CS542 1 Schema Refinement Chapter 19 (part 1) Functional Dependencies

CS542 2 The Evils of Redundancy  Redundancy is at the root of several problems associated with relational schemas:  redundant storage,  insert/delete/update anomalies  Main refinement technique:  decomposition  Example: replacing ABCD with, say, AB and BCD,  Functional dependeny constraints utilized to identify schemas with such problems and to suggest refinements.

CS542 3 The Evils of Redundancy  Redundancy is at the root of several problems associated with relational schemas:  redundant storage,  insert/delete/update anomalies  Functional dependeny constraints utilized to identify schemas with such problems and to suggest refinements.  Main refinement technique:  decomposition  Example: replacing ABCD with, say, AB and BCD,  Decomposition should be used judiciously:  Is there reason to decompose a relation?  What problems (if any) does the decomposition cause?

CS542 4 Insert Anomaly sNumbersNamepNumberpName s1Davep1prof1 s2Gregp2prof2 Student Question : How do we insert a professor who has no students? Insert Anomaly: We are not able to insert “valid” value/(s)

CS542 5 Delete Anomaly sNumbersNamepNumberpName s1Davep1MM s2Gregp2ER Student Question : Can we delete a student that is the only student of a professor ? Delete Anomaly: We are not able to perform a delete without losing some “valid” information.

CS542 6 Update Anomaly sNumbersNamepNumberpName s1Davep1MM s2Gregp1MM Student Question : How do we update the name of a professor? Update Anomaly: To update a value, we have to update multiple rows. Update anomalies are due to redundancy.

CS542 7 Functional Dependencies (FDs)  A functional dependency X Y holds over relation R if, for every allowable instance r of R: t1 r, t2 r, ( t1 ) = ( t2 ) implies ( t1 ) = ( t2 )  Given two tuples in r, if the X values agree, then the Y values must also agree.

CS542 8 FD Example sNumbersNameaddress 1Dave144FL 2Greg320FL Student Suppose we have FD sName  address for any two rows in the Student relation with the same value for sName, the value for address must be the same i.e., there is a function from sName to address

CS542 9 Note on Functional Dependencies  Similar as for constraints, an FD is a statement about all allowable relations :  Must be identified based on semantics of application.  Intuition:  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!

CS Keys + Functional Dependencies  Assume K is a candidate key for R  What does this imply about FD between K and R?  It means that K R !  Does K R require K to be minimal ?  No. Any superkey of R also functionally implies all attributes of R.

CS Example: Constraints on Entity Set  Consider relation obtained from Hourly_Emps:  Hourly_Emps ( ssn, name, lot, rating, hrly_wages, hrs_worked )  Notation :  We denote relation schema by its attributes: SNLRWH  This is really the set of attributes {S,N,L,R,W,H}.  Some FDs on Hourly_Emps:  ssn is the key: S SNLRWH  rating determines hrly_wages : R W

CS Problems Caused by FD  Problems due to Example FD :  rating determines hrly_wages : R W

CS Example  Problems due to R W :  Update anomaly : Can we change W in just the 1st tuple of SNLRWH?  Insertion anomaly : What if we want to insert an employee and don’t know the hourly wage for his rating?  Deletion anomaly : If we delete all employees with rating 5, we lose the information about the wage for rating 5! Hourly_Emps rating (R) determines hrly_wages (W)

CS Same Example  Problems due to R W ! Hourly_Emps2 Wages Solution : 2 smaller tables instead of one big one ! Hourly_Emps

CS Same Example  Problems due to R W ! Hourly_Emps2 Wages Will 2 smaller tables be better than one big one? Update anomaly: Can we change W in just the 1st tuple of SNLRWH? Insertion anomaly: What if we want to insert an employee and don’t know the hourly wage for his rating? Deletion anomaly: If we delete all employees with rating 5, we lose information about wage for rating 5?

CS Reasoning About FDs  Given some FDs, we can usually infer additional FDs:  ssn did, did lot implies ssn lot  An FD f is implied by a set of FDs F if f holds whenever all FDs in F hold.  = closure of F is the set of all FDs that are implied by F.  Armstrong’s Axioms (X, Y, Z are sets of attributes):  Reflexivity : If X Y, then Y X  Augmentation : If X Y, then XZ YZ for any Z  Transitivity : If X Y and Y Z, then X Z  These are sound and complete inference rules for FDs!

CS Reasoning About FDs  Given some FDs, we can usually infer additional FDs:  ssn did, did lot implies ssn lot

CS Properties of FDs  Consider A, B, C, Z are sets of attributes Armstrong’s Axioms:  Reflexive (also trivial FD): if A  B, then A  B  Transitive : if A  B, and B  C, then A  C  Augmentation : if A  B, then AZ  BZ These are sound and complete inference rules for FDs! Additional rules (that follow from AA):  Union : if A  B, A  C, then A  BC  Decomposition : if A  BC, then A  B, A  C

CS Example : Reasoning About FDs Example: Contracts( cid,sid,jid,did,pid,qty,value ), and:  C is the key: C CSJDPQV  Project purchases each part using single contract: JP C  Dept purchases at most one part from a supplier: SD P Inferences:  JP C, C CSJDPQV imply JP CSJDPQV  SD P implies SDJ JP  SDJ JP, JP CSJDPQV imply SDJ CSJDPQV

CS Inferring FDs  What for ?  Suppose we have a relation R (A, B, C)  and functional dependencies : A  B, B  C, C  A  What is a key for R?  We can infer A  ABC, B  ABC, C  ABC.  Hence A, B, C are all keys.

CS Closure of FDs  An FD f is implied by a set of FDs F if f holds whenever all FDs in F hold.  = closure of F is set of all FDs that are implied by F.  Computing closure of a set of FDs can be expensive.  Size of closure is exponential in # attrs!

CS Reasoning About FDs (Contd.)  Computing the closure of a set of FDs can be expensive. (Size of closure is exponential in # attrs!)  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 attribute closure of X (denoted ) wrt F: Set of all attributes A such that X A is in There is a linear time algorithm to compute this.  Check if Y is in  Does F = {A B, B C, C D E } imply A E?  i.e, is A E in the closure ? Equivalently, is E in ?

CS Reasoning About FDs (Contd.)  Instead of computing full closure F+ of a set of FDs  Too expensive  Typically, we just need to know if a given FD X Y is in closure of a set of FDs F.  Algorithm for efficient check:  Compute attribute closure of X (denoted ) wrt F: Set of all attributes A such that X A is in There is a linear time algorithm to compute this.  Check if Y is in X+. If yes, then X  Y in F+.

CS Algorithm for Attribute Closure  Computing the closure of set of attributes {A1, A2, …, An}, denoted {A1, A2, …, An} + 1.Let X = {A1, A2, …, An} 2.If there exists a FD B1, B2, …, Bm  C, such that every Bi  X, then X = X  C 3.Repeat step 2 until no more attributes can be added. 4.Output X + = {A1, A2, …, An} +

CS Example : Inferring FDs  Given R (A, B, C), and FDs A  B, B  C, C  A,  What are possible keys for R ?  Compute the closure of attributes:  {A} + = {A, B, C}  {B} + = {A, B, C}  {C} + = {A, B, C}  So keys for R are,,

CS Another Example : Inferring FDs  Consider R (A, B, C, D, E)  with FDs F = { A  B, B  C, CD  E }  does A  E hold ?  Rephrase as :  Is A  E in the closure F+ ?  Equivalently, is E in A+ ?  Let us compute {A} +  {A} + = {A, B, C}  Conclude : E is not in A+, therefore A  E is false

CS Recap: So Far.  Functional Dependencies : Relationships across Attributes of Relations  Redundancy : Arises due to certain relationships (FDs) holding.  So far : Reasoning with FDs.  Approach : Establish certain “normal forms” with respect to dependencies