Func8onal Dependencies

Transcription

1 ICS 321 Data Storage & Retrieval Func8onal Dependencies Prof. Lipyeow Lim Informa8on & Computer Science Department University of Hawaii at Manoa Lipyeow Lim - - University of Hawaii at Manoa 1

2 Example: Movies1 What are the keys for this rela8on? What if you ignore the column starname? Can starname be a key? Lipyeow Lim - - University of Hawaii at Manoa 2

3 Func8onal Dependency A func8onal dependency X - > Y holds over rela8on R if, for every allowable instance r of R: for all tuples t1,t2 in r, π X (t1) = π X (t2) implies π Y (t1) = π Y (t2) i.e., given two tuples in r, if the X values agree, then the Y values must also agree. (X and Y are sets of avributes.) An FD is a statement about all allowable instances. Must be iden8fied based on seman8cs of applica8on. Given some allowable instance r1 of R, we can check if it violates some FD f, but we cannot tell if f holds over R! K is a candidate key for R means that K - > R However, K - > R does not require K to be minimal! Lipyeow Lim - - University of Hawaii at Manoa 3

4 Keys & Superkeys A set of one or more avributes {A 1, A 2,... A n } is a key for a rela8on R if : 1. Those avributes func1onally determine all other avributes of the rela8on. 2. No proper subset of {A 1, A 2,... A n } func8onally determines all other avributes of R a key must be minimal. When a key consists of a single avribute A, we o`en say that A ( rather than {A} ) is a key. Superkey : a set of avributes that contain a key. Lipyeow Lim - - University of Hawaii at Manoa 4

5 FD Example: Movies1 What are the FDs for this rela8on? What are the keys for this rela8on? Can starname be a key? Lipyeow Lim - - University of Hawaii at Manoa 5

6 Reasoning about FDs Given some FDs, we can usually infer addi8onal FDs: ssn - > deptid, deptid - > building implies ssn - > building T implies S, or S follows from T Every rela8on instance that sa8sfies all the FDs in T also sa8sfies all the FDs in S S is equivalent to T The set of rela8on instances sa8sfying S is exactly the same as the set sa8sfying T Alterna8vely, S implies T AND T implies S Lipyeow Lim - - University of Hawaii at Manoa 6

7 Armstrong s Axioms Let X, Y, Z are sets of avributes: Reflexivity If X is a subset of Y, then Y - > X Augmenta1on If X - > Y, then XZ - > YZ for any Z Transi1vity If X - > Y and Y - > Z, then X - > Z These are sound and complete inference rules for FDs! Lipyeow Lim - - University of Hawaii at Manoa 7

8 Example: Armstrong s Axioms Hourly_Emps SSN Name Lot Ra8ng Hourly_Wages Hours_worked Aishoo Smiley Smethurst Guldu Madayan Reflexivity: If X is a subset of Y, then Y - > X SNLR is a subset of SNLRWH, SNLRWH - > SNLR Augmenta1on: If X - > Y, then XZ - > YZ for any Z S - > N, then SLR - > NLR Transi1vity: If X - > Y and Y - > Z, then X - > Z S - > R, R - > W, then S - > W Lipyeow Lim - - University of Hawaii at Manoa 8

9 Two More Rules Firstname Lastname DOB Address Telephone John Smith Sep Honolulu,HI Union / Combining If X Y and X Z, then X YZ Eg. FLD A and FLD T, then FLD AT Decomposi;on / Spli<ng If X YZ, then X Y and X Z Eg. FLD AT, then FLD A and FLD T Trivial FDs Right side is a subset of Le` side Eg. F F, FLD FD Does XY Z imply X Z and Y Z? Lipyeow Lim - - University of Hawaii at Manoa 9

10 Closure Implica;on: An FD f is implied by a set of FDs F if f holds whenever all FDs in F hold. f=a C is implied by F={ A B, B C} (using Armstrong s transi8vity) Closure F + : the set of all FDs implied by F Algorithm: start with F + =F keep adding new implied FDs to F + by applying the 5 rules ( Armstrong s Axioms + union + decomposi8on) Stop when F + does not change anymore. Lipyeow Lim - - University of Hawaii at Manoa 10

11 Example: Closure Firstname Lastname DOB Street CityState Zipcode Telephone John Smith Sep East West Rd. Honolulu,HI Given FLD is the primary key and C Z Find the closure: Start with { FLD FLDSCZT, C Z } Applying reflexivity, { FLD F, FLD L, FLD D, FLD FL, FLD LD, FLD DF, FLDSCZT FLD, } Applying augmenta8on, { FLDS FS, FLDS LS, } Applying transi8vity Applying union Applying decomposi8on Repeat un8l F + does not change Lipyeow Lim - - University of Hawaii at Manoa 11

12 AVribute Closure Compu8ng the closure of a set of FDs can be expensive. (Size of closure is exponen8al in # avrs!) Typically, we just want to check if a given FD X Y is in the closure of a set of FDs F. An efficient check: Compute aeribute closure of X (denoted X + ) wrt F: Set of all avributes A such that X A is in F + There is a linear 8me algorithm to compute this. Check if Y is in X + Does F = {A B, B C, C D E } imply A E? i.e, is A E in the closure F +? Equivalently, is E in A +? Lipyeow Lim - - University of Hawaii at Manoa 12