Topological Queries on Graph-structured XML Data: Models and Implementations

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1 Topologicl Queries on Grph-structured XML Dt: Models nd Implementtions Hongzhi Wng, Jinzhong Li, nd Jizhou Luo Astrct In mny pplictions, dt is in grph structure, which cn e nturlly represented s grph-structured XML. Existing queries defined on tree-structured nd grph-structured XML dt minly focus on sugrph mtching, which cn not cover ll the requirements of querying on grph. In this pper, new kind of queries, topologicl query on grph-structured XML is presented. This kind of queries consider not only the structure of sugrph ut lso the topologicl reltionship etween sugrphs. With existing sugrph query processing lgorithms, efficient lgorithms for topologicl query processing re designed. Experimentl results show the efficiency of implementtion lgorithms. Keywords XML, Grph Structure, Topologicl query. I. INTRODUCTION In mny pplictions, dt cn e modeled s directed grph nd grph-structured XML cn e nturlly used to descrie grph-structured dt. For exmple, Figure I shows grph structure representing reltionships etween uthors nd cdemic ppers. Query processing on grph-structured dt rings new chllenges. Some query processing techniques hve een presented such s [6], [11], [7], [3].However, current techniques for grph-structured dt hve limittions. An notle one is tht current reserch on querying grph-structured dt minly focus on sugrph mtching without considering topologicl reltionship etween sugrphs. Topologicl reltionship of sugrphs cn e defined s the reltionship of sugrphs in position. For exmple, in Figure 2(), sugrph (1,1,2,2) hs overlpping reltionship with sugrph (1,1,2,c1) in ecuse they shre some nodes. In some pplictions of querying sugrphs on lrge grph, topologicl reltionship cn e used to descrie the constrint of sugrph more sutly. An exmple of the ppliction of topologicl constrints on the grph in Figure I is shown in Exmple 1. Exmple 1: Retrieve the conferences shring t lest one pper with sme title nd uthors with journl. Such query use topologicl reltionship overlpping s constrint. Processing this query in the grph in Figure I is to find sugrphs contining conference nd shring common prts of uthor informtion with some sugrph contining informtion out journl. Processing topologicl query is lso chllengele prolem. Current techniques of query processing on grph dtse cn e clssified into two kinds. Hongzhi Wng is with the School of Computer Science nd Technology, Hrin Institute of Technology, Hrin, emil: wngzh@hit.edu.cn Jinzhong Li is with the School of Computer Science nd Technology, Hrin Institute of Technology, Hrin, emil: lijzh@hit.edu.cn Jizhou Luo is with the School of Computer Science nd Technology, Hrin Institute of Technology, Hrin, emil:luojz@hit.edu.cn Mtch sugrph with structurl constrint on lrge grph [3], [13]. Such techniques retrieve sugrphs mtching given sugrph. Such techniques cn only solve suprolem of processing topologicl query. Additionlly, in some cses, the result of sugrph query is very lrge. If such sugrph is just constrint, only some fetures of intermedite results re useful. Therefore, it is not efficient for topologicl query processing. Select grphs with some feture in lrge set of grphs [14], [15], [8]. For topologicl query, the gol is to find some sugrph is lrge grph stisfying some constrint. Therefore, such techniques cnnot e pplied to topologicl query processing directly. Current techniques cn not e pplied directly on topologicl query processing. For processing topologicl queries efficiently, we present series of lgorithms sed on intervl sed leling scheme. Even sugrph mtching cnnot e voided during the processing, we only store useful fetures s intermedite results nd filter s mny useless nodes s possile. The contriutions of this pper cn e summrized s follows: Topologicl query, novel type of queries on grph, is presented. As we know, this is the first pper of considering topologicl reltionship etween sugrphs in query processing on grph dt. We design series of efficient lgorithms for processing topologicl queries with vrious topologicl constrints. With feture extrcting, mssive storge of intermedite results is voided. Experimentl results show tht our methods use smll extr cost of sugrph query processing nd hve good sclility. This pper is orgnized s follows. In Section II, some ckground knowledge is presented. In Section III, the definitions of topologicl query nd vrious topologicl reltionships re presented. We design topologicl query processing lgorithms in Section IV. Experimentl results re shown in Section V. In Section VI, we give n overview of work relted to this pper. We drw the conclusions in Section VII. II. PRELIMINARIES In this section, we riefly introduce the grph-structured XML model nd some techniques used in this pper. A. Dt Model With IDREF etween two elements representing their reference reltionship [12], XML dt cn e modeled s lelled 721

2 i conference confernece Journl s nme pper pper nme pper pper nme pper pper n 1 title uthor title uthor n 2 title uthor title uthor n 3 title uthor title uthor nme ddress editor nme ddress PCmemer nme ddress Chirmn t 1 t 2 t 3 t 4 t 1 t 4 p1 1 p 2 2 p 3 3 Fig. 1. An Exmple of Grph Structure digrph: elements nd ttriutes re mpped into nodes of the grph; directed nesting nd reference reltionships re mpped into edges in the grph. B. Rechiility Leling Scheme for Grph-structured XML Dt The gol of leling scheme of XML dt is to represent the structurl reltionship so tht the reltionship etween nodes in n XML grph cn e judged y their lels quickly. With suitle leling scheme, structurl query processing cn e efficient. In this pper, we focus on rechility leling scheme, which is used to judge the rechility reltionship etween nodes in n XML grph. We use n extension version of rechility leling scheme presented in [9]. In this leling scheme, ech node of grph is ssigned postid nd some intervls. For two nodes nd in grph, if nd only if.postid is contined y one of intervls ssocited with, so tht the rechility reltionship etween two nodes in the grph cn e judged y their lels without other informtion. The steps of encoding grph G is: 1) A DAG D is generted from G y contrcting ech strongly connected components (SCC for rief) in G into node. 2) Find n optiml tree-cover T of the DAG D generted in the first step. An id nd n intervl is ssigned ech node in T. For node n, its id is its postorder. In tts intervl, [x, y], y is the postorder of n nd x is the smllest postorder numer of ll n s descendnts in T. Note tht during the trversl, when n node n c contrcted from SCC is met, its numer of postorder is incresed y numer c, the numer of nodes in this SCC in G insted of 1. Suppose the postorder of the node met efore n c is pc. The id of this node is n series of numers of pc +1, pc +2,, pc + numer c. 3) Exmine ll the nodes of D in the reverse topologicl order. At ech node n, copy nd merge, if possile, ll the intervls of its out-going nodes in D to its code. 4) All nodes in the sme SCC in G re ssigned to sme intervls s the node n c contrcted from this strongly connected grph in D. Ech of the postid of n c in D is ssigned to node in this SCC. For exmple, the rechility coding of grph in Figure 2() is shown in Figure 2(). For ech node in Figure 2(), the numer fter colon is its id nd series of intervls fter id re the intervl set of this node. The id of c2 is contined in intervl [0, 2] of 3. By such reltionship, we cn judge tht 3 reches c2. III. CONCEPTS OF TOPOLOGICAL QUERIES In this section, we will give the concept of topologicl query. Intuitively speking, topologicl query is to use topologicl reltionship to descrie grph. To retrieve some grphs with some topologicl constrint with grph in some schem, we present six kinds of topologicl reltionships: connecting, connected y, disjoint, overlpping, contining nd contined y. At first, We give the intuitions of these six reltionships. Then we will gives forml definition of these reltionships nd topologicl constrints sed on them. A grph g 1 hs connecting reltionship with nother grph g 2 mens tht there is pth from some node in g 1 to some node in g 2. For exmple, in Figure 2(), sugrph (1,1) is connecting sugrph (2,c2) ecuse oth 2 nd 1 cn rech c2. A grph g 1 hs connected y reltionship with nother grph g 2 mens tht there is pth from some node in g 2 to some node in g 1. For exmple, in Figure 2(), sugrph (2,c2)is connected y sugrph (1,1). A grph g 1 hs disjoint reltionship with nother grph g 2 mens tht no node is shred y g 1 nd g 2. For exmple, in Figure 2(), sugrphs (2,c2)nd (1,1) re disjoint since they shres no node. A grph g 1 hs overlpping reltionship with nother grph g 2 mens tht g 1 nd g 2 shres t lest one node. For exmple, in Figure 2(), sugrphs (2,c2)nd (3,c2) re overlpping since they shres node c2. A grph g 1 hs contining reltionship with nother grph g 2 mens tht g 2 is sugrph of g 1. A grph g 1 hs contined y reltionship with nother grph g 2 mens g 1 is sugrph of g

3 r1 1:8,[0,8] r1:11,[0,11] :6,[0,6] 2:7,[0,2][7,7] 3:9,[1,3][9,9] 4:10,[4,5][10,10] 2 c1 c2 3 4 c3 () Exmple of Grph 2:0,[0,0] c1:1,[1,1] c2:2,[2,2] 3:3,[3,3] 4:4,[4,4] c3:5,[5,5] () Coding of Exmple Grph Fig. 2. An Exmple of Rechiility Leling Scheme The grph structure of XML dt cn e represented s leled, directed grph G =(V,E) with V the set of nodes nd E the set of edges. There is function tg : V TAG where TAG is the set of tgs nd tg(v) is the tg of v. Since topologicl query is sed on sugrph query, we will define sugrph query t first, then define topologicl reltions. Bsed on these concepts, topologicl query is defined formlly. Definition 1 (Sugrph Query): A grph Q =(V,E) is sugrph query, where there re two functions tg : V TAG nd rel : E AXIS. TAG is the set of tgs nd AXIS is the set of node reltions such s prent-child(pc for rief) or ncestor-descendnt(ad for rief). In Definition 1, the set of AXIS descries the reltionship etween two nodes. Bsed on the definition of XPth [4], there re 13 kinds of reltionship. Here we only discuss two xis in common use, PC nd AD. In grph G, two nodes n 1 nd n 2 V G hs PC reltionship if there is edge (n 1,n 2 ) E G. In grph G, two nodes n 1 nd n 2 V G hs AD reltionship if there is pth from n 1 to n 2 in G. Definition 2 (Result of Sugrph Query): The result of sugrph query Q =(V Q,E Q ) over grph G is R G,Q = {g equivlent function f g : V g V Q nd totl nd single vlued function p g : V g V G, such tht n V g,tg(n) =tg(f g (n)) = tg(p g (n))nd e =(n 1,n 2 ) E g,p g (n 1 ),p g (n 2 ) stisfy the reltionship rel(f(n 1 ),f(n 2 ))nd if n 1 n 2 then f g (n 1 ) f g (n 2 ), p g (n 1 ) p g (n 2 ).}. g R G,Q is represented s g = G Q. An exmple of sugrph query is shown in Exmple 2. Exmple 2: A sugrph query on dt in Figure 2() is shown in Figure 3(). The direction of ech edge is from top to ottom. Single line represents PC-reltionship nd doule line represents AD-reltionship. This query represents sugrph with n r node s source. This node must hve PC reltionship with n node nd AD reltionship with node. Such node nd node must hve PC reltionship etween the sme c node. The results of this sugrph query include (r1, 2, 1, c1), (r1, 2, 1, c2), (r1, 3, 1, c1),(r1, 3, 1, c2) nd (r1, 4, 1, c3). Definition 3 ( Connecting Reltionship): Two leled, directed grphs g 1 = G Q 1 hs Connecting reltionship with g 2 = G Q 2 in grph G if n 1 V g1 n 2 V g2, in grph G there is pth from p g1 (n 1 ) to p g2 (n 2 ). Definition 4 ( Connected Reltionship): Two leled, directed grphs g 1 = G Q 1 hs Connected y reltionship with g 2 = G Q 2 in grph G if n 1 V g1 n 2 V g2, in grph G there is pth from p g2 (n 2 ) to p g1 (n 1 ). Definition 5 (Disjoint Reltionship): Two leled, directed grphs g 1 = G Q 1 hs Disjoint reltionship with g 2 = G Q 2 Fig. 3. r c () Exmple Queries () c in grph G if n 1 V g1 n 2 V g2, p g1 (n 1 ) p g2 (n 2 ). Definition 6 (Overlpping Reltionship): Two leled, directed grphs g 1 = G Q 1 hs Overlpping reltionship with g 2 = G Q 2 in grph G if n 1 V g1 n 2 V g2, p g1 (n 1 )=p g2 (n 2 ). Definition 7 (Contining Reltionship): Two leled, directed grphs g 1 = G Q 1 hs contining reltionship with g 2 = G Q 2 in grph G if n 2 V g2 n 1 V g1, p g1 (n 1 )= p g2 (n 2 ). Definition 8 (Contined y Reltionship): Two leled, directed grphs g 1 = G Q 1 hs Contined y reltionship with g 2 = G Q 2 in grph G if n 1 V g1 n 2 V g2, p g1 (n 1 )=p g2 (n 2 ). g 1 hs topologicl reltionships r with g 2 cn e represented s g G,r g 2 Definition 9 (Topologicl Query): A topologicl query Q is triple (G 1,G 2, reltion), where G 1 nd G 2 re sugrph queries nd reltion is one of connecting, connected y, disjoint, overlpping, contining nd contined y. Definition 10: The result of topologicl query Q=(G 1, G 2, reltionship) over grph G is R G,Q = {g g = G G 1 g 2 (g 2 G g 2 = G G 2 g G,reltionship g 2 )}, where type is some topologicl reltionship type. An exmple in Exmple 3 is to illustrte the concept of topologicl result. Exmple 3: A topologicl query Q = (G 1,G 2, Overlpping ) with G 1 in Figure 3() nd G 2 in Figure 3(). Q is to retrieve the sugrphs mtching G 1 nd hve t lest one node in some sugrph mtching G 2. The results include (r1, 2, 1, c1), (r1, 2, 1, c2), (r1, 3, 1, c1) nd (r1, 3, 1, c2). (r1, 4, 1, c3) is not result of Q. It is ecuse tht it hs none node overlpping ny sugrph mtching G 2. IV. EVALUATION OF TOPOLOGICAL QUERIES In this section, we present lgorithms for the evlution of topologicl queries. We find tht topologicl reltionship cn e clssified into two clsses, query-relted nd queryunrelted. Query-relted reltionship mens tht just from the (c) (d) c (e) 723

4 query the reltionship etween the results cn e judged. Such reltionships includes Contining nd Contined y. For nother four topologicl reltionships, the reltionship etween sugrphs cnnot e judged just from the query directly ut hve to e judged from the results of sugrph queries. A topologicl query with query-relted reltionship re clled query-relted query. Otherwise, it is clled query-unrelted query. For query-relted query, the sugrph query G cn e generted from the topologicl query Q nd the result of Q cn e otined directly from the result of G. To process query-unrelted query, we design n uniform frmework with different filters for deferent reltionship. In this section, t first, the strtegy of processing queryrelted queries is presented. Then we will introduce the lgorithms for processing query-unrelted queries. This pper focuses on topologicl query processing. As their su-queries, the processing of sugrph query is eyond the scope of this pper. Existing sugrph query processing lgorithms [3], [13] re used s function. A. Processing Query-Relted Topologicl Queries In this susection, we present the processing strtegy for query-relted topologicl queries. Definition 11 (Query-Relted Reltionship): A topologicl reltionship r is query-relted reltionship if for queryrelted query Q =(G 1,G 2,r), the topologicl reltionship r etween two sugrphs mtching G 1 nd G 2 cn e judged directly from the query G 1 nd G 2 without generting the result. The topologicl reltionships of Contining nd Contined y cn e judged directly from the query. It is ecuse for two sugrph queries G 1 nd G 2 on G, ifg 2 is sugrph of G 1, ny result g 1 of G 1 must contin result of G 2. Otherwise, if some g 1 does not contin ny result of G 2 there must e some node in G 2 tht cn not e mtched y ny node of g 1. Then the corresponding node in G 1 cn not e mtched y ny node of g 1. Then g 1 is not result of G 1. Therefore, just from the query, Contining nd Contined y reltionship cn e judged from the query. Bsed on the feture of query-relted reltionship, topologicl query Q =(G 1,G 2,type) with reltionship Contining cn e process with Algorithm 1. Algorithm 1 Query contining (G, Q) check whether Q.G 2 is sugrph of Q.G 1 if Q.G 2 is sugrph of Q.G 1 then return QuerySugrph(G, Q.G 1) else return φ In this lgorithm, if Q.G 2 is sugrph of Q.G 1, ny result of Q.G 1 must contin sugrph mtching Q.G 2. Otherwise, ny result of Q.G 1 does not contin result mtching Q.G 2. A topologicl query Q =(G 1,G 2,type) with reltionship Contined y cn e process with Algorithm 2. In Algorithm 2, checking whether Q.G 1 is sugrph of Q.G 2 cn use sugrph mtching lgorithm nd ll the Algorithm 2 Query contined (G, Q) check whether Q.G 1 is sugrph of Q.G 2 if Q.G 2 is sugrph of Q.G 1 then G =QuerySugrph(G, Q.G 2) for ech g in G do S =sugrps of g mtching Q.G 1 S = S S return S else return φ sugrphs mtching Q.G 1 in Q.G 2 re found nd ech is given n identify. Note tht one node in Q.G 2 my e ttched multiple identifies. The step otining sugrph of g mtching Q.G 1 is different from sugrph mtching. It is ecuse during the step of checking whether Q.G 1 is sugrph of Q.G 2, the nodes of Q.G 1 relted to Q.G 2 is identified nd during projection, nodes corresponding to the query node with the sme identify cn e extrcted directly. We use n exmple in Exmple 4 to show the processing of Contined y topologicl query. Exmple 4: For topologicl query Q =(G 1, G 2, Contined y ) on grph in Figure 2() with G 1 in Figure 3(d) nd G 2 in Figure 3(c). At first, the reltionship etween G 1 nd G 2 is checked nd G 1 is sugrph of G 2 nd there re two sugrph mtching G 1 in G 2 : the left-top one is defined s group1 nd the right-ottom one is defined s group2. After processing sugrph query G 2, result (1,1,2,2) is otined with (1,1) mtching group1 nd (2,2) mtching group2. Thus,(1,1) nd (2,2) re two results of Q. Complexity Anlysis Since this pper focuses on topologicl query processing, during nlysis, we tke sugrph query processing lgorithm s n orcle nd use Cost M(q,G) to represent the cost of processing sugrph q on grph G. Bsed on Algorithm 4, the time complexity of Contining topologicl query processing only depends on the time of sugrph query nd is t most Cost M(G,Q.G1) + Cost M(Q.G1,Q.G 2). Bsed on Algorithm 5, the time complexity of Contined y is Cost M(G,Q.G2) + Cost M(Q.G2,Q.G 1) + O( Q.G 1.V result size ), where result size is the numer of finl results. It is ecuse esides the cost of grph mtching, extr cost is the genertion of sugrphs mtching Q.Q 1 from the sugrphs mtching Q.Q 2 nd with identity, such cost is just the cost of generting ll finl results. B. Frmework of Processing Query-unrelted Topologicl Queries In this susection, we will present the frmework of topologicl query evlution. The evlution strtegies for 4 Queryunrelted topologicl queries shre the sme frmework. The frmework cn e descrie in Algorithm 3. Algorithm 3 Query(G, Q) G filter = QuerySugrph(G, Q.G 2) S filter =GenerteFilters(G filter, Q.type) G result = QuerySugrphWithFilter(G, Q.G 1,S filter ) return G result In the frmework, for grph G nd topologicl query Q =(G 1,G 2,type), t first, G 2 is processed nd then from the 724

5 results of G 2, the set of filters re generted. At lst, sugrph query G 1 is processed, during the processing of G 1, the filters generted in the lst step is pplied to filter the results not stisfying the topologicl constrint. Note tht in order to void storing intermedite results, sme filters cn e generted when one result mtching G 2 nd such result is not necessry to e stored. The enefit of shring the sme frmework is tht topologicl query with the logicl comintion of multiple topologicl constrints cn e processed with the comintion of filters. C. Algorithms of Processing Query-unrelted Topologicl Queries In this susection, we will present lgorithms of processing query-unrelted topologicl queries sed on the frmework presented in Section IV-B. Connecting For processing topologicl query with topologicl reltionship connecting, rechiility leling scheme is used. As introduced in Section II-B, ech node is ttched to n id nd list of intervls. For topologicl query Q = (G 1,G 2, connecting ), sorted list L is mintined s filter. When sugrph g 2 mtching G 2 is generted, id of ech node g is inserted to L. When sugrph g 1 mtching G 1 is generted, ll intervls of g 1 s nodes re used to vlidte g 1.If n intervl [x, y] contins some id in L, it mens some node of g 1 cn rech node in some sugrph mtching G 2. Then g 1 is result of Q. The lgorithm is sketched in Algorithm 4. Algorithm 4 Query connecting (G, Q) G filter = QuerySugrph(G, Q.G 2) for ech g G filter do for ech node n g do insert n.id to L G pr = QuerySugrph(G, Q.G 1) for ech grph ging pr do for ech node n g do for ech intervl [x, y] in n.intervls do if id L, x id y then dd g to G result nd stop the processing of current g return G result To ccelerte the inserting nd serching, L is implemented with inry serch tree with insert nd delete cost o(logn), where n is the numer of nodes. Complexity Anlysis The run time of connecting lgorithm cn e estimted s following. Cost = Cost M(G,Q.G1) + Cost M(G,Q.G2) + cost insertl Q.G 2.V result ( M(G, Q.G 2 )) + cost serchl result intervl (M(G, Q.G 1 )), where result (M(G, Q.G 2 )) nd result intervl (M(G, Q.G 1 ) re the numer of results of M(G, Q.G 2 ) nd the totl numer of intervls in the result of M(G, Q.G 1 ), respectively. Since the size of L is t most G.V, cost insertl nd cost serchl re oth t most o(log G.V ). Connected y Processing topologicl query Q(G 1,G 2, Connected y ) uses intervls of nodes in ll the results of G 2. All the intervls re mintined in n ordered list L. When result g of G 1 is generted, L is serched for id of ech node in g to find the intervl [x, y] stisfying x id y. If such intervl cn e found, it mens tht some node in g is connected y some node in sugrph s result of G 2 nd g is ccepted s result. The pseudo code of processing Connected y topologicl query is shown in Algorithm 5. Algorithm 5 Query connected (G, Q) G filter = QuerySugrph(G, Q.G 2) for ech g G filter do for ech node n in g do insert ll the intervl in n.intervls to L G pr = QuerySugrph(G, Q.G 1) for ech grph ging pr do for ech node n g do if [x, y] L, x n.id y then dd g to G result nd stop the processing of current g return G result For efficient inserting nd serching, L is in order of x vlue nd mintined with inry serch tree. In order to reduce the spce cost of L, overlpping nd nested intervls cn e merged. Since the judgement condition is existing intervl contining id, the merge of overlpping nd nested intervls will not ffect the result. The merging is performed online. When n intervl [x, y] is inserted into L, the merging strtegies re shown s follows. If n intervl [, ] in L is found stisfying x y, then [x, y] will not e inserted. If n intervl [, ] in L is found stisfying x y, then [x, y] is inserted to L nd [, ] is delete from L. If two intervls [, ] nd [c, d] in L re found stisfying x 1 nd c y +1, then [, ] nd [c, d] re replced y [, d]. If n intervl [, ] in L is found stisfying x +1 y, then [, ] is replced y [, y]. If n intervl [, ] in L is found stisfying x y +1, then [, ] is replced y [x, ]. With merging strtegies, ll intervls in L re not overlpping or nesting. When result g of G 1 is generted, L is serched for id of ech node in g to find the intervl [x, y] with lrgest x vlue mong the intervls with x vlue smller thn id. If such intervl cn e found nd id stisfies x id y, g is considered s result. An exmple in Exmple 5 is used to illustrte the processing of Connected y query. Exmple 5: For topologicl grph Q =(G 1, G 2, Contined y ), where G 1 is the sugrph query in Figure 3(d) nd G 2 is the sugrph query in Figure 3(e). At the first step, sugrph query G 2 is processed nd then results (2, c1), (2, c2), (3, c1), (3, c2) nd (4, c3) re returned. Then intervls of ech nodes in result re dded to the filter. For 2, [0, 2] nd [7, 7] re dded to filter set L. Forc1 nd c2, no dditionl intervl is dded to L since their intervls [1, 1] nd [2, 2] re contined in existing intervl in L, [0, 2]. When 3 is considered, intervl [1, 3] is merged to [0, 2] in L nd [0, 3] tkes the plce of [0, 2] in Lr. For 4, [4, 5] is merged with [0, 3] nd [0, 5] tkes the plce of [0, 3] in L. [10, 10] is lso required to merged with [9, 9]. After processing ll nodes of 725

6 sugrphs mtching G 2, intervls in L include [0, 5],[7, 7] nd [9, 10]. During the process of G 1, when sugrph mtching G 1 such s (1,1) is generted, for ech id of its nodes, L is serched. For (1,1), no intervl in L contins ny of the ids of1 nd 1. So(1,1) is not result for Q. For(2,1), since the id of 1 is contined in the intervl [0,5], it is result of Q. Complexity Anlysis The run time of connecting lgorithm cn e estimted s Cost = Cost M(G,Q.G1) + Cost M(G,Q.G2) + cost insertl Q.G 2.V result intervl (M(G, Q.G 2 )) + cost serchl Q.G 1.V result(m(g, Q.G 1 )), where result intervl (M(G, Q.G 2 )) nd result(m(g, Q.G 1 )) is the totl numer of intervls in the results of M(G, Q.G 2 ) nd numer of results of M(G, Q.G 1 ), respectively. With our merging strtegy nd the property of the leling scheme, the numer of intervls re t most G.V. Therefore, cost insertl nd cost serchl re oth t most o(log G.V ). Overlpping nd Disjoint The filter for topologicl query Q =(G 1,G 2,type) with type overlpping or disjoint is set S of ids of nodes in the sugrphs mtching G 2. The difference of processing these two types of topologicl queries is the filter condition. For sugrph g mtching G 1, the filter condition of Overlpping topologicl query is tht for some node n in g, id equlling to n.id in S cn e found. While the filter condition of Disjoint topologicl query is tht n.id cn not e found in S for ll nodes n in g. The description of the lgorithm to process overlpping is in Algorithm 6. The lgorithm for disjoint is similr except during the filtering step, the grph g with ids of ll nodes not in S re considered s finl result. Algorithm 6 Query Overlpping (G, Q) G filter = QuerySugrph(G, Q.G 2) for ech g G filter do for ech node n g do insert n.id to S G pr = QuerySugrph(G, Q.G 1) for ech grph ging pr do for ech node n g do if n.id cn e found in S then dd g to G result nd stop the processing of current g return G result For efficient inserting nd serching, S cn e implemented with hsh set. Complexity Anlysis The run time of overlpping nd disjoint lgorithms re similr nd cn e estimted s following. Cost = Cost M(G,Q.G1) + Cost M(G,Q.G2) + cost inserts Q.G 2.V result(m(g, Q.G 2 )) + cost serchs Q.G 1.V result(m(g, Q.G 1 )), where result(m(g, Q.G 2 )) nd result(m(g, Q.G 1 )) re the numer of results of M(G, Q.G 2 ) nd numer of results of M(G, Q.G 1 ), respectively. With hsh set s dt structure, cost inserts nd re oth o(1). cost serchs V. EXPERIMENTAL RESULTS In this section, we present the experimentl results nd nlysis of our lgorithms for processing topologicl queries. TABLE I STATISTICS OF THE XMARK DATASETS fctor Size 11.3M 22.8M 34.0M 45.3M 56.2M #Nodes #Edges A. Experimentl Setup All of our experiments were performed on PC with Pentium 2.4GMHz, 512M memory nd 40G IDE hrd disk. The OS is WindowsXP Professionl. We implemented our system using Microsoft Visul C We implemented ll our lgorithms in this pper. We implemented sugrph query processing lgorithms presented in[13] s the sugrph query processing module nd emedded the filter genertion nd result filtering in result genertion step of sugrph query processing module. The sugrph query processing lgorithm in [13] is disk-sed lgorithm. we fix lock size 1K nd uffer size 32K. The dtset we tested is the XMrk enchmrk dtset [10]. It cn e modeled s grph nd hs firly complicted schem, with circles s sugrphs nd mny nodes with multiple prents. We choose fctor 0.1, 0.2, 0.3, 0.4, 0.5 to generte XMrk documents. Their prmeters re shown in Tle I For topologicl queries, we choose three groups of grphs, groupu1: (G u11,g u12 ), groupu2: (G u21,g u22 ) nd groupu3: (G u31,g u32 )for Connecting, Connected, Overlpping nd Disjoint nd three groups of grphs groupr1: (G r11,g r12 ), groupr2: (G r21,g r22 ) nd groupr3: (G r31,g r32 ) for Contining nd Contined y. The grphs re shown in Figure V, where doule line represents ADreltionship etween nodes. G u32 nd G r21 re the sme grphs. For ech group for query-unrelted queries, we design four queries on them with four topologicl reltionship, respectively. In ech group, the former sugrph is used s G 1 nd the ltter sugrph s G 2 in the form of topologicl query Q =(G 1,G 2,type). For ech group for query-relted queries with sugrph query G 1 nd G 2, we design two queries (G 1,G 2, Contining ) nd (G 2,G 1, Contined y ) on them. Since the cse tht G 1 does not contin G 2 cn only processed y compring the two queries nd the cse is trivil, in ech group, G 2 is sugrph of G 1. B. Comprison In this susection, we will present the comprison results etween our lgorithms nd grph mtching. For queryunrelted query, We compre the processing time of queryunrelted queries with the sum of processing time of the two sugrph queries in the topologicl query. The results is shown in Figure 5(). Since the three group of sugrph queries for query-relted queries ll include one sugrph contining nother, we lso compre the processing time of query-relted queries with the lrger one of two sugrph queries in ech group. The results re shown in Figure 5(). From the results, it cn e seen tht compring with the time of processing 726

7 ctegory open_uction closed_uction open_uction seller uyer seller idder item description old emph keyword old emph keyword keyword emph () G u11 () G u12 (c) G u21 (d) G u22 (e) G u31 closed_uction uthor site item open_uction wtch open_uction uthor wtch old emph ctegory old emph ctegory old emph keyword edge eduction edge eduction idder open_uction (f) G u32 nd G r21 (g) G r12 (h) G r21 (i) G r22 (j) G r32 (k) G r32 Fig. 4. Query Grphs () Comprison: Query-Unrelted () Comprison: Query-Relted (c) Sclility: GroupU1 (d) Sclility: GroupU2 (e) Sclility: GroupU3 (f) Sclility: Query-Relted Fig. 5. Experimentl Results sugrph queries, the extr costs of our lgorithms re very smll. C. Sclility In this susection, we will present the results of sclility of our lgorithms. We rnge XMrk dt from 10M to 50M. The results of query-unrelted queries re shown in Figure 5(c), Figure 5(d) nd Figure 5(e). The results of queryrelted queries re shown in Figure 5(f). Note tht the run time of Figure 5(c), Figure 5(e) nd Figure 5(f) re in log scle. From the results, the run time of our lgorithm increses little fster thn linerly ut slower thn exponentilly with dt size. It is ecuse tht our lgorithm increses with the size of prtil results of the sugrph query in topologicl constrint nd finl results. The size of prtil results my increse fster thn linerly ecuse the numers of nodes mtching ech query node in sugrph query increses out linerly with the size of XML document nd so tht the in some cses the increse of the numer of results my e the production of the increse time of nodes mtching every query node. 727

8 VI. RELATED WORK In this section, we give n overview of previous work relted to this pper. Relted work cn e clssified into two kinds. One is the model nd representtion of query on grph, the other is query processing lgorithms on grph. Existing query lnguges for XML [2] re only sed on tree structure without considering grph fetures. Even though Lorel [1] considers IDREF, it considered pth s sic unit insted of grph. The disdvntge of using pth s the sic unit is tht circle nd topologicl reltionships re difficult to e represented. There re lso severl query lnguges designed for descriing recursion reltionship [5] nd grph mtching [6], [11]. They focus on the description of query in the form of lelled grph without complex structurl restrictions nd topologicl restrictions. Query lnguges relted to RDF [7] cn e used to represent query in the form of grph. But current query lnguges do not consider topologicl restrictions. Currently, existing work of querying grph-structured XML minly focus on structurl query of sugrph/sutree mtching in grph [16], [3], [13]. None of them considers the prolem of topologicl query processing. Another kind of work on querying grph is to retrieve grph stisfying some condition from lrge set of grphs. Such work includes [14], [15], [8]. Such works consider the structurl fetures of grph. However, none of them considers the topologicl fetures of grph. VII. CONCLUSIONS In this pper, topologicl query, new kind of query for grph-structured XML dt, is presented. This query is to retrieve sugrph with some given topologicl reltionship with some other sugrph in grph-structured dt. We define six types of topologicl reltionship etween sugrphs to e used s the topologicl constrint. We give the definitions nd evlution lgorithms of topologicl queries. Experimentl results show tht our implementtion lgorithms use smll extr cost of sugrph query nd scles with the size of results of sugrph query in it. Future work includes efficient index uilding nd holistic lgorithms to process topologicl queries. [8] H. He nd A. K. Singh. Closure-tree: An index structure for grph queries. In ICDE, pge 38, [9] H. V. J. Rkesh Agrwl, Alexnder Borgid. Efficient mngement of trnsitive reltionships in lrge dt nd knowledge ses. In Proceedings of the 1989 ACM SIGMOD Interntionl Conference on Mngement of Dt (SIGMOD 1989), pges , Portlnd, Oregon, My [10] A. Schmidt, F. Ws, M. L. Kersten, M. J. Crey, I. Mnolescu, nd R. Busse. XMrk: A enchmrk for XML dt mngement. In Proceedings of 28th Interntionl Conference on Very Lrge Dt Bses (VLDB 2002), pges , [11] L. Sheng, Z. M. Özsoyoglu, nd G. Özsoyoglu. A grph query lnguge nd its query processing. In Proceedings of the 15th Interntionl Conference on Dt Engineering, Mrch 1999, Sydney, Austrili, pges IEEE Computer Society, [12] J. P. Tim Bry, C. M. Spererg-McQueen, nd F. Yergeu. Extensile mrkup lnguge (xml) 1.0 (third edition). In W3C Recommendtion 04 Ferury 2004, [13] H. Wng, W. Wng, X. Lin, nd J. Li. Sugrph join: Efficient processing sugrph queries on grph-structured xml document. In WAIM, pges 68 80, [14] X. Yn, P. S. Yu, nd J. Hn. Grph indexing: A frequent structure-sed pproch. In SIGMOD Conference, pges , [15] X. Yn, P. S. Yu, nd J. Hn. Sustructure similrity serch in grph dtses. In SIGMOD Conference, pges , [16] V. J. T. Zogrfoul Vgen, Mirell Mour Moro. Twig query processing over grph-structured xml dt. In Proceedings of the Seventh Interntionl Workshop on the We nd Dtses(WeDB 2004), pges 43 48, Hongzhi Wng Hongzhi Wng received his PHD in computer science from Hrin Institute of Technology in He is lecturer of Deprtment of Computer Science nd Technology, Hrin Institute of Technology. His reserch re is XML dt mngement nd informtion integrtion. Jinzhong Li Jinzhong Li received his PHD in computer science from Hrin Institute of Technology in He is professor of Deprtment of Computer Science nd Technology, Hrin Institute of Technology. His reserch re includes prllel dtse, sensor network, dt mining, dt wrehouse, compressed dtse nd XML dt mngement. REFERENCES [1] S. Aiteoul, D. Quss, J. McHugh, J. Widom, nd J. L. Wiener. The lorel query lnguge for semistructured dt. Int. J. on Digitl Lirries, 1(1):68 88, [2] A. Bonifti nd S. Ceri. Comprtive nlysis of five xml query lnguges. SIGMOD Record, 29(1):68 79, [3] L. Chen, A. Gupt, nd M. E. Kurul. Stck-sed lgorithms for pttern mtching on dgs. In VLDB, pges , [4] J. Clrk nd S. DeRose. XML pth lnguge (XPth). In W3C Recommendtion, 16 Novemer 1999, [5] I. F. Cruz, A. O. Mendelzon, nd P. T. Wood. A grphicl query lnguge supporting recursion. In SIGMOD Conference, pges , [6] R. H. Güting. Grphd: Modeling nd querying grphs in dtses. In J. B. Bocc, M. Jrke, nd C. Zniolo, editors, VLDB 94, Proceedings of 20th Interntionl Conference on Very Lrge Dt Bses, Septemer 12-15, 1994, Sntigo de Chile, Chile, pges Morgn Kufmnn, [7] P. Hse, J. Broekstr, A. Eerhrt, nd R. Volz. A comprison of rdf query lnguges. In Interntionl Semntic We Conference, pges , Jizhou Luo Jizhou Luo received his PHD in computer science from Hrin Institute of Technology in He is lecturer of Deprtment of Computer Science nd Technology, Hrin Institute of Technology. His reserch re is compressed dtse. 728