A Tractable, Approximate, Combinatorial 3D rigidity characterization

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1 A Trtl, Approximt, Comintoril D riiity rtriztion Mr Sitrm Yon Zou Jun 0, 00 Astrt Tr is no known, trtl, rtriztion o D riiity o sts o points onstrin y pirwis istns or D istn onstrint rps. W iv omintoril pproximt rtriztion o su rps wi w ll moul-riiity, wi n trmin y polynomil tim loritm. W sow tt tis proprty is nturl n roust in orml sns. Riiity implis moul-riiity, n moulriiity siniintly improvs upon t nrliz Lmn r-o-rom or nsity ount. Spiilly, rps ontinin nns or ins [8] r not moul-rii, wil t nrliz Lmn ount woul lim riiity. T loritm tt ollows rom our rtriztion o moul-riiity ivs omplt omposition o non moul-rii rps into its mximl moul-rii surps. To put t rsult in prsptiv, it soul not tt, prior to t rnt loritm o [] tr ws no known polynomil tim loritm or otinin ll mximl surps o n input onstrint rp tt stisy t nrliz Lmn ount, spiilly wn ovronstrints or runnt onstrints r prsnt. T nw mto s n implmnt in t FRONTIER [], [5], [8], [9] opnsour D omtri onstrint solvr n s mny usul proprtis n prtil pplitions [0], [], [], [], []. Spiilly, t mto is us or onstrutin so-ll omposition-romintion (DR) pln or D omtri onstrint systms, wi is ruil to t t xponntil omplxity o solvin t (sprs) polynomil systm otin rom t ntir omtri onstrint systm. T DR-pln uis t lri-numri solvr y nsurin tt only smll susystms r vr solv. T nw, pproximt rtriztion o D riiity prmits FRONTIER to l wit r lrr lss o D onstrint systms ( lss qut or most pplitions) tn ny otr urrnt omtri onstrint solvr. Kywors: Comintoril Riiity, Vritionl omtri onstrint solvin, Cylil n D omtri onstrint systms, Domposition o omtri onstrint systms, Unronstrin n Ovronstrin systms, Dr o From nlysis, Constrint rps. Introution A D istn onstrint rp is wit rp wit vrtis rprsntin point ojts in D n s rprsntin istn onstrints twn t points. T wit o vrtx is, rprsntin its positionl rs o rom (o), n t wit o is, rprsntin t numr o rs o rom t onstrint rmovs. T onstrints n writtn s qurti qutions in vrils rprsntin t oorints o t points For xmpl, istn onstrint o twn two points (x, y, z ) n (x, y, z ) in D is writtn s (x x ) + (y y ) + (z z ) =. T rsultin D istn onstrint systm is si to nrilly rii, i it s t most initly mny inonrunt solutions (i.., its solution st, t orrsponin lri vrity, is init moulo rottions n trnsltions) in t nri s (i.., wn nri, lrilly inpnnt st o vlus is osn or t istn onstrints). Tus t proprty o nri riiity o istn onstrint systm - in inpnnt o t tul istn vlus - is in t proprty o t unrlyin istn onstrint rp lon. (W ll t orrsponin onstrint rp rii). On woul xpt purly omintoril rtriztion (n orrsponin loritm) or trminin riiity o istn onstrint rps. Wil Lmn s torm [9] ivs su rtriztion or D istn onstrint rps, no su rtriztion s n provn or D, ltou svrl onjturs Univrsity o Flori; Work support in prt y NSF Grnt CCR , NSF Grnt EIA orrsponin utor: sitrm@is.ul.u

2 xist [8]. A (rl) solution or min or rliztion o istn onstrint systm is somtims ll rmwork. Tr is rtriztion o riiity o istn onstrint rp usin so ll ininitsiml riiity o rmworks n t ssoit riiity mtrois [8]. Tis rtriztion ssrts ull nri rnk o so-ll riiity mtrix (its ntris r vtors trmin y t oorint positions o t onstrin pirs o points, xprss s intrmints). Howvr, tis rtriztion os not yil polynomil tim loritm or trminin riiity o istn onstrint rp. In t, non o t omintoril rtriztion onjturs pprs to trnslt to polynomil tim loritm. Hr, w opt irnt tk. W iv omintoril pproximt rtriztion o D riiity, wi w ll moul-riiity, wi n trmin y polynomil tim loritm. W sow tt tis proprty is nturl n roust in orml sns. Riiity implis moul-riiity, n moul-riiity siniintly improvs upon t nrliz Lmn r-o-rom or nsity ount. Spiilly, rps ontinin nns or ins [8] r not moul-rii, wil t nrliz Lmn ount woul lim riiity. Mor prisly: rii moul-rii nrliz Lmn or o rii (ontins nns n ins) T loritm tt ollows rom our rtriztion o moul-riiity s numr o usul proprtis. Mny o ts r s on t t tt t loritm ivs omplt omposition o non moul-rii rps into its mximl moul-rii surps. T nw mto s n implmnt in t FRONTIER [], [5], [8], [9] opnsour D omtri onstrint solvr (00 n 00 vrsions) n s mny prtil pplitions: omtri onstrint systms r us s suint, miniml, itl rprsnttions o omtri omposits in mny ontxts inluin mnil omputr i sin, rootis, molulr molin n tin omtry For rnt rviws o t xtnsiv litrtur on omtri onstrint solvin s,., [, 8, 6]. Most o t omtri onstrint solvrs so r l wit D onstrint systms, ltou som o t nwr ppros inluin [,, 6, 7] [5,, ] [0,,, 5], xtn to D onstrint systms. Ts onstrint solvrs v n ompr wit rspt to vrious orml prormn msurs in [6]. T nw, pproximt rtriztion o D riiity prmits FRONTIER to l wit r lrr lss o D onstrint systms ( lss qut or most pplitions) tn ny otr urrnt onstrint solvr. Most omtri onstrint solvrs rly on rursivly omposin t input onstrint systm into smll, nrilly rii susystms prior to solvin. Ts susystms r to n o-t-sl lri-numri solvr, n t solutions r romin to nrt solution to t ntir onstrint systm. Tis ompositionromintion (DR) pln (in in Stion.) is ruil to t t xponntil omplxity o solvin t (sprs) polynomil systm otin rom t ntir omtri onstrint systm, y uiin t lrinumri solvr tivly: only smll susystms r vr solv. It is lso ruil tt t DR-pln nrt iintly, rtinly in polynomil tim. As rsult, DR-plnnrs r usully rp loritms tt omintorilly trmin riiity o t susystms in t DR-pln. For t DR-pln to tivly ui t solvr, t susystms in t DR-pln n to rii so tt tir solutions n romin. Howvr, i t DR-plnnr soul rr ossionlly y lsly limin riiity, tis rror will tt urin t tul solvin pross. In Stion. w iv t si kroun on omtri onstrint rps, nri riiity, nrliz Lmn or o nlysis, DR-plns n tir si proprtis. T rtriztion o moul-riiity (n t orrsponin loritm) prsnt in Stion uil upon n ltrnt rtriztion (n orrsponin Frontir vrtx loritm) o [] o nrliz Lmn or o riiity or D omtri onstrint rps tt is s on DR-plns. Tis rtriztion s mny usul proprtis [], [], [], [0], [], [] wi r inrit y t nw moul-riiity rtriztion, spiilly us it ivs polynomil tim loritm or trminin omplt omposition into mximl o rii surps, i t rp is not o rii. Most importntly, t loritm works in t prsn o runnt or ovronstrints. Erlir rp loritms su s [9] or trminin o riiity rly on t rmovl o ovronstrints. Wil [0] provi mto or rmovin ovronstrints witout mkin t wol rp o unronstrin, tis oul mk o rii surps unronstrin. Hn mtos su s [9] o not provi omplt ompositions.. Constrint Grps, Drs o From, DR-Plns Gomtri onstrint rps r nrliztion o t istn onstrint rps to wi t nw rsults o tis ppr r rstrit. Howvr, t onpts in tis n t nxt stions pply to nrl onstrint rps

3 s wll. A omtri onstrint rp G = (V, E, w) is wit rp wit n vrtis (rprsntin omtri ojts) V n m s (rprsntin onstrints) E; w(v) is t wit o vrtx v n w() is t wit o, orrsponin to t numr o rs o rom vill to n ojt rprsnt y v n numr o rs o rom (os) rmov y onstrint rprsnt y rsptivly. Svrl D istn onstrint rps wos vrtis o wit (rprsntin points) n s o wit (rprsntin istn ) n oun in Fiurs,,,, 0. A surp A G tt stisis w(v) () A w() + D v A is ll ns, wr D is imnsion-pnnt onstnt, to sri low. Funtion (A) = A w() v A w(v) is ll nsity o rp A. Its mnitu is lso ll t nrliz Lmn or o ount, sin it is nturl nrliztion o Lmn s torm [9] tt ivs omintoril rtriztion o riiity or D istn onstrint rps. T onstnt D is typilly ( ) + wr is t imnsion. T onstnt D pturs t rs o rom o rii oy in imnsions. For D ontxts n Eulin omtry, w xpt D = n or D ontxts D = 6, in nrl. I w xpt t rii oy to ix wit rspt to lol oorint systm, tn D = 0. Nxt, w iv som purly omintoril proprtis o onstrint rps s on nsity. Ts will ltr sown to rlt to proprtis o t orrsponin onstrint systms. A ns rp wit nsity stritly rtr tn D is ll ovronstrin. A rp tt is ns n ll o wos surps (inluin itsl) v nsity t most D is ll o wllonstrin. A rp G is ll o wll-ovronstrin i it stisis t ollowin: G is ns, G s tlst on o ovronstrin surp, n s t proprty tt on rplin ll o ovronstrin surps y o wllonstrin surps (in ny mnnr), G rmins ns. A rp tt is o wllonstrin or o wll-ovronstrin is si to o rii or o lustr. A ns rp is miniml i it s no ns propr surp. Not tt ll miniml ns surps r o lustrs ut t onvrs is not t s. A rp tt is not o lustr is si to o unronstrin. I ns rp is not miniml, it oul in t n o unronstrin rp: t nsity o t rp oul t rsult o min surp o nsity rtr tn D. A trivil rp is ny rp tt rus (y rsolvin inins) to sinl point in D or to ix or vril lnt lin smnt in D. All ts trivil rps v rottionl symmtris n r o rii; t ormr two ss r truly rii n r o ovronstrin. To isuss ow t rp torti proprtis s on r o rom (o) nlysis sri ov rlt to orrsponin proprtis o t orrsponin onstrint systm, w n to introu t notion o nriity. Formlly w us t notion o nriity o., []. A proprty is si to ol nrilly or polynomils,..., n i tr is nonzro polynomil P in t oiints o t i su tt tis proprty ols or ll,..., n or wi P os not vnis. Tus t onstrint systm E is nrilly rii i tr is nonzro polynomil P in t oiints o t qutions o E - or t prmtrs o t onstrint systm - su tt E s t most initly mny zros moulo rottions n trnsltions, wn P os not vnis. For xmpl, i E onsists o istn onstrints, t prmtrs r t istns. Evn i E s no ovrt prmtrs, i., i E is m up o onstrints su s inins or tnnis or prpniulrity or prlllism, E in t s in prmtrs pturin t xtnt o inin, tnny, t., wi w onsir to t prmtrs o E. (Gnrilly ovronstrin systms v no zros wn P os not vnis n nrilly unronstrin systms v ininitly mny zros, i.., non-zro-imnsionl lri vrity; ot nrilly wllonstrin n nrilly ovronstrin systms r si to nrilly rii; t lttr r somtims rr to s runntly rii).. Inquy o nrliz Lmn or o nlysis A nrilly rii systm lwys ivs o lustr, ut t onvrs is not lwys t s. In t, tr r o wll-onstrin lustrs wos orrsponin systms r not nrilly rii n r in t nrilly not rii. T root us o ts mislssiitions is t prsn o in pnnt onstrints tt nnot oun y o ount.

4 Fiur : D onstrint systm rwn on D nvs ilustrtin t nns prolm; orrsponin onstrint rps v vrtis o wit n s o wit ; s txt or xplntion Consir or xmpl t Fiurs,,, rlt to t so-ll nns prolm in D, wi is typ o onstrint pnn ourrin in lr lss o D istn onstrint rps, ltou tis ttion is nontrivil. A o nlysis o t D onstrint systm in Fiur (top) woul rport t lt n rit susystms (P, P, P, P, P 5 n P, P 6, P 7, P 8, P 5 rsptivly) n t wol systm to o wllonstrin lustrs. Fiur (ottom) s t sm numr o onstrints, ut o nlysis woul rport ot tt t lt surp s o ovronstrin n t wol s o unronstrin. Fiur (lt) lso s t sm numr o onstrints n is o rii lustr. Howvr, wil t lt n rit susystms r (in t) nrilly rii, t wol systm is nrilly ovronstrin. In wll-in sns, tis us y onstrint pnn. On t otr n, wn rstrit to onsistntly ovronstrin situtions (tos ois o istns - su s in tis xmpl - tt r urnt to mit solution), t systm in Fiur (lt) is nrilly unronstrin, ltou t systm on Fiur (rit) is nrilly wllonstrin. In t, onstrint systm is nrilly ovronstin i t ommon ovrlp o ny sust o its o wllonstrin lustrs is o unronstrin. T ov nns is spil s o tis. Howvr, t o nlysis is inurt only in t nns sitution. Anotr stnr xmpl, in imnsions t rp K 7,6 rprsntin istns is miniml ns, n n o rii lustr, ut it os not rprsnt nrilly rii systm. Howvr, s mntion rlir, in imnsions, orin to Lmn s torm [9] i ll omtri ojts r points n ll onstrints r istn onstrints twn ts points tn ny miniml ns o rii lustr rprsnts nrilly rii systm. In t, tr is no known, trtl rtriztion o nri riiity o istn systms or or ir imnsions, s purly proprtis o t onstrint rp. In t vn in D, wil Lmn s torm [9] omintorilly rtrizs nri riiity o point n istn systms, tr r no known omintoril rtriztions o riiity, wn otr onstrints sis istns r involv. For xmpl, in t s o nl onstrints in D: lin smnts wit inin onstrints orm trinl wit *-* = 6 (rsp. *6-* = 9) rs o rom. It woul ppr tt to mk it wllonstrin, w n introu nl onstrints ( o wi rmovs o). But in t, tis woul mk it nrilly ovronstrin.. T n or omposition: o DR-plns n tir proprtis T ovrwlmin ost o solvin omtri onstrint systm is t siz o t lrst susystm tt is solv usin irt lri/numri solvr. Tis siz itts t prtil utility o t ovrll onstrint solvr, sin t tim omplxity o t onstrint solvr is t lst xponntil in t siz o t lrst su

5 Fiur : Moiitions to D systm in Fiur : o wllonstrin (DR-pln s sinl sour, top) n unronstrin ( DR-pln s mny sours, ottom); s txt or xplntion susystm. T DR-plnnr is rp loritm tt outputs omposition-romintion pln (DR-pln) o t onstrint rp. In t pross o omintorilly onstrutin t DR-pln in ottom up mnnr, t st i, it lots o rii surp S i in t urrnt onstrint rp G i, n uss n strt simpliition o S i to to rt trnsorm onstrint rp G i+. Domposition loritms s on onstrint rps v n propos sin t rly 90 s s on ronition o surp pttrns su s trinls [7, 5, 6, ] [0, ]; n s on Mximum Mtin [7, ]. Howvr, prior to [6], t DR-plnnin prolm n pproprit prormn msurs or t plnnrs wr not ormlly in. Tt ppr lso ivs tl omprin min typs o DR-plnnrs, wit rspt to ts prormn msurs. A susqunt ppr [7] prsnts t rmwork o DR-plnnr s on nrliz o nlysis (yon ttin spii pttrns) tt woul optimiz ts prormn msurs. T omplt o-s DR-plnnr, ll t Frontir vrtx DR-plnnr, s on tis rmwork, lon wit proprtis, proos n pplitions is prsnt in []. Ts r skt in Stion n orm t strtin point o t nw rtriztion o moul-riiity n t orrsponin loritm prsnt in Stion. Formlly, DR-pln o onstrint rp G is irt yli rp (DAG) wos nos rprsnt rii surps in G, n s rprsnt ontinmnt. T lvs or sinks o t DAG r ll t vrtis (primitiv lustrs) o G. T roots or sours r ll t mximl rii lustrs o G. In prtil DR-pln, t lst onition my not ol. Tr oul mny DR-plns or G. S Fiur. Not tt t inition o DR-plns is roust (s typ o Cur-Rossr proprty) in tt ny prtil DR-pln n xtn to otin DR-pln or G. I.., i t DR-pln is in uilt ottom up, ny onstrution pt will l to vli DR-pln. A o DR-pln is on wr o t lustrs is only rquir to o rii, n t roots r rquir to ll t mximl o rii lustrs o G. On n in prtil o DR-pln nloously: t o DR-pln inition is lso roust in tt ny prtil o DR-pln xtns to o DR-pln. An optiml (o) DR-pln is on tt minimizs t mximum n-in. T siz o (o) rii lustr in (o) DR-pln is its n-in (it rprsnts t siz o t orrsponin susystm, on its il lustrs r solv). All proprtis in or DR-plns trnsr s prormn msurs o t DR-plnnrs or DR-plnnin 5

6 Fiur : o ovronstrin lustrs in wll (sinl sour in DR-pln) n unronstrin (multipl sours in DR-pln) rps; s txt or xplntion 6

7 C0 C6 C7 i C C6 C7 C C C C0=S(D(G)) C C9 6 C5 8 C Fiur : D istn onstrint rp G n DR-pln; ll vrtis rprsnt points n v wit n s rprsnt istns n v wit loritms. It is sown in [], tt t prolm o inin t optiml DR-pln o onstrint rp is NPr, n pproximility rsults r sown only in spil ss. Nonpproximility rsults r not known. Tis is t s vn wn on is only intrst in o DR-pln. Howvr, most DR-plnnrs mk o ois urin omputtion (sy t orr in wi vrtis r onsir) n w n sk ow wll (los to optiml) t st omputtion pt o su DR-plnnr woul prorm (on t worst s input). W ll tis t st-oi pproximtion tor o t DR-plnnr. As w sll s in t nxt stion, oo (o) DR-pln is ruil not only or solvin iiny, ut or trminin (o) riiity o t input onstrint rp, s wll s or unronstrint ttion n ompltion [], [], is inispnsl or nvition o t solution sp, [], [], or lin wit ovronstrints, [0] n or iintly uptin t onstrint systm []. All o ts proprtis n oun in [9], [8]. A w otr proprtis o DR-plns r o intrst. W woul lik t wit i., numr o lustrs in t DR-pln to smll, prrly t most ui in t siz o G: tis rlts t omplxity o t DR-plnnr. Dtrminin o riiity vi omplt, mximl ompositions: T Frontir Vrtx Aloritm (FA) DR-Plnnr In tis stion, w irst iv n ltrntiv rtriztion o o riiity wi trnslts to usul proprty o o DR-plns ll o ompltnss. (W omit proos). Tn w skt rlvnt proprtis o Frontir vrtx DR-plns n t orrsponin DR-plnnr (FA DR-plnnr) [7, ] wi ollows tis rtriztion. Lt C omtri onstrint rp. Tn Q = {C,..., C m }, st o o rii propr surps o C, is omplt, mximl, o rii omposition o C i t ollowin ol. I tr is mximl, o rii propr surp o C tn it must ontin on o t C i in Q. Furtrmor, Q soul stisy on o t ollowin. Cs : m = n C n C intrst on nontrivil surp n tir union inus ll o C Cs : E o t C i s is nrly mximl wit rspt to t st Q in t ollowin sns: t only o rii propr surps o C tt stritly ontin C i intrst ll t otr surps C j, j i on nontrivil surps; T nxt torm ivs n ltrnt rtriztion o o riiity. Torm. Lt C omtri onstrint rp n Q = {C,..., C m }, omplt, mximl, o rii omposition o C. Tn C is o rii i n only i ( ) S Aj o ( (C i )) D, S Q wr (rll) D is t numr o os o rii oy, n Aj-o(C i ) is itr t numr o os (ntion o nsity) o C i i C i is trivil; or simply D i C i is nontrivil. Not tt i Cs ols, tn C is utomtilly o rii - in t, t irst proprty o Q is runnt. C i S 7

8 t t t s 0 0 s s 0 * Fiur 5: From Lt. Constrint rp G wit wit istriution. D is ssum to 0 (systm ix in oorint systm); A orrsponin low in iprtit G. Anotr possil low. Initil low ssinmnt tt rquirs ristriution T nxt lmm xplins t trtility o tis mto o trminin o riiity. Lmm. I C is not o rii, tn only Cs in t Dinition pplis. Furtrmor, Cs implis tt no pir o C i intrst on mor tn trivil surp. Tus (usin simpl Rmsy torti rumnt), m is t most O(n ), wr n is t numr o vrtis C. Furtrmor, t omputtion o t inlusion-xlusion ormul in Lmm. tks O(n ) tim. Tis ls to roust proprty o DR-plns usin wi t rtriztion n trnslt to n loritm. A o DR-pln P or omtri onstrint rp G is o omplt i t st Q o il lustrs o vry o lustr C in P is omplt, mximl, o rii omposition o C. Prtil o omplt DR-plns r in nloously s in Stion., n just s or, ny prtil o-omplt) DR-pln or onstrint rp G n xtn to o-omplt) DR-pln or G. T Frontir Vrtx DR-pln (FA DR-pln) Not. Trouout tis stion, unlss otrwis mntion, lustr mns o lustr, rii mns o rii, n DR-pln mns o DR-pln. Intuitivly, n FA DR-pln is uilt y ollowin two stps rptly:. Isolt lustr C in t urrnt rp G i (wi is lso ll t lustr rp or low rp or rsons tt will lr low). Ck n nsur omplt, mximl, o rii omposition o C.. Simpliy C into T (C), trnsormin G i into t nxt lustr rp G i+ = T (G i ) (t romintion stp)... Isoltin Clustrs T isoltion loritm, irst ivn in [, ] is moii inrmntl ntwork mximum low loritm. T ky routin is t istriution o n (s t DR-plnnr psuoo in t Appnix o Prt II) in t onstrint rp G. For, w try to istriut t wit w() + D + to on or ot o its npoints s low witout xin tir wits, rrr to s istriutin t. S DistriutE in t psuoo in Prt II, Appnix. Tis is st illustrt on orrsponin iprtit rp G : vrtis in on o its prts rprsnt s in G n vrtis in t son prt rprsnt vrtis in G; s in G rprsnt inin in G. As illustrt y Fiur 5, w my n to ristriut (in n umntin pt). I w r l to istriut ll s, tn t rp is not ns. I no ns surp xists, tn t low s loritm will trmint in O(n(m + n)) stps n nnoun tis t. I tr is ns surp, tn tr is n wos wit plus D + nnot istriut (s r istriut in som orr, or xmpl y onsirin vrtis in som orr n istriutin ll s onntin nw vrtx to ll t vrtis onsir so r). It n sown tt t sr or t umntin pt wil istriutin tis mrks t rquir ns rp. It n lso sown tt i t oun surp is not ovronstrin, tn it is in t miniml. I it is ovronstrin, [, ] iv n iint loritm to in miniml (non-trivil, i on xists) o lustr insi it. Tn [] ivs mto to nsur omplt, mximl, o rii omposition o C. 8

9 7 i C 0 9 C C Fiur 6: From lt: FA s simpliition o rp ivin DR-pln in Fiur ; lustrs r simplii in tir numr orr: C is simplii or C7 t... Clustr Simpliition Tis simpliition ws ivn in [7, 5]. T oun lustr C intrts wit t rst o t onstrint rp trou its rontir vrtis; i.., t vrtis o t lustr tt r jnt to vrtis not in t lustr. T vrtis o C tt r not rontir, ll t intrnl vrtis, r ontrt into sinl or vrtx. Tis or is onnt to rontir vrtx v o t simplii lustr T (C) y n wos wit is t t sum o t wits o t oriinl s onntin intrnl vrtis to v. Hr, t wits o t rontir vrtis n o t s onntin tm rmin unn. T wit o t or vrtx is osn so tt t nsity o t simplii lustr is D, wr D is t omtry-pnnt onstnt. Tis is importnt or provin mny proprtis o t FA DR-pln: vn i C is ovronstrin, T (C) s ovrll wit is tt o wllonstrin rp, (unlss C is rottionlly symmtri n trivil, in wi s, it rtins its o or wit). Tnilly, T (C) my not wllonstrin in t pris sns: it my ontin n ovronstrin surp onsistin only o rontir vrtis n s, ut its ovrll o ount is tt o wllonstrin rp. Fiur 6 illustrts tis itrtiv simpliition pross nin in t inl DR-pln o Fiur.. T Frontir Vrtx Aloritm (FA DR-plnnr) T lln mt y FA is tt it provly mts svrl omptin rquirmnts. Spiilly, it ivs o omplt DR-pln. T rp trnsormtion prorm y t FA lustr simpliition is sri ormlly in [7, 5] tt provi t voulry or provin rtin proprtis o FA tt ollow irtly rom tis simpliition. Howvr, otr proprtis o FA rquir tils o t tul DR-plnnr tt nsurs tm, n r rily skt r. Not: til psuoo o t FA DR-plnnr (t xistin vrsion, s wll s inorportin t moul-riity loritm o tis ppr) n oun in [5], [8]. T psuoo s n implmnt s prt o t ownlol, opnsour FRONTIER omtri onstrint solvr [], [5], [8], [9]. T si FA loritm is s on n xtnsion o t istriut routin or s (xplin ov) to vrtis n lustrs in orr or t isoltion loritm to work t n ritrry st o t plnnin pross, i., in t lustr or low rp G i. First, w rily sri tis si loritm. Nxt, w skt t prts o t loritm tt nsur ruil, intr-rlt proprtis o t output DR-pln: () nsurin o ompltnss; () or unronstrin rps: outputin omplt st o mximl lustrs s sours o t DR-pln; () ontrollin wit o t DR-pln to nsur polynomil tim loritm. Tr tstruturs r mintin. T urrnt low or lustr rp, G i t urrnt DR-pln (tis inormtion is stor ntirly in t irril strutur o lustrs t t top lvl o t DR-pln), n 9

10 lustr quu, wi is t top-lvl lustrs o t DR-pln tt v not n istriut so r, in t orr tt ty wr oun (s low or n xplntion o ow lustrs r istriut). W strt wit t oriinl rp (wi srvs s t lustr or low rp initilly, wr t lustrs r sinlton vrtis). T DR-pln onsists o t l or sink nos wi r ll t vrtis. T lustr quu onsists o ll t vrtis in n ritrry orr. T mto DistriutVrtx (s psuoo o Prt II, Appnix) istriuts ll s (lls DistriutE) onntin t urrnt vrtx to ll t vrtis onsir so r. Wn on o t s nnot istriut n miniml ns lustr C is isovr, its simpliition T (C) (sri ov) trnsorms t low rp. T lows on t intrnl s n t or vrtx r inrit rom t ol lows on t intrnl s n intrnl vrtis. Noti tt unistriut wits on t intrnl s simply isppr. T unistriut wits on t rontir s r istriut (witin t lustr) s wll s possil. Howvr, unistriut wits on t rontir s (s twn rontir vrtis) my still rmin i t rontir portion o t lustr is svrly ovronstrin. Ts v to lt wit rully. (S isussion on lin wit t prolms us y ovronstrints low.) T nw lustr is introu into t DR-pln n t lustr quu. Now w sri t mto DistriutClustr Assum ll t vrtis in t lustr quu v n istriut (itr ty wr inlu in ir lvl lustr in t DR-pln, or ty il to us t ormtion o lustr n ontinu to top lvl no o t DR-pln, ut v isppr rom t lustr quu). Assum urtr tt t DR-pln is not omplt, i.., its top lvl lustrs r not mximl. T nxt lvl o lustrs r oun y istriutin t lustrs urntly in t lustr quu. Tis is on y illin up t ols or t vill rs o rom o lustr C in istriut y D units o low. T PusOutSi mto sussivly onsirs inint on t lustr wit npoint outsi t lustr. It istriuts ny unistriut wit on ts s + xtr wit unit on o ts s. It n sown tt i C is ontin insi lrr lustr, tn tlst on su lustr will oun y tis mto on ll t lustrs urrntly in t lustr quu v n istriut. T nw lustr oun is simplii to iv nw low rp, n ts in t lustr quu, n t DR-pln s sri ov. Evntully, wn t lustr quu is mpty, i., ll oun lustrs v n istriut, t DR-pln s top lvl lustrs r urnt to t omplt st o mximl o rii surps o t input onstrint rp. S [] or orml proos. Not: Trouout, in t intrst o orml lrity, w lv out o, ut ily tiv uristis tt in simpl lustrs y voiin ull-l low. On su xmpl is ll squntil xtnsions wi utomtilly rts lrr lustr ontinin lustr C n vrtx v provi tr r tlst D s twn C n v. Ts n sily inorport into t low s loritm, provi rtin si invrints out istriut s is mintin (s low). Tis omplts t sription o t kon o t si FA DR-plnnr. Nxt w onsir som tils nsurin t proprtis () () ov... Ensurin o ompltnss First w intuitivly xplin wy o ompltnss is ruil proprty. In Fiur 7, tr C n C r oun, wn C is istriut, C n C woul pik up s lustr, ltou ty o not orm lustr. T prolm is tt t ovronstrin surp W intrsts C on trivil lustr, n W itsl s not n oun. H W n oun or C ws istriut, W woul v n simplii into wllonstrin surp n tis mislssiition woul not v ourr. It s n sown in [] tt tis typ o mislssiition n voi (W n or to oun tr C is oun), y mintinin tr invrints. T irst two r sri r. T tir is ily rlt to proprty () n is sri in t nxt sustion. T irst is t ollowin invrint: lwys istriut ll unistriut s onntin nw oun lustr C (or t lst istriut vrtx tt us C to oun), to ll t vrtis istriut so r tt r outsi t lustr C. Unistriut wit on s insi C r lss ruil: i ty om intrnl s o t lustr, tn tis unistriut wit ispprs wn C is simplii into wllonstrin lustr; tr is lso simpl mto o trtin unistriut wit on rontir s so tt ty lso o not us prolms - t mto n proo n oun in []). T son invrint tt is usul or nsurin o ompltnss is tt or ny lustr in t DR-pln, no 0

11 W W C C i j Fiur 7: Finin W irst will prvnt o mislssiition: Lt D xmpl, Rit D xmpl. C C C C C C C C C C C C C C, Fiur 8: Ensurin Clustr Minimlity: E is st o ssntil lustrs tt must prsnt in ny sust o t ilrn o C tt orm lustr. In tis s, E itsl orms lustr. C is lustr m up o propr sust o t lst o C s ilrn propr sust o tlst o its il lustrs orms lustr. W ll tis proprty lustr minimlity. FA nsurs tis usin nrliztion o t mto Miniml o [, ] wi ins miniml ns surp insi ns surp lot y DistriutVrtx n DistriutE. S Fiur 8. On lustr C is lot n s ilrn C,..., C k, or k, rursiv mto lusmin rmovs on lustr C i t tim (rplin rlir rmovls) rom C n ros t low insi t low rp rstrit to C, or C s simpliition. I propr sust o tlst C j s orms lustr C, tn t lusmin loritm is rpt insi C n trtr in C in, rplin t st o il lustrs o C tt r insi C y sinl il lustr C. I inst no su lustr is oun, tn t rmov lustr C i t ssntil. I.., it lons to vry sust o C s ilrn tt orms lustr. Wn t st o lustrs itsl orms lustr E (usin o ount), tn lusmin is ll on C in wit nw il lustr E rplin ll o C s ilrn insi E... () Finin omplt st o mximl lustrs in unronstrin rps Wil t DR-plnnr sri so r urnts tt t trmintion, top lvl lustrs o t DR-pln r mximl. It lso urnts tt t oriinl rp is o unronstrin only i tr is mor tn on top lvl lustr in t DR-pln. Howvr, in orr to urnt tt ll t mximl lustrs o n unronstrin rp ppr s top lvl lustrs o t DR-pln, w us t osrvtion tt ny pir o su lustrs intrst on surp tt rus (on inin onstrints r rsolv) into trivil surp ( sinl point in D or sinl in D). Tis ouns t totl numr o su lustrs n ivs simpl mto or inin ll o tm. On t DR-plnnr trmints wit st o mximl lustrs, otr mximl lustrs r oun y simply prormin Pusoutsi o units on vry vrtx (in D) or vry vrtx n (in D), n ontinuin wit t oriinl DR-plnnin pross until it trmints wit lrr st o mximl lustrs. Tis is prorm or vrtx in D n in D wi urnts tt ll mximl lustrs will oun. S [] or proos... () Controllin wit o t DR-pln FA ivs linr oun on DR-pln wit y mintinin t ollowin invrint o t lustr or low rp: vry pir o lustrs in t low rp (top lvl o t DR-pln) t ny st intrst on t most trivil

12 C C5 i C C i Fiur 9: Prvnt umultion o lustrs surp. FA os tis y rptly prormin oprtions tim nw potntil lustr is isolt. T irst is n nlrmnt o t oun lustr. In nrl, nw oun lustr N is nlr y ny lustr D urrntly in t low rp, i tir nonmpty intrstion is not rottionlly symmtri or trivil surp. In tis s, N nitr ntrs t lustr rp nor t DR-Pln. Only N D ntrs t DR-pln, s prnt o ot D n t otr ilrn o N. It is sy to s tt t sizs o t susystms orrsponin to ot N D n N r t sm, sin D woul lry solv. For t xmpl in Fiur 9, wn t DR-pln ins t lustr C tr C, t DR-plnnr will in tt C n nlr y C T DR-plnnr orms nw lustr C s on C n C n puts C into t lustr quu, inst o puttin C to lustr quu. T son oprtion is to itrtivly omin N D wit ny lustrs D, D,... s on nonmpty ovrlp tt is not rottionlly symmtri or trivil. In tis s, N D D, N D D D t. ntr t DR-pln s stirs, or in, ut only t sinl lustr N D D D..... ntrs t lustr rp tr rmovin D, D, D.... Oours, ot o ts prosss r istint rom t oriinl low istriution pross tt lots lustrs. Moul-Riiity: Crtriztion n Aloritm W iv rursiv inition o D moul-riiity (lon wit inition o moul-omplt DR-plns) n sow tt it is nturl n roust rtriztion. Tn w skt n xtnsion o t FA loritm in orr to trmin moul-riiity y onstrutin moul-omplt DR-plns. W ollow wit numr o xmpls o rps tt r o rii ut not moul-rii. Lt C D istn onstrint rp. Lt E, C,..., C k propr surps o C. W sy tt C,..., C k r,c E (r: implis riiity o) i y mkin C,..., C m omplt rps (y in itionl s), E oms rii. Anloously, w in n m y ssrtin o riiity n moul-riiity (to in low) s t rit n si o t implition, rsptivly. Lt C D istn onstrint rp. C is moul-rii i: Bs s: it is trivil n o rii. Or t ollowin ols. Lt Q = {C,..., C m } ny omplt, mximl, moul omposition o C. Tis is in s ollows. Lt φ m,c, t trnsitiv losur o t mpty st unr m,c, i.., i tr is propr surp E o C s.t. itr it is moul-rii, or tr is som st o moul-rii propr surps C,..., C k o C s.t. C,..., C k m,c, E, tn E lons to φ m,c,. Lt Q ny sust {C,..., C m } o φ m,c, s.t. Cs : m = ; C n C intrst on nontrivil surp n tir union inus ll o C. Or, t ollowin ols. Cs : Any mximl surp in φ m,c, must ontin on o t C i in Q. E o t C i s is nrly mximl wit rspt to t st Q in t ollowin sns: t only lmnts o φ m,c, tt stritly ontin C i intrst ll t otr surps C j, j i on nontrivil surps.

13 Tn C is moul-rii i ( ) S Aj o ( (C i )) D, S Q wr (rll) D is t numr o os o rii oy, n Aj-o(C i ) is itr t numr o os (ntion o nsity) o C i i C i is trivil; or simply D i C i is nontrivil. Osrvtion. Evry moul-rii rp is o rii; n vry rii rp is moul-rii. C i S T nxt lmm sows t trtility o t ov rtriztion. Lmm. Lt Q = {C,..., C m } ny st o propr surps o C tt orm omplt, mximl, moul omposition o C. Tis implis tt i m >, tn no pir o C i intrst on mor tn trivil surp. Tus m is t most O(n ), wr n is t numr o vrtis C. Tus, t omputtion o t inlusion-xlusion ormul in Dinition tks O(n ) tim. Usin t ov lmm, t ollowin inition sows t us o so-ll moul-omplt DR-plns to iintly trmin moul riiity. A moul DR-pln P or D istn onstrint rp G is prtil orr wr no rprsnts surp in φ m,g, or G itsl, i G is moul rii. T orrin is y ontinmnt. Ts nos r ll moul lustrs (to ruilly irntit rom moul-rii surps o G, wi w ll inrnt moul lustrs). T lvs r t oriinl vrtis o G. E no in t sudr-pln root t no C rprsnts surp in φ m,c, or C itsl, i C is moul-rii, or n inrnt moul lustr. I G is moul-rii, tr is sinl sour lustr; i G is not moul-rii, t roots or sours orm is omplt, mximl, moul omposition o G. A prtil moul DR-pln os not v to stisy t onitions on t roots or sours. A moul DR-Pln is moul-omplt i t st Q o il lustrs o vry moul lustr C in P is omplt, mximl, moul omposition o C. A moul DR-pln is typilly in to ontin itionl inormtion y inorportin notr prtil orr ll t solvin priority orr, wi is onsistnt wit t moul DR-pln s DAG orr, ut oul mor rin. T intnt is tt moul-riiity o moul lustrs tt ppr ltr in t orr pn on lustrs tt ppr rlir. I.., t orrin rlts t numr o pplitions o m rquir to in moul lustr. T nxt torm sows tt moul-riiity is roust. I.., t orr o ottom-up onstrution o moul(- omplt) DR-plns is immtril, typ o Cur-Rossr proprty. Torm. I rp G is moul-rii, tn vry prtil, moul(-omplt) DR-pln or G n xtn to moul(-omplt) DR-pln. Moiition o FA to trmin moul riiity T ov isussion tivly lys out trtl mto or trminin moul-riiity y omputin moul-omplt DR-plns ottom up. Tis is on y xtnin t o-omplt DR-plnnr FA ivn in t prvious stion s ollows. First not tt y usin FA w urnt no ls ntivs, sin moul-rii implis o-rii. W now skt ow to limintin o-rii rps tt r not moul-rii. T FA DRplnnr runnin on n input rp G uss DistriutClustr (low) on t urrnt st S o o rii lustrs to isolt o lustr nit C, n trtr onstruts omplt, mximl, o rii omposition o it, tr wi it is wtr nw o-rii lustr C s n oun, usin t o-rii rtriztion o Stion. Tis is t ky point o xtnsion. W us t nloy twn tis o-rii rtriztion n t moul-rii rtriztion t t innin o tis Stion. Inutivly, w n ssum tt t urrnt st S onsists o moul-rii surps o G or inrnt moul lustrs. T mto o onstrution o omplt, mximl, moul omposition Q o t nit (inrnt moul) lustr C n on witout onstrutin φ m,c,, y onstrutin squn o Q i s o wi stisis t ov onitions on Q, ut wit rspt to φ m,c,i (i., losur wit rspt to i pplitions o t m oprtion). Tis squn rs ix point t Q. W iv xmpls tt illustrt t us o moul-riiity.

14 j C i C5 C C C C C C6 C C C5 C C C Fiur 0: Exmpls wr moul-riiity ts o riiity. S txt or xplntion. In Fiur 0 Top Lt: t rp is o rii, ut not moul-rii, s sn y t omplt mximl moul omposition sown. Top Rit: moul-rii, ut no pir o inrnt moul lustrs sown orms moulrii surp, ty o orm o-rii surps. Bottom: not moul-rii ut o rii; omplt mximl moul omposition n solvin priority orrs s ollows: t pir C, C is n inrnt moul lustr C 5 ut tt C 5 n solv only tr C is solv; I.., or t virtul (, ) is, C n C woul not pik up totr s lustr nit. Similrly, it will lso trmin tt C 5, C orm lustr C 6, ut solvin priority orr sown. Bottom Rit: moul-omplt DR-pln or lt onstrint systm wit sours or roots: C 6 n C. Fiur sows lssi rp rom [, 5], wit ins, wi is not moul-rii ut is o rii. A omplt mximl, moul omposition is sown. T mil lustr C is not n inrnt moul lustr, ltou C n C r. Opn Prolm A qustion tt immitly riss is to rlt t rtriztion ivn r to riiity mtrois n stnr onjturs on omintoril riiity rtriztions or D [8]. Rrns [] S. Ait-Aoui, R. Jou, n D. Milui. Rution o onstrint systms. In Compurpis, ps 8 9, 99. [] B. Brurlin. Construtin tr-imnsionl omtri ojt in y onstrints. In ACM SIG- GRAPH. Cpl Hill, 986.

15 i C j C k. Fiur : Clssi Hin xmpl: not moul-rii, ut o rii [] D. Cox, J. Littl, n D. O S. Usin lri omtry. Sprinr-Vrl, 998. [] Hnry Crpo. Struturl riiity. Struturl Topoloy, :6 5, 979. [5] Hnry Crpo. T ttrrl-otrl truss. Struturl Topoloy, 7:5 6, 98. [6] I. Fuos. Gomtri Constrint Solvin. PD tsis, Puru Univrsity, Dpt o Computr Sin, 995. [7] I. Fuos n C. M. Homnn. A rp-onstrutiv ppro to solvin systms o omtri onstrints. ACM Trnstions on Grpis, 6:79 6, 997. [8] Jk E. Grvr, Briitt Srvtius, n Hrmn Srvtius. Comintoril Riiity. Grut Stuis in Mt., AMS, 99. [9] B. Hnrikson. Conitions or uniqu rp rliztions. SIAM J. Comput., :65 8, 99. [0] C Homn, M Sitrm, n B Yun. Mkin onstrint solvrs mor usl: t ovronstrint prolm. to ppr in CAD, 00. [] C. M. Homnn, A. Lomonosov, n M. Sitrm. Finin solvl susts o onstrint rps. In Smolk G., itor, Sprinr LNCS 0, ps 6 77, 997. [] C. M. Homnn, A. Lomonosov, n M. Sitrm. Gomtri onstrint omposition. In Brurlin B. n Rollr D., itors, Gomtri Constr Solvin n Appl, ps 70 95, 998. [] Cristop M. Homnn, Anrw Lomonosov, n Mr Sitrm. Finin solvl susts o onstrint rps. In Constrint Prormmin 97 Ltur Nots in Computr Sin 0, G. Smolk E., Sprinr Vrl, Linz, Austri, 997. [] Cristop M. Homnn, Anrw Lomonosov, n Mr Sitrm. Gomtri onstrint omposition. In Brurlin n Rollr E.s, itors, Gomtri Constrint Solvin. Sprinr-Vrl,

16 [5] Cristop M. Homnn, Anrw Lomonosov, n Mr Sitrm. Plnnin omtri onstrint ompositions vi rp trnsormtions. In AGTIVE 99 (Grp Trnsormtions wit Inustril Rlvn), Sprinr ltur nots, LNCS 779, s Nl, Surr, Mun, ps 09, 999. [6] Cristop M. Homnn, Anrw Lomonosov, n Mr Sitrm. Domposition o omtri onstrints systms, prt i: prormn msurs. Journl o Symoli Computtion, (), 00. [7] Cristop M. Homnn, Anrw Lomonosov, n Mr Sitrm. Domposition o omtri onstrints systms, prt ii: nw loritms. Journl o Symoli Computtion, (), 00. [8] G. Krmr. Solvin Gomtri Constrint Systms. MIT Prss, 99. [9] G. Lmn. On rps n riiity o pln skltl struturs. J. Enr. Mt., : 0, 970. [0] R. Ltm n A. Milit. Conntivity nlysis: tool or prossin omtri onstrints. Computr Ai Dsin, 8:97 98, 996. [] Anrw Lomonosov n Mr Sitrm. Grp loritms or omtri onstrint solvin. In sumitt, 00. [] A. Milit n C. R. A krnl or omtri turs. In ACM/SIGGRAPH Symposium on Soli Molin Fountions n CAD/CAM Applitions. ACM prss, 997. [] J. J. Oun, M. Sitrm, B. Moro, n A. Arr. Frontir: ully nlin omtri onstrints or tur s sin n ssmly. In strt in Proins o t ACM Soli Molin onrn, 00. [] J. Own. In D-u ommril omtri onstrint solvin sotwr. [5] J. Own. Alri solution or omtry rom imnsionl onstrints. In ACM Symp. Foun. o Soli Molin, ps 97 07, Austin, Tx, 99. [6] J. Own. Constrints on simpl omtry in two n tr imnsions. In Tir SIAM Conrn on Gomtri Dsin. SIAM, Novmr 99. To ppr in Int J o Computtionl Gomtry n Applitions. [7] J.A. Pon. Molin mto or sortin pnnis mon omtri ntitis. In US Stts Ptnt 5,5,90, Ot 99. [8] M Sitrm. Frontir, n opnsour omtri onstrint solvr: loritms n rittur. monorp, in prprtion, 00. [9] M Sitrm. Grp s omtri onstrint solvin: prolms, prorss n irtions. In Dutt, Jnrn, n Smi, itors, AMS-DIMACS volum on Computr Ai Dsin, 00. [0] M Sitrm n M Anj-Mknn. A omtry n tnsrity s virus ssmly ptwy mol. sumitt, vill upon rqust, 00. [] M Sitrm, A Arr, Y Zou, n N Korswrn. Solution mnmnt n nvition or omtri onstrint systms. sumitt, vill upon rqust, 00. [] M Sitrm, J Oun, n A Arr. Eiint unronstrin ompltions, upts n on lin solution o nrl omtri onstrint rps. sumitt, vill upon rqust, 00. [] M Sitrm, J Ptrs, n Y Zou. Solvin miniml, wllonstrin, omtri onstrint systms: omintoril optimiztion o lri omplxity. sumitt to ADG 00, vill upon rqust, 00. [] M Sitrm n Y Zou. Mixin turs n vritionl onstrints in. sumitt, vill upon rqust, 00. [5] Mr Sitrm. Frontir, opnsour nu omtri onstrint solvr: Vrsion (00) or nrl systms; vrsion (00) or n som systms; vrsion (00) or nrl n systms. In ttp:// sitrm, ttp://