=C ; yields nothing. • BC +=BC ; yields nothing.

25. Example, Continued • Resulting FD’s: A ->B, A ->C, and B ->C. • Projection onto AC : A ->C. • Only FD that involves a subset of {A,C }.

26. A Geometric View of FD’s • Imagine the set of all instances of a particular relation. • That is, all finite sets of tuples that have the proper number of components. • Each instance is a point in this space.

27. Example: R(A,B) {(1,2), (3,4)} {(5,1)} {} {(1,2), (3,4), (1,3)}

28. An FD is a Subset of Instances • For each FD X -> A there is a subset of all instances that satisfy the FD. • We can represent an FD by a region in the space. • Trivial FD : an FD that is represented by the entire space. • Example: A -> A.

29. Example: A -> B for R(A,B) A -> B {(1,2), (3,4)} {(5,1)} {} {(1,2), (3,4), (1,3)}

30. Representing Sets of FD’s • If each FD is a set of relation instances, then a collection of FD’s corresponds to the intersection of those sets. • Intersection = all instances that satisfy all of the FD’s.

31. Instances satisfying A->B, B->C, and CD->A Example A->B B->C CD->A

32. Implication of FD’s • If an FD Y -> B follows from FD’s X1 -> A1,…, Xn -> An , then the region in the space of instances for Y -> B must include the intersection of the regions for the FD’s Xi-> Ai . • That is, every instance satisfying all the FD’s Xi-> Ai surely satisfies Y -> B. • But an instance could satisfy Y -> B, yet not be in this intersection.

33. Example B->C A->C A->B

34. Normalization: Anomalies • Goal of relational schema design is to avoid anomalies and redundancy. • Update anomaly : one occurrence of a fact is changed, but not all occurrences. • Deletion anomaly : valid fact is lost when a tuple is deleted.

35. Example of Bad Design Drinkers(name, addr, beersLiked, manf, favBeer) Data is redundant, because each of the ???’s can be figuredout by using the FD’s name -> addrfavBeer andbeersLiked -> manf.

36. This Bad Design AlsoExhibits Anomalies • Update anomaly: if Janeway is transferred to Intrepid,will we remember to change each of her tuples? • Deletion anomaly: If nobody likes Bud, we lose trackof the fact that Anheuser-Busch manufactures Bud.

37. Boyce-Codd Normal Form • We say a relation R is in BCNF: if whenever X ->A is a nontrivial FD that holds in R, X is a superkey. • Remember: nontrivial means A is not a member of set X. • Remember, a superkey is any superset of a key (not necessarily a proper superset).

38. Example • Drinkers(name, addr, beersLiked, manf, favBeer) • FD’s: name->addr favBeer, beersLiked->manf • Only key is {name, beersLiked}. • In each FD, the left side is not a superkey. • Any one of these FD’s shows Drinkers is not in BCNF

39. Another Example • Beers(name, manf, manfAddr) • FD’s: name->manf, manf->manfAddr • Only key is {name}. • name->manf does not violate BCNF, but manf->manfAddr does.

40. Decomposition into BCNF • Given: relation R with FD’s F. • Look among the given FD’s for a BCNF violation X ->B. • If any FD following from F violates BCNF, then there will surely be an FD in F itself that violates BCNF. • Compute X+. • Not all attributes, or else X is a superkey.

41. Decompose R Using X -> B • Replace R by relations with schemas: • R1 = X+. • R2 = (R – X+) U X. • Project given FD’s F onto the two new relations. • Compute the closure of F = all nontrivial FD’s that follow from F. • Use only those FD’s whose attributes are all in R1 or all in R2.

42. Decomposition Picture R1 R-X+ X X+-X R2 R

43. Example • Drinkers(name, addr, beersLiked, manf, favBeer) • F = name->addr, name -> favBeer, beersLiked->manf • Pick BCNF violation name->addr. • Close the left side: {name}+ = {name, addr, favBeer}. • Decomposed relations: • Drinkers1(name, addr, favBeer) • Drinkers2(name, beersLiked, manf)

44. Example, Continued • We are not done; we need to check Drinkers1 and Drinkers2 for BCNF. • Projecting FD’s is complex in general, easy here. • For Drinkers1(name, addr, favBeer), relevant FD’s are name->addr and name->favBeer. • Thus, name is the only key and Drinkers1 is in BCNF.

45. Example, Continued • For Drinkers2(name, beersLiked, manf), the only FD is beersLiked->manf, and the only key is {name, beersLiked}. • Violation of BCNF. • beersLiked+ = {beersLiked, manf}, so we decompose Drinkers2 into: • Drinkers3(beersLiked, manf) • Drinkers4(name, beersLiked)

46. Example, Concluded • The resulting decomposition of Drinkers : • Drinkers1(name, addr, favBeer) • Drinkers3(beersLiked, manf) • Drinkers4(name, beersLiked) • Notice: Drinkers1 tells us about drinkers, Drinkers3 tells us about beers, and Drinkers4 tells us the relationship between drinkers and the beers they like.

47. Third Normal Form - Motivation • There is one structure of FD’s that causes trouble when we decompose. • AB ->C and C ->B. • Example: A = street address, B = city, C = zip code. • There are two keys, {A,B } and {A,C }. • C ->B is a BCNF violation, so we must decompose into AC, BC.

48. We Cannot Enforce FD’s • The problem is that if we use AC and BC as our database schema, we cannot enforce the FD AB ->C by checking FD’s in these decomposed relations. • Example with A = street, B = city, and C = zip on the next slide.

49. Join tuples with equal zip codes. street city zip 545 Tech Sq. Cambridge 02138 545 Tech Sq. Cambridge 02139 An Unenforceable FD street zip 545 Tech Sq. 02138 545 Tech Sq. 02139 city zip Cambridge 02138 Cambridge 02139 Although no FD’s were violated in the decomposed relations, FD street city -> zip is violated by the database as a whole.

50. 3NF Let’s Us Avoid This Problem • 3rd Normal Form (3NF) modifies the BCNF condition so we do not have to decompose in this problem situation. • An attribute is prime if it is a member of any key. • X ->A violates 3NF if and only if X is not a superkey, and also A is not prime.