A Linear-Space Top-down Algorithm for Tree Inclusion Problem

Size: px

Start display at page:

Download "A Linear-Space Top-down Algorithm for Tree Inclusion Problem"

Gordon Wilkinson
5 years ago
Views:

1 A Linr-Spc Top-own Algorithm for Tr Inclusion Prolm Yngjun Chn 1, Yiin Chn 2 Dpt. Appli Computr Scinc, Univrsity of Winnipg, Cn 1 y.chn@uwinnipg.c, 2 chnyiin@gmil.com Astrct W consir th following tr-mtching prolm: Givn ll, orr trs P n T, cn P otin from T y lting nos? Dlting no v ntils rmoving ll gs incint to v n, if v hs prnt u, rplcing th gs from u to v y gs from u to th chilrn of v. Th st known lgorithm for this prolm ns O( T lvs(p) ) tim n O( lvs(p) min{d T, lvs(t) } + T + P ) spc, whr lvs(t) (rsp. lvs(p)) stns for th st of th lvs of T (rsp. P), n D T (rsp. D P ) for th hight of T (rsp. P). In this ppr, w prsnt n fficint lgorithm tht rquirs O( T lvs(p) ) tim n O( T + P ) spc. I. INTRODUCTION Lt T tr n v no iffrnt of root in T with prnt no u. Dnot y lt(t, v) th tr otin from T y rmoving th no v. Th chilrn of v com th chilrn of u s illustrt in Fig. 1. T: lt(t, v) c () f () illustrtion. Th orr tr inclusion prolm ws initilly introuc y Knuth [10], whr only sufficint conition for this prolm is givn. Th tr inclusion hs n suggst s n importnt primitiv for xprssing quris on structur ocumnt tss [4, 5, 6, 12]. A structur ocumnt ts is consir s collction of prs trs tht rprsnt th structur of th stor txts n th tr inclusion is us s mns of rtriving informtion from thm. Anothr ppliction of th orr tr mtching is th vio contnt-s rtrivl. Accoring to Rui t l. [14], vio cn succssfully compos into hirrchicl tr structur, in which ch no rprsnts scn, group, shot, frm, ftur, n so on. Espcilly, such tr is n orr on sinc th tmporl orr is vry importnt for vio. In ition, th orr tr mtching cn lso ppli in th scn nlysis, th computtionl iology (such s th RNA structur mtching [11]), s wll s in th t mining (such s th tr mining [15]). np t n v s vp np v f Th stunt rs t j n intnsivly Fig. 1 Th ffct of rmoving no from tr vp th intrsting ook () Givn two orr ll trs P n T, cll th pttrn n th trgt, rspctivly. An intrsting prolm is: Cn w otin pttrn P y lting som nos from trgt T? Tht is, is thr squnc v 1,..., v k of nos such tht for T 0 = T n T i+1 = lt(t i, v i+1 ) for i = 0,..., k - 1, w hv T k = P? If this is th cs, w sy, P is inclu in T [9]. Such prolm is cll th tr inclusion prolm. Orr ll trs ppr in vrious rsrch fils, incluing progrmming lngug implmnttion, nturl lngug procssing, n molculr iology. As n xmpl [9], consir qurying grmmticl structurs s shown in Fig. 2(), which is th prs tr of nturl lngug sntnc. On might wnt to loct, sy, thos sntncs tht inclu vr phrs contining th vr rs n ftr it noun ook follow y ny vr. This is xctly th sntncs whos prs tr cn otin y lting som nos from th tr shown in Fig. 2(). S Fig. 2() for v rs n ook v () Fig. 2 Illustrtion for orr tr inclusion This prolm hs n th ttntion of much rsrch. Kilpläinn n Mnnil [9] prsnt th first polynomil tim lgorithm using O( T P ) tim n spc. Most of th ltr improvmnts r rfinmnts of this lgorithm. In [13], Richtr gv n lgorithm using O( α(p) T + m(p, T) D T ) tim, whr α(p) is th lpht of th lls of P, m(p, T) is th siz of st cll mtchs, fin s ll th pirs (v, w) P T such tht ll(v) = ll(w), n D T (rsp. D P ) is th pth of T (rsp. P). Hnc, if th numr of mtchs is smll, th tim complxity of this lgorithm is ttr thn O(T P ). Th spc complxity of th lgorithm is O( α(p) T + m(p, T)). In [3], mor sophistict lgorithm ws prsnt using O( T lvs(p) ) tim n O( lvs(p) min{d T, lvs(t) } + T + P ) spc. In [1], n fficint vrg cs lgorithm ws iscuss. Its vrg tim complxity is O( T /12/$ IEEE 2127

2 + C(P, T) P ), whr C(P, T) rprsnts th numr of T s nos tht hv n xmin uring th inclusion srch. Howvr, its worst tim complxity is still O( T P ). In [2], nothr ottom-up lgorithm is propos. Th tim complxity of th lgorithm is oun y min O( T lvs(p) ) O( lvs(t) lvs(p) loglog lvs(p) + lvs(p)) O( T P /(log T ) + T log T ) But it is clim tht th lgorithm ns only O( T + P ) spc. A crful nlysis rvls tht th spc complxity of th lgorithm is th sm s tht of [3]. In th lgorithm, t structur EMB(v) for ch v in P is us to rcor p occurrncs of P[v] in T. It is of siz O( lvs(t) ) in th worst cs. EMB(v) is gnrt rcursivly n works in wy similr to th concpt of shll iscuss in [3]. So th nlysis of shll pplis to EMB(v) s. In our rlir work [7], top-own lgorithm ws propos with O( T + P ) spc rquirmnt. Howvr, its tim complxity is not polynomil, s shown in [11]. In this ppr, w rvisit this issu n prsnt nw topown lgorithm to rmov ny runncy of [10]. Th tim complxity of th nw on is oun y O( T lvs(p) ). Although th tim complxity of our lgorithm is comprl to Chn s lgorithm [3], it is mor fficint thn Chn s sinc in Chn s lgorithm ch no in T will chck ginst, sis som intrnl nos, ll th lf nos in P. But in our lgorithm, no in T my chck so mny tims only whn som conitions r stisfi. Mor importntly, our lgorithm ns only linr spc O( T + P ). Th tr inclusion prolm on unorr trs is NPcomplt [9] n not iscuss in this ppr. II. BASIC DEFINIION W concntrt on ll trs tht r orr, i.., th orr twn silings is significnt. Tchniclly, it is convnint to consir slight gnrliztion of trs, nmly forsts. A forst is finit orr squnc of isjoint finit trs. A tr T consists of spcilly signt no root(t) cll th root of th tr, n forst <T 1,..., T k >, whr k 0. Th trs T 1,..., T k r th sutrs of th root of T or th immit sutrs of tr T, n k is th outgr of th root of T. A tr with th root t n th sutrs T 1,..., T k is not y <t; T 1,..., T k >. Th roots of th trs T 1,..., T k r th chilrn of t n silings of ch othr. Also, w cll T 1,..., T k th siling trs of ch othr. In ition, T 1,..., T i-1 r cll th lft siling trs of T i, n T i-1 th immit lft siling tr of T i. Th root is n ncstor of ll th nos in its sutrs, n th nos in th sutrs r scnnts of th root. Th st of scnnts of no v is not y sc(v). A lf is no with n mpty st of scnnts. Somtims w trt tr T s th forst <T>. W my lso not th st of nos in forst F y V(F). For xmpl, if w spk of functions from forst G to forst F, w mn functions mpping th nos of G onto th nos of F. Th siz of forst F, not y F, is th numr of th nos in F. Th rstriction of forst F to no v with its scnnts sc(v) is cll sutr of F root t v, not y F[v]. Lt F = <T 1,..., T k > forst. Th prorr of forst F is th orr of th nos visit uring prorr trvrsl. A prorr trvrsl of forst <T 1,..., T k > is s follows. Trvrs th trs T 1,..., T k in scning orr of th inics in prorr. To trvrs tr in prorr, first visit th root n thn trvrs th forst of its sutrs in prorr. Th postorr is fin similrly, xcpt tht in postorr trvrsl th root is visit ftr trvrsing th forst of its sutrs in postorr. W not th prorr n postorr numrs of no v y pr(v) n post(v), rspctivly. Using prorr n postorr numrs, th ncstorship cn sily chck. If thr is pth from no u to no v, w sy, u is n ncstor of v n v is scnnt of u. In this ppr, y ncstor ( scnnt ), w mn propr ncstor (scnnt), i.., u v. Lmm 1 Lt v n u nos in forst F. Thn, v is n ncstor of u if n only if pr(v) < pr(u) n post(u) < post(v). Proof. S Exrcis in [10] (pg 347). Similrly, w chck th lft-to-right orring s follows. Lmm 2 Lt v n u nos in forst F. v is si to to th lft of u if thy r not rlt y th ncstor-scnnt rltionship n u follows v whn w trvrs F in prorr. Thn, v is to th lft of u if n only if pr(v) < pr(u) n post(v) < post(u). Proof. Th proof is trivil. In th following, w us th postorr numrs to fin n orring of th nos of forst F givn y v p v iff post(v) < post(v ). Also, v p v iff v p v or v = v. Furthrmor, w xtn this orring with two spcil nos 7 p v p 6. Th lft rltivs, lr(v), of no v V(F) is th st of nos tht r to th lft of v n similrly th right rltivs, rr(v), r th st of nos tht r to th right of v. Th following finition is u to [9]. Dfinition 1 Lt F n G ll orr forsts. W fin n orr ming (ϕ, G, F) s n injctiv function ϕ: V(G) V(F) such tht for ll nos v, u V(G), i) ll(v) = ll(ϕ(v)); (ll prsrvtion conition) ii) v is n ncstor of u iff ϕ(v) is n ncstor of ϕ(u), i.., pr(v) < pr(u) n post(u) < post(v) iff pr(ϕ(v)) < pr(ϕ(u)) n post(ϕ(u)) < post(ϕ(v)); (ncstor conition) iii) v is to th lft of u iff ϕ(v) is to th lft of ϕ(u), i.., pr(v) < pr(u) n post(v) < post(u) iff pr(ϕ(v)) < pr(ϕ(u)) n post(ϕ(v)) < post(ϕ(u)). (Siling conition) If thr xists such n injctiv function from V(G) to V(F), w sy, F inclus G, F contins G, F covrs G, or sy, G cn m in F. Fig. 3 shows n xmpl of n orr inclusion. Lt P n T two ll orr trs. An ming ϕ of P in T is si to root-prsrving if ϕ(root(p)) = root(t). If thr is root-prsrving ming of P in T, w sy tht th root of T is n occurrnc of P. Fig. 3() lso shows n xmpl of root prsrving ming. Accoring to [9], rstricting to root-prsrving 2128

3 ming os not los gnrlity. In fct, wht cn foun y th top-own lgorithm to iscuss is rootprsrving tr ming. Throughout th rst of th ppr, w rfr to th ll orr trs simply s trs. III. ALGORITHM DESCRIPTION Lt T = <t; T 1,..., T k > (k 0) tr n G = (q 0) forst. W hnl G s tr P = , whr p v rprsnt virtul no, mtching ny no in T. Not tht vn though G contins only on singl tr it is consir to forst. So virtul root is. Thrfor, ch no in G, xcpt th virtul no, hs prnt. Consir no v in G = with chilrn v 1,..., v j. W us pir <i, v> (i j) to rprsnt n orr forst contining th first i sutrs of v: <G[v 1 ],..., G[v i ]>. If v is p v, or no on th lft-most pth in P 1, <i, v> is cll lft cornr of G. Espcilly, <i, p v > is lft cornr, rprsnting th first i trs in G: P 1,..., P i. In ition, δ(v) rprsnts link from no v to th lft-most lf no in G[v], s illustrt in Fig. 4. Lt v lf no in G. δ(v ) is fin to link to v itslf. So in Fig. 5, w hv δ(v 1 ) = δ(v 2 ) = δ(v 3 ) = v 3. W lso not y δ -1 (v ) st of nos x such tht for ch v x δ(v) = v. Thrfor, in Fig. 5, δ -1 (v 3 ) = {v 1, v 2, v 3 }, δ -1 (v 4 ) = {v 4 }, n δ -1 (v 5 ) = {v 5 }. Th out-gr of v in tr is not y (v) whil th hight of v is not y h(v), fin to th numr of gs on th longst ownwr pth from v to lf. Th hight of lf no is st to 0. As with [7], w rrng two functions: top-own(t, G) n ottom-up(t, G) to chck tr inclusion, whr T is tr, n T n G r two forsts. Howvr, iffrnt from [7], ch of th two functions rturns lft cornr <i, v> of G with th following proprtis: Lt v th lft-most lf in G[v]. If i > 0, it shows tht th first i sutrs of v in G cn m in T (or in T ), n for ny i > i, <i, v> cnnot m in T (or in T ), n for ny v s ncstor u δ -1 (v ) thr xists no j > 0 such tht <j, u> is l to m in T (or in T ). If i = 0, v is th lft-most lf in G, inicting tht no lft cornr of G cn m in T (or in T ). Fig. 3: () Th tr on th lft cn inclu in th tr on th right y lting th nos lll :,, n ; () th ming corrsponing to (). () () In this sns, w sy, <i, v> is th highst n wist lft cornr which cn m in T (or in T ). P: δ(v 2) δ(v 1) v 3 v 1 v 2 v 5 v 4 Fig. 4 A pttrn tr W notic tht if v = p v n i > 0, it shows tht P 1,..., P i cn inclu. In [7], oth top-own(t, G) n ottom-up(t, G) rturn n intgr i to inict tht T ms th first i trs in G. Although our lgorithm follows th rrngmnt of [7], th min i is quit iffrnt. It is not ncssry to rfr to [7] to unrstn th following iscussion. If th trgt is tr n th pttrn is forst, w cll th function top-own. If oth th trgt n th pttrn r forsts, w cll th function ottom-up. But uring th computtion, thy will cll from ch othr. In top-own(t, G), w n to hnl two css. Cs 1: G = ; or G = (q > 1), ut T P 1 + P 2. In this cs, to fin th highst n wist lft cornr <i, v> tht cn m in T = <t; T 1,..., T k >, th following chckings will conuct: i) If t is lf no, w will chck whthr ll(t) = ll(δ(p 1 )), whr p 1 is th root of P 1. If it is th cs, rturn <1, prnt of δ(p 1 )>. Othrwis, rturn <0, δ(p 1 )>. ii) If T ), whr is forst of th sutrs of p 1. Th rturn vlu of top-own(t, ) is us s th rturn vlu of top-own(t, G). iii) If T P 1 n h(t) h(p 1 ), w furthr istinguish twn two sucss: ll(t) = ll(p 1 ). In this cs, w will cll ottomup(<t 1,..., T k >, ). ll(t) ll(p 1 ). In this cs, w will cll ottomup(<t 1,..., T k >, ). In oth css, ssum tht th rturn vlu of ottom-up( ) is <i, v>. W n to prform furthr chcking: - If ll(t) = ll(v) n i = (v), th rturn vlu of top-own(t, G) is st to <1, v s prnt>. - Othrwis, th rturn vlu of top-own(t, G) is th sm s <i, v>. Cs 2: G = (q > 1), n T > P 1 + P 2. In this cs, w will cll ottom-up(<t 1,..., T k >, G). Assum tht th rturn vlu of ottom-up(<t 1,..., T k >, G) is <i, v>. Th following chckings will continully conuct. iv) If v p 1 s prnt, chck whthr ll(t) = ll(v) n i = (v). If so, th rturn vlu of top-own(t, G) will st to <1, v s prnt>. Othrwis, th rturn vlu of topown(t, G) is th sm s <i, v>. v) If v = p 1 s prnt, th rturn vlu of top-own(t, G) is th sm s <i, v>. 2129

4 Th following is forml scription of th lgorithm. In th procss, ch no t in T is ssocit with t structur, rfrr to s κ(t). Initilly, ch κ(t) is st to φ. Ech tim cll of th form top-own(t[t], G ) rturns lft cornr <i, v>, κ(t) will chng to <i, v>, whr G is forst m up of st of sutrs root rspctivly t st of conscutiv chil nos (strting from spcific chil to th lst chil) of crtin no in G. This vlu is minly us in ottom-up( ) to voi runncy. Howvr, for simplicity, in th following lgorithm κ(t) is not xplicitly rprsnt. function top-own(t, G) input: T = <t; T 1,..., T k >, G = . output: <i, v> spcifi ov. gin 1. if (q = 1 or T P 1 + P 2 ) 2. thn { lt P 1 = ; (*Cs 1*) 3. if t is lf thn { { lt δ(p 1 ) = v; (*Cs 1 - (i)*) 4. if ll(t) = ll(v) thn rturn <1, v s prnt> ls rturn <0, v>; } 5. if ( T ); (*Cs 1 - (ii)*) 6. if ll(t) = ll(p 1 ) (*Cs 1 - (iii)*) 7. thn { if p 1 is lf thn {v := p 1 s prnt; i := 1;} 8. ls {<i, v> := ottom-up(<t 1,..., T k >, ); 9. if ll(t) = ll(v) n i = (v) thn {v := v s prnt; i := 1; } 10. } 11. ls <i, v> := ottom-up(<t 1,..., T k >, ); (*If ll(t) ll(p 1 ), cll ottom-up( ).*) 12. rturn <i, v>; 13. } 14. ls {<i, v> := ottom-up(<t 1,..., T k >, G); (*Cs 2*) 15. if v p 1 s prnt thn (*Cs 2 - (iv)*) 16. if (ll(t) = ll(v)) n i = (v) thn rturn <1, v s prnt>; 17. rturn <i, v>; (*Cs 2 - (v)*) 18. } n In th ov lgorithm, w first chck whthr q = 1 or T P 1 + P 2 (s lin 1). If it is th cs w hv Cs 1 n thn lins 2-13 r xcut. In this procss, ll th thr sucss (i), (ii), n (iii) r chck. If q > 1 n T > P 1 + P 2, w hv Cs 2 n lins will crri out, in which w first cll ottom-up(<t 1,..., T k >, G). Dpning on its rturn vlu, (vi) or (v) is conuct. ottom-up(t, G) is sign to hnl th cs tht oth T n G r forsts m up of st of sutrs root t nos tht r conscutiv silings in T n P, rspctivly. Lt T = <T 1,..., T k > n G = . Dnot y t l th root of T l (l = 1,..., k). Dnot y p j th root of P j (j = 1,..., q). In ottom-up(t, G), w will mk sris of clls topown(t l, < P,..., P q >), whr l = 1,..., k, j 1 = 1, n j 1 j 2... j l j h q (for som h k), controll s follows. 1. Two inx vrils l, j r us to scn T 1,..., T k n P 1,..., P q, rspctivly. (Initilly, l is st to 1, n j is st to 0.) Thy lso inict tht hs n succssfully m in <T 1,..., T l >. 2. Lt th rturn vlu of top-own(t l, ). If v l = p 1 s prnt, st j to j + i l. Othrwis, j is not chng. St l to l + 1. Go to (2). 3. Th loop trmints whn ll T l s or ll P j s r xmin. If j > 0 whn th loop trmints, ottom-up(t, G) rturns <j, p 1 s prnt>, inicting tht T contins P 1,..., P j. Othrwis, j = 0, inicting tht vn P 1 lon cnnot m in ny T l (l {1,..., k}). Howvr, in this cs, w n to continu to srch for highst n wist lft cornr <i, v> in G, which cn m in T. This is on s scri low. i) Lt ,..., th rturn vlus of topown(t 1, ),..., top-own(t k, ), rspctivly. Sinc j = 0, ch v l δ -1 (v ) (l = 1,..., k), whr v is th lft-most lf in P 1. ii) If ch i l = 0, rturn <0, lft-most lf of P 1 >. Othrwis, thr must som v l s such tht i l > 0. W cll such no non-zro point. Fin th first non-zro point v f with chilrn w 1,..., w s such tht v f is not scnnt of ny othr non-zro point. Thn, w will chck <T f+1,..., T k > ginst <P[ w + 1 ],..., P[w s ]>. Lt x (0 x s - i ) f numr such tht <P[ w + 1 ],..., P[ w + x ]> cn m in <T f+1,..., T k >. Th rturn vlu of ottom-up(t, G) shoul st to < + x, v f >. In th ottom-up procss, κ(t) cn us to voi runnt computtion. Concrtly, ch tim for w mk cll of th form top-own(t l, ), w will clcult function κ-chcking(t l, p j ) fin low to trmin whthr this cll cn skipp ovr, whr t l n p j r th roots of T l n P j, rspctivly. function κ-chcking(t, p) input: t - no in T; p - no in G. output: φ or <i, v> spcifi ov. gin 1. if κ(t l ) φ thn { 2. lt κ(t l ) = <i, v>; 3. if i = 0 thn rturn φ; 4. if i > 0, δ(v) = δ(p), n p is qul to v s first chil or n ncstor of v s first chil 5. thn rturn <i, v>; 6. if i > 0, δ(v) = δ(p), n p is scnnt of v s first chil 7. thn rturn <(p s prnt), p s prnt>. 8. ls rturn φ. n Only whn κ-chcking(t l, p j ) rturns φ, top-own(t l, ) will crri out. Othrwis, w us th vlu of κ- chcking(t l, p j ) s th rturn vlu of top-own(t l, ). In trms of th ov iscussion, w rrng nw suprocur to chck T l ginst forst , oing th sm work s th top-own procss ut with κ-chcking(t l, p j ) ing us to voi unncssry chckings. function top-own-κ(t, ) input: T - tr; - forst. output: <v, i> spcifi ov. gin 1. if κ-chcking(t, p 1 ) = φ 2130

5 thn <i, v> := top-own(t, ) 2. ls <i, v> = κ-chcking(t, p 1 ); 3. rturn <i, v>; n In th following lgorithm, w us top-own-κ( ), inst of top-own( ), to chck tr ginst forst. function ottom-up(t, G) input: T = <T 1,..., T k >, G = output: <i, v> spcifi ov. gin 1. l := 1; j := 0; 2. whil (j < q n l k) o (*min chcking*) 3. { := top-own-κ(t l, ) 4. if (v l = p 1 s prnt n i l > 0) thn j := j + i l ; 5. l := l + 1; } 6. if j > 0 thn rturn <j, p 1 s prnt>; 7. if for ll s i l = 0 thn rturn <0, lft-most lf in G> 8. ls { lt v f th first non-zro point such tht it is not scnnt of ny othr non-zro point; 9. lt w 1,..., w s th chilrn of v f ; 10. l := f + 1; j := ; 11. whil (j < s n l k) o (*supplmnt chcking*) 12. { := top-own-κ(t l, <G[w j+1 ],..., G[w s ]>); 13. if (v l = v f n i l > 0) thn j := j + i l ; 14. l := l + 1; } 15. rturn <j, v f >; 16. } n In ottom-up(t, G), w hv two whil-loops: on from lin 2 to 5 n nothr from lin 11 to 14. In th first whilloop, w chck <T 1,..., T k > ginst , rfrr to s th min ottom-up chcking (or simply th min chcking). In this chcking, ch T l is chck on y on, y rptly clling top-own-κ(t l, ) (lin 3), y which κ- chcking(t l, p j+1 ) is us to rmov runncy (s lins 1-2 in top-own-κ( )). In th scon whil-loop, w o supplmnt chcking. This is crri out only whn th following two conitions r stisfi (s lins 6 n 7): (1) j = 0, n (2) Thr xists t lst non-zro point (rturn vlu of top-own-κ(t l, ) such tht i l > 0. W rfr to ths two conitions s th supplmnt chcking conition. Lt v f th first non-zro point such tht v f is not scnnt of ny othr non-zro point. Lt w 1,..., w s th chilrn of v f. In th supplmnt chcking, w will chck <T f+1,..., T k > ginst <G[ w + 1 ],..., G[w s ]> (s lins ) computtion to voi ny uslss ffort, high prformnc is chiv. Th tim complxity of th nw lgorithm is oun y O( T lvs(p) ) whil th spc rquirmnt is oun y O( T + P ), whr T n P r trgt n pttrn tr, rspctivly. REFERENCES [1] L. Alonso n R. Schott. On th tr inclusion prolm. In Procings of Mthmticl Fountions of Computr Scinc, pgs , [2] P. Bill n I.L. Gørtz, An Orr Tr Inclusion Algorithm Bs on Dynmic Tr Lling, in Proc.32th Intl. Colloquium on Automt, Lngugs n Progrmming, Lctur Nots in Computr Scinc, vol. 3580, 2005, pp [3] W. Chn. Mor fficint lgorithm for orr tr inclusion. Journl of Algorithms, 26: , [4] Y. Chn n Y.B. Chn, Sutr Rconstruction, Qury No Intrvls n Tr Pttrn Qury Evlution, Journl of Informtion Scinc n Enginring 28, (2012). [5] Y. Chn n L. Zou, n Unorr tr mtching n orr tr mtching: th vlution of tr pttrn quris, Int. J. Informtion Tchnology, Communictions n Convrgnc, Vol. 1, No. 3, 2011, pp [6] Y. Chn, A Nw Algorithm for Twig Pttrn Mtching, in: Proc. of Int. Conf. on Entrpris Informtion Systms (ICEIS 2007), IEEE, Funchl-mir, Portugl, Jun 2007, pp [7] Y. Chn n Y.B. Chn, A Nw Tr Inclusion Algorithm, Informtion Procssing Lttrs 98(2006) , Elsvir Scinc B.V. [8] H.L Chng n B.F Wng, On Chn n Chn's nw tr inclusion lgorithm, Informtion Procssing Lttrs, 2007, Vol. 103, 14-18, Elsvir Scinc B.V. [9] P. Kilpläinn n H. Mnnil. Orr n unorr tr inclusion. SIAM J. Comput, 24: , [10] D.E. Knuth, Th Art of Computr Progrmming, Vol. 1 (1st ition), Aison-Wsly, Ring, MA, [11] R.B. Lyngs, M. Zukr & C.N.S. Prsn, Intrnl loops in RNA sconry structur priction, in Procings of th 3r nnul intrntionl confrnc on computtionl molculr iology (RECOMB), (1999). [12] H. Mnnil n K.-J. Räih, On Qury Lngugs for th p-string t mol, in Informtion Molling n Knowlg Bss (H. Kngsslo, S. Ohsug, n H. Jkol, Es.), pp , IOS Prss, Amstrm, [13] Thorstn Richtr. A nw lgorithm for th orr tr inclusion prolm. In Procings of th 8th Annul Symposium on Comintoril Pttrn Mtching (CPM), in Lctur Nots of Computr Scinc (LNCS), volum 1264, pgs Springr, [14] Y. Rui, T.S. Hung, n S. Mhrotr, Constructing tl-of-contnt for vios, ACM Multimi Systms Journl, Spcil Issu Multimi Systms on Vio Lirris, 7(5): , Spt [15] M. Zki, Efficintly mining frqunt trs in forst. In Proc. of KDD, IV. CONCLUSION In this ppr, nw lgorithm is propos to improv th lgorithm iscuss in [7]. Th min i hin it is to lt ny suprocur cll rturn pir to inict sutr (suforst) ming whil in [7], only singl intgr is rturn to inict whthr whol forst (or th first svrl sutrs of th forst) is m y th corrsponing trgt sutr. Togthr with simpl t structur ssocit with ch no in th trgt tr to trnsfr th rsult otin in prvious stp to th nxt stp 2131

CSI35 Chapter 11 Review

CSI35 Chapter 11 Review 1. Which of th grphs r trs? c f c g f c x y f z p q r 1 1. Which of th grphs r trs? c f c g f c x y f z p q r . Answr th qustions out th following tr 1) Which vrtx is th root of c th tr? ) wht is th hight