Optimally Cutting a Surface into a Disk

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1 Opimlly Cuing Surfe ino Dik Jeff Erikon Sriel Hr-Peled Univeriy of Illinoi Urn-Chmpign hp:// {jeffe,riel} Sumied o Diree & Compuionl Geomery: July 2, 2002 Ar We onider he prolem of uing e of edge on polyhedrl mnifold urfe, poily wih oundry, o oin ingle opologil dik, minimizing eiher he ol numer of u edge or heir ol lengh. We how h hi prolem i NP-hrd, even for mnifold wihou oundry nd for punured phere. We lo derie n lgorihm wih running ime n O(g+k), where n i he ominoril omplexiy, g i he genu, nd k i he numer of oundry omponen of he inpu urfe. Finlly, we derie greedy lgorihm h oupu O(log 2 g)-pproximion of he minimum u grph in O(g 2 n log n) ime. A preliminry verion of hi pper w preened he 18h Annul ACM Sympoium on Compuionl Geomery [EHP02]. See hp:// pu/hem.hml for he mo reen verion of hi pper. Prilly uppored y Slon Fellowhip, NSF CAREER wrd CCR , nd NSF ITR grn DMR Prilly uppored y NSF CAREER wrd CCR

2 Opimlly Cuing Surfe ino Dik 1 1 Inroduion Severl ppliion of hree-dimenionl urfe require informion ou he underlying opologil ruure in ddiion o he geomery. In ome e, we wih o implify he urfe opology, o filie lgorihm h n e performed only if he urfe i opologil dik. Appliion when hi i imporn inlude urfe prmeerizion [Flo97, SdS00] nd exure mpping [BVIG91, PB00]. In he exure mpping prolem, we wih o find oninuou nd inverile mpping from he exure, uully wo-dimenionl rengulr imge, o he urfe. Unforunely, if he urfe i no opologil dik, no uh mp exi. In uh e, he only feile oluion i o u he urfe o h i eome opologil dik. (Hker e l. [HAT + 00] preen n lgorihm for direly exure mpping model wih he opology of phere, where he exure i lo emedded on phere.) Of oure, when uing he urfe, one would like o find he e poile u under vriou oniderion. For exmple, one migh wn o u he urfe o h he reuling urfe n e exured mpped wih minimum diorion [Flo97, GGH02, SdS00]. To our knowledge, ll previou pprohe for hi uing prolem eiher rely on heurii wih no quliy gurnee [GGH02, She02, SF02] or require he uer o perform hi uing eforehnd [Flo97, PB00]. Lzru e l. [LPVV01] preened nd implemened wo lgorihm for ompuing nonil polygonl hem of n orienle urfe of omplexiy n nd wih genu g, in ime O(gn), implifying n erlier lgorihm of Veger nd Yp [VY90]. Compuing uh hem require finding 2g yle, ll ping hrough ommon epoin in M, uh h uing long hoe yle rek M ino opologil dik. Sine hee yle mu hre ommon poin, i i ey o find exmple where he overll ize of hoe yle i Ω(gn). Furhermore, hoe yle hre everl edge nd re viully unifying. For mo ppliion, ompuing nonil hem i overkill. I i uully uffiien o find olleion of edge whoe removl rnform he urfe ino opologil dik. We ll uh e of edge u grph; ee Figure 1 for n exmple. Cu grph hve everl dvnge. Fir, hey re omp. Trivilly, ny u grph onin mo n edge of he urfe meh, muh le hn ny nonil hem in he wor e, lhough we expe i o e muh mller in prie. Seond, i i quie ey o onru u grph for n rirry polyhedrl urfe in O(n) ime, uing redh-fir erh of he dul grph [DS95], or imply king mximl e of edge whoe omplemen i onneed [LPVV01]. Finlly, he u grph h n exremely imple ruure: ree wih O(g) ddiionl edge. A uh, i hould e eier o mnipule lgorihmilly hn oher repreenion. For exmple, Dey nd Shipper [DS95] derie f lgorihm o deermine wheher urve i onrile, or wo urve re homoopi, uing n rirry u grph ined of nonil hem. In hi pper, we inveige he queion of how find he e uh uing of urfe, reriing ourelve o u long he edge of he given meh. Speifilly, we wn o find he mlle ue of edge of polyhedrl mnifold urfe M, poily wih oundry, uh h uing long hoe edge rnform M ino opologil dik. We lo onider he weighed verion of hi prolem, where eh edge h n rirry non-negive weigh nd we wn o minimize he ol weigh of he u grph. The mo nurl weigh of

3 2 Jeff Erikon nd Sriel Hr-Peled d e d e e e d d Figure 1. A u grph for wo-holed oru nd i indued (non-nonil) polygonl hem. n edge i i Euliden lengh, u we ould lo ign weigh o ke prolem-peifi oniderion ino oun. For exmple, if we wn o ompue exure mpping for peifi viewpoin, we ould mke viile edge more expenive, o h he minimum u grph would minimize he numer of viile edge ued in he u. Our lgorihm do no require he edge weigh o ify he ringle inequliy. We how h he minimum u grph of ny polyhedrl mnifold M wih genu g nd k oundry omponen n e ompued in n O(g+k) ime. We lo how h he prolem i NP-hrd in generl, even if g or k i fixed. Finlly, we preen imple nd effiien greedy pproximion lgorihm for hi prolem. Our lgorihm oupu u grph whoe weigh i for O(log 2 g) lrger hn opiml, in O(g 2 n log n) ime. 1 If g = 0, he pproximion for i exly 2. A ool in our pproximion lgorihm, we lo derie effiien lgorihm o ompue hore nd nerly-hore nonrivil yle in mnifold; we elieve hee lgorihm re of independen inere. 2 Bkground Before preening our new reul, we review everl ueful noion from opology nd derie reled reul in more deil. We refer he inereed reder o Hher [H01], Munkre [Mun00], or Sillwell [Si93] for furher opologil kground nd more forml definiion. For reled ompuionl reul, ee he reen urvey y Dey, Edelrunner, nd Guh [DEG99] nd Veger [Veg97]. 2.1 Topology A 2-mnifold wih oundry i e M uh h every poin x M lie in neighorhood homeomorphi o eiher he plne IR 2 or loed hlfplne. The poin wih only hlfplne neighorhood oniue he oundry of M; he oundry oni of zero or more dijoin 1 To implify noion, we define log x = mx{1, log 2 x }.

4 Opimlly Cuing Surfe ino Dik 3 irle. Thi pper will onider only omp mnifold, where every infinie equene of poin h onvergen uequene. The genu of 2-mnifold M i he mximum numer of dijoin non-epring yle γ 1, γ 2,..., γ g in M; h i, γ i γ j = for ll i nd j, nd M \ (γ 1 γ g ) i onneed. For exmple, phere nd di hve genu 0, oru nd Möiu rip hve genu 1, nd Klein ole h genu 2. A mnifold i orienle if i h wo diin ide, nd non-orienle if i h only one ide. Alhough mny geomeri ppliion ue only orienle 2-mnifold (primrily eue non-orienle mnifold wihou oundry nno e emedded in IR 3 wihou elfinereion) our reul will pply o non-orienle mnifold well. Every (omp, onneed) 2-mnifold wih oundry i hrerized y i orieniliy, i genu g, nd he numer k of oundry omponen [FW99]. A polyhedrl 2-mnifold i onrued y gluing loed imple polygon edge-o-edge ino ell omplex: he inereion of ny wo polygon i eiher empy, verex of oh, or n edge of oh. We refer o he omponen polygon h fe. (Sine he fe re loed, every polyhedrl mnifold i omp.) For ny polyhedrl mnifold M, he numer of verie nd fe, minu he numer of edge, i he Euler hrerii χ of M. Euler formul [Epp01] implie h χ i n invrin of he underlying mnifold, independen of ny priulr polyhedrl repreenion; χ = 2 2g k if he mnifold i orienle, nd χ = 2 g k if he mnifold i non-orienle. Euler formul implie h if M h v verie, hen M h mo 3v 6 + 6g edge nd mo 2v 4 + 4g k fe, wih equliy for orienle mnifold where every fe nd oundry irle i ringle. We le n 6v g k denoe he ol numer of fe, edge, nd verie in M. The 1-keleon M 1 of polyhedrl mnifold M i he grph oniing of i verie nd edge. We define u grph G of M ugrph of M 1 uh h M\G i homeomorphi o dik. 2 The dik M \ G i known polygonl hem of M. Eh edge of G pper wie on he oundry of polygonl hem M \ G, nd we n oin M y gluing ogeher hee orreponding oundry edge. Finding u grph of M wih minimum ol lengh i lerly equivlen o o finding polygonl hem of M wih minimum perimeer. Any 2-mnifold h o-lled nonil polygonl hem, whoe ominoril ruure depend only on he genu g, he numer of oundry omponen k, nd wheher he mnifold i orienle. 3 The nonil hem of n orienle mnifold i (4g + 3k)-gon wih ueive edge leled x 1, y 1, x 1, ȳ 1,..., x g, y g, x g, ȳ g, z 1, e 1, z 1,..., z k, e k, z k ; for non-orienle mnifold, he nonil hem i (2g + 3k)-gon wih edge lel x 1, x 1,..., x g, x g, z 1, e 1, z 1,..., z k, e k, z k. 2 Cu grph re generlizion of he u lou of mnifold M, whih i eenilly he geodei medil xi of ingle poin. 3 Aully, here re everl differen wy o define nonil hem; he one deried here i merely he mo ommon. For exmple, he nonil hem for n oriened urfe wihou oundry ould lo e leled x 1, x 2,..., x 2g, x 1, x 2,..., x 2g.

5 4 Jeff Erikon nd Sriel Hr-Peled Every pir of orreponding edge x nd x i oriened in oppoie direion. Gluing ogeher orreponding pir in he indied direion reover he originl mnifold, wih he unmhed edge e i forming he oundry irle. For mnifold M wihou oundry, redued polygonl hem i one where ll he verie re glued ino ingle poin in M; nonil hem of mnifold wihou oundry re redued. We emphize h he polygonl hem onrued y our lgorihm re neiher neerily nonil nor neerily redued. 2.2 Previou nd Reled Reul Dey nd Shipper [DS95] derie n lgorihm o onru redued, u no neerily nonil, polygonl hem for ny ringuled orienle mnifold wihou oundry in O(n) ime. Eenilly, heir lgorihm onru n rirry u grph G y deph-fir erh, nd nd hen hrink pnning ree of G o ingle poin. (See lo Dey nd Guh [DG99].) Veger nd Yp [VY90] developed n lgorihm o onru nonil hem in opiml O(gn) ime nd pe. Two impler lgorihm wih he me running ime were ler developed y Lzru e l. [LPVV01]. The edge of he polygonl hem produed y ll hee lgorihm re (poily overlpping) ph in he 1-keleon of he inpu mnifold. We will modify one of he lgorihm of Lzru e l. o onru hor nonrivil yle nd u grph. Very reenly, Colin de Verdiére nd Lzru onider he prolem of opimizing nonil polygonl hem [CL02]. Given nonil polygonl hem for ringuled oriened mnifold M, heir lgorihm onru he hore nonil hem in he me homoopy l. Surpriingly (in ligh of our Theorem 3.1) heir lgorihm run in polynomil ime under ome mild umpion ou he inpu. A yprodu, hey lo oin polynomil-ime lgorihm o onru he minimum-lengh imple loop homoopi o given ph. Surfe prmeerizion i n exremely ive re of reerh, hnk o numerou ppliion uh exure mpping, remehing, ompreion, nd morphing. For mple of reen reul, ee [AMD02, EDD + 95, Flo97, SF02, GGH02, LSS + 98, LPRM02, She02, SCOGL02, SdS00, ZKK01] nd referene herein. In mo of hee work, urfe of high genu re prmeerized y uing hem ino everl (poily overlpping) phe, eh homeomorphi o dik, eh wih epre prmeerizion. A reen exepion i he work of Gu e l. [GGH02], whih ompue n iniil u grph in O(n log n) ime y running hore ph lgorihm on he dul of he mnifold meh, ring from n rirry eed ringle. Eenilly he me lgorihm w independenly propoed y Seiner nd Fiher [SF02]. One urfe h een u ino dik (or everl dik), furher (opologilly rivil) u re uully neery o redue diorion [GGH02, She02, SCOGL02]. Mny of hee lgorihm inlude heurii o minimize he lengh of he u in ddiion o he diorion of he prmeerizion [GGH02, LPRM02, She02], u none wih heoreil gurnee. All of our lgorihm re ulimely ed on Dijkr ingle-oure hore ph lgorihm [Dij59, Tr83]. Mny previou reul hve ued Dijkr lgorihm or one of

6 Opimlly Cuing Surfe ino Dik 5 i oninuou generlizion [KAB95, MMP87, Ti95] o diover inereing opologil ruure in 2-mnifold, uh u grph [GGH02, SF02], mll hndle ( opologil noie ) [GW01], exure le [LPRM02], onour ree [AE98, LV99], nd Ree grph [HSKK01, SF02]. 3 Compuing Minimum Cu Grph i NP-Hrd In hi eion, we prove h finding minimum u grph of ringuled mnifold i NP-hrd. We onider wo verion of he prolem. In he weighed e, he mnifold i umed o e polyhedrl urfe in IR 3 nd we wn o ompue he u grph whoe ol Euliden lengh i mll poile. In he unweighed e, we re ompue he u grph wih he minimum numer of edge; he geomery of he mnifold i ignored enirely. Boh reduion re from he reiliner Seiner ree prolem: Given e P of n poin from m m qure grid in he plne, find he hore onneed e of horizonl nd veril line egmen h onin every poin in P. Thi prolem i NP-hrd, even if m i ounded y polynomil in n [GJ77]. Our reduion ue he Hnn grid of he poin, whih i oined y drwing horizonl nd veril line hrough eh poin, lipped o he ounding ox of he poin. A le one reiliner Seiner ree of he poin i ue of he Hnn grid [Hn66]. Theorem 3.1. Compuing he lengh of he minimum (weighed or unweighed) u grph of ringuled punured phere i NP-hrd. Proof: Fir onider he weighed e, where he weigh igned o eh edge of he mnifold i i Euliden lengh. Le P e e of n poin in he plne, wih ineger oordine eween 1 nd m. We onru punured phere in O(n 2 ) ime follow. Aume h P lie on he xy-plne in IR 3. We modify he Hnn grid of P y repling eh erminl wih qure of widh 1/2n, roed 45 degree o h i verie lie on he neighoring edge. Thee qure will form he punure. We hen h in under eh fe f of he modified Hnn grid, y joining he oundry of f o lighly led opy of f on he plne z = n 2. We lo h in of deph n o he oundry of he enire modified Hnn grid. The ide fe of eh in re rpezoid. The in re pered o h djen in inere only on he modified Hnn grid. Tringuling hi urfe rirrily, we oin polyhedrl phere M wih n punure nd overll omplexiy O(n 2 ). See Figure 2. Le G e minimum weighed u grph of M. We eily oerve h G onin only long edge from he modified Hnn grid nd onin le one verex of every punure. Thu, he edge of G re in one-o-one orrepondene wih he edge of reiliner Seiner ree of P. For he unweighed e, we modify he originl m m ineger grid ined of he Hnn grid. To ree punured phere, we reple eh erminl poin wih mll dimond ove. We hen fill in eh modified grid ell wih ringulion, hoen o h he hore ph eween ny wo poin on he oundry of ny ell y on he oundry of h ell; hi require onn numer of ringle per ell. The reuling mnifold M h

7 6 Jeff Erikon nd Sriel Hr-Peled () () () Figure 2. () A e of ineger poin. () The modified Hnn grid. () A u-wy view of he reuling punured phere. omplexiy O(m 2 ). By induion, he hore ph eween ny wo poin on he modified grid lie enirely on he grid. Thu, ny miniml unweighed u grph of M onin only edge from he modified grid. I follow h if he minimum unweighed u grph of M h r edge, he lengh of ny reiliner Seiner ree of P i exly r. We n eily generlize he previou proof o mnifold wih higher genu, wih or wihou oundry, oriened or no, y hing mll ringuled ori or ro-p o ny ue of punure. Theorem 3.2. Compuing he lengh of he minimum (weighed or unweighed) u grph of ringuled mnifold wih oundry, wih ny fixed genu or wih ny fixed numer of oundry omponen, i NP-hrd. 4 Compuing Minimum Cu Grph Anywy We now derie n lgorihm o ompue he minimum u grph of polyhedrl mnifold in n O(g+k) ime. For mnifold wih onn Euler hrerii, our lgorihm run in polynomil ime. Our lgorihm i ed on he following hrerizion of he minimum u grph he union of hore ph. A rnh poin of u grph i ny verex wih degree greer hn 2. A imple ph in u grph from one rnh poin or oundry poin o noher, wih no rnh poin in i inerior, i lled u ph. Lemm 4.1. Le M e polyhedrl 2-mnifold, poily wih oundry, nd le G e minimum u grph of M. Any u ph in G n e deompoed ino wo equl-lengh hore ph in M 1.

8 Opimlly Cuing Surfe ino Dik 7 Proof: Le G e n rirry u grph of M, nd onider u ph eween wo (no neerily diin) rnh poin nd of G. Le e he midpoin of hi ph, nd le nd denoe he uph from o nd from o, repeively. Noe h my lie in he inerior of n edge of M 1. Finlly, uppoe i no he hore ph from o in M 1. To prove he lemm, i uffie o how h G i no he hore u grph of M. Le e he rue hore ph from o. Clerly, i no onined in G. Wlking long from o, le e he fir verex whoe following edge i no in G, nd le e he fir verex in G whoe preeding edge i no in G. (Noe h nd my e joined y ingle edge in M\G.) Finlly, le σ e he rue hore ph from o. Equivlenly, σ i he fir mximl uph of whoe inerior lie in M \ G. See Figure 3. σ σ' Figure 3. If he dhed ph from o i horer hn, hen he u grph n e horened y uing long σ nd regluing long σ. The uph σ u M \ G ino wo mller dik. We lim h ome uph σ of eiher or pper on he oundry of oh dik nd i longer hn σ. Our lim implie h uing M\G long σ nd regluing he piee long σ give u new polygonl hem wih mller perimeer, nd hu new u grph horer hn G. See Figure 3 for n exmple. We prove our lim y exhuive e nlyi. Fir onider he e where he mnifold M i orienle. We n udivide he enire oundry of he dik M \ G ino ix ph leled oneuively,, γ,, ᾱ, δ. Here, ᾱ nd re he orreponding opie of nd in he polygonl hem. Beue M i orienle, nd ᾱ hve oppoie orienion, do nd. Eiher or oh of γ nd δ ould e empy. See he lower lef pr of Figure 3.

9 8 Jeff Erikon nd Sriel Hr-Peled δ γ δ γ δ γ () () () δ γ δ γ δ γ (d) (e) (f) Figure 4. Six e for he proof of Lemm 4.1 for orienle mnifold; ll oher e re refleion of hee. In eh e, ome uph of or pper on he oundry of oh u-dik. The uph σ n ener he inerior of he dik M \ G from four of hee ix ph (,, ᾱ, nd ) nd leve he inerior of he dik hrough ny of he ix ph. Suppoe σ ener he inerior of M \ G from ; he oher hree e re ymmeri. Figure 4 how he ix eenilly differen wy for σ o leve he inerior of M \ G. In eh e, we eily verify h fer uing long σ, ome uph σ of eiher or i on he oundry of oh dik. Speifilly: (i) If σ leve hrough or ᾱ (ee Figure 4() nd 4(), repeively), hen σ i he uph of from o. Sine oh σ nd σ hve he me endpoin nd σ i hore ph, σ mu e longer hn σ. (ii) If σ leve hrough or (ee Figure 4(f) nd 4(d), repeively), hen σ i he uph of from o. Indeed, ρ = [, ] σ i he hore ph from o, nd uh σ ρ < [, ] = σ. (Here denoe ph onenion, nd [x, y] denoe he uph of from x o y.) (iii) If σ leve hrough γ (ee Figure 4(e)), hen σ =. Clerly, σ < = = σ. (iv) Finlly, if σ leve hrough δ (ee Figure 4()), hen σ i he uph of from o. Clerly, σ < [, ] σ.

10 Opimlly Cuing Surfe ino Dik 9 δ γ δ γ δ γ δ γ δ γ δ γ Figure 5. Six ddiionl e for he proof of Lemm 4.1 for non-orienle mnifold; ll oher e re refleion or roion of hee. In eh e, ome uph of or pper on he oundry of oh u-dik. If M i non-orienle, he ph ould pper eiher wih he me orienion or wih oppoie orienion on he oundry of he dik M \ G. If he orienion re oppoie, he previou e nlyi pplie immediely. Oherwie, he oundry n e udivided ino ix ph leled oneuively,, γ,,, δ. Wihou lo of generliy, σ ener he inerior of M \ G from nd leve hrough ny of hee ix ph. The ix e re illured in Figure 5. Agin, we eily verify h in eh e, ome uph σ of eiher or i on he oundry of oh dik. We omi furher deil. For ny u grph G of mnifold M, we define he orreponding redued u grph Ĝ follow. Fir we remove ny opologilly rivil u; h i, we repeedly remove ny edge wih verex of degree 1 h i no on he oundry of M. We hen ugmen he u grph y dding ll he oundry edge of M. Finlly, we onr eh mximl ph hrough degree-2 verie ino ingle edge. The reuling redued u grph Ĝ i 2-edge onneed, nd eh of i verie h degree le 3. Every verex of Ĝ i eiher rnh poin or oundry poin of G, nd every edge of Ĝ orrepond o eiher u ph or oundry ph in G. However, in generl, no ll rnh poin nd u ph in G re repreened in Ĝ. Lemm 4.2. Le M e polyhedrl 2-mnifold wih genu g nd k oundry omponen. Any redued u grph Ĝ of M i onneed, h eween mx{1, k} nd 4g +2k 2 verie, nd h eween g + mx{0, 2k 1} nd 6g + 3k 3 edge.

11 10 Jeff Erikon nd Sriel Hr-Peled Proof: Le G e he u grph orreponding o Ĝ, fer ll he rivil u hve een removed. The oundry of he polygonl hem M \ G n e priioned ino u ph nd oundry ph, eh orreponding o n edge in Ĝ. Thu, Ĝ i onneed. Le v nd e denoe he numer of verie nd edge in Ĝ, repeively. If ny verex in Ĝ h degree d 4, we n reple i wih d 3 rivlen verie nd d 3 new edge of lengh zero. Thu, in he wor e, every verex in Ĝ h degree exly 3, whih implie h 3v = 2e. Sine Ĝ i emedded in M wih ingle fe, Euler formul implie h v e + 1 = χ = 2 2g k if M i orienle, nd v e + 1 = χ = 2 g k if M i non-orienle. I follow h v 4g + 2k 2 nd e 6g + 3k 3, limed. On he oher hnd, Ĝ h le one verex on eh oundry omponen of M, nd le one verex even if M h no oundry, o v mx{1, k}. Thu, Euler formul implie h 2 2g k = v e + 1 mx{1, k} e + 1 if M i orienle, or equivlenly, e 2g + mx{0, 2k 1}. Similrly, if M i non-orienle, Euler formul implie h e g + mx{0, 2k 1}. Our minimum u grph lgorihm exploi Lemm 4.1 y ompoing poenil minimum u grph ou of O(g + k) hore ph. Unforunely, ingle pir of node in M ould e joined y 2 Ω(n) hore ph, in 2 Ω(g+k) differen ioopy le, in he wor e. To void hi ominoril exploion, we n dd rndom infinieiml weigh ε w(e) o eh edge e. The Iolion Lemm of Mulmuley, Vzirni, nd Vzirni [MVV87] implie h if he weigh w(e) re hoen independenly nd uniformly from he ineger e {1, 2,..., n 2 }, ll hore ph re unique wih proiliy le 1 1/n; ee lo [CRS95, KS01]. 4 We re now finlly redy o derie our minimum u grph lgorihm. Theorem 4.3. The minimum u grph of polyhedrl 2-mnifold M wih genu g nd k oundry omponen n e ompued in ime n O(g+k). Proof: We egin y ompuing he hore ph eween every pir of verie in M in O(n 2 log n) ime y running Dijkr ingle-oure hore ph lgorihm for eh verex [Dij59, Joh77], reking ie uing rndom infinieiml weigh deried ove. One hee hore ph nd midpoin hve een ompued, our lgorihm enumere y rue fore every poile u grph h ifie Lemm 4.1 nd 4.2, nd reurn he mlle uh grph. Eh u grph i peified y e V of up o 4g + 2k 2 verie of M, e E of up o 6g + 3k 3 edge of M, rivlen muligrph Ĝ wih verie V, nd ignmen of edge in E o edge in Ĝ. Eh edge (v, w) of Ĝ i igned unique edge e E o define he orreponding u ph in M. Thi u ph i he onenion of he hore ph from v o e, e ielf, nd he hore ph from e o w. If he midpoin of hi u ph i no in he inerior of e, we delre he u ph invlid, ine i viole Lemm 4.1. (Beue 4 Alernely, if we hooe w(e) uniformly from he rel inervl [0, 1], hore ph re unique wih proiliy 1. Thi my ound unreonle, u rell h no polynomil-ime lgorihm i known o ompre um of qure roo of ineger in ny model of ompuion h doe no inlude qure roo primiive operion [Blö91]. Thu, o ompue Euliden hore ph in geomeri grph wih ineger verex oordine, we mu eiher ume ex rel rihmei or (grudgingly) ep ome pproximion error [GGJ77].

12 Opimlly Cuing Surfe ino Dik 11 Figure 6. From lef o righ: non-epring, eenil u epring, nd rivil yle on 2-mnifold. hore ph eween verie re unique, he midpoin of ny u ph in he miniml u grph mu lie in he inerior of n edge.) If ll he u ph re vlid, we hen hek h every pir of u ph i dijoin, exep poily heir endpoin, nd h removing ll he u ph from M leve opologil dik. Our rue-fore lgorihm onider O(n 4g+2k 2 ) differen verex e V, O(n 6g+3k 2 ) differen edge e E, mo ( ) (4g+2k 2) 2 6g+3k 2 differen grph Ĝ for eh verex e, nd mo (6g + 3k 2)! differen ignmen of edge in eh grph Ĝ o edge in eh edge e E. Thu, n O(g+k) poenil u grph re onidered logeher. The vlidiy of eh poenil u grph n e heked in O(n) ime. 5 Finding Shor Nonrivil Cyle A ep owrd effiienly ompuing pproxime minimum u grph, we develop lgorihm o ompue hore nd nerly-hore nonrivil yle in rirry 2-mnifold, poily wih oundry. Alhough our mo effiien pproximion lgorihm for u grph require only pproxime hore nonrivil yle on mnifold wihou oundry, we elieve hee lgorihm re of independen inere. We diinguih eween wo ype of nonrivil imple yle. A imple yle γ in M i non-epring if M \ γ h only one onneed omponen. A imple yle γ in M i eenil if i i no onrile o poin or ingle oundry yle of M. Every nonepring yle i eenil, u he onvere i no rue. Formlly, non-epring yle re homologilly nonrivil, nd eenil yle re homoopilly nonrivil. See Figure 6. A n e een immediely from Figure 6, i i no poile o deermine wheher yle i non-epring, eenil, or rivil y exmining only lol neighorhood. Dey nd Shipper [DS95] derie n lgorihm o deermine wheher n rirry irui i onrile in O(n) ime; heir lgorihm egin y ompuing n rirry u grph of he mnifold. Forunely, we hll ee horly, we n implify heir lgorihm oniderly when he given yle i imple. 5.1 Shore Cyle We egin y deriing how o find he hore nonrivil yle hrough given verex. Our lgorihm ue ominion of Dijkr ingle-oure hore ph lgorihm [Dij59] nd

13 12 Jeff Erikon nd Sriel Hr-Peled modifiion of he nonil polygonl hem lgorihm of Lzru e l. [LPVV01]. Our lgorihm i imilr o he pproh ken y Gukov nd Wood [GW01] o find nd remove mll hndle from geomeri model reonrued from noiy d; ee lo [WHDS02]. The lgorihm of Lzru e l. uild onneed ue S of ringuled mnifold wihou oundry, ring wih ingle ringle nd dding new ringle on he oundry of S one ime. If new ringle inere he oundry of S in more hn one omponen, he lgorihm hek whih of he following hree e hold: (1) M \ S i onneed; (2) neiher omponen of M \ S i dik; or (3) one omponen of M \ S i dik. In he finl e, he lgorihm dd he dik omponen o S nd oninue erhing he oher omponen of M \ S. If we run hi lgorihm unil eiher e (1) or e (2) hold, he ol running ime i O(n). See Lzru e l. [LPVV01] for furher deil. Fir we derie righforwrd generlizion of he lgorihm ued y Lzru o deermine he ruure of M \ S. Lemm 5.1. Le M e onneed polyhedrl 2-mnifold M whoe genu g nd numer of oundry omponen k re known. Any imple yle γ in M 1 n e lified nonepring, eenil u epring, or rivil in O(m) ime, where m i he omplexiy of he mller omponen of M \ γ. In priulr, if γ i non-epring, he running ime i O(n). Proof: We perform wo imulneou deph-fir erhe, ring from eiher ide of ny verex of γ. If eiher erh mee verex lredy viied y he oher, γ i non-epring yle. The running ime in hi e i rivilly O(n). Converely, if one erh hl wihou meeing he oher, γ i epring yle. Le C e he mller omponen of M \ γ. To deermine wheher γ i eenil, we ompue he Euler hrerii nd he numer of oundry omponen of C, whih we denoe y χ(c) nd k(c), repeively. (Thi n e done on he fly during he deph-fir erh phe.) Le χ denoe he Euler hrerii of M. The yle γ i onrile if nd only if one of he following ondiion hold. ˆ C i dik, or equivlenly, χ(c) = 1. ˆ C i n nnulu, or equivlenly, χ(c) = 0 nd k(c) = 2. ˆ M \ C i dik, or equivlenly, χ(c) = χ 1. The equivlene follow from he inluion-exluion formul χ(a B) = χ(a) + χ(b) χ(a B). ˆ M \ C i n nnulu, or equivlenly, χ(c) = χ(m) nd k(c) = k(m). If none of hee ondiion hold, hen γ i eenil. The ol running ime if γ i epring yle i O(m), where m i he omplexiy of C. Noe h we n implify hi lgorihm lighly if he mnifold h no oundry, ine in h e neiher omponen of M \ γ i n nnulu. We will ue hi implifiion in our pproxime u grph lgorihm.

14 Opimlly Cuing Surfe ino Dik 13 Lemm 5.2. Le u e verex of polyhedrl 2-mnifold M, poily wih oundry. The hore eenil yle in M 1 h onin u n e ompued in O(n log n) ime. Proof: We find he hore eenil yle hrough u y imuling irulr wve expnding from u. Whenever he wve ouhe ielf, eiher we hve he hore eenil yle hrough u, or one omponen of he wve ound dik in M nd we n oninue expnding he oher omponen. We modify he lgorihm of Lzru e l. [LPVV01] follow. Fir, S i no longer e of ringle u more generl onneed ue of verie, edge, nd fe of M. Iniilly, S onin only he oure verex u. Seond, we ue Dijkr lgorihm o deermine he order for edge o e dded. We dd fe o S only when ll i verie hve een dded o S, eiher direly or pr of noher fe. We run he lgorihm from Lemm 5.1 whenever S i no longer imply onneed, h i, when we dd new edge vw wih oh endpoin on he oundry of S. If M\S i dionneed, we oninue only if one omponen of M \ S i dik or n nnulu, n e heked uing Lemm 5.1. In h e, we dd he dik or nnulu omponen of M \ S o S, dird he verie of h omponen from he Dijkr prioriy queue, nd oninue erhing in he oher omponen. If M \ S i onneed, or if neiher omponen i dik or n nnulu, we hve found he hore eenil yle hrough u, oniing of he hore ph from u o v, he edge vw, nd he hore ph from w o u. Dijkr lgorihm require O(n log n) ime. Eh ime we find rivil yle, we pend O(m) ime nd dird dik wih omplexiy le m. Thu, he ol ime pen performing yle lifiion nd minining he wvefron e S i O(n). Thu, he ol running ime of our lgorihm i O(n log n). Running hi lgorihm one for every verex of M immediely give u he hore eenil yle. Corollry 5.3. Le M e polyhedrl 2-mnifold, poily wih oundry. The hore eenil yle in M 1 n e ompued in O(n 2 log n) ime. A imple modifiion of our lgorihm llow u o find hore non-epring yle in he me ympoi ime. Lemm 5.4. Le u e verex of polyhedrl 2-mnifold M, poily wih oundry. The hore non-epring yle in M 1 h onin u n e ompued in O(n log n) ime. Proof: The only hnge from he previou lgorihm i h if we diover n eenil epring yle, we oninue reurively in oh omponen of M\S. The o of Dijkr lgorihm i ill O(n log n), u we now mu pend exr ime in he yle-lifiion lgorihm of Lemm 5.1. A efore, he ol ime pen finding rivil yle i O(n), ine we n hrge he erh ime o he dirded omponen. Le T (n, g) denoe he ol ime pen finding epring eenil yle. Thi funion ifie he reurrene T (n, g) T (m, h) + T (n m, g h) + O(m),

15 14 Jeff Erikon nd Sriel Hr-Peled where m n/2 i he omplexiy of he mller omponen of M \ S nd h i i genu. The e e of he reurrene i T (n, 1) = 0, ine every eenil yle on genu-1 urfe i non-epring. Similr reurrene pper in he nlyi of oupu-eniive plnr onvex hull lgorihm [BS97, CSY97, KS86, Wen97], uggeing h he oluion o our reurrene i T (n, g) = O(n log g). Indeed, we n prove hi y induion follow. Suppoe T (n, g) T (m, h) + T (n m, g h) + m for ome onn. We lim h T (n, g) n lg g. The induive hypohei implie h T (n, g) m lg h + (n m) lg(g h) + m mx (m lg h + (n m) lg(g h)) + m. 1 h g 1 A imple ppliion of derivive implie h he righ hnd ide of hi inequliy i mximized when h = mg/n. Thu, T (n, g) m lg mg (n m)g + (n m) lg + m n n = n lg g + m lg m + (n m) lg(n m) n lg n + m. Sine m n/2 nd n m n, we n implify hi inequliy o T (n, g) n ln g + m lg(n/2) + (n m) lg n n lg n + m = n lg g, ompleing he proof. Thu, he ol ime pen in he yle-lifiion phe of our lgorihm i O(n log g). Sine g n, hi i domined y he o of minining he Dijkr prioriy queue. Corollry 5.5. Le M e n polyhedrl 2-mnifold, poily wih oundry. The hore non-epring yle in M 1 n e ompued in O(n 2 log n) ime. 5.2 Nerly-Shore Cyle A we will rgue in he nex eion, ompuing hor nonrivil yle i he olenek in our pproxime u grph lgorihm. Forunely, ex minimum yle re no neery for our reul. We n peed up our u grph lgorihm, wihou ignifinly inreing he pproximion for, y erhing for nonrivil yle mo wie long he hore. Our pproximion lgorihm ume h he mnifold M h no oundry; forunely, we hll ee in he nex eion, hi i uffiien for our purpoe. Our pproximion lgorihm work follow. Fir, we ompue e of hore ph (in f, u grph) h inere every non-epring yle in he mnifold M. Then we onr eh hore ph π in hi e o poin nd find he hore nonrivil yle hrough h poin, deried y Lemm 5.2 nd 5.4.

16 Opimlly Cuing Surfe ino Dik 15 Lemm 5.6. Le π e hore ph eween wo verie in polyhedrl 2-mnifold M, nd le γ e he hore eenil (rep. non-epring) yle in M 1 h inere π. In O(n log n) ime, one n ompue n eenil (rep. non-epring) yle γ in M uh h γ 2 γ. Proof: Le M e he mnifold oined y onring he hore ph π o ingle verex v. Beue π h no yle, M h he me opologil ype M. Le γ e he hore eenil (rep. non-epring) yle in M h pe hrough v. Clerly, γ γ. We n ompue hi yle in O(n log n) ime y Lemm 5.2 (rep. Lemm 5.4). We onru yle γ in M y onening wo ph nd, where onin he edge of γ nd i he uph of π eween he endpoin of. The equene of edge onrion h rnform M o M lo rnform γ o γ. Hene, γ i n eenil yle of M. Beue i uph of hore ph, i ully he hore ph eween he endpoin of, o = γ. I follow h γ = + 2 γ 2 γ. Thi lemm ugge nurl lgorihm for finding hor nonrivil yle: Compue e of hore ph h inere every non-epring yle (nd hu every eenil yle), nd hen run he lgorihm from Lemm 5.6 for eh ph in hi e. Lemm 5.7. Le M e polyhedrl 2-mnifold wihou oundry. In O(n log n) ime, one n ompue e Π of O(g) hore ph on M 1 uh h every non-epring yle (nd hu every eenil yle) in M 1 inere le one ph in Π. Proof: We ompue u grph G follow. Fir we ompue hore-ph ree T from n rirry iniil verex v uing Dijkr lgorihm. We hen ompue n rirry pnning ree T of he dul of M \ T, h i, he grph whoe verie re fe of M nd whoe edge join pir of fe h hre ommon edge no in T. Anlyi imilr o Lemm 4.2 implie h here re O(g) edge h do no pper in T nd whoe dul edge o no pper in T. Cll hi e of unlimed edge E. Le Π e he e of O(g) hore ph from v o he endpoin of E; hee ph re ll in T. Finlly, le G = Π E. We eily oerve h M \ G i opologil dik, o G i u grph. I follow h every non-epring yle in M 1 inere G. Sine every verex of G i lo verex of ome ph in Π, every non-epring yle in M 1 inere le one ph in Π. Noie h hi lgorihm doe no work if M h oundry, ine he dul grph of M \ T ould e dionneed. Corollry 5.8. Le M e polyhedrl 2-mnifold wih genu g nd no oundry, nd le γ e i hore eenil (rep. non-epring) yle. In O(gn log n) ime, one n ompue n eenil (rep. non-epring) yle γ in M 1 uh h γ 2 γ. Proof: We onru e Π of O(g) hore ph, le one of whih i gurneed o inere γ, deried in he previou lemm. Then for eh ph π Π, we onr π o poin nd find he hore nonrivil yle hrough h poin in O(n log n) ime, deried y Lemm 5.6.

17 16 Jeff Erikon nd Sriel Hr-Peled 6 Approxime Minimum Cu Grph We now derie imple polynomil-ime greedy lgorihm o onru n pproxime minimum u grph for ny polyhedrl mnifold M. To hndle mnifold wih oundry, i will e onvenien o onider he following implified form. Given mnifold M wih genu g nd k oundry omponen, he orreponding punured mnifold (M, P ) oni of mnifold M wih he me genu M u wihou oundry, nd e P of k poin in M, lled punure. To onru M, we onr every oundry omponen of M o ingle poin, whih eome one of he punure in P. 5 If ny verex of M h muliple edge o he me oundry omponen, M onin only he edge wih mlle weigh, reking ie uing he Iolion Lemm ove. If M h no oundry, hen M = M nd P =. Our gol now i o ompue he minimum u grph of M h ouhe every punure in P ; heneforh, we ll hi imply he minimum u grph of (M, P ). Thi reduion i moived y he following rivil oervion. Lemm 6.1. The minimum u grph of ny polyhedrl 2-mnifold M h he me lengh he minimum u grph of (M, P ). Our pproximion lgorihm work follow. We repeedly u long hor nonrivil yle unil our urfe eome olleion of punured phere, onne he punure on eh omponen y uing long minimum pnning ree, nd finlly (if neery) reglue ome previouly u edge o oin ingle dik. The reuling u grph i ompoed of ue of he edge of he hor nonrivil yle nd ll he edge of he minimum pnning fore. 6.1 Uing Shor Nonrivil Cyle The fir omponen of our lgorihm i urouine o ompue pproximely hore nonrivil yle, deried y Corollry 5.8. A required y h lgorihm, he inpu mnifold M h no oundry; he punure re ompleely ignored. The following rgumen rele he lengh of he hore nonrivil yle o he lengh of he minimum u grph. Lemm 6.2. Le G e ny u grph of polyhedrl 2-mnifold M wih genu g nd no oundry. The hore yle in G onin O((log g)/g) of he ol lengh of G. Proof: Fir onider he redued u grph Ĝ, onrued y repeedly onring ny edge wih verex of degree le hn hree, in Seion 4. Every verex in Ĝ h degree le 3. Wihou lo of generliy, ume h every verex in Ĝ h degree exly 3, pliing eh high-degree verex ino ree of degree-3 verie if neery, in he proof of Lemm 4.2. A righforwrd ouning rgumen implie h ny rivlen grph whoe 5 We ould imule hi onrion y rifiilly igning every oundry edge of M weigh of zero, lhough hi would require few imple hnge in our lgorihm.

18 Opimlly Cuing Surfe ino Dik 17 girh (minimum yle lengh) i mu hve le /2 2 verie if i odd, nd le 2 2 /2 2 verie if i even [Big98]. By Lemm 4.2, Ĝ h mo 4g 2 verie, o Ĝ mu hve yle ˆγ wih mo 2(lg g + 1) = O(log g) edge. Sring wih Ĝ0 = Ĝ, we induively define equene of redued grph Ĝ1, Ĝ2,... follow. For eh i > 0, le ˆγ i denoe he hore yle in Ĝi 1. We oin Ĝi y reduing he grph Ĝi 1 \ ˆγ i, or equivlenly, removing he verie of ˆγ i nd ll heir edge, nd hen onring ˆγ i nery lengh-2 ph o ingle edge. Our erlier rgumen implie h eh yle ˆγ i h mo 2(lg g + 1) edge. Thu, for eh i, we hve E(Ĝi) = E(Ĝi 1) 6(lg g + 1). Lemm 4.2 implie h he originl redued u grph Ĝ h le g edge, o we n repe hi proe le g/6(lg g + 1) ime. Le γ i denoe he yle in he originl u grph G orreponding o Ĝi. By our onruion, γ i nd γ j re dijoin for ll i j, o we hve e of le g/6(lg g + 1) dijoin yle in G. A le one of hee yle h lengh mo 6(lg g + 1)/g = O((log g)/g) ime he ol lengh of G. Sine every yle in he minimum u grph i non-epring, nd herefore eenil, we immediely hve he following orollry. Corollry 6.3. For ny polyhedrl 2-mnifold M wih genu g nd no oundry, oh he lengh of he hore non-epring yle nd he lengh of he hore eenil yle re mo O((log g)/g) ime he lengh of he minimum u grph of M. 6.2 Punure-Spnning Tree The eond omponen of our u grph lgorihm i urouine o ompue he minimum punure-pnning ree of punured mnifold (M, P ), h i, he minimum pnning ree of he punure P in he hore-ph meri of M 1. Lemm 6.4. The minimum punure-pnning ree of ny punured polyhedrl 2-mnifold (M, P ) n e ompued in O(n log n) ime. Proof: We imule Krukl minimum pnning ree lgorihm [Kru56, Tr83] y dding hore punure-o-punure ph one ime, in inreing order of lengh. To ompue he hore ph, we imulneouly propge wvefron from ll k punure uing Dijkr lgorihm. Whenever wo wvefron (i.e., wo growing hore-ph ree) ollide, we dd new edge o he evolving minimum pnning ree nd merge hoe wo wvefron. To implemen hi lgorihm effiienly, we minin he wvefron in union-find d ruure. The reuling running ime i O(n log n). Thi i eenilly he lgorihm propoed y Tkhhi nd Muym [TM80] o ompue pproxime Seiner ree in rirry grph. The me lgorihm w lo reenly ued y Sheffer [She02] nd y Lévy e l. [LPRM02] o ompue u grph, where urfe feure wih high diree urvure ply he role of punure.

19 18 Jeff Erikon nd Sriel Hr-Peled Lemm 6.5. The lengh of he minimum punure-pnning ree of ny punured polyhedrl 2-mnifold (M, P ) i mo wie he lengh of ny u grph of (M, P ). Proof: The minimum Seiner ree of P i he ugrph of M 1 of minimum ol weigh h inlude every poin in P. Sine ny u grph of (M, P ) mu ouh every punure, no u grph i horer hn hi minimum Seiner ree. On he oher hnd, he minimum pnning ree of P h mo wie he lengh of he minimum Seiner ree [KMB81, TM80]. 6.3 Anlyi We now hve ll he omponen of our greedy u grph lgorihm. A ny ge of he lgorihm, we hve punured mnifold (M, P ). Our lgorihm repeedly u long hor non-epring yle of M, uing Corollry 5.8. Thi u ree one or wo new oundry irle, whih we ollpe o new punure. When he mnifold i redued o olleion of punured phere, we u long he minimum punure-pnning ree of eh omponen uing he lgorihm in Lemm 6.4. Eh non-epring yle u redue he genu of M y 1. Thi immediely implie h our lgorihm perform exly g yle u, o he overll running ime i g O(gn log n) + O(n log n) = O(g 2 n log n). For ny grph X, le X denoe i ol lengh. Le G denoe he minimum u grph of (M, P ). Le (M i, P i ) denoe he punured mnifold fer g i yle u hve een performed, o M i h genu i, nd le G i denoe he minimum u grph of M i, ignoring he punure P i. Sine ollping edge nno inree he minimum u grph lengh, we hve G i G g G for ll i. Le γ i denoe he hor non-epring yle of M i found y Corollry 5.8. (We eily oerve h ny u grph of M i mu inere hi yle.) Lemm 6.2 nd Corollry 5.8 imply h γ i O((log i)/i) G i for ll i nd j. Summing over ll g u, we onlude h he ol lengh of ll yle u i mo g O((log i)/i) G i = O(log 2 g) G. i=1 Similrly, Lemm 6.5 implie h he minimum punure-pnning fore h lengh mo 2 G. Finlly, regluing previouly u edge o oin ingle dik only redue he lengh of he finl u grph. Thu, he finl u grph ompued y our lgorihm h lengh mo O(log 2 g) G. Theorem 6.6. Given polyhedrl 2-mnifold M wih genu g nd k oundry omponen, n O(log 2 g)-pproximion of i minimum u grph n e onrued in O(g 2 n log n) ime. Cuing long hor eenil yle ined of hor non-epring yle led o exly he me ympoi running ime nd pproximion ound, lhough he lgorihm nd

20 Opimlly Cuing Surfe ino Dik 19 i nlyi re lighly more omplied. For purpoe of nlyi, we n divide he lgorihm ino phe, where in he ih phe, we u long hor eenil yle of every omponen of he mnifold h h genu i. Eenil yle u n epre he mnifold ino muliple omponen, u ine eh omponen mu hve nonrivil opology, he lgorihm perform mo g 1 epring u. A he end of he lgorihm, if neery, we reglue long ome previouly u edge o oin ingle opologil dik. We refer o he erlier verion of hi pper for furher deil [EHP02]. 7 Open Prolem We hve developed new lgorihm o ompue ex nd pproxime miniml u grph for mnifold urfe wih rirry genu nd rirry oundry omplexiy. Our pproximion lgorihm i priulrly imple. Our reul ugge everl open prolem, he mo oviou of whih i o improve he running ime nd pproximion for of our lgorihm. I he minimum u grph prolem fixed-prmeer rle [DF99]? Th i, n we ompue ex minimum u grph in ime f(g, k) n O(1) for ome funion f? The imilriy o he Seiner prolem offer ome hope here, ine he minimum Seiner ree of k node in n n-node grph n e ompued in O(3 k n + 2 k n 2 + n 3 ) ime [DW71, HW94]. The pproximion lgorihm of Theorem 6.6 i omewh indire. I ompue hor u grph y repeedly ompuing reonle u grph nd hen exring hor nonrivil yle h iner wih hi u grph. I i nurl o onjeure h one n ompue uh hor u grph direly, reuling in fer lgorihm. In priulr, we onjeure h n pproximely minimum u grph n e ompued in O(gn log n) ime. How well n we pproxime he minimum u grph in nerly-liner ime? There re everl imple heurii o ompue good u grph in O(n log n) ime, uh he dul hore-ph lgorihm ued y Gu e l. [GGH02] nd y Seiner nd Fiher [SF02], nd he lgorihm deried in he proof of Lemm 5.7. How well do hee lgorihm pproxime he minimum u grph? More generlly, i here imple, pril, O(1)-pproximion lgorihm, like he minimum pnning ree pproximion of Seiner ree? In f, i migh e h our lgorihm provide uh n pproximion, our urren nlyi eem o e fr from igh. Unforunely, he generl Seiner ree prolem i MAXSNP-hrd [BP89], o n effiien (1 + ε)-pproximion lgorihm for rirry ε > 0 eem unlikely. Severl uhor hve poined ou ppren rdeoff eween he quliy of prmeerizion nd he lengh of he required u grph; ee, for exmple, Sorkine e l. [SCOGL02]. How hrd i i o ompue he (pproximely) hore u grph required for prmeerizion whoe diorion i le hn ome given limi? Converely, how hrd i i o (pproximely) minimize he diorion of prmeerizion, given n upper ound on he permied lengh of he urfe u? The omplexiy of hee prolem lerly depend he whih diorion meure i ued, u we expe lmo ny vrin of hi prolem o e NP-hrd.

21 20 Jeff Erikon nd Sriel Hr-Peled Finlly, n our ide e pplied o oher ueful fmilie of urve on mnifold, uh homology generor (fmilie of 2g yle h inere in g pir) nd pn deompoiion (mximl e of pirwie dijoin eenil yle [Thu97])? Aknowledgmen. We would like o hnk Herer Edelrunner for n enlighening iniil onverion. We re lo greful o Nog Alon, Seven Gorler, John Hr, Benjmin Sudkov, Kim Whileey, nd Zoë Wood for helpful diuion. Referene [AE98] [AMD02] [Big98] [Blö91] Ulrike Axen nd Herer Edelrunner. Audiory More nlyi of ringuled mnifold. In H.-C. Hege nd K. Polhier, edior, Mhemil Viulizion, pge Springer-Verlg, Pierre Alliez, Mrk Meyer, nd Mhieu Derun. Inerive geomery remehing. In Pro. SIGGRAPH 02, o pper, Normn Bigg. Conruion for ui grph wih lrge girh. Ele. J. Comin., 5:A1, J. Blömer. Compuing um of rdil in polynomil ime. In Pro. 32nd Annu. IEEE Sympo. Found. Compu. Si., pge , [BP89] Mrhll Bern nd Pul Plmn. The Seiner prolem wih edge lengh 1 nd 2. Inform. Pro. Le., 32(4): , [BS97] [BVIG91] [CL02] [CRS95] B. K. Bhhry nd S. Sen. On imple, pril, opiml, oupu-eniive rndomized plnr onvex hull lgorihm. J. Algorihm, 25: , Chki Benni, Jen-Mr Vézien, Gérrd Igléi, nd André Gglowiz. Pieewie urfe flening for non-diored exure mpping. In Thom W. Sedererg, edior, Compuer Grphi (SIGGRAPH 91 Proeeding), volume 25, pge , Èri Colin de Verdiére nd Frni Lzru. Opiml yem of loop on n orienle urfe. In Pro. 43rd Annu. IEEE Sympo. Found. Compu. Si., pge o pper, Sureh Chri, Pnkj Rohgi, nd Arvind Srinivn. Rndomne-opiml unique elemen iolion wih ppliion o perfe mhing nd reled prolem. SIAM J. Compu., 24(5): , [CSY97] T. M. Chn, J. Snoeyink, nd C. K. Yp. Priml dividing nd dul pruning: Oupu-eniive onruion of 4-d polyope nd 3-d Voronoi digrm. Diree Compu. Geom., 18: , 1997.

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