A Bit-Parallel Tree Matching Algorithm for Patterns with Horizontal VLDC s

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1 A Bit-Prllel Tree Mthin Alorithm or Ptterns with Horizontl VLDC s Hisshi Tsuji 1, Akir Ishino 1, nd Msyuki Tked 1,2 1 Deprtment o Inormtis, Kyushu University 33, Fukuok , Jpn 2 SORST, Jpn Siene nd Tehnoloy Aeny (JST) {h-tsuji, ishino, tked}@i.kyushu-u..jp Astrt. The tree pttern mthin prolem is, iven two leled trees P nd T, respetively lled pttern tree nd tret tree, to ind ll ourrenes o P within T. Mny studies hve een undertken on this prolem or oth the ses o ordered nd unordered trees. To relize lexile mthin, kind o vrile-lenth-don t-re s (VLDC s) hve een introdued. In prtiulr, the pth-vldc s pper in XPth, lnue or ddressin prts o n XML doument. In this pper, we introdue horizontl VLDC s, eh mthes sequene o trees whose root nodes re onseutive silins in ordered trees. We ddress the tree pttern mthin prolem or ptterns with horizontl VLDC s. In our settin, the tret tree is iven s ted sequene suh s XML dt strem. We present n lorithm tht solves the prolem in O(mn) time usin O(mh) spe, where m nd n re the sizes o P nd T, respetively, nd h is the heiht o T. We dopt the it-prllel tehnique to otin prtilly st lorithm. 1 Introdution Semistrutured dt, in prtiulr XML douments, hs emered reently nd hs een widely spred. Tree pttern mthin plys entrl role in queryin suh semistrutured dt. Let N nd Σ e disjoint inite sets o symols. One strtion o XML douments would e ordered leled trees suh tht the internl nodes re leled with elements rom N nd the leves re leled with elements rom Σ. The prolem we ddressed in this pper is to ind ll ourrenes o pttern tree in tret tree, where the tret tree is n ordered leled tree hvin two kinds o lels s desried ove, nd the pttern tree is lmost the sme s the tret tree exept tht the lels o leves re either onstnt symols in Σ or vrile symols in V = {x 1,x 2,...}. Exmples o pttern nd tret trees re displyed in Fi. 1. The pentons leled with vriles in the pttern tree re met nodes, eh o whih is repled with (possily empty) sequene o trees. Sine the enerl se o this prolem is NP-hrd, we restrit our ttention to the lss o reulr pttern trees suh tht every vrile ours t most one in pttern tree. We note tht the vriles in reulr pttern tree t s M. Consens nd G. Nvrro (Eds.): SPIRE 2005, LNCS 3772, pp , Spriner-Verl Berlin Heideler 2005

2 A Bit-Prllel Tree Mthin Alorithm 389 h x 1 x 3 h h x 2 x 4 Fi. 1. A pttern tree is displyed on the let nd tret tree is displyed on the riht, where the symols,,h re rom N,ndthesymols,, re rom Σ. The symols x 1,x 2,x 3,x 4 in the pttern tree re vriles rom V. The tret tree hs one ourrene o the pttern tree, the root o whih is indited y n rrow. The vriles x 1,x 2,x 3,x 4 re respetively repled with the sequenes o trees surrounded with roken lines. vrile-lenth-don t res (VLDC s in short), similrly to the strin mthin se. Zhn, et l. introdued in [20] the notions o pth-vldc s nd umrell- VLDC s, nd the ormer ppers in XPth, lnue or ddressin prts o n XML doument. The pth-vldc s t s VLDC s in the vertil diretion, rom the root to le. On the ontrry, the vriles o our prolem t s VLDC s in the horizontl diretion, rom the let to the riht. To our est knowlede, this is the irst reserh delin with the horizontl VLDC s in tree mthin. It should e stted tht Kilpeläinen llowed loil vriles to e lels o leves in pttern trees in Chpter 6 o [9]. We note tht only tree is sustituted or vrile in his settin, while sequene o trees is sustituted or vrile in our settin. The notion o internl vriles ws introdued y Shoudi, et l. in [18], whih re repled with ritrry trees. In this pper we present n online lorithm solvin the prolem in O(mn) time usin O(mh) spe, where m nd n re the sizes o P nd T, respetively, nd h is the heiht o T. We then dopt the it-prllel tehnique to otin prtilly st lorithm. 2 An Overview o Tree Pttern Mthin 2.1 Vrious Notions o Ourrene o Pttern Tree The nestor reltion on the nodes o tree is the relexive trnsitive losure o the prent reltion. The let-to-riht order in n ordered tree T, denoted y T, is prtil order on the nodes o T deined s ollows: u T v i u = v or the lowest ommon nestor o u nd v hs two hildren u nd v suh tht u nd v re nestors o u nd v, respetively, nd u is let silin o v. Proposition 1. For ny two distint nodes u, v o n ordered tree T,eithero the ollowin sttements holds: (1) u is n nestor o v or vie vers; nd (2) u T v or vie vers.

3 390 H. Tsuji, A. Ishino, nd M. Tked Deinition 1 (ourrene o pttern tree). An ordered leled tree P is sid to our in n ordered leled tree T i there exists n injetion ϕ rom the nodes o P to the nodes o T whih stisies the ollowin onditions. (C1) ϕ preserves lels : For ny node u in P,thelelou is identil to the lel o ϕ(u). (C2) ϕ preserves the nestor reltion : For ny nodes u, v in P, u is n nestor o v in P i nd only i ϕ(u) is n nestor o ϕ(v) in T. (C3) ϕ preserves the let-to-riht order : For ny nodes u, v in P, u P v in P i nd only i ϕ(u) T ϕ(v) in T. Kilpeläinen nd Mnnil [10] ddresses the tree pttern mthin prolem with the ove notion o pttern ourrene (reerred to s the ordered tree inlusion prolem) nd presents n O( P T ) time nd spe lorithm, sin on the dynmi prormmin tehnique. The notion miht e too enerl. Replin the onditions (C2) nd/or (C3) with stroner onditions ives restrited notion o ourrene. For exmple, the ondition (C2) n e strenthened s ollows. (C2 ) ϕ preserves the prent reltion : For ny nodes u, v in P, u is the prent o v in P i nd only i ϕ(u) is the prent o ϕ(v) int. It is esy to see tht (C2 ) implies (C2). I ϕ stisies oth (C2 ) nd (C3), then silins in P re mpped to silins in T, nd the order o silins is preserved. Tht is: (C3 ) ϕ preserves the order o silins: For ny nodes u, v in P, u is let silin o v in P i nd only i ϕ(u) is let silin o ϕ(v) int. We note tht (C2 ) nd (C3) hold i nd only i (C2 ) nd (C3 ) hold. In [3] the tree pttern mthin or the omintion o (C1), (C2 ), nd (C3 ) is disussed, nd n lorithm is iven tht runs in O( T l(p )) time ter O( T + P Σ ) time nd spe preproessin nd with O( T + P Σ ) extr spe, where l(p ) denotes the numer o leves in P. The ondition (C3 ) n e strenthened s ollows. (C3 ) ϕ preserves the order nd djeny o silins : For ny nodes u, v in P, u is n immedite let silin o v in P i nd only i ϕ(u) is n immedite let silin o ϕ(v) int. (C3 ) ϕ preserves the numerin o silins : For ny node u o P, u is the i-th hild o its prent in P i nd only i ϕ(u) isthei-th hild o its prent in T. The notion o ourrene implied y (C1), (C2 ), nd (C3 ) is lled ompt ourrene [3]. In Fi. 2, the pttern ourrene on the let is ompt, while the pttern ourrene on the riht is not ompt. The works [8,11,6,4,5,12,13] re devoted to serhin or ompt ourrenes. Fi. 2 illustrtes tree ourrenes.

4 A Bit-Prllel Tree Mthin Alorithm 391 Fi. 2. The pttern tree on the let hs two ourrenes within the tret tree on the riht, whih re indited y rrows. The let ourrene is ompt while the riht one is not ompt. 2.2 VLDC s in Strins nd in Trees For while we turn to the se o strin mthin, not tree mthin. Let e VLDC tht mthes ny strin over n lphet Σ. A VLDC pttern is strinoverσ { }. For instne, is VLDC pttern tht mthes ny strin o the orm uv with u, v Σ.Thesustrin pttern mthin nd the susequene pttern mthin re speil ses o the VLDC pttern mthin in whih the ptterns re restrited to the orm 1 2 k nd to the orm 1 2 k, respetively, where i Σ or i =1,...,k (k >0). In the se o tree pttern mthin, there n e seen two types o strins : One is strin o lels spelled out y pth rom the root to le (vertil strins), nd the other is strin o lels spelled out y let-to-riht sequene o silins (horizontl strins). The onditions (C2) nd (C2 ), respetively, n e rerded s the susequene mthin nd the sustrin mthin or the vertil strins. The onditions (C3 ), (C3 ), nd (C3 ), respetively, n e viewed s the susequene mthin, the sustrin mthin, nd the preix mthin or the horizontl strins. Thus, introduin VLDC s into tree pttern mthin in the vertil nd in the horizontl diretions enerlize our prolem. As VLDC s or vertil strins, the notion o pth-vldc s ws introdued in [20]. The pth-vldc s pper in XPth, lnue or ddressin prts o XML douments, where they re denoted y //. However, VLDC s or horizontl strins hs een not disussed to our est knowlede. In the ollowin setion, we introdue horizontl VLDC s into the tree pttern mthin. 3 Pttern with Horizontl VLDC s It would e most suitle to deine the ptterns with horizontl VLDC s s speil se o the hedes [17]. The expressions (xy()z) nd ((x)y)(x) zh() re exmples o the hedes, where,, re onstnt symols, nd x, y, z re vrile symols. The hedes resemle the irst order terms, ut the rities o untion symols re ree. The hedes re lso lled orests [19] nd ordered orests [1] nd rerded s dt strutures suited or representin semistrutured dt suh s XML douments [7,14,15].

5 392 H. Tsuji, A. Ishino, nd M. Tked From now on, we express hede (x()) s [ x [ ] ]. The deinition o hedes ollows. Let Σ e inite set o onstnt symols, nd let V = {x 1,x 2,...} e ountle set o vrile symols. Let N e set o nmes, ndletb L = { [ N}nd B R = { ] N}, respetively. The elements in B L (resp. B R ) re lled the let rkets (resp. the riht rkets). We ssume Σ N =. Deinition 2. The hedes re reursively deined s ollows. The empty strin ε is hede. A onstnt symol Σ is hede. Avrilesymolx V is hede. I N nd h is hede, then [ h ] is hede. I h 1 nd h 2 re hedes, then the ontention h 1 h 2 is hede. A hede is sid to e round i it ontins no vriles. We denote y H nd y H G the sets o hedes nd round hedes, respetively. A sustitution is mppin rom V to H speiied y x 1 := h 1,...,x k := h k (h 1,...,h k H). Note tht the empty sustitution is llowed here. A sustitution is nturlly extended to the domin H. Deinition 3 (HedeMthin). Given hede p nd round hede t, determine whether there exists sustitution θ with pθ = t. The strin version o HedeMthin in whih the input hedes ontin no rket symols is identil to the memership prolem or pttern lnues [2], whih is known to e NP-omplete. Thus HedeMthin is NP-hrd. Ahedeh is reulr i every vrile ours t most one within h. Oneinterestin restrition o HedeMthin would e ReulrHedeMthin deined s ollows. Deinition 4 (ReulrHedeMthin). Given reulr hede p nd round hede t, determine whether there exists sustitution θ with pθ = t. The strin version o the ove prolem is known s the VLDC pttern mthin nd is solvle in liner time. We shll onsider vrint o the ove prolem. A hede p is sid to e suhede o nother hede t i there exist vrile x nd hede h with x suh tht t = hθ or sustitution θ = {x := p}. Deinition 5 (ReulrHedeSerhin). Given reulr hede p nd round hede t, ind ll suhedes t o t suh tht pθ = t or some sustitution θ. We onentrte on ReulrHedeSerhin. In the next setion we present n lorithm or solvin this prolem.

6 A Bit-Prllel Tree Mthin Alorithm Alorithm or ReulrHedeSerhin 4.1 Bsi Ide Consider the reulr hede P =[ x[ ] ]. I only the strins over Σ n e sustituted or x, the lnue o P is L =[ Σ [ ] ]. The lnue L is reulr. Fi. 3 shows n NFA eptin the lnue Σ L, where the rs leled with denote stte trnsition y n ritrry symol Σ. We note tht in the stte-trnsition dirm the rs leled with let rkets re depited oin to the lower-riht diretion nd the rs leled with riht rkets re depited oin to the upper-riht diretion. Suh hierrhil illustrtion will e needed in desriin our lorithm. Fi. 3. NFA uilt rom the pttern hede P =[ x[ ] ] In relity, ritrry round hedes re sustituted or vriles. We need mehnism or skippin not only symol in Σ ut lso round hede in the orm [ n h ] n (n N,h H G ) t sel-loop leled with o the NFA. Exmple 1. Consider the move o the NFA in Fi. 3 runnin on the round hede T 1 =[ [ ] [ ] ]. We wnt to skip t stte 2 the hede [ ],whihis sustrin einnin t position 3 o T 1, so tht the NFA epts T 1 ter redin the lst symol ]. Exmple 2. Consider the move o the NFA in Fi. 3 or T 2 =[ [ [ ] ] ].I we llow the NFA to skip the third symol [ o T 2 t stte 2, then the NFA is in inl stte 6 just ter redin the seond symol ] rom the lst. This leds lse detetion o the pttern. 4.2 The Alorithm Denote the NFA or pttern hede P s mentioned ove y M P =(Q, Σ B L B R,δ,Q 0,F) where: Q = {0, 1,...,m} is the set o sttes; δ : Q (Σ B L B R ) 2 Q is the stte trnsition untion;

7 394 H. Tsuji, A. Ishino, nd M. Tked Fi. 4. Copies o NFA M P, eh indin ourrenes o P =[ x[ ] ] t nodes o the orrespondin depth in tret tree Q 0 = {0} is the set o initil sttes; nd F = {m} is the set o inl sttes. Here, m is the numer o symols in the round hede otined rom P y removin ll vriles. For ny suset S o Q nd ny symol in Σ B L B R, let δ(s, ) = δ(q, ). q S We wnt to ind ourrenes o P t ny position o ny depth in tret hede. We rete opies o M P or ll possile depths in the tret hede, nd simulte the moves o ll the opies. See Fi. 4. Nive method would e, or every opy o M P, to store the tive sttes s set vrile nd simulte the nondeterministi stte trnsitions y updtin the vlue o the set vrile. Sine we hve to updte the set vriles or ll possile depths or eh o the symols in the tret hede, the simultion is time onsumin. The key ide in overomin this prolem is to prllelize the stte trnsitions o multiple opies o M P in depthwise mnner. Nmely, ll the sttes o the opies o M P re lssiied into roups ordin to their depths, nd the tive sttes in eh roup re stored into the orrespondin set vrile. We use s stk n rry S o sets so tht S[d] stores the tive sttes o depth d. Initilly, set the vrile depth to 0. The lorithm reds the symols o the tret hede rom the let to the riht, nd lters the vlue o depth ordinly. Tht is, it inrements depth y one when redin let rket [ n, derements depth y one when redin riht rket, nd do nothin when redin symol o Σ. From the tive sttes in S[depth] or old vlue o depth nd rom the input symol, we ompute the set o tive sttes or the new vlue o depth y usin the stte trnsition untion δ nd store them into S[depth] orthenewvlueo depth.

8 A Bit-Prllel Tree Mthin Alorithm 395 Input: A round hede T = T [1..n] (T [i] Σ B L B R)Cnd the NFA M P or reulr hede P. Output: All ourrenes o P within T. Method: ein Let LoopSttes e the set o sttes hvin sel-loops in M P ; depth := 0; S[depth] := ; or j := 1to n do := T [j]; i B L then S[depth + 1] := δ(s[depth],) {0}; (1) depth := depth + 1; else i Σ then S[depth] := δ(s[depth],); (2) else // B R S[depth 1] := δ(s[depth],) (S[depth 1] LoopSttes); (3) depth := depth 1; i S[depth] ontins inl stte then Report n ourrene o P t position j o T ; end. Fi. 5. Alorithm or ReulrHedeSerhin A mehnism or skippin hedes t the sttes with loops n e relized s ollows. Suppose tht the urrent vlue o depth is d, ndsttes hs loop nd is tive, i.e. s S[d]. I the next input symol is let rket [ n, then the vlue o S[d] remins without lterntion nd the lorithm oes to the lower diretion with inrementin depth y one. The vlue o S[d] is never hned (nd thereore ontins s) until it returns to the sme depth d y redin the orrespondin riht rket ] n. Amon the tive sttes stored in S[d], the ones with loops should remin tive. The lorithm is summrized s in Fi. 5. The move o the lorithm serhin T =[ h [ [ ] [ ] ] ] h or P =[ x[ ] ] is displyed in Fi. 6. The trees orrespondin to P nd T re,respetively,shownontheletndonthe riht o Fi. 7. Theorem 1. ReulrHedeSerhin is solved in O(mn) time usin O(mh) spe, where m nd n re the sizes o the pttern hede nd the tret hede, respetively, nd h is the heiht o the tret hede. 4.3 Eiient Implementtion y Bit-Prllel Tehnique Now we exploit the it-prllel tehnique [16] to otin n eiient implementtion o our lorithm. The set S[depth] Q = {0, 1,...,m} or every depth is

9 396 H. Tsuji, A. Ishino, nd M. Tked Fi. 6. Move o the lorithm, where the pttern hede is [ x[ ] ] nd the tret hede is [ h [ [ ] [ ] ] ] h. The illed nd empty irles men the tive nd intive sttes, respetively. The retnle indites S[depth], the set o tive sttes t the urrent depth. The sttes upper thn the urrent depth re sleepin in the stk S. represented with (m + 1)-it inteer. The stte trnsition untion δ is relized s ollows. For eh symol in Σ B L B R, the msk vetor Msk() ={i 1 i m, P [i] =} is uilt, where P denotes the round hede otined y removin vriles rom the iven pttern P.The set LoopSttes is lso represented s n (m + 1)-it

10 A Bit-Prllel Tree Mthin Alorithm 397 h x Fi. 7. Trees desried y the reulr hede P =[ x[ ] ] (on the let) nd y the round hede T =[ h [ [ ] [ ] ] ] h (on the riht). An ourrene o P in T is emphsized with thik lines. inteer. Then, or ny S Q nd or ny Σ B L B R, The set o the next sttes δ(s, ) n e otined s ((S <<1)&Msk()) (S&LoopSttes), where <<C&C, respetively, mens the it-shit, the itwise AND, nd the itwise OR opertions. The omputtion o δ(s[depth],) t lines (1),(2),(3) o the lorithm o Fi. 5 n e repled with omintion o the it-shit nd the itwise loil opertions s mentioned ove. The runnin time o the lorithm is then O( m/w n), nd is O(n) im + 1 is t most the lenth w o omputer word. 5 Conlusion We ddressed the prolem o indin reulr hede P within round hede T, nd presented n eiient lorithm sin on the it-prllelism. The reulr hedes n e rerded s tree ptterns with horizontl VLDC s. We note tht Chuve [3] dels with tree pttern mthin with notion o ourrenes in whih the prent reltion nd the order nd djeny o silins re preserved. The prolem ddressed in this pper is enerliztion o the one ddressed in [3]. Reerenes 1. P. C. Amoth, T. R. nd P. Tdeplli. Ext lernin o tree ptterns rom queries nd ounterexmples. In Proeedins o COLT 98, pes , D. Anluin. Findin ptterns ommon to set o strins. J.Comput.Sys.Si., 21:46 62, C. Chuve. Tree pttern mthin with more enerl notion o ourrene o the pttern. Inorm. Proess. Lett., 82: , R. Cole nd R. Hrihrn. Tree pttern mthin nd suset mthin in rndomized O(n lo 3 n)-time. In STOC 97, pes 66 75, R. Cole, R. Hrihrn, nd P. Indyk. Tree pttern mthin nd suset mthin in deterministi O(n lo n)-time. In SODA 99, pes , 1999.

11 398 H. Tsuji, A. Ishino, nd M. Tked 6. M. Duliner, Z. Glil, nd E. Men. Fster tree pttern mthin. J. ACM, 41(2): , F. N. Geert Jn Bex, Sestin Mneth. A orml model or n expressive rment o xslt. In Proeedins o CL 2000 (LNAI 1861), pes , C. M. Homnn nd M. J. O Donnell. Pttern mthin in trees. J. ACM, 29(1):68 95, P. Kilpeläinen. Tree Mthin Prolems with Applitions to Strutured Text Dtses. PhD thesis, Dept. o Computer Siene, University o Helsinki, P. Kilpeläinen nd H. Mnnil. Ordered nd unordered tree inlusion. SIAM J. Comput., 24(2): , S. R. Kosrju. Eiient tree pttern mthin. In FOCS 89, pes IEEE Comput. So. Press, F. Luio nd L. Pli. An eiient lorithm or some tree mthin prolems. Inorm. Proess. Lett., 39(1):51 57, F. Luio nd L. Pli. Approximte mthin or two milies o trees. Inormtion nd Computtion, 123(1): , M. Murt. Trnsormtion o douments nd shems y ptterns nd ontextul onditions. In Proeedins o Doument Proessin 96 (LNCS 1293), pes , M. Murt. Dt model or doument trnsormtion nd ssemly (extended strt). pes , G. Nvrro nd M. Rinot. Flexile pttern mthin in strins: Prtil on-line serh lorithms or texts nd ioloil sequenes. Cmride University Press, Cmride, M. Nivt nd H. Ait-Ki. On reonizle sets nd tree utomt. In Resolution o Equtions in Aleri Strutres T. Shoudi, T. Uhid, nd T. Miyhr. Polynomil time lorithms or indin unordered tree ptterns with internl vriles. In FCT 01, pes , M. Tkhshi. Generliztions o reulr sets nd their pplition to study o ontext-ree lnues. Inormtion nd Control, 27:1 36, K. Zhn, D. Shsh, nd J. T. L. Wn. Approximte tree mthin in the presene o vrile lenth don t res. J. Alorithms, 16(1):33 66, 1994.