4.1.1 Design quality. 4.Normalization: Quality of relational designs. Informal guidelines (2) Informal guidelines (1)

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1 .Normalization: Quality of relational designs. Funtional Dependenies.. Design quality.. Update anomalies.. Funtional Dependenies: definition.. Properties of Funtional Dependenies. Normal forms.. Informal introdution.. Normal Forms and FDs.. Normal forms (NF, NF, BCNF, MV NF).. Lossless join and dependeny preservation.. Multivalued dependenies and NF. Algorithms for finding Normal Forms. Normal Forms: Critial review.. Design quality What is a good oneptual model? Usually many alternatives. No lear guidelines, est pratie. Wanted: Formal methods for omparing designs But Use ommon sense! Simple prolems have simple solutions! "Design is an art ut a siene" Lit: Kemper/Eikler: hap 6; Garia-Molina/Ullman/Widom: hap. ff.; Elmasr/Navathe: hap 0-DBS-NF- Informal guidelines () Avoid redundanies: CREATE TABLE employee ( p# serial, name VARCHAR(0),... qualifiation VARCHAR(0), PRIMARY KEY (p#, qualifiation)); More than one qualifiation, primitive values p# is not a key. Informal guidelines () Avoid modeling more than one ojet from reality in one entity / relation CREATE TABLE Experiment ( id SERIAL PRIMARY KEY, responsile_sientist VARCHAR(0), institute VARCHAR (0), phone INT, purpose VARCHAR(00), start TIMESTAMP, endtime TIMESTAMP, result INT) Dum design! 0-DBS-NF- 0-DBS-NF-.. Update Anomalies Redundanies may ause "anomalies" Deletion of a row may delete all data aout a different ojet Update of an attriute may ause update on many rows Insertion may e diffiult / impossile, sine data are missing Update anomalies: Examples CREATE TABLE Experiment ( id SERIAL PRIMARY KEY, responsile_sientist VARCHAR(0), institute VARCHAR (0), phone INT, purpose VARCHAR(00), start TIMESTAMP, endtime TIMESTAMP, result INT) DELETE FROM Experiment WHERE result < 0 Tale definition using SQL / DDL (PostgresSQL) "update anomaly": deletion, update or insertion anomaly Data on experimenter may e lost! 0-DBS-NF- 0-DBS-NF-6

2 Update anomaly Example update Experiment set phone = 778 where resonsile_sientist = '-Lüdensheid' all those experiment tuples the experimentor of whih was '-Lüdensheid' have to e hanged update anomaly What is an insertion anomaly?.. Funtions, Funtional dependenies Important formal onept: Funtions Funtional dependenies (FD) generalize key onept FDs used to formalize integrity onstraints on attriutes and relationships Experiment (id, responsile_sientist, institute, phone,...result) {responsile_sientist} -> {institute} is a funtion whih expresses the onstraint: "The same responsile_person annot have different affiliations (institutes)" 0-DBS-NF-7 0-DBS-NF-8 Formal notation of RDM Relation: R dom(a) x dom(a) x x dom(an) Attriute set: Σ(R) = {a, a,, an} = R A, signature Tuple: r R Degree of R: numer of attriutes Relation Shema: R(a, a,, an) Dataase shema: set of relation shemas different notations in use Funtional Dependenies and keys Property of a key : at most one row for eah value Given a key, i.e. values for the key attriutes, then the value of eah a Σ(R) is unique or all values are undefined. eah suset of Σ(R) is funtional dependent on K Terminology: Relation = tale ( file) ut relations are sets. tales may have dupliate entries Tuple = row, reord Attriute = field (omponent) 0-DBS-NF-9 fname Tina Anna Carla Student name Maus mueller@... mei@... piep@... matrno key For a given key value, there is a unique value for eah attriute e.g. {matrno} {fname} 0-DBS-NF-0 Funtional Dependenies Funtional and key dependenies: onstraints (invariants) of the appliation domain "Funtional dependeny" onstraints have to e identified during requirements engineering like all onstraints. Ultimate goal: DBS monitors ompliane with DB state. Experiment (id, responsile_sientist, institute, phone,...result) Motivation for Normal Forms Suppose we an find a relational shema whih only has keyindued funtional dependenies (FD) (and "trivial" ones like {a,} -> {} ) How an we effiiently hek the DB state after an update with respet to FD? Do they still hold? A "good" shema avoids Update anomalies Costly hek of funtional dependenies after update What has to e done, when a new experiment is inserted? 0-DBS-NF- 0-DBS-NF-

3 Funtional Dependeny: : Definition Def.: Funtional Dependenies (FDs) Let A = Σ(R)* = {a,,,...a i,..} e the attriute set of a relation R, e, e' tuples of R, and X, Y A, r, r' R Y is funtionally dependent on X ( written: X Y) : ( xi X ) r.xi = r.xi ( yi Y ) e.yi = e.yi As we know: invariants are independent of the partiular dataase state They must hold at all times, i.e. they restrit the valid states of the dataase.. FD Properties Trivial funtional dependeny X Y Y X Augmentation Z A=Σ(R), X Y XZ YZ Transitivity X,Y,Z A=Σ(R), X Y, Y Z X Z Proof? Notation XY -> Z means X Y -> Z * Σ(R) : Attriute set of relation R 0-DBS-NF- 0-DBS-NF- Implied and inferred FD A funtional dependeny Y Z is alled implied y a set F= {F,, Fn} of funtional dependenies, if Y Z an e proven from F. A funtional dependeny Y Z an e inferred y a set of inferene rules R = {r, rm} from set F = {F,, Fn} of funtional dependenies if Y Z an e onstruted y a finite numer of syntati transformations of F aording to rules ri Armstrong inferene rules Given a set of FDs, find all implied FD's A sound, omplete, minimal set (Armstrong axioms): Y X X Y (I: inlusion) {X Y, Y Z} X Z (T: transitivity) {X Y} XZ YZ (A: augmentation) Sound: Only implied FDs are onstruted y the inferene rules Complete: Every implied FD will e produed y a finite numer of inferenes 0-DBS-NF- 0-DBS-NF-6 Types of Funtional Dependenies Given shema signature Σ(R) = {a,,an} Primary key P = {p, p e } Set of andidate keys C = { {k,..,k f },,{k',,k' g }} Def.: K = P» C are alled prime (or key) attriutes i.e. attriutes elonging to any andidate key. S(R) \ K : non-prime (or non-key) attriutes Types of funtional dependenies:. Key dependenies. Partial dependenies on one of the andidate keys expl.: {p#} -> {name} sine key is {p#,qualifiation} Types of FDs. Dependenies among non-key attriutes expl.: {responsile_sientist} -> {institute}. Dependenies among attriutes of different andidate keys Roadmap Funtional dependenies may ause "update anomalies" Update anomalies ause troules find relational shema without "anomalies" in ase of update Define "Normal forms" for relations whih do not show (all) anomalies Given a set of funtional dependenies, find algorithm whih generates a relational shema in some normal form 0-DBS-NF-7 0-DBS-NF-8

4 . Normal Forms.. Definitions Def.: First normal form A relation is in NF : all attriutes are single valued and atomi Customer (_id, name,, {phone}, ) (,'Miller',, {76, 7899}, ) ontradits key dependeny property: every attriute is funtionally dependent on every andidate key Seond Normal Form (NF) Def.: R is in seond normal form (NF), : X Σ( R ), a Σ ( R ) : a X a is not a prime attriute X -> a X is a key or a superset of a key ut not a proper suset of any key of R This means: No non-prime attriute funtionally depends on only a part of a key ("No partial dependeny") Building(No, roomno, rseats, Adr, NoRooms, ) 0-DBS-NF-9 0-DBS-NF-0 Third Normal Form (NF) More on NF Def.: R is in third normal form (NF) : X Σ( R ), a Σ ( R ) : a X X -> a X ontains a key or a is prime There is no funtional dependeny etween non-prime attriutes (or attriute sets). Proof? Experiment (id, responsile_sientist, institute, phone,...result) ut: {responsile_sientist} {institute} not in NF Equivalent definition: R is in NF : no non-prime attriute depends transitively on a key. A non-prime attriute y is transitively dependent on a key K, if K X and X y and not X K Notation: K X y Experiment (id, responsile_sientist, institute,..,result) 0-DBS-NF- 0-DBS-NF- NF and NF Design quality and Normal Forms R is in NF R is in NF Proof: Suppose R not in NF exists andidate key K and X K and non-prime attriute a and X a (i.e. exists partial dependeny) Sine K is a key K X a, ontradition to NF Experiment (id, responsile_sientist, institute,..,result) Indiates the modeling of two different 'real world entities' as one relation Split into two relations: Experiment (id, purpose, start, responsile,..,result); Experimentor ( p_id, intitute, phone,..) 0-DBS-NF- 0-DBS-NF-

5 DESIGN QUALITY: what do we have?. Key dependenies NF. No partial dependenies on andidate keys NF. No dependenies among non-key attriutes NF. Dependenies among attriutes of different andidate keys?? there is at least one more Normal Form whih exludes FDs etween prime attriutes. Beyond third NF Dependenies among key attriutes Example (*): R(p, o, s, n) with keys {o,s,n} and {p,s,n}, FD p -> o R in NF, ut transitive dependeny involving key attriute o: {p,s,n} -> p -> o Proposition: If R is in NF and X is a proper suset of a andidate key K and X a for some attriute a a K Corrolar: If X a is a nontrivial FD of a relation in NF, and X not a superkey X and a elong to different andidate keys 0-DBS-NF- (*) Interpretation e.g.: PLZ, Ortsteil, Straße, Nummer (in Germany) Annahme: (Ort, Straße, Nr), (PLZ, Straße, Nr) eindeutig0-dbs-nf-6 Boye Codd Normal Form (BCNF) Def.: A relation R is in BCNF : if there is a non-trivial dependenies X a then X is a superkey of R Equivalent to: X a (i) trivial or (ii) X ontains a key of R Ovious onsequene: BCNF NF NF and BCNF NF more important in pratie than BCNF Partial dependenies of andidate keys infrequent R relation in NF and andidate keys have only one attriute eah R is in BCNF R in NF and at most one andidate key has more than one attriute R is in BCNF Proof? 0-DBS-NF-7 0-DBS-NF-8 BCNF vs NF Last proposition useful in many pratial situation: If a relation R has a multi-attriute key and a unique identifier (e.g. a sequene numer) then NF implies BCNF e.g. Customer ( ID, name, ity, street, no, disount, ) has keys {ID} and {name, ity, street, no}.. Lossless property and preserving dependenies Normalization (y deomposition) Given relation R having shema Σ(R) and FD = {X Y X,Y Σ(R)} set of FDs, Find a set R,..., R n of relations in NF / BCNF suh that: Σ(R) =» Σ(R i ) For eah f = X Y FD there exists R i suh that X Y Σ(R i ) "Dependeny preserving" R an e reonstruted from R i, i=..n 0-DBS-NF-9 "Lossless" 0-DBS-NF-0

6 Joining relations When relation R has een split into relations R, R,..., R n, reonstrution of R from R,..., R n y means of the join operator Join operation (natural join): onatenate those tupels of R and S whih have same name and same value. Eliminate the redundant attriute. Lossless property Criterion for preserved information (losslessness) : R R R n = R R a R = = "Natural join" a 0-DBS-NF- a R = a = a R 0-DBS-NF- Lossless joins But a FD = {a ->, -> } In general: Deomposition of R into R and R is lossless, if Σ(R) Σ(R) Σ(R) or Σ(R) Σ(R) Σ(R) a = a Lossless joins Lossless deomposition and keys Σ(R) Σ(R) -> Σ(R) or Σ(R) Σ(R) -> Σ(R) The ommon attriute(s) of R and R are a key (or a superset of a key) of R or R Funtional dependenies are transformed into key dependenies Invariane property expressed y FDs may now e heked y heking the primary key property - effiiently done y every DBS 0-DBS-NF- 0-DBS-NF- BCNF and NF BCNF does not always guarantees oth the lossless property and dependeny preservation R(p, o, s, n) with keys {o,s,n} and {p,s,n}, FD p -> o Normalisation to BCNF: R (p,s,n) and R( p,o) Dependeny (o,s,n) -> p is lost Consequene: Normalization to NF is the est to ahieve in general.. Multivalued dependenies and NF Example Hoies(name,affiliation,hoy) B TUB B skiing Assumption: a person an have one or more affiliations, and one or more hoies, e.g {.} {,} These rows must exist (MVD restrition) Two multivalued attriutes: affiliation, hoies, oth dependent on name. Introdue redundany MVD defines whih tuples must exist. 0-DBS-NF- 0-DBS-NF-6 6

7 MVD Example with MV attriutes: {, } {,} MVD are invariants on the existene of tuples if multivalued attriutes are represented in NF Def.: MVD (multivalued dependeny) Let R = (a, y, ), is multivalued dependent on a (a ->>) if for eah value v of a {v} X (π y (σ a=v R)) X (π (σ a=v R)) R MVD: example {''} X {''} X {'skiing', ''} Person {''} X {'TU'} X {''} Person {''} X {'',} X {'',''} Person B TUB B skiing 0-DBS-NF-7 0-DBS-NF-8 Fourth Normal Form Def.: Let A, B Σ(R); R is in Fourth Normal Form if for every MVD A ->> B (i) B A or (ii) B = Σ(R) \ A or (iii) A ontains a key Example not in NF, hek(!) Normalized representation: TUB B skiing Normal forms: summary Normal forms are quality riteria for dataase design. Important: NF NF Exotis: BCNF, NF (and higher!) NF / NF formalize the asi design priniple: "Never mix up different real world entities into a single design ojet (e.g. entity)" NF / NF already defined for ERM, sine FDs are given (result of requirement analysis, just like key dependenies). 0-DBS-NF-9 0-DBS-NF-0. Finding Normal Forms Invariants of appliation domain have to e made expliit during requirements analysis e.g. A sientist has at most one affiliation her institute" A region-id is unique within a ountry A person has exatly one date of irth Formalization Funtional Dependenies Wanted: algorithm produing "good" relational shemas from the set DEP of all FDs FDs and Normal Forms Given a set of dependenies DEP there are two approahes: Synthesis Set up relations in suh a way, that All attriutes are onsumed The relations are in normal form Deomposition For a given set of relations find those whih are not normalized with respet to DEP and deompose them into normalized relations 0-DBS-NF- 0-DBS-NF- 7

8 Deomposition: eliminate FDs Given Σ(R) = U and DEP the set of FDs Algorithm DECOMP(R): (i) Find the set of keys K: K U DEP or K U DEP + (DEP+ set of all implied dependenies) (ii) Eliminate all transitive dependenies y splitting reursively: {if K Y -> a is a transitive FD in R k, split R k into R i, R j Σ(R i ) = Σ(R k ) \ {a}, Σ(R j ) = Y {a} } (iii) If no more relations R k with transitive dependeny exit else for all R k DECOMP(R k ) Synthesis Disadvantage of deomposition: ineffiient (e.g. determination of keys) produes more relations than neessary Synthesis Given relation R and set of FDs DEP Find a anonial set MIN of FDs whih "overs" DEP and is minimal. Construt normalized Relations R k from MIN with» Σ(R k ) = Σ(R) 0-DBS-NF- 0-DBS-NF- Finding a anonial set of FDs Finding a minimal over () Given a set of FDs DEP and a relational shema R - Find a minimal set MIN suh that DEP MIN + - Find a relational shema in NF, from whih R an e losslessly reonstruted MIN is alled minimal over of DEP Definitions X Y MIN + iff X Y an e proven from the FDs MIN e.g. {a, }+ = {a,, a } MIN is minimal iff for every FD f (MIN \ f) + MIN + {a d,, d e }+ = {a d,, a d, d e, a e} } Important first step: Given a set of attriutes X, determine all attriutes (losure of X) whih an e funtionally determined y X? Def.: Closure of attriute set X with respet to the set DEP of FDs is the largest set Y of attriutes suh that X Y DEP + 0-DBS-NF- 0-DBS-NF-6 Closure of attriute set X I = 0; X[0] = X; /* integer I, attr. set X[0] */ REPEAT /* loop to find larger X[I] */ I = I + ; /* new I */ X[I] = X[I-]; /* initialize new X[I] */ FOR ALL Z->W in DEP /* loop on all FDs Z ->W in DEP*/ IF Z X[I] /* if Z ontained in X[I] */ THEN X[I] = X[I] W; /* add attriutes in W to X[I]*/ END FOR /* end loop on FDs */ UNTIL X[I] = X[I-]; /* loop till no new attriutes*/ RETURN X = X[I] ; /* return losure of X */ Rule used: X -> YZ and Z -> W then X -> YZW Proof? Finding a anonial set Algorithm for determining a minimal over in polynomial time Step : Normalization Replae eah FD X Y of DEP in whih Y ontains more than one attriute, y FDs with one attriute on the right hand side DEP = {a -> d, a -> e} {a ->, a -> d, a -> e} 0-DBS-NF-7 0-DBS-NF-8 8

9 Finding a anonial set () Step : Remove redundant FDs f is redundant, if (DEP \ {f} ) + = DEP + For eah f = X Y DEP : if Y (X+) using only FDs DEP \{f} then f is redundant else not redundant { d, d e, ef a, f, a} + = {,d,e,f,a} {a} FD f= -> a is redundant Expliit derivation not using f: -> d, d -> e -> e, -> f -> ef, ef -> a -> a Finding a anonial set () Step : Minimize left hand side For of eah FD f = X Y DEP, a X DEP'= {DEP \ f } { X \ {a} Y} if {DEP}+ = (DEP') + then replae DEP y DEP' end_for If a FD f has een minimized repeat step {d -> a, -> e, e -> } + = {d -> a, -> e, e -> } + 0-DBS-NF-9 0-DBS-NF-0 Finding a anonial set () Step : Unify left hand side Applying the union rule " X Y X Z X YZ " {d -> a, d -> e, d -> f} unified: {d -> ae, d -> f} The remaining set of Funtional Dependenies is the minimal over of the original set DEP of FDs Synthesis algorithm Normalization prolem: Given a relation R in NF and a set DEP of FD Find a lossless, dependeny preserving deomposition R,,R k, all of them in NF Synthesis Algorithm Find minimal over MIN of DEP; For all X -> Y in MIN define a relation RX with shema Σ(RX) = X Y Assign all FDs X' -> Y' with X' Y' Σ(RX) to RX If none of the synthesized relations RX ontains a andidate key of R then introdue a relation Rkey whih ontains a andidate key of R Remove relations RY where: Σ(RY) Σ(RX) 0-DBS-NF- 0-DBS-NF- Normal Forms: Critial review Should relations e always normalized? Yes : makes invariant heking easy, no update anomalies No: Why should we normalize if there are no updates? Customer( u_id, name, fname, zipcode, ity, street, no) No reason to normalize into e.g.: Cu(u_Id, name, fname) and CuAdr(u_Id, zipcode, ity, street, no) if only one address per ustomer and updates are infrequent Consider ost of joins / updates How expensive are selets whih need joins eause of normalization? Updates whih ause anomalies? 0-DBS-NF- ER modeling and Normal Forms ER and Normal Forms: Two different mehanisms to design a dataase sheme ER more intuitive, NF uses algorithms ER-models often already in NF Use normalization as a omplementary design method Set up ER model Transform to relations Normalize eah non normalized relation if the tradeoff of join proessing and updating redundant data suggests to do so 0-DBS-NF- 9