Tree Pattern Aggregation for Scalable XML Data Dissemination

Transcription

1 Tree Pttern Aggregtion or Slle XML Dt Dissemintion Chee-Yong Chn, Wenei Fn, Psl Feler, Minos Grolkis, Rjeev Rstogi Bell Ls, Luent Tehnologies Astrt With the rpi growth o XML-oument tri on the Internet, slle ontent-se issemintion o XML ouments to lrge, ynmi group o onsumers hs eome n importnt reserh hllenge. To inite the type o ontent tht they re intereste in, t onsumers typilly speiy their susriptions using some XML pttern speiition lnguge e.g., XPth). Given the lrge volume o susriers, system slility n eiieny mnte the ility to ggregte the set o onsumer susriptions to smller set o ontent speiitions, so s to oth reue their storgespe requirements s well s spee up the oumentsusription mthing proess. In this pper, we provie the irst systemti stuy o susription ggregtion where susriptions re speiie with tree ptterns n importnt sulss o XPth expressions). The min hllenge is to ggregte n input set o tree ptterns into smller set o generlize tree ptterns suh tht: 1) given spe onstrint on the totl size o the susriptions is met, n 2) the loss in preision ue to ggregtion) uring oument iltering is minimize. We propose n eiient tree-pttern ggregtion lgorithm tht mkes eetive use o oument-istriution sttistis in orer to ompute preise set o ggregte tree ptterns within the llotte spe uget. As prt o our solution, we lso evelop severl novel lgorithms or tree-pttern ontinment n minimiztion, s well s lest-upper-oun omputtion or set o tree ptterns. These results re o interest in their own right, n n prove useul in other omins, suh s XML query optimiztion. Extensive results rom prototype implementtion vlite our pproh. 1 Introution XML extensile Mrkup Lnguge) [16] hs eome the ominnt stnr or t enoing n exhnge Currently on leve rom Temple University n supporte in prt y NSF Creer Awr IIS Current ilition: Institut EURECOM, Sophi Antipolis, Frne Permission to opy without ee ll or prt o this mteril is grnte provie tht the opies re not me or istriute or iret ommeril vntge, the VLDB opyright notie n the title o the pulition n its te pper, n notie is given tht opying is y permission o the Very Lrge Dt Bse Enowment. To opy otherwise, or to repulish, requires ee n/or speil permission rom the Enowment. Proeeings o the 28th VLDB Conerene, Hong Kong, Chin, 2002 on the Internet, inluing e-business trnstions in oth Business-to-Business B2B) n Business-to-Consumer B2C) pplitions. Given the rpi growth o XML tri on the Internet, the eetive n eiient elivery o XML ouments hs eome n importnt issue. Consequently, there is growing interest in the re o XML ontent-se iltering n routing e.g., [4]), whih resses the prolem o eetively ireting high volumes o XML-oument tri to intereste onsumers se on oument ontents. Unlike onventionl routing, where pkets re route se on limite, ixe set o ttriutes e.g., soure/estintion IP resses n port numers), ontent-se routing is se on generl ptterns o the oument ontents, whih is signiintly more lexile n emning. Consumers typilly speiy their susriptions, initing the type o XML ontent tht they re intereste in, using some XML pttern speiition lnguge e.g., XPth [15]). For eh inoming XML oument, ontent-se router mthes the oument ontents ginst the set o susriptions to ientiy the su)set o intereste onsumers, n routes the oument to them. Thus, in ontent-se routing, the estintion o n XML oument is generlly unknown to the t prouer, n is ompute ynmilly se on the oument ontents n the tive set o susriptions. Eetive support or slle, ontent-se XML routing is ruil to enling eiient n timely elivery o relevnt XML ouments to lrge, ynmi group o onsumers. Given the lrge volume o potentil onsumers, system slility n eiieny mte the ility to juiiously ggregte the set o onsumer susriptions to smller set o ontent speiitions. The gol, o ourse, is to oth reue the susriptions storge spe requirements e.g., so tht the routing tle its in min memory), s well s spee up the iltering o inoming XML tri. For instne, ore router in B2B pplition my hoose to ggregte susriptions se on geogrphil lotion, ilition, or omin-speii inormtion e.g., teleommunitions). Susription ggregtion essentilly involves ggregting n initil set o susriptions into smller set suh tht ny oument tht mthes some susription in lso mthes some susription in. However, sine there is typilly loss o preision ssoite with suh ggregtion, the ouments mthe y the ggregte set is, in generl, superset o those mthe y the originl set. As result, oument my e route to onsumers who hve not susrie to it, thus resulting in n inrese in the mount o unwnte

2 Bh CD SONY ) p CD Bh ) p CD Bh ) p Bh ) p CD Bh CD SONY Clssil Jzz Pop e) T Figure 1: Exmple Tree Ptterns n XML Doument Tree. oument tri. In orer to voi suh spurious orwring o ouments, it is esirle to minimize the numer o suh lse mthes i.e., minimize the loss in preision) with respet to the given spe onstrint or the ggregte susriptions. So r, there hs only een limite work on susription ggregtion, minly or very simple susription moels. For exmple, in [12], eh susription is set o ttriute-preite ') pirs e.g., GE! #"$ &% ), n n ggregte susription is llowe to ontin wilr vlues, initing the entire set o omin vlues or ertin ttriutes. 1 In this pper, we provie the irst systemti stuy o the susription ggregtion prolem where susriptions re speiie using the muh more expressive moel o tree ptterns. Tree ptterns represent n importnt sulss o XPth expressions tht oers nturl mens or speiying tree-struture onstrints in XML n LDAP pplitions []. Compre to erlier work se on ttriute/preite-se susriptions, eetively ggregting tree-ptterns poses muh more hllenging prolem sine susriptions involve oth ontent inormtion noe lels) s well s struture inormtion prent-hil n nestor-esennt reltionships). Briely, our tree pttern ggregtion prolem n e stte s ollows: Given n input set o tree ptterns n spe onstrint, ggregte into smller set o generlize tree ptterns tht meets the spe onstrint, n or whih the loss in preision ue to ggregtion is minimize. Exmple 1.1 Consier the two similr tree-pttern-se susriptions n + shown in Figure 1, where, mthes ny oument with root element lele CD tht hs oth su-element lele SONY s well s su-element with n ritrry lel) tht in turn hs su-element lele Bh n + mthes ny oument tht hs some element lele CD with suelement lele Bh. Here the noe lele - wilr) mthes ny lel, while the noe lele.. esennt) mthes some possily empty) pth. The XML oument / shown in Figure 1e) mthes or stisies) 0 ut not + euse the su-element lele Bh in 1 Due to spe onstrints, more etile overview o relte work n e oun in the ppenix. / oes not hve prent element lele CD. For eiieny resons, one might wnt to ggregte the set o tree ptterns! + into single tree pttern. Two exmples o ggregte tree ptterns or1,'!0+ re 2 n 0 in Figure 1) sine ny oument tht stisies or + lso stisies oth 2 n. Although oth 2 n 0 hve the sme numer o noes, 2 is intuitively more preise thn 0 with respet to1! + sine 2 preserves the nestor-esennt reltionship etween the CD n Bh elements s require y n +. Inee, ny XML oument tht stisies 2 lso stisies n thus we sy tht ontins 02 ). 4 To the est o our knowlege, our work is the irst to ress this timely susription ggregtion prolem or XML t issemintion. Our min ontriutions n e summrize s ollows. 5 We stuy the properties o tree ptterns n evelop eiient lgorithms or eiing tree pttern ontinment, minimizing tree pttern, n omputing the most preise ggregte i.e., the lest upper oun ) or set o ptterns. Our results re not only interesting in their own right, ut lso provie solutions or speil ses o our tree pttern ggregtion prolem. 5 We propose novel, eiient metho tht exploits orse sttistis on the unerlying istriution o XML ouments to ompute preise set o ggregte ptterns within the llotte spe uget. Speiilly, our sheme employs the oument sttistis to estimte the seletivity o tree pttern, whih is lso use s mesure o the pttern s preiseness. Thus, our ggregtion prolem reues to tht o ining ompt set o ggregte ptterns with miniml loss in seletivity, or whih we present greey heuristi. 5 We emonstrte experimentlly the eetiveness o our pproh in omputing spe-eiient n preise set o ggregte tree ptterns. The useulness o our results on tree ptterns n their ggregtion is not limite to ontent-se routing, ut lso extens to other pplition omins suh s the optimiztion o XML queries involving tree ptterns n the proessing/issemintion o susription queries in multist environment [9] where ggregtion n e use to reue server lo n network tri). Further, our work n results re omplementry to reent work on eiient inexing strutures or XPth expressions [2, 6]. The ous o this erlier reserh is to spee up oument iltering with given set o XPth susriptions using pproprite inexing shemes. In ontrst, our work ouses on eetively reuing the volume o susriptions tht nee to e mthe in orer to ensure slility given oune storge resoures or routing. Clerly, our tehniques n e use s pre-proessing step or the inexes o [2, 6] when hr onstrints on the size o the inex must e met. Due to spe limittions, the proos o ll theoretil results n e oun in the ull version o this pper [5].

3 2 Prolem Formultion 2.1 Deinitions A tree pttern is n unorere noe-lele tree tht speiies ontent n struture onitions on n XML oument. More speiilly, tree pttern hs set o noes, enote y, where eh noe in hs lel, enote y, whih n either e tg nme, - wilr tht mthes ny tg), or.. the esennt opertor). In prtiulr, the root noe hs speil lel.. We use 0! to enote the sutree o roote t, reerre to s su-pttern o. Some exmples o tree ptterns re epite in Figure 2. To eine the semntis o tree pttern, we irst give the semntis o su-pttern!#, where is not the root noe o. Rell tht XML ouments re typilly represente s noe-lele trees, reerre to s XML trees. Let / e n XML tree n e noe in /. We sy tht / stisies!# t noe, enote y /!#, i the ollowing onitions hol: 1) i! is tg, hs hil noe lele suh tht or eh hil noe o,!/!! 2) i -, hs hil noe lele with n ritrry tg suh tht or eh hil noe o, /!! n ) i.., hs esennt noe possily ) suh tht or eh hil o, /!!. We next eine the semntis o tree ptterns. Let / e e tree pttern with n XML tree with root, n root!. We sy tht / stisies, enote y /", i or eh hil noe o #, 1) i is tg, is lele with n or eh hil noe o,!/ $ %! here! speiies the tg o ) 2) i -, my hve ny lel n or eh hil noe o, / &!! ) i.., hs esennt noe possily ' ) suh tht /, where / is the sutree roote t, n is ientil to!# exept tht is the lel or the root noe inste o! ). Oserve tht is trete ierently rom the rest o the noes o. The motivtion ehin this is illustrte y in Figure 2, whih speiies the ollowing: or ny XML tree / stisying, its root must e lele with n moreover, it must ontin two onseutive elements somewhere. This nnot e expresse without our speil root lel s tree ptterns o not llow union opertor). Exmple 2.1 Consier the tree pttern, in Figure 2. An XML oument / stisies, i its root element stisies ll the ollowing onitions: 1) its lel is 2) it must hve hil element with n ritrry tg, whih in turn hs hil element with lel n ) it must hve esennt element whih hs oth -hil element n n -hil element. Thus, essentilly speiies existentil) onjuntive onitions on XML ouments. It shoul e note tht ouments stisying my hve tgs/sutrees not mentione in. For instne, the root element o / my hve -hil element, n the -elements o / my hve -esennt elements. 4 A tree pttern is si to e onsistent i n only i there exists n XML oument tht stisies. We only onsier onsistent tree ptterns in our work. Further, the tree ptterns eine ove n e nturlly generlize to ommote simple onitions ' n preites e.g., GE n ). To simpliy the isussion, we o not onsier suh extensions in this pper. It is worth mentioning tht tree pttern n e esily onverte to n equivlent XPth expression [15] in whih eh su-pttern is expresse s onition/quliier [5]. Thus, our tree ptterns re grph representtions o lss o XPth expressions, whih re similr to the tree ptterns tht hve een stuie or XML queries e.g., [, 17]). It is tempting to onsier using lrger rgment o XPth to express susription ptterns. However, it turns out tht even mil generliztion o our tree ptterns e.g., with the ition o union/isjuntion opertors) les to muh higher omplexity onp-hr or eyon) or si opertions suh s ontinment omputtion e.g., see [10]). A tree pttern ) is si to e ontine in nother tree pttern, enote y )+, i n only i or ny XML tree /, i / stisies ) / lso stisies. I )+, we reer to s the ontiner pttern n ) s the ontine pttern. We sy tht n ) re equivlent, enote y -,.), i /0) n )-. This einition n e generlize to sets o tree ptterns: set o tree ptterns is ontine in nother set o tree ptterns, enote y 1, i or eh 2, there exists 2 suh tht. Continment or su-ptterns is eine similrly. The size o tree pttern, enote y 4, is simply the rinlity o its noe set. For exmple, reerring to Figure 2, 65 n + ' Prolem Sttement The tree pttern ggregtion prolem tht we investigte in this pper n now e stte s ollows. Given set o tree pttern susriptions n spe oun 9 on the totl size o the ggregte susriptions, ompute set o tree ptterns tht stisies ll o the ollowing three onitions: C1) : i.e., is t lest s generl s ), C2) <&=?>A@ B >C D-9 i.e., is onise ), n C) is s preise s possile, in the sense tht there oes not exist nother set o tree ptterns tht stisies the irst two onitions n. Clerly, the tree pttern ggregtion prolem my not neessrily hve unique solution sine it is possile to hve two sets n tht stisy the irst two onitions ut E n FE. Thereore, we nee to evise some mesure to quntiy the gooness o nite solutions in terms o oth their oniseness s well s preiseness. With respet to oniseness, we re intereste in miniml tree ptterns tht o not ontin ny reunnt noes. More preisely, we sy tht tree pttern is minimize i or ny tree pttern suh tht,, it is the se tht GHDI J. With respet to preiseness, it n e

4 E E E ) p ) p ) p ) p x y e) p e ) p g) p g h) p h i) p i Figure 2: Exmples o Tree Ptterns. shown tht the ontinment reltionship on the universe o tree ptterns tully eines lttie. In prtiulr, the notions o upper oun n lest upper oun re o relevne to the ggregtion prolem n, thereore, we eine them ormlly here. An upper oun o two tree ptterns n ) is tree pttern suh tht n ), i.e., or ny XML tree /, i / or / 6) /. The lest upper oun LUB) o n ), enote y ), is n upper oun o n ) suh tht, or ny upper oun o n ),. One gin, we generlize the notion o LUBs to set o tree ptterns. An upper oun o is tree pttern, enote y /, suh tht : or every :2. The LUB o, enote y, is n upper oun o suh tht or ny upper oun o,. Clerly, i is n ggregte tree pttern or set o tree ptterns i.e.,. ), is n upper oun o. Oserve tht, i is the LUB o, is the most preise ggregte tree pttern or. In t, it n e shown tht exists n is unique up to equivlene or ny set o tree ptterns [5] thus, it is meningul to tlk out s the most preise ggregte tree pttern. Exmple 2.2 Consier gin the tree ptterns in Figure 2. Oserve tht +, 02 n sine + % 02, 0+ is not minimize pttern. In t, exept or,+, ll the tree ptterns in Figure 2 re minimize ptterns. Note tht 02 euse the root noe o oes not hve tg- hil noe n 2 euse there exists no noe in 2 tht is prent noe o oth tg--noe n tg--noe. Oserve tht 0 n 2 0 i.e., is n upper oun o n 2. However, 2 sine we hve nother tree pttern,, whih is n upper oun o n 2 suh tht. Inee, 2 with! 8 2. Note, however, tht the size o n LUB is not neessrily lwys smller thn the size o its onstituent ptterns. For exmple, 2 ut % 02. Note tht is n upper oun o1 '0+!02!!. 4 We onlue this setion y presenting some itionl nottion use in this pper. For noe in tree pttern, we enote the set o hil noes o in y 0!. We lso eine prtil orering on noe lels suh tht i n re tg nmes, 1) -.'. n 2) i. Given two noes n, 0 is eine to e the lest upper oun o their lels! n s ollows: 65 9!%'& ),+:1! #"$!%'& ),+.-0/214 7 =>= 9!%'& ),+:19!%'& ),/21<- i )?9!%'& ),+:1 =>= 1 86 i )?9!%'& ),/214 =@= 1<- or otherwise. For exmple, # - n -.'.!... For nottionl onveniene, we reer to noe in tree pttern s n A -B ' i CA, n reer to s tg-noe i 2 E Computing the Most Preise Aggregte In this setion, we onsier speil se o our tree pttern ggregtion prolem, nmely, when the ggregte set onsists o single tree pttern n there is no spe onstrint. For this se, we provie n lgorithm to ompute the most preise ggregte tree pttern i.e., LUB) or set o tree ptterns. Some o the lgorithms given in this setion re lso key omponents o our solution or the generl prolem, whih is presente in the next setion. Given two input tree ptterns n ), Algorithm LUB in Figure omputes the most preise ggregte tree pttern or ) i.e., the LUB o n ) ). It trverses n ) top-own n omputes the tightest ontiner su-ptterns or eh pir o su-ptterns!# n ) )! enountere, where n re noes in n ), respetively. The tightest ontiner su-ptterns o n ) re set D o su-ptterns suh tht: 1) D onsists o ontiner su-ptterns 2 o n ), i.e., or ny XML oument / n ny element in /, i!/ or /.) / or eh 2ED n, 2 Note tht su-pttern o tree ptterns F n G is n upper-oun o F n G, n we use these two terms interhngely.

5 ë h )- >1 Algorithm LUB Input: n re tree ptterns. Output: A tree pttern representing the LUB o n. 1) i ) return 2) i ) return % +.-0/ ) Initilize +!&! ) 1<- /"!&#!, )$>1 4) Let +#%'&&) n /%'&&) enote the root noes o n, resp. 5) or eh + /#),+0%'&1&) -2#1,+.- o 6) or eh /" /#),/ %'&1&) - 1 % 4, /5#76#849 :849 o),+.- / - % 4 1 7) 8) Crete tree pttern with root noe lel =4< n the set o hil = su-ptterns Y >?A@CBEDGF HEIJ>1K'L LNMPO QERSO TU?4@CBDVF HEIJTWK'LNLNMO XR ), 1 9) return Z\[E]\[EZ^[#_#` % 4 1 Algorithm + / ),+.- / - LUB SUB Y Input:, re noes in tree ptterns % /, % 4 respetively),, % 4 is +.- / 2-imensionl rry suh tht, %PN>& & is the),+.- set#1 o & ontiner ),/ -N 1 su-ptterns o n Y. Output: ) % / Y % / >1. 1) i Y 2) return ) % /5 %PN>& & ),/ -N 1 %PN>& & ),+.-#1 1 ) else i 4) return ) & ),+.-$ 1 %PN>& & Y ),+.- 1 %PN>& & ),/ - >1 1 5) else i 6) return & ),/ -'>1 Y 7) else 8) Initilize + g e e, e 9) or eh / /#),+.-2 1,+.- V#),/ -1>1 o 10) or eh 60849,+\- :E849),+ - / o 11) e + % 1 /#), ) or eh,+.- o 1) e e / :E849),+ - / - % 4 1 h V#),/ -1>1 14) or eh,+\- 6#849 :849 ),+.- o / % ) e e h, 16) Let e the pttern with root noe lel n set o hil sutree ptterns e 17) Let =>= e the pttern with root noe lel n set o hil sutree ptterns e 18) Let =@= e the pttern with root noe lel n set o hil sutree ptterns e 19) return % /5. -! #"$!%'& ),+.-0/21 Figure : Lest-Upper-Boun Computtion Algorithm. 2) D is tightest in the sense tht or ny other set o ontiner su-ptterns D o n ) tht stisies onition 1), ny XML oument / n ny element in /, i!/ + or eh 2 D / or ll 2 D. Intuitively, D is olletion o onitions impose y oth n ) suh tht i / stisies or ) t, / lso stisies the onjuntion o these onitions t. We now show how the LUB or n ) n e ompute rom the tightest ontiner su-ptterns. Let n e the roots o ptterns n ), respetively. Note tht oument / tht stisies lso stisies, or eh 2!, the restrition o to the root noe n only 0!. Consequently, oument / tht stisies or ) must lso stisy the pttern onsisting o root noe with lel ) whose hilren re the tightest ontiner suptterns or eh pir!# n )!, where 2 # 1! n /2 )!. This pttern is thus n LUB o n ). The min suroutine in our LUB omputtion Algorithm LUB SUB) omputes the tightest ontiner suptterns o n ) s ollows. I ) resp. 0) ), resp. ) ) is the tightest ontiner supttern otherwise, the tightest ontiner su-ptterns re set > < o su-ptterns, whih re eine in the ollowing mnner. The root noe o is lele with 0 n the hil sutrees o re the tightest ontiner su-ptterns o eh hil sutree o n eh hil sutree o )!. Intuitively, the root o orrespons to the roots o n ) with lel equl to the lest upper oun o tht o n ) ). In other wors, preserves the positions o the orresponing noes in n ). However, this position-preserving generliztion is not suiient sine n ) my hve ommon suptterns t ierent positions reltive to their roots. For exmple, 2 n in Figure 2 hve ommon su-pttern roote t n -noe tht hs oth -hil n -hil, ut this pttern is lote t ierent positions reltive to the roots o 2 n. To pture these o-position ommon su-ptterns, we nee to ompute n. The hil sutrees o re the tightest ontiner su-ptterns o ) itsel n eh hil sutree o n the lel o the root noe o is.'. to ommote ommon su-ptterns t ierent positions reltive to the roots o n ). Similrly, the root noe o hs lel.., n the hil sutrees o re the tightest ontiner su-ptterns o itsel n eh hil sutree o ). By omputing the tightest ontiner su-ptterns reursively, the lgorithm omputes the LUB o the input tree ptterns n ). By inution on the strutures o n ), we n show the ollowing result [5]. Proposition.1: Given two tree ptterns n ), Algorithm LUB )! omputes ). 4 Exmple.1 Given 2 n in Figure 2, Algorithm LUB returns, whih is inee 2. To help explin the omputtion o, we use the nottion i to reer the B noe in some tree pttern) tht is lele, where eh olletion o noes shring the sme lel re orere se on their pre-orer sequene or exmple, in, we use..^j n.'.ak to reer to the letmost n rightmost.. -noes, respetively. Algorithm LUB SUB invoke y Algorithm LUB) irst extrts the position preserving tightest ontiner su-ptterns or j!02 n 0! #, whih yiels the su-pttern j in Steps 9 11). Note tht the root noe o j! is lele euse oth the root noes o j! 2 n 0! re lele. The su-ptterns Ẅl! 2 n, however, hve quite ierent strutures n thus position-preserving ttempt to extrt their ommon su-ptterns only yiels

6 ! & & & j. In prtiulr, the ommon su-pttern onsisting o n -noe with oth -hil-noe n -hil-noe is not pture y the ove proess euse they our t ierent positions reltive to the root noes o Ẅl! 2 n. To extrt suh o-position ommon su-ptterns, Algorithm LUB SUB ompres j 2 with n!, s well s ompres # with Ẅl 2 in Steps 12 15). Inee, this yiels..0k! whih hs.'. -root sine this ommon su-pttern ours t ierent positions reltive to the root noes o j! 2 n 0! #. It shoul e mentione tht oth.'.j! n.. l! re lso proue y the o-position proessing, s Algorithm LUB SUB reursively proesses the su-pttern l 02 with n! #, respetively. Finlly, the lgorithm removes the reunnt noes in the result tree pttern y using minimiztion lgorithm whih will e expline shortly) to generte the LUB. 4 It is strightorwr to show tht our LUB opertor, onsiere s inry opertor, is ommuttive n ssoitive, i.e., j Ul Ul j n j l k j l k. As result, Algorithm LUB n e nturlly extene to ompute the LUB o ny set o tree ptterns. We next explin the etils o the two uxiliry lgorithms use in Algorithm LUB. Algorithm LUB nees to hek the ontinment o tree ptterns, whih is implemente y Algorithm CONTAINS in Figure 4. Given two input tree ptterns n ), the lgorithm etermines i )%. It mintins two-imensionl rry 4,, whih is initilize with 4 #< B,! to inite tht 2/ n 2/ )! hve not een ompre otherwise, 4 #< 2 suh tht 4 #< ' 0 i n only i? )! +!#. Clerly, ) i n only i F, #, where! n enote the root noes o n ), respetively. The min suroutine in our ontinment lgorithm is Algorithm CONTAINS SUB. Astrtly, CONTAINS SUB trverses n ) top-own n uptes 4, 0 or eh pir o noes 2. n 2. ' )! visite s ollows. Let n )! enote 0! n? )!, respetively. I 4 #< hs lrey een ompute i.e., 4, 0 E B,! ), its vlue is returne. Otherwise, our lgorithm etermines whether ), s ollows. I! E.., 4, 0 i n eh hil sutree o ontins some hil sutree o. Otherwise, i.'., two itionl onitions nee to e tken into ount. This is euse unlike - -noe or tg-nme-noe,.. -noe in ontiner tree pttern n lso e mppe to possily empty) hin o noes in ontine tree pttern. For exmple, onsier the tree ptterns n in Figure 2. Note tht, n the.'. -noe in is not mppe to ny noe in in the sense tht woul still e ontine in i the.'. -noe )- >1 Algorithm CONTAINS Input: n re two tree ptterns. Output: Returns i :$! & 04!A +.-0/5. otherwise. 1) Initilize +!&! )#1<- /" #!&,! )$>1 2) Let ) +#%1&1&) /#),+ n /%1&1&) enote the root noes o n %1&1&) -$ 1$ 1, resp. ) i,+.- 4) return N 5) else ),+ 6) return CONTAINS SUB %'&&) - / %'&1&) -! 1 04! 1 Algorithm + / ),+.- / - CONTAINS SUB Input:, re noes in tree ptterns!a +.- /5C - :$! & - 4!A +.-0/ Output: 4!A +.-0/.,1, respetively), is 2-imensionl rry suh tht eh. ) 1) i 2) return + 04!4 +.- / ) i is le noe in ) 4) )?9!%'& ),/21!A +.- /5# )?9!%'& ),/21 :% & ),+:1 1 5) else i :% & ),+:1 1!A +.- /5#! & 6) 7) else 8)!A +.- /5# : :E849 ),+ -0/ ]^[E] >>S?A@ ) BDGF H I> O QR T>S?A@CBEDGF HEIJTUO XR 4!A +.-0/ :$! & 1 )?9!% & ),+:14 9) i =@=!A +.- /5. n 10) >>S?A@CBEDGF ) : :E849),+ - / - 04 HEIJ> O QR ]^[E] 4!A +.-0/ :$! & 1 )?9!% & ),+:14 11) i =@= n!a +.- /5. : :E849 ),+.-0/ 12) ]^[E] T >?A@ BDGF H IJTUO XR!A +.-0/ 1) return Figure 4: Tree-Pttern Continment Algorithm. 04! 1 1! ! 1 in 0 is elete. On the other hn, or the tree ptterns n in Figure 2, n the.'. -noe in is mppe to oth the - - n -noes in in the sense tht -.'. n..)!. These two itionl senrios re hnle y Steps 10 n 12 in Algorithm CONTAINS SUB: Step 10 ounts or the se where.'. -noe itsel) is mppe to n empty hin o noes, n Step 12 or the se where.. -noe itsel) is mppe to nonempty hin. Note tht in Steps 8 n 12, the expres- > ) sion i "! $# %'&)+,)-).0/, < 4, returns i )! 65. By inution on the strutures o n, we n show the ollowing result. Proposition.2: Given two tree ptterns n ), Algo- ) 9 4 rithm +7,)-).8/,+1 )! etermines i in 4+: )C time. The qurti time omplexity o our tree-pttern ontinment lgorithm is ue to, mong other things, the t tht eh pir o su-ptterns in n ) is heke t most one, euse o the use o the 4, rry. To simpliy the isussion, we hve omitte rom Algorithm CON- TAINS ertin sutle etils tht involve tree ptterns with

7 l hins o.. - n - -noes. Suh ses require some itionl pre-proessing to onvert the tree pttern to some nonil orm, ut this oes not inrese our lgorithm s time omplexity. To ensure tht our tree ptterns re onise, we nee to ientiy n eliminte reunnt noes in them. Given tree pttern, minimize tree pttern equivlent to n e ompute using reursive lgorithm MIN- IMIZE. Strting with the root o, our minimiztion lgorithm perorms the ollowing two steps to minimize the su-pttern 0! roote t noe in : 1) For ny 2 0!, i!!!!, elete! rom 0! n, 2) For eh 2!# tht ws not elete in the irst step), reursively minimize!. The omplete etils n e oun in [5]. Proposition.: Algorithm MINIMIZE minimizes ny tree pttern in 9 4 time. 4 Proposition.4: For ny minimize tree ptterns n,, i i.e., they re synttilly equl). 4 Given the low omputtionl omplexities o CON- TAINS n MINIMIZE, one might expet tht this woul lso e the se or Algorithm LUB. Unortuntely, in the worst se, the size o the minimize) LUB o two tree ptterns n e exponentilly lrge see [5] or etile nlysis). Our implementtion results, however, emonstrte tht our LUB lgorithm exhiits resonly low vergese omplexity in prtie. 4 Seletivity-se Aggregtion Algorithm While the LUB lgorithm presente in the previous setion n e use to ompute single, most preise ggregte tree pttern or given set o ptterns, the size o the LUB my e too lrge n, thereore, my violte the speiie spe onstrint 9 on the totl size o the ggregte susriptions Setion 2.2). Thus, in orer to it our ggregtes within the llotte spe uget, we relx the requirement o single preise ggregte y permitting our inste o solution to e set 1 j l?? single pttern), suh tht eh pttern ) 2 is ontine in some pttern F2. O ourse, we lso require tht provie the tightest ontinment or ptterns in or the given spe onstrint Setion 2.2) tht is, the numer o XML ouments tht stisy some tree pttern in ut not, is smll. A simple mesure o the preiseness o is its seletivity, whih is essentilly the rtion o iltere XML ouments tht stisy some pttern in. Thus, our ojetive is to ompute set o ggregte ptterns whose seletivity is very lose to tht o. Clerly, the seletivity o our tree ptterns is highly epenent on the istriution o the unerlying olletion o XML ouments enote y ). It is, however, inesile to mintin the etile istriution o streming XML ouments or our ggregtion the spe requirements woul e enormous! Inste, our pproh is se on uiling onise synopsis o on-line i.e., s ouments re streming y), n using tht synopsis to estimte pproximte) tree-pttern seletivities. At high level, our ggregtion lgorithm itertively omputes set tht is oth seletive n stisies the spe onstrint, strting with i.e., the originl set o ptterns), n perorming the ollowing sequene o steps in eh itertion: 1. Generte nite set o ggregte tree ptterns onsisting o ptterns in n LUBs o similr pttern pirs in. 2. Prune eh pttern in y eleting/merging noes in in orer to reue its size.. Choose nite 2 to reple ll ptterns in F tht re ontine in. Our nite-seletion strtegy is se on mrginl gins [14]: The selete nite is the one tht results in the minimum loss in seletivity per unit reution in the size o to the replement o ptterns in y ). ue Note tht our pruning step Step 2) ove mkes nite ggregte ptterns less seletive in ition to eresing their size). Thus, y repling ptterns in y ptterns in, we re eetively trying to reue the size o F y giving up some o its seletivity. In the ollowing susetions, we esrie in more etil our lgorithm or omputing. We egin y presenting our pproh or estimting the seletivity o tree ptterns over the unerlying oument istriution, whih is ritil to hoosing goo replement nite in Step ove. 4.1 Seletivity Estimtion or Tree Ptterns The Doument Tree Synopsis. As mentione ove, it is simply impossile to mintin the urte oument istriution i.e., the ull set o streming ouments) in orer to otin urte seletivity estimtes or our tree ptterns. Inste, our pproh is to pproximte y onise synopsis struture, whih we reer to s the oument tree. Our oument tree synopsis or, enote y /, ptures pth sttistis or ouments in, n is uilt on-line s XML ouments strem y. The oument tree essentilly hs the sme struture s n XML tree, exept or two ierenes. First, the root noe o / hs the speil lel. Seon, eh non-root noe in / hs requeny ssoite with it, whih we enote y 0 ). Intuitively, i j. l. : : :. i is the sequene o tg nmes on noes long the pth rom the root to exluing the lel or the root), 0 ) represents the numer o ouments / in tht ontin pth with tg sequene j. l. :": :1. i originting t the root o /. The requeny or the root noe o / is set to, the numer o ouments in. As XML ouments strem y, / is inrementlly mintine s ollows. For eh rriving oument /, we / / / / irst onstrut the skeleton tree / or oument. In the skeleton tree /, eh noe hs t most one hil with given tg. / is uilt rom y simply olesing two hilren o noe in i they shre ommon tg. Clerly, y trversing noes in in top-own shion, n olesing

8 x x e) Doument Tree 2. x x ) Compresse Doument Tree ) T1 ) T2 ) T ) Skeleton tree or T1 x x g) p1 h) p2 x i) p Figure 5: Exmple Douments, Skeleton Tree, Doument Tree, n Ptterns. hil noes with ommon tgs, we n onstrut / rom / in single pss using n event-se XML prser). As n exmple, Figure 5) epits the skeleton tree or the XML-oument tree in Figure 5). Next, we use / to upte the sttistis mintine in our oument tree synopsis / s ollows. For eh pth in /, with tg sequene sy j. l. : : :. i, let e the lst noe on the orresponing unique) pth in /. We inrement 0 ) y. Figure 5e) shows the oument tree with noe requenies) or the XML trees / j, / l, n / k in Figure 5) to ). Note tht it is possile to urther ompress / y using tehniques similr in spirit to the methos employe y Aoulng et l. [1] or summrizing pth trees. The key ie is to merge noes with the lowest requenies n store, with eh merge noe, the verge o the originl requenies or noes in / tht were merge. This is illustrte in Figure 5) or the oument tree in Figure 5e), n with the lel use to inite merge noes. Due to spe onstrints, in the reminer o this susetion, we only present solutions to the seletivity estimtion prolem using the unompresse tree /. However, our propose methos n e esily extene to work even when / is ompresse [5]. We shoul note here tht our seletivity estimtion prolem or tree ptterns iers rom the work o Aoulng et l. [1] in two importnt respets. First, in [1], the uthors onsier the prolem o estimting seletivity or only simple pths tht onsist o -noe ollowe y tg noes. In ontrst, we estimte seletivities o generl tree ptterns with rnhes, n - or -noes ritrrily istriute in the tree. Seon, we re intereste in seletivity t the grnulrity o ouments, so our gol is to estimte the numer o XML ouments tht mth tree pttern inste, [1] resses the seletivity prolem t the grnulrity o iniviul oument elements tht re isovere y pth. It is esy to see tht these re two very ierent estimtion prolems. Seletivity Estimtion Proeure. Rell tht the seletivity o tree pttern is the rtion o ouments / in tht stisy. By onstrution, our / synopsis gives urte seletivity estimtes or tree ptterns omprising single hin o tg-noes i.e., with no or ). However, otining urte seletivity estimtes or ritrry tree ptterns with rnhes,, n is, in generl, not possile with / summries. This is euse, while / ptures the numer o ouments ontining single pth, it oes not store oument ientities. As result, or pir o ritrry pths in tree pttern, it is impossile to etermine the ext numer o ouments tht ontin oth pths or ouments tht ontin one pth, ut not the other. Our estimtion proeure solves this prolem, y mking the ollowing simpliying ssumption: The istriution o eh pth in tree pttern is inepenent o other pths. Thus, we estimte the seletivity o tree pttern ontining no.. or - lels, simply s the prout o the seletivities o eh root to le pth in the pttern. For ptterns ontining.. or -, we onsier ll possile instntitions or.'. n - with element tgs, n hoose s our pttern seletivity the mximum seletivity vlue over ll instntitions. This is similr to the einition o uzzy opertor in uzzy logi [1].) We illustrte our seletivity estimtion methoology in the ollowing exmple. Exmple 4.1 Consier the prolem o estimting the seletivities o the tree ptterns shown in Figures 5g) to i) using the oument tree shown in Figure 5e). The totl numer o ouments,, is. Clerly, the numer o ouments stisying pttern j whih onsists o single pth, n e estimte urtely y ollowing the pth in / n returning the requeny or the -noe t the en o the pth) in /. Thus, the seletivity o j is. whih is urte sine only ouments / l n / k stisy j. Estimting the numer o ouments ontining pttern l, however, is somewht more triky. This is euse there re two pths with tg sequenes.!. n.!.. in / tht mth Ul orresponing to instntiting with n. ). Summing the requenies or the two -noes t the en o these pths gives us n nswer o 4 whih over-estimtes the numer o ouments stisying Yl only ouments /Yl n / k stisy Ul ). To voi oule-ounting requenies, we estimte the numer o ouments stisying l to e the mximum n not the sum) o requenies over ll pths in / tht mth l. Thus, the seletivity o l is estimte s.. Finlly, the seletivity o k is ompute y onsiering ll possile instntitions or n, n hoosing the one with the mximum seletivity. The two possile instntitions or tht result in non-zero seletivities re n,.!, n - n e instntite with either or or.., n or or.'.,.!. Choosing.'. n -& results in the mximum seletivity sine the prout o the seletivities o pths.! ). n,.. is mximum, n is equl to. $:#... 4 Algorithm SEL epite in Figure 6), invoke with input prmeters $ root o pttern ) n root o / ), omputes the seletivity or n ritrry tree

9 ) ) : ` ` ` E & h Algorithm + + SEL, ) Input: is & noe in tree pttern, % 4 +.-NN is noe in. Output: & % 4 +.-NN. 1) i & % 4 is lrey +.- N ompute) 2) return :% & ) 1 9!%'& ),+:1 ) else i ) 4) return + & % N. 5) else i is >&:) le) 1 = 6) return 7) or eh hil + 5 /#),+.-$#1 &,+.- o 8) > )?4@CBDVF HEI O R : 6 ),+ - 1 & 9) 9!%'& ),+:1 >?A@ BDGF =>= & H I> O QR > 10) i & ) 11) > 6 ),+ - 1 >?A@ BDGF H I> O QR & & - & 12) > & 1) > )?4@CBDVF HEI O R : 6 ),+.-N 1 & & - & 14) > & % # & 15) return Figure 6: Tree Pttern Seletivity Estimtion Algorithm. pttern in 9 / :# 4 time. In the lgorithm, or noes 2 n 2 /, 0 stores the seletivity o the su-pttern!# with respet to the sutree o / roote t noe. This seletivity is estimte similr to the seletivity or pttern, exept tht we now onsier ll instntitions o 0! otine y instntiting.'. n - with element tgs), n the seletivity o eh instntition is ompute with respet to s the root inste o the root o /. For instne, suppose tht is the -noe in k in Figure 5i)), n is the hil -noe o the -noe in / in Figure 5e)). Then, the seletivity o!# k with respet to is essentilly the prout o the seletivity o pths.- n. with respet to noe, whih is :.. Thus, 0.. #. Our gol is to ompute For pir o noes n, Algorithm SEL omputes # rom vlues or the hilren o n. Clerly, i E Steps -4 o the lgorithm), every pth in!# egins with lel ierent rom n thus the seletivity o eh o the pths is. I n is le Steps 5-6), we simply instntite i.'. or ) with, giving seletivity o 0 ).. On the other hn, i is n internl noe o, in ition to instntiting! with, we lso nee to ompute, or every hil 2 o, the instntition or!2! tht hs the mximum seletivity with respet to some hil 2 o. Sine 2 2' is the seletivity o '2 with respet to 2, the prout "! # & 2 2 or the hilren '2 o gives the seletivity o 0! with respet to. Finlly, i!..,.. n e simply B, 0, in whih se the seletivity o!# with respet to is ompute s esrie in Step 11, or.. is instntite to sequene onsisting o ollowe y 2, where 2 is the hil o suh tht the seletivity o!# with respet to 2 is mximize Step 1). Oserve tht, in Steps 8 n 1, i hs no "! # "8&? evlutes to. 4.2 Tree Pttern Aggregtion Algorithm We re now rey to present our greey heuristi lgorithm or the tree pttern ggregtion prolem eine in Setion 2.2 whih is, in generl, n #%$ -hr lustering prolem [5]). As esrie erlier, to ggregte n input set o tree ptterns into spe-eiient n preise set, our lgorithm Algorithm AGGREGATE in Figure 7) itertively prunes the tree ptterns in y repling smll suset o tree ptterns with more onise upper-oun ggregte pttern, until stisies the given spe onstrint. During eh itertion, our lgorithm irst genertes smll set o potentil nite ggregte ptterns, n selets rom these the lolly) est nite pttern, i.e., the nite tht mximizes the gin in spe while minimizing the expete loss in seletivity. ) -'& 1 Algorithm AGGREGATE Input: is set o tree ptterns, & is spe onstrint. Output: A set o tree ptterns Y suh tht n Q#?!) >+,. 1) Initilize ) Y & 1 2) while Q#?-) >. 0/ o 4-5#8 ) - 1<-W ) #8 ]A` ) <! <- -N 4) :9 ]A` >= 6?7 5) 21Uh 9 6) Selet & & ), 1 suh tht is mximum 7) 6 8) return Y Figure 7: Tree Pttern Aggregtion Algorithm. Cnite Genertion. We now explin the proess or generting the nite set in Steps 5 o Algorithm AGGREGATE. To reue the size o iniviul nite ptterns o the orm or ), eh nite is prune y invoking Algorithm PRUNE etils in [5]). Given n input pttern n spe onstrint B, Algorithm PRUNE prunes to smller tree pttern suh tht & n D B. The lgorithm trets tg-noes s more seletive thn - - n.. -noes, n thereore tries to prune wy - - n.. -noes eore the tg-noes. Speiilly, the lgorithm irst prunes the - - n.'. -noes in y 1) repling eh jent pir o non-tg-noes #< with single.. -noe, i is the only hil o, n 2) eliminting sutrees tht onsist o only non-tg-noes. I the tree pttern is still not smll enough ter the pruning o the nontg-noes, we strt pruning the tg-noes. There re two wys to reue the size o tree pttern y one noe. The irst is to elete some le noe in, n the seon is to ollpse two noes n into single.. -noe, where!. n 0! >. To help selet goo le noe to elete or, pir o noes to ollpse), we mke use o the seletivity o the tg nmes. More speiilly, we use our oument tree synopsis / to estimte the totl numer o ourrenes o tg nme in the oument olletion, n hoose the tgs with higher totl requenies whih re less seletive) s nites or pruning.

10 : ` 6 ` Cnite Seletion. One the set o nite ggregte ptterns hs een generte, we nee some riterion or seleting the est nite to insert into. For this purpose, we ssoite eneit vlue with eh nite ggregte pttern 2, enote y >B 0J?, se on its mrginl gin [14] tht is, we eine >B 0J? s the rtio o the svings in spe to the loss in seletivity. More ormlly, i o using over- /!/2 K L LNM,, n = K'L LNM represent the root noes o, /, n 0J? is equl to: Q 0O Q#?!)> 6 6 ),+ K'LNLNM -' %'&&) 1 Q 0O Q#?!)> : 6 ),+ Q K'L L M -' %1&1&) 1 Note tht we ompute the seletivity loss y ompring the seletivity o the nite ggregte pttern with tht o the lest seletive pttern ontine in it. This gives goo pproximtion o the seletivity loss in ses when the ptterns ) 2 use to generte re similr n overlp in the oument tree /. The nite ggregte pttern with the highest eneit vlue is hosen to reple the ptterns ontine in it in Steps 6 7). 5 Experimentl Stuy To veriy the eetiveness o our tree pttern ggregtion lgorithms, we hve onute n extensive perormne stuy using rel-lie DTDs n lrge numers o tree ptterns. Our results inite tht our propose ggregtion tehniques hieve signiint reutions in the numer s well s totl size o tree ptterns with miniml loss in seletivity. 5.1 Experimentl Teste n Methoology Our generl methoology or evluting the eetiveness o pttern ggregtion lgorithm is s ollows. Given lrge input set o tree ptterns n spe onstrint 9, we use to ompute set o ggregte ptterns or, where & n B > G D 9 our spe onstrint is expresse in terms o numer o noes, sine ptterns n e ritrrily lrge). We mesure the loss in preision when using inste o to ilter XML ouments. Oserve tht when 9, ontins single ontiner pttern.. ). To mesure the loss in preision o the ggregte set, we use suset o representtive set o XML ouments, suh tht no oument in mthes ny tree pttern in our initil pttern set. The reson, o ourse, is tht XML ouments tht mth re lso gurntee to mth, so they re unlikely to et our preisionloss mesurements. As eomes less preise, some ouments in will e erroneously reporte s mthes. e the numer o ouments in tht mth the loss in preision o over n e estimte An ggregtion lgorithm is oviously more eetive i remins smll s <&=@ B >C G ereses. XML Douments. We use two rel-lie DTDs to generte our XML oument t set. The irst one, the Extensile Hypertext Mrkup Lnguge XHTML) DTD [7], is reormultion o HTML s n XML pplition n is rguly the oument type most wiely use over the Internet. The XHTML DTD version 1.0) ontins 5 5 elements with 5#5 ttriutes. The seon DTD, the News Inustry Text Formt NITF) DTD[8], is supporte y most o the worl s mjor news genies. The NITF DTD version 2.5) ontins elements with ttriutes. We generte our t set o XML ouments using IBM s XML Genertor tool [11]. Both the XHTML n NITF DTDs ontin reursive strutures, whih n e neste to proue XML ouments with ritrry numer o levels. We e the option o generting ouments skewe oring to Zip istriution [18], where some tg nmes pper more requently thn others, s is generlly the se with rel-lie t. For eh eh DTD n eh skew vlue, we generte two isjoint sets o XML ouments with pproximtely noes n levels on verge. The irst set orrespons to the olletion o XML ouments use to onstrut the oument tree / or seletivity estimtion the seon set is use to mesure the loss in preision o the ggregtion lgorithms. Both sets were generte with the sme prmeters, n thus n e expete to hve similr istriutions. In eh experiment, we use the omine XML ouments or oth the XHTML n NITF DTDs, i.e., we use totl o ouments or the oument tree /, n ierent) ' ouments or mesuring the loss in preision. XPth Expressions. To generte the set o tree ptterns, we implemente n XPth expression genertor tht tkes DTD s input n retes set o vli XPth expressions se on set o prmeters tht ontrol: 1) the mximum height o the tree ptterns 2) the proilities n o hving wilr - or esennt.. opertor t noe o tree pttern ) the proility o hving more thn one hil t given noe n 4) the skew B o the Zip istriution use or seleting element tg nmes. For eh DTD n eh skew vlue B ) ', we generte set o tree ptterns with n. Eh experiment ws run with tree ptterns rom oth the XHTML n NITF DTDs, i.e., ' tree ptterns whih mounte to more thn ' noes. Algorithms. We ompre two ierent ggregtion lgorithms in our experiments. The irst nive ) lgorithm, PRUNE, is se on simple noe pruning n works s ollows. At eh itertion, it selets tree pttern rom with the lrgest numer o tg-noes, ollpses multiple - - n.. -noes, n eletes prunle noe i.e., le noe or noe lote next to.. -noes) with the highest requeny i.e., lest seletive) in the oument tree /. I there is lrey tree pttern ientil to the prune pttern, the uplite is remove rom. The lgorithm itertes until the spe onstrint is stisie. The seon lgorithm, AGGR, is our greey tree pttern ggregtion lgorithm rom Figure 7) with oth nite genertion n seletion se on mximizing the eneit). Our experiments were onute on 866 MHz Intel Pentium III

11 Seletivity Loss %) Prune θ D =0) Prune θ D =1) Prune θ D =2) Aggr θ D =0) Aggr θ D =1) Aggr θ D =2) Seletivity Loss %) Prune θ S =0) Prune θ S =1) Prune θ S =2) Aggr θ S =0) Aggr θ S =1) Aggr θ S =2) Seletivity Loss %) Prune θ D =θ S =0) Prune θ D =θ S =1) Prune θ D =θ S =2) Aggr θ D =θ S =0) Aggr θ D =θ S =1) Aggr θ D =θ S =2) Numer o Noes x1,000) ) Vrying ) ) Numer o noes x1,000) ) Vrying ) ) Numer o noes x1,000) ) Vrying ) n Figure 8: Evlution o the Aggregtion Algorithms. mhine with MB o min memory running Linux. Both lgorithms omplete the ggregtion o ' eliminte erly. tree ptterns in pproximtely minutes. 5.2 Experimentl Results We irst ompre the perormne o the two ggregtion lgorithms y vrying the skew or element tgs in the XML ouments n in the XPth expressions. We rn the experiments with no skew, with skewe XML ouments, with skewe XPth expressions, n with skew in oth the XML ouments n XPth expressions. In the lst se, we skew the istriution or element nmes in the opposite iretion pplying the sme skew to oth the XML ouments n XPth expressions woul yiel similr results s with no skew). The experimentl results re shown in Figures 8), 8), n 8), where the spe onstrint, expresse in terms o the numer o noes, is vrie long the -xis, n the -xis inites the oserve loss in seletivity or given spe onstrint, i.e., the perentge o XML ouments tht re erroneously reporte s mthes. We lso mesure the eneits o ggregtion in terms o iltering perormne, using the XTrie mthing lgorithm esrie in [6]. Sine the ost o iltering in XTrie grows linerly with the numer o XPth expressions, we expet to oserve signiint improvement in iltering spee s the rinlity o ereses. Non-skewe worklo. When neither the XML t nor the tree ptterns ontin skew i.e., B ), the AGGR lgorithm n ggregte tree ptterns up to o their originl size with only loss in preision the results or non-skewe t re reporte in ll grphs o Figure 8). In ontrst, the preision o PRUNE lgorithm strts to egre muh sooner, n the loss in preision rehes lmost t o the initil spe. The etter perormne o AGGR n e ttriute to three min tors: 1) the upper oun omputtion genertes goo nites with ew noes n little loss in preision, 2) the seletivity-se heuristis help to etet n isr nites tht orrespon to ptterns with low seletivity i.e., requently ourring or given DTD), n ) the overing omputtion enles reunnt tree ptterns to e Skewe XML ouments. Rel-worl XML ouments re generlly not uniormly istriute mong the vli XML t or given DTD. When XML ouments re skewe Figure 8)), we oserve tht the eetiveness o the AGGR lgorithm inreses. The reson or this is tht, s t eomes more skewe, the XML ouments ten to orm lusters with ouments within luster eing more similr thn those in ierent lusters this, in turn, improves the ury o seletivity estimtion. The PRUNE lgorithm lso eneits rom the skew lthough to lesser extent) euse o its requeny-se pruning heuristi. Skewe tree ptterns. We lso oserve signiint improvement in our ggregtion lgorithm when the element nmes o tree ptterns re skewe Figure 8)). Inee, the skew inues lustering o ptterns suh tht similr tree ptterns re groupe into the sme luster, whih onsequently inreses the proportion o ptterns tht evelop ontinment reltionships. This permits the ggregtion lgorithm to reue the size o with miniml loss o seletivity, y omputing tighter upper oun ptterns n isring overe ptterns. Skewe worklo. The two ggregtion lgorithms perorm est when oth the XML t n the tree ptterns re skewe in ierent iretions Figure 8)). With high skew vlues, there is little overlp etween the element nmes o the XML ouments n the tree ptterns, n AGGR remins highly seletive with only ew hunres noes. The PRUNE lgorithm lso exhiits signiint improvements n mintins seletivity even ter the originl numer o noes re reue to less thn thir. Filtering spee. As mentione previously, the ost o mthing tree ptterns ginst inoming XML ouments is proportionl to the numer o tree ptterns. Sine AGGR genertes nites y omputing upper ouns, the nites over more ptterns, n s result, the numer o ptterns in shrinks ster with AGGR. Figure 9 shows tht the verge iltering time per oument ereses ster s spe is inrese) or AGGR thn or the PRUNE lgorithm. Our ggregtion lgorithm is thereore more ee-

12 l Filtering Time ms) Prune Aggr Numer o noes x1,000) Figure 9: Filtering spee. tive oth in terms o seletivity s well s iltering spee. 6 Relte Work To the est o our knowlege, our tree pttern ggregtion prolem is novel prolem tht hs not een stuie in erlier work. In ontrst to the lt ptterns previously stuie in the ontext o ggregting ttriute-preite-se susriptions [12], our pper ouses on hierrhil ptterns, whih re more omplex s tree ptterns onsist o oth t ontents n struture) n require more sophistite ggregtion tehniques. A relte re is the work on query merging to reue t issemintion osts o query susriptions in multist environment [9]. The motivtion or query merging is to merge multiple similr queries into single, more generl query so s to reue the worklo o the server n possily the mount o tri etween the server n its lients. However, the prolem omin onsiere in [9] ouses on geogrphil queries represente s retngles) urthermore, the issue o spe onstrint is not relevnt there. Some orms o tree ptterns hve een stuie s queries or XML t [, 17]. In prtiulr, minimiztion lgorithms or these ptterns hve een evelope in orer to optimize pttern queries. The tree ptterns in [] ier rom ours in two spets. On the one hn, the tree ptterns o [] o not llow - -noes wilrs) whih, s mentione in Setion, give rise to sutle prolems in the presene o.. -noes esennts) when ontinment o tree ptterns is onsiere. On the other hn, they support seletion o set o oument noes s the result o pttern query, whih we o not onsier sine wht mtters or our susription ggregtion ontext is whether or not oument mthes susription the tul set o oument noes tht mthes susription is not relevnt. Beuse o these ierenes, the minimiztion lgorithm o [] hs n 9 B omplexity in ontrst to our 9 B omplexity. Similrly, the work in [17] stuies ierent lss o tree ptterns n their minimiztion lgorithm is only known to e in polynomil time. 7 Conlusions We hve provie the irst systemti stuy o tree pttern ggregtion, n importnt prolem in uiling nextgenertion, slle XML issemintion systems. The min hllenge is to ggregte n input set o tree ptterns into smller set suh tht: 1) given spe onstrint on the totl size o the ptterns is met, n 2) the loss in preision ue to ggregtion) is minimize. We hve propose n eiient ggregtion lgorithm tht mkes eetive use o oument-istriution sttistis in orer to ompute preise set o ggregte tree ptterns within the llotte spe uget. Further, some o our lgorithmi results re o interest in their own right, n n prove useul in other omins, suh s XML query optimiztion. Extensive results rom prototype implementtion hve veriie the eetiveness o our pproh. Reerenes [1] A. Aoulng, A. Almeleen, n J. Nughton. Estimting the seletivity o xml pth expressions or internet sle pplitions. In Pro. 27th Intl. Con. on Very Lrge Dtses VLDB 2001), Septemer [2] M. Altinel n M.J. Frnklin. Eiient iltering o xml ouments or seletive issemintion o inormtion. In Pro. 26th Intl. Con. on Very Lrge Dtses VLDB 2000), pges 5 64, Septemer [] S. Amer-Yhi, S. Cho, L.V.S. Lkshmnn, n D. Srivstv. Minimiztion o Tree Pttern Queries. In Pro. o SIG- MOD, pges , Snt Brr, Cliorni, My [4] A. Crznig n A.L. Wol. Content-se Networking: A New Communition Inrstruture. NSF Workshop on Inrstruture or Moile n Wireless Systems, Otoer [5] C.-Y. Chn, W. Fn, P. Feler, M. Grolkis, n R. Rstogi. Tree Pttern Aggregtion or Slle XML Dt Dissemintion. Bell Ls Teh. Memornum, Ferury [6] C.-Y. Chn, P. Feler, M. Grolkis, n R. Rstogi. Eiient Filtering o XML Douments with XPth Expressions. In Pro. o the 18th Intl. Con. on Dt Engineering, Sn Jose, Cliorni, Ferury [7] Worl Wie We Consortium. The SGML/XML We Pge. Jnury [8] R. Cover. The SGML/XML We Pge. open.org/over/sgml-xml.html, Deemer [9] A. Crespo, O. Buyukkokten, n H. Gri-Molin. Query Merging: Improving Query Susription Proessing in Multist Environment. IEEE Trns. on Knowlege n Dt Engineering. To pper. [10] A. Deutsh n V. Tnnen. Continment o Regulr Pth Expressions uner Integrity Constrints. In Pro. o Intl. Workshop on Knowlege Representtion meets Dtses KRDB), [11] A.L. Diz n D. Lovell. XML Genertor. lphworks.im.om/teh/xmlgenertor, Septemer [12] L. Opyrhl et l. Exploiting IP Multist in Content-se Pulish-Susrie Systems. In Pro. o Intl. Con. on Distriute Systems Pltorms Milewre), [1] R. Fgin. Comining Fuzzy Inormtion rom Multiple Systems. In Pro. o the 15th ACM Symp. on Priniples o Dtse Systems, Montrel, Quee, June [14] B. Fox. Disrete Optimiztion Vi Mrginl Anlysis. Mngement Siene, 1): , Novemer [15] WC. XML Pth Lnguge XPth) [16] WC. Extensile Mrkup Lnguge XML) [17] P. T. Woo. Minimizing Simple XPth Expressions. In Pro. o Intl. Workshop on the We n Dtses WeDB), Snt Brr, Cliorni, My [18] G.K. Zip. Humn Behviour n Priniple o Lest Eort. Aison-Wesley, Cmrige, Msshusetts, 1949.