Revisiting Decomposition Analysis of Geometric Constraint Graphs

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1 Rvisitin Domposition Anlysis o Gomtri Constrint Grps R. Jon-Arinyo A. Soto-Rir S. Vil-Mrt J. Vilpln-Pstó Univrsitt Politèni Ctluny Dprtmnt Llnuts i Sistms Inormàtis Av. Dionl 647, 8, E Brlon [rort, tonis, ss, josp]@lsi.up.s ABSTRACT Gomtri prolms in y onstrints n rprsnt y omtri onstrint rps wos nos r omtri lmnts n wos rs rprsnt omtri onstrints. Rution n omposition r tniqus ommonly us to nlyz omtri onstrint rps in omtri onstrint solvin. In tis ppr w irst introu t onpt o iit o onstrint rp. Tn w iv nw ormliztion o t omposition loritm u to Own. Tis nw ormliztion is s on prsrvin t iit rtr tn on omputin trionnt omponnts o t rp n is simplr. Finlly w pply tr ompositions to prov tt t lss o prolms solv y t ormliztions stui r n otr ormliztions rport in t litrtur is t sm. Ctoris n Sujt Dsriptors I.3.5 [Computr Grpis]: Computtionl Gomtry n Ojt Molin Gomtri Aloritms, Lnus, n Systms; F.2.2 [Anlysis o Aloritms n Prolm Complxity]: Nonnumril Aloritms n Prolms Gomtril Prolms n Computtions Gnrl Trms Tory, loritms Kywors Constrint solvin, omtri onstrints, rp-s onstrint solvin 1. INTRODUCTION Gomtri prolms in y onstrints n rprsnt y omtri onstrint rps wos nos r Prmission to mk iitl or r opis o ll or prt o tis work or prsonl or lssroom us is rnt witout provi tt opis r not m or istriut or proit or ommril vnt n tt opis r tis noti n t ull ittion on t irst p. To opy otrwis, to rpulis, to post on srvrs or to ristriut to lists, rquirs prior spii prmission n/or. SM 02, Jun 17-21, 2002, Srrükn, Grmny. Copyrit 2002 ACM /02/ $5.00. omtri lmnts n wos rs rprsnt omtri onstrints. For pplition wit potntilly lr onstrints systms, t iiny o t loritms or solvin t systm t n is n importnt issu. To i on t suitility o ivn onstrint solvin mto, its orrtnss must prov n t lss o prolms t mto n solv soul rtriz. Mny ttmpts to provi nrl, powrul n iint mtos or solvin systms o omtri onstrints v n rport in t litrtur. For n xtnsiv rviw in omtri onstrint solvin rr to Fuos [6] n Durn [3]. Amon t xistin mtos w ous on two tniqus ommonly us to nlyz omtri onstrint rps in omtri onstrint solvin, nrilly known s omposition n rution, rsptivly. Mor spiilly w r intrst in omposition n rution wr t nlysis is s on irt omtri intrprttion. Tr r otr ppros. S or xmpl Homnn t l. [7] or low-s omposition loritm. In [15], Own sri top-own loritm or omputin omposition o n ritrry rp. T loritm rursivly splits t rp into split omponnts, [8]. T loritm trmints wn t rps nnot split urtr. At t n o t nlysis t oriinl rp s n ompos into st o trinls. Fuos n Homnn, [6], rport on rp-onstrutiv ppro to solvin systms o omtri onstrints. T mto is s on n nlysis o t onstrint rp tt rivs squn o onstrution stps tt squntilly pls t omtri lmnts in t prolm wit rspt to otr. T nlysis s two prts. T irst prt is ottom-up rution nlysis wr stp in t squn orrspons to positionin tr rii omtri ois tt pirwis sr omtri lmnt, point or lin. T son prt is top-own omposition nlysis tt prous squn o ompositions tt orrspon to rvrs squn o rii omtri ois. In tis ppr w rormult t loritm rport y Own in [15] to solvin omtri onstrint prolms s on t omposition nlysis o t onstrint rp. First w introu t onpt o iit ssoit wit onstrint rp. T iit msurs t istn twn ivn onstrint rp n wll-onstrin rp inu y t sm st o nos. Tis onpt llows us to voi

2 t n or omputin trionnt omponnts yilin onptully simplr loritm. Tn w rll t tr omposition o onstrint rp, tool tt is usul to onptully nlyz onstrint rps. Finlly tr ompositions r ppli to rtriz t lss o prolms solv y t omposition nlysis stui r n to prov tt irnt ormliztions solv t sm lss o prolms. Stion 2 rviws si onpts rom rp tory n omtri onstrint rps. Stion 3 ls wit omposition nlysis. First w rll Own s loritm, tn w prsnt t nw ormliztion o t loritm. Stion 4 prsnts t tr omposition o onstrint rp. Stion 5 is vot to rtriz t lss o prolms solv y t omposition nlysis stui r n isusss t quivln o irnt ormliztions. W los wit ri summry in Stion PRELIMINARIES In tis stion w rll si trminoloy o rp tory, t onpt o omtri onstrint rp ssoit to omtri prolm in y onstrints, n som initions rlt to omtri onstrint rps. 2.1 Grp Conpts First w rll som si trminoloy o rp tory tt will us in t rst o t ppr. For n xtnsiv trtmnt s [2] n [8]. A rp G = (V, E) is si to onnt i vry vrtx is onnt to vry otr vrtx y t lst on pt o s. W sy tt vrtx o onnt rp G is n sprtion vrtx (rtiultion vrtx) i y rmovin, t rp splits into two or mor isonnt surps. I is n rtiultion no in G, tn tr r two vrtis u n v irnt rom su tt is on vry pt onntin u n v. A rp wit no rtiultion vrtis is ll ionnt. I u n v r ritrry irnt vrtis o ionnt rp G, tn tr r t lst two irnt pts in G onntin tm. A onnt rp n uniquly ompos into ionnt omponnts y splittin it t sprtion vrtis. Ao t l., [1], rport pt irst loritm tt iintly omputs su omposition. Lt n two vrtis in ionnt rp G. T s o G n ivi into sprtion lsss E 1, E 2,..., E n in s ollows, [8]. Two s r in t sm sprtion lss E i i tr is pt usin ot s n not ontinin or xpt, possily, s npoints. I t two vrtis n ivi t s into mor tn two sprtion lsss, tn t pir {, } is sprtion pir (rtiultion pir) o G. Morovr, i {, } ivis t s into two sprtion lsss, ontinin mor tn on, tn {, } is lso sprtion pir. A trionnt rp is rp wit mor tt two vrtis wit no sprtion pirs. In trionnt rp tr r t lst tr isjoint pts twn vry pir o non jnt vrtis. Lt {, } sprtion pir in t rp G tt inus t sprtion lsss E 1, E 2,..., E n. Assum 1 m n n lt E = m i=1 Ei n E = n i=m+1 Ei su tt E 2 n E 2. Tn w will rr to t rps G = (V (E ), E ) n G = (V (E ), E ) s t sprtin rps o G. T rps n G 1 = (V (E ), E {(, )}) G 2 = (V (E ), E {(, )}) r ll split rps o G. T (, ) is ll to not t split n is ll virtul. Assum tt t rp G n its split rps r rursivly split until otinin rps tt nnot split urtr. T st o ts rps ins t st o split omponnts o G. Not tt t split omponnts r trionnt rps n tt y mrin t split omponnts w rovr t oriinl rp. Hoprot n Trjn, [8], n Millr n Rmnrn, [14], rport on loritms to iintly omput sprtin rps n split omponnts o rp. 2.2 Gomtri Constrint Solvin n Grps In tis ppr w onsir onstrint prolms in y ivin st o omtri lmnts lik points, lins, lin smnts, irls n irulr rs, lon wit st o rltionsips, ll onstrints, lik istn, nl, inin n tnny twn ny two omtri lmnts. As xplin y Fuos, [4] n y Mt, [13], w my trnsorm t omtri onstrint prolm into on wr only points n lins wit pirwis istn n nl onstrints n to onsir. T omtri onstrint prolm n o s onstrint rp G = (V, E), wr t rp vrtis V r t omtri lmnts wit two rs o rom n t rp s E r t omtri onstrints, nlin on r o rom, [15, 6]. Fiur 1 sows omtri prolm in y onstrints n t omtri onstrint rp ssoit. A nssry onition or omtri onstrint prolm to solvl is tt t ssoit onstrint rp must wll-onstrin. Comintoril proprtis o wllonstrin rps v n rtriz y Lmn, [11]. Tnilly, t notion o wll-onstrin rp n ormliz s ollows, [6]. Dinition 1. Lt G = (V, E) omtri onstrint rp. 1. G is struturlly ovr-onstrin i tr is n inu surp wit m V vrtis n mor tn 2m 3 s. 2. G is struturlly unr-onstrin i it is not struturlly ovr-onstrin n E < 2 V G is struturlly wll-onstrin i it is not struturlly ovr-onstrin n E = 2 V 3. Not tt t ovr-onstrint s nls t sitution wr t sm rp is ovr-onstrin in on prt n unr-onstrint in notr t t sm tim. 3. DECOMPOSITION ANALYSIS First w rily rll t Own s loritm. Tn w iv nw ormliztion or t Own s loritm wi is simplr n tt will us in t ollowin stions.

3 A L AC 2 D C L AB L BC α B un Own(G) SC := SplitComponnts(G) S := or in SC o i Ruil() tn S := S Own(Ru()) ls S := S {} i on rturn S n 1 Fiur 2: Own s nlysis loritm. D L AC A 2 C L AB 1 α L BC Fiur 1: A omtri onstrint prolm n ssoit onstrint rp. 3.1 Own s Aloritm Own in [15] introu omtri onstrint solvin tniqu s on top-own nlysis o t omtri onstrint rp ssoit wit omtri prolm. T loritm s two stps. In irst stp, t loritm omputs t st o split omponnts S o t ivn rp G, [8]. Ts split omponnts r itr trinls or omplx trionnt rps, tt is, rps wit mor tn tr s. As omput, t omplx split omponnts r no urtr omposl. To ovrom tis prolm, in son stp t omplx split omponnts r trnsorm, i possil, y rmovin rom t rp on o t virtul s introu in t irst stp. Not tt virtul s r lwys inint to sprtion pirs. Tn t irst stp is rursivly ppli to t trnsorm split omponnts. T loritm trmints wn t rps nnot split urtr. I ny trionnt rp wit mor tn tr vrtis rmins, t prolm nnot solv qurtilly, tt is, t unrlyin qutions v r ir tn two. At t n o t nlysis, t oriinl rp s n ompos into st o trinls wos s r itr oriinl s or virtul s. I untion SplitComponnts(G) omputs t split omponnts o G, untion Ruil() ks wtr split omponnt soul urtr suivi, n untion Ru() rmovs unn virtul s o rp, Own s loritm n writtn s sown in Fiur 2. T nlysis pross ollow y Own s nlysis loritm is illustrt in Fiur 3. Virtul s r sown in B s lins. 3.2 T Nw Formliztion To ompos rp, Own s mto uss t loritm or inin trionnt omponnts rport y Hoprot n Trjn in [8], wi is s on prsrvin rp onntivity. As rsult, t split omponnts nrt y t omposition inlu xtr virtul s. To rursivly pply t omposition pross, Own s loritm must rmov ts xtr virtul s. In wt ollows w will prsnt n loritm to ompos onstrint rp in trionnt rps wit xtly tr vrtis, tt is, trinls. T loritm is s on ivi n onqur strty wi prsrvs t onstrint rp proprty o in wll-onstrin. T rsultin loritm is onptully simpl n sy to implmnt. As in [15] n [6], t loritm will s on suiviin t onstrint rp into two sprtin rps inu y sprtion pir. Wit t im o lrly sttin suivision ritrion, w strt y ivin som initions n rivin proprtis wi rlt wll-onstrin rps wit tir sprtin rps. Dinition 2. Lt G = (V, E) omtri onstrint rp. W in t Diit untion ssoit wit G y Diit(G) = (2 V 3) E T untion Diit omputs t irn twn t numr o s n or onstrint rp to wllonstrin n its tul numr o s. Not tt i G is not ovr-onstrin, Diit(G) 0 Lmm 1. Lt G onstrint rp n G n G sprtin rps. Tn Diit(G) = Diit(G ) + Diit(G ) 1 Proo. By inition, Diit(G) = (2 V 3) E ). Sin G n G r sprtion rps o G, tn V = V + V 2 n E = E + E. Tror, Diit(G) = 2( V + V 2) 3 ( E + E ) = (2 V 3 E ) + (2 V 3 E ) 1 = Diit(G ) + Diit(G ) 1

4 G SPLIT v 2 v 2 v 1 v 1 REDUCE REDUCE v 1 SPLIT SPLIT v 3 v 5 v 4 v5 v 1 v 4 v 3 Fiur 3: Own s loritm omputtion ppli to n xmpl rp.

5 Lmm 2. Lt G wll-onstrin rp n G n G sprtin rps. Tn i Diit(G ) > Diit(G ), G is unr-onstrin n G is wll-onstrin. Proo. Sin G is wll-onstrin, Diit(G) = 0 n sprtion rps, G n G, r not ovr-onstrin, tt is, Diit(G ) 0 n Diit(G ) 0. From Lmm 1 Diit(G) = Diit(G ) + Diit(G ) 1. Tus Diit(G ) + Diit(G ) = 1. Tn, Diit(G ) = 1 n Diit(G ) = 0, wi mns tt G is unr-onstrin n G wll-onstrin. Dinition 3. Lt G wll-onstrin onstrint rp n G n G sprtin rps. T moii split rps, G 1 n G 2, o G r in s ollows. I Diit(G ) > Diit(G ) tn G 1 = (V (E ), E {(, )}) n G 2 = G Lmm 3. Lt G = (V, E) onstrint rp n, G 1 = (V 1, E 1) n G 2 = (V 2, E 2) moii split rps. Tn Diit(G) = Diit(G 1) + Diit(G 2). Proo. Now E = E 1 + E 2 1. Lmm 1. Apply proo o Dinition 4. Lt G omtri onstrint rp. An s-tr S o G is inry tr o rps su tt: 1. t root is t rp G, 2. or no G in S its sutrs r root in t moii split rps G 1 n G 2 o G, n 3. t lvs r itr trinls or trionnt rps. As w will s in Stion 5 w r intrst in rps or wi tr r s-trs wos l nos r trinls us w know ow to solv t ssoit omtri onstrint prolm, [9, 15]. Dinition 5. W sy tt onstrint rp G is s-tr omposl i tr is n s-tr su tt its root is G n ll its lvs r trinls. Lt Trionnt(G) untion tt tsts wtr rp s sprtion pir, SprtinGrps(G) untion tt omputs t sprtin rps o G, (Rll tt sprtin rps o not inlu virtul s), n AVirtulE(G) untion tt s virtul inint to t sprtion pir us to omput t split rp G. Tn t omposition nlysis loritm s on prsrvin iits o rps n writtn s sown in Fiur 4. T input to t loritm is rp G ssoit to omtri onstrint prolm. T output is s-tr S wos root is G. Not tt i G is s-tr omposl t rsultin s-tr omposs G into trinls n t prolm is solv. Fiur 5 illustrts t viour o t nw omposition nlysis loritm ppli to t xmpl rp in Fiur 3. Not tt now only tos virtul s tt r stritly nssry to kp t iit proprty r inlu in t moii split rps, tror voiin t n or rp trnsormtion. Wn on o t sprtion lsss is sinl, it is inint to t vrtis in t sprtion pir. In tis s w prov t ollowin rsult. un Anlysis(G) i Trionnt(G) tn S := BinryTr(G, nulltr, nulltr) ls G 1,G 2 := SprtinGrps(G) i Diit(G 1) > Diit(G 2) tn G 1 := AVirtulE(G 1) ls G 2 := AVirtulE(G 2) i S := BinryTr(G, Anlysis(G 1), Anlysis(G 2)) i rturn S n Fiur 4: Nw loritm or omposition nlysis. Lmm 4. Lt G = (V, E) wll-onstrin omtri onstrint rp n {, } sprtion pir su tt (, ) E. Tn t sprtion rp wi ontins t (, ) is wll-onstrin. Proo. Lt {, } t sprtion pir in t rp G = (V, E) n E 1,..., E n 1, E n t sprtion lsss, wr E n ontins just t (, ). Lt E = m i=1 Ei n E = n 1 i=m+1 Ei. su tt E 2 n E 2. W v E = E + E + E n = E + E + 1 V = V (E ) + V (E ) 2 G wll-onstrin mns tt 2 V 3 E = 0. Sustitutin E n V y t xprssions ov n rrrnin trms (2 V (E ) 3 E ) + (2 V (E ) 3 E ) 2 = 0 Tr r two irnt ss. First lt (2 V (E ) 3 E ) = (2 V (E ) 3 E ) = 1 Sin lsss E i r roup ritrrily, ssum tt E = E E n. Tn V (E ) = V (E ) n E = E + 1. Tn t sprtion rps r G = (V (E ), E ) n G = (V (E ), E ). Tus t iit o t sprtion rp G = (V (E ), E ) is (2 V (E ) 3 E ) = 2 V (E ) 3 ( E + 1) = 2 V (E ) 3 E 1 = 0 Tror t sprtion rp G ontins (, ), is wll-onstrin n, sin Diit(G ) < Diit(G ), no virtul is to it. T son s ls to ontrition. Witout loss o nrlity, lt (2 V (E ) 3 E ) = 0, tt is, G is wllonstrin. Lt (2 V (E ) 3 E ) = 2. n ssum in E = E E n. Tis woul rsult in t sprtion rp G G in ovr-onstrin, tt is G woul ovr-onstrin wi is ontrition. Lmms 1, 2 n 3 lon wit Lmm 4 prov tt t loritm prsrvs t iit o t input rp.

6 G v 1 v 2 v 3 v 3 v4 v 5 Fiur 5: Domposition nlysis nrt y t nw loritm on t xmpl rp in Fiu 3.

7 v C C 2 Fiur 6: Grp wit r two vrtx v. C C 2 C 1 C3 Fiur 8: Lt: A st C. Rit: A st omposition o C. C 3 C 1 V 2 V3 Fiur 7: Suiviin rp into tr surps y usin tr vrtis, n. 3.3 Suivision Pttrn T loritm ivn in t prvious stion nlyzs onstrint rp y omposin it into two split rps inu y sprtion pir. Howvr, tr is notin ssntil in tis suivision mto. To, [16], rport on mto wr rps r suivi y isoltin vrtis o r two rom tir niors. In t, tis is prtiulr s o omposin trou sprtion pirs. Fiur 6 illustrts rp wit r two vrtx v. Not tt its niors, n, r sprtion pir. Tis suivision mto is rtr limit ut n stistorily omin wit otr mtos, lik tos in [8] or in [14], to omput mor nrl rp suivisions. Anotr mto suivis rp into tr surps y sltin tr vrtis su tt y rmovin tm t rp splits into tr onnt omponnts. S Fiur 7. W o not know ny iint loritm to slt t tr vrtis ut ty n lwys omput y usin rut or ppro. 4. TREE DECOMPOSITION In tis stion irst w in t onpt o st omposition tt rrs to wy o prtitionin ivn strt st. Tn w in t onpt o tr omposition o rp. Tis tool will us in Stion 5 to prov tt svrl onstrutiv omtri onstrint solvin mtos solv t sm lss o prolms. Dinition 6. Lt C st wit, t lst, tr irnt mmrs, sy,,. Lt {C 1, C 2, C 3} tr susts o C. W sy tt {C 1, C 2, C 3} is st omposition o C i 1. C 1 C 2 C 3 = C, 2. C 1 C 2 = {}, Fiur 9: Lt: Grp. Rit: St omposition o t rp. 3. C 1 C 3 = {} n 4. C 2 C 3 = {}. W sy tt {,, } r t tiv lmnts o t st omposition. Fiur 8 sows st n possil st omposition. Nxt w in t onpt o st omposition o rp, illustrt in Fiur 9. Dinition 7. Lt G = (V, E) rp. Lt V () not t vrtis in V tt r t npoints o E. Lt {V 1, V 2, V 3} tr susts o V. Tn {V 1, V 2, V 3} is st omposition o G i it is st omposition o V n or vry in E, V () V i or som i, 1 i 3. Rouly spkin, st omposition o rp G = (V, E), is st omposition o t st o vrtis V su tt os not rk ny in E. Fiur 10 lt sows rp G = (V, E) n Fiur 10 rit sows st omposition o V wi is not st omposition o G us vrtis inint to (, ) os not lon to ny st in t prtition. Lmm 5. Lt {V 1, V 2, V 3} st omposition o rp G n lt V 1 V 2 = {} n V 1 V 3 = {}. I V 1 > 2, tn {, } is sprtion pir o G. Proo. T surps o G inu y V i, or 1 i 3, v isjoint sts o s. By Dinition 6 V 1 (V 2 V 3) = {, }. Tus, rmovin {, } isonnts G. Tror {, } is sprtion pir. V 1

8 V 2 V 3 Fiur 10: Lt: Grp. Rit: St omposition wit rokn. V 1 {, } {, } {,,,,, } {,,, } {,, } {, } {,, } {, } {, } {, } {, } {, } {, } Fiur 12: Tr omposition o t rp in Fiur 11. Fiur 11: Colltion o st ompositions o t rp in Fiur 9. To los tis stion, w in t onpt o tr omposition o rp. Dinition 8. Lt G = (V, E) rp. A 3-ry tr T is tr omposition o G i 1. V is t root o T, 2. E no V V o T is t tr o xtly tr nos, sy {V 1, V 2, V 3 }, wi r st omposition o t surp o G inu y V, n 3. E l no ontins xtly two vrtis o V. A rp or wi tr is tr omposition is tr omposl rp. Fiur 11 sows olltion o st ompositions rursivly nrt or t tr omposl rp o Fiur 9. T orrsponin tr omposition is sown in Fiur 12. By Dinition 8, ll lvs o tr omposition T o rp G v rinlity two. 5. DOMAIN OF CONSTRUCTIVE GE- OMETRIC CONSTRAINT SOLVING TECHNIQUES Jon-Arinyo t l. sow in [10] tt t lss o tr omposl rps rtrizs t omin o two onstrutiv omtri onstrint solvin tniqus: rution n omposition nlysis s sri rsptivly in [6], n [10]. Hr, w will s tt t omposition nlysis stui in Stion 3 n rtriz lso y t xistn o tr omposition n tt it s t sm omin s t rution n omposition nlysis ov mntion. In wt ollows w will only onsir onstrint rps G ssoit wit wll-onstrin prolms. In ts onitions, s-trs r inry trs wos root is G, t otr nos r sprtion rps wit rspt to som sprtion pir o t prnt no n, t lvs r trinls or trionnt rps wit no rtiultion pirs. Aorin to t numr o virtul s in t trinls in t l nos o n s-tr, w lssiy tm in our irnt typs. S t son olumn in Tl Domin Crtriztion To rtriz t omposition nlysis stui in Stion 3 w prov two lmms. Lmm 6. I rp G is tr omposl, tn G is s-tr omposl. Proo. Assum tt T is tr omposition o G. W sll pro y inution on t strutur o T, [12]. Rr to Tl 1. Inution s: Lt G = (V, E) rp su tt V = {,, } n E = {(, ), (, ), (, )}. T tr T in t tir olumn o Tl 1 is tr omposition o G. Tn t tr in t ourt olumn is s-tr wos root is rp G = G 0 wit just on no rprsntin t trinl {,, }. Inution ypotsis: Lt G surp o G. I G is tr omposl tn G is s-tr omposl.

9 n G 0 Tr-omposition T S-tr S {,, } 0 G 0 {, } {, } {, } {,, } C G 1 1 {, } C 1 {, } {, } S 1 G 0 {,, } C G 2 2 S 2 G 1 {, } C 1 {, } C 2 {, } S 1 G 0 G 3 {,, } C S 3 G 2 3 S 2 G 1 {, } C 1 {, } C 2 {, } C 3 S 1 G 0 Tl 1: Typs o intrior nos in tr omposition n t quivlnt s-tr omposition.

10 un FromTrToS-Tr(T ) G 0 := ComputTrinl(T ) S := BinryTr(G 0, NullTr, NullTr) n := NumrOVirtulEs(G 0) or j in 1 to n o T j := Sutr(T, j) S j := FromTrToS-Tr(T j) G j := Root(S j) G j := MrGrps(G j, G j 1) S := BinryTr(G j, S j, S) n rturn S n Fiur 13: Computin s-tr S rom tr omposition T. Inution stp: I {C 1, C 2, C 3} is st omposition o C n {,, } t tiv lmnts, w v tt C = C {,, }, C 1 = C 1 {, }, C 2 = C 2 {, } n C 3 = C 3 {, }. Lt G rp n T tr omposition o G su tt its root is {,, } C n t roots o its sutrs r {, } C 1, {, } C 2 n {, } C 3. Assum tt C 1 n C 2 = C 3 =. Lt G 1 t surp inu y {, } C 1 in G. Lt T 1 t tr omposition o G 1. By Lmm 5, (, ) is sprtion pir o G tus G 1 is sprtion rp o G. Buil t otr sprtion rp G 2 s t rp G 2 = (V 2, E 2) wit V 2 = {,, } n E 2 = {(, ), (, )}. By Dinition 1, G 2 is unronstrin, tus y Lmm 2, G 1 is wll-onstrin. Now uil t moii split rps o G s G 0 = (V 2, E 2 {(, )}) n G 1. Sin G 1 is tr omposl, y t inution ypotsis it is s-tr omposl. Tror tr is s-tr, sy S 1, wos root is G 1. Hn t inry tr wos root is G n wos sutrs r G 0 n S 1 is s-tr. Tror G is s-tr omposl. Applyin t sm prour or ss C 2 n C 3 omplts t proo. I untion ComputTrinl(T ) omputs t trinl ssoit wit no o tr omposition, n untion MrGrps(G 1, G 2) ruils rp rom its moii split rps, Fiur 13 sows n loritm tt, s on Lmm 6, omputs s-tr S rom tr omposition T o rp G. Lmm 7. I rp G is s-tr omposl, tn G is tr omposl. Proo. Assum tt S is s-tr wos root is G. Ain w sll pro y inution on t strutur o S. Rr to Tl 1. Inution s: Lt G = (V, E) rp su tt V = {,, } n E = {(, ), (, ), (, )}. T s-tr S o G is tt ivn in t ourt olumn o Tl 1. Tn t tr ivn in t tir olumn is tr omposition o G. Inution ypotsis: Lt G surp o G. I G is s-tr omposl tn G is tr omposl. Inution stp: Lt S s-tr wos root is G n wos sutrs r G 0 n S 1. Assum tt G 0 = (V 0, E 0) wit V 0 = {,, } n E 0 = {(, ), (, )} {(, )} n lt G 1 = (V 1, E 1) t root o t s-tr S 1. By Dinition 4, G 0 n G 1 r t moii split rps o G wit rspt to t sprtion pir {, }. Sin G 1 is s-tr omposl, y inution ypotsis it is tr omposl. Tror tr is tr omposition T 1 o G 1. Buil tr omposition T su tt its sutrs r T 1, {, } n {, }. S Tl 1. Sutrs {, } n {, } sr t no. Sin {, } is sprtion pir o G, V 0 V 1 = {, }. But V 0, tus / V 1, {, } V 1 = {}, n {, } V 1 = {}. Tror T is tr omposition o G. Applyin t sm prour or ss in rows tr n our in Tl 1 omplts t proo. 5.2 Domin quivln Now, w will s tt t lss o s-tr omposl omtri onstrint rps n t lss o tr omposl rps r t sm. In otr wors, omtri onstrint prolm xprss y mns o omtri onstrint rp is solvl y Own s tniqu i n only i t rp is tr omposl. Sin w prov in [10] tt omtri onstrint rp is solvl y rution nlysis, [5], i n only i t rp is tr omposl, tis implis tt Own s tniqu n rution nlysis v t sm omin n tt its omin n rtriz y t lss o tr omposl rps. Torm 1. Lt G = (V, E) omtri onstrint rp. T ollowin ssrtions r quivlnt: 1. G is tr omposl. 2. G is s-tr omposl. 3. G is solvl y rution nlysis. 4. G is solvl y omposition nlysis. Proo. Jon-Arinyo t l. prov in [10] t quivln o ssrtions 1, 3 n 4. Lmm 6 provs tt 1 implis 2 n Lmm 7 provs tt 2 implis 1. Tis torm sows tt t onstrutiv mtos onsir r v t sm omin, tt is, ty solv t sm lss o prolms. Howvr, o tm xiits irnt proprtis lik iiny n vior wit rspt to unr-onstrin prolms, s [6], [10] n [15]. 6. SUMMARY W v introu t onpt o iit o onstrint rp. Bs on tis onpt, w v prsnt nw ormliztion or omposition nlysis loritm. W v prov its orrtnss. T i o prsrvin just t onstrint rp iit vois t n or nrl loritms to omput trionnt omponnts. Tus t rsultin omposition nlysis loritm is onptully simplr. W v us t tr omposition s nrl tool or omposition nlysis o onstrint rps. Spiilly, w v ppli it to prov tt irnt omposition nlysis ormliztions solv t sm lss o omtri onstrint prolms.

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