Normaliza)on and Func)onal Dependencies

Transcription

1 Normaliza)on and Func)onal Dependencies 1NF and 2NF Redundancy and Anomalies Func)onal Dependencies A9ribute Closure Keys and Super keys 3NF BCNF Minimal Cover Algorithm 3NF Synthesis Algorithm Decomposi)on of Tables

2 1NF A9ribute values are atomic This part is assumed for any rela)onal database Object-rela)onal extensions to the rela)onal model might violate 1NF depending on your defini)on of atomic Some)mes 1NF includes the requirement that a table has a primary key

3 2NF A table T is in 2NF If there are no non-trivial dependencies, X -> A, that lie in T, where X is a proper subset of a key and A is not a prime a9ribute No non-prime a9ribute is func)onally dependent on a proper subset of a key Some)mes this is phrased as no par)al key dependencies exists in the table

4 Redundancy and Anomalies Consider combining all Library tables into one table What a9ribute(s) could be the primary key for the table? Redundancy The name of an author will appear in many places (once for each loan of a copy wri9en by the author) Update Anomaly If the customers name changes it must be changed in many places Delete Anomaly If all loans for a copies of books wri9en by an author are deleted all informa)on about the author is lost. Why? Insert Anomaly A new author cannot be added to the database un)l at least one loan for a copy of a book wri9en by the author is added. Normaliza)on (3NF and BCNF) reduces redundancy and eliminates the anomalies described above.

5 Func)onal Dependencies In the following let le9ers late in the alphabet represent sets of a9ributes and le9ers early in the alphabet represent individual a9ributes Func)onal Dependencies (X -> A or X -> Y) are constraints on the data that can be entered into the database If the FD, X -> A, holds for a database then if t1 and t2 are tuples that contain the a9ributes X and a9ribute A (and possibly other a9ributes) and if the tuples have the same values for a9ributes X they must have the same value for a9ribute A

6 Func)onal Dependencies (FDs) FDs can entail or imply other FDs (Armstrong s Axioms) Reflexivity: if Y is a subset of X then X -> Y Augmenta)on: if X -> Y then XZ -> YZ Transi)vity: if X -> Y and Y -> Z then X -> Z Union: if X -> A and X -> B then X -> AB Decomposi)on: if X -> AB then X -> A and X -> B The closure of a set of FDs, F, is designated by F + Two FD sets, F and G, are equivalent iff F + = G + Equivalency of two FD sets can be shown by showing that the FDs in F are implied by the FDs in G and the FDs in G are implied by the FDs in F

7 A9ribute Closure Find all a9ributes dependent on a par)cular set of a9ributes. The closure of a set of a9ributes, X, is designated by X +

8 A9ribute Closure Algorithm Under FD Set F closure := X; // since X X + repeat old := closure; if there is an FD Z V in F such that Z closure then closure := closure V un,l old = closure If T closure then X T is implied by F

9 Problem Let R = {A, B, C, D, E, F} Let the FD set be ABF C CF B CD A BD AE C F B F Find the closure of ABC

10 Keys and Super Keys A set of a9ributes, X, in a super key for a table T if X T and X -> T Another way of saying this is that T X + A set of a9ributes, X, in a key for a table T if it has the super key property and no proper subset of X has the super key property

11 Problem Let R = {A, B, C, D, E, F} Let the FD set be ABF C CF B CD A BD AE C F B F Is ABF a super key for R? Is ABD a super key for R? What a9ribute must be part of any key for R?

12 3NF A table T is in 3NF if for all non-trivial dependencies, X -> A, that lie in T, X is a super key or A is a prime a9ribute An FD is a 3NF violator for table T if it is a non-trivial dependency, X -> A, that lies in the T where X is not a super key and A is not a prime a9ribute. A prime a9ribute is an a9ribute that is part of some key

13 BCNF A table T is in BCNF if for all non-trivial dependencies, X -> A, that lie in T, X is a super key An FD is a BCNF violator for table T if it is a non-trivial dependency, X -> A, that lies in T where X is not a super key.

14 Create 3NF Tables Iden)fy all a9ributes, R, and FDs, F A table containing all a9ributes in R is called the universal table The designers must work with the customers to iden)fy R and F The FDs in F represent real world constraints of the data that can be entered into the database Create a minimal cover FD set, G, from F Apply the 3NF synthesis algorithm using the FD set G and the set of a9ributes R

15 Minimal Cover Set A minimal cover set, G, of an FD set F is an FD set such that G is equivalent to F No FD can be removed from G to create a smaller FD set equivalent to F No FD in G can have an a9ribute removed from the FD to create a smaller FD set equivalent to F Minimal cover sets are not unique

16 Minimal Cover Algorithm for FD Set F Step 1: Make all RHS single a9ributes Use decomposi)on of RHS on all FDs Step 2: Remove redundant a9ributes from LHS G = F repeat old = G for each XB -> A G if X -> A is implied by G (i.e. A X + ) then G = G {XB -> A } {X -> A} un)l old == G (i.e. keep going un)l G does not change)

17 Minimal Cover Algorithm for FD Set F Step 3: Remove redundant FDs from G (G was produced in step 2) H = G repeat old = H For each X -> A H if H is equivalent to H {X -> A} (i.e. A X + where X + is found using FD set H {X -> A}) then H = H {X -> A} un)l old == H (i.e. keep going un)l H does not change)

18 Minimal Cover Algorithm for FD Set F Step 4: Combine FDs that have the same LHS Use the Union rule Some)mes this step is considered part of the 3NF synthesis algorithm

19 Problem Let R = {A, B, C, D, E, F} Let the FD set be ABF C CF B CD A BD AE C F B F Find a minimal cover FD set.

20 3NF Synthesis Algorithm Input: Set of a9ributes R and FDs F Step 1: Create a minimal cover for F called G Step 2. For each FD in G create a table. Call the tables T 1,T 2, Step 3: If none of the T i contain a super key for the universal table create a new table containing the a9ributes of a key for the universal table

21 Problem Let R = {A, B, C, D, E, F} Let the FD set be ABF C CF B CD A BD AE C F B F Create a set of 3NF tables from R and the FD set.

22 Decomposi)on of Tables Lossless Decomposi)on A decomposi)on of T into T1 and T2 is a lossless if and only if T1 T2 -> T1 or T1 T2 -> T2 A decomposi)on of T into T1 and T2 is a lossless if for every valid T (valid rela)ve to the FDs) T = T1 Natural Join T2

23 Decomposi)on of Tables Dependency Preserving Decomposi)on Let T 1 and T 2 be a decomposi)on of T with FD set F Let F 1 and F 2 be the FDs from F + that lie in T 1 and T 2 respec)vely The decomposi)on is dependency preserving if and only if F + = F 1 F 2

24 Decomposi)on of Tables The 3NF synthesis algorithm is equivalent to a series of lossless, dependency preserving decomposi)ons into a set of 3NF tables A lossless decomposi)on of the universal table into a set of BCNF tables is possible but the decomposi)on might not be dependency preserving

25 Decomposi)on of Tables To remove a 3NF or BCNF violator through decomposi)on do the following Let T contain a9ributes X, a9ributes Y and a9ribute A Let X -> A be violator that lies in T Decompose T into T1 and T2 where T1 contains a9ributes X and a9ribute A and T2 contains a9ributes X and a9ributes Y The decomposi)on is lossless because X = T1 T2 and X is a super key for T1

26 Problem Let R = ABCDEFGH Let the FD set be A E BE D AD BE BDH E AC E F A E B D H BG F CD A

27 Problem Find keys for the universal table Create a minimal cover FD set Create a set of 3NF tables If any of the tables are not in BCNF decompose them into BCNF tables

28 Problem Universal table {A,B,C,D,E,G,H,K,L,M} FDs ABE -> CK AB -> D C -> BE EG -> DHK D -> L DL -> EK KL -> DM

29 Problem Find keys for the universal table Create a minimal cover FD set Create a set of 3NF tables If any of the tables are not in BCNF decompose them into BCNF tables