Nearest Keyword Search in XML Documents

Transcription

1 Neares Keyword Search in XML Documens Yufei Tao Savros Papadopoulos Cheng Sheng Kosas Sefanidis Deparmen of Compuer Science and Engineering Chinese Universiy of Hong Kong New Terriories, Hong Kong {aoyf, savros, csheng, ABSTRACT This paper sudies he neares keyword (NK) problem on XML documens. In general, he daase is a ree where each node is associaed wih one or more keywords. Given a node q and a keyword w, an NK query reurns he node ha is neares o q among all he nodes associaed wih w. NK search is no only useful as a sandalone operaor bu also as a building brick for imporan asks such as XPah query evaluaion and keyword search. We presen an indexing scheme ha answers NK queries efficienly, in erms of boh pracical and wors-case performance. The query cos is provably logarihmic o he number of nodes carrying he query keyword. The proposed scheme occupies space linear o he daase size, and can be consruced by a fas algorihm. Exensive experimenaion confirms our heoreical findings, and demonsraes he effeciveness of NK rerieval as a primiive operaor in XML daabases. Caegories and Subjec Descripors H3. [Conen analysis and indexing]: Indexing mehods General Terms Theory Keywords Neares keyword, XPah, keyword search, group seiner ree. INTRODUCTION We consider he problem of neares keyword (NK) search on XML documens. The daase is a ree T wih undireced edges. Each node is associaed wih one or more keywords. Define he disance beween wo nodes as he number of edges in he (unique) pah linking hem. Given a node q in T and a keyword w, an NK query finds he neares w-neighbor of q, namely, he node having he minimum disance o q among all he nodes associaed wih w. To illusrae, Figure shows par of an XML documen, where all nodes have been encoded wih he Dewey code [34] for easy reference. An elemen node has is ype displayed in brackes, while he oher nodes are value nodes. Given a node q = 0000 and a Permission o make digial or hard copies of all or par of his work for personal or classroom use is graned wihou fee provided ha copies are no made or disribued for profi or commercial advanage and ha copies bear his noice and he full ciaion on he firs page. To copy oherwise, o republish, o pos on servers or o redisribue o liss, requires prior specific permission and/or a fee. SIGMOD, June 6, 0, Ahens, Greece. Copyrigh 0 ACM //06...$0.00. keyword w = guard, an NK query reurns node 000, whose disance oq is 4 (edges). I is he neares guard-neighbor of q.. Moivaion NK queries can serve as he building brick o ackle some imporan problems in XML daabases, as elaboraed below. For convenience, we assign each value node o a ype, whose name concaenaes is paren s ype and he sring Val (e.g., node 000 has ype nameval, and so does node 000). Every node carries is ype as a keyword. In addiion, each value node has is value as anoher keyword. For example, node 000 has wo keywords: is ype nameval, and is value Lakers. XPah query evaluaion. NK search gives a new mehodology for efficienly solving a class of XPah queries. An example is: Q: Find he names of all s ha originaed from Maryland, bu are in a eam of he wes division. The XPah saemen ofqcan be expressed as a wig paern [4, 5] shown in Figure a, where a single-lined (double-lined) edge represens paren-child (ancesor-descenden) relaionship. The goal is o find occurrences of he paern in he daa ree, and for each occurrence, oupu he value a he posiion of Val (signified as underlined). Figure b demonsraes such an occurrence in Figure, from which he oupu is he value Blake of node There are wo ineresing facs abou he paern Q in Figure a, wih respec o he daa of Figure : Le q be he node in an occurrence corresponding o he node Maryland of Q. The ype of q is fromval. The neares wes-neighbor of q mus have disance exacly 6 o q. For example, q is node 000 in Figure b, and is neares wes-neighbor is node 00. Le q be any fromval node ha carries he word Maryland, bu is no in any occurrence, i.e., he eam of q is in he eas division. The neares wes-neighbor of q mus have disance greaer han 6 o q, noicing ha he neighbor mus come from a eam differen from ha of q. For example, le q be node 000 in Figure. Is neares wesneighbor is node 00, which has disance 8 oq. The above facs enable us o processqvia NK search as follows. We enumerae all he fromval nodes ha conain Maryland. For each such node q, find is neares wes-neighbor. If he neighbor rerieved has a disance greaer han 6 o q, i is ignored. Oherwise, we have found an occurrence, from which he Val should be oupu. The Val node can be found wih anoher NK query, which obains he neares Val-neighbor of q. Group seiner ree rerieval. Keyword search has emerged as a new paradigm of inquiring XML daabases. I enables a user o

2 <league> 0 <name> <division> 00 0 <eam> 0 <s> 0 <name> <division> 00 0 <eam> 0 <s> 0 <eam> Lakers 000 wes 00 <> 00 <> <posiion> <from> <> 0 <> <posiion> <from> <> Nes 000 eas 00 <> 00 <> <posiion> <from> <> Blake 0000 guard 000 Maryland Walon guard 00 Michigan 00 Figure : A sample of he NBA daase Smih 0000 forward 000 Maryland 000 <eam> 0 0 <division> <> wes <> <from> Val Maryland (a) Twig paern for Q (b) An occurrence in Figure Figure : An XPah query example specify only a few keywords as he query, insead of complying wih a rigorous synax. Is advanage is ha he user does no need o learn any query language like XPah, and neiher does s/he need o be aware of he daa schema. The disadvanage, however, is ha wha should be he query resul becomes heavily dependen on he applicaion backdrop. This has riggered he proposiions of a variey of resul semanics, among which an inuiive one is o reurn he group seiner ree (GST) [, 0, 3, 5]. More specifically, given a se of query keywords {w,...,w l }, a GST is a ree ha (i) conains all he query keywords in he exs of is nodes, and (ii) has he fewes edges among such rees. For example, Figure 3 presens he GST for{lakers, Blake, guard} on he daa of Figure. GST compuaion is known o be NP-hard (even if he daase is a ree) [7]. Forunaely, as discussed laer, NK queries provide an elegan way o exrac a good approximae soluion, namely, a ree ha saisfies requiremen (i), and has more edges han he GST by only a small facor.. Conribuions This paper presens he firs sudy on he NK problem. We propose an indexing scheme ha can answer any NK query in O(logN w) ime, wheren w is he number of nodes associaed wih he query keyword w. The scheme consumes space linear o he size of he daase. Somewha surprising is he fac ha, despie he complicaion of he underlying heory, our access mehod can be implemened as merely a number of binary rees. All he resuls also hold in disk-oriened environmens, where each binary ree is simply replaced wih a B-ree. Accordingly, he query cos is O(log B N w) I/Os, where B is he size of a disk block. The proposed index also leads o rigorous resuls on he usefulness of he NK operaor. Specifically: We heoreically esablish he fac ha a large class of XPah queries can be reduced o NK search (in a way similar o how he query of Figure a was answered earlier). Our algorihm for processing his query class enjoys a wors-case ime complexiy ha is irrelevan o he number of elemens whose ypes appear as an inernal node of Q. No previous soluion is known o have his feaure (as surveyed in Secion 5). We give a fas soluion o finding an approximae GST wih Figure 3: The group seiner ree of {Lakers, Blake, guard} in Figure an aracive qualiy guaranee (which is acually opimal if he query has only wo keywords). We achieve he running ime of O(N minllogn max), plus he cos of oupuing he resuling ree, where l is he number of keywords in he query,n min is he number of nodes carrying he rares query keyword (i.e., he one appearing he leas imes in he XML documen), andn max conversely is he number of nodes carrying he mos frequen query keyword. Besides confirming our heoreical findings, our experimenaion also demonsraes he effeciveness of he NK operaor on real XML documens. In paricular, we show ha XPah queries like he one in Figure a can be processed via NK search wih performance comparable o or beer han ha of he exising approaches. Furhermore, for XML keyword search, our NK-based algorihm discovers high-qualiy approximae GSTs in real ime. Roadmap. The res of he paper is organized as follows. Secion clarifies he problem definiion and several echnical preliminaries. Secion 3 elaboraes on he proposed soluions for NK search. Secion 4 discusses he applicaions of NK queries in XPah evaluaion and keyword search. Secion 5 reviews he previous work relaed o ours. Secion 6 conains exensive experimenaion o evaluae he effeciveness and efficiency of our echniques. Finally, Secion 7 concludes he paper wih a summary of our findings.. PRELIMINARIES For each node u in he daa ree T, we use W(u) o represen he se of keywords associaed wih u. For simpliciy, assume ha W(u) has a leas one keyword. Define he lengh of a pah in T as he number of edges i conains. Denoe by u,v he disance beween wo nodes u, v, namely, he lengh of he pah connecing u and v. Le U(w) be he se of nodes in T ha include word w (hence, N w = U(w) ). Given a node q and a keyword w, he resul of an NK query is a node u U(w) such ha u,q v,q v U(w). We denoe u, he neares w-neighbor of q, as NN(q,w).

3 O(N). Assume ha we wan o perform a level-on-pah operaion o find he level-l node from u o v (which is a descenden of u). We find in O(logN) ime he predecessor of rank(v), among all he ranks indexed in he level-l binary ree. The node whose rank equals ha predecessor is exacly wha we are looking for Figure 4: A running example Figure 4 shows an example T, where each node is labeled an ineger. Someimes we may refer o a node by is label direcly, when he meaning is clear. For insance, as node carries a single keyword, we wrie W() = {}. Similarly, as also appears in nodes 5, 9 and 3,U() = {,5,9,3}. The keywords of he oher nodes are omied for clariy. Given q = node 7 and w =, an NK query reurns node, namely, NN(7, ) =. Le N be he number of nodes in T, and K he oal number of keywords in all he nodes (couning a word wice if i appears in wo nodes), i.e., K = u W(u). Noe ha T requires Ω(K) space o sore. In oher words, linear cos should be inerpreed as O(K), insead of O(N). We label he levels of T in a op-down manner, seing he roo a level 0. Denoe by level(u) he level of a node u, which is also he number of edges on he pah from he roo ou. In Figure 4, all he leaf nodes are a level 4. Also, we use sub(u) o represen he subree of u. In he sequel, we review several basic resuls useful in our echnical discussion. Inerval encoding. For each node u of T, define is rank, denoed asrank(u), o be he sequence number ofuin he pre-order raversal of T. We associae u wih an inerval R(u) = [x,y], where x is he rank ofu, andyis he larges rank of he nodes insub(u). In Figure 4, he label of each node indicaes is rank direcly. As an example, he inerval R(0) associaed wih node 0 is [0,6]. The inervals defined his way have several properies commonly uilized in managing XML daa: For any wo nodesuandv,r(u) conainsr(v) if and only if u is an ancesor of v. In oher words, he ancesor-descenden relaionship of u and v can be verified in consan ime. The inervals of he nodes a he same level of T mus be disjoin. In Figure 4, for insance, he nodes a level have disjoin inervals R(3) = [3, 9], R(0) = [0, 6], R(8) = [8,4], andr(5) = [5,3]. The above properies allow us o solve he so-called level-onpah queries efficienly. Le u be an ancesor of v; given a level l [level(u), level(v)], a level-on-pah query finds he level-l node on he pah from u ov. For example, if u (v) is node (5), a level-on-pah query wih l = rerieves node 0. LEMMA. T can be pre-processed ino a srucure ha occupies O(N) space, such ha any level-on-pah query can be answered in O(log N) ime. The srucure can be buil in O(N logn) ime. PROOF. We manage he nodes of T of each level separaely. Specifically, creae a binary ree o index he ranks of he nodes a he same level (i.e., here are as many rees as he number of levels in T ). As each node appears in only one ree, he overall space is Subree NK search. NK search is easy if aenion is resriced o he subree of he query node. Formally, given a node q and a keyword w, a subree-nk query finds he node u wih he smalles disance o q, among all nodes in sub(q) ha are associaed wih w. We refer ouas he subree neares w-neighbor of q. For insance, le q be node 7 in Figure 4; he subree-nk query wih w = reurns node 3. Noe ha he (global) neares -neighbor of node 7 is in fac node. LEMMA. T can be pre-processed ino a srucure ha occupies O(K) space, such ha any subree-nk query can be answered ino(logn w) ime. The srucure can be buil ino(k logk) ime. PROOF. Le us firs review a relaed resul. Le S be a se of numbers in he real domain R. Each number x S is associaed wih a weigh in R. Given an inerval I, a range-min query finds he number ha has he minimum weigh among all he numbers in S I. We can indexs wih an SB-ree [36] ha useso( S ) space, and solve any range-min query ino(log S ) ime. The ree can be buil in O( S log S ) ime. We can conver subree-nk search o he range-min problem. Lewbe he keyword of concern. Consruc S o include he ranks of he nodes in U(w). Each rank is associaed wih a weigh ha equals he level of he corresponding node. Subree-NK search wih node q is equivalen o a range-min query on S wih inerval R(q). We sele he problem wih an SB-ree in O(logN w) query ime. The ree occupies O(N w) space and can be buil in O(N w logn w) ime. The SB-rees of all keywords require O( wnw) = O(K) space in oal. The overall consrucion ime is O( w (NwlogNw)) = O(KlogK). Boh Lemmas and will be needed o analyze he consrucion cos of he proposed srucure. Lowes common ancesor (LCA). We use lca(u, v) o denoe he LCA of nodes u,v in T (e.g., lca(0,6) is node 7 in Figure 4). In general, he disance of wo nodes can be calculaed in consan ime, once heir LCA has been idenified, as can be seen from he following equaion: u,v = (level(u) level(z))+(level(v) level(z)) where z = lca(u,v). LCA compuaion has been horoughly sudied. Harel and Tarjan [8] were he firs o observe ha he problem can be seled opimally in consan ime using linear space. Their srucure, however, is raher heoreical and difficul o implemen. To remedy he drawback, several (much) simpler srucures [,, 3] have been developed, keeping he same space and query performance. As a corollary, we can obain u,v of anyu,v in consan ime. 3. NEAREST KEYWORD SEARCH We pre-process he daa ree T by building a separae srucure for each disinc keyword w ha appears in T. This is reminiscen of he invered index, which also has an invered lis dedicaed o each w. Insead of a simple lis, however, our srucure for w is a binary ree consruced in a more sophisicaed manner.

4 3. Overview We concenrae on NK queries wih a specific keyword w, as he srucure is idenical for all keywords. The erm neares w- neighbor will be abbreviaed as neares neighbor (NN), when no ambiguiy arises. Accordingly, we simplify noaion NN(u, w) o NN(u). A sraighforward soluion o answering an NK query is o perform a breah firs raversal (BFT) saring from q. Namely, he BFT explores he nodes of T in ascending order of heir disances o q, and sops as soon as i encouners a node associaed wih w. This approach is efficien only if he NN ofq is close, and may end up visiing a large number of nodes oherwise. Alernaively, we can pre-compue he NN of every node in T. Each query can be answered in consan ime, because we can simply reurn he (pre-compued) NN of he query node q. This approach, however, has he severe drawback ha, he pre-compuaion incurs Ω(N) space for every keyword appearing int. The number of disinc keywords can be easily Ω(N). In his case, he space complexiy of he above approach isω(n ), which is prohibiively large in pracice. The chief observaion owards reducing he space is ha, many nodes of T have he same NN, hus raising he hope ha we could capure hem collecively wih much less informaion. Recall ha each node can be uniquely idenified by is rank, while he ranks of all nodes come from he rank domaind = [,N] (e.g.,d = [,3] in Figure 4). We can always pariion D ino a se I of disjoin inervals such ha, for each inerval I I, he nodes wih ranks in I have he same NN, which can be associaed wihi. Given an NK query wih node q, we can solve i by idenifying he (only) inerval I ha covers rank(q), and reurning he NN associaed wih I. This can be easily achieved by indexingi wih a binary ree, which consumes O( I ) space and has query cos O(log I ). Figure 5 illusraes he conens of a possible I for he daa of Figure 4 when he keyword w of concern is. ranks node is he NN for any node in rank inerval [,3] Figure 5: A ree Voronoi pariion We refer o I as a ree Voronoi pariion (TVP) of w. An immediae issue is wheher a smalli always exiss. Forunaely, we will show in Secion 3. ha here is definiely an I wih size O(N w), where N w is he size of U(w) (i.e., he number of nodes in T carrying w). Furhermore, he size of O(N w) is asympoically igh because I needs o be a leas N w every node in U(w) apparenly finds iself as he NN. Anoher imporan issue is how o compue I efficienly. Naively, one could firs compue he NN of every node in T, and hen go over he nodes in ascending order of heir ranks, merging consecuive nodes ino an inerval if heir NNs are he same. This approach, however, enails Ω(N) ime, which would render he oal pre-compuaion cos (for all keywords) prohibiively expensive in pracice. In Secion 3.3, we will give a significanly faser algorihm o produce I ino(n w logn w) ime. 3. TVP characerisics This subsecion will esablish our firs main resul: THEOREM (TVP THEOREM). For any keyword w appear (a) CT() (b) ECT() 8 3 Figure 6: Compac and exended compac rees ing int, here is a ree Voronoi pariioni wih size less han8n w, where N w is he number of nodes int associaed wihw. Le us sar he proof by inroducing he compac ree of w, denoed as CT(w). Firs, all he nodes of U(w) are in CT(w), and ermed he daa nodes. Second, a non-daa node u belongs o T, if and only if here are a leas wo child nodes of u whose subrees conain a daa node. We call u a branching node. Consider w = in Figure 4. There are four daa nodes, 5, 9, 3, and wo branching nodes, 3. Node 7, for example, is no a branching node because only one of is child nodes (i.e., node 8) has a daa node in is subree. Le S be he se of all daa and branching nodes. We form CT(w) by drawing an edge from each node u S o is lowes ancesor in S. Figure 6a shows he CT() for he daa of Figure 4. Node 5, for insance, is conneced o node 3, because among all he daa and branching nodes, node 3 is he lowes ancesor of node 5. LEMMA 3. CT(w) has a mosn w nodes. PROOF. Each branching node mus have a leas wo child nodes inct(w). AsCT(w) has a mosn w leaf nodes, he oal number of branching nodes canno be more hann w. Consider any edge(u,v) inct(w). Le us walk, in he daa ree T, along he pah fromuov. As we go, monior henn(z) of he node z being visied, and coun how many changes in NN(z) here are in oal. Call each of hose changes an NN-change on(u,v). As an example, consider edge (, 3) in he CT() of Figure 6a. Now we walk from node o node 3 in he T of Figure 4. Along he pah here is only a single NN-change, which happens as we move from node 7 o node 8 (i.e., NN(7) =, bu NN(8) = 3). The nex lemma gives an imporan fac: LEMMA 4. There can be a mos one NN-change on each edge of CT(w). PROOF. Le(u,v) be an edge ofct(w). The removal of(u,v) cus CT(w) ino wo conneced componens. Le C u (C v) be he componen including u (v). Denoe by P he pah from u o v in T. Suppose ha u (v ) is he daa node in C u (C v) closes o u (v). We claim ha, for any node z onp,nn(z) mus be eiheru or v. In fac, for any node u C u, i holds ha z,u = z,u + u,u z,u + u,u = z,u. Similarly, for any node v C v, we have z,v z,v. Therefore, excepu andv, no oher daa node inc u C v can be he NN of z. Hence, if here were a leas wo NN-changes on (u,v), here would have o be hree nodes z,z,z 3 on P such ha z was on he pah from z o z 3, bu NN(z ) = NN(z 3) NN(z ). I is rivial o show ha such a scenario canno happen. Nex, CT(w) is augmened wih he nodes where NN-changes occur. Consider any edge (u,v) of CT(w) on which here is an

5 NN-change. Le z be he firs node on he u-o-v pah in T such ha NN(z) NN(u). If z is differen from v, we add i o CT(w), breaking (u,v) ino wo edges (u,z) and (z,v). Denoe by ECT(w) he resuling ree, afer applying such ransformaion on all edges of CT(w) wih NN-changes. In case roo(t ) is no already in ECT(w), we add i as he paren of he curren roo of ECT(w). The final ECT(w) is called he exended compac ree of w. Le us illusrae he ransformaion wih edge (,3) in he CT() of Figure 6a (i.e., u =,v = 3). As menioned earlier, on he pah from node o node 3 in Figure 4, here is an NNchange as we cross from node 7 o node 8. Hence, z = 8, and accordingly, (,3) is broken ino wo edges (,8) and (8,3) in he exended compac ree ECT(), as shown in Figure 6b. We are ready o generae a ree Voronoi pariion I of w. I suffices o invoke he following algorihm for every node u of ECT(w) in urn (ordering does no maer): algorihm voronoiinv(u) /* u is a node in ECT(w) */. S = {R(u)}. for each child node v of u in ECT(w) do 3. I he (only) inerval in S covering R(v) 4. break I ino inervals I,R(v),I /* I (I ) is he par of I o he lef (righ) of R(v) */ 5. removei froms, and add I,I 6. add o I he inervals in S, afer associaing hem wih NN(u) For example, le u be node in he ECT() of Figure 6b. A Line, voronoiinv ses S = {R()} = {[,3]}. Since node has wo child nodes in ECT(), he for-loop in Lines - 5 is execued wice. The firs ime rims R() = [,6] away from [,3], afer which S = {[,],[7,3]}. The second execuion cus R(8) = [8,4] ou of [7,3], leaving S = {[,],[7,7],[5,3]}. Line 6 adds all hree inervals of S o I, afer associaing hem wih NN() =, indicaing ha node is he NN of any node (whose rank falls) in hose inervals. LEMMA 5. Applying voronoiinv o all nodes of ECT(w) creaes a ree Voronoi pariioni wih less han8n w inervals. PROOF. We sar by proving ha he inervals of I are disjoin, and heir union covers he rank domain D. Observe ha he inervals insered in I a Line 6 are disjoin wih he R(v) of any child node v of u, where u is he inpu o he curren execuion of voronoiinv. None of hose inervals can overlap wih he inervals added o I by he execuion of voronoiinv invoked wih v (which only adds inervals wihin R(v)). On he oher hand, every value x D is covered by an inerval in he final I. Such an inerval is insered in I by running voronoiinv wih he lowes node u in ECT(w) whose R(u) covers x. We proceed o show ha, for each inerval I I, he NN associaed wihi is indeed he NN of all nodes ini. Consider any node u inect(w). Denoe is child nodes inect(w) asv,...,v f for some f 0. For each j f, cuing R(v j) ou of R(u) a Line 4 effecively removessub(v j) fromsub(u) (recall hasub(.) represens he subree of a node). Lesub (u) be he se of nodes in sub(u), bu no in he sub(v j) of anyj. I suffices o prove ha all nodes insub (u) have he same NN asu. For each edge(u,v j) of ECT(w), define P(u,v j) as he pah int from node u o v j, bu excluding nodev j. Each nodez sub (u) is eiher (i) onp(u,v j) for some j, or (ii) has an ancesor z in T ha is on P(u,v j) for some j. In he former case, NN(z) mus be NN(u) by he way ECT(w) is consruced. In he laer case, NN(z) = NN(z ), whilez is a node of case (i), implyingnn(z) = NN(u) as well. I remains o bound he size of I. By Lemma 3, CT(w) has a mos N w edges. As each of hem may generae wo edges in ECT(w), he number of edges in ECT(w) is a mos 4N w 4 + = 4N w 3, including he (poenial) exra edge due o roo(t ). In voronoiinv, each edge breaks D ino a mos more inervals. Therefore, he final I is bounded above by + (4N w 3) = 8N w 5. The proof of Theorem is hus compleed. 3.3 Finding he minimum TVP This subsecion complees our discussion of NK search by elaboraing an algorihm for compuing a TVP I of a keyword w wih he minimum size. In fac, he main idea of he algorihm has been menioned in Secion 3., as summarized below: algorihm compuetvp(w). build CT(w). build ECT(w) from CT(w) 3. I = 4. for each node u in ECT(w) do 5. voronoiinv(u) /* a he end of he for-loop, I < 8N w */ 6. merge consecuive inervals in I ha are associaed wih he same NN 7. reurn I Nex, we clarify he deails of Lines and, because voronoiinv has been presened in he previous subsecion. Consrucion of CT(w). We consruc CT(w) in wo seps: firs collec he se S of nodes in CT(w), and hen connec hem properly o mee he definiion of CT(w). The firs sep applies he algorihm below o compue S: algorihm collecctnodes(u(w)). S = U(w). sor he nodes of U(w) in ascending order of ranks 3. for each pair of consecuive nodes u, v do 4. S = S lca(u,v) We claim ha he finals includes all he branching nodes. Lez be any branching node. By definiion, i has a leas wo child nodes ha have a daa node in heir subrees, respecively. Le u,u be he wo lef-mos ones of hose child nodes, wih u being he lef-mos one. Denoe by v (v ) he daa node wih he larges (smalles) rank in he subree of u (u ). By he propery of preorder ranking,v andv consiue a pair of consecuive daa nodes in he rank domain. Hence, z will be discovered as lca(v,v ). Recall ha each node inct(w) should be conneced o is lowes ancesor (if any) among all he nodes inct(w). We achieve he purpose using a sackj. Specifically, we process he nodes ofs in ascending order of heir ranks, and mainain CT(w) for he nodes already seen. A each momen, J keeps he righ-mos roo-o-leaf pah of he curren CT(w). Nodes of he pah are pushed in J in he same order hey are scanned, i.e., wih he leaf (roo) a he op (boom). algorihm compuectedges(w). sor he nodes of S in ascending order of ranks. J = 3. while S do 4. u he firs node of S; remove u from S

6 5. keep popping he op node v of J as long as R(v) is disjoin wih R(u) 6. if J is no empy hen 7. add an edge beween u and he op node of J 8. push u in J Le us illusrae he algorihm using he se of nodes in hect() of Figure 6a. Here, S = {,,3,5,9,3}. Node is he firs scanned, and direcly pushed oj. For node, Line 5 has no effec, while Lines 6-7 add an edge beween nodes and. J = {,} a his ime (wih node a he op). Similarly, he scanning of nodes 3 and 5 creae edges (,3) and(3,5) respecively, afer whichj = {5,3,,}. Nex, he algorihm comes o node 9. Line 5 pops node 5 ou of J because R(5) is found o be disjoin wih R(7). Line 6 hen spawns anoher edge (3, 9) in CT(). Finally, he handling of node 3 pops nodes 9, 3,, and adds one more edge (,3). The CT() now becomes final as S has been exhaused. Consrucion ofect(w). Le(u,v) be an edge inct(w), wih u being an ancesor of v. Lemma 4 saes ha here can be a mos one NN-change on (u,v). In case no NN-change exiss, (u,v) is kep direcly in ECT(w). Oherwise, we should creae wo edges (u,z), (z,v) in ECT(w), where z is he firs node (on he u-o-v pah int ) such ha NN(u) NN(z). Our algorihm for building ECT(w), named compueect, processes he edges (u, v) of CT(w) in ascending order of level(u). I keeps he invarian ha, a he ime (u,v) is o be processed, we have already deermined NN(u). I is easy o see ha NN(v) can only be eiher NN(u) or NN sub (v), where NN sub (v) is he subree neares w-neighbor of u (see Secion ). Anoher key o he algorihm is ha, if NN(v) urns ou o be differen from NN(v), he node z spliing (u,v) can be idenified by a level-on-pah query. This is because he level l of z can be calculaed as: δ +level(u)+level(v) l = () where δ = v,nn(v) u,nn(u). algorihm compueect(ct(w)). sor he edges (u, v) of CT(w) in ascending order of level(e); E he sored lis. find he NN of he roo of CT(w) wih a subree NK query 3. whilee do 4. (u,v) he firs edge of E; remove i frome 5. NN sub (v) he subree neares w-neighbor ofv 6. NN(v) whichever of NN(u) and NN sub (v) ha is closer o v 7. if NN(u) = NN(v) hen 8. creae edge (u,v) in ECT(w) 9. else 0. z he resul of a level-on-pah query using (u,v) and l (Equaion ). creae edges (u,z) and (z,v) in ECT(w) We demonsrae he algorihm by explaining how o derive he ECT() in Figure 6b from he CT() in Figure 6a. Firs, Line arranges he edges of CT() in he order of E = {(,),(,3),(,3),(3,5),(3,9)}. Line obains NN() =. The subsequen execuion examines each edge of E in urn. The firs one, (, ), is easy o handle because NN() = NN() =. Hence, (, ) is included in ECT() direcly. Consider he second edge (,3) of E. As NN(3) = 3 is differen from NN(), Line 0 calculaes l = ( +0+4)/ =, and issues a levelon-pah query o find he level- node z on he pah from node o node 3 in he daa ree of Figure 4. The z rerieved is node 8. Line hen adds edges (,8) and (8,3) ino ECT(). The res of he algorihm proceeds in he same manner. Analysis. Line of compuetvp invokes collecctnodes and compuectedges, boh of which erminae in O(N w logn w) ime. The same complexiy applies o a Line, which execues compueect. Lines 4-5 essenially spend consan ime on each edge of ECT(w), and hence, incur only O(N w) cos. Line 6 apparenly requires O(N w) ime. Therefore, he overall complexiy of compuetvp iso(n w logn w). THEOREM. T can be pre-processed ino a srucure ha occupies O(K) space, such ha any NK query can be answered in O(logN w) ime, where N w is he number of nodes in T carrying he query keyword. The srucure can be buil ino(k logk) ime. PROOF. The resul follows from he discussion in Secion 3., Theorem, he analysis of his subsecion, and he fac ha (i) he oal consrucion cos of he srucures of all keywords is O( w (NwlogNw)) = O(K logk), and (ii) he space of all hese srucures iso( w Nw) = O( u W(u) ) = O(K). When B-rees, insead of binary rees, are deployed, he space and query complexiies of our srucure are O(K/B) disk blocks ando(log B N w) I/Os, respecively, whereb is he size of a block. 4. NEAREST KEYWORD SEARCH AS AN OPERATOR In he inroducion, we oulined why NK search can be deployed as a primiive operaor o suppor oher asks efficienly. Secions 4. and 4. elaborae his for XPah query answering and group seiner ree compuaion, respecively. 4. XPah evaluaion This subsecion aims a a heoreical jusificaion ha many XPah queries can be reduced o NK search. We consider queries ha can be represened as a wig paernqas follows. Each inernal node of Q gives an elemen ype. A leaf node is designaed as he oupu, indicaing he informaion solicied (exension o muliple oupu nodes is rivial). Every oher, non-oupu, leaf node carries a keyword, which imposes a predicae ha mus hold on each occurrence of Q in he daa ree T. See Figure a for an example. Wihou loss of generaliy, we assume ha Q is compaible wih he daa schema; oherwise, i can be rejeced by sandard synax checking wih he DTD. We will prescribe wo condiions whose saisfacion guaranees he success of reducingqo a se of NK queries. These condiions are conservaive, in he sense ha one may sill carry ou reducion even if he condiions do no hold (as shown in he experimens). We will see ha he class of reducible queries can be processed by a new algorihm wih an aracive wors-case performance bound. Type-sequence condiion. Le P be any pah of T saring from roo(t ). Define he ype sequence of P as he ordering of he node ypes encounered as we walk along P. For insance, if P is he pah from node 0 o node 00 in Figure, is ype sequence is (league, eam, name). CONDITION. For each node u of he same ype, he roo-o-u pah int has he same ype sequence. In Figure, for example, he pah from he roo o every node has ype sequence (league, eam, s, ).

7 eam division divisionval Val from fromval Figure 7: Type paern of he query in Figure a Anchor condiion. As he nex condiion is more complex, we firs explain he idea using an example. Consider Q o be he paern in Figure a. Le us examine is ype paern, which is also a ree, and resuls from replacing each node of Q wih is ype, as shown in Figure 7. Denoe he ype paern of Q as ype(q). Le us fix a leaf node, fromval, inype(q) as he anchor, denoed as anc. Then, find he LCA, called a criical LCA, of anc and every oher leaf, namely, (i) clca = eam, which is he LCA of anc and leaf = divisionval, and (ii) clca =, which is he LCA of anc and leaf = Val. Each pair (clca i,leaf i) decides a criical se represened as cse i, which includes all he ypes on he pah from clca i o leaf i. Tha is, cse = {eam, division, divisionval}, and cse = {,, Val}. In he T of Figure, for each ype-clca i node u, sub(u) (i.e., he subree of u) has a mos one node of each ype in cse i. To be specific, le us inspec clca = eam. The subree of a eam node can have a mos one node of ype eam, division, and divisionval, respecively. Similarly, his is also rue for clca, i.e., every node can have in is subree a mos one,, and Val node, respecively. In his case, we say ha Q is unambiguous. No all he choices of anc would make Q unambiguous. For example, i is no hard o see ha Q is no unambiguous if divisionval is seleced as he anchor. The second condiion requires: CONDITION. A leas one choice of anchor makes Q unambiguous. To grasp he inuiion behind he condiion, consider again he sraegy explained in Secion. for processing he query Q of Figure a. Recall ha we used he fromval nodes carrying Maryland as he query nodes q for NK search. Insead, le us choose q as he divisionval nodes associaed wih wes. Tha is, for each such q, say node 000, he algorihm finds is neares Maryland-neighbor (i.e., node 000) in he se of fromval nodes, and hen checks wheher he neighbor has disance 6 o q. The answer is yes, so he presence of an occurrence has been deeced. The problem, however, is ha we can no longer find he oupu value Blake by idenifying he neares Val-neighbor of q. This is because muliple Val nodes have an idenical (smalles) disance 6 o q and, hence, he NK search wih Val as he query word may happen o reurn a node (e.g., 000) ha is no describing he same as node 000. We poin ou ha, alhough he definiions and noaions leading o Condiion were inroduced wih an example, hey can be generalized in a sraighforward manner o arbiraryq. Reducibiliy. The heorem below formally esablishes he reducion from XPah evaluaion o NK search. THEOREM 3. An XPah query can be reduced o NK search under Condiions and. PROOF. As in Secion., we se he ype of each nodeuint as a keyword inu. Each value node has is value as an exra keyword. An XPah queryqis processed wih he following algorihm: algorihm xpah(q). for each node q in T having ype anc do. for each non-oupu leaf node u of Q do 3. w he keyword of u 4. v he resul of NN(q,w) among all nodes in T of he same ype as u /* This can be done wih minor exension o our echnique in Secion 3. For example, one simple soluion is o prefix each word wih he ype of he node conaining i. Such a prefix is added o w, which auomaically resrics he search o he nodes of he designaed ype. */ 5. if q, v he correc value as in an occurrence hen 6. mark q as pruned 7. break for 8. if q has no been pruned hen 9. w he ype of he oupu node of Q 0. v NN(q,w). if q, v = he correc value as in an occurrence hen. repor he value of v Le us refer o he ype-anc node in an occurrence of Q as an anchor node. Denoe byf be he number of leaf nodes inype(q) oher han anc. Lines - of xpah are collecively called an ieraion. We will prove ha (a) he oupu value in every occurrence is repored by xpah, and (b) every value repored is indeed he oupu value of an occurrence. Proof of saemen (a). Le occ be any occurrence wih q as is anchor node. Denoe byu i he ype-leaf i node inocc ( i f). Eachu i mus be rerieved by he NK-query a Line 4 (or 0) in he ieraion for q, and pass he if-condiion a Line 5 (or ). The ieraion repors he oupu value of occ a Line. Proof of saemen (b). Under Condiion, i suffices o consider Q wih single-lined edges only, because any double-lined edge can be expanded ino a se of single-lined edges wihou affecing he query resul. Condiion also allows us o focus on Q (i) ha has only a single node, or (ii) whose roo has a leas wo child nodes. If roo(q) has a single child, we can remove he roo while sill obaining he same resul. Assume ha xpah repors a value in an ieraion wih q as he anchor node. Line 4 or 0 mus have feched a ype-leaf i node u i ( i f). Le occ be he ree ha (i) is rooed a he LCA of u,...,u f,q, and (ii) includes (only) he edges on he pah from he LCA o each of u,...,u f,q. The res of he proof argues ha occ is an occurrence. Towards his, we consider ree ype(occ), which is obained by replacing each node ofocc wih is ype. Our goal is o show ha ype(occ) andype(q) are exacly he same. Condiion implies ha all nodes of a ype mus be a he same level. Given a node ype z, we use seq(z) o represen he ype sequence of he pah in he daa ree T ha goes from he roo (of T ) o an arbirary ype-z node u (recall ha seq(z) is unique regardless of u). Also, suppose ha clca,...,clca f are in he opdown order. As clca i seq(anc) for each i [,f], q has a (unique) ypeclca i ancesor int, which we represen asv i. A key observaion is hau i mus be in he subree ofv i. Oherwise, q,u i would have been a leas more han he correc value in an occurrence, noicing ha he pah from q o u i would need o go ouside sub(v i), and hen evenually descend o he level where ype-leaf i nodes are. Fromype(Q), we know ha clca i is he las common ype in seq(leaf i) and seq(anc), implying ha v i mus be he LCA of u i and q, and belong oocc. Hence, he roo of occ has ype clca. Le P Q (or P occ) be he pah in ype(q) (or ype(occ)) from he roo o anc. The earlier analysis indicaes ha P Q is ideni-

8 cal o P occ. Nex we prove ha ype(occ) and ype(q) are he same in he oher pars as well. Le us walk down P Q and P occ synchronously, and sop as soon as encounering a clca i (of any i [,f]) in boh pahs. LeT Q (ort occ) be he subree ofclca i in ype(q) (orype(occ)), removing he subree rooed az, wherez is he child node of clca i ha is an ancesor of anc. DefineS = {j clca j = clca i} (i is possible for several criical LCAs o coincide on one node). Le z be any ype in he cse j of anyj S. Under Condiion, here is a unique ype-z node in he subree of v i in T. Therefore, each ype in T Q (T occ) can appear only once. For each leaf j (j S), define a ype-sequence s j ha equals he suffix ofseq(leaf j) saring fromclca i. Regarding each ype as a symbol, boht Q andt occ are in fac a rie of he same se of srings {s j j S}. The above reasoning oft Q andt occ is independen ofi. We hus have esablished he equivalence of ype(q) and ype(occ). Remark. We ouline he main ideas on how o verify Condiions and efficienly, while leaving he complee deails o he full paper. We regard each rule of he DTD as having he form e s, where e is an elemen ype and s a sring specifying he possible child elemen ypes of e. Combine muliple rules whose lef hand sides are he same ype. Denoing bys he se of resuling rules, we can show ha Condiion is saisfied if and only if every elemen ype appears on he righ hand side (RHS) of exacly one rule ins. To check Condiion, we resor o a schema reest, where each node is an elemen ype e, whose child nodes are he elemen ypes in he RHS of rulee s. Recall ha every elemen ypecinsmay carry a sar (e.g.,e c means ha here can be muliple insances of c below e direcly). In his case, he edge (in ST ) from e o c is a sar edge; oherwise, i is a non-sar edge. Given a query Q, ST allows us o verify Condiion as follows. Firs pick a leaf node of ype(q) as he anchor anc. Perform he following for every oher leaf node u: idenify he LCA v of anc and u, and examine if he pah from v o u in ST has a sar edge. If he answer is no for all u, we asser ha Q saisfies Condiion. Oherwise, pick a differen anchor and repea he above process. If all anchors have been aemped and Q has no been confirmed o saisfy Condiion, we conclude ha i violaes he condiion. Clearly, he ime of Condiion- checking depends only on he schema, is irrelevan o he size of he XML documen, and hence, ypically accouns for only a fracion of he oal query cos. Noe ha Condiion () is a consrain on he daa (queries). I is worh menioning ha he wo condiions are saisfied by many real daases and meaningful queries. In paricular, we noe ha all he daases experimened in [4, 6, 0, 34, 35] and a leas one in [7, 5, 6, 0, 7, 8, 9, 30, 3, 37] saisfy Condiion (see also our experimens). We poin ou ha simple aliasing can ofen be carried ou o make a daase saisfy Condiion. For example, in he well-known DBLP daase, an auhor elemen may be under an aricle or inproceedings elemen. To mee Condiion, we can rename he auhor of former ype as a-auhor, and ha of he laer ype as i-auhor. Remark. Denoe byleaf,...,leaf f he leaf nodes ofqha are no he anchor anc. By Theorem, algorihm xpah erminaes in O(N anc f i= logn leaf i ) ime, wheren x is he number of nodes in he daa ree T fulfilling he predicae implied by he node x in ype(q). For example, for he Q of Figure a, he execuion ime is O(N Maryland(logN wes +logn Val)). In general, he ime complexiy of xpah is independen on he number of nodes int whose ypes are inernal nodes of ype(q). This is a unique performance characerisic ha is no shared by any previous soluion. 4. Finding approximae group seiner rees This secion discusses he GST problem as moivaed in Secion. The daase is a ree T as described in Secion. A query specifies a se of keywords QS = {w,...,w l }. Recall ha U(w i) is he se of nodes in T associaed wih w i ( i l). Any l nodes u,...,u l, where u i U(w i) for each i, uniquely deermines a minimum connecing ree (MCT) [0] M as follows. M has he LCA of allu,...,u l as is roo, and includes (all and only) he edges on he pah int from he LCA o every u i. Referring o (u,...,u l ) as a vecor, he GST problem is equivalen o discovering he vecor ha minimizes he number of edges in M. This problem is NP-hard [7]. As an example, assume T o be he daa ree in Figure, and w = Lakers, w = Blake, w 3 = guard (l = 3). Given u = node 000,u = node 0000, andu 3 = node 000, Figure 3 demonsraes he corresponding MCT M. Noice ha he roo of M is he LCA of u, u and u 3. We assume ha w,...,w m have been arranged in such a way ha U(w )... U(w l ). The following algorihm exracs an MCT wih a small number of edges. algorihm approxgst(w,...,w l ). d min. for each node u U(w ) do 3. for i o l do 4. u i NN(u,w i ) 5. d l i= u,u i 6. if d < d min hen 7. remember (u,...,u l ) as he bes vecor 8. d min = d 9. reurn he MCT M deermined by he bes vecor The for-loop in Lines -8 enumeraes each node u U(w ). Lines 3-4 find he neares w i-neighbor of u for every oher query keyword w i. The resuling neighbors, ogeher wih u, deermine an MCT. The key of he algorihm is o measure he qualiy of he ree as he sum of he disance from u o each rerieved neighbor (see Line 5). The algorihm evenually reurns he bes ree according o his qualiy meric. We say ha an MCTM is a c-approximae GST if i has a mos c imes more edges han he GST. Formally, ifcos(m) represens he number of edges inm, i holds ha cos(m) c cos(m ), where M is he (opimal) GST. LEMMA 6. The oupu of approxgst is an (l )- approximae GST. PROOF. For each i [,l], le u i be a node in M associaed wih keyword w i (if here are several such nodes, u i can be any of hem). Define d = l i= u,u i. Since every edge of M is included a mos once in he pah from u o each u i ( i l), i holds ha: d (l ) cos(m ). () Le M apx be he oupu of approxgst. The d min in he sequel refers o he final d min of approxgst. Noice ha {u, NN(u,w ),...,NN(u,w l )} is a se{u,...,u l } ha mus have been inspeced a Line 5 of approxgst. In oher words: d min l u,nn(u,w i) d. (3) i= where he second is because u,nn(u,w i) u,u i.

9 Le {û,...,û l } be he se {u,...,u l } ha deermines M apx a Line 7 of approxgst. Then, d min = l i= û,ûi. Every edge of M apx is used a leas once in he union of he pahs from û oû,...,û l, respecively. Hence: cos(m apx) d min. (4) Combining Inequaliies -4 gives cos(m apx) (l ) cos(m ). By Theorem, he execuion ime of approxgst is bounded byo(n minllogn max), plus he cos of oupuing he ree, where N min = U(w ) and N max = U(w l ). Remark 3. Someimes i is useful o reurn k MCTs wih small cos, where k is a user-specified parameer. In his case, we mainain hek bes vecors currenly found, as opposed o only he op-. This can be achieved wih minor modificaion o approxgst. 5. RELATED WORK NK search has no been sudied previously. In he sequel, we review he exising work on oher opics relaed o our discussion. The firs opic is he processing of holisic wig joins, where he goal is o enumerae all occurrences of a wig paern. The exising algorihms can be classified as sequenial or indexed. A sequenial algorihm [4, 5, 8, 7, 5,, 30, 3, 37] scans synchronously he nodes, whose ypes appear in he query paern, in ascending order of heir ranks. The drawback of hese algorihms is ha hey mus access every such node a leas once, even hough i does no paricipae in any occurrence. Moivaed by his, an indexed algorihm [4, ] uilizes a daa srucure ha can be used o efficienly rerieve a paricular ancesor/descenden of a node. Such an abiliy allows he algorihm o skip many nodes no involved in any occurrence and, herefore, o erminae much earlier. In his paper, we do no aemp o aack general holisic wig joins. Insead, our focus is o improve he efficiency of hose XPah queries ha can be processed wih NK search. As far as hese queries are concerned, our mehod significanly ouperforms all he soluions in he sequenial caegory because (similar o indexed algorihms) i only needs o access a small number of nodes ha may form an occurrence. Regarding he indexed caegory, he comparison is more suble, mainly because he algorihms of [4, ] are heurisic in naure, and are no accompanied by any nonrivial complexiy analysis. We will experimenally compare our echnique agains he sae of he ar, TSGeneric + [], of ha caegory. Noeworhily, an advanage of our algorihm xpah (Secion 4.) is ha, in pracice, is cos can be accuraely esimaed by a query opimizer. This is no possible for [4, ], as heir behavior is sensiive o he daa disribuions. We noe ha no exising algorihm has he performance characerisic pinpoined in Remark. The only approach ha comes close o having he characerisic is TJFas [30], which is a sequenial algorihm ha needs o scan he nodes maching only he leaves of he query paern. Unforunaely, he exended dewey codes ha TJFas relies on can be as long as he heigh of he XML ree. The ime of reading a node is proporional o he lengh of is exended dewey code. The lengh, unforunaely, is linear o he number of ancesors of he node, which can be asympoically idenical o he oal number of nodes in he ree in he wors case. I is also worh menioning ha here exis some oher mehods [9, 4], jus like ours, ha are designed for cerain special classes of XPah queries. Anoher relaed opic is keyword search in XML daabases. A bulk of he exising research [6, 0, 6, 0, 7, 8, 9, 35] explores various semanics of query resuls ha is suiable for differen scenarios. In his work, we showed he applicabiliy of NK search o he GST semanics [, 0, 3, 5]. This choice does no imply our preference of GST; in fac, he poenial applicaion of he NK operaor o he oher semanics is an exciing direcion for fuure work. As menioned before, he GST problem on rees is NP-hard, and remains so even if he goal is changed o finding an O(log ǫ n)- approximae soluion for arbirarily smallǫ > 0 [7], wherenis he number of nodes in he daa ree. The bes known approximaion raio achievable in polynomial ime is O(lognlogl) [4], where l is he number of query keywords (and can be as large as n). The algorihm of [4], which is based on linear programming, is highly heoreical and no appropriae for pracical implemenaion. In he daabase area, research on GST compuaion (e.g., [3,, 9, 0,, 5]) is largely moivaed by he observaion ha he value oflcan ofen be regarded as a consan in realiy. In his case, he l approximaion raio guaraneed by our soluion (Lemma 6) can be (much) lower han O(lognlogl). A small l also considerably shrinks he search space for discovering he GST. This fac is leveraged in [0] o enumerae all he MCTs (defined in Secion 4.), and hereby, evenually come across he GST (which is also an MCT). While he mehod of [0] works on rees only, he approaches reviewed below apply o general graphs. Algorihms for compuing approximae GSTs (faser han exracing he GST wih [0]) are presened in [3, ]. Somewha surprisingly, here is a dynamic-programming algorihm [] for finding he exac GST, in even less ime han he approximae mehods of [3, ]. The above approaches do no rely on pre-compuaion, whereas he BLINKS sysem [9] leverages a sophisicaed access mehod consruced in advance o produce l-approximae GSTs efficienly. The mos serious drawback of BLINKS, however, is ha i occupies Ω(n 4/3 ) space, where n is he number of nodes in he graph. For n a he order of millions, Ω(n 4/3 ) amouns o duplicaing he daabase 00 imes, which is oo expensive in many environmens. Furhermore, he query algorihm of BLINKS is ad-hoc and has no ineresing wors-case performance bound. Noe ha our resul in Lemma 6 in fac dominaes he performance of BLINKS, i.e., we guaranee a slighly beer approximaion raio, wih a linear space access mehod and wors-case efficien query ime. I should be noed, however, ha he improvemen is made possible by uilizing properies of a ree (recall ha BLINKS deals wih general graphs). The work of [3, 5] considers he seiner ree problem, which is a special case of he GST problem where here can be only a single node in he daa ree associaed wih each query keyword. Noe ha, while he seiner ree problem is NP-hard on graphs, i is no on rees, as is clear from he way ha MCT is buil from a vecor in Secion 4.. Finally, in he special case where only one disinc keyword exiss, NK rerieval is similar o neares neighbor search on spaial neworks. In ha problem, he daa consis of a graphgand a ses of poins, each locaed a a node ofg. Given a nodeqofg, a query reurns he poin ins ha has he smalles shores-pah disance o q. This problem has been well sudied in various seings wih differen performance goals (see [, 6, 3, 33] and he references herein). The exising soluions (i) perform Dijksra-like expansion from he query poin (e.g., []), (ii) pre-compue he answer for each node of G (e.g., [3]), or (iii) rely on specialized srucures ha leverage properies of spaial daa (e.g., [6, 33]). In our con- Given a se of keywords, he GST in a graph is he minimum ree (i) whose nodes and edges come from he underlying graph, and (ii) ha has he minimum number of edges among all rees saisfying (i). When he graph is a ree, his definiion degeneraes ino he one in Secion 4.. This can be derived from Theorem 3 of [9] by observing ha, b N b +BP n /B +B, which is Ω(n 4/3 ).