Extended Multiset-Table-Algebra Operations

Transcription

1 Iteratoal Joural of Itellget Coputg esearch IJIC) Volue 7 Issue Jue 06 Exteded Multset-Table-lgebra Operatos Irya Glusho Nzhy Gogol State Uversty Urae bstract The paper s focused o soe theoretcal questos of the Database Theory Multset table algebra s cosdered The oto of a table specfed usg the oto of a ultset or bag) sgature of ultset table algebra s flled up wth ew operatos such as er ad outer jos se-jo ad aggregate operatos foral atheatcal seatcs of these operatos s defed The specal eleet s serted the uversal doa for defe of the outer jo Itroducto The relatoal data odel s owadays wdespread use as database scetfc research so ad practce I ts foral defto orgally proposed by E Codd the relatoal odel s based o sets of tuples e t does ot allow duplcate tuples a relato [] There are ay applcatos the ost pecular feature of whch s ultplcty ad repeatablty data For exaple these are socologcal polls of dfferet populato groups calculatos o DN ad others Coercal relatoal database systes are alost varably based o ultsets stead of sets I other words tables are geeral allowed to clude duplcate tuples For exaple the data odel of SQL s relatoal ature as well as the relevat operatos However ule relatoal algebra the tables apulated by SQL are ot relatos but rather ultsets The reaso for ths pecularty s twofold [] Frst ths s due to a practcal reaso sce SQL tables ay be very large duplcate elato ght becoe a bottleec for the coputato of the query result Secod SQL exteds the set of query operators by eas of aggregate fuctos whose operads are geeral requred to be ultsets of values So aturally there s a eed to expad possbltes of relatoal databases due to use of ultsets or bags) Ths proble was also cosdered [-8] However ths questo requres specfcato ad exteso because the specfed wors the due atteto st pad to operatos of er ad outer jos sejo ad aggregate operatos of ultset table algebra Multset Basc Deftos Let s troduce the foral defto of ultsets ters of oograph [5] Defto ultset wth bass U s a fucto U { } where U s a arbtrary set a classcal Cator s uderstadg) Lets agree a ultset wth bass U { d d } to wrte dow as { d d } where s a uber of duplcate of the eleet d і the ultset so d ) Let be a ultset wth bass U do Here do s the rage of defto of ultset as a fucto Defto characterstc fucto of ultset s a fucto D {0 } the values of whch are specfed by the followg pecewse schea d) f d do d ) 0 else; for all d D where D s the uverse of eleets of ultset bases Defto 3 epty ultset s a ultset a characterstc fucto of whch s a costat fucto value of whch s everywhere equal zero Defto 4 ra of fte ultset s a su of duplcate eleets of ts bass where 0 ddo d) ; Let s troduce a bary relato cluso over ultsets Defto 5 Multset s cluded ultset ) f U U d d U & ) d d Drectly fro defto follows that ths relato s a partal order The -ultsets are the ultsets whose rage of values s a epty set or a sgle-eleet set {} These ultsets are the aalogues of ordary sets alogs of stadard set-theoretc operatos ad operatos whch are usg peculartes of ultsets ad therefore caot be useful for abstract) sets are defed ters of characterstc fuctos oograph [5] There are operatos of ultset uo ll tersecto ll dfferece \ ll whch Copyrght 06 Ifoocs Socety 7

2 Iteratoal Joural of Itellget Coputg esearch IJIC) Volue 7 Issue Jue 06 buld ultsets of geeral vew The Cartesa product of ultset the operato Dst ) whch buld -ultset ad aalog of a full age for ultsets are defed too 3 Multset Table lgebra Basc Deftos og the two sets that are cosdered s the set of attrbutes ad D s the uversal doa Defto 6 arbtrary fte) set of attrbutes s called the schee Defto 7 tuple of the schee s a oal set o par D The projecto of ths oal set for the frst copoet s equal to I other words a tuple of the schee s a fucto s D The set of all tuples o schee s desgated as S ) ad the set of all tuples s desgated as S S ) Defto 8 table s a par where the frst copoet s a arbtrary ultset bass of whch ) s a arbtrary set partcular fte) of tuples of the schee ad other copoet s a schee of the table be the frst copoet of the par Let e s the set Thus a certa schee s ascrbed to every table The set of all table o schee s desgated as ) ad the set of all table s desgated as ) The otato Occ s ) deotes the uber of duplcate tuple s the ultset Lets agree a ultset to wrte dow as { s s } where ) { s s Occ ) ad } s s a bass of the ultset Defto 9 The ultset table algebra s the algebra P where s the set of all tables P \ t } s the sgature { ll ll ll p X P X p P are the sets of paraeters The operatos of sgature P are defed [6] Theore y expresso over ultset table algebra ca be replaced by equvalet to h expresso whch uses oly operatos of selecto jo projecto uo dfferece ad reag Proof To prove ths stateet we wll show that operatos of tersecto ad actve copleet ca be expressed through the operatos oted forulato of theore Ideed the followg equaltes hold ) ll \ ll \ ll \ ll C ; ) C where { } { } { }{ } { }{ } { ad } s a schee of the table see for exaple [5 0]) 4 Exteded Operatos The exteded operatos clude er ad outer jos sejo aggregate operatos Lets cosder these operatos oe at a te 4 Ier Jos There are four ds of er jo operatos Cartesa jo atural jo jo usg attrbutes ad jo o predcate p Let s defe the Defto 0 The Cartesa Jo of table o schee ad table o schee oreover of the for Cj s a bary paraetrc operato Cj ) ) ) where ) ) Cj The bass of the ultset s defed by follow ) { s s s s ) s ) s s s )} The uber of duplcates s gve by the followg forula Occ s ) Occ s ) Occ s ) where s ) ad s s s Exaple Cosder the two tables ad are show Table ad Table respectvely Let s { B } ad { D E } F Table ψ B C a a c a а с b b a Table ψ D E F a a а a a c a a c a a c C Copyrght 06 Ifoocs Socety 7

3 Iteratoal Joural of Itellget Coputg esearch IJIC) Volue 7 Issue Jue 06 If we decded to obta a Cartesa jo of these tables the result would be as show Table 3 Ths result ca also be preset as Cj {{ А a В a С c { D a E a F a } { А a В a 6 С c D a E a F c } { А b В b С a D a E a F a } { b B b C a D a E a 3 F c } } where { B C E F} Table 3 ψ Cj ψ B C D E F a a c a a а a a c a a а b b a a a а b b a a a c b b a a a c b b a a a c D Defto The Ier Natural Jo of table o schee ad table o schee s a bary paraetrc operato wrtte as whose value s the table o schee cosstg of all the uos of copatble tuples of put tables Hece ) ) ) where ) ) I other words each tuple of s pared wth each tuple of regardless of whether t s a duplcate or ot The bass of the ultset s defed by follow ) { s s s s ) s ) s s s s s )} The relato s a bary relato of copatblty of tuples df s s s s ad s s the restrctos of tuple s o the schee [5] The uber of duplcates s gve by the followg forula Occ s s ) Occ s ) Occ s ) The top dex specfes the cout of the tuple the table where s ) ad s s s Exaple Cosder the two tables ad are show Table 4 ad Table 5 respectvely Let s { B } ad { B } D C Table 4 ψ B C a b c a c a a c a b b b Table 5 ψ B D a b а a c a c b а c b а If we decded to obta a Ier Natural Jo of these tables the result ca be preset as {{ А a В b С c D a } { А a В с С a D a } } where { B C } D Defto The Ier Jo usg of table o schee ad table o schee oreover s a bary paraetrc operato of the for ) ) ) where ) ) Moreover all are parwse dfferet ad If put tables have also other geeral attrbutes whch dffer fro before jo they eeds to be reaed The bass of the ultset s defed by follow ) { s s s s ) s ) s ) s ) s s s )} The uber of duplcates s gve by the followg forula Occ s ) Occ s ) Occ s ) where s ) ad s s s Exaple 3 Cosder the two tables ad are show Table 4 ad Table 5 Lets fd a Ier Jo usg of these tables Both Copyrght 06 Ifoocs Socety 73

4 Iteratoal Joural of Itellget Coputg esearch IJIC) Volue 7 Issue Jue 06 put tables have aother geeral attrbute B We ust to reae t before usg operato We tae the attrbute B of the table ad reae t to E s result we wll get the table {{ a E b D a } { a E c D a } { с E b D a } } where { А Е D} The the result of the Ier Jo usg ca be preset as {{ a B b C c E b D a } { a B b C c E c D a } { a B c C a E b D a } { a B c C a E c D a } } where { B C Е D} Let p S S { true false} be а partal bary predcate o the set of all tuples S such that s s s s do p p s s true s s ) Defto 3 The Ier Jo o predcate p of table o schee ad table o schee s a bary partal paraetrc operato of the for ) ) ) p p do The rage of defto of ths operato s ) p { ) do p} The bass of the ultset s defed by follow ) { s ss s ) s ) p s ) s true s s s)} ad s a geeralzed equalty strog Kleee s equalty) [9] The uber of duplcates s gve by the followg forula Occ s s ) Occ s ) Occ s ) where s ) ad s s s Exaple 4 Cosder the two tables ad are show Table 4 ad Table 5 respectvely Lets fd a Ier Jo o predcate p of these tables ad def p s s ) true s ) a s ) a s B) c s B) c ) s where s ) The result ca be preset as p А a В c С a D a } } {{ where { B C D} Lets ote the followg obvous fact The jo s exteso of aother arbtrary er jo operato the followg sese t Cj t t t t t t t t t t t p The values of these operatos the left parts of these two equaltes ad cluso ust be defed Defto 4The Sejo of table o schee ad table o schee s a bary paraetrc operato wrtte as whose value s the table o schee cotag tuples of the frst table whch are cluded the er atural jo of put tables Thus ) ) ) = where ) ) The bass of the ultset s defed by follow ) { s s ) s s ) s s)} The uber of duplcates s gve by the followg forula Occ s ) Occ s ) where s ) Exaple 5 Cosder the two tables ad are show Table 4 ad Table 5 respectvely If we decded to obta a Sejo of these tables the results ca be preset as = {{ А a В b С a } { А a В с С a } } = {{ В b А a D a } { А a В с a } D } So sejo s ot coutatve 4 Outer Jo We ca lose forato whe usg the er jo operatos because the tuples whch are ot copatble wll ot be represeted the output table The outer jo operatos use whe t s ecessary to cosder the tuples of put tables whch ddt get to result of the er jo operatos Let be a specal eleet of the uversal doa D used to deote abset values the output table Let s be a costat tuple o schee e s { } Copyrght 06 Ifoocs Socety 74

5 Iteratoal Joural of Itellget Coputg esearch IJIC) Volue 7 Issue Jue 06 There s oe logcal schee for defto of the outer jo operatos [5] Let ) ) ) be soe partal bary operato o the set of all tables ad for all tables do Lets otce that the operatos are such p Cj We fx two tables fro rage of defto of the operato The the table taes the followg for ll Cosder the table The bass of the ultset s defed by follow ) { s s ) s s ) s s } The uber of duplcates s gve by the followg forula Occ s ) Occ s ) where s ) Cosder the table The bass of the ultset s defed by follow ) { s s ) s s ) s s } The uber of duplcates s gve by the followg forula Occ s ) Occ s ) where s ) I other words the tuples of the table are used forato of result of the jo operato ad tuples of the table are ot used We obta a represetato of the table replacg the roles of the tables ad the presetato of the table Lets otce that f the operato cocdes wth the operato the the table s the se-jo of the tables ad = e There are four ds of the outer jos operatos whch are duced of the er jo operato outer left jo outer rght jo outer full jo ad uo jo Let s defe the Cosder the followg er atural jos { s \ } \ \ where ) { s s s )} \ Occ s ) Occ s ) s ) s s ad s \ { s \ } \ \ where ) { s \ )} s s Occ s ) Occ s ) s ) s s s \ Defto 5 The Outer Left Jo operato s a partal bary operato of the for ) ) ) where l dol do ad \ \ ll { s } \ l \ Defto 6 The Outer ght Jo operato s a partal bary operato of the for ) ) ) where r dor do ad \ \ ll { s } \ r \ Defto 7 The Outer Full Jo operato s a partal bary operato of the for ) ) ) where f do f do ad f \ { s \ ll } \ ll ll \ { s \ } \ \ Defto 8 The Outer Uo Jo operato s a partal bary operato of the for ) ) ) where ad do do { s \ } \ \ ll ll { s \ } \ \ Copyrght 06 Ifoocs Socety 75

6 Iteratoal Joural of Itellget Coputg esearch IJIC) Volue 7 Issue Jue 06 Exaple 6 Cosder the two tables ad show Table 4 ad Table 5 The results of Outer Jo operatos would be as follows {{ a B b C c l D a } { a B c C a D a } { b B b C b D } }; r {{ a B b C c D a } { a B c C a D a } { с B b C D а } }; f {{ a B b C c D a } { a B c C a D a } { b B b C b D } { с B b C D а } }; {{ b B b C b D } { с B b C D а } } 43 ggregate Operatos The fve types of aggregate operatos dscussed ths artcle are SUM VEGE MXIMUM MINIMUM COUNT The aggregate operatos trasfor a fte table to a table wth sgle tuple ad sgle attrbute Cosder the table ) where s a fte ultset ad Let be a ultset of colu wth attrbute of table whch cotas all eleets cludg duplcates The ) { d s s ) d s)} { d { d } } s a aalogue of { } actve doa of the attrbute [5] The uber of duplcates of eleet d ) s gve by the А followg forula d) А } ) Occ{ d } { Occ s ) s ) s А) d D D Let { А) be a faly of all ultsets bases of whch are the fte subsets of the set D Here D D s a subset of the uversal doa Let Nu s a uercal subset of the uversal doa D that s closed uder addto Exted the set Nu by the specal eleet We wll ot exted the operato of addto to the case where at least oe of the arguets s Let s defe the aggregate operatos t frst the fve aggregate fuctos cout su average axu u are defed o a fte ultset ad the these fuctos are trasferred to the tables Defto 9 The aggregate operato Su by the attrbute of the fte table o schee s a uary paraetrc operato of the for Su ) { }) Su { } Su ) where ) The Su ) fucto s appled to a colu wth attrbute the table the result obtaed s the su of every value occurrece I addto values do t udertae atteto ad t s assued that the colu cotas oly data of uerc type Thus Su Nu Nu f ) ; Su ) f ) { }; d d) f ) \{ } d )\{} So we have Su{ }) Su d d d d f all eleets dffer fro I the case of the epty table Su { } we have here Exaple 7 Let Su Su Su 8 be the Table 6 The { } В6 { } С6 { } В В С С Table 6 ψ А В С Let be a lear order o the uversal doa D The top dex specfes that the table clude the tuple Su ) } oly oce e { {{ Su ) } } s {}-ultset Copyrght 06 Ifoocs Socety 76

7 Iteratoal Joural of Itellget Coputg esearch IJIC) Volue 7 Issue Jue 06 Defto 0 The aggregate operato M by the attrbute of the fte table o schee s a uary paraetrc operato of the for M ) { }) M { } M ) where ) The M ) fucto s appled to a colu wth attrbute the table the result obtaed s the u value aog values of I addto values do t udertae atteto Thus M D D f ) ; M ) f ) { }; { d d ) \{ }} f ) \{ } We have M ) M{ }) d d { d d } M f all eleets d dffer fro I the case of the epty table we have M { } here Exaple 8 Let M M M be the Тable 6 The { } В0 { } С { } В В С С Defto The aggregate operato Max by the attrbute of the fte table o schee s a uary paraetrc operato of the for Max ) { }) Max { } Max ) where ) The Max ) fucto s appled to a colu wth attrbute the table the result obtaed s the axu value aog values of I addto values do t udertae atteto Thus Max D D f ) ; Max ) f ) { }; ax{ d d ) \ { }} f ) \ { } We have Max ) Max{ }) d d ax{ d d } Max f all eleets d dffer fro I the case of the epty table Max { } we have here Exaple 9 Let Max Max Max { } В3 { } С3 { } В В С С be the Тable 6 The Defto The aggregate operato by the attrbute of the fte table o schee s a uary paraetrc operato of the for ) { }) { } ) where ) The ) fucto s appled to a colu wth attrbute the table result obtaed s the cout of all values of whch dffer fro the D Thus {0 } ) d) Put by defto that d )\ {} the su of a epty set of eleets s equal to zero So we have ) 0 { }) 0 d d f all eleets d dffer fro I the case of the epty table { } 0 we have here Exaple 0 Let 4 be the Тable 6 The { } В5 { } С4 { } В В С С We assue that a uercal subset Nu of the uversal doa D s closed uder the partal operato) dvso operato / Nu Nu Nu We wll detere the dvso operato so that whe the frst arguet s equal to the fucto accepts value Defto 3 The aggregate operato vg by the attrbute of the fte table o schee s a uary paraetrc operato of the for vg ) { }) vg { } vg where Copyrght 06 Ifoocs Socety 77

8 Iteratoal Joural of Itellget Coputg esearch IJIC) Volue 7 Issue Jue 06 ) The vg fucto s appled to a colu wth attrbute the table the result obtaed s the arthetc ea of values whch dffer fro Thus Nu vg Nu ad have Su ) vg ) We ) Su ) vg ) ) 0 Su{ }) vg{ }) { }) vg{ d d d ) Su{ d d }) }) { d d }) f all eleets d dffer fro I the case of the epty table we have vg { } here Exaple Let vg be the Тable 6 The { } 6 vg В В { В} 5 6 vg С С { С} 4 Defto 4 The aggregate operato ) by the attrbute of the fte table o schee s a uary paraetrc operato of the for *) ) { }) { } ) where ) ad s the ra of the ultset The operato ) fds the uber of tuples the table I the case of a epty table we have { ) } 0 { } here Exaple Let 5 be the Table 6 The { } ) { } С5 { } В5 В ) В С ) С 5 Coclusos I ths paper the ultset table algebra s cosdered The cocept of the table s specfed usg cocept of a ultset The sgature of the ultset table algebra s flled up wth ew operatos such as er ad outer jo sejo ad aggregate operatos For each operatos are defed a bass of the resultg table ad uber of duplcates of every tuple The specal eleet s serted the uversal doa for a defe of outer operatos It should also be oted that a paraeter of aggregate operatos s ot ecessarly oly a sgle attrbute; t also ca be soe fucto of the tuples 6 efereces [] EF Codd elatoal Model for Large Shad Data Bas Coucatos of the CM vo3 No6 970 pp [] G Lapert M Melchor ad M Zaella O Multsets Database Systes Multset Processg Matheatcal Coputer Scece ad Molecular Coputg Pots of Vew uber 35 Lecture Notes Coputg Sce Berl Sprger-Verlag 00 pp 47-5 [3] Paul WPJ Grefe olf de By Mult-Set Exteded elatoal lgebra Foral pproach to a Practcal Issue Proceedgs of the 0th Iteratoal Coferece o Data Egeerg ICDE February Housto TX US 994 pp [4] H Garca-Mola JD Ulla J Wdo Database SystesThe Coplete Boo Pretce Hall Upper Saddle ver New Jersey 009 [5] V edo J Broa D Buy S Polaov elato Database elato lgebras ad SQL-slar Laguages Kyv 00 Uraa) [6] DB Buy IM Glusho Exteded of Table lgebr Multset Table lgebra Moder scetfc research ad ther practcal applcato volj309 rtcle CID Nuber 6 May 03 [7] Slberschatz H Korth S Sudarsha Database Syste Cocepts McGraw-Hll 0 [8] I Glusho Foral Matheatcal Seatcs of dvaced Operatos of Multset Table lgebra Proceedgs of the 7th Iteratoal Coferece o Iforato Techology a l-zaytooah Uversty of Jorda 05 Р Copyrght 06 Ifoocs Socety 78

9 Iteratoal Joural of Itellget Coputg esearch IJIC) Volue 7 Issue Jue 06 [9] N Cutlad Coputablty troducto to recursve fucto theory Cabrdge Uversty Press Lodo 980 [0] B Petrovsy Space of sets ad ultsets Moscow 003 ussa) Copyrght 06 Ifoocs Socety 79