Opimlly Cuing Surfe ino Dik Jeff Erikon Sriel Hr-Peled Univeriy of Illinoi Urn-Chmpign {jeffe,riel}@.uiu.edu hp://www..uiu.edu/ {jeffe,riel} ABSTRACT 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. Cegorie nd Suje Deripor F.2.2 [Anlyi of Algorihm nd Prolem Complexiy]: Nonnumeril Algorihm nd Prolem Geomeril prolem nd ompuion; G.2.m [Diree Mhemi]: Miellneou Cominoril opology Generl Term Algorihm Keyword ompuionl opology, polyhedrl 2-mnifold, polygonl hem, u grph, NP-hrdne, pproximion 1. INTRODUCTION Severl ppliion of hree-dimenionl urfe require informion ou underlying opologil ruure in ddiion o geomery. In ome e, we wih o implify he See hp://www..uiu.edu/ jeffe/pu/hem.hml for he mo reen verion of hi pper. Prilly uppored y Slon Fellowhip, NSF CAREER wrd CCR-0093348, nd NSF ITR grn DMR-0121695. Permiion o mke digil or hrd opie of ll or pr of hi work for peronl or lroom ue i grned wihou fee provided h opie re no mde or diriued for profi or ommeril dvnge nd h opie er hi noie nd he full iion on he fir pge. To opy oherwie, o repulih, o po on erver or o rediriue o li, require prior peifi permiion nd/or fee. SoCG 02, June 5-7, 2002, Brelon, Spin. Copyrigh 2002 ACM 1-58113-504-1/02/0006... 5.00. urfe opology, o filie lgorihm h n e performed only if he urfe i opologil dik. Appliion when hi i imporn inlude urfe prmeerizion [14, 29] nd exure mpping [2, 28]. In he exure mpping prolem, one wih o find oninuou 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. [18] 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 [14, 29]. To our knowledge, ll urren pprohe for hi uing prolem eiher rely on heurii wih no quliy gurnee or require he uer o perform hi uing eforehnd [14, 28]. The prolem of ju uing urfe ino opologil dik i no rivil y ielf. Lzru e l. [25] 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 [34]. 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 [9], or imply king mximl e of edge whoe omplemen i onneed [25]. 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 [9] derie f lgorihm
e e e d d d Figure 1. A u grph for wo-holed oru nd i indued (nonnonil) polygonl hem. 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 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. We pln o implemen hi lgorihm nd preen experimenl reul in he ner fuure. 1 To implify noion, we define log x = mx{1, log 2 x }. d e 2. BACKGROUND 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 [21], Munkre [27], or Sillwell [30] for furher opologil kground nd more forml definiion. For reled ompuionl reul, ee he reen urvey y Dey, Edelrunner, nd Guh [7] nd Veger [33]. 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 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 1, 2,..., g in M; h i, i j = for ll i nd j, nd M \ ( 1 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 elf-inereion) 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 [15]. 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 [13] 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 10 + 10g 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, 2 Cu grph re generlizion of he u lou of mnifold M, whih i eenilly he geodei medil xi of ingle poin.
he numer of oundry omponen k, nd wheher he mnifold i orienle. 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. 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. Dey nd Shipper [9] 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 [8].) Veger nd Yp [34] 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. [25]. 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 nd eenil yle nd nerly-miniml u grph. 3. COMPUTING MINIMUM CUT GRAPHS IS NP-HARD In hi eion, we prove h finding minimum u grph of ringuled mnifold i NP-hrd. We onider wo exreme e: oundry u no genu, nd genu u no oundry. 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 [17]. 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 [20]. Theorem 3.1. Compuing he lengh of he minimum (weighed or unweighed) u grph of ringuled punured phere i NP-hrd. Proof: 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. () () Figure 2. () A e of ineger poin. () The modified Hnn grid. () A u-wy view of he reuling punured phere. 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 2 +1 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 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. COMPUTING MINIMUM CUT GRAPHS ANYWAY In hi eion, we 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. 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 of 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 hore ph from o, nd le τ e he hore ph from o. Thu, τ i he fir mximl uph of whoe inerior lie in M \ G. See Figure 3. The uph τ u M \ G ino wo mller dik. We lim h ome uph τ of eiher or pper on he oundry of oh dik nd i longer hn τ. Our lim implie h uing M \ G long τ nd regluing he piee long τ 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,,,, ᾱ,. 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 nd ould e empy. See he lower lef pr of Figure 3. The uph τ n ener he inerior of he dik M \ D from four of hee ix ph (,, ᾱ, nd ) nd leve he inerior of he dik hrough ny of he ix ph. Suppoe τ ener he inerior of M \ G from ; he oher hree e re ymmeri. Figure 4 how he ix eenilly differen wy for τ o leve he inerior of M \ D. In Figure 3. If he dhed ph from o i horer hn he equllengh ph nd in he u grph, hen he u grph n e horened y uing nd regluing. eh e, we eily verify h fer uing long τ, ome uph τ of eiher or i on he oundry of oh dik. Speifilly, if τ leve hrough or ᾱ, hen τ o e he uph of from o. If τ leve hrough or, hen τ i he uph of from o. If τ leve hrough, hen τ =. Finlly, if τ leve hrough, hen τ i he ue of from o. 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,,,,,. Wihou lo of generliy, τ 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 τ 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 h follow. We fir ugmen he u grph y dding ll he oundry edge of M. Nex, we remove every verex of degree one nd i only edge, mking he grph 2-edge-onneed. Finlly, we reple eh
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. 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.
mximl ph hrough degree-2 verie wih ingle edge, o h eh verex in he redued u grph Ĝ 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 re repreened in Ĝ. Lemm 4.2. Any redued u grph Ĝ of mnifold M h mo 4g + 2k 2 verie nd 6g + 3k 3 edge. Proof: 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. 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 [26] 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 [6, 23]. 3 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) y running Dijkr ingle-oure hore ph lgorihm for eh verex [10, 22], 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, 3 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[5]. 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 [16]. we delre he u ph invlid, ine i viole Lemm 4.1. (Beue 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 verex e V, n 4g+2k 2 differen n (4g+2k 2) 6g+3k 2 differen edge e E, 2 6g+3k 2 differen grph Ĝ, nd (6g + 3k 2)! differen edge ignmen. 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. APPROXIMATE MINIMUM CUT GRAPHS In hi eion, we 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. 4 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 =. Thi reduion i moived y he following rivil oervion. Lemm 5.1. The minimum u grph of M h he me lengh he minimum u grph of M h inlude every punure in P. A imple yle in M i eenil if i doe no ound dik or n nnulu. In erm of punured mnifold, yle in M i eenil if i doe no ound dik onining le hn wo punure in P. Our pproximion lgorihm work follow. We repeedly u long hor eenil 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 eenil yle nd ll he edge of he minimum pnning fore. Before we derie our lgorihm in more deil, we fir derie how o exeue eh of he uing operion nd how he lengh of eh ype of u ompre o he minimum u grph lengh. 5.1 Shore Eenil Cyle We now derie n lgorihm o ompue he hore eenil yle in he 1-keleon M 1 of polyhedrl 2- mnifold M. Alhough our mo effiien pproximion 4 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.
lgorihm for u grph doe no ue require he hore eenil yle, we elieve hi lgorihm i of independen inere. Our lgorihm ue ominion of Dijkr ingle-oure hore ph lgorihm [10] nd modifiion of he nonil polygonl hem lgorihm of Lzru e l. [25]. The lgorihm of Lzru e l. uild onneed ue S of ringuled mnifold, ring wih ingle ringle nd dding new ringle one ime on he oundry. 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 fir wo e, S onin n eenil yle. In e (1), he hek run in O(n) ime; in he oher wo e, he running ime of he hek i proporionl o he ize of he mller omponen of M \ S. In e (3), 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. [25] for furher deil. Lemm 5.2. Le u e verex of polyhedrl 2-mnifold M. 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. in hree wy. 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 Lzru opology hek when S i no longer imply onneed, h i, when we dd new edge vw wih oh endpoin on he oundry of S. (In he unweighed e, our lgorihm ehve imilrly o he wve rverl lgorihm of Axen nd Edelrunner [1].) Our hird hnge i h if M\S i dionneed, we oninue only if one omponen of M\S i dik or n nnulu. 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. Oherwie, 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. Alogeher, Dijkr lgorihm require O(n log n) ime. By our erlier diuion, he ol ime pen modifying he wve e S i only O(n). Thu, he ol running ime of our lgorihm i O(n log n). Running hi lgorihm one for every verex of M give u he following: Corollry 5.3. Le M e n polyhedrl 2-mnifold. The hore eenil yle in M 1 n e ompued in O(n 2 log n) ime. The following lemm rele he lengh of he hore eenil yle o he lengh of he minimum u grph. Lemm 5.4. Le G e ny u grph of 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 girh (minimum yle lengh) i mu hve le 3 2 2 /2 2 verie if i odd, nd le 2 2 /2 2 verie if i even [4]. By Lemm 4.2, Ĝ h mo 4g 2 verie, o Ĝ mu hve yle ˆ 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 ˆ i denoe he hore yle in Ĝi 1. We oin Ĝi y reduing he grph Ĝi 1 \ ˆ i, or equivlenly, removing he verie of ˆ i nd ll heir edge, nd hen onring ˆ i nery lengh-2 ph o ingle edge. Our erlier rgumen implie h eh yle ˆ i h mo 2(lg g + 1) edge. Thu, for eh i, we hve E(Ĝ i ) = E(Ĝ i 1 ) 6(lg g + 1). Sine he originl redued u grph Ĝ onin le g edge, i follow h we n repe hi proe le g/6(lg g + 1) ime. Le i denoe he yle in he originl u grph G orreponding o Ĝi. By our onruion, i nd 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. Corollry 5.5. For ny 2-mnifold M wih genu g nd no oundry, he lengh of he hore eenil yle i mo O((log g)/g) ime he lengh of he minimum u grph. 5.2 Nerly-Shore Eenil Cyle A we will rgue horly, ompuing hor eenil yle i he olenek in our pproxime u grph lgorihm. Forunely, ex minimum eenil yle re no neery. We n peed up our u grph lgorihm, wihou ignifinly inreing he pproximion for, y erhing for n eenil yle mo wie long he hore. Our pproximion lgorihm work follow. Fir, we ompue e of hore ph (in f, u grph) h inere every eenil yle in he mnifold M. Then we onr eh hore ph π in hi e o poin, nd find he hore eenil yle hrough h poin, deried y Lemm 5.2. Lemm 5.6. Le π e hore ph eween wo verie in polyhedrl 2-mnifold M, nd le e he hore eenil yle in M 1 h inere π. In O(n log n) ime, one n ompue n eenil yle in M uh h 2.
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 e he hore eenil yle in M h pe hrough v. Clerly,. We onru yle in M y onening wo ph nd, where onin he edge of nd i he uph of π eween he endpoin of. The equene of edge onrion h rnform M o M lo rnform o. Hene, i n eenil yle of M. Beue i uph of hore ph, i ully he hore ph eween he endpoin of, o =. I follow h = + 2 2. Thi lemm ugge nurl lgorihm for finding hor eenil yle: Compue e of hore ph h inere every eenil yle, nd hen run he lgorihm from Lemm 5.6 for eh ph in hi e. Lemm 5.7. In O(gn log n) ime, one n ompue e Π of O(g) hore ph on M 1 uh h every eenil yle in M 1 inere le one ph in Π. Proof: We ue noher vrin of he lgorihm of Lzru e l. [25], repling he imple redh-fir erh wih Dijkr hore ph lgorihm. The u grph G ompued y our lgorihm oni of hore-ph ree T from he ring verex v, plu n ddiionl e E of O(g) edge. Le Π e he e of O(g) hore ph from v o he endpoin of E, nd le G = Π E. We eily oerve h G i lo u grph; h i, M \ G i opologil dik. Thu, every eenil yle in M inere G. Sine every verex of G i lo verex of Π, every eenil yle inere le one ph in Π. Corollry 5.8. Le M e polyhedrl 2-mnifold wih genu g nd no oundry, nd le e i hore eenil yle. In O(gn log n) ime, one n ompue n eenil yle of M uh h 2. 5.3 Punure-Spnning Tree A eond omponen of our u grph lgorihm i ompuing he minimum punure-pnning ree of punured mnifold (M, P ): he minimum pnning ree of he punure P in he hore-ph meri of M 1. Lemm 5.9. 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 Prim minimum pnning ree lgorihm y dding hore punure-o-punure ph one ime in inreing order of lengh [31]. 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). Lemm 5.10. The lengh of he minimum punure-pnning ree of ny punured 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 [24, 31]. 5.4 Greedy Algorihm Anlyi We now hve ll he omponen of our greedy u grph lgorihm. A ny ge of he lgorihm, we hve (poily dionneed) punured mnifold (M, P ). Our lgorihm repeedly u long hor eenil yle of M, uing he lgorihm of 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 5.9. Finlly, we reglue ome u edge o oin ingle opologil dik. Eh eenil yle u i eiher epring u, whih rek omponen of M ino wo mller omponen, eh wih non-rivil opology, or reduing u, whih deree he genu of ome omponen of M y 1. The lgorihm perform mo g 1 epring u nd exly g reduing u. Thu, he overll running ime of our lgorihm i (2g 1) O(gn log n) + O(n log n) = O(g 2 n log n). I remin only o nlyze how well our greedy u grph pproxime he rue minimum u grph G. For ny grph X, le X denoe i ol lengh. We pli he lgorihm ino phe numered from g down o 1. In he ih phe, we u long he hore eenil yle in every omponen of he mnifold whoe genu i exly i. Some phe my inlude no u. Le (M i, P i ) denoe he punured mnifold he eginning of he ih phe, nd le G i denoe he union of he minimum u grph of i omponen. Sine ollping edge nno inree he minimum u grph lengh, we hve G i G g = G for ll i. Le M ij denoe he jh omponen of M i wih genu i; le G ij denoe i minimum u grph; le ij denoe he hor eenil yle found y Corollry 5.8. (We eily oerve h ny u grph of M ij mu inere hi yle.) Lemm 5.4 nd Corollry 5.8 imply h ij O((log i)/i) G ij for ll i nd j. Thu, we n ound he ol lengh of ll u in phe i follow. j ij j O((log i)/i) G ij O((log i)/i) G i Summing over ll g phe, we onlude h he ol lengh of ll yle u i mo g i=1 O((log i)/i) G i = O(log 2 g) G. Similrly, Lemm 5.10 implie h he minimum punurepnning 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 5.11. 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. 6. OPEN PROBLEMS 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. We pln o implemen hi lgorihm nd preen experimenl reul in fuure verion of hi he pper. 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 [11]? 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 [12, 19]. How well n we pproxime he minimum u grph in nerly-liner ime? More generlly, i here imple, pril, O(1)-pproximion lgorihm, like he MST pproximion of Seiner ree? Unforunely, he generl Seiner ree prolem i MAXSNPhrd [3], o n effiien (1 + ε)-pproximion lgorihm for rirry ε > 0 eem unlikely. The pproximion lgorihm of Theorem 5.11 i omewh indire. I ompue hor u grph y repeedly ompuing reonle u grph nd hen exring hor eenil 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. 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 [32])? Aknowledgmen. We would like o hnk Herer Edelrunner for n enlighening iniil onverion. We re lo greful o Nog Alon, John Hr, Benjmin Sudkov, nd Kim Whileey for helpful diuion. Referene [1] U. Axen nd H. Edelrunner. Audiory More nlyi of ringuled mnifold. Mhemil Viulizion, 223 236, 1998. Springer-Verlg. [2] C. Benni, J.-M. Vézien, G. Igléi, nd A. Gglowiz. Pieewie urfe flening for non-diored exure mpping. Compuer Grphi 25:237 246, 1991. Pro. SIGGRAPH 91. [3] M. Bern nd P. Plmn. The Seiner prolem wih edge lengh 1 nd 2. Inform. Pro. Le. 32(4):171 176, 1989. [4] N. Bigg. Conruion for ui grph wih lrge girh. Ele. J. Comin. 5:A1, 1998. [5] J. Blömer. Compuing um of rdil in polynomil ime. Pro. 32nd Annu. IEEE Sympo. Found. Compu. Si., 670 677, 1991. [6] S. Chri, P. Rohgi, nd A. Srinivn. Rndomneopiml unique elemen iolion wih ppliion o perfe mhing nd reled prolem. SIAM J. Compu. 24(5):1036 1050, 1995. [7] T. Dey, H. Edelrunner, nd S. Guh. Compuionl opology. Advne in Diree nd Compuionl Geomery, 109 143, 1999. Conemporry Mhemi 223, Amerin Mhemil Soiey. [8] T. K. Dey nd S. Guh. Trnforming urve on urfe. J. Compu. Sy. Si. 58:297 325, 1999. [9] T. K. Dey nd H. Shipper. A new ehnique o ompue polygonl hem for 2-mnifold wih ppliion o null-homoopy deeion. Diree Compu. Geom. 14:93 110, 1995. [10] E. W. Dijkr. A noe on wo prolem in onnexion wih grph. Numerihe Mhemik 1:269 271, 1959. [11] R. G. Downey nd M. R. Fellow. Prmeerized Complexiy. Monogrph in Compuer Siene. Springer- Verlg, 1999. [12] S. Dreyfu nd R. Wgner. The Seiner prolem in grph. Nework 1:195 207, 1971. [13] D. Eppein. Seveneen proof of Euler formul: V E + F = 2. The Geomery Junkyrd, My 2001. hp: //www.i.ui.edu/ eppein/junkyrd/euler/. [14] M. S. Floer. Prmerizion nd mooh pproximion of urfe ringulion. Compu. Aided Geom. Deign 14(4):231 250, 1997. [15] G. K. Frni nd J. R. Week. Conwy ZIP proof. Amer. Mh. Monhly 106:393 399, 1999. hp://new. mh.uiu.edu/zipproof/. [16] M. R. Grey, R. L. Grhm, nd D. S. Johnon. The omplexiy of ompuing Seiner miniml ree. SIAM J. Appl. Mh. 32:835 859, 1977. [17] M. R. Grey nd D. S. Johnon. The reiliner Seiner ree prolem i NP-omplee. SIAM J. Appl. Mh. 32:826 834, 1977. [18] S. Hker, S. Angenen, A. Tnnenum, R. Kikini, G. Spiro, nd M. Hlle. Conforml urfe prmeerizion for exure mpping. IEEE Trn. Viuliz. Compu. Grph. 6(2):181 187, 2000. [19] M. Hlle nd T. Wrehm. A ompendium of prmeerized ompplexiy reul. SIGACT New 25(3):122 123, 1994. hp://we..mun./ hrold/ W hier/ompendium.hml. [20] M. Hnn. On Seiner prolem wih reiliner dine. SIAM J. Appl. Mh. 14:255 265, 1966. [21] A. Hher. Algeri Topology. Cmridge Univeriy Pre, 2001. hp://www.mh.ornell.edu/ hher/. [22] D. B. Johnon. Effiien lgorihm for hore ph in pre nework. J. Ao. Compu. Mh. 24(1):1 13, 1977. [23] A. Klivn nd D. A. Spielmn. Rndomne effiien ideniy eing of mulivrie polynomil. Pro. 33rd Annu. ACM Sympo. Theory Compu., 216 223, 2001. [24] L. Kou, G. Mrkowky, nd L. Bermn. A f lgorihm for Seiner ree. A Inform. 15:141 145, 1981. [25] F. Lzru, M. Pohiol, G. Veger, nd A. Verrou. Compuing nonil polygonl hem of n orienle ringuled urfe. Pro. 17h Annu. ACM Sympo. Compu. Geom., 80 89, 2001.
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