Answering Queries with Acyclic Lineage Expressions over Probabilistic Databases

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1 Answeing Queies wih Acclic Lineage Exessions ove obabilisic Daabases Baa Kenig Technion Isael Insiue of Technolog Avigdo Gal Technion Isael Insiue of Technolog Ofe Sichman Technion Isael Insiue of Technolog ABSTRACT This wok exends he class of lineage exessions of queies ove ule indeenden obabilisic daabases fo which evaluaion can be efomed in TIME. We define a new chaaceizaion of lineage exessions, called γ-acclic, and esen a mehod o comue he obabili of such exessions in TIME. The mehod is based on he juncion ee message assing algoihm and alies boh o conjuncive queies wihou self joins and, unde ceain consains, also o union of such queies. Caegoies and Subjec Descios H..m [Daabase Managemen]: Miscellaneous-Daabase Theo Geneal Tems Algoihms, Theo Kewods obabilisic daabases, Read-once funcions, obabilisic gahical models. INTRODUCTION Answeing queies ove obabilisic daabases has dawn much aenion in he daabase communi in ecen eas. A model of ule-indeenden (o ulelevel semanics) obabilisic daabases was inoduced b Cavallo and iaelli [4] and was exensivel discussed in he lieaue, e.g., [, 6]. Accoding o his model, each ule is annoaed b an exisence obabili >, meaning i aeas in a ossible wold wih obabili, indeendenl of ohe ules. This defines a obabili disibuion ove all ossible daabase insances. Que evaluaion ove ule-indeenden obabilisic daabases is # -had even fo simle conjuncive queies wihou self-joins. Handling his comlexi, he lieaue in obabilisic daabases can be boadl caegoized ino wo aoaches, namel exensional and inensional. In he fome aoach, he que oeaos ae augmened o enable comuing he esul s obabili. Fo examle, using Table, conside a ojecion ove aibue Y.B. To have a esul ule e, eihe e, c o e, g needs o exis. In he exensional aoach, ojecion would be augmened o comue he esul ule s obabili. In his case he obabili of e, c e, g is ( )( ). If a que can be comued in an exensional manne, hen i is said o have a safe lan and he esul obabiliies ma be calculaed efficienl [6, 7, 5]. Howeve, no all queies can be comued exensionall because safe lans ma ield incoec obabiliies when alied o non-hieachical queies [5, 7]. A B a b f e (a) R B C b c e c e g (b) Y C D c d g h (c) T Table : obabilisic Daabase wih vaiables In he inensional aoach he esul ules ae associaed wih a Boolean fomula, emed a lineage exession []. Using lineage exessions, an answe o a que ove a obabilisic daabase can be eesened as a Boolean exession, whee each vaiable eesens a ule (see Table ). Vaious infeence algoihms can be used o comue he esul ule obabiliies, eihe exacl [, 7] o aoximael []. Ro e al. [, ] showed a olnomial ime algoihm fo comuing he esuls of a Boolean conjuncive que wihou self-joins (CQ ) whose lineage exession can be ansfomed o a ead-once Boolean exession [9]. Examle. Conside he ule-indeenden obabilisic daabase of Table and he Boolean conjuncive que Q() : R (x, ), Y (, z) () The Boolean exession, eesening he lineage of Q ove he daabase is () Exession has an equivalen ead-once exession: Fo que Q ( ) () Q() : R (x, ), Y (, z), T (z, w) (4)

2 he Boolean exession, eesening is lineage ove he daabase is (5) I was shown [, 4] ha his exession does no have an equivalen ead-once fom. In his wok we show a class of lineage exessions whose obabili can be comued in TIME. We ovide a new chaaceizaion of lineage exessions called γ-acclic lineage exessions and esen a mehod ha enables comuing he obabili of exessions of his fom in TIME. We base his chaaceizaion on he noion of acclic daabase schemes, inoduced in he 8s b Beei e al. [] and shown o be consisen in he sense of being a ojecion of a univesal elaion. γ-acclic daabase schemes eesen a esiced fom of acclic daabase schemes ha enables osing simle, high-level SQL queies []. Acclici has also been alied o he oblem of evaluaing conjuncive queies (CQs), showing ha acclic conjuncive queies can be evaluaed efficienl [5]. Jus like acclic daabase schemes and acclic conjuncive queies, acclic lineage exessions ae defined using esicions on hei esecive hegah. The conibuions of his ae is summaized as follows: We inoduce a new class of lineage exessions, γ- acclic lineage exessions, whose obabili can be comued in TIME. We ovide a TIME algoihm, based on he juncion ee message assing algoihm [6], fo comuing he obabili of γ-acclic lineage exessions. We fomall show ha hee ae γ-acclic lineage exessions ha ae no ead-once exessions, hus exending ou caabiliies o comue queies ove obabilisic daabases in TIME. We show ha he oosed aoach suies no onl he CQ class bu also a esiced se of queies fom he UCQ class. The es of he ae is oganized as follows: Secion inoduces backgound on juncion ees and hegah acclici. Lineage acclici is esened in Secion. In Secion 4 we esen he exension of evious woks o a esiced se of UCQ. Secion 5 esens he message assing algoihm. In Secion 6 we show he elaionshis beween ead-once and γ-acclic lineage exessions. We conclude in Secion 7.. RELIMINARIES In his secion we eview wo noions ha will be used heavil in his wok. Juncion ees ae inoduced in Secion. and Secion. discusses hegah acclici. Secion. inoduces a secial case of juncion ees, he disjoin bach juncion ee. A chod is an edge connecing wo non-consecuive nodes in a ccle o ah. A gah G(V, E) is chodal o iangulaed if i does no conain an codless ccles. Discoveing whehe a gah G(V, E) is chodal can be efomed in ime O( V E ) using maximum cadinali seach [4]. A iangulaion of a gah G(V, E) is a (ossibl em) se of edges E such ha E E = and he gah G (V, E E ) is chodal. Also, a 4 denoes a codless ah wih fou veices and hee edges. Fo nodes in a codless ah, hee ae no edges beween non-successo nodes. A gah is consideed o be 4- fee if i does no conain a 4 ah. A ee decomosiion is a maing of a gah ino a ee, whee each node in he ee eesens a se of veices in he oiginal gah. The widh of a ee decomosiion is he size of he lages node minus one. The eewidh of a gah is he smalles widh of a ee decomosiion fo i. A gah ha is a ee has a eewidh of. Also, he eewidh of a chodal gah is he size of he lages clique minus.. Juncion Tees obabilisic Gahical Models (GMs) efe o a se of aoaches fo eesening and easoning abou lage join obabili disibuions [8]. A GM is usuall eesened using a dieced o undieced gah in which nodes eesen andom vaiables and edges eesen diec deendencies beween hem. An examle of a dieced GM is given in Figue a. A known eesenaive of GMs ae dieced gahical models, emed Basian newoks, ha encode deendenc elaionshis using a dieced acclic gah. (a) GM i i i (b) Moalized GM,,,i,,,i,,,i (c) Juncion ee Figue : GM, moalizaion and juncion ee Infeence in GMs efes o he ask of answeing queies ove he obabili disibuion descibed b he gah. The geneal oblem of infeence in GMs is #-comlee [8], neveheless, cuen infeence algoihms can be ve efficien fo some gahical models. Mos algoihms designed fo infeence in GMs exloi he indeendencies encoded ino he gahical

3 sucue o efom he infeence efficienl. Exac obabilisic infeence algoihms un in ime ha is exonenial in he eewidh [8, 6]. Theefoe, fo mos cases, bounded eewidh imlies acabili in gahical models. In his wok we inoduce a mehod o comue he obabili of lineage exessions in ime ha is olnomial in he eewidh. One of he well-known infeence algoihms is he juncion-ee algoihm [6]. A juncion ee is a ee, T (V, E), which coesonds o a gah, as follows. The nodes in he ee eesen maximal cliques C,..., C m in he moalized and chodal gah, whee moalizaion is obained b connecing all aens of a given node and doing he diecion of he edges (e.g., Figue b). Chodali is obained b adding edges as needed. Noe ha he examle gah obained afe moalizaion (Figue b) is chodal and in his case no edges need o be added o he gah o consuc he juncion ee. Also, he eewidh of his gah is he size of he lages clique minus, in ou case (see Figue c). A juncion ee saisfies he juncion ee oe (also known as he unning inesecion oe): if a vaiable X aeas in wo nodes of he ee X C i, X C j hen X is esen in all nodes on he (single) ah beween nodes i and j. Chodali is a necessa condiion fo he exisence of he juncion ee oe [6]. I is a common acice o augmen each edge (i, j) E wih a seaao node S ij = C i C j conaining he inesecion of he neighboing cliques. Afe unning a message assing algoihm (exlained below) on a juncion ee ceaed fom a GM, each node conains he maginal obabili disibuion of he vaiables in i. This enables o obain maginal obabiliies b using local oeaions ove vaiables in a clique. definiion. Le X be a se of andom vaiables. A faco F is a funcion fom he se of ossible values of X o [, ]. In ou seing, andom vaiables coesond o ules in he obabilisic daabase (see Table ). Heeinafe we shall use a abula noaion o eesen a faco, whee a ow eesens a secific value assignmen and a column eesens a andom vaiable. Theefoe, given a se of andom vaiables X = (X, X,..., X n ) we denoe b F X [j, k] (o siml F [j, k] wheneve he vaiable se is clea fom he conex) he value of X k in he jh assignmen. The assignmen eesened b he jh en is denoed b F [j, ], and is obabili is denoed b F [j, ]. Le X X, we denoe b F [j, X ] he assignmen o vaiables X in he jh assignmen. Finall, given an assignmen X = x, we denoe b F [x, ] he obabili coesonding o his assignmen in faco F. X X X X X X X X X X X Table : Faco fo he join disibuion of X, X, X Examle. Conside he faco F [, ] in Table, eesening he join disibuion of boolean andom vaiables X, X, X whee X = X X. X and X ae indeenden, assigned wih obabiliies X, X, esecivel, and wih obabiliies X, X, esecivel. Using ou noaion, F [, ] = and F [, ] = [,, ]. Also, F [, ] = X X and F [, {X, X }] = [, ]. Finall, F [{X =, X =, X = }, ] = X X. Each faco in he GM, which coesonds o a io o condiional disibuion, is assigned o a node in he juncion ee ha conains he vaiables in he faco. Facos assigned o a common node ae mulilied o ceae he node s oenial, Ϝ C, which is also a faco. The invaian mainained b he juncion ee algoihm is ha he join disibuion among GM vaiables is he oduc of clique oenials divided b seaao oenials. Theefoe, given an assignmen X = x: (X, X,..., X n ) = Ϝ C [x, ]... Ϝ Cm [x, ] Ϝ S [x, ]... Ϝ Sm [x, ] (6) A he algoihm ouse, he seaao oenials ae iniialized o and he clique oenials o he oduc of he condiional and io obabili disibuions assigned o hem. This invaian is mainained also a he end of he algoihm when he clique and seaao nodes conain hei maginal disibuions. The message assing algoihm is a seies of udaes o he clique oenials ha ae based on he disibuions in he clique s neighbos. A he end of he message assing algoihm, each clique conains he maginal disibuion of is vaiables. Theefoe, neighboing cliques mus agee on he maginal disibuion of hei inesecion (o seaao node) [6]. The ocess sas wih choosing a oo node fom which he messages ae fis sen ouwad, and hen colleced back inwad. Afe eve ai of neighboing cliques C i, C j exchange wo messages, one in each diecion, boh he clique and seaao nodes conain hei maginal disibuions and his disibuion is consisen ove all nodes shaing a common se of vaiables. The message assing algoihm maginalizes he ables eesening he oenials in he cliques. As a esul, he numbe of oeaions gows as he numbe of enies in he able. The numbe of such enies is exonenial in he numbe of membes in he clique. This leads o a comlexi of O(N D k ) fo he juncion ee message assing algoihm, whee N is he numbe of nodes in he ee, D is he size of he vaiables domain (in he case of Boolean exessions D = ), and k is he size of he lages clique in he ee. This

4 means ha he message assing algoihm uns in ime ha is exonenial in he GM s eewidh. Moeove, consucing a juncion ee whose widh is equal o is eewidh is known o be N-had [6] in geneal due o he hadness of finding an oimal iangulaion fo a GM.. Hegah Acclici A hegah H = (V, E) is a genealizaion of a gah whee V is he se of nodes and he se of edges E is a se of non-em subses of V. Edges in a hegah ae emed heedges. The dual G (V, E ) of a hegah H = (V, E) is a gah whose veices coesond o he heedges of H, i.e., V = E, and (v, v ) E iff v v in H. Each edge in he dual gah, G, is augmened wih he inesecion of he nodes a is ends. The imal gah G(H) = (V, E G ) coesonding o a hegah H is he gah whose veices ae hose of H and whose edges ae he se E G = {(u, v) : {u, v} V, e E, {u, v} e}, namel all ais of nodes ha occu ogehe in some heedge of H. I is woh noing ha boh he dual gah and he imal gah ae egula gahs and no hegahs. A hegah is confomal if eve clique in is imal gah G(H) is conained in a heedge of H. In he conex of Boolean exessions, he em confomal is elaed o he em nomal, used in evious woks [, ]. The noion of acclici is well known in he sud of elaional daabases, wih man desiable oeies [, 5]. Thee ae hee foms of acclici in hegahs, emed α-alicclic, β-acclic and γ-acclic [], wih he following subsumion elaionshi: α-acclic β-acclic γ-acclic. definiion (α-acclici). A hegah H is α-acclic if H is confomal and is imal gah G(H) is chodal. Definiion [, ] enables ecognizing β-acclic hegahs b defining β-cclic hegahs. definiion (β-cclici). A hegah is β- cclic iff hee is a sequence (S, x,..., S m, x m, S m ) s..:. x,..., x m ae disinc nodes in he hegah.. S,..., S m ae disinc edges and S m = S.. m (a leas hee heedges ae involved). 4. i [, m], x i is in S i and S i and in no ohe S j. Definiions 4 and 5 ovide equivalen condiions fo ecognizing γ-acclic hegahs [, ]. definiion 4 (γ-acclici). A hegah is γ- acclic if i has no ai (E, F ) of incomaable, nondisjoin heedges such ha in he hegah ha esuls b emoving E F fom eve heedge, wha is lef of E is conneced o wha is lef of F. definiion 5 (Sufficien cond. fo γ-cclici). A hegah H is γ-cclic if i conains a leas one of he following configuaions:. A ue ccle. A se of heedges S, S,..., S m, S m is called a ue ccle if m, and if wheneve i j hen S i S j iff S i and S j ae neighbos.. Thee exis a se of hee heedges E, F, G s.. E F G, (E G)\F and (F G)\E (ohe inesecions beween, E, F and G ma also occu). I was exemlified ha γ-acclic elaional schemes enable osing simle queies []. Fuhemoe, γ-acclici allows geae flexibili in que oimizaion. Finall, hee exis olnomial ime algoihms fo ecognizing boh β and γ acclic hegahs [, ].. Disjoin Banch Juncion Tees Given an α-acclic hegah H, a juncion ee can be ceaed fom is chodal and nomal imal gah G(H). Duis [9] inoduced juncion ees wih disjoin banches (dbj fo sho) and showed ha if a hegah is γ-acclic hen i has a dbj [9] ooed a an node (heedge). Definiion 6 chaaceizes he dbj oeies ha enable he efficien comuaion we show in his wok. Le T denoe a dieced ee wih oo. Recalling ha each node of he juncion ee eesens a clique in he co-occuence gah (see Secion.), we denoe b C he membes of node. definiion 6. Le T be a juncion ee wih oo and childen,,..., l, which ae he oos of subees T,..., T l, esecivel. A juncion ee T is a dbj if:. T conains a single node,, i.e., T =, o. The following wo condiions joinl hold: (a) i j, C i C j = (b) T i {T, T,..., T l }, T i ecusivel comlies wih he condiions and. Definiion 6 is equivalen o he definiion esened b Duis [9]. Secion A of he aendix shows his equivalence and esens he algoihm fo consucing he juncion ee in a olnomial ime.. LINEAGE ACYCLICITY We now elae beween he foms of acclici in hegahs and lineage exessions in he conex of obabilisic daabases. Oiginall, he vaious foms of acclici enabled chaaceizing daabase schemes wih desiable oeies [, ]. Hee, we al he noions of acclici o lineage exessions ahe han schemaa. Le f denoe he lineage exession, as deived b he que engine. Fo queies of class CQ and UCQ f is monoone since all lieals ae osiive and onl conjuncions and disjuncions ae used. An imlican of f is a se of lieals such ha wheneve he ae ue, f is ue as well. A ime imlican of f is a

5 minimal imlican ha canno be educed. A DNF conaining onl ime imlicans is efeed o as an iedundan DNF and is denoed as f IDNF []. The co-occuence gah G co (V, E) of f, has as is nodes f s vaiables (denoed va(f)) and an edge exiss beween wo nodes (u, v) E iff he boh occu in he same ime imlican of f. A ovenance gah is a dieced acclic gah (DAG) whose nodes ae labeled b vaiables o oeaion smbols, disjuncion ( ), and conjuncion ( ). A dieced edge u v in his gah signifies ha he ule coesonding o u is comued using he ule coesonding o v in eihe a join (i.e., u is labeled wih ) o a ojecion (i.e., u is labeled wih ). A ovenance gah fo ead-once lineage exessions can be efficinl ceaed [4,, ]. The DNF exession of f can be modeled as a hegah H f (V, E), whee each lieal coesonds o a node in a gah and each ime imlican coesonds o a heedge. definiion 7 (Lineage exession acclici). A lineage exession f is α/β/γ-acclic if H f is α/β/γacclic, esecivel. The ovenance gahs we efe o in his wok as well as he lineage exession, f, coesond o he que lan used o comue he que. The que engine ma oduce an (no necessail DNF) lineage exession. Exanding f o is DNF fom ma esul in an exoneniall lage fomula. Theefoe, Definiion 7 canno be alied diecl and H f emains unknown in geneal. Howeve, we can check if H f is acclic b suding is imal gah G(H f ). I is eas o see ha he imal gah G(H f ) is acuall he co-occuence gah G co of f. Ro e al. [] inoduces a mehod which enables consucing he co-occuence gah of f wihou having o go hough is IDNF eesenaion. This mehod oeaes diecl on f s ovenance gah. Fo Boolean exessions esuling fom CQ and a esiced se of UCQ (Secion 4), we can al his mehod o build he equied imal gah G(H f ) in TIME. If he imal gah is no chodal, hen is coesonding hegah canno be α-acclic and heefoe canno be β o γ-acclic eihe. Howeve, if i is chodal we need o es he hegah fo confomali. In ou seing his means ha eve maximal clique in he imal gah G(H f ) (o co-occuence gah G co ) coesonds o a ime imlican in f. This would equie ieaing ove all of he maximal cliques in G(H f ), which in non-chodal gahs is an N-comlee oblem. Howeve, in chodal gahs his ask can be achieved b using maximum cadinali seach [4], and can be efomed in ime O(nm) whee n is he numbe of veices and m is he numbe of edges in he gah [8]. In ode fo he imal gah G(H f ) o be confomal, each maximum clique in G(H f ) should coesond o a ime imlican in f s DNF fom. Algoihm shows a simle ocedue fo esing whehe a se of lieals foms a ime imlican. This means ha fo lineage exessions whose imal gah is chodal and confomal (i.e., coesonding hegah is α- acclic), he coesonding hegah can be easil Algoihm : Tesing ime imlican Inu: Monoone lineage exession f; se of lieals Ouu: ue if he se of lieals fom a ime imlican of f, false ohewise : Define assignmen φ: { ue φ() = f alse ohewise : if f(φ) = false hen : eun false { does no saisf f and heefoe canno be a ime imlican} 4: end if 5: fo all l do 6: Define assignmen φ l : { ue \l φ l () = f alse ohewise 7: if f(φ l ) = ue hen 8: eun f alse { is no minimal} 9: end if : end fo : eun ue consuced. An assignmen is a maing φ : va(f) {ue, false}. We will denoe b f(φ) he esul of f unde assignmen φ. We will sa ha φ φ if { va(f) : φ () = ue} { va(f) : φ () = ue}. Since we assume ha f is monoone, hen if an assignmen φ does no saisf f, hen none of is subses will saisf f eihe. oosiion. Algoihm euns ue fo aamees (f, ) iff is a ime imlican of f. oof. B definiion, assigning ue o he lieals in and false o he es should saisf f, and is minimal in he sense ha none of is subses saisf f. Since f is monoone i is sufficien o check he nonsaisfiabili of subses of size o veif ha is indeed minimal. If eihe he saisfiabili equiemen o he minimali equiemen ae violaed, he algoihm euns f alse. oosiion. The comlexi of Algoihm is O(n ), ielding a oal comlexi of O(n ) fo esing whehe a lineage exession wih a chodal imal gah is confomal. oof. The loo in lines 5- uns imes, and unde he assumion ha esing an assignmen φ is O(n) hen he comlexi is indeed O(n ). Ou goal is o ulimael comue he obabili ha f =. We achieve his b consucing he juncion ee based on f s imal gah G(H f ). Juncion ees ae associaed wih GMs, and we devoe he es of his secion o descibing he dieced GM behind f s juncion ee and imal gah. The obabili (f = ) = (f = ). f = iff each and eve one of is ime imlicans is false. We can view he esul of

6 each imlican in f IDNF as a vaiable in a Baesian ne whose aens ae he membes of he ime imlican. The oveall esul can be eesened b e anohe vaiable j ha is he disjuncion ove hese esul vaiables. Fo examle, le f =. We denoe f s value b j, and he esul of is ime imlicans b i, i and i, esecivel. The baesian ne coesonding o his exession is given in Figue b. Seing f o false (j = ) ses i = i = i = as well. This inoducion of evidence o he baesian ne causes he emoval of hese esul vaiables fom he newok, while limiing he se of configuaions ossible fo he vaiables. Afe he emoval of hese esul vaiables and aling moalizaion, we aive a a gah ha is ecisel f s co-occuence gah, o he imal gah coesonding o is hegah, H f.. Roadma Afe validaing ha f is chodal and confomal and heefoe α-acclic, we need o calculae he obabili ha i is saisfied. We ovide now a quick oadma o he wa we do i, using Figue fo illusaing how o geneae one daa sucue fom anohe. f denoes he que s lineage, G co (f) denoes is co-occuence gah [], H f denoes f s IDNF hegah and G of H f is dual. G(H f ) denoes H f s imal gah. If f is α-acclic, a juncion ee can be consuced based on he imal gah G(H f ) a hand ahe han consucing H f fis [6]. Howeve, i is known ha obabilisic infeence in juncion ees is exonenial in is widh [6]. In ou case, since f is α-acclic, is imal gah G(H f ) is chodal and is widh is he same as is eewidh (Secion.). Lineage exessions coesonding o γ-acclic hegahs enable he ceaion of a dbj. In Secion 5 we inoduce an algoihm, alicable o dbjs, which evaluaes he equied obabili in ime O(n k ) whee n = va(f) and k is G(H f ) s ee s eewidh. 4. FROM CQ TO UCQ We devoe his secion o exend he e of queies fo which ou new esuls al, beond hose shown in [, ]. Le Q = CQ CQ... CQ m be he union of conjuncive queies wihou self joins, i.e., Q UCQ. We denoe b els(cq j ) he elaions used in a ceain que CQ j. Q UCQ is a no-subsumion UCQ, denoed UCQ, if fo each ai of conjuncive queies {CQ i, CQ j } UCQ, els(cq j ) els(cq i ) and els(cq i ) els(cq j ). We now exend he scoe of a lemma, used b Ro e al. [], fo he case of UCQ. Lemma. Le f be a lineage exession of a que Q UCQ. The DNF geneaed b exanding f using onl he disibuivi ule is in fac he IDNF (Iidundan DNF) of f u o idemoenc (i.e., eeiion of he same ime imlican is allowed). oof. Le g be he DNF geneaed fom f b aling disibuivi eeaedl. Eve imlican in g will esul fom some CQ i Q. Due o he absence of self-joins, eve imlican in g has exacl one ule fom eve elaion in he que. Since hee is no subsumion elaionshi beween he elaions fo each ai of conjuncive queies in Q, hen he se of vaiables in one imlican canno be a sic subse of he se of vaiables in he ohe. Secificall, absoion (e.g., x xz = x) does no al. Theefoe, g is he IDNF of f (u o commuaivi and associaivi). The es of he lemmas inoduced b Ro e al. [] when showing he coecness of he co-occuence gah comuaion, eihe el on he no self join assumion (common o boh he CQ class and he UCQ class) o el, diecl o indiecl, on he equivalen of Lemma. Theefoe, we can conclude ha he coecness of he co-occuence gah comuaion, as esened b Ro e al., alies o he UCQ class as well. 5. EFFICIENT MESSAGE ASSING IN DIS- JOINT BRANCH JUNCTION TREES In his secion we inoduce an efficien message assing algoihm fo disjoin banch juncion ees. Secion 5. ovides an illusaing examle of he algoihm. Faco eesenaion and faco ojecion ae discussed in Secion 5.. Secion 5. inoduces he message assing algoihm. 5. Illusaing Examle We fis moivae and exlain he algoihm aoach using a simle examle. (a) imal/cooccuence gah i= i= j= (b) Baesian ne Figue : Illusaion fo Examle Examle. Conside que Q, esened ealie (Q() : R (x, ), Y (, z), T (z, w)), ove he insance in Table. The lineage of he que is j =, as in Eq.. The co-occuence gah coesonding o he que is given in Figue a. I is eas o see ha his lineage exession is no ead-once since i has a 4: (,,, ). Le us denoe b i =, i=

7 [] If α-acclic, [] lineage f G co(f)/g(h f ) H f If α-acclic, [] G of H f γ-acclic, [6] γ-acclic: [9] Sec. 5 Juncion ee dbj (f = ue) Figue : Roadma Illusaion i =, and i =. We ae ulimael ineesed in calculaing he obabili: (j = ue) = (j = false). If j = false hen we know ha i = i = i = false. These values can be seen as inoducion of evidence in a Baesian newok, illusaed in Figue b. Afe moalizaion (see Secion.), he newok is chodal and nomal and heefoe has a juncion ee deiced in Figue 4a wih node {,, } seleced as he oo. The ke o efficien obabili comuaion lies in elacing he eesenaion of he ossible configuaions of he node membes. Fom a eesenaion whose size is exonenial in he numbe of node membes, o one which is linea in his numbe. See, fo examle Figues 4a-4c, whee aseisks eesen wildcad assignmens. Hee, he numbe of enies in each faco is exacl he node s cadinali. The message assing is esened in Figues 4b-4c, and will be demonsaed in deail in Secion 5.. Afe he comleion of he message assing algoihm, he oo node conains he maginal obabili of is enies (see secion.). Theefoe, all he enies of he oo node s faco ae added in ode o obain he equied obabili. 5. Faco Reesenaion and ojecion Given an acclic lineage exession, we know ha is co-occuence (o imal) gah is chodal and nomal. Theefoe, each maximal clique in his gah coesonds o exacl one DNF ime imlican of he lineage s iedundan DNF fom. As a esul, each node in he coesonding juncion ee conains a faco ha eesens a single DNF ime imlican of he lineage. We will efe o hese as DNF facos. Fo each vaiable X in he exession we define a base faco, F b X. Base facos conain exacl wo enies, wih values, and hei aoiae obabiliies X, X, esecivel. Each base faco will be assigned o exacl one node in he juncion ee. Conside some DNF ime imlican d, conaining m lieals, d = X X... X m. The obabili of d = is comued as follows: (d = ) = (X = ) (X =, X = )... (X =,..., X m =, X m = ). (7) The m summands in Eq. 7 ceae a muual exclusive and exhausive se of configuaions. This means ha eve assignmen o he vaiables X,..., X m ha ields d = coesonds o exacl one of he summands. Fo illusaion, conside Table ove X,..., X m. The values ae iniialized o. X X X k Table : Faco Table The aseisks in he able eesen wildcad assignmens, as given in Eq. 8, using he ule indeendence assumion in he ansiion fom he hid o he fouh line of he equaion. Infomall, once we know ha X j =, hen he imlican s value is false egadless of he values of he ohe lieals in he imlican. A he beginning of he algoihm, each node s faco able conains onl aial infomaion egading he obabiliies of is vaiables. This is based on he assignmen of he base facos o he DNF facos in he ee. Fo examle, a faco able ove vaiables X, X, X a he beginning of he algoihm is given in Table 4. Fo ease of exosiion, we use (X = ) = X and (X = ) = X. Also, fo he sake of illusaion, we assume ha he base faco FX b is assigned o a diffeen node (DNF faco).

8 (X =,..., X i =, X i =, X i =,..., X k = ) = (X =,...X i =, X i =, X i = x i,..., X k = x k ) = x i,...,x k {,} x i,...,x k {,} (X i = x i,..., X k = x k X =,..., X i =, X i = ) (X =,..., X i =, X i = ) = (X = )... (X i = ) (X i = ) (X i = x i,..., X k = x k ) x i,...,x k {,} (8) X X X X X X X Table 4: Faco Table wih aial base facos Message assing in he juncion ee algoihm can be seen as a ojecion ove he able eesening he faco. The onl diffeence is ha he column of he esul able should conain he sum of obabiliies of he coesonding enies in he oiginal faco. In he classic seing, in which each en in he faco eesens a single configuaion of he vaiables (and heefoe he size of he faco is exonenial in he numbe of vaiables), he obabiliies of he enies wih common values in he ojeced aibues ae siml added. This is no he case fo linea sized facos used in ou seing. Definiion 8 fomalizes his noion of ojecion in ou seing, and Examle 4 demonsaes i. definiion 8 (faco ojecion). Le F X X be a faco ove vaiables X X whee X = {X,..., X m } and X = {X,..., X l }. The ojecion of F ove he vaiables in X, denoed F X = X F X X, is a new faco conaining onl vaiables X. The obabili column in F X is comued as follows: F X [j, ] = F X X [i, ] i [, X X ]:F X X [i,x]=f X [j, ] ojecion ma be alied o faco F X X unde he following condiions:. The vaiables X, ojeced ou of he faco F X X, aea afe he vaiables X.. The base facos coesonding o he vaiables X ae included in faco F X X befoe he ojecion oeaion can be alied. Examle 4. Conside Table 4 and he faco F X,X,X ove he vaiable se {X, X, X }. We sa b ojecing ou he vaiable X. Condiion of Definiion 8 is saisfied. As fo Condiion, X s obabili, X, is alead available in he faco, and heefoe F X,X = X,X F X,X,X = X X X X X X ojecing ou X fom F X,X equies muliling in X s base faco o saisf Condiion. Theefoe, F X = X F X,X = X X X ( X X X ) 5. Algoihm Desciion We denoe b C i he se of vaiables in node i of he juncion ee, and b C i is cadinali. The faco of node i is denoed b F i. In his secion we use he faco noaion defined in Secion. B denoes he se of vaiables in node, fo which base facos have been assigned wih : B = {X : X C, F b X is assigned o }. C \ B is he se of vaiables whose base facos ae no assigned o. We denoe b childen(i) and (i) he childen and aen of node i in he juncion ee, esecivel. A message beween node i and node j, µ i,j (C i C j ) is a faco ove he wo nodes inesecion. The numbe of enies in µ i,j (C i C j ) is C i C j (including he en conaining all ones). Algoihm eceives as a aamee a aial ode ove he vaiables in he juncion ee T. We denoe b vas( ) he se of vaiables ove which is defined. Algoihm : Message assing: Iniial Call Inu: dbj (see Definiion 6) T wih oo, consuced fom a γ-acclic lineage exession f. Ouu: (f = false) : Call Algoihm wih aamees: T and =. : Reun C j= F [j, ]. The seudocode of he message assing algoihm ove linea sized facos is given in algoihms and. Afe he iniial call o Algoihm (Line of Algoihm ), he algoihm efoms a seies of ecusive calls o udae he obabiliies in he node facos of he juncion ee. This is he aallel of he fis hase in he classic message assing algoihm [6], whee messages ae assed fom he nodes owads he oo. Since we ae onl ineesed in he obabili ha he lineage exession is false we do no conduc a second hase ha udaes each node of he ee o ulimael conain he coec maginal obabiliies.

9 ,,,,,, (a) Juncion ee, befoe message assing,,,,,, ) (, = µ (b) Juncion ee, afe assing s message,,,,,, ) (, = µ (c) Juncion ee, afe assing nd message Figue 4: Examle

10 Algoihm : Message assing: Main ocedue Inu: dbj T wih oo and a aial ode. Ouu: faco F wih coec obabiliies : if hen : Define an ode ove C s..:. C C () aea befoe C \C (). The ode of vaiables C vas( ) comlies wih. : else 4: define an abia ode ove vaiables C 5: end if 6: Udae accoding o he ses above. 7: Define a linea-sized faco F based on. 8: fo j o C do 9: F [j, ]. {iniialize faco enies} : fo X B do : F [j, ] F [j, ] (X = F [j, {X}]) {Mulil in all base facos assigned o he node} : end fo : end fo 4: fo all i childen() do 5: Recusivel call he algoihm on subee T i wih oo i and (udaed) odeing. 6: µ i, (C C i ) C C i (F i ) {ojec on he childen s faco o ge he message} 7: fo j o C C i do 8: M i, [j]. {ieae ove enies in he message} 9: fo X ((C C i ) \ B i ) do : M i, [j] M i, [j] (X = µ i, [j, {X}]) : end fo : end fo : ob µ i, [ C i C, ] M i, [ C i C ] {iniialize ob accoding o en [,,...,]} 4: fo k C i C o do 5: ob ob µ i, [k, ] M i, [k] {udae ob} 6: µ i, [X =,..., X k =, X k =,..., X Ci C =, ] ob 7: end fo 8: end fo 9: fo j o C do : fo all i childen() do : F [j, ] F [j, ] µ i, [F [j, C i C ], ] : end fo : end fo Each message fom a node i o is aen, (i), is a ojecion on faco F i ove he vaiables C i C (i). Accoding o he definiion of ojecion (Definiion 8), vaiables C i C (i) should aea befoe C i \ C (i) in he faco able eesenaion. Theefoe, Lines -5 of Algoihm define an ode ove he vaiables in he oo node ha was given as a aamee (C ) such ha ojecion ove vaiables C C () is made ossible. If is in fac, he oo of he juncion ee, i has no aen node and an abia ode ove he vaiables C is deemined. In Examle, Figue 4a, he oo node conains odeed vaiables {,, }. In he faco fo he child node wih vaiables {,, }, comes befoe and because he message beween his node and is aen is ove vaiable. Likewise, in he faco wih vaiables {,, }, aeas befoe and. The ode is udaed in line 6. Lines 7- iniialize a faco fo node based on he odeing ha was udaed in lines -5 and accoding o he base facos assigned o his node. In Examle, Figue 4a, he base facos fo vaiables and ae assigned o he oo node, while he base faco fo is assigned o he middle node (wih vaiables {,, }). The assignmen of a base faco o a node udaes he obabili of he enies in he node s faco as efomed in line of he algoihm. In examle, Figue 4a, he base faco fo, F b, is assigned o node {,, }, udaing he obabili column of he second and hid enies b muliling wih and, esecivel. Lines 4-8 iniiae a ecusive call on he childen of. In Line 6, he messages fom all of he childen of ae colleced. Each one of he messages µ i, (C i C ) eceived b he node in line 6 conains C i C enies, which fom an exhausive and muual exclusive se of configuaions. Fo examle, conside he message µ, ( ) fom node {,, } o node {,, } (Figue 4b). The en whee = in Figue 4b was added o he message because i will be used b node {,, }, bu i is no a of he ojecion ove he faco of node {,, }. Such enies ae calculaed in lines 7-7 of he algoihm. Thee ae exacl C i C such enies added o he message which coesond o aial sums ove he enies in he oiginal message µ i, (C i C ). As in he case of ojecion, in ode o add he obabiliies of he enies in he messages, he aoiae base facos need o be mulilied in befoe he addiion can ake lace. Fo examle, in Figue 4b, he base faco fo, F b, is no a of he faco of node {,, }, and heefoe no a of he oiginal message (conaining onl enies whee = and = ). Howeve, in ode o augmen he faco wih he en =, he enies coesonding o = and = need o be added. In ode fo he esuling obabili o be coec he value of he obabili column coesonding o = is mulilied b and he value fo he second en is mulilied b. The obabili values fo hese enies ae hen added o give he coec obabili in he en fo

11 = in Figue 4b. Lines 9- udae F accoding o he messages eceived fom is childen. The loo beginning in line 9 ieaes ove each of he enies of F, and he inne loo (lines -) ieaes ove he messages he node has eceived fom is childen. Fo each of he enies in F, is obabili is udaed accoding o he messages eceived fom he childen of (line ). We now move on o ove he coecness of algoihms and fo disjoin banch juncion ees. Theoem. Le T be a dbj wih oo, consuced fom a γ-acclic lineage exession. Afe unning Algoihm on T, F [j, ], j [, C ] conains he maginal obabili coesonding o he configuaions eesened b he jh en of his faco. oof. Fis, we noe ha he odeing defined in lines -5 of Algoihm can alwas be achieved. vas( ) C can onl conain vaiables ha belong o an anceso of. Since T is a ee, is conneced o is ancesos onl via (). Theefoe, due o he juncion ee oe, vas( ) C C () vas( ) (C \ C () ) = (vas( ) C ) \ (vas( ) C ) = Theefoe, he se of vaiables ha should aea las in F, C \C (), is no a a of vas( ), and heefoe no conadicion wih can occu. The oof is b inducion on he size of he ee T. Le C = {X, X,..., X l }. Fo he base case ( T = ), he lineage exession, f, conains a single ime imlican, f = X... X l. All base facos ae mulilied ino he faco F. Assuming, w.l.g., ha he ode of he vaiables is X, X,..., X l, he jh en of he faco (j l) is F [j] = X =, X =,..., X j =, X j =, X j =,..., X l = and F [j, ] = (X = ) (X j = ) (X j = ) which is he coec obabili fo his en. Since he enies in he faco fom an exhausive and muual exclusive se of configuaions, adding hei coesonding obabili columns will esul in he obabili (f = ). Le us assume he coecness of he algoihm fo ees of size T n and ove he heoem fo ees of size T = n. Le us look a he disconneced subees ceaed b emoving he oo fom T. We ae lef wih ooed subees T, T,..., T l whose sizes ae less o equal n. Fo each ai of subees T i, T j, {i, j} [, l], i j, T i T j =. Ohewise, due o he unning inesecion oe of juncion ees, C i C j in conadicion o i being a dbj. B he inducion hohesis, he facos F,..., F l conain he coec obabiliies in hei obabili column. The message fom child i o is µ (C i, i C ) = C i C (F i ). Due o he inducion hohesis and he coecness of he ojecion oeaion (see Definiion 8), we assume he coecness of he message facos. Since he subees ae indeenden, conaining disjoin ses of vaiables, he oveall obabili can be calculaed b siml muliling he obabiliies in he aoiae enies in messages µ,,..., µ l, wih he obabili values in faco F. To jusif he mulilicaion oeaion in line, we show ha he equied infomaion is available in he messages µ,,..., µ l, eceived b he oo. Le us look a en F [j] = {X =,..., X j =, X j =, X j =,..., X l = } The message equied fom child i, µ i, (C i C ), whee C i C = {X i,..., X im } ae odeed accoding o hei lacemen in faco F i, ma ake one of he following foms: X i =, X i =,..., X im = : Accoding o he wa he facos (and hei ojecions) ae sucued µ i, (C i C ) = C i C (F i ) conains his en. X i =, X i =,..., X ij =, X ij =..., X im = : Accoding o he wa he facos (and hei ojecions) ae sucued µ i, (C i C ) = C i C (F i ) conains his en. X i =,..., X ij =, X ij =,..., X im = : The obabili fo his en is calculaed in lines -7 of he algoihm. Is value is C i C k=i j µ i, [k, ] M i, [k]. This comuaion is coec based on he inducion hohesis which leads o he coecness of he enies in he faco F i, he coecness of he ojecion oeaion, leading o he coecness of he obabiliies in he enies of µ i, and he muual exclusiveness of he enies which enables adding hem (afe he mulilicaion wih he missing base facos, M i, [k]). Ohe foms ae imossible due o he odeing induced b i s aen ( i ) =. To summaize, due o he subees indeendence, he coecness of he messages assed, and he availabili of he needed message values, he oveall obabili is comued coecl. Theoem. The comlexi of algoihms and on a disjoin banch juncion ee of size n is O(n k MAX ) whee k MAX = MAX i=..n C i. oof. The loo in lines 8- is efomed in O( C ) since B C. Fo he same eason, he loo in lines 7- is efomed in O(( C C i ) ), bu since subsumion canno occu in he juncion ee, C > C C i, we aive again a unime of O( C ). The loo in lines 4-7 akes ime O( C C i ). A node in he ee eceives messages fom all of is neighbos, exce is aen in he algoihm. Since he childen ceae a aiion of he vaiables in he node, hen he numbe of childen can be a mos C. The numbe of enies fo which he obabili is udaed is exacl C, heefoe he oal unime is n i= O( C ) = O(n (kmax )).

12 6. γ-acyclic AND READ-ONCE LINEAGE EXRESSIONS UCQ CQ Co-occuence gah conains codless ccles of lengh 4 γ acclic RO Co-occuence gah does no conain codless ccles Figue 5: Read-once and γ-acclic elaions In his secion we show he elaion, illusaed gahicall in Figue 5, beween γ-acclic and ead-once lineage exessions. Examle ovides a lineage exession ha is γ-acclic bu no ead-once (Eq. ). As i uns ou, hee ae also ead-once exessions ha ae no γ-acclic. As an examle, conside he following ead-once lineage exession: f = ab bc cd da = (a c)(b d) This exession is ead-once bu no γ-acclic because i has a codless ccle of lengh 4. Acuall, in his case, f is no even α-acclic because is imal gah is no chodal. Theoem saes ha such ead-once lineage exessions ae ecisel hose ha ae no γ-acclic. Theoem. Le f be a ead-once lineage exession. f is γ-acclic iff is co-occuence (o imal) gah G(H f ) does no conain codless ccles of lengh 4. oof. ( ) If f is γ-acclic hen is imal gah canno conain codless ccles, secificall no hose of lengh 4. ( ) Assume, b conadicion, ha f is γ-cclic. Accoding o Definiion 4, his means ha f s hegah conains wo heedges E, F s.. E F = Q and emoving Q fom all heedges esuls in E and F saing conneced in he esuling hegah H. Conside he shoes ah beween E and F in H, whee x E E \ F, x F F \ E E, x E, S, x, S, x,..., S m, x E, F We illusae his ah in Figue 6a using boldface edges. Such a ah esuls in a codless ah conaining m nodes in H f s imal gah, e.g., Figue 6b. Since f is ead-once, G(H f ) does no conain a 4, heefoe m. We seaae ino cases: Case : m =, Figue 6c. In his case hee is some heedge S {x E, x F }, and hus he edge (x E, x F ) aeas in G(H f ). As a esul G(H f ) conains a clique wih nodes {x E, x F } Q. Le S be he maximal clique in G(H f ) ha conains nodes {x E, x F } Q. Since f is ead-once and heefoe confomal [4], hen hee mus be a heedge conaining he nodes in S. Since f eesens a boolean exession in iedundan DNF fom, hen E S and F S, heefoe x E E, x F F s. x E, x F S. Secificall, i is imoan o noe ha hee is no edge beween x E and x F in G(H f ). If hee wee, hen we aive a a conadicion of S being a maximal clique because x E is conneced o {x E } Q (hough heedge E) and o x F, leading o a clique conaining nodes {x E, x F, x E } Q subsuming S. Using he same easoning, hee is no edge beween x F and x E in G(H f ). This means ha he following ah aeas in G(H f ): x E, x E, x F, x F. If hee is an edge beween x E and x F, hen we aive a a conadicion ha G(H f ) does no conain codless ccles of lengh 4. If no such edge exiss, hen he ah is a 4, conadicing he fac ha f is ead-once. Case : m =, Figue 6d. In his case hee exis wo disinc heedges S, S, S S = W on he ah fom E o F. This imlies ha x E and x F do no belong o a common heedge and due o f s confomali, no edge can exis beween hem in G(H f ). If hee exis wo nodes, q Q and w W such ha no edge (q, w) exiss in G(H f ), hen he following ccle is codless: (q, x E, w, x F, q), conadicing ou assumion ha G(H f ) does no conain chodless ccles. Ohewise, eve wo nodes in Q and W shae an edge in G(H f ), his means ha he nodes Q W {x E } fom a clique in G(H f ). Le S be he maximal clique conaining nodes Q W {x E }. Since f eesens a boolean exession in iedundan DNF fom, hen E S, heefoe x E E \ S. I is imoan o noe ha hee mus exis some w W such ha (x E, w) is no an edge in G(H f ). Ohewise, we aive a a conadicion of S being a maximal clique because x E is conneced o all nodes in W and o all nodes in Q {x E }, hough heedge E, foming a clique conaining nodes Q W {x E } {x E } subsuming S. This esuls in he following codless ah in G(H f ): x E, x E, w, x F. If hee is an edge beween x E and x F in G(H f ), hen he ccle x E, x E, w, x F, x E is chodless binging us o a conadicion ha G(H f ) does no conain codless ccles. Ohewise, he ah x E, x E, w, x F is a 4 conadicing he fac ha f is ead-once. Theoem shows ha an ead-once lineage exession whose co-occuence gah does no conain codless ccles, o is a foes, is also γ-acclic. Eq. in Examle is one such examle. We summaize he elaionshi beween γ-acclic and ead-once lineage exessions in Table 5. γ-acclic ead-once Examle abc bde = b(ac de) - abc bde dfg - ab bc cd da = (a c)(b d) - - ab bc cd de ae Table 5: Read-onl and γ-acclic elaion examles 7. CONCLUSIONS AND FUTURE RESEARCH We have esened a new class of lineage exessions of queies ove ule indeenden obabilisic daabases

13 Examle 5. Conside he lineage exession f(a, b, c, e, f, g) = abcfh bcg abd aef fhi (a) Examle ah in H (b) ah in G(H f ) (c) Case, m = (d) Case, m = Figue 6: ahs beween heedges E and F fo which evaluaion can be efomed in TIME. We have ovided a se of algoihms of low olnomial daa comlexi and shown fomall he elaionshi beween he new γ-acclic class and he ead-once class esened befoe. An oen quesion is whehe hee is a lage se of lineage exessions whose obabili can be calculaed efficienl wihou eewidh esicion. Some β-acclic exessions have dbj eesenaions, e ohes do no, as demonsaed in Examle 5. An ineesing quesion is whehe hee is an efficien algoihm fo chaaceizing β-acclic exessions fo which a dbj eesenaion exiss. b,c,g e a f d c i (a) Co-occuence gah b,c a,b a,b,c,f,h b h a,f b a,b,d a a,e,f f f,h,i (b) Dual gah Figue 7: β-acclic lineage exession examle f,h g The co-occuence and dual gahs fo his exession ae esened in Figue 7. Noe ha his exession is no ead-once due o he 4 in Figue 7a. B looking a he dual gah in Figue 7b we can see ha i does no have a β-ccle (see Definiion ) and heefoe is β- acclic. Howeve, egadless of he node chosen as oo in Figue 7b, he esuling juncion ee is no a dbj. I is woh noing ha he mehod esened in Secion 5 is alicable also given he co-occuence gah of he fomula s CNF exession, whee an edge beween wo nodes in he gah exiss iff he aea ogehe in some CNF ime imlican. Cuenl, hee is no algoihm o consuc such a co-occuence gah in TIME. Anhe fuue eseach diecion would seek chaaceisics of juncion ees ha ield algoihms ha enable efficien comuaion of he equied obabili. One final eseach diecion involves exending Algoihm o handle ees ha do no have disjoin banches. ehas hee exiss a diffeen wa o analze he comlexi of he obabili comuaion on such ees. 8. REFERENCES [] C. Beei, R. Fagin, D. Maie, A. Mendelzon, J. Ullman, and M. Yannakakis. oeies of acclic daabase schemes. In STOC, ages 55 6, 98. [] C. Beei, R. Fagin, D. Maie, and M. Yannakakis. On he desiabili of acclic daabase schemes. J. ACM, :479 5, Jul 98. [] O. Benjelloun, A. Sama, A. Halev, M. Theobald, and J. Widom. Daabases wih unceain and lineage. VLDB Jounal, 7():4 64, 8. [4] R. Cavallo and M. iaelli. The heo of obabilisic daabases. In VLDB, ages 7 8, 987. [5] N. Dalvi, K. Schnaie, and D. Suciu. Comuing que obabili wih incidence algebas. In ODS, ages 4, New Yok, NY, USA,. ACM. [6] N. Dalvi and D. Suciu. Efficien que evaluaion on obabilisic daabases. In VLDB, ages , 4. [7] N. Dalvi and D. Suciu. Efficien que evaluaion on obabilisic daabases. VLDB Jounal, 6:5 544, Ocobe 7. [8] R. Deche. Consain ocessing. Elsevie Mogan Kaufmann,. [9] D. Duis. Some chaaceizaions of gamma and bea-acclici of hegahs. Technical eo, Univesi ais Dideo, 8. [] R. Fagin. Acclic daabase schemes (of vaious degees): A ainless inoducion. In CAA, ages 65 89, 98.

14 [] R. Fagin. Degees of acclici fo hegahs and elaional daabase schemes. J. ACM, :54 55, 98. [] N. Fuh and T. Rölleke. A obabilisic elaional algeba fo he inegaion of infomaion eieval and daabase ssems. ACM Tansacions on Infomaion Ssems, 5: 66, 994. [] D. R. Fulkeson and O. A. Goss. Incidence maices and ineval gahs. acific J. Mah., 5:85 855, 965. [4] M. Golumbic, A. Minz, and U. Roics. Facoing and ecogniion of ead-once funcions using cogahs and nomali. In DAC, Las Vegas, NV, USA, June. [5] A. K. Jha and D. Suciu. Knowledge comilaion mees daabase heo: comiling queies o decision diagams. In ICDT, ages 6 7,. [6] M. Jodan and C. Bisho. An inoducion o obabilisic gahical models. used b emission in MIT couse 6.867,. [7] B. Kanagal and A. Deshande. Lineage ocessing ove coelaed obabilisic daabases. In SIGMOD, ages ,. [8] D. Kolle and N. Fiedman. obabilisic Gahical Models: inciles and Techniques (Adaive Comuaion and Machine Leaning). The MIT ess, Augus 9. [9] I. Newman. On ead-once boolean funcions. In M. aeson, edio, Boolean Funcion Comlexi, ages 5 4. Cambidge Univesi ess, 99. [] D. Oleanu, J. Huang, and C. Koch. Aoximae confidence comuaion in obabilisic daabases. In ICDE, ages 45 56,. [] S. Ro, V. educa, and V. Tannen. Fase que answeing in obabilisic daabases using ead-once funcions. In ICDT, ages 4,. []. Sen, A. Deshande, and L. Geoo. db: managing and exloiing ich coelaions in obabilisic daabases. VLDB Jounal, 8(5):65 9, Oc. 9. []. Sen, A. Deshande, and L. Geoo. Read-once funcions and que evaluaion in obabilisic daabases. VLDB, ():68 79,. [4] R. E. Tajan and M. Yannakakis. Addendum: Simle linea-ime algoihms o es chodali of gahs, es acclici of hegahs, and selecivel educe acclic hegahs. SIAM J. Comu., 4():54 55, 985. [5] M. Yannakakis. Algoihms fo acclic daabase schemes. In VLDB, ages 8 94, 98. AENDIX A. DISJOINT BRANCH JUNCTION TREE CONSTRUCTION This secion esens he algoihm fo consucing a dbj fom a γ-acclic lineage exession [9]. Befoe his, we ove he equivalence beween Definiion 6 and he definiion of Duis: A juncion ee T of hegah H is a dbj if he heedges of H belonging o diffeen banches of T ae disjoin. Theoem 4. Definiion 6 and dbjs ae equivalen. oof. We ecall ha each node of he juncion ee coesonds o a se of vaiables fom one o moe heedges of hegah H. A dbj equies ha nodes on diffeen banches of he ee ae disjoin. Le T be a dbj. Theefoe, fo each node A in T wih childen A,..., A m, V a(a i ) V a(a j ) =, fo each i, j [, m], i j. Theefoe comling o Definiion 6. Now, le T coml o Definiion 6. Le us assume, b conadicion, ha i is no a dbj, i.e, he same vaiable Z aeas in wo nodes A, B on diffeen banches. Since T is a juncion ee, i comlies o he juncion ee oe, hus Z mus aea in he leas common anceso of nodes A, B, LCA(A, B) (such a leas common anceso exiss because T is a ee). Since A, B ae on diffeen banches, hen node LCA(A, B) has a leas wo childen wih a common vaiable, Z, in conadicion o Definiion 6. We denoe b f he lineage exession, and b H(V, E) is hegah. Afe veifing ha f is α and hen γ- acclic we consuc H s dual gah G (V, E ), which is he inu o Algoihm 4. Algoihm 4: Disjoin banch juncion ee ceaion Inu: G (V, E ), Roo Ouu: A disjoin banch juncion ee T wih oo. : T : Remove node fom G and le G (V, E ) be he gah induced b V (V \ ). : Le G (V, E ),..., G l (V l, E l ) be he se of conneced comonens ha esul fom se. 4: fo j o l do 5: Define i agmax v Vi C v C {find he node wih he lages inesecion wih he oo} 6: T T (, i ) 7: T i Call Algoihm 4 wih aamees G(V i, E i ) and oo i 8: end fo 9: Reun T Algoihm 4 ecusivel adds edges o he foes T unil a ee, ooed a, is fomed. The main a of he algoihm occus in he loo in lines 4-8. The loo ieaes ove all conneced comonens fomed when emoving, along wih is adjacen edges, fom G (se

15 ). Fo each such conneced comonen a single edge is added o T, which connecs o he node, in his comonen, having he lages inesecion wih. In line 7 he algoihm is called ecusivel on his node. Theoem 5. Algoihm 4 euns a disjoin banch juncion ee. oof. We need o show ha Algoihm 4 euns a ee ha saisfies boh he juncion ee oe and he condiions of Definiion 6. The oof is b inducion on he numbe of nodes in he gah G (V, E ). Fo he base case ( V = ), he algoihm euns a ee wih a single node, which saisfies boh Definiion 6 and he juncion ee oe. Le us assume he coecness of he algoihm fo gahs G (V, E ) s.. V n and ove he coecness fo a gah wih n nodes. We fis show ha he esuling ee saisfies he condiions in Definiion 6. B he inducion hohesis, each ecusive call ceaes such a ee T i, i [, l]. Theefoe, we onl need o show ha fo each ai of nodes i, j, s.. i, j [, l] and i j, C i C j =. If we assume he cona hen we ge a conadicion o he fac ha G i, G j ae wo diffeen conneced comonens. We now ove he juncion ee oe fo he ee T. We need o ove ha fo each v V he se of nodes in T ha conain v fom a conneced subee. Le us look a some v V, hee ae hee cases:. i [, l] s.. v V i and v / C. Since V i n, hen he inducion hohesis alies o i, and heefoe he nodes ha conain v in G i fom a conneced subee in T i. Fuhemoe, since v / C hen v / V j fo all j i (ohewise we ge a conadicion o he fac ha G i, G j ae diffeen conneced comonens). Theefoe, he nodes ha conain v fom a conneced comonen in T as well.. v C and i [, l] s.. v V i. This means ha v belongs o a single node in T, hence he juncion ee oe iviall alies o i.. No wo nodes on he ah inesec, ohewise we ge a conadicion o is minimali. Theefoe, we have a ue ccle of lengh a leas fou ha conains nodes, i, u and he nodes on ah (since is lengh is a leas, i conains a leas one node). We have aived, again, a a conadicion o he γ-acclici oe. To conclude, v C i and hence he juncion ee oe alies o he consuced ee T. Theoem 6. The un-ime comlexi of Algoihm 4 is O( V ). oof. In each ieaion ove a gah wih V = n nodes, exacl n nodes ae esed fo hei inesecion wih he cuen oo. This leads o an oveall comlexi of (n i) = O(n ) i=n i=. v C and i [, l] s.. v V i. If v C i hen b he inducion hohesis he nodes in V i ha conain v ae conneced in subee T i. Since v C and j i v / V j, hen he nodes in V ha conain v ae conneced in T. Le us assume b conadicion ha v / C i. Since v V i, le us look a some node u V i ha conains v. If C i C u = Q hen v / Q, heefoe emoving Q fom each node in G (V, E ) kees nodes i and u conneced hough he oo, in conadicion o he γ-acclici oe (Definiion 4). This is because v (C C u ) \ Q and C C i = W Q. Ohewise, W Q W C u W {v} C u C, heefoe W = (C C i ) (C C u ) = Q in conadicion o he fac ha i was chosen b he algoihm in line 5. If C i C u = hen le us look a he shoes ah ha connecs hem in G i. Since C i C u =,