Compression vs Queryability - A Case Study

Transcription

1 Compression vs Queryility - A Cse Stuy Siv Annthrmn To ite this version: Siv Annthrmn. Compression vs Queryility - A Cse Stuy. Dgstuhl Seminr 08621, Jun 2008, Dgstuhl, Germny , Struture-Bse Compression of Complex Mssive Dt. < <inri > HAL I: inri Sumitte on 22 Jn 2010 HAL is multi-isiplinry open ess rhive for the eposit n issemintion of sientifi reserh ouments, whether they re pulishe or not. The ouments my ome from tehing n reserh institutions in Frne or ro, or from puli or privte reserh enters. L rhive ouverte pluriisiplinire HAL, est estinée u épôt et à l iffusion e ouments sientifiques e niveu reherhe, puliés ou non, émnnt es étlissements enseignement et e reherhe frnçis ou étrngers, es lortoires pulis ou privés.

2 Compression vs Queryility - A Cse Stuy Siv Annthrmn LIFO, Université Orléns (Frne) e-mil: siv@univ-orlens.fr Astrt Some ompromise on ompression is known to e neessry, if the reltive positions of the informtion store y semi-struture ouments re to remin essile uner queries. With this in view, we ompre, on n exmple, the query-frienliness of XML ouments, when ompresse into strightline tree grmmrs whih re either regulr or ontext-free. The queries onsiere re in limite frgment of XPth, orresponing to type of ptterns; eh suh query efines nturlly non-eterministi, ottom-up query utomton tht runs just s well on tree s on its ompresse g. Keywors: Tree utomt, Tree Grmmrs, Dgs, XML ouments, Queries. 1 Introution Strutures over gs inste of over trees hve een wiely use in orer to optimize lgorithms. Tree utomt (TA) re mong the si tools employe for querying XML ouments (e.g., [10, 11, 16, 17]); on the other hn, the notion of ompresse XML oument hs een introue in [2, 9, 14], n possile vntge of using g strutures for the mnipultion of suh ouments hs een rought out in [14]. It is legitimte then to investigte the possiility of using utomt running iretly over gs inste of over trees, for querying ompresse XML ouments. Unfortuntely however, the Dg Automt (DA), efine s nturl extension of tree utomt in [5] s ottom-up tree utomt running on gs, nnot iretly serve suh purpose. The reson is tht the lss of their lnguges efine s the set of gs epte uner their ottom-up runs is lgerilly ill-ehve ([1]): lthough eterministi ottom-up TA runs extly like on g or on its unompresse tree, the set of gs epte y non-eterministi DA oes not represent, in generl, regulr tree lnguge. Thus if the notion of eptne is not pte ppropritely, the lnguges of DAs (woul) form strit superlss of the lss of regulr tree lnguges. Note, on the other hn, tht the nswers to MSO-efinle queries on semi-struture trees re known to e regulr tree lnguges, f. [17, 19]. So, for suitle efinition of eptne. ottom-up Dgstuhl Seminr Proeeings Struture-Bse Compression of Complex Mssive Dt

3 run of non-eterministi TA, on g, hs to e ouple in generl with n on the fly eterminiztion of the TA long the run. Severl oservtions re in orer t this point, efore we proee. First, we re onerne with queries on XML ouments, n the trees moeling suh ouments re unrnke, i.e., the symols t the noes n hve ritrrily mny rguments; so the utomt we wnt to employ for querying shoul lso e unrnke. A seon oservtion is tht, fully or prtilly ompressing n unrnke tree into DAG is not mjor issue: suffies to represent it s strightline regulr tree grmmr (SLR), e.g., s in [3]. In the sequel, therefore, we shll onsier the terms oument n g to e synonymous; n y tree, we shll men g with opying fully llowe, i.e., with no ompression t ll. It must e pointe out here tht higher egree of ompression of rnke tree n e hieve y representing it s strightline, ontext-free, tree grmmr (SLCF), f. e.g., [3, 4]; ut it is not ler if the requirement on the se lphet to e rnke n e roppe. Now, ptterns (f. [15], n Setion 3 elow) re often onveniently use to speify queries on XML ouments; n they n e nturlly visulize s ottom-up, non-eterministi, query utomt; the full evlution of suh queries i.e., ess to the informtion store, long with their respetive reltive positions is quite possile on ouments ompresse uner SLR, without ny nee for eompression; while on oument ompresse uner SLCF, heking for query stisfiility n e heke effiiently, ut full query evlution ppers to e more intrite. Severl mehnisms hve een propose in the literture for querying (ompresse, unrnke) ouments. The one propose in [2] is priori for query stisfiility, n employs tree-like utomt; it n e either top-own, or ottom-up, on gs. Tht of [7] is mixture of top-own n ottom-up nlysis, se on the SLR vision, suitle for query evlution on gs, for frgment of Core XPth, ut it is not hr to exten it little frther. [17] proposes very generl query mehnism nme query utomton (QA), whih is 2-wy tree utomton with speifie seleting sttes: seletion lel 1 is ttriute to some speifie sttes, n 0 or to the others; this works only for trees, however. For the view presente elow, our onern will e limite to lss of queries tht re expressile insie restrite XPth formt; they orrespon to the ptterns using the / n // of XPth, s efine in [15], possily with some filters (or rnhes, s ws lle there). Besies eing simple, this limittion will lso hve the vntge tht suh queries n e visulize s usul, ottomup, non-eterministi tree utomt with speifie seleting sttes (Note: this, inientlly, will lso llow us to onsier n-ry queries). This pper is struture s follows. The nottion n the preliminries re given in Setion 2. In Setion 3, we efine the notion of ptterns (s in [15]), n show how to visulize them, nturlly, s ottom-up query utomt (QA); we lso show, on n exmple, how to evlute the query orresponing to given pttern p on given oument t, vi ottom-up run on t of the QA ssoite with p, if the oument is ompresse uner SLR; on the sme exmple, we will 2

4 lso show tht, if the oument gets ompresse s SLCF, the query is not tht esy to evlute (lthough it n e teste for stisfiility). A finl remrk efore losing this setion: lthough we note ove tht ottom-up runs of non-eterministi TA on gs n e prolemti, no suh omplition rises tully for the QAs efine y the ptterns we onsier here; so, we o not nee to eterminize the QA on the fly, long its evluting runs. Moreover, it n e shown tht the nswer to the query efine y ny given pttern on given SLR-ompresse oument t, will e the sme s when the query is evlute on the unompresse tree equivlent of t. 2 Nottion n Preliminries We ssume given n unrnke se lphet Σ. Trees n gs over Σ re efine s usul, eh of their noes ring nme tht is symol from Σ. (Forml efinitions o not seem neee.) A first exmple to show wht we men y fully or prtilly ompresse gs is the following: f f f Tree Fully Compresse Prtilly Compresse The most elegnt n effiient wy to istinguish etween these 3 formts of the sme oument (sme informtion, ut store with more or less optimiztion) is to ssoite to eh formt t, nonilly, strightline regulr tree grmmr (SLR) G t : nonil in the sense tht G t reognizes extly t, n nothing else. The three respetive SLR for the ove 3 formts re s follows: X 0 f(x 1,X 2,X 3,X 4 ) X 1 X 2 X 3 X 4 X 0 f(y 1,Y 1,Y 2,Y 1 ) Y 1 Y 2 X 0 f(z 1,Z 1,Z 2,Z 3 ) Z 1 Z 2 Z 3 The notions of hil, prent, esenent, nestor re ll efine, in nturl mnner, on the set of noes of ny given g. A noe is root (resp. lef) for g, iff it hs no prent (resp. hil). All our gs will e ssume roote, i.e., to hve unique root noe. It is then esy to ssoite set of positions to ny given noe on ny given g, suh tht the following hols: A g is tree iff the set of positions of ny of its noes is singleton. (Note: noe on g n hve more thn one prents.) 3

5 We lso nee the the notion of ottom-up tree utomton over n unrnke lphet; to filitte unerstning, we first rell the notion over rnke lphet; the efinition is esily extene to the unrnke se. Definition 1 A ottom-up tree utomton (TA) over rnke lphet Σ is tuple (Σ,Q,F,Δ), whereq is finite non-empty set of sttes, F Q is the set of finl (or epting) sttes, n Δ is set of trnsition rules of the form: f(q 1,..., q k ) q, wheref Σ is of rnk k, nq 1,...,q k,q Q. Now, trnsition f(q 1,...,q k ) q n lso e written s f(q 1...q k ) q, where q 1...q k is seen s wor in Q, tht hs to e of length = rnk(f) in the rnke se. So the extension is esy to the unrnke se: suffies to efine the trnsitions to e of the form f(ω) q, whereω Q,nf Σ. A TA is si to e ottom-up eterministi iff whenever there re two trnsition rules of the form f(ω) q, f(ω ) q, with q q, we hve neessrily ω ω = ; otherwise it is si to e non-eterministi. We lso gree to enote the trnsitions of the form f( ) q simply s f q, nrefertothems initil trnsitions. Exmple. We ome now to our seon exmple: to the right of the figure elow is the tree formt of oument, of whih the fully ompresse g formt is to the left. Over the unrnke signture {, f, g} we onsier the ottom-up TA A, with the following trnsitions: p, p, q, (p) q, (q) p, g(qq ) q, g(pq) p, f(qpq) q fin, f(pq ) q fin, with Q = {p, q, q fin }, q fin eing the unique epting stte. f f g g g It is not hr to hek tht there is n epting run of the TA on the tree to the right. An if we wnt to run the TA iretly on the g to the left, we my hve to eterminize the run on the fly, s n when we move up, in generl. This n e one more elegntly, n more nturlly, y seeing the g s its SLR, n y seeing its proutions s well s the trnsitions of the TA s rewrite rules, n using innermost rewriting. For instne, the SLR for 4

6 the g to the left ove is s follows: X 0 f(x 1,X 1,X 2 ) X 1 g(x 3,X2) X 3 (X 2 ), X 2 We get y innermost rewriting: X 2 = {p, q}, X 3 = (X 2 ) {p, q}, X 1 g({p, q}{p, q}), n finlly: X 0 f({p, q}{p, q}{p, q}). An we en up y heking tht q fin is in the set f({p, q}{p, q}{p, q}). This is how one woul proee, in generl, for eiing eptne uner the ottom-up runs of non-eterministi TAs on gs. However, to ny query efine y pttern of limite formt tht we shll e onsiering elow, we shll nturlly ssoite ottom-up non-eterministi TA, lle its query utomton (QA); for suh QA, the seleting runs n e ompute more iretly (without on the fly eterminiztions). 3 Query Automt for Ptterns We heneforth ssume known the usul notions n terminology of XML, n of XPth whih is lnguge generlly employe for querying XML ouments. For simpliity, we only onsier unry queries, i.e., queries tht return set of noes n/or the t tthe to these noes. We re intereste here in limite su-lss of XPth expressions whih n lso e seen nturlly s non-eterministi ottom-up QA. These expressions re ll representle s ptterns with seletion lels with the help of the symols /, //, of XPth, long with those from the (unrnke) se lphet Σ, n the filter expressions of XPth. For instne, the XPth expression // [//][] is represente s the following pttern, where,,, re in the se lphet, is the wilr, n s is the seletion symol: * s Anoeu on ny given oument t is n nswer to the query of this pttern, iff: - the root noe of t ers the nme, nu is esennt of the root, -nu hs two hilren noes n, the hil nme itself hving esennt. 5

7 The XPth expressions tht we re intereste in here, n e efine formlly in terms of simple grmmrs; for instne, s in [15], one n efine them y: P ::= σ. P/P P//P P [P ] where σ Σ, is the wilr of XPth, n. stns for the urrent noe position. To eh suh expression, one n ssoite pttern tree (pttern, for short), s in the ove exmple (f. [15] for etils). A pttern is essentilly (roote) tree with usul eges, plus some istinguishe oule-eges (s in the ove exmple), referre to s esennt-eges; the noes re nme from Σ { }; in ition, to extly one of the noes is ssigne seletion symol: 1ors. (Note: we re only onerne here with unry queries.) An essentil ifferene etween the usul trees n ptterns is tht the set of outgoing noes from noe, on pttern, re not orere. (Thus, in the ove pttern exmple, it is not require tht the -hil, of the noe u to e selete, e to the left of its -hil.) A point we wnt to rive in now, is tht to ny pttern n e ssoite QA in nturl mnner. We shll o tht only on n exmple, with whih we will ontinue in the next setion. Consier, for instne, the following pttern: s This pttern represents the following XPth expression: //[]//[.//], whih is short for: //[hil : ]//[eennt : ]. The QA ssoite with this pttern is the ottom-up non-eterministi TA over the lphet {,,, }, with Q = {q in,q 0,q 1,q 2,q 3 q } s its set of sttes, q s the epting stte, q 1 s the seleting stte, n the following trnsitions: q in (Q q in Q ) q in q 0 (Q q in Q ) q 0 (Q q 0 Q ) q 0 (Q q 0 Q ) q 1 (Q q 1 Q ) q 1 (Q q in Q ) q 2 (Q q 1 Q q 2 Q ) q 3 (Q q 2 Q q 1 Q ) q 3 (Q q 3 Q ) q 3 (Q q 3 Q ) q (Q q Q ) q (The Q stns for ny string over the set Q.) The semntis of the trnsitions re s follows: t ny lef noe other thn -nme we woul e in stte q in ; when we reh n -noe we woul e in stte q 0 ;whenwereh-noe ove 6

8 we woul e in stte q 1 ;t-noe, in generl we woul e in q 2, ut if it is ove -noe, we woul e in q 3 ; finlly, t ny noe stritly ove -noe rehe t stte q 3, we woul e in the epting stte q. (Note: query utomton returns selete noe only on n epte oument.) 4 Querying uner SLR or uner SLCF We show here, riefly, how the query //[hil : ]//[eennt : ] ne evlute on the following oument : t = ((, ),((, ),((, ),((, ),(, ))))), uner the runs of the QA efine in the previous setion. The exmple hs een orrowe from [3]; s ws shown there, the following SLCF with S, A, B(y),C s non-terminls, S eing the xiom gives nie ompression of this tree: S B(B(C)) B(y) (C, (C, y)) C (A, A) A Oserve first tht our QA, if run on the tree t, woul selet the suterm ((, ),(, )), the position of whih (on t) n e eue y following the pth from the -noe to whih is ssigne the seleting stte q 1,totheroot. Now, the g to the left of the figure ove is the fully ompresse g formt for t; n it is not hr to hek tht our QA n run ottom-up just s well on this g (without neeing ny eterminiztion on the fly), s on the tree t; the g will e epte, n the -noe to the ottom-right will e ssigne the seleting stte q 1 ; the set of positions on g eing well-efine, one eues esily the position(s) of the selete noe: it orrespons here to the pth forme of the full rrows in the g to the right. It is esy then to output the su-g t tht position (or its SLR), s the nswer to our query. On the other hn, our QA n lso run on the SLCF ompresse formt given ove for the oument t; we will then get the following: 7

9 - there is n innermost rewrite erivtion from the xiom S to the stte q ; -none of the intermeiry terms thus erive gets selete. In other wors, we n onlue tht the query efine y the pttern oes hve n nswer, somewhere. It oul lso e possile to sy extly where, using some itionl n more intrite syntx, se on the BPLEX lgorithm of [3, 4]. But, lthough BPLEX is ottom-up multiplex, it is not ler if it n output the ext SLCF grmmr orresponing to the nswer of the query; more preisely, it oes not seem simple to relte the SLCF grmmr of the nswer to the SLCF of the given oument. Before we lose this setion, few oservtions re perhps in orer: i) Severl of the usul XPth se queries on XML ouments or their skeletons n e visulize s pttern trees. ii) Bottom-up unrnke query utomt n e erive nturlly, from suh ptterns. iii) These unrnke query utomt re (not esy to eterminize, ut they) run ottom-up on ompresse gs, just s esily s on their orresponing unompresse trees; n the nswer sets otine orrespon. iv) Compression se on SLR, n the ottom-up evlution tehnique esrie ove for queries efine y our ptterns, n oth ontinue to funtion without ny mjor moifitions on ouments tht re sujet to rule-se ess ontrol poliies (RBAC). It is not ler if the sme hols for SLCF-se ompression. 5 Conlusion We hve trie to show tht lthough ompression uner SLR ensures t est only single exponentil spe optimiztion, it hs more query-frienly ehvior on severl fronts, thn ompression se on SLCF. We hve lso shown tht noneterministi, ottom-up, tree utomt re nturl n intuitive nites for fully evluting ertin lss of queries in the XPth formt, visulizle s ptterns, even on ouments ompresse s DAGs (it is tully the formultion s pttern tht gives the lue for eriving n utomton orresponing to the query). The pttern-se view for queries seems to hve other vntges s well: for instne, n-ry queries n e hnle quite nturlly in tht set up. Moreover, the ptterns of the formt stuie here n lso e iretly efine in terms of suitle grmmr; this is of help in hnling other prolems, suh s tht of pttern ontinment, vi rewrite tehniques (f. e.g., [8]). Referenes [1] S. Annthrmn, P. Nrenrn, M. Rusinowith, Closure Properties n Deision Prolems of Dg Automt, In Informtion Proessing Letters, 94(5): , [2] P. Bunemn, M. Grohe, C. Koh, Pth queries on ompresse XML. In Pro. of the 29th Conf. on VLDB, 2003, pp , E. Morgn Kufmnn. 8

10 [3] G. Bustto, M. Lohrey, S. Mneth, Grmmr-Bse Tree Compression. EPFL Teh. Report IC/2004/80, [4] G. Bustto, M. Lohrey, S. Mneth, Effiient Memory Representtion of XML Douments. In Pro. DBPL 05, LNCS 3774, pp , Springer-Verlg, [5] W. Chrtonik, Automt on DAG Representtions of Finite Trees, Tehnil Report MPI-I , Mx-Plnk-Institut für Informtik, Srrüken, Germny. [6] H. Comon, M. Duhet, R. Gilleron, F. Jquemr, D. Lugiez, S. Tison, M. Tommsi, Tree Automt Tehniques n Applitions, [7] B. Fil, S. Annthrmn, Automt for Positive Core XPth Queries on Compresse Douments, In Pro. of the Int. Conf. LPAR-13, pp , LNAI 4246, Springer-Verlg, Novemer [8] B. Fil-Kory, A Rewrite Approh for Pttern Continment, In Pro. WADT 2008, Pis, June [9] M.Frik,M.Grohe,C.Koh, Query Evlution of Compresse Trees, InPro.of LICS 03, pp , IEEE, [10] G. Gottlo, C. Koh, Moni Queries over Tree-Struture Dt, In Pro. of LICS 02, pp , IEEE. [11] G. Gottlo, C. Koh, Moni Dtlog n the Expressive Power of Lnguges for We Informtion Extrtion, In Journl of the ACM, 51(1):12 28, [12] G. Gottlo, C. Koh, R. Pihler, L. Segoufin, The omplexity of XPth query evlution n XML typing In Journl of the ACM 52(2): , [13] W. Mrtens, F. Neven, On the omplexity of typeheking top-own XML trnsformtions, In Theoretil Computer S., 336(1): , [14] M. Mrx, XPth n Mol Logis for Finite DAGs. In Pro. of TABLEAUX 03, pp , LNAI 2796, [15] G. Miklu, D. Suiu, Continment n Equivlene for Frgment of XPth, In Journl of the ACM, 51(1):2-45, [16] F. Neven, Automt Theory for XML Reserhers, In SIGMOD Reor 31(3), Septemer [17] F. Neven, T. Shwentik, Query utomt over finite trees, In Theoretil Computer Siene, 275(1 2): , [18] A. Potthof, S. Seiert, W. Thoms, Noneterminism versus eterminism of finite utomt over irete yli grphs, In Bull. Belgin Mth. Soiety, 1: , [19] J.W. Thther, J.B. Wright, Generlize finite utomt theory with n pplition to eision prolem of seon-orer logi, In Mth. Syst. Theory, 2(1):57 81,