Efficient Processing of Ordered XML Twig Pattern

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1 Effcent Processng of Ordered XML Twg Pattern Jaheng Lu, Tok Wang Lng, Tan Yu, Changqng L, and We N School of computng, Natonal Unversty of Sngapore {lujahen, lngtw, yutan, lchangq, nwe}@comp.nus.edu.sg Abstract. Fndng all the occurrences of a twg pattern n an XML database s a core operaton for effcent evaluaton of XML queres. Holstc twg jon algorthm has showed ts superorty over bnary decompose based approach due to effcent reducng ntermedate results. The exstng holstc jon algorthms, however, cannot deal wth ordered twg queres. A straghtforward approach that frst matches the unordered twg queres and then prunes away the undesred answers s obvously not optmal n most cases. In ths paper, we study a novel holstc-processng algorthm, called OrderedTJ, for ordered twg queres. We show that OrderedTJ can dentfy a large query class to guarantee the I/O optmalty. Fnally, our experments show the effectveness, scalablty and effcency of our proposed algorthm. 1 Introducton Wth the rapdly ncreasng popularty of XML for data representaton, there s a lot of nterest n query processng over data that conforms to a tree-structured data model([1],[5]). Effcent fndng all twg patterns n an XML database s a major concern of XML query processng. Recently, holstc twg jon approach has been taken as an effcent way to match twg pattern snce ths approach can effcently control the sze of ntermedate results([1],[2],[3],[4]). We observe that, however, the exstng work on holstc twg query matchng only consdered unordered twg queres. But XPath defnes four ordered axes: followng-sblng, precedng-sblng, followng, precedng. For example, XPath: //book/text/followng-sblng::chapter s an ordered query, whch fnds all chapters n the dataset that are followng sblngs of text whch should be a chld of book. We call a twg query whch cares the order of the matchng elements as an ordered twg query. On the other hand, we denote a twg query that does not consder the order of matchng elements as an unordered query. In ths paper, we research how to effcently evaluate an ordered twg query. To handle an ordered twg query, navely, we can use the exstng algorthm (e.g. TwgStack[1]/TwgStackLst[5]) to output the ntermedate path solutons for each ndvdual root-leaf query path, and then merge path solutons so that the fnal solutons are guaranteed to satsfy the order predcates of the query. Although exstng algorthms are appled, such a post-processng approach has a serous dsadvantage: many ntermedate results may not contrbute to fnal answers. K.V. Andersen, J. Debenham, and R. Wagner (Eds.): DEXA 2005, LNCS 3588, pp , Sprnger-Verlag Berln Hedelberg 2005

2 Effcent Processng of Ordered XML Twg Pattern 301 Motvated by the recent success n effcent processng unordered twg queres holstcally, we present n ths paper a novel holstc algorthm, called OrderedTJ, for ordered twg queres. The contrbuton of ths paper can be summarzed as follows: 1. We develop a new holstc ordered twg jon algorthm, namely OrderedTJ, based on the new concept of Ordered Chldren Extenson (for short OCE). Wth OCE, an element contrbutes to fnal results only f the order of ts chldren accords wth the order of correspondng query nodes. Thus, effcent holstc algorthm for ordered-twgs can be leveraged. 2. If we call edges between branchng nodes and ther chldren as branchng edges and denote the branchng edge connectng to the n th chld as the n th branchng edge, we analytcally demonstrate that when the ordered-twg contans only ancestor-descendant relatonshp from the 2nd branchng edge, OrderedTJ s I/O optmal among all sequental algorthms that read the entre nput. In other words, the optmalty of OrderedTJ allows the exstence of parent-chld relatonshps n non-branchng edges and the frst branchng edges. 3. Our expermental results show that the effectveness, scalablty and effcency of our holstc twg algorthms for ordered twg pattern. The remander of the paper s organzed as follows. Secton 2 presented related work. The novel ordered twg jon algorthm s descrbed n Secton 3. Secton 4 s dedcated to our expermental results and we close ths paper by concluson and future work n Secton 5. 2 Related Work Wth the ncreasng popularty of XML data, query processng and optmzaton for XML databases have attracted a lot of research nterest. There s a rch set of lterature on matchng twg queres effcently. Below, we descrbe these lteratures wth the notce that the exstng work deals wth only unordered twg queres. Zhang et al.([9]) proposed a mult-predcate merge jon (MPMGJN) algorthm based on (DocId, Start, End, Level) labelng of XML elements. The later work by Al- Khalfa et al.([7]) gave a stack-based bnary structural jon algorthm. Dfferent from bnary structural jon approaches, Bruno et al.([1]) proposed a holstc twg jon algorthm, called TwgStack, to avod producng a large ntermedate result. However, the class of optmal queres n TwgStack s very small. When a twg query contans any parent-chld edge, the sze of useless ntermedate results may be very large. Lu et al.([5]) propose a new algorthm called TwgStackLst. They use lst data structure to cache lmted elements to dentfy a larger optmal query class. TwgStackLst s I/O optmal for queres wth only ancestor-descendant relatonshps n all branchng edges. Recently, Jang et al.([3]) researched the problem of effcent evaluaton of twg queres wth OR predcates. Chen et al.([2]) researched the relatonshp between dfferent data partton strateges and the optmal query classes for holstc twg jon. Lu et al.([6]) proposed a new labelng scheme called extended Dewey to effcently process XML twg pattern.

3 302 J. Lu et al. 3 Ordered Twg Jon Algorthm 3.1 Data Model and Ordered Twg Pattern We model XML documents as ordered trees. Fgure 1(e) shows an example XML data tree. Each tree element s assgned a regon code (start, end, level) based on ts poston. Each text s assgned a regon code that has the same start and end values. XML queres make use of twg patterns to match relevant portons of data n an XML database. The pattern edges are parent-chld or ancestor-descendant relatonshps. Gven an ordered twg pattern Q and an XML database D, a match of Q n D s dentfed by a mappng from the nodes n Q to the elements n D, such that: () the query node predcates are satsfed by the correspondng database elements; and () the parent-chld and ancestor-descendant relatonshps between query nodes are satsfed by the correspondng database elements; and () the orders of query sblng nodes are satsfed by the correspondng database elements. In partcular, wth regon encodng, gven any node q Q and ts rght-sblng r Q(f any), ther correspondng database elements, say e q and e r n D, must satsfy that e q.end<e r.start. The answers to query Q wth n nodes can be represented as a lst of n-ary tuples, where each tuple (t 1,t 2,.t n ) conssts of the database elements that dentfy a dstnct match of Q n D. Fgure 1(a) shows three sample XPath and Fgure 1(b-d) shows the correspondng ordered twg patterns for the data of Fg 1(e). For each branchng node, we use a symbol > n a box to mark ts chldren ordered. Note that n Q3, we add book as the root of the ordered query, snce t s the root of XML document tree. For example, the query soluton for Q3 s only <book 1, chpater 2, ttle 2, related work, secton 3 >. But f Q3 were an unordered query, secton 1, secton 2 also would nvolve n answers. XPah of Q1: //chapter/secton/precedngsblng::ttle XPah of Q2: /book/author/followngsblng::chpater/ttle/followngsblng::secton XPah of Q3: //chapter[ttle="related work"]/followng::secton (a) Xpath expressons chapter > ttle secton (b) Q1 book > author chapter > ttle (c) Q2 secton book > chapter secton ttle "Related work" (d) Q3 (2,3,2) author 1 chapter 1 (1,25,1) book 1 (4,10,2) (11,17,2) chapter2 (18,24,2) chapter 3 (5.7,3) (8.9,3) (12,14,3) (15.16,3) (19,21,3) (22,23,3) ttle 1 secton 1 ttle 2 secton 2 ttle 3 secton 3 (6.6,4) (13,13,4) (20,20,4) "Introducton" "Related work" "Algorthm" (e) XML document Fg. 1. (a) three XPaths (b)-(d) the correspondng ordered twg query (e) an XML tree 3.2 Algorthm In ths secton, we present OrderedTJ, a novel holstc algorthm for fndng all matches of an ordered twg pattern aganst an XML document. OrderedTJ makes the extenson of TwgStackLst algorthm n the prevous work [5] to handle ordered twg pattern. We wll frst ntroduce data structures and notatons to be used by OrderedTJ.

4 Effcent Processng of Ordered XML Twg Pattern 303 Notaton and data structures. An ordered query s represented wth an ordered tree. The functon, return chld nodes whch has parent-chld or ancestor-descedant relatonshps wth n, respectvely. The self-explanng functon returns the mmedate rght sblng node of n (f any). There s a data stream T n assocated wth each node n n the query twg. We use C n to pont to the current element n T n. We can access the values of C n by, and. The cursor can advance to the next element n T n wth the procedure. Intally, C n ponts to the frst element of T n. Our algorthm wll use two types of data structures: lst and stack. We assocate a lst L n and a stack S n for each node of queres. At every pont durng computaton: the nodes n stack S n are guaranteed to le on a root-leaf path n the database. We use to denote the top element n stack S n. Smlarly, elements n each lst L n are also strctly nested from the frst to the end,.e. each element s an ancestor or parent of that followng t. For each lst L n, we declare an nteger varable, say p n, as a cursor to pont to an element n L n. Intally, p n =0, whch ponts to the frst element of L n. a > b c d e 1 b 1 a 1 c 1 e2 d 1 e 1 b 1 a 1 c 1 d 1 b a > c c 1 a 1 a 2 b 1 (a) Query (b) Doc1 (c) Doc2 (a) Query (b) Data Fg. 2. Illustraton to ordered chld extenson Fg. 3. Optmalty example The challenge to ordered twg evaluaton s that, even f an element satsfes the parent-chld or ancestor-descendant relatonshp, t may not satsfy the order predcate. We ntroduce a new concept, namely Ordered Chldren Extenson (for short, OCE), whch s mportant to determne whether an element lkely nvolves n ordered queres. DEFINITION 1(OCE) Gven an ordered query Q and a dataset D, we say that an element e n (wth tag n Q) n D has an ordered chldren extenson (for short OCE), f the followng propertes are satsfed: () for n ADRChldren( n) n Q (f any), there s an element e n (wth tag n ) n D such that e n s a descendant of e n and e n also has OCE; () for n PCRChldren( n) n Q (f any), there s an element e (wth tag n) n the path e n to e n such that e s the parent of e n and e n also has OCE; () for each chld n of n and m = rghtsblng(n ) (f any), there are elements e n and e m such that e n.end < e m.start and both en and e have OCE. Propertes () and () dscuss the ancestor-descendant and parent-chld relatonshp respectvely. Property () manfests the order condton of queres. For example, see m

5 304 J. Lu et al. the ordered query n Fg 2(a), n Doc 1, a 1 has the OCE, snce a 1 has descendants b 1,d 1, chld c 1 and more mportantly, b 1,c 1,d 1 appear n the correct order. In contrast, n Doc2, a 1 has not the OCE, snce d 1 s the descendant of c 1, but not the followng element of c 1 (.e. c 1.end d 1.start ). Algorthm OrderedTJ () 01Whle ( end() ) 02 q act = getnext(root 03 f (sroot(q act ) empty( S parent ( qact ))) cleanstack(q act, getend(q act ) 04 movestreamtostack(q act, S q act 05 f (sleaf(q act )) 06 showpathsolutons ( S,getElement(q act ) 07 else proceed( T 08 mergeallpathsolutons; Functon end() 01 return n subtreesnodes( root): sleaf ( n) eof ( C n q act Procedure cleanstack (n,actend) 01 whle ( empty( S n ) and (topend(s n )< actend)) do pop(s n Procedure movestreamtostack(n,s n ) 01 f((getend(n)< top(s rghtsblng(n) ).start) //check order 02 push getelement(n) to stack S n 06 proceed(n Procedure proceed(n) 01 f (empty(l n )) advance(t n 02 else L n.delete(p n 03 p n = 0; //move p n to pnt to the begnnng of L n Procedure showpathsolutons(s m,e) 01 ndex[m]=e 02 f (m == root ) //we are n root 03 Output(ndex[q 1 ],,ndex[q k ]) //k s the length of path processed 04 else //recursve call 05 for each element e n S parent(m) 06 f e satsfes the correspondng relatonshp wth e 07 showpathsolutons(s parent(m), e ) q act Fg. 4. Rocedure OrderedTJ Algorthm OrderedTJ. OrderedTJ, whch computes answers to an ordered query twg, operates n two phases. In the frst phase (lne 1-7), the ndvdual query rootleaf paths are output. In the second phase (lne 8), these solutons are merged-joned to compute the answers to the whole query. Next, we frst explan getnext algorthm whch s a core functon and then presents the man algorthm n detals. getnext(n)(see Fg 5) s a procedure called n the man algorthm of OrderedTJ. It dentfes the next stream to be processed and advanced. At lne 4-8, we check the condton () of OCE. Note that unlke the prevous algorthm TwgStackLst[5], n lne 8, we advance the maxmal (not mnmal) element that are not descendants of the

6 Effcent Processng of Ordered XML Twg Pattern 305 current element n stream T n, as we wll use t to determne sblng order. Lne 9-12 check the condton () of OCE. Lne 11 and 12 return the elements whch volate the query sblng order. Fnally, lne check the condton () of OCE. Now we dscuss the man algorthm of OrderedTJ. Frst of all, Lne 2 calls getnext algorthm to dentfy the next element to be processed. Lne 3 removes partal answers that cannot be extended to total answer from the stack. In lne 4, when we nsert a new element to stack, we need to check whether t has the approprate rght sblng. If n s a leaf node, we output the whole path soluton n lne 6. Algorthm getnext (n) 01 f (sleaf(n)) return n; 02 for all n n chldren(n) do 03 g = getnext(n f (g n ) return n ; 04 nmax = max arg n ( ) ( ) chldren n getstart n ; 05 nmn = mn arg n ( ) ( ) chldren n getstart n ; 06 Whle (getend(n) < getstart(n max )) proceed(n 07 f (getstart(n) >getstart(n mn )) 08 return max arg n ( ) ( ( ) ( )) ( ) chldren n getstart n getstart n getstart n ; > 09 sort all n n chldren(n) by start values; // assume the new order are n 1,n 2,,n k 10 for each n (1 n) do //check chldren order 11 f (n' n ) return n ; 12 else f ((>1) (getend(n -1 )>getstart(n )) return n MoveStreamToLst(n, n max 14 for n n PCRchldren(n) //check parent-chld relatonshp 15 f ( e' L such that e s the parent of n C n ) 16 f (n s the frst chld of n) 17 Move the cursor of lst L q to pont to e ; 18 else return n ; 19 return n; Proceudre MoveStreamToLst(n,g) 01 delete any element n L n that s not an ancestor of getelement(n 02 whle C n.start < getstart(g) do f C n.end>getend(g) L n.append(c n 03 advance(t n ) Procedure getelement(n) 01 f ( empty( L n )) return L n.elementat(p n 02 else return C n ; Procedure getstart(n) 01 return the start attrbute of getelement(n Procedure getend(n) 01 return the end attrbute of getelement(n Fg. 5. Functon GetNext n the man algorthm OrderedTJ EXAMPLE 1. Consder the ordered query and data n Fg 1(d) and (e) agan. Frst of all, the fve cursors are (book 1, chapter 1, ttle 1, related work, secton 1 ). After two calls of getnext(book), the cursors are forwarded to (book 1, chapter 2, ttle 2, related work, secton 1 ). Snce secton 1.start=6<chapter 2.start=9, we return secton (n lne 11 of getnext) and forward to secton 2. Then chapter 2.end=15> secton 2.start=13. We

7 306 J. Lu et al. return secton agan (n lne 12 of getnext) and forward to secton 3. Then chapter 2.end=15<secton 3.start=17. The followng steps push book 1 to stack and output the ndvdual two path solutons. Fnally, n the second phase of man algorthm, two path solutons are merged to form one fnal answer 3.3 Analyss of OrderedTJ In the secton, we show the correctness of OrderedTJ and analyze ts effcency. Some proofs are omtted here due to space lmtaton. DEFINITION 2 (head element e n ) In OrderedTJ, for each node n the ordered query, f Lst L n s not empty, then we say that the element ndcated by the cursor p n of L n s the head element of n, denoted by e n. Otherwse, we say that element C n n the stream T n s the head element of n. LEMMA 1. Suppose that for an arbtrary node n n the ordered query we have getnext(n)=n. Then the followng propertes hold: n has the OCE. Ether (a) n=n or (b) parent(n) does not have the OCE because of n (and possbly a descendant of n ). LEMMA 2. Suppose getnext(n)=n returns a query node n the lne 11 or 12 of Algorthm getnext. If the current stack s empty, the head element does not contrbute to any fnal soluton snce t does not satsfy the order condton of query. LEMMA 3. In Procedure movestreamtostack any element e that s nserted to stack S n satsfy the order requrement of the query. That s, f n has a rght-sblng node n n query, then there s an element e n n stream T n such that e n.start >e n.end. LEMMA 4. In OrderedTJ, when any element e s popped from stack, e s guaranteed not to partcpate a new soluton any longer. THEOREM 1. Gven an ordered twg pattern Q and an XML database D. Algorthm OrderedTJ correctly returns all answers for Q on D. Proof:[sketch] Usng Lemma 2, we know that when getnext returns a query node n n the lne 11 and 12 of getnext, f the stack s empty, the head element e n does not contrbute to any fnal solutons. Thus, any element n the ancestors of n that use e n n the OCE s returned by the getnext before e n. By usng lemma 3, we guarantee that each element n stack satsfy the order requrement n the query. Further. By usng lemma 4, we can mantan that, for each node n n the query, the elements that nvolve n the root-leaf path soluton n the stack S n. Fnally, each tme that n =getnext(root) s a leaf node, we output all soluton for e n (lne 6 of OrderedTJ). Now we analyze the optmalty of OrderedTJ. Recall that the unordered twg jon algorthm TwgStackLst([5]) s optmal for query wth only ancestor-descendant n all branchng edges, but our OrderedTJ can dentfy a lttle larger optmal class than TwgStackLst for ordered query. In partcular, the optmalty of OrderedTJ allows the exstence of parent-chld relatonshp n the frst branchng edge, as llustrated below.

8 Effcent Processng of Ordered XML Twg Pattern 307 EXAMPLE 2. Consder the ordered query and dataset n Fg 3. If the query were an unordered query, then TwgStackLst([5]) would scan a 1, c 1 and b 1 and output one useless soluton (a 1,c 1 ), snce before we advance b 1 we could not decde whether a 1 has a chld tagged wth b. But snce ths s an ordered query, we mmedately dentfy that c 1 does not contrbute to any fnal answer snce there s no element wth name b before c 1. Thus, ths example tells us that unlke algorthms for unordered query, OrderedTJ may guarantee the optmalty for queres wth parent-chld relatonshp n the frst branchng edge. THEOREM 2. Consder an XML database D and an ordered twg query Q wth only ancestor-descendant relatonshps n the n th (n 2) branchng edge. The worst case I/O complexty of OrdereTJ s lnear n the sum of the szes of nput and output lsts. The worst-case space complexty of ths algorthm s that the number of nodes n Q tmes the length of the longest path n D. text > bold keyword (a) Q1 descrpton text > > text partlst bold keyword emph S NP VP VBN (b) Q2 (c) Q3 (d) Q4 (e) Q5 (f) Q6 PP > IN NP S > > NN PP DT PRP_DOLLAR_ Fg. 6. Sx tested ordered twg queres (Q1,2,3 n XMark; Q4,5,6 n TreeBank) 4 Expermental Evaluaton 4.1 Expermental Setup We mplemented three ordered twg jon algorthms: straghtforward -TwgStack ( for short STW), straghtforward-twgstacklst (STWL) and OrderedTJ. The frst two algorthms use the straghtforward post-processng approach. By post-processng, we mean that the query s frst matched as an unordered twg (by TwgStack[1] and TwgStackLst[5], respectvely) and then we merge all ntermedate path solutons to get the answers for an ordered twg. We use JDK 1.4 wth the fle system as a smple storage engne. All experments were run on a 1.7G Pentum IV processor wth 768MB of man memory and 2GB quota of dsk space, runnng wndows XP system. We used two data sets for our experments. The frst s the well-known benchmark data: XMark. The sze of fle s 115M bytes wth factor 1.0. The second s a real dataset: TreeBank[8]. The deep recursve structure of ths data set makes ths an nterestng case for our experments. The fle sze s 82M bytes wth 2.4 mllon nodes. For each data set, we tested three XML twg queres (see Fg 6). These queres have dfferent structures and combnatons of parent-chld and ancestor-descendant edges. We choose these queres to gve a comprehensve comparson of algorthms.

9 308 J. Lu et al. Evaluaton metrcs. We wll use the followng metrcs to compare the performance of dfferent algorthms. () Number of ntermedate path solutons Ths metrc measures the total number of ntermedate path solutons, whch reflects the ablty of algorthms to control the sze of ntermedate results. () Total runnng tme Ths metrc s obtaned by averagng the total tme elapsed to answer a query wth sx consecutve runs and the best and worst performance results dscarded. STW STWL Or der e dtj STW STWL Or der edtj STW STWL OrderedTJ Execut on t me( second) Q1 Q2 Q3 XMar k dat aset (a) XMark Execut on t me( second) Q4 Q5 Q6 Tr eebank dat aset (b) TreeBank Execut on t me (seconds) XMark factor (c) varyng data sze Fg. 7. Evaluaton of ordered twg pattern on two datasets Table 1. The number of ntermedate path solutons Query Dataset STW STWL OrderedTJ Useful solutons Q1 XMark Q2 XMark Q3 XMark Q4 TreeBank Q5 TreeBank Q6 TreeBank Performance Analyss Fgure 7 shows the results on executon tme. An mmedate observaton from the fgure s that OrderedTJ s more effcent than STW and STWL for all queres. Ths can be explaned that OrderedTJ output much less ntermedate results. Table 1 shows the number of ntermedate path solutons. The last column shows the number of path solutons that contrbute to fnal solutons. For example, STW and STWL could output 500% more ntermedate results than OrderedTJ (see XMark Q2). Scalablty. We tested queres XMark Q2 for scalablty. We use XMark factor 1(115MB) 2(232MB) 3 (349M) and 4(465M). As shown n Fg 7(c), OrderedTJ scales lnearly wth the sze of the database. Wth the ncrease of data sze, the beneft of OrderedTJ over STW and STWL correspondngly ncreases. Sub-optmalty of OrderedTJ. As explaned n Secton 3, when there s any parentchld relatonshp n the n th branchng edges (n 2), OrderedTJ s not optmal. As shown n Q4,Q5 of Table 1, none of algorthms s optmal, snce all algorthms output some useless solutons. However, even n ths case, OrderedTJ stll outperforms STW and STWL by outputtng less useless ntermedate results.

10 Effcent Processng of Ordered XML Twg Pattern 309 Summary. Accordng to the expermental results, we draw two conclusons. Frst, our new algorthm OrderedTJ, could be used to evaluate ordered twg pattern because they have obvous performance advantage over the straghtforward approach: STW and STWL. Second, OrderedTJ guarantee the I/O optmalty for a large query class. 5 Concluson and Future Work In ths paper, we proposed a new holstc twg jon algorthm, called OrderedTJ, for processng ordered twg query. Although the dea of holstc twg jon has been proposed n unordered twg jon, applyng t for ordered twg matchng s nontrval. We developed a new concept ordered chld extenson to determne whether an element possbly nvolves n query answers. We also make the contrbuton by dentfyng a large query class to guarantee I/O optmal for OrderedTJ. Expermental results showed the effectveness, scalablty, and effcency of our algorthm. There s more to answer XPath query than s wthn the scope of ths paper. Consder an XPath query: //a/followng-sblng::b, we cannot transform ths query to an ordered twg pattern, snce there s no root node n ths query. Thus, algorthm OrderedTJ cannot be used to answer ths XPath. In fact, based on regon code (start,end,level), none of algorthms can answer ths query by accessng the labels of a and b alone, snce a and b may have no common parent even f they belong to the same level. We are currently desgnng a new labelng scheme to handle such case. References 1. N. Bruno, N. Koudas, and D. Srvastava. Holstc Twg Jons: Optmal XML pattern matchng. In Proc. of the SIGMOD, pages T. Chen J. Lu, and T. W Lng On boostng holsm n XML twg pattern matchng usng structural ndexng technques In Proc. of the SIGMOD 2005 To appear 3. H. Jang, H. Lu, W. Wang, Effcent Processng of XML Twg Queres wth OR-Predcates, In Proc. of the SIGMOD pages H. Jang, et al. Holstc twg jons on ndexed XML documents. In Proc. of the VLDB, pages , J. Lu, T. Chen and T. W. Lng Effcent Processng of XML Twg Patterns wth Parent Chld Edges: A Look-ahead Approach In Proc. of CIKM, pages , J. Lu et. al From Regon Encodng To Extended Dewey: On Effcent Processng of XML Twg Pattern Matchng In Proc. of VLDB, 2005 To appear 7. S. Al-Khalfa et. al Structural jons: A prmtve for effcent XML query pattern matchng. In Proc. of the ICDE, pages , Treebank 9. C. Zhang et. al. On supportng contanment queres n relatonal database management systems. In Proc. of the SIGMOD, 2001.