A Space-Economic Representation of Transitive Closures in Relational Databases

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1 A Sp-Eonomi Rprsnttion o Trnsitiv Closurs in Rltionl Dtss Ynjun Chn* Dpt. Appli Computr Sin, Univrsity o Winnip, 515 Port Av. Winnip, Mnito, Cn R3B E9 yhn@uwinnip. ABSTRACT A omposit ojt rprsnt s irt rph (irph or short) is n importnt t strutur tht rquirs iint support in CAD/CAM, CASE, oi systms, sotwr mnmnt, w tss, n oumnt tss. It is umrsom to hnl suh ojts in rltionl ts systms whn thy involv nstor-snnt rltionships (or sy, rursiv rltionships). In this ppr, w prsnt nw noin mtho to ll irph, whih rus th ootprints o ll prvious strtis. This mtho is s on tr llin mtho n th onpt o rnhins tht r us in rph thory or inin th shortst onntion ntworks. A rnhin is surph o ivn irph tht is in t orst, ut ovrs ll th nos o th rph. On th on hn, th propos noin shm hivs th smllst sp rquirmnts mon ll prviously pulish strtis or ronizin rursiv rltionships. On th othr hn, it ls to nw lorithm or omputin trnsitiv losurs or DAGs (irt yli rph) in O( ) tim n O(n ) sp, whr n rprsnts th numr o th nos o DAG, th numrs o th s, n th DAG s rth. In ition, this mtho n xtn to yli irphs n is spilly suitl or rltionl nvironmnt. Kywors: Dirt Grphs, Trs, Trnsitiv Closurs, Brnhins, Grph Enoin 1. INTRODUCTION It is nrl opinion tht rltionl ts systms r inqut or mnipultin omposit ojts tht ris in novl pplitions suh s w n oumnt tss [3, 1, 13, 35, 49], CAD/CAM, CASE, oi systms n sotwr mnmnt [8, 8, 43]. Espilly, whn rursiv rltionships r involv, it is umrsom to hnl thm in rltionl tss, whih sts urrnt rltionl systms r hin th nvitionl ons [31]. A omposit ojt n nrlly rprsnt s irt rph (irph). For xmpl, in CAD ts, omposit ojt orrspons to omplx sin, whih is ompos o svrl susins [8]. Otn, susins r shr y mor thn on hihr-lvl sins, n st o sin hirrhis thus orms irt yli rph (DAG). As nothr xmpl, th * Th uthor is support y NSERC (453) (Nturl Sins n Eninrin Counil o Cn). ittion inx o sintii litrtur, rorin rrn rltionships twn uthors, onstruts irt yli rph. As thir xmpl, w onsir th tritionl orniztion o ompny, with vril numr o mnr-suorint lvls, whih n rprsnt s tr hirrhy. In rltionl systm, omposit ojts must rmnt ross mny rltions, rquirin joins to thr ll th prts. A typil pproh to improvin join iiny is to quip rltions with hin pointr ils or ouplin th tupls to join [11]. Th so-ll join inx is nothr uxiliry ss pth to mitit this iiulty [46, 47]]. Also, svrl vn join lorithms hv n sust, s on hshin n lr min mmory. In ition, irnt kin o ttmpts to ttin ompromis solution is to xtn rltionl tss with nw turs, suh s lustrin o omposit ojts, y whih th ontnt orin kys o nstor pths r stor in primry ky (s [3, 33, 39] or til sription). Anothr xtnsion to rltionl systm is nst rltions (or NF rltions, s,.., [18]). Althouh it n us to rprsnt omposit ojts without sriiin th rltionl thory, it surs rom th prolm tht surltions nnot shr. Morovr, rursiv rltionships nnot rprsnt y simpl nstin us th pth is not ix. Finlly, utiv tss n ojt-rltionl tss n onsir s two quit irnt xtnsions to hnl this prolm [9, 37, 15]. In this ppr, w isuss nw noin pproh to pk nstor pths in rltionl nvironmnt. Th min i o this mtho is tr llin, y mns o whih w ssoit h no v with pir o intrs (α, β) suh tht i v, nothr no ssoit with (α, β ), is snnt o v, som rithmtil rltionship twn α n α, s wll s β n β n trmin. Thn, suh rltionships n us to in ll snnts o no, n th rursiv losur w.r.t. tr n omput vry iintly. This mtho n nrliz to DAGs or irphs ontinin yls y omposin rph into sris o trs, or whih th pproh sri ov n mploy. As w n s ltr, nw mtho or omputin rursion iintly in rltionl nvironmnt n vlop s on ths thniqus. In t, it is nw lorithm to hnl this prolm with rprsnttion o trnsitiv losurs irnt rom tritionl ons. It ns only O( ) tim n O(n ) sp, whr is th rth o th rph, in to th lst numr o isjoin pths tht ovr ll th nos o rph. This omputtionl omplxity is ttr thn ny xistin mtho or this prolm, inluin th rph-s lorithms [0, 1, 34, 36, 38], th rph noin [1,, 6, 7, 14, 16, 46, 46] n th mtrix-s lorithms [4, 3, 44, ISSN: SYSTEMICS, CYBERNETICS AND INFORMATICS VOLUME 4 - NUMBER 5 1

2 45]. (S Stion 6 or til omprison.) Th rminr o th ppr is orniz s ollows. Stion spiis th prolm to solv. Stion 3 introus th onpt o tr llin, n pplis it to simpl tr-strutur rursion hirrhis. Stion 4 lorts th thniqu to ritrry rursion pttrn. In Stion 5, w rss how to hn lls whn nw no or nw is insrt into DAG. In Stion 6, w isuss n ompr th rlvnt work. Stion 7 rports tst rsults. Finlly, short onlusion is st orth in Stion 8.. TASK DEFINITION W onsir omposit ojts rprsnt y irph, whr nos stn or ojts n s or prnt-hil rltionships, stor in inry rltion. In mny pplitions, th trnsitiv losur o irph ns to omput, whih is in to ll nstor-snnt pirs. Lt G = (V, E) irt rph (irph or short). Dirph G* = (V, E*) is th rlxiv, trnsitiv losur o G i (v, w) E* i thr is pth rom v to w in G. For xmpl, th rph shown in Fi. 1() is th trnsitiv losur o th rph shown in Fi. 1(). A lot o rsrh hs n on on this issu. Amon thm, th smi-niv [10] n th lorithmi [46] r typil lorithmi solutions. Anothr min pproh is th mtriliztion o trnsitiv losur, ithr prtilly or ompltly [1,, 5, 6, 7, 49]. Th implmnttion o th trnsitiv losur lorithms in rltionl nvironmnt hs riv xtnsiv ttntion, inluin prormn n th pttion o th tritionl lorithms [, 4, 5, 6, 19, 6, 43]. Th mtho propos in this ppr n hrtriz s prtil mtriliztion mtho. Givn no, w wnt to omput ll its snnts iintly s on spiliz t strutur. Th ollowin is typil strutur to ommot prt-suprt rltionships [17]: - Prt(Prt-i,...) - Conntion(Prnt-i, Chil-i,...) whr Prnt-i n Chil-i r oth orin kys, rrrin to Prt-i. In orr to sp up th rursion vlution, w ll ssoit h no with pir o intrs, whih hlps to roniz nstor-snnt rltionships. In th rst o th ppr, th ollowin thr typs o irphs will isuss. (i) (ii) v 3 v 5 () v 1 v v4 Tr hirrhy, in whih th prnt-hil rltionship is o on-to-mny typ, i.., h no hs t most on prnt. Dirt yli rph (DAG), whih ours whn th rltionship is o mny-to-mny typ, with th rstrition tht prt nnot su/suprprt o itsl (irtly or in- v 3 v 1 v 5 () Fi. 1. A DAG n its trnsitiv losur v v4 irtly). (iii) Dirt yli rph, whih ontins yls. Ltr w ll us th trm rph to rr to th irt rph, sin w o not isuss non-irt ons t ll. 3. TREE LABELING AND RECURSION COMPUTATION In this stion, w isuss how to ll tr to sp up th omputtion o rursion in rltionl nvironmnt. Consir tr T. By trvrsin T in prorr, h no v will otin numr (it n n intr or rl numr) pr(v) to ror th orr in whih th nos o th tr r visit. In similr wy, y trvrsin T in postorr, h no v will t nothr numr post(v). Ths two numrs n us to hrtriz th nstor-snnt rltionships s ollows. Proposition 1. Lt v n v two nos o tr T. Thn, v is snnt o v i pr(v ) > pr(v) n post(v ) < post(v). Proo. S Exris.3.-0 in [30]. I v is snnt o v, thn w know tht pr(v ) > pr(v) orin to th prorr srh. Now w ssum tht post(v ) > post(v). Thn, orin to th postorr srh, ithr v is in som sutr on th riht si o v, or v is in th sutr root t v, whih ontrits th t tht v is snnt o v. Thror, post(v ) must lss thn post(v). Th ollowin xmpl hlps or illustrtion. Exmpl 1. S th pirs ssoit with th nos o th rph shown in Fi.. Th irst lmnt o h pir is th prorr numr o th orrsponin no n th son is its postorr numr. With suh lls, th nstor-snnt rltionships n sily hk. For instn, y hkin th ll ssoit with inst th ll or, w s tht is n nstor o in trms o Proposition 1. Not tht s ll is (, 4 ) n s ll is (5, ), n w hv < 5 n 4 >. W lso s tht sin th pirs ssoit with n o not stisy th onition ivn in Proposition 1, must not n nstor o n vi vrs. (3, 1) Dinition 1. (ll pir susumption) Lt (p, q) n (p, q ) two pirs ssoit with nos u n v. W sy tht (p, q) is susum y (p, q ), not (p, q) (p, q ), i p > p n q < q. Thn, u is snnt o v i (p, q) is susum y (p, q ). To th st o our knowl, this ni proprty hs nvr n utiliz in th ts rsrh or in th sutypin tsts to roniz th nstor-snnt rltionships. Althouh in th rltiv numrin [40] th postorr numrs r us, it works in quit irnt wy. With this strty, h no v is ll with two numrs: th lrst n th smllst postorr numrs in th sutr root t v. It is mor omplit llin pross thn th tr llin shown ov. Th rn-omprssion [] is nrliztion o th rltiv numrin to DAGs. Its sp r (, 4) (7, 6) (6, 5) h (4, 3) (5, ) (1, 7) Fi.. Llin tr SYSTEMICS, CYBERNETICS AND INFORMATICS VOLUME 4 - NUMBER 5 ISSN:

3 quirmnt is on O(n ) n it is irnt rom our nrl strty to isuss in th nxt stion. In [49], to sp up vlution o ontinmnt quris, no v (whih rprsnts n lmnt in oumnt tr) is ssoit with tripl (in, n, lvl), whr in is in t prorr numr, n is th lrst prorr numr in th sutr root t v, n lvl is th pth, t whih v is lot. Thror, it is lso quit irnt rom our mtho. Aorin to th tr llin isuss ov, th rltionl shm to hnl rursion n onsists o only on rltion o th ollowin orm: No(No_i, ll_pir,...), whr ll_pir is us to ommot th prorr n th postorr numrs o th nos o rph, not ll_pir.prorr n ll_pir.postorr, rsptivly. Thn, to rtriv th snnts o no x, w issu two quris s low. Q 1 : SELECT ll_pir FROM No WHERE No_i = x Lt th ll pir otin y vlutin Q 1 y. Thn, th son qury is o th ollowin orm: Q : SELECT * FROM No WHERE ll_pir.prorr > y.prorr n ll_pir.postorr < y.postorr From this, w n s tht with th tr llin, th rursion w.r.t. tr n hnl onvnintly in rltionl nvironmnt. 4. GENERALIZATION Now w isuss how to roniz th nstor-snnt rltionships w.r.t. nrl strutur: DAG or rph ontinin yls. First, w rss th prolm o DAGs in 4.1. Thn, yli rphs will isuss in is vot to th omputtion o trnsitiv losurs. 4.1 Rursion w.r.t. DAGs Wht w wnt is to pply th thniqu isuss ov to DAG. To this n, w stlish rnhin o th DAG s ollows. Dinition. (rnhin [T4]) A surph B = (V, E ) o irph G = (V, E) is ll rnhin i it is yl-r n inr(v) 1 or vry v V. Clrly, i or only on no r, inr (r) = 0, n or ll th rst o th nos, v, inr (v) = 1, thn th rnhin is irt tr with root r. Normlly, rnhin is st o irt trs. Now, w ssin vry sm ost (.., lt ost () = 1 or vry ). W will in rnhin or whih th sum o th osts, E For xmpl, th trs shown in Fi. 3() r mximl rnh ( ), is mximum. in o th rph shown in Fi. 3() i h hs sm ost. Assum tht th mximl rnhin or G = (V, E) is st o trs T i with root r i (i = 1,..., m). W introu virtul root r or th rnhin n n r r i or h T i, otinin tr G r, ll th rprsnttion o G. For instn, th tr shown in Fi. 3() is th rprsnttion o th rph shown in Fi. 3(). Usin Trjn s lorithm or inin optimum rnhins [4], w n lwys in mximl rnhin or irt rph in O( E ) tim i th ost or vry is qul to h othr. Thror, th rprsnttiv tr or DAG n onstrut in linr tim. By trvrsin G r in prorr, h no v will otin numr pr(v); n y trvrsin G r in postorr, h no v will t nothr numr post(v). Ths two numrs n us to roniz th nstor-snnt rltionships o ll G r s nos s isuss in Stion 3. () () () Fi. 3. A DAG n its rnhin In G r (or som G), no v n onsir s rprsnttion o th sutr root t v, not T su (v); n th pir (pr, post) ssoit with v n onsir s pointr to v, n thus to T su (v). (In prti, w n ssoit pointr with suh pir to point to th orrsponin no in G r.) In th ollowin, wht w wnt is to onstrut pir squn: (pr 1, post 1 ),..., (pr k, post k ) or h no v in G, rprsntin th union o th sutrs (in G r ) root rsptivly t (pr j, post j ) (j = 1,..., k), whih ontins ll th snnts o v. In this wy, th sp ovrh or storin th snnts o no is rmtilly ru. Ltr w will shown tht pir squn ontins t most O() pirs, whr is th rth o G. (Th rth o irph is in to th lst numr o th isjoint pths tht ovr ll th nos o th rph.) Exmpl. Th rprsnttiv tr G r o th DAG G shown in Fi. 3() n ll s shown in Fi. 4(). Thn, h o th nrt pirs n onsir s rprsnttion o som sutr in G r. For instn, pir (3, 3) rprsnts th sutr root t in Fi. 4(). (1, 8) (, 5) (7, 7) (3, 3) (6, 4) (8, 6) (4, 1) (5, ), 5 4, 1 5, 6, 4 7, 7 3, 3 4, 1 5, 6, 4 8, 6 4, 1 5, 4, 1 5, 6, 4 () T su () T su () T su () T su () T su () T su () T su () T su () T su () T su () T su () T su () () Fi. 4. Tr llin n illustrtion or trnsitiv losur rprsnttion r T su () T su () T su () ISSN: SYSTEMICS, CYBERNETICS AND INFORMATICS VOLUME 4 - NUMBER 5 3

4 I w n onstrut, or h no v, pir squn s shown in Fi. 4(), whr it is stor s link list, th snnts o th nos n rprsnt in n onomil wy. Lt L = (pr 1, post 1 ),..., (pr k, post k ) pir squn n h (pr i, post i ) is pir llin v i (i = 1,..., k). Thn, L orrspons to th union o th sutrs T su (v 1 ),..., n T su (v k ). For xmpl, th pir squn (4, 1)(5, )(6, 4)(8, 6) ssoit with in Fi. 4() rprsnts union o 4 sutrs: T su (), T su (), T su () n T su (), whih ontins ll th snnts o in G. Th qustion is how to onstrut suh pir squn or h no v so tht it orrspons to union o som sutrs in G r, whih ontins ll th snnts o v in G. First, w noti tht y llin G r, h no in G = (V, E) will initilly ssoit with pir s illustrt in Fi. 5. Tht is, i no v is ll with (pr, post) in G r, it will initilly ll with th sm pir (pr, post) in G. To omput th pir squn or h no, w sort th nos o G topoloilly, i.., (v i, v j ) Ε implis tht v j pprs or v i in th squn o th nos. Th pirs to nrt or no v r simply stor in link list A v. Initilly, h A v ontins only on pir prou y llin G r. W sn th topoloil squn o th nos rom th innin to th n n t h stp w o th ollowin: Lt v th no in onsir. Lt v 1,..., v k th hilrn o v. Mr A v with h A vl or th hil no v l (l = 1,..., k) s ollows. Assum A v = p 1 p... p n = q 1 q... q h, s shown in Fi. 6. Assum tht A vl oth A v n Avl r inrsinly orr. (W sy pir p is lrr thn nothr pir p, not p > p i p.pr > p.pr n p.post > p.post.) A v : W stp throuh oth A v n (, 5) (7, 7) (3, 3) (6, 4) (8, 6) (4, 1) (5, ) Fi. 5. Grph llin p 1 p p A v i : q 1 q q h Fi. 6. link lists ssoit with nos in G A vl rom lt to riht. Lt p i n q j th pirs nountr. W ll mk th ollowin hkins. (1) I p i.pr > q j.pr n p i.post > q j.post, insrt q j into A v tr p i-1 n or p i n mov to q j+1. () I p i.pr > q j.pr n p i.post < q j.post, rmov p i rom A v n mov to p i+1. (*p i is susum y q j.*) (3) I p i.pr < q j.pr n p i.post > q j.post, inor q j n mov to q j+1. (*q j is susum y p i ; ut it shoul not rmov A vi rom.*) (4) I p i.pr < q j.pr n p i.post < q j.post, inor p i n mov to p i+1. (5) I p i = p j n q i = q j, inor oth (p i, q i ) n (p j, q j ), n mov to (p i+1, q i+1 ) n (p j+1, q j+1 ), rsptivly. In trms o th ov isussion, w hv th ollowin lorithm to mr two pir squns tothr. Alorithm pir-squn-mr(a 1, A ) Input: A 1 n A - two link lists ssoit with v 1 n v. Output: A - moii A 1, otin y mrin A into A 1, ontinin ll th pirs in A 1 n A with ll th susum pirs rmov. in 1 p irst-lmnt(a 1 ); q irst-lmnt(a ); 3 whil p nil o{ 4 whil q nil o{ 5 i (p.pr > q.pr p.post > q.post) thn 6 {insrt q into A 1 or p; 7 q nxt(q);} (*nxt(q) rprsnts th pir nxt to q in A.*) 8 ls i (p.pr > q.pr p.post < q.post) thn 9 {p p; (*p is susum y q; rmov p rom A 1.*) 10 rmov p rom A 1 ; 11 p nxt(p );} (*nxt(p ) rprsnts th pir nxt to p in A 1.*) 1 ls i (p.pr < q.pr p.post > q.post) thn 13 {q nxt(q);} (*q is susum y p; mov to th nxt lmnt o q.*) 14 ls i (p.pr < q.pr p.post < q.post) thn 15 {p nxt(p);} 16 ls i (p.pr = q.pr p.post = q.post) 17 thn {p nxt(p); q nxt(q);} 18 i p = nil q nil thn {tth th rst o A to th n o A 1 ;} n Th ollowin xmpl hlps or illustrtion. Exmpl 3. Assum tht A 1 = (7, 7)(11, 8) n A = (4, 3)(8, 5)(10, 11). Thn, A = pir-squn-mr(a 1, A ) = (4, 3)(7, 7)(10, 11). Fi. 7 shows th ntir mrin pross. A 1 : A : p (7, 7)(11, 8) (4, 3)(8, 5)(10, 11) q p (4, 3)(7, 7)(11, 8) (4, 3)(8, 5)(10, 11) q (4, 3)(7, 7)(11, 8) (4, 3)(8, 5)(10, 11) In h stp, th A 1 -pir point y p n th A -pir point y q p q p = nil (4, 3)(7, 7)(10, 11) (4, 3)(8, 5)(10, 11) p (4, 3)(7, 7)(11, 8) (4, 3)(8, 5)(10, 11) () () () () q () Fi. 7. An ntir mrin pross A q 4 SYSTEMICS, CYBERNETICS AND INFORMATICS VOLUME 4 - NUMBER 5 ISSN:

5 r ompr. In th irst stp, (7, 7) in A 1 will hk inst (4, 3) in A (s Fi. 7()). Sin (4, 3) is smllr thn (7, 7), it will insrt into A 1 or (7, 7) (s Fi. 7()). In th son stp, (7, 7) in A 1 will hk inst (8, 5) in A. Sin (8, 5) is susum y (7, 7), w mov to (10, 11) in A (s Fi. 7()). In th thir stp, (7, 7) is smllr thn (10, 11) n w mov to (11, 8) in A 1 (s Fi. 7()). In th ourth stp, (11, 8) in A 1 is hk inst (10, 11) in A. Sin (11, 8) is susum y (10, 11), it will rmov rom A 1 n p oms nil (s Fi. 7()). In this s, (10, 11) will tth to A 1 (s lin 18 o Alorithm pir-squn-mr( )), ormin th rsult A = (4, 3)(7, 7)(10, 11) (s Fi. 7()). Fi. 8 is pitoril illustrtion o th rsult o mrin A into A 1. A 1 : A : (4, 3) (7, 7) (8, 5) (10, 11) (11, 8) rnhin Fi. 8. Illustrtion o mrin two pir squns From this, w n s tht A 1 rprsnts th union o two sutrs rsptivly root t (7, 7) n (10, 11) whil A rprsnts th union o thr sutrs root t (4, 3), (8, 3) n (10, 11), rsptivly. Th rsult A ot y mrin A into A 1 is th union o thr sutrs root t (4, 3), (7, 7) n (10, 14), rsptivly (s th sh rrows in Fi. 7). In th ollowin, w stlish svrl propositions to lriy th proprtis o th ov lorithm. Proposition. Lt A 1 n A two pir squns sort in inrsin orr. Lt A th rsult otin y mrin A into A 1 usin Alorithm pir-squn-mr( ). Thn, A is lso sort inrsinly. Proo. Durin th xution o th lorithm, som pirs my rmov rom A 1 n som pir o A my insrt into A 1. Oviously, rmovin pir rom A 1 will not hn th orrin o A 1. Lt q pir o A insrt into A 1. It my on t lin 6 or t lin 18. I it is on t lin 6, thr must pir p in A 1 suh tht p > q. Consir th pir p or p. W hv p < q; othrwis, q will insrt or p or will not insrt into A 1 t ll. In this s, th proposition hols. I q is insrt into A 1 y xutin lin 18, ll th pirs in A 1 must us up or th lin 18 is rri out. W noti tht t this momnt, ll th pirs in A 1 r inrsinly orr n smllr thn ll th rminin pirs in A, whih r oriinlly in inrsin orr. Thror, in this s, th proposition hols, too. Proposition 3. Lt A 1 n A two pir squns sort in inrsin orr. Lt A th rsult otin y mrin A into A 1 usin Alorithm pir-squn-mr( ). I v is no in sutr o G r, whih is root t som no ll with pir in A, thn thr must pir in A suh tht th sutr root t it A: ontins v. Proo. Assum tht v is in sutr root t u ll with (pr, post) tht pprs in A. I (pr, post) pprs in A, th proposition hols. Suppos tht (pr, post) os not ppr in A. In this s, thr must pir (pr, post ) in A 1, whih susums (pr, post). Noti tht (pr, post ) nnot susum y ny pir in A sin it susums (pr, post). Othrwis, w will hv pir (pr, post ) in A suh tht pr < pr n post > post. But w hv (pr, post) < (pr, post ) or (pr, post) > (pr, post ). In th ormr s, w hv pr < pr < pr. It ontrits th t tht (pr, post ) susums (pr, post). In th lttr s, w hv post > post > post. It lso ontrits th t tht (pr, post ) susums (pr, post). Thror, (pr, post ) will ppr in A. Sin v is in th sutr root t (pr, post), it must in th sutr root t (pr, post ). Thus, th proposition hols. Proposition 4. Lt A 1 n A two pir squns sort in inrsin orr. Lt A th rsult otin y mrin A into A 1 usin Alorithm pir-squn-mr( ). I v is no in sutr o G r, whih is root t som no ll with pir in A 1, thn thr must pir in A suh tht th sutr root t it ontins v. Proo. Similr to Proposition 3. Proposition 5. Th tim omplxity o Alorithm pir-squnmr is oun y O( A 1 + A ). Proo. Durin th xution o th lorithm, h pir in A 1 n A is visit t most on. Bs on th mrin oprtion isuss ov, th pir squns or ll th nos in DAG n omput s ollows. Alorithm ll-squn-nrtion in 1 Lt v n, v n-1,..., v 1 th topoloil squn o th nos o G; or i rom n ownto 1 o 3 {lt,..., th hil nos o v i ; 4 or j rom 1 to k o 5 ll pir-squn-mr(a i, ); 6 } n v i1 v ik Proposition 6. Th sp omplxity o Alorithm ll-squnnrtion is oun y O(n ), whr is th rth o G n n is th numr o th nos o G. Proo. In th lorithm, h no v is ssoit with link list A v. W lim tht th siz o A v is oun y. Assum tht A v ontins + 1 pirs tht r irnt rom h othr. Thn, thr must xist two pirs p n q so tht p susums q or vi vrs. Thror, on o thm will rmov. Thus, th sp n or Alorithm ll-squn-nrtion is oun y O(n ). Proposition 7. Th tim omplxity o Alorithm ll-squnnrtion is oun y O( ), whr is th rth o G n is th numr o th s o G. A ij ISSN: SYSTEMICS, CYBERNETICS AND INFORMATICS VOLUME 4 - NUMBER 5 5

6 Proo. In th out or-loop o th lorithm, n stps r prorm. In h stp, i mr oprtions r m or h v i, whr i rprsnts th outr o v i. Thror, th tim spnt or h stp is O( i ). Th whol tim omplxity is thus O( ) = O( ). Sin th omposition n th llin o DAG n only O() tim n O() sp, th whol tim omplxity o our lorithm is oun y O( ) orin to Proposition 7, n th sp omplxity is oun y O(n ) orin to Proposition 6. This omputtionl omplxity is suprior to ny xistin ons. W ompr our lorithm with th rlvnt work in Stion 6. Proposition 8. Lt v no in G. Any snnt u o v must in sutr o G r root t no ll with pir in A v onstrut y Alorithm ll-squn-nrtion. proo. Assum tht v n v n-1... v 1 is topoloil squn o G. W prov th proposition y inution on th orinl numr l in th topoloil squn. Bsis. Whn l = 1, v n is l no in G n its link list ontins only on ll ssoit with v n. Th proposition hols. Hypothsis. Suppos tht whn m k th proposition hols. Tht is, h link list A i ssoit with v i (i = n,..., n - k) ontins ll th pirs ovrin ll th snnts o v i. Consir l = k + 1. Aorin to th proprty o th topoloil squn, ll th hil nos o v n-k-1 must ppr in {v n, v n-1,..., v n-k }. Thn, rom lins 3-6 o Alorithm ll-squn-nrtion, s wll s Proposition, 3 n 4, w n s tht th link list A n-k-1 ssoit with v n-k-1 must ontin ll th pirs ovrin ll th snnts o v n-k-1. It omplts th proo. Exmpl 4. A possil topoloil squn or th rph shown in Fi. 3() is:. Th pirs ssoit with thm r: (4, 1), (5, ), (6, 4), (3, 3), (8, 6), (, 5) n (7, 7), rsptivly. Thy r otin y llin th tr shown in Fi. 4(). Applyin Alorithm ll-squn-nrtion to this squn, w will prou link list or h o thm s shown in Fi. 9. topoloil orr 4, 1 5, 4, 1 5, 6, 4 3, 3 4, 1 5, 6, 4 8, 6, 5 4, 1 5, 6, 4 7, 7 W n stor physilly th ll pir or h no, s wll s its ll pir squn prou usin Alorithm ll-squn-nrtion. Conrtly, th rltionl shm to hnl rursion w.r.t. DAG n stlish in th ollowin orm: No(No_i, ll, ll_squn,...), Fi. 9. link lists rprsntin pir squns whr ll n ll_squn r us to ommot th ll pirs n th ll pir squns ssoit with th nos o rph, rsptivly. Thn, to rtriv th snnts o no i x, w issu two quris. Th irst qury is similr to Q 1 : Q 3 : SELECT ll_squn FROM No WHERE No_i = x. Lt th ll squn otin y vlutin Q 3 y. Thn, th son qury will o th ollowin orm: Q 4 : SELECT * FROM No WHERE φ(ll, y), whr φ(p, s) is ooln untion with th input: p n s, whr p is pir n s pir squn. I thr xists pir p in s suh tht p p (i.., p.pr > p.pr n p.post < p.post), thn φ(p, s) rturns tru; othrwis ls. W not tht h xution o φ untion ns only O(lo s ) tim. This n on s ollows. Assum s = p 1 p... A v 1 p, whih is stor in two rrys n : on or ll th p i prs n th othr or ll th p i posts. To hk whthr p is susum y p i, w mk inry srh o to in th lrst i suh tht p i pr < p pr. Thn, w srh rom th 1st to th ith lmnt to in whthr thr xists j suh tht p j post < p post. I it is th s, p is susum y som pir in s; othrwis not. Oviously, th tim omplxity o this pross is O(lo s ). Sin s is oun y th rth o th rph, this mtho rquirs only O(n lo ) tim to in ll th snnts o ivn no. 4. Rursion w.r.t. yli rphs Bs on th mtho isuss in th prvious sustion, w n sily vlop n lorithm to omput rursion or yli rphs. First, w us Trjn s lorithm or intiyin stronly onnt omponnts (SCCs) to in th yls o yli rph [41] (whih ns only O(n + ) tim). Thn, w tk h SCC s sinl no (i.., onns h SCC to no) n trnsorm yli rph into DAG. Nxt, w hnl th DAG s isuss in 4.1. In this wy, howvr, ll nos in n SCC will ssin th sm pir (n th sm pir squn). For this rson, th mtho or omputin th rursion t som no x shoul slihtly hn. In th ollowin, Q 5 is th sm s Q 3, ut Q 6 is irnt rom Q 4. Q 5 : SELECT ll_squn FROM No WHERE No_i = x Lt th ll squn otin y vlutin Q 5 y. Thn, th son qury will o th ollowin orm: Q 6 : SELECT * FROM No WHERE γ(ll, y), whr γ(t, s) is ooln untion with th input: t n s, whr t is pir n s pir squn. I thr xists pir t in s suh tht t.pr t.pr n t.post t.post, thn γ(t, s) rturns tru; othrwis ls. W not tht γ( ) is slihtly irnt rom φ( ) in in 4.1. A v 1 A v A v 6 SYSTEMICS, CYBERNETICS AND INFORMATICS VOLUME 4 - NUMBER 5 ISSN:

7 In Q 6, ny two nos in th sm SCC r onsir to th snnts o h othr. Finlly, w point out tht th tim omplxity o omputin rursion w.r.t. yli rph is still O( ) sin Trjn s lorithm runs in O(n + ) tim [41]. 4.3 Aout omputtion o trnsitiv losurs Th lorithm isuss in 4.1 is in ssn nw strty with rprsnttion o trnsitiv losurs quit irnt rom th tritionl mthos. As mntion in 4.1, h pir in pir squn ssoit with no in G n onsir s pointr to sutr in G r. (In prti, h pir n ssoit with physil pointr s illustrt in Fi. 10().) Thn, th union o ll th sutrs point to y th pirs in pir squn ssoit with no v ontins ll th snnts o v. 4, 1 5,. 4, 1 5, 6, 4 3, 3 4, 1 5, 6, 4 8, 6, 5 4, 1 5, 6, 4 7, 7 () For n intuitiv xplntion, s th pir (, 5) ssoit with in Fi. 10(). It is pointr to th no in th rnhin, initin tht ll th nos in th sutr root t (in th rnhin) r th snnts o in th oriinl rph. As nothr xmpl, s th pir squn or tht ontins 4 pirs, rprsntin 4 pointrs s shown in th iur. From th oriinl rph shown in Fi. 3(), w s tht th union o th sutrs point to y ths 4 pointrs ontins rlly ll th snnts o. In this wy, w n only O(n ) sp to stor trnsitiv losur, whr is th rth o th rph. As isuss in 4.1, th tim spnt to nrt th pir squns or ll th nos in rph is O( ). In tritionl mtho, howvr, th trnsitiv losur o rph is rprsnt y st o lists n h list ontins ll th snnts o som no (s Fi. 10()) or y n n ooln mtrix Thror, th sp rquirmnt or storin trnsitiv losur is O(n ). 5. INCREMENTAL CHANGES TO GRAPHS In ynmi nvironmnt, th t in ts n rquntly hn. With stti noin, th romputtion o lls is n, whih is normlly vry tim onsumin. Howvr, in our mtho, in no to DAG ns only O() tim or lultin its ll. Ain n is mor iiult sin it my us O(n ) s in th trnsitiv losur o rph (s [7]). In our mtho, it ns O() tim in th st s n O(n ) tim in th worst s. But or n onsutiv insrtions o s, it tks only O(n ) tim. Thror, th vr tim or insrtin n is O(). In th ollowin, w isuss th no insrtion r : : : [] [] [,, ] : [,, ] : [,,, ] : : [,,,, ] [,,,, ] () Fi. 10. link lists rprsntin pir squns n th insrtion in 5.1 n 5., rsptivly. 5.1 No insrtion W noti tht ll n rl numr, not lwys n intr. Btwn two onsutiv intrs, w hv ininitly mny rl numrs. Thn, whn nw no is insrt into rph, w n ssin rl numr to no s its prorr or postorr numr, whih will not twist th mhnism isuss in th prin stion. Howvr, ininitly mny numrs r not implmntl. Dpnin on irnt pplitions, w st irnt numrs o iits to th riht o th iml point or th possil insrtion. It n onsir s n xtnsion to th i isuss in [], whih lvs ps twn h pir o sussiv postnumrs or insrtin nos. In th ollowin, w irst isuss th no insrtion into tr. Thn, th no insrtion into DAG is rily outlin. - No insrtion into tr First, w onsir two ss o no insrtion into tr. 1. A nw no v is insrt into tr T s irt riht silin o som no u.. A nw no v is insrt into tr T s prnt o som no u. In ths two ss, th pir (pr, post) ssoit with v is lult s ollows. Lt th pir ssoit with u (p, q). Lt th pir ssoit with th no s prin u (orin to th prorr numrin) (p s, q s ). Lt th pir ssoit with th no t nxt to s (orin to th postorr numrin) (p t, q t ). W hv p p s q t q pr = p s , n post = q t Oviously, i h no kps pointr to its prssor (orin to th prorr numrin) n pointr to its sussor (orin to th postorr numrin), this oprtion ns only onstnt tim. Now w onsir nothr two ss tht r ul to s 1 n, rsptivly. 3. A nw no v is insrt into tr T s irt lt silin o som no u. 4. A nw no v is insrt into tr T s hil o som no u n th prnt o on o u s hilrn. Lt th pir ssoit with u (p, q). Lt th pir ssoit with th no s prin u (orin to th postorr numrin) (p s, q s ). Lt th pir ssoit with th no t nxt to s (orin to th prorr numrin) (p t, q t ). W hv p t p q q s pr = p , n post = q No insrtion into DAG To lult th pir n th pir squn or nw no insrt into DAG, w hv to trmin whr to insrt th no in th orrsponin rnhin. This n on y hkin th prnt o th nw no, whih is n sy tsk sin th inormtion on its prnt must spii. Two ss will onsir: 1. Th no is insrt s hil o som no in th rnhin.. Th no is insrt twn n (u, v) (i.., it is insrt s hil o u n th prnt o v.) ISSN: SYSTEMICS, CYBERNETICS AND INFORMATICS VOLUME 4 - NUMBER 5 7

8 For oth ss, w omput th pir or th nw no usin th mtho isuss ov; ut yt w n to mr th pir squn o its hil with its pir or s. Th tim omplxity is O(). 5. E insrtion Now w onsir th insrtion o n into DAG G. Lt = (u, v). Lt A u n A v th ll squns or u n v, rsptivly. Th ollowin lorithm hns th lls o G whn is insrt into it. Alorithm insrtin-on-(g, ) in ll pir-squn-mr(a v, A u );(*pir-squn-mr(a v, A u ) ws isuss in Stion 3.*) i A v is not hn thn rturn; ls {Lt v n, v n-1,..., v 1 th topoloil squn o th nos o G; Assum v = v l or som l; or i rom l ownto 1 o(*hn th lls o th nos lon th topoloil orr.*) {lt,..., th hil nos o v i ; v i1 v ik A ij or j rom 1 to k o * i is not hn thn ll pir-squnmr(a i, Aij ); ** i thr r onsutiv ll squns not hn (* rprsnts th rth o G.*) thn rturn; } n This lorithm is similr to th Alorithm ll-squn-nrtion. Th min irn is th lin mrk with *, whr whthr A ij is hn is hk or pir-squnmr(a i, A ij ) is invok. Anothr irn is th lin mrk with **, whr w hk whthr thr r onsutiv ll squns not hn ( rprsnts th rth o G). I it is th s, th lorithm trmints. Rll tht th tim omplxity o Alorithm pir-squn-mr( ) is O(). Thror, w n only O() tim or insrtin n in th s tht A v is not hn tr A u is mr into it. In th worst s, w will xut th lorithm pir-squn-mr( ) l tims (i v is th lth no in th topoloil squn o G). For h o ths nos, w will mr ll thos hn pir squns o its hilrn with its pir squn. (S th lin mrk * in th ov lorithm.) Sin t most n pir squns my hn, th tim omplxity is oun y O(n ). W noti tht insrtin m (m > 1) s onsutivly into DAG G os not rquir mor tim thn insrtin only on. This is us tr hnin th ll squns or ll th nos involv, w n to sn th topoloil squn o G only on. Thror, th tim omplxity o insrtin m s n into G is oun y O(m ). min{ mn, } In prtiulr, i m = n, th tim omplxity is O(n ). 6. RELEVANT WORK AND COMPARISON In th pst svrl s, th trnsitiv losur o rph hs xtnsivly rsrh n lot o irnt strtis hv n propos, suh s th rph-s lorithms [0, 1, 34, 36, 38], th rph noin [1,, 7, 14, 16, 7, 43, 48] n th mtrixs mthos [4, 3, 44, 45]. In th ollowin, w minly isuss th rph noin n th mtrix-s mthos sin thy uniormly outprorm th rph-s mthos. - Grph noin Prhps th most lnt lorithm or noin is rltiv numrin [40] (lso ll Shurt s numrin), y mns o whih h no v is ssoit with n intrvl [q v, p v ] to ilitt th hkin o nstor-snnt rltionships, whr p v is th postorr numr o v n q v is th lst postorr numr o th sutr root t v. This mtho ws nrliz to DAGs y Arwl, Bori n Jish [] into wht is ll rn-omprssion. In this mtho, h no v is ssoit with postorr numr p v n two rrys A v n B v suh tht i thr xists nothr no u stisyin th ollowin onition or som i: A u [i] < p v < B u [i], v is snnt o u. Sin th siz o A v (or B v ) is on th orr O(n), th sp ovrh o this noin is O(n ). To nrt suh rrys, ll th s hv to ss. Thus, th tim ovrh o this mtho is O(n ). In ition, y lvin ps twn h pir o sussiv postorr numrs, insrtin no n on iintly. Howvr, or nw no, its rrys hs to rt n thror, t lst, O(n) tim is n. To insrt n is mor iiult sin th t struturs or ll th snnts hv to romput, whih ns in th worst s O(n ) tim. Finlly, O(n) tim is n or pth hkin sin oth th rrys ssoit with no will snn. Bomml n Bk [7] improvs th mtho propos in [] only y sttin suitl ps. Thror, thir mtho hs th sm omputtionl omplxitis s []. In [7], Jish isuss irnt rph noin, y mns o whih, or ny pir o nos u n v with lls (i, j) n (k, j), i i < k, thr xists irt pth rom u to v. Sin or ll th nos on h pth (ll hin in [7]) suh lls r stlish, th sp ovrh is O( n), whr is th numr o pths ovrin ll th nos o rph. Sin th pths my not isjoin (i.., no my ppr on mor thn on pth), th numr o th lls ssoit with no is on th orr O(n) in th worst s. Th tim omplxity o this mtho is O(n ) n it ns O( n) tim to insrt n. Finlly, insrtin no ns t lst O(n) tim sin or h pth whr th no pprs th lls hv to rt, n th lls or ll th snnts s wll s ll th nstors hv to hn. In th mtho propos y Aim [1], no v is ssoit with two intr squns o th sm lnth: p = p 1,..., p k n s = s 1,..., s k, whr k is th numr o th isjoin pths ovrin th rph. Eh p i in p is th numr o th nos on pth P i, whih r th prssors o v whil h s j in s is qul to P j - su j (v) + 1, whr P j is th numr o nos on P j n su j (v) is th numr o th sussors o v on P j. With suh squns ssoit with th nos, th pth hkin n on in O(k) tim. Howvr, this mtho mintins th trnsitiv losur o 8 SYSTEMICS, CYBERNETICS AND INFORMATICS VOLUME 4 - NUMBER 5 ISSN:

9 rph in O(k + n min{n, }) tim n O(k n) sp; n ns O(n ) tim to moiy rlvnt su j (v) s whn insrtin n. In [43], Tuhol isuss n intrstin it-squn noin. In this mtho, h no v is ssoit with n intrvl (l, h), whr l n h r two sinturs h onsistin o it strin. Ths it strins r onstrut in suh wy tht i th intrvl ssoit with snnt o v is (l, h ), thn l l n h h hol. In th s o DAGs, th sp n tim ovrh o this mtho r oun rsptivly y O( n) n O( ), whr is th lnth o sintur. Howvr, th lnth o sintur is snsitiv to th hiht o rph (whih is in to th lonst pth in rph) n th outrs o th nos. Thror, in th worst s, insrtin no or n ns to hn ll th sinturs o th rph, lin to n O( ) tim omplxity. Oviously, th pth hkin only ns O() tim. Th min prolm o this mtho is tht th rph omposition is not lrly in, whih is n to ll rph usin sinturs. Bsis th ov mthos, svrl othr noins hv n vlop in th ontxt o prormmin lnus to sp up typ-sutyp hkins, suh s Cohn s noin [16], Pk- Enoin (lso ll PE-Enoin) [14] n PQ-Enoin [48]. Cohn s noin n us only or tr struturs; n PE- Enoin is it-vtor s strty. Th lnth o it-vtor is on O(n), whr n is th numr o th typs (sutyps) in hirrhy. Th PE-Enoin ws propos y Ziin n Gil in 001 [48], whih ssoits h no v with n rry o intrs A v n n intrvl [l v, h v ] suh tht or ny nstor u o v w hv l u < A v [i] < h u or som i. Th lnth o th rry is shortn y omposin rph into slis in suh wy tht thr is only n ntry in th rry or h sli. This noin is s on th onpt o PQ-trs, t strutur propos in [9], whih is us to tst or th onsutiv 1 s proprty in inry mtris. Howvr, sin th siz o sli is oun y onstnt, th lnth o th rry is on O(n/), whr rprsnts vr numr o nos in sli. Thror, th sp ovrh o th PQ- Enoin is on O(n /). Th tim ovrh shoul O(n /). Insrtin n n insrtin no r iiult sin in th worst s, th slis hv to rin, lin to tim omplxity o O(n /). In trms o th nlysis o [48], pth hkin ns O(lon) tim. From th isussion in Stion 3, w n s tht our lorithm is nw rph noin irnt rom ny on outlin ov n ns only O( ) tim n O(n ) sp to omput th trnsitiv losur o rph. Th tim or Insrtin no is oun y O(). In ition, w n only O(lo) tim or pth hkin n O(m ) tim or insrtin m s sus- n min{ mn, } sivly. Tl 1 shows th omputtionl omplxitis o ll th min rph noin lorithms. Tl 1. Comprison o omputtion omplxitis. - Mtrix-s mthos In th omputin rph thory, th trnsitiv losur hs n lon n intrstin topi. Mny mthos hv n propos s on th mtrix rprsnttion, suh s th mthos isuss in [4, 3, 44, 43]. In suh mthos, trnsitiv losur is mintin s mtrix M with M[i, j] = 1 initin tht thr xists pth rom no i to no j in G. Amon ll ths lorithms, th most intrstin is Wrrn s [45]. It improvs th tim omplxity o Wshll s [45] rom O(n 3 ) to O(n ). Sin th trnsitiv losur o rph is stor in mtrix, th tim or pth hkin is oun y onstnt. Howvr, th sp ovrh is O(n ). In ition, vn pplyin th wll-known st-union lorithm, insrtin m s sussivly ns O(mα(m +, n) + + n) tim, whr α(x, y) is vry slowly rowin untion. Ain no is lso iiult sin n n mtrix will hn to (n + 1) (n + 1) on. Thror, it ns t lst O(n) tim. Arwl Bori Bomml Ziin Jish Bk Jish Am Tuhol Gil our rph llin tim O(n ) O(n ) O(n ) O(k + n min{n, }) O( ) O(n /) O( ) sp pth hk O(n ) O(n) O(n ) O(n) O( n) O(n) O(k n) O(k) O( n) O() O(n /) O(lon) O( n) O(lo) insrt n O(n ) O(n ) O( n) O(n ) O( ) O(n /) O((n/min{m, n}) ) insrt no O(n) O(n) O(n) O(n) O( ) O(n /) O() Th mtho propos y Irki n Ktoh [4] mintins th trnsitiv losur o rph in th sm omputtion omplxity s Wrrn s. Howvr, it ns only O(n 3 ) tim to insrt q s sussivly y usin spil it oprtion. Whn th orr o m is lrr, O(n 3 ) is ttr thn O(mα(m +, n) + + n). In Itlino s mtho [5], n n n mtrix o pointrs is us. Eh o thm points to spnnin tr. Thror, mor tim n sp thn Wrrn s r n. Th ol o Itlino s ws to t ttr mortiz tim omplxity or two oprtions: insrtin n n hkin pth (i suh pth xists, rturn it). Usin Itlino s t strutur, h o th ov two oprtions n on in O(n) mortiz tim. Th mtho isuss in [3] is mor omplit strty. Inst o omputin th trnsitiv losur o G = (V, E), this mtho omputs th trnsitiv losur o G s rution G - = (V, E - ), whr E - is in s ollows: E - = E / {(u, v) (u, v) (u, v) E n thr is pth P rom u to v with P }. Oviously, E - E. Thror, th omputtion o th trnsitiv losur o G - ns lss tim thn G. But th tim omplxity is still on th orr o O(n ). In ition, urin th omputtion, two mtris r us n thus mor sp thn Wrrn s is n. Th tim or insrtin m s sussivly is O(m n). Tl shows th omputtionl omplxitis o ll th min mtrix-s lorithms. Tl. Comprison o omputtion omplxitis Irki n L Poutr n Wrrn Ktoh Itlino Luwn tim sp pth hk O(n ) O(n ) O(1) O(n ) O(n ) O(1) O(n ) O(n ) O(n) O(n ) O(n ) O(1) insrt m s O(mα(m +, n) + + n) O(n 3 ) O(m n) O(m n) insrt no O(n) O(n) O(n) O(n) ISSN: SYSTEMICS, CYBERNETICS AND INFORMATICS VOLUME 4 - NUMBER 5 9

10 7. EXPERIMENT RESULTS W hv implmnt tst in C++, with our own ur mnmnt (with irst-in-irst-out rplmnt poliy) n B+tr strutur. Th omputr ws Intl Pntium III, runnin stnlon. W hv tst two mthos: th mtho s on sinturs (Tuhol s), n th mtho s on rph llin (th on isuss in this ppr). W hos Tuhol s mtho or tstin sin it hs st sp omplxity ovr th othrs. W us two strutur typs: Trs n DAGs, n msur th physil I/O quot s wll s th pu tim. W i not hk yli irphs sin in [43] no orml mtho ws sust to hnl this s usin sinturs. In t, vn or DAGs, it ws not xpliitly isuss in [43] how to ompos irph into st o spnnin trs (s in in tht rtil). W o or th spii t sttin sri in [43]; ut it is not nrl strty. Alon with [43], w hv tst th ollowin thr ss: (1) orst o 18 trs with thr hilrn pr nonl; iht lvls, nos, n 590 onntions; () orst o 18 trs with our hilrn pr nonl; iht lvls, nos, n onntions; (3) DAG o 640 roots with thr hilrn pr nonl; two prnts pr nonroot, iht lvls, 3155 nos n onntions. For th two mthos tst, th t ils r sin littl it irntly s shown in Fi. 11. In oth th t ils, no is rprsnt y no intiir tht is in t n intr rprsnt s it squn. Th il or tstin Tuhol s mtho ontins, or h no, 3 or 64 its or its sintur plus 4 its or th lvl, t whih th no pprs. In this il, only th low vlu ( sintur) o th intrvl ssoit with no is stor; n th hih vlu o th intrvl n lult usin th low vlu n th orrsponin lvl numr. Th il or tstin th rph-llin mtho ontins prorr numr n postorr numr or h no. Eh o thm is 18 or its lon. For th tsts, th ur is o 50 ps n h p is o siz 4096 yts. il strutur us or tstin Tuhol s mtho: 4 its 8 its n its 4 its no intiir sintur lvl il strutur us 4 its 8 its m its m its or tstin rph-lllin mtho: no intiir pr-numr post-numr or s (1) n (3): n = 3, m = 18 or s (): n = 64, m = Fi. 11. Fil struturs Fi. 1 shows th tst rsults o s (1). For this tst, h o th sinturs or Tuhol s is 3 its lon whil h o th prorr (postorr) numrs or th rph-llin tks 18 its. From this, w n s tht in th s o trs, i th sinturs o Tuhol s mtho hv th sm lnth s th lls o th rphllin mtho, thy hv lmost th sm prormn. Howvr, i th sinturs om lonr, th prormn o Tuhol s rs. S Fi. 13 or illustrtion. This iur shows th tst rsults o s (). For this tst, h sintur is 64 its lon whil h prorr (postorr) numr tks its. Espilly, thr is shrp inrs o p sss o Tuhol s mtho rom lvl 6 onwrs. It is us in ths ss, th umult impt o lon sinturs oms vint. p ss p ss Th tst rsults o s (3) r thr in Tl 1. In this tst, ll th snnts o root (3041 nos on vr) r rtriv. strutur Tuhol s rph-llin lvl Tuhol s rph-llin Tl 3: Tst rsults o DAGs no ss tim (s.) Fi. 1. Tst rsults o s (1) Tuhol s p pu tim sss (s.) rph-llin p pu tim sss (s.) DAG From Tl 3, w n s tht in th s o DGAs, Tuhol s mtho is muh wors thn ours. It is us DAG hs to ompos into sris o spnnin trs to us Tuhol s mtho. For h ompos tr, th sinturs ssoit with th nos n us to hk snnts; ut th hil sintur o h ss onntion must hk. I it is outsi th [Low, Hih] rn o th sinturs o th urrnt tr, thn th hil lons to nothr tr, n nw qury is issu inst it. This ls to lot o xtr p sss. Howvr, or th rphllin mtho, th rph omposition is utiliz only or th nrtion o pir squns, no xtr p ss is us. But w noti tht in th DAG s, lthouh th numr o p ss o this mtho is not muh lrr thn th s o trs, th tim irn o ths two ss is rltivly i. It is us or DAG h no is ssoit with squn o ll pirs n h hk o ll pir squns ns mor tim. Fi. 14 shows th tst rsults or th trs with irnt outrs. Th orst ontins 18 trs h o 380 nos. W hn th outr o th nos or h run n just th sinturs so tht h tim th sinturs hv irnt lnth. In ontrst, th ll lnth o our lorithm rmins unhn. This rrnmnt is rsonl sin th lnth o th nos sinturs t lvl (in tr) pns on th lrst outr t this lvl tim (s.) Tuhol s rph-llin lvl lvl lvl Fi. 13. Tst rsults o s () 10 Tuhol s rph-llin 10 SYSTEMICS, CYBERNETICS AND INFORMATICS VOLUME 4 - NUMBER 5 ISSN:

11 I thr is no t lvl hs lr outr, th sinturs or ll th nos t tht lvl must st vry lon orin to th sintur onstrution propos in [43]. Howvr, th lnth o lls us in ours pns only on th lls vlus. Th lrst ll is qul to th numr o th nos o rph. Finlly, w mphsiz tht th mi mrit o th propos mtho os not only onsists in spin-up th rursion in rltionl nvironmnt, ut lso in th nw rprsnttion o rursiv losurs, whih ls to ttr omputtionl omplxity thn ny xistin strtis or this prolm. tim (s.) Tuhol s rph-llin outr Fi. 14. Tst rsults 8. CONCLUSION In this ppr, nw llin thniqu hs n propos. Usin this thniqu, th rursion w.r.t. tr hirrhy n omput without join oprtions. Th propos mtho n xtn to omput th rursion w.r.t. DAGs n rphs ontinin yls. Givn DAG, w n lwys in mximl rnhin s its rprsnttion, whih is in t tr (or orst) n thus n lwys ll. To in th rursiv losur o th DAG, w onstrut squn o ll pirs or h no, whih n on in topoloil orr o th DAG in O( ) tim n O(n ) sp, whr n rprsnts th numr o th nos o th DAG, th numrs o th s, n th DAG s rth. To omput th rursion w.r.t. yli rph, w irst us Trjn s lorithm [41] to in ll its SCCs; n thn onns h SCC onto sinl no, rsultin in DAG. Sin Trjn s lorithm ns only O(n + ) tim, th tim omplxity or inin th squns o ll pirs or ll nos o yli rph is thortilly oun y O( ). Finlly, w noti tht th squns o ll pirs r in t nw rprsnttion o th trnsitiv losur o rph with muh lss sp ovrh thn ny xistin strty. Thror, th mtho isuss in this ppr n onsir s nw mtho or omputin trnsitiv losurs with optiml tim n sp omplxitis. REFERENCES [1] S. Aim, On Inrmntl Computtion o Trnsitiv Closur n Gry Alinmnt, in: Pro. 8th Symp. Comintoril Pttrn Mthin,. Alrto Apostolio n Jotun Hin, 1997, pp [] R. Arwl, A. Bori n H.V. Jish, Eiint mnmnt o trnstiv rltionships in lr t n knowl ss, Pro. o th 1989 ACM SIGMOD Intl. Con. on Mnmnt o Dt, Oron, 1989, pp [3] S. Aitoul, S. Clut, V. Christophis, T. Milo, G. Morkott, J. Simon, Qurin oumnts in ojt tss, Int. J. Diitl Lirris, Vol. 1, No. 1, April 1997, pp [4] R. Arwl, S. Dr, H.V. Jish, Dirt trnsitiv losur lorithms: Dsin n prormn vlution, ACM Trns. Dts Syst. 15, 3 (Spt. 1990), pp [5] R. Arwl n H.V. Jish, Mtriliztion n Inrmntl Upt o Pth Inormtion, in: Pro. 5th Int. Con. Dt Eninrin, Los Anls, 1989, pp [6] R. Arwl n H.V. Jish, Hyri trnsitiv losur lorithms, In Pro. o th 16th Int. VLDB Con., Brisn, Austrli, Au. 1990, pp [7] M.F. vn Bomml n T.J. Bk, Inrmntl Enoin o Multipl Inhritn Hirrhis Supportin Ltti Oprtions, Linkopin Eltroni Artils in Computr n Inormtion Sin, [8] J. Bnrj, W. Kim, S. Kim n J.F. Grz, Clustrin DAG or CAD Dtss, IEEE Trns. on Knowl n Dt Eninrin, Vol. 14, No. 11, Nov. 1988, pp [9] K.S. Booth n G.S. Lukr, Tstin or th onsutiv ons proprty, intrvl rphs, n rph plnity usin PQ-tr lorithms, J. Comput. Sys. Si., 13(3): , D [10] F. Bnihon n R. Rmkrishnn, An Amturs Introution to Rursiv Qury Prossin Strtis, in: Pro. ACM SIGMOD Con., Wshinton D.C., 1986, pp [11] M. Cry t l., An Inrmntl Join Atthmnt or Strurst, in: Pro. 16th VLDB Con., Brisn, Austrli, 1990, pp [1] Y. Chn, K. Arr, Lyr Inx Struturs in Doumnt Dts Systms, Pro. 7th Int. Conrn on Inormtion n Knowl Mnmnt (CIKM), Bths, MD, USA: ACM, 1998, pp [13] Y. Chn n K. Arr, Cominin Pt-Trs n Sintur Fils or Qury Evlution in Doumnt Dtss, in: Pro. o 10th Int. DEXA Con. on Dts n Exprt Systms Applition, Florn, Itly: Sprinr Vrl, Spt pp [14] L. Crlli, J. Donhu, M. Jorn, B. Klsow, n G. Nlson, Th Moul-3 typ systm, Pro. o 16th Simposium on Prinipls o Prormmin Lnus, POPL 89, ACM SIGPLAN, Austin, Txs, Jn. 1989, pp [15] Y. Chn, On th Grph Trvrsl n Linr Binry-hin Prorms, IEEE Trnstions on Knowl n Dt Eninrin, Vol. 15, No. 3, My 003, pp [16] N.H. Cohn, Typ-xtnsion tsts n prorm in onstnt tim, ACM Trnstions on Prormmin Lnus n Systms, 13:66-69, [17] R.G.G. Cttll n J. Skn, Ojt Oprtions Bnhmrk, ACM Trns. Dts Systms, Vol. 17, no. 1, pp. 1-31, 199. [18] P. Dm t l., A DBMS Prototyp to Support Extn NF Rltions: An Intrt Viw on Flt Tls n Hirrhis, Pro. ACM SIGMOD Con., Wshinton D.C., 1986, pp [19] S. Dr n R. Rmrkrishnn, A Prormn Stuy o Trnsitiv Closur Alorithm, in Pro. o SIGMOD Int. Con., Minnpolis, Minnsot, USA, 1994, pp [0] J. Dzikiwiz, An Alorithm or Finin th Trnsitiv Closur o Dirph, Computin 15, 75-79, ISSN: SYSTEMICS, CYBERNETICS AND INFORMATICS VOLUME 4 - NUMBER 5 11

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