Hash-based Subgraph Query Processing Method for Graph-structured XML Documents

Transcription

1 Hsh-bse Subgrph Query Proessing Metho for Grph-struture XML Douments Hongzhi Wng Hrbin Institute of Teh. Jinzhong Li Hrbin Institute of Teh. Jizhou Luo Hrbin Institute of Teh. Hong Go Hrbin Institute of Teh. ABSTRACT When XML ouments re moele s grphs, mny reserh issues rise. In prtiulr, there re mny new hllenges in query proessing on grph-struture XML ouments beuse tritionl query proessing tehniques for tree-struture XML ouments nnot be iretly pplie. This pper stuies the problem of struturl queries on grphstruture XML ouments. A hsh-bse struturl join lgorithm, HGJoin, is first propose to hnle rehbility queries on grph-struture XML ouments. Then, it is extene to the lgorithms to proess struturl queries in form of biprtite grphs. Finlly, bse on these lgorithms, strtegy to proess subgrph queries in form of generl DAGs is propose. Anlysis n experiments show tht ll the lgorithms hve high performne. It is notble tht ll the lgorithms bove n be slightly moifie to proess struturl queries in form of generl grphs. 1. INTRODUCTION XML hs beome the e fto stnr for informtion representtion n exhnge over the Internet. In mny pplitions, n XML oument nees to be moele s grph more nturlly thn tree. For exmple, the XML oument of the reltionship of publitions n uthors pts to grph struture sine one pper my hve more thn one uthor n one uthor my hve more thn one pper. A frgment of suh informtion is shown in Fig 1. Obviously, the grph-struture XML oument n be represente in tree struture by upliting the element with more thn one inoming pths. But it will result in reunny. If the informtion in Fig 1 is represente with tree-struture XML oument, the element uthor will Support by the Key Progrm of the Ntionl Nturl Siene Fountion of Chin uner Grnt No ; the Ntionl Grn Funmentl Reserh 973 Progrm of Chin uner Grnt No.2006CB303000; the Ntionl Nturl Siene Fountion of Chin uner Grnt No n No Permission to mke igitl or hr opies of portions of this work for personl or lssroom use is grnte without fee provie tht opies Permission re not me to opy or istribute without fee for ll or profit prt of or this ommeril mteril is vntge grnte provie n tht tht the opies opies ber re not this me notie or n istribute the full for ittion iret ommeril on the first vntge, pge. the Copyright VLDB opyright for omponents notie n of the this title work of the owne publition by others n thn its te VLDB pper, n notie is given tht opying is by permission of the Very Lrge Dt Enowment must be honore. Bse Enowment. To opy otherwise, or to republish, to post on servers Abstrting with reit is permitte. To opy otherwise, to republish, or to reistribute to lists, requires fee n/or speil permission from the to post on servers or to reistribute to lists requires prior speifi publisher, ACM. permission n/or fee. Request permission to republish from: VLDB 08, August 24-30, 2008, Aukln, New Zeln Publitions Dept., ACM, In. Fx +1 (212) or Copyright 2008 VLDB Enowment, ACM /00/00. permissions@m.org. be uplite. onferene ICDE proeeings proeeing title t 1 bib rtile journls journl nme volume TKDE 17(2)title t 2 uthor rtile nme n 1 emil Figure 1: An Exmple of Grph-struture XML Among the queries on grph-struture XML ouments, the subgrph queries re wiely use. A subgrph query on grph-struture XML ouments (subgrph query for short) is to retrieve the subgrphs mthing the grph in the query. For instne, subgrph query on the grphstruture XML oument in Fig 1 is to retrieve the nmes of uthors with publitions both in proeeings n journls. Another subgrph query on the XML oument in Fig 1 is to retrieve the nme of the journl with n uthor who publishe ppers in the onferene ICDE. Suh queries re iffiult to represent with tritionl tree-struture queries. It is big hllenge to proess subgrph queries effiiently. All the four kins of tritionl methos of proessing struturl queries on tree-struture XML ouments, struturl join bse methos[2], holisti Twigjoin bse methos[3], struturl inex bse methos[14, 12] n subsequene mthing bse methos[23, 19], nnot be use to proess subgrph queries. The struturl join bse methos n the holisti Twigjoin bse methos both epen on the lbelling sheme speilly for tree-struture XML ouments. The enoing sheme of the grph-struture XML ouments is totlly ifferent from tht of the tree-struture XML ouments. As result, they nnot be use to proess subgrph queries. The struturl inex bse methos of proessing struturl queries on tree-struture XML ouments require tht the size of the inex must be very smll. However, the inies of the grph-struture XML ouments re very lrge in generl. Thus, the struturl inex bse methos e 1 PVLDB '08, August 23-28, 2008, Aukln, New Zeln Copyright 2008 VLDB Enowment, ACM /08/08 478

2 nnot be use to proess the subgrph queries effiiently. The subsequene mthing bse methos require tht the tree-struture XML ouments n the queries on the ouments must be onverte into sequenes before query proessing. It is iffiult to overt grph-struture XML oument into sequene so tht it is not esy to proess subgrph queries using the subsequene mthing bse methos. A few methos hs been propose to proess some kins of subgrph queries on XML t in form of some speil kins of grphs. A metho, lle StkD, is presente in to [4] to proess twig queries on DAG-struture t. It is moifition of holisti TwigJoin bse metho. However, StkD fouses on tree-struture twig queries n is not suitble for queries in form of omplex grphs. Aitionlly, when there re mny eges in the grph, StkD shoul mintin very lrge t struture. In this se, it nees very huge min memory spe, whih is not prtil n it beomes ineffiient. Another moifition of the holisti TwigJoin-bse metho is presente in [27] to proess twig queries on grph-struture t. However, it only works on kin of speil grphs, i.e. st-plnr grphs [11], but not suitble for other grphs. In summry, urrent methos nnot proess generl subgrph queries effetively or effiiently. To proess generl subgrph queries effetively n effiiently, new metho bse on lbeling sheme [17] is propose in this pper. The resons of hoosing the lbeling sheme [17] re s following. It ontins only intervls n ientifitions (is). All intervls n is re numeril vlues so tht there is n orer on them, whih mkes query proessing esier. It is omptible with the jent lbeling sheme so tht it n be use to proess queries with both rehbility n jent eges. We will isuss this in etils in Setion 6. By slightly moifition, the lbeling sheme n be use to proess grphs with irles. Our propose metho is esigne step by step. First, hsh-bse join lgorithm, HGJoin, is propose for proessing rehbility queries on grph-struture XML. Seon, the HGJoin lgorithm is extene to the IT-HGJoin n T-HGJoin lgorithms to proess the rehbility queries with multiple nestors or multiple esennts. Then, Bi- HGJoin, the ombintion of IT-HGJoin n T-HGJoin, is esigne to proess queries in form of omplete biprtite grphs. Finlly, bse on ll the bove lgorithms, the metho for proessing subgrph queries in form of DAGs is propose. Without losing generlity, this pper will fous on subgrph queries in form of DAG with only rehbility reltionships on eges for the onveniene of isussion. With slight moifition, the metho n be use to proess ny generl subgrph queries. The ontributions of this pper re s follows: Bse on the rehbility sheme in [17], fmily of hsh-bse join lgorithms is presente s bsi opertors of subgrph query proessing. For struturl queries in form of generl grphs, n effiient metho is presente. The bsi ie is to split query into biprtite subqueries eh of whih n be proesse by some HGJoin lgorithm. In orer to fin effetive splitting strtegy, ost-bse query optimiztion strtegy is presente with some elertion strtegies. The extensive experimentl results show tht the propose lgorithms outperform the existing lgorithms n our query splitting strtegy is effetive n effiient. The rest of this pper is orgnize s follows: Setion 2 introues some bkgroun knowlege. Setion 3 presents the bsi version of HGJoin lgorithm for rehbility query with one nestor n one esennt. Setion 4 illustrte the lgorithms for queries in form of biprtite grphs. The strtegy of proessing queries in form of generl DAGs is propose in Setion 5. The extensions of our metho for the generl subgrph queries re isusse in Setion 6. Experimentl results n nlysis re shown in Setion 7. Relte work is isusse in Setion 8 n Setion 9 onlues this pper. 2. PRELIMINARIES In this setion, the bkgroun n nottions use in this pper re presente. 2.1 Grph-struture XML Dt n Queries With IDREF-ID in n XML oument representing referene reltionship, n XML oument n be onsiere s tgge irete grph. Elements n ttributes in n XML oument is mppe to the noes in grph. Direte nesting reltionships n referene reltionships in n XML oument is mppe to the eges in grph. For exmple, n XML oument is shown in Fig 2() n its grph struture is shown in Fig 2(b). In grph-struture XML oument, struturl queries re efine bse on the struturl reltionship between noes. In the grph struture G of n XML oument, two noes n b stisfy jent reltionship if n only if n ege from to b exists in G; two noes n b stisfy rehbility reltionship if n only if pth from to b exists in G. A rehbility query is to retrieve ll pirs of noes (n,n ) in G where n hs tg, n hs tg n n n n stisfy rehbilty reltionship in G. For exmple, the result of rehbility e in the grph in Fig 2(b) inlues (,e 1), (, e 2), (,e 3). Ajent queries n be efine similrly. The ombintion of multiple rehbility or jent reltionships forms subgrph queries. A subgrph query is tgge irete grph Q={V, E, tg, rel}, where V is the noe set of Q; E is the ege set of Q; the funtion tg : V T AG is the tg funtion (T AG is the set of tgs); the funtion rel : E AXIS shows the struturl reltionships between the noes in the query (AXIS is the set of reltionships between noes; P C AXIS represents jent reltionship; AD AXIS represents rehbility reltionship). The irete grph (V,E) is lle the query grph. If b E n rel(b) = P C, then b is lle n jent ege. If b E n rel(b) = AD, then b is lle rehbility ege. In V, the noes without inoming eges re lle soures n the noes without outgoing 479

3 < i="1 > < b i="b1" > < i="1" f="f1"/ > < i="2" > < f i="f1"/ > </> < i="3" f="f1" ="1"/ > < /b > < i="1" > < e i="e1" ="1" ="2" ="3"/ > < e i="e2" ="1" ="2" ="3"/ > < e i="e3" ="1" ="2" ="3"/ > < / > < / > () An XML Frgment b1 8, [0,8] 1 9, [0,9] (b) The Grph of Fig 2() 1 7, [0,7] noes to generte the post-orer of eh noe. Note tht uring the trversl, when noe n C generte from strongly onnete omponent C G is esse, if the post orer of lst esse noe is p, then p + 1, p + 2,, p + V re ssigne to n C (where V C is the set of noes in C). Then, eh noe n T is ssigne number i n n intervl [x,i], where i is the post orer of n; x is the smllest post orer of esennts of n in T. All the noes in D re trverse in the reverse topologil orer. When noe n is rehe, the intervl sets of n s hilren in D re opie to tht of n. Intersete intervls in the intervl set of n re merge , [2,2] 1,[0,1] 3 3, [0,7] e1 4, [0,7] e2 5, [0,7] e3 6, [0,7] [0,0] n C, its intervl set is tht of n C; its i is one of the is of n C. Eh noe in C hs ifferent i. 0, [0,0] f1 () The Rehbility oe of Fig 2() Figure 2: An Exmple of Grph-struture XML () f (b) 3 1 f1 () 1 Figure 3: Exmple Queries eges re lle sinks. For simpliity, subgrph query is represente s Q = (V, E). The result of subgrph query Q = {V Q, E Q} on grph G = {V G, E G} is R G,Q = {g = (V g, E g) V g V G n bijetive mp f g : V g V Q n injetive mp p g : V g V G stisfying n V g, tg(n) = tg(f g(n)) = tg(p g(n)); 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))}. For exmple, Fig 3(b) shows subgrph query, in whih single line represents n jent ege n ouble line represents rehbility ege. The result of suh query on the grph shown in Fig 2(b) is the grph in Fig 3(). e () 2.2 Rehbility Lbelling Sheme The lbelling sheme for grph-struture XML oument is to juge the rehbility reltionship between ny two noes in n XML oument without retrieving other informtion, suh tht subgrph queries n be proesse effiiently. The lbelling sheme use in this pper is n extension [24] of tht in [17]. The rehbility lbelling sheme n be generte in the following steps: Eh strongly onnete omponent in G is ontrte to one noe to onvert G to DAG D. An optimum tree overing T of the DAG D is foun. A epth-first trversl from the root of T esses ll When suh steps re finishe, eh noe n in G is ssigne number n.i n set of intervls I n. In [24], it is prove tht noe rehes noe b if n only if b.i belongs to some intervl of I. For exmple, the lbelling sheme of the grph in Fig 2(b) is shown in Fig 2(). Sine the i of f1 is in the intervl [0,0] of 2, it n be juge tht 2 n f1 stisfy rehbility reltionship. In orer to retrieve pirs of noes stisfying rehbility query bse on rehbility lbelling sheme, the following storge strtegy is pplie. For eh tg t T AG of n XML oument, two lists, t.alist n t.dlist, re store. N t is the set of noes with tg t. t.dlist is the list of the elements in set {n.i n N t} sorte inrementlly. t.alist is the elements in set {(vl, n.i) n N t, [x, y] I n, vl = x or vl = y} sorte by the first item. From the thir step of enoing, for eh noe n, I n hs no overlpping intervls. Therefore, tg(n).alist hs the following property whih is the bse of the lgorithms in the following setions. Proposition 1. For noe n, ll the elements in tg(n). Alist with the seon item equls to n.i form n orere list (vl i1, n.i), (vl i2, n.i),, (vl ik, n.i). In suh n orere list, l s u r v k o not exist suh tht both [vl is, vl ir ] n [vl iu, vl iv ] belong to I n. Aitionlly, in orer to juge whether ll the intervls of one noe hs been proesse, for eh noe n, tuple (null, i) is inserte next to the lst tuple of tg(n).alist with the seon item equls to n.i, where null represents empty. To proess the queries with preites n buil-in funtions, s the preproessing step, the lbelling shemes re filtere with the preites n buil-in funtions before the proessing of subgrph query. In orer to nlyze the omplexity of lgorithms in this pper, N = mx t T AG {n V g tg(n) = t}. The set of the seon items of ll elements in Alist is ID Alist. Obviously, for eh Alist n Dlist, ID Alist N, Dlist N. 3. HASH-BASED JOIN ALGORITHM FOR GRAPH-STRUCTURED XML (HGJOIN) In this setion, hsh-bse join lgorithm is presente to proess rehbility queries in form of on grphstruture XML t bse on the rehbility lbelling sheme. 480

4 Algorithm 1 HGJoin =Alist.begin =Dlist.begin while Alist.en AND Dlist.en o if vl i then if i / H then H.insert(i ) =.nextnoe else if vl < i then H.elete(i ) =.nextnoe else for i H o ppen (i, i ) to OutputList =.nextnoe; else for i H o ppen (i, i ) to OutputList =.nextnoe; b 11 (b 11 ) ( 21 ) ( 21 ) (b 31 ) 22 ( 31 ) 11 ( 12 ) 31 ( 11 ) ( 12 ) b 12 (b 21 ) 22 e e 1 e ( 21 ) Figure 4: Exmple t Bse on the storge strtegy, suppose Alist = tg().alist n Dlist = tg().dlist. Intuitively, for ny i j Dlist, if two tuples in Alist, (x, i i) n (y,i i), stisfy x i i y n [x, y] is n intervl of the lbelling sheme of some noe, then (i i, i j) belongs to the result set. Proposition 1 ssures tht the query n be proesse with snning Alist n Dlist lterntely only one. During the sn, hsh tble H is use to store is of noes stisfying the onition tht for eh noe n, the strt point x of some intervl in I n hs been snne while orresponing en point y is not met. Proportion 1 shows tht before suh y is met, none of the strt points of other intervls in I n will be met. When orresponing y is met, n.i will be remove from H. The step is shown s following. At first, ursors n re ssigne to Alist n Dlist, respetively. During the sn of Alist, if the urrent tuple (vl,i ) stisfies vl i n i H, it mens tht the en position of some intervl is met n suh n intervl is impossible to ontin i. Sine the elements in Dlist is in inrementl orer, n suh n intervl is impossible to ontin other unsnne elements in Dlist. So i is remove from H n is upte. If vl = i n i H or vl > i, it mens tht i is ontine in some intervl with vl s the en point. In suh n instne, prtil results re outputte n the sn is swithe to Dlist. During the sn of Dlist, if i < vl, or vl = i n i H, then bse on the properties of elements in H, for i H, (i, i ) is outputte n is upte; if i > vl, or vl = i but i / H, it mens tht vl is possibly the strt position of n intervl ontining i n the sn is swithe to Alist. The pseuo-oe of HGJoin is shown in Algorithm (b 41 ) ( 32 ) ( 31 ) e 3 ( 41 ) (b 51 ) b e () b (b) Figure 5: Query Exmples Exmple 1. In Fig 4, we give the intuition of Alist n Dlist, where, b,, n e re tgs. The elements with tg re 1, 2, 3 n 4. Eh intervl of i is represente by line segment n enote by ij. The strt point of the line segment is the strt position of the intervl n the en point of the line segment is the en position of the intervl. The position of element i in the intervl is represente by irle. For exmple, 1.i is in the en position of intervl 12 n 2.i is in the mile of the intervl 21; other symbols hve the sme menings. For rehblity query b, orresponing Alist hs n orere list with first items sorte in form of (the en point) of ij, i.i n Dlist={b 1.i, b 3.i, b 2.i}. After Alist n Dlist snne with HGJoin lgorithm, the outputte result is ( 1.i, b 1.i), ( 1.i, b 3.i), ( 2.i, b 3.i), ( 1.i, b 2.i) in orer. Complexity Anlysis The time ost of HGJoin lgorithm hs three prts, the ost of opertions of H, the ost of isk I/O n the ost of result outputting. The time ost is Cost = Alist (ost 2 I + ost D) + result ost out + ( Alist entry A + Dlist entry D ) ost B B I/O with eh item orresponing to eh prt, where the ost of insertion n eletion of H one re ost I n ost D, respetively, entry A n entry D re the sizes of eh tuple in Alist n Dlist, respetively, B is the size of isk blok, ost I/O is the ost of essing eh isk blok n ost out is the ost of outputting one tuple. The spe ost of HGJoin lgorithm is the spe ost of the hsh tble H. Therefore, the spe omplexity is the lrgest size of H uring the lgorithm. In the worst se, ll the elements in ID Alist re in H. Therefore, the spe ost of HGJoin is no more thn N. 4. EXTENSIONS OF HGJOIN HGJoin n be extene to proess some speil ses of subgrph queries. For subgrph query Q=(V, E, tg, rel), if V = V s {}, / Vs, n E = {(, ) V s}, then Q is n IT-query. If V = V s, / V s, n E = {(, ) V s}, then Q is T-query. In this setion, we will present two extensions of HGJoin lgorithm, IT-HGJoin n T-HGJoin to proess IT-query n T-query, respetively. 4.1 Algorithm for IT-queries In this setion, HGJoin is extene to proess IT-queries. For n IT-query Q, suppose its soures re 1,, k n its sink is. Let Alist i = tg( i).alist, Dlist = tg().dlist, i 1, 2,, k. Similr s HGJoin lgorithm, IT-HGJoin lgorithm sns Alist 1,, Alist k n Dlist lterntively one n obtins the results of n IT-query. During the sn, hsh tble H i is ssigne to eh Alist i, the funtion of whih is the sme s H in HGJoin lgorithm. In the lgorithm, ursors l 1, l 2,, l k point to the urrent snne position of Alist 1, Alist 2,, Alist k, respetively. A ursor l points to the urrent snne position of Dlist. The lgo- 481

5 Algorithm 2 IT-HGJoin for i = 1 to k o o i = Alist i.begin = Dlist.begin while i Alist i.en o for i=1 to k o while vl i < i o if i i H i then H i.insert(i i ) else H i.elete(i i ) i = i.next while vl i = vl AND i i / H i o H i.insert(i i ) i = i.next if none of the hsh tbles re not empty then OutputList = OutputT uples(h 1,, H k, i ) for eh i o while vl i = vl AND i i H i o H i.elete(i i ) i = i.next =.next rithm sns Alist 1, Alist 2,, Alist k in turn. i li s in pirs (vl li, i li ) stisfying vl li i l n i li / H i re inserte to H i n i li s in pirs (vl li, i li ) stisfying vl li i l n i li H i re remove from H i. After Alist k is proesse, the sn is swithe to Dlist. Suh proessing is similr s the Dlist sn in HGJoin lgorithm. At tht time, the noe orresponing to eh i in ny H i is n nestor of the noe orresponing to i l. If ny of H i is not empty, it mens tht with its nestors, the noe with i l mthes the esenent in the query. Suh subgrphs re outputte s prtil results. Sine eh tuple in H 1 H 2 H k orrespons to eh group of ifferent nestors of i l, ll the tuples in H 1 H 2 H k {i l } shoul be outputte. In the step of result output, funtion OutputTuples(H 1,, H k, i ) is invoke. After suh prtil results re outputte, the sn is swithe to Alist 1,, Alist k. IT-HGJoin lgorithm is shown in Algorithm 2. In the implementtion of IT-HGJoin lgorithm, the size of H 1 H 2 H k my be very lrge. In orer to store prtil results effiiently, lteny proessing strtegy is pplie. Tht is, H 1,, H k n orresponing i l re store respetively. The Crtesin proution is not performe until suh prtil result will be use. Complexities Anlysis Obviously, the worst spe omplexity is kn. Similr s the nlysis of HGJoin, the time omplexity Cost = Alist i (ost 2 I + ost O) + result ost out + ( Alist i entry A ) ost I/O + Dlist entry D B B 4.2 Algorithm for T-queries In this setion, n lgorithm for proessing T-queries is presente. In T-query Q, the soure is n sinks re 1,, k. Let Alist=tg().Alist, Dlist i=tg( i).dlist, i {1, 2,, k}. Sine in Q, the soure hs multiples esennts 1, k n in the rehbility lbelling sheme, ll the noes with tgs tg( 1), tg( k ) o not belong to the sme intervl of noe with tg(), in orer to obtin the result of Q, ll results of rehbility query i where i {1,, k} shoul be obtine n the join opertion is performe on suh intermeite results. HGJoin n be pplie to proess rehbility queries i. For the interest of effiieny, the k wy sns of HGJoin lgorithm re ombine. Tht is, uring the sns on Alist, Dlist 1,, Dlist k re proesse together. Suh tht ll intermeite results n be obtine by snning ll lists only one. In orer to mke join opertion effiient, hsh tble IHT i is ssigne to eh Dlist i. When buket in some IHT i is full, the intermeite results in suh buket re sorte bse on the first items (the i vlue of the noe mthing ) n written out to the isk. When the intermeite results re obtine, ll tuples in form of (i, i 1) IHT 1,, (i, i k )IHT k re joine to generte tuple, (i, i 1,, i k ), the prtil result of ITquery. Obviously, the ost of join opertion inreses fst with IHT i. In orer to reue the ost of join, the size of IHT i shoul be erese uring join. The strtegy in T-Join lgorithm is tht uring the sn of Alist, when the en sign (null,i) (see Setion 2.2) is met, it mens tht the following steps of the sn will not generte intermeite result in form of (i, ). Therefore, the join opertion n be performe on urrent IHT 1,,IHT k to generte ll results in form of (i, i 1,, i k ). Then the tuples with form (i, ) re elete from IHT 1,,IHT k n orresponing isk bloks re merge. Even though the bove strtegy n minimize the size of IHT i uring join opertion, frequent join opertions will mke n lgorithm ineffiient. Aitionlly, fter eh join opertion, the number of isk bloks to be merge is very limite. In orer to mke it more effiient, the join lteny strtegy is pplie in T-HGJoin lgorithm. Tht is, uring T-HGJoin lgorithm, n nestor tble A with fixe size is mintine. During the sn of Alist, when en sign (null,i) is met, i is inserte to A. When A is full or the sn of Alist is finishe, the join opertion is performe on urrent IHT 1,, IHT k to generte prtil query results with form (i,,, ) where i is the i of ny element in A. Then intermeite results with form (i, ) re elete n for eh buket in ny IHT i, mergble isk bloks re merge. In orer to reue the spe ost of prtil results storge, the lteny proessing strtegy similr s IT-HGJoin n lso be pplie. Exmple 2. The proessing of the T-query in Fig 5(b) on the t in Fig 4 is onsiere. For esy isussion, suppose tht eh buket n only ontin two tuples n eh IHT uses hsh funtion hsh(x)=x mo 2. It mens tht eh IHT hs only two bukets. The size of nestor tble A is 2. When T-HGJoin lgorithm esses tuple (null, 1.i), generte intermeite results re shown in Fig 6(). 1.i is inserte to A. At tht time, A is not full, so the join opertion is not performe. When the tuple (null, 3.i) is snne, the intermeite results re shown in Fig 6(b). 3.i is inserte to A n A is full, so the join opertion is performe on the intermeite results with only 1 n 3 n results ( 1, b 1, 2, 2), ( 1, b 2, 2, 2) n ( 1, b 3, 2, 2) re generte. After join, ll intermeite results in IHTs relte to 1 n 3 re elete. Current intermeite result is shown s Fig 6(). Complexities Anlysis The time ost of T-HGJoin inlues 4 prts, the ost of opertions on H, the isk I/O ost of essing Alist n ll Dlists, the ost of proessing intermeite results n the ost of outputting finl results. In the worst se, sine eh element in Alist is the nestor of ny element in eh Dlist i, IHT i N 2. Therefore, the worst time ost of T-HGJoin is Cost = Alist (ost 2 I + ost O)+( Alist entry A + Dlist i entry D ) ost B B I/O +(k N 2 482

6 Memory -b bukets 1 b 1 1 b 3 2 b 3 - bukets bukets Memory Disk -b bukets - bukets - bukets 3 b 4 2 b b 2 2 b b 1 1 b 3 Memory Disk IHT b IHT IHT 2 b b () (b) () Figure 6: Intermeite Results in ifferent Steps of Exmple 2 ost I +2 k ( entry IHT N 2 n B b ) ost I/O +[( entry IHT N 2 B n b 1) + ( N 2 n b ) k ost join] N) + result F ost out, eh item of whih orrespons the ost of eh prt. The ost nlysis of the former two n lst prts is similr s tht of HGJoin. In the thir item, ost join is the ost of generting one output tuple. The min memory ost of T-HGJoin lgorithm inlues the spe for the hsh tble H uring the sn of Alist n min memory use to store intermeite results in IHT i. With the symbols isusse bove, the min memory spe ost of T-HGJoin is N + k n b B in the worst se. 4.3 Algorithm for Biprtite Queries In this setion, the proessing lgorithm for biprtite subgrph queries is presente. At first, the lgorithm for the biprtite subgrph queries in speil se tht ll esennts shre the sme nestor (CBi for brief) is presente n then tht of biprtite subgrph queries is presente. Suppose tht the soures of CBi query re 1,, m n the soures re 1,, n. Let Alist i=tg( i).alist, Dlist j=tg( j).dlist, i {1,, m}, j {1,, n}. In the lgorithm, ursors l 1,, l m points to the urrent snne position of Alist 1,, Alist m, respetively. t 1,, t n points to the urrent snne positions of Dlist 1,, Dlist n, respetively. Similr s IT-HGJoin, the lgorithm ssigns hsh tble H i for eh soure i. Similr s T-HGJoin, the lgorithm ssigns hsh tble IHT j for the intermeite results orresponing to eh sink j. Bi-HGJoin lgorithm inlues two lterntive steps. In the first step, similr s IT-HGJoin, the lgorithm sns Alist 1,, Alist m in turn n inserts i li in the pir (vl li, i li ) stisfying vl li min(i tj ) (1 j n) n i li / H i to H i. When Alist m is proesse, the lgorithm swithes to proess the Dlist j with the smllest i tj. If the hsh tbles H 1,,H m is not empty, eh tuple (h 1,, h m, i tj ) H 1 H m {i tj } is inserte to the buket with hsh vlue hsh(h 1,,, h m) of IHT j. The seon step is similr s the join of intermeite results in T-HGJoin. When in eh Alist i, the en sign (null, h i) of h i is met, where 1 i m, the ombintion of nestor (h 1,, h m) is inserte into the nestor tble A. When A is full or ll sns on Alists hve been finishe, for eh ombintion (h 1,, h m) in A, the bukets with hsh vlue hsh(h 1,, h m) in IHT 1,, IHT n re joine to generte prtil query result with form (h 1,, h m, i 1,, i n). Then orresponing intermeite results re elete n isk bloks in suh bukets re merge. When the join opertion is finishe, the first step resumes. The bove steps re repete until ll elements in ny Alist i hve been snne. Suh lgorithm nnot proess generl biprtite subgrph b e () f b e f g (b) b e f () f g () Figure 7: An Exmple for Complex Biprtite Query queries. Sine sinks my hve ifferent soures, it is iffiult to fin hsh funtion to ssure effetive exeution of join opertion. Intuitively, suh problem n be proesse with following steps. Bse on the ost moel presente in Setion 5.1, query is split to some CBi subqueries. Then Bi-HGJoin is invoke to proess CBi subqueries to obtin intermeite results. At lst, intermeite results re joine together to obtin finl query results. For exmple, the biprtite subgrph query shown in Fig 7(b) n be split to CBi subqueries in Fig 7() n Fig 7(). When intermeite results re obtine, the equl join is performe on f elements to obtin finl results. 5. DAG SUBGRAPH QUERY EVALUATION In this setion, we present hsh-bse evlution strtegy for struturl queries in form of DAGs. The iret proessing of generl DAG subgrph query requires not only lrge min memory spe but lso lrge isk spe. The effiieny is ffete. Hene the strtegy presente in this setion oes not proess generl subgrph queries iretly but lso splits DAG subgrph query to some CBi-queries. Eh CBi subquery is proesse to obtin intermeite results. Then, join opertions re exeute to obtin finl results. Suh join is performe with sort-merge lgorithms. For exmple, to proess the query in Fig 8(), it is split to the subquerires: q 11 in Fig 8(b), q 12 in Fig 8() n rehbility query e. Then lbelling shemes with tgs tg(b) n tg() re filtere with intermeite results of q 11. Then subquery q 12 is proesse on filtere lbelling shemes to obtin intermeite results. The rehbility query is proesse on filtere t to obtin intermeite results. At lst, the join opertion is performe on suh three groups of intermeite results to obtin finl results. Obviously, the key of the bove strtegy is how to split the query n onstrut the query pln. A query pln n be moelle s DAG D=(V, E), where eh noe in V represents n opertion (possible opertions inlues HGJoin, IT-HGJoin, T-HGJoin n Bi-HGJoin, Filter n sort-merge opertions). The results of the opertion in rrow til is the input of opertion in rrow he. For exmple, in Fig 8(), T HGJoin {(,b),(,)} represents tht 483

7 b () e b (b) q 11 b e () q 12 Bi-HGJoin {(,b), (,)} Filter b IT-HGJoin{(b,),(,)} () Filter HGJoin(,e) Figure 8: Exmple of Query Pln T-HGJoin lgorithm is performe on the lbelling shemes with tg tg(), tg(b) n tb(). F ilter represents tht the lbelling shemes with tg tg() re filtere to eliminte the lbelling shemes whih re not in the intermeite results of T HGJoin {(,b),(,)}. Other opertions hve the sme menings. At first, we esign the ost moel. Then query pln genertion lgorithm n query optimiztion elerte strtegy re presente. 5.1 The Cost Moel The query pln of subgrph query hs multiple hoies. In orer to generte n effiient query pln, query optimizer is require. As the bse of the query optimiztion, the ost moel is presente in this setion. For eh opertion in the query pln, its ost hs two prts, exeution time n require min memory. The former represents the exeution effiieny of query pln n the ltter is the min memory spe whih is require uring query pln exeution. The estimtion of the ost of sort-merge join opertion hs been stuie extensively n the time ost of HGJoin n IT-HGJoin n be estimte s the time omplexity in Setion 3 n Setion 4.1, respetively. This setion fouses on the ost moel of T-HGJoin n Bi-HGJoin. The ost moel of T-HGJoin is similr s time nlysis in Setion 4.2. With intermeite size estimtion tehniques[15, 16], for ID Alist n i {1, 2,, k}, P,i, the number of tuples relte to in the intermeite results of join opertion of Alist n Dlist i, n be estimte. Aitionlly, suh tehnique n be use to estimte S,i, the number of isk bloks of the intermeite results relte to in IHT i t the time when (null,) is met in Alist. Therefore, intermeite results relte to n be estimte s S = i 1,..,k S,i. When intermeite results relte to re joine, S,1,,S,k times of isk bloks istribute in IHT 1,, IHT k re require to be esse with Nest loop metho, respetively. In suh step, the number of isk bloks to be esse is estimte s NL = i 1,..,k S,i. Bse on suh estimtions, the time ost of T-HGJoin is estimte s Cost = Alist (ost 2 I +ost O)+( Alist entry A + B Dlisti entry D ) ost B I/O +ost I k i=1 seletivity(a, Di)+ 2 ID Alist S ost I/O + A NL + resultf ostout, where sel(a, D i) is the result number of join on A n D i. The estimtion of Bi-HGJoin is similr s T-HGJoin. The number of intermeite result genertion with the join on Alist 1,, Alist m n Dlist is enote by sel(a 1,, A m, D i). The number of intermeite results relte to the ombintion of noes 1,, m( i ID Alisti is enote by S 1, m. The number of isk bloks for joining intermeite results with 1,, m is enote by NL 1,, m. The estimtion of these prmeters is similr s those of T- HGJoin. The time of Bi-HGJoin opertion n be estimte b (e) s Cost = Alist i (ost 2 I + ost O) + ( Alist entry A + B Dlisti entry D ) ost B I/O +ost I n i=1 sel(a1,, Am, Di)+ 2 i A i S 1, m ost I/O + i A ji NL 1,, m + result F ost out. The min memory require by n opertion inlues fixe min memory n lterble min memory. The fixe min memory is the buffer for the intermeite result. The size of suh prt equls to fixe vlue fixe mm. The lterble prt is the min memory of the hsh tbles orresponing to Alists uring query proessing. The size of suh prt is liner with the mximum number of nestors of esennts uring query proessing. nestor represents the number of nestors in Alist(s) of Dlist. entry H is the size of t element in hsh tble H. The spe ost of the opertion is ost mm = fixe mm + mx Dlist ( nestor ) entry H. 5.2 Algorithms for Query Pln Genertion In this setion, the proess of query exeution is represente by stte grph n then optiml query pln is generte s the shortest pth genertion lgorithm on the stte grph. Suppose (V Q,E Q) is the query grph of subgrph query Q (m= E Q ). The irete grph G = (V G, E G ) is the stte grph of the subgrph query Q, where V G = {g g = (V g, E g) n E g E Q}. There is irete ege from g V G to g V G if n only if subquery Q gg = (V gg, E gg ) belonging to one of rehbility query, T-query, IT-query n CBi-query exists with E g E gg = E g. The noe in G representing the query grph (V Q, E Q) is lle the strt stte of G, enote by g 0. The noe (V Q, φ) in G is lle en stte of G, enote by g m. Other elements in V G re lle intermeite sttes of G. Obviously, in G, ny pth g 0=g i1,g i2,, g ik = g m (k m) from g 0 to g m orrespons to proessing ourse of the query Q. Suh ourse proesses subqueries Q gij g ij+1 1 j< k) step by step. The query pln esribing suh ourse is generte with following steps. For the first ege in the pth g i1 g i2, n opertion O E (where E = E gi1 g i2 ) is onstrute bse on the opertion type O {HGJoin, T HGJoin, IT HGJoin, Bi HGJoin} of Q gi1 g i2. It is suppose tht the query pln for g 0=g i1,g i2,,g ij hs been generte n the olletion of sets of intermeite results is enote by B j. The ege g ij g ij+1 in the pth is onsiere. At first, bse on the opertion type O {HGJoin, T HGJoin, IT HGJoin, Bi HGJoin} of Q gij g ij+1, n opertion O E is e to the query pln (where E = E gi1 gi 2 ). Then eh noe V gi1 g i2 is onsiere. If some set of intermeite results exists in B j, then n opertion F ilter is e to the query pln. An ege from orresponing intermeite result set is e to F ilter n nother ege from F ilter is e to O E. Then intermeite results set of O E is e to B j. At tht time, if there is mergble intermeite results set in B j, then n opertion sort-merge is e to the query pln n n ege from orresponing intermeite is e result set to new e sort-merge opertion. At the sme time, the unmerge intermeite results set is elete from B j n the merge intermeite result set is inserte. The bove steps re repete until no intermeite result sets n be merge. Let B j+1 = B j. The query pln of other eges is generte until ll the eges in the pth re onsiere. During the genertion of query pln for the pth g 0 = 484

8 g i1,g i2,,g ik =g m, group of opertions re e to the query pln for g ij g ij+1. wg ij g ij+1, the sum of time ost of suh opertion group, n the mximum spe ost w re onsiere. If w is not lrger thn vilble min memory spe, then let the weight of ege g ij g ij+1 equl to w gij gi j+1. Otherwise, suh group of opertions is infesible n the weight of ege g ij g ij+1 equls to +. In suh wy, ny ege gg in G is unique, sine whtever the pth to g is, the olletion of sets of intermeite results sets re the sme. Therefore, the opertion e to gg is the sme. From the bove isussion, it n be seen tht pth from g 0 to g m in the weighte query grph G =(V G,E G ) orrespons to query pln with the weighte length s the ost of the query pln. Therefore, the genertion of the optiml query pln is to fin the shortest pth from g 0 to g m in G. Suh problem n be solve with Dijkstr lgorithm [1]. For the interests of spe, etils of the lgorithm is omitte. From the onstrution of G, it n be known tht V G = 2 m. Sine the time omplexity of Dijkstr lgorithm with n noes in the grph is O(n 2 ), the time omplexity for the genertion of the shortest pth with Dijkstr lgorithm is O(2 2m ). Note tht m is the number of eges in the query grph, for the grph with smller grphs suh lgorithm is effiient. 5.3 Query Optimiztion Aelertion In setion 5.2, when the size of G is lrge, both the time n spe omplexity of optiml query pln genertion with Dijkstr lgorithm will be very lrge. In this setion, some elertion rules re presente. In the following isussion, the weighte length of the urrent shortest pth from g 0 to g is enote by w g with initil vlue +. One g is rehe in the lgorithm, w g is upte. Sine Dijkstr lgorithm uses Best-first expning strtegy, when g is hosen to expen, the shortest pth from g 0 to g is obtine. Bse on suh property, the following rules n be obtine. The former n hlt the lgorithm to reue the run time. The ltter will elete the sttes impossibly belonging to the shortest pth to reue the spe ost. Rule 1 In the Dijkstr lgorithm, if the selete stte g equls to g m or w g w gm, then the urrent shortest pth from g 0 to g m is outputte n the lgorithm hlts. Rule 2 In the Dijkstr lgorithm, if the selete stte is g n eh outgoing ege gg of g stisfies w g + w gg > w g (w gg is the weight of the ege gg ), then g is elete from the t struture of the Dijkstr lgorithm. Proposition 2. Rule 1 n Rule 2 will not ffet the result of query optimiztion. Intuitively, the time ost of exeuting omplex opertion iretly is often smller thn the sum of the time ost of the series of simple opertion split from suh opertion. For exmple, CBi-subquery n be split to some T-subqueries or IT-subqueries, but the sum of the run time of these subqueries is often lrger thn the run time of exeution of CBi iretly. Therefore, in the Dijkstr lgorithm, the sttes with ll esennts expne n be neglete to elerte query optimiztion. This is Rule 3. Rule 3 For urrent stte g n gg E G in Dijkstr lgorithm, if some esennt stte of g hs been inserte into the t struture, then the expning of g will not be performe. For hil stte g of g, the seletivity of g is efine s the seletivity of the opertions orresponing to the ege gg. When the urrent stte g hosen to expn with Dijkstr lgorithm hs multiple hilren sttes, only the hilren with higher seletivities re hosen. So tht query optimiztion will be elerte. Then we hve the following rule. Rule 4 For eh expnsion stte g in Dijkstr lgorithm, ll the hilren of g re sorte by the seletivities. When the expnsion is performe for C times or ll the hilren hve been proesse, the expnsion of g is finishe. Even though Rule 3 n Rule 4 will result in non-optiml query pln. These two rules n reue the time omplexity of query optimiztion effetively. Proposition 3. With Rule 3 n Rule 4, the omplexity of Dijkstr lgorithm is O(C 2 m ) in the worst se. 6. DISCUSSIONS In this setion, we present the isussions bout two vritions to mke our query proessing metho to support subgrph queries in form of generl grphs. One is how to mke the metho to support subgrph query in form of grphs with irles. The other is how to mke the metho to support subgrph queries with jent reltionships. 6.1 Evlute Queries with Cirles In this setion, we present strtegy to pt the fmily of HGJoin to support subgrph queries with irles. Suh strtegy is bse on the feture tht ll rehbility lbelling sheme of the noes in the sme strongly onnete omponent(scc for brief) shre the sme intervl sets. The bsi ie is to ientify ll the SCCs in the noes in the grph relte to the query with lbelling shemes. An i is ssigne to eh SCC n suh i is lso ssigne to the noes belong to suh SCC. All eges in the SCC in the query re elete. Eh seprte prt of the query is proesse iniviully. Then the results of these prts re joine together with the i of SCC bse on the noes orresponing to the query noes in the sme SCC. 6.2 Evlute Queries with Ajent Eges Subgrph queries with jent eges n be proesse in the lgorithms similr to the fmily of HGJoin. An jent lbelling shem is ssigne to eh noe. The genertion of jent lbelling sheme is tht for eh noe with posti i, intervl [i,i] is ssigne to eh of its prents. The benefit of suh sheme is tht the jugement of jent reltionship is sme s tht of rehbility lbelling sheme so tht the proessing tehniques for subgrphs with only rehbility reltionships n be pplie to evlute struturl queries with jent eges. If query noe n s n nestor hs both rehbility n jent outgoing eges, n shoul be split to two query noes with only rehbility n jent outgoing eges, respetively. It is beuse ifferent intervls re use to juge rehbility n jent reltionships. Consiering only outgoing eges is beuse the jugements of rehbility n jent reltionship use the sme postis. When the query proessing metho is pplie, for the query noe with rehbility outgoing eges, intervls in rehbility sheme re use, while for query noe with jent outgoing eges, intervls in jent sheme re use. The lgorithm is sme s the orresponing one in the fmily of HGJoin. 485

9 Tble 1: Sttistis of the XMrk Dtsets ftor Size(M) #Noes(K) #Eges(K) Num(M) Tble 2: The Qulity of Query Pln Query OPT MAX MIN AVG XMQC XMQC EXPERIMENTAL EVALUATION In this setion, we present the results n nlysis of prt of our extensive experiments on the lgorithms in this pper. 7.1 Experimentl Setup Experiments were run on Pentium 3GHz with 512M memory. We implemente ll our lgorithms in this pper. The tset we teste is the XMrk benhmrk [20]. It n be moele s grph with omplite shem n irles. Douments re generte with ftors 0.1 to 2.5. Their prmeters re shown in Tble 1, where Num is the number of numbers in the lbelling sheme in the storge. In orer to evlute the lgorithms on grphs in vrious forms, we lso use syntheti t generte with 2 prmeters: the number of noes with eh tg (noe number for brief) n n the probbility between noes with two ifferent tgs(ege probbility for brief) p. The t set generte in this wy is lle the rnom tset. All the grphs in the rnom tset hve 8 tgs in orer {A, B, C, D, E, F, G, H}. The grphs in the rnom tset my be DAGs or generl grph (GG for brief). For GG, the probbility between eh pir of noes with ny tg is p. For DAG, only the probbility of n existing ege from noe with smller tg to noe with lrger tg is p but the probbility of the eges in inverte iretion is 0. We use run time s the mesure of lgorithms (RT for brief). For queries on XMrk, we hoose two queries for eh lgorithm, one ontins ll the noes not in the irle with smller seletivity, the other one ontins some noes in the irle with lrger seletivity. The queries for HGJoin re XMQS1:text emph n XMQS2:person bol. Queries for IT-HGJoin n T-HGJoin re in Fig 9(), Fig 9(b) n Fig 9(), Fig 9(), respetively. For the omprison with the lgorithm in [4], we lso esign omplex twig queries XMQW1 n XMQW2 shown in Fig 9() n Fig 9(). In orer to stuy the performne of query optimiztion eeply, we esign two omplex struture queries XMQC1 n XMQC2 in Fig 9(i) n Fig 9(j), respetively. Sine Bi-HGJoin lgorithm is the ombintion of IT-HGJoin n T-HGJoin n its fetures re represente by the experiments of these two lgorithms, ue to spe limittion, the experimentl results speilly for Bi-HGJoin re not shown. In orer to mke query grphs ler, without onfusion, in the query grphs we use rrows to represent AD reltionships. Some of these queries re from rel instnes while some of them re syntheti. For exmple, XMQS1 represents the query for retrieving the text n the emph prt belonging to it n XMQT1 is to retrieve the text with ll emph, bol n keywors in it. For queries on the rnom tset, sine the seletivity is minly etermine by ege probbility, we hoose one query for eh lgorithm. The query for HGJoin is RQS: A E. Queries for IT-HGJoin n T-HGJoin re in Fig 9(e) n Fig 9(f). Twig query n omplex query re RQW n RQC, shown in Fig 9(k) n Fig 9(l). Sine the query proessing methos for the queries in form of irle or with jent reltionships re the extensions of tht for DAG queries, the fetures of these lgorithms re similr. Due to spe limittion, the experimentl results of suh queries re not shown in this pper. 7.2 Comprisons For omprison, we implemente stkd lgorithm in [4]. The omprison is performe on 10M XMrk t n rnom XML oument in form of DAG n generl grph with 4096 noes n ege probbility of 0.1, 0.8, 0.4, representing sprse eges, ense eges n the ensity of ege between those two instnes. Note tht in suh se, the ege probbilities 0.1, 0.4 n 0.8 orrespon to the rtio of the numbers of eges n the numbers of noes of 250, 922 n The results re shown in Fig 10 From the results, it n be seen tht the effiieny of our lgorithm outperforms StkD signifintly, espeilly when the ensity of eges is lrge. For rnom grphs with high ege ensity, the ifferene is the most signifint. It hs two resons. The first is tht when the ege of grph is ense, one intervl my be shre by mny noes; our metho n proess sme intervls of noes with sme tg together while StkD proesses them seprtely. The seon is when noes hve mny intervls, stkd hs to mintin very lrge t struture, the opertions on whih is ostly. 7.3 The Effetiveness of Query Optimiztion To vlite the effetiveness of the query optimiztion, we hek the qulity of query plns n the effiieny of query optimiztion. In the experiments in this setion, we fixe the vilble memory to 1M n performe query optimiztion on the 50M XMrk oument The Qulity of Query Pln To vlite the qulity of query plns, we ompre the exeution time of the query pln generte by the optimizer with those of 10 rnom query plns. The results re shown in Tble 2, where the unit of time is ms n OPT is the exeution time of query pln optimize with rule4 with C = 4. The mximl, miniml n verge run time of 10 rnomly generte query plns re shown in the olumns of MAX, MIN n AVG, respetively. From the results, the query optiml strtegy lwys vois the worst query pln n obtins better query pln thn rnom plns o The Effiieny of Query Optimiztion To hek the effiieny of query optimiztion, we ompre the optimiztion time of XMQC1 n XMQC2 with vrious elertion rules. The result is shown in Tble 3, where timei represents the optimiztion time of the optimiztion with Rule i, respetively. EXE-Time is the run time of query pln obtine by query optimizer with rule4. Here in rule4, C is set to be 4. The unit of time is ms. From the results, it n be seen tht our rules n reue query optimiztion time effetively n ompring with query pln exeution 486

10 text bol keywor person item bier text person A B C A emph tegory emph keywor bol ress emph wth D B C D () XMQI1 (b) XMQI2 () XMQT1 () XMQT2 (e) RQI (f) RQT site lose_ution A wth person phone person item eution open_ution tegory keywor text buyer person seller site person item open_ution A B C D B C D E F G bol keywor (g) XMQW1 bol (h) XMQW2 emph ity ge (i) XMQC1 bol emph tegory (j) XMQC2 E F G H (k) RQW H (l) RQC Figure 9: Query Set () XMrk (b) Rnom DAG () Rnom Generl Grph Figure 10: Results of Comprison Tble 3: Effiieny of Query Optimiztion Query time1 time2 time3 time4 EXE-Time XMQC XMQC time, optimize query optimiztion time is very smll. 7.4 Chnging Prmeters Slbility The slbility experiment is to test the run time with the oument in the sme shem but with vrious size. In our experiments, for XMrk, we hnge the ftor from 0.5 to 2.5 n the results re shown in Fig 11() n Fig 11(b). Run time of Fig 11() is in log sle. From the results, the run time of XMQS1, XMQS2, XMQI1, XMQI2 n XMQW2 hnges lmost linerly with the t size. When t size gets lrger, the proess times of XMQT1, XMQT2, XMQW1, XMWC1 n XMWC2 inrese fst. It is beuse mjor prts of these queries re relte to some irle or biprtite subgrph in XML t. The results of suh prt re s the Crtesin proution of relte noes n the number of results n intermeite results inrese in the power of the number of query noes. Therefore, the proessing time inreses fster thn linerly. Sine the time omplexity is relte to the size of results, it is inherent. For the rnom tset, experiments on DAG were performe. Noe number ftors re hnge with fixe ege probbilities 0.1. The results re shown in Fig 11(). Note tht run time n noe number xes of Fig 11() re in log sle. For the sme reson isusse in the lst prgrph, the query proessing time of RQS, RQI, RQT n RQW inreses fster thn linerly with noe number. The run time of RQC oes not inrese signifintly with noe number beuse with query optimizer, RQC is performe bottom-up n the seletivity of subquery ({E, F, G}, H) is very smll. As result, the query proessing time inreses fster thn linerly only when the size of finl results inreses fster. For the rnom tset, we performe experiments on DAGs n hnge ege probbilities from 0.1 to 1.0 with fixe noe number The results re shown in Fig 11(), it shows tht the run time of RQS, RQI, RQT n RQW oes not hnge signifintly with the number of eges. It is beuse when the eges beome ense, more intervls re opie to nestors n the intervls of ll noes tren to be the sme. Bse on our t preproessing strtegy, sme intervls re merge. Therefore, the query proessing time of these queries oes not hnge lot. RQC is n exeption. When the ege probbility hnges from 0.2 to 0.3, the run time hnges signifintly. It is beuse RQC is omplex n only when the ensity of eges rehes threshol, the number of results beomes lrge. We lso performe experiments on generl grphs with 487