Calculating Exact Transitive Closure for a Normalized Affine Integer Tuple Relation

Transcription

1 Clulg E Trsve Closure for Normlzed Affe Ieger Tuple elo W Bele*, T Klme*, KTrfuov** *Fuly of Compuer See, Tehl Uversy of Szze, lme@wpspl, bele@wpspl ** INIA Sly d Prs-Sud Uversy, ordrfuov@rfr Absr: A pproh o lule he e rsve losure of prmeerzed d ormlzed ffe eger uple relo s preseed A relo s ormlzed whe desrbes grphs of he h opology oly A proedure of he ormlzo s hed The e rsve losure lulo s bsed o resolvg sysem of reurree equos beg formed from he pu d oupu uples of ormlzed relo The pproh perms for lulg e rsve losure for relo whe he osrs of hs losure re represeed by boh ffe d o-ler forms Emples of lulg he e rsve losure of ormlzed ffe eger uple relos re preseed Keywords: ffe eger uple relo, e rsve losure, Presburger Arhme, reurree equo Iroduo My problems redue o ompug rsve losures of grphs, for emple, dbse problems, prsg uom ompler osruos, opmzg ompler problems: redud syhrozo removl [], esg he legly of ero reorderg rsformos [], ompug losed form epressos for duo vrbles[], ero spe slg d ode geero [,] Grphs be represeed dffere wys Oe of possble represeos s bsed o uple relos I hs pper, we osder he lss of prmeerzed d ormlzed ffe eger uple relos whose osrs oss of ffe equles d equles Suh relos desrbe fe grphs There re los of ehques for ompug rsve losures for fe grphs, bu o our bes owledge, ehques for ompug he posve rsve losure of prmeerzed ffe eger uple relo, h desrbes fe grphs, were he sube of he vesgo of few ppers oly [] The soluo, preseed pper [] does o perm lulg e rsve losure whe s osrs

2 re preseed by o-ler forms, oly ppromos be ppled for oe represeo of rsve losure The gol of hs pper s o prese ehque permg us o lule he e rsve losure of prmeerzed d ormlzed ffe eger uple relo whe he osrs of hs losure re represeed by boh ffe d o-ler forms Bgroud The rsve losure of dreed grph G=V, E) s grph H=V, F) wh edge v, w) F f d oly f here s ph from v o w G I s very esy o he f here s ph from ode v o ode w G usg he rsve losure H of grph G There s ph G f d oly f here s edge v, w) H A grph be represeed wh eger uple relo whose dom osss of eger -uples d whose rge osss of eger -uples, for some fed d The followg s emple of relo from -uple o -uples [] [, ] The geerl form of eger uple relo s s follows[] :,,, m s F,,, y, y,, y where he F s re ouos of ffe equles d equles o he pu vrbles,,,, he oupu vrbles y, y,, y, he eselly qufed vrbles,,,, d symbol oss Ths relo be wre equvlely s he uo of umber of smpler relos, eh of whh be desrbed usg sgle ou[]:,,, y, y,, y,,, s m F m I s evde h eger uple relo desrbes he orrespode grph where he pu d oupu uples of he relo represe veres of he grph whle ffe equles d equles desrbe he edges of he grph edge ess f orrespode ffe equles d equles re hoored for gve pr of veres represeed wh pu d oupu uples) There es he wo relos reled o rsve losure: posve rsve losure, +, d rsve losure, *= + I, where I s he dey relo, I [] [] : osr s for Fgure shows grphs represeed wh he followg relo := {[][+]: <= <= }, where =, s well s relos + d *, respevely

3 + * Fgure elo, s posve rsve losure, +, d rsve losure, *, respevely The e rsve losure + of relo be preseed by he followg formul From he formul bove s ler h o lule +, we frsly lule d he relo + by mg he formul bove eselly qufed, e, ddg he qufero o he osrs of Normlzo of relo The pproh preseed hs pper dels oly wh ormlzed relos elo s ormlzed f ssfes he followg odos: elo s o preseed s uo of wo or more smpler relos,, =,,,, where s he umber of relos, wh pu d oupu uples, IN, OUT, suh h here es l, m [,] suh h IN l IN m or OUT l OUT m Eh soure/deso vere desrbed by he relo hs he ely oe orrespode deso/soure vere All equles d equles wh he relo osrs re resolved Codos d ogeher guree h ormlzed relo desrbes grphs of he h opology oly, whle odo perms us o smplfy he proess of buldg reurree equos for lulg see Seo 4) If relo does o ssfy y of he odos bove, we should pply he ormlzo proedure preseed below If relo s represeed s uo of wo or more smpler relos whose pu d oupu uples re dffere, he he ed wh sglg he flure of

4 he ormlzo If relo s represeed s he uo of wo or more relos wh he sme pu d oupu uples, he form sgle relo whose osr s he dsuo of he osrs of ll relos Che wheher eh soure/deso vere desrbed by he relo yelded sep,, hs he ely oe orrespode deso/soure vere, e, for eh soure/deso vere, s/d, s dom), d rge ) here ess he ely oe deso/soure d/s, for hs purpose, we mus he wheher here es suh s d s h s s d s) s ) s well s suh d d d h d d d d ) d ) If he verfed odo s o rue, he ry o remove redud edges desrbed by pplyg well-ow-ehques [] d repe he heg proess Whe suess, go o sep, oherwse he ed wh sglg he flure of he ormlzo esolve ll equos beg oed he osrs of, le = be he soluo o hese equos, subsue ll he pperes of for he pu d oupu uples of s well s s osrs 4 esolve ll equles he relo osrs Le us osder he followg emple of [, ] [, [, ] [, ] : ] : 5 5 Afer sep of he ormlzo proedure, we ge [, ] [, ] : 5 O 5 Afer eeug seps -4, he ormlzed relo s of he form [,] [,] : 4 O 49 4 Clulg relos d + for ormlzed relo If he szes of he pu d oupu uples of relo re dffere or he se dom) rge) s empy, whh mes h eh deso vere desrbed by s o y soure vere he sme me, we wre for for hs pper dels oly wh sgle ormlzed relo, our fuure reserh we ed o vesge lulg he power of uo of mulple relos 4

5 Oherwse g o ou h ormlzed relo desrbes grphs of he h opology oly, we form sysem of reurree equos desrbg he sme grphs h relo does For relo beg represeed he form "! " " b " # b!! b! " " d # d osrs!! d o, we wre he followg sysem of reurree equos "! " " b " # b!! b! " " d # d!! d ) d he sysem defg l vlues of uow veor " "!! For emple, for he relo [] [ ] ), we form he followg equo The soluo o s where s he l vlue of A sysem of reurree equos ) wh l odos ) be resolved by mes of y well-ow solver, for emple, suh s Mple [5], Mhem [6], Mm [7], MuPAD [8], PUS [9] For rryg ou our epermes, we,,, hose Mhem Gve soluo,,, o sysems ) d ),,,, s veor defg l vlues of veor for =), we where wre he form,,,,,,,,, :,,, dom),,,,,,,,, rge ),,,,,,, ) $ 4,,, d =,,, where,,, of ) re he pu d oupu uples, respevely; he osrs ) me he followg osrs mposed o he pu uple of, osrs defg vlues of he oupu uple of o he bss of he soluo of sysems ) d ) hey deped o d ), 5

6 osrs mposed o he oupu uple of o he se rge ), : he rge of mus belog 4 osrs mposed o he oupu uple of : relo desrbes he sme rsve relos h relo does Beuse + s uow whle we form, we rewre osr 4 he form:, % : UVD), 4) where UVD Ulme Vere Desos ) deoes se of vere desos h re o y soure vere; % s he ego operor; s he esee qufer operor Cosr 4) mes h here does o es y edge bewee y wo dso hs lyg log sgle le he grphsee Fgure ) Fgure elo desrbes wo hs log eh sgle le he grph To dsover wheher osr 4) s eessry, we use he proedure followg Clule se UVS dom) rge), Clule se UVD rge) dom), Clule Hull UVS UVD), 6

7 4 Form relo r whose pu d oupu uples re he sme s hose of relo whle s dom d rge belog o HullUVS UVD), e, relo r desrbes oly ouous hs wh Hull UVS UVD) 5 Che wheher r, f so he desrbes ouous hs wh Hull UVS UVD ), oherwse oe or more hs desrbed by re o ouous wh Hull UVS UVD) Le us llusre he proedure bove by mes of he followg emple of [] [ ] : :,4#, 4# ) The resuls of he proedure seps re he followg UVS [] [] : : 4# ), UVD [] : : 4 # 4 ), Hull UVS UVD ) [] :, 4 r [] [] :, 5 r The fl resul mes h relo Le us ow demosre lulg mus lude osr 4) for he followg emple of [, ] [, ]: Ths relo requres he ormlzo Applyg he ormlzo proedure, we ge he ormlzed relo below [, # ] [, 4 ] : 4 4 The proess of formg relos d + ludes he followg seps Form he followg sysem of reurree equos # 4 # wh he followg sysem for l vlues of uows The soluo o he sysems bove produed by Mhem s s follows,, # # # # ) 7

8 Clule se dom) [, ] : lph: lph 4 4 ) Clule se rge) [, ] : lph: 4lph & & ) 4 Clule se UVS [, ] : [, ] : 5 Clule se UVD [, ] : [, ] : [, ] : : # 4 4 ) : 4# 4 4 ) : 4 : 4 4 ) : 4 4 ) 6 Che wheher osr 4) s eessry For hs purpose, we fulfl he followg lulos Clule HullUVS uo UVD) Clule relo r ) [, ] : r [, ] [, ]: 4,, Clule 7 Form relo r, e, osr 4) s o eessry ordg o formul ),, [, ] [, ]: $ : # # 4 4 ),, #& # # #,,, : # # 4 ) 8 Form relo + by mg he relo bove eselly qufed 8

9 ,, [, ] [, ]: : $ : # # 4 4 ),, #& # # #,,, : # # 4 )) 5 Emples of lulg + Applyg he proposed pproh, we hve luled relos d + for he relos preseed Tble I s worh o oe h ll relos from he ble re o-uform d y from well-ow lgorhms d ools do o perm for lulg d + for hese relos N Tble elos, elo [, ] [, ] : 5 [, ] [, ]:, [,# ] [, ]: 5 4 [,# ] [, ]: 5 5 [] [] : :,4#, 4# ) Tble preses he resuls of lulg relos + for he relos preseed Tble 9

10 4,,,, [,] [, ]: : $ 5 #,,, :# )) [, ] [, ]:,,, [, ] [, ]: : $ : 5 ),, # # 4 # # ),,,, : # # 4, 4, )),, [,] [, ]: : $ : 5 ),, # # # 4 # # ),,,, : # 4#, 4# 4,)) 5, [ ] [ ]: : $ :,4#,4 # ),, :,4#,4 # 4) % : : 4#4 ))) 4 6 eled wor To our bes owledge, oly pper [] preses severl ehques for lulg he rsve losure of prmeerzed ffe eger uple relo The m lmo of hose ehques s hey o produe he e rsve losure of relo whe he osrs of hs losure re represeed by oler forms I hs se, he uhors of pper [] demosre how ppromos of rsve losure be luled The pproh preseed hs pper

11 llows for lulg he e rsve losure of relo whe s osrs re boh ffe d o-ler No-ler osrs re produed s soluo o reurree equos beg formed from he pu d oupu uples of relo 7 Coluso We preseed pproh permg for lulg he e rsve losure of ormlzed ffe eger uple relo whe he osrs of hs losure re represeed o oly by ffe forms bu lso by o-ler oes Ths opes he door for elrgg he usge of rsve losure felds where s requred, for emple, for resg prllelsm erg from sequel progrms by mes of opmzg omplers Our fuure reserh s o develop pprohes permg for lulg he rsve losure for uo of ormlzed relos eferees W Kelly, W Pugh, E osser, T Shpesm, Trsve lousure of fe grphs d s pplos, Lguges d Complers for Prllel Compug, 995 W Kelly, V Mslov, W Pugh, E osser, T Shpesm, D Woo, The Omeg lbrry erfe gude, Tehl epor CS-T-445, Dep of Compuer See, Uversy of Mryld, College Pr, Mrh 995 Beles A, Bele W, S Pero P, 7 Erg Corse- Gred Prllelsm Progrm Loops wh he Slg Frmewor I Proeedgs of ISPDC 8 4 Bele W, Beles A, Płows M, S Pero P, 8 Fdg syhrozo-free prllelsm represeed wh rees of depede operos I Proeedgs of ICAPP8 5 hp://wwwmplesofom 6 hp://wwwwolfrmom 7 hp://mmsoureforgee 8 hp://wwwmupdom 9 Bgr e l, The uom soluo of reurree relos I Ler reurrees of fe order wh os oeffes, Qudero 4, Dprmeo d Mem, Uvers d Prm, Ily,