Unfolding Orthogonal Polyhedra with Quadratic Refinement: The Delta-Unfolding Algorithm

Transcription

1 Unfolding Orhogonal Polyhedra wih Quadraic Refinemen: The Dela-Unfolding Algorihm The MIT Faculy ha made hi aricle openly available. Pleae hare how hi acce benefi you. Your ory maer. Ciaion A Publihed Publiher Damian, Mirela, Erik D. Demaine, and Robin Flaland. Unfolding Orhogonal Polyhedra wih Quadraic Refinemen: The Dela- Unfolding Algorihm. Graph and Combinaoric 30, no. 1 (January 2014): hp://dx.doi.org/ / Springer-Verlag Verion Auhor' final manucrip Acceed Tue Mar 13 19:29:46 EDT 2018 Ciable Link hp://hdl.handle.ne/1721.1/86067 Term of Ue Creaive Common Aribuion-Noncommercial-Share Alike Deailed Term hp://creaivecommon.org/licene/by-nc-a/4.0/

2 Graph and Combinaoric manucrip No. (will be inered by he edior) Unfolding Orhogonal Polyhedra wih Quadraic Refinemen: The Dela-Unfolding Algorihm Mirela Damian Erik D. Demaine Robin Flaland Received: dae / Acceped: dae Abrac We how ha every orhogonal polyhedron homeomorphic o a phere can be unfolded wihou overlap while uing only polynomially many (orhogonal) cu. By conra, he be previou uch reul ued exponenially many cu. More preciely, given an orhogonal polyhedron wih n verice, he algorihm cu he polyhedron only where i i me by he grid of coordinae plane paing hrough he verice, ogeher wih Θ(n 2 ) addiional coordinae plane beween every wo uch grid plane. Keyword general unfolding, grid unfolding, grid refinemen, orhogonal polyhedra, genu-zero 1 Inroducion One of he major unolved problem in geomeric folding i wheher every polyhedron (homeomorphic o a phere) ha an unfolding [3, 10]. In general, an unfolding coni of cuing along he polyhedron urface uch ha wha remain flaen ino he plane wihou overlap. Convex polyhedra have been known o unfold ince a lea he 1980 [10, Sec ]. A recen breakhrough for nonconvex polyhedra i he unfolding of any orhogonal polyhedron (homeomorphic o a phere) [6]. A polyhedron i E. D. Demaine parially uppored by NSF CAREER award CCF M. Damian Dep. of Compuing Science, Villanova Univeriy, 800 Lancaer Avenue, Villanova, PA 19085, USA. mirela.damian@villanova.edu. E. D. Demaine Compuer Science and Arificial Inelligence Laboraory, Maachue Iniue of Technology, 32 Vaar S., Cambridge, MA 02139, USA. edemaine@mi.edu R. Flaland Dep. of Compuer Science, Siena College, 515 Loudon Road, Loudonville, NY 12211, USA. flaland@iena.edu

3 2 Mirela Damian e al. orhogonal if all of i edge are parallel o a coordinae axi, and hu all edge and face mee a righ angle. While very general, a diadvanage of hi unfolding algorihm i ha he cuing i inefficien, making exponenially many cu in he wor cae, reuling in an unfolding ha i long and hin ( epilon hin ). In hi paper, we how how o unfold any orhogonal polyhedron uing only a polynomial number of cu. Grid refinemen. To more preciely quanify he cu required by an unfolding, everal model of allowed cu have been propoed. See [9, 10, 11] for urvey. For convex polyhedra, he major unolved goal i o ju cu along he edge (which implie a linear number of cu) [10, ch. 22]. For nonconvex polyhedra, however, hi goal i unaainable, even when he polyhedron i opologically convex [3] or i orhogonal [4]. A imple example of he laer i a mall box on op of a larger box. More generally, deciding wheher an orhogonal polyhedron ha an edge unfolding i rongly NP-complee [1]. For orhogonal polyhedra, i eem mo naural o conider orhogonal cu. The malle exenion from edge unfolding eem o be grid unfolding (a concep implici in [4]), where we lice he polyhedron wih all axi-aligned plane ha pa hrough a lea one polyhedron verex, and allow cuing along all lice line. Even wih hee addiional edge, few nonrivial ubclae of orhogonal polyhedra are known o have grid unfolding: orhoube [4], orhoack compoed of orhogonally convex lab [8], and well-eparaed orhoree [5]. On he negaive ide, here are four orhogonal polyhedra wih no common grid unfolding [2]. The nex exenion beyond grid unfolding i grid refinemen k, which addiionally lice wih k plane in beween every grid plane (a above), and allow cu along any edge of he refined grid. Wih conan grid refinemen, a few more clae of orhogonal polyhedra have been uccefully unfolded: orhoack [4], and Manhaan ower [7]. The breakhrough wa he dicovery ha arbirary orhogonal polyhedra (homeomorphic o a phere) unfold wih finie grid refinemen [6]. Unforunaely, he amoun of grid refinemen i exponenial in he wor cae (hough polynomial for well-balanced polyhedra). For hi reaon, he unfolding algorihm wa called epilon-unfolding. Our reul. We how how o modify he epilon-unfolding algorihm of [6] o reduce he refinemen from wor-cae exponenial (2 Θ(n) ) o wor-cae quadraic (Θ(n 2 )), while ill unfolding any orhogonal polyhedron (wih n verice) homeomorphic o a phere. We call our algorihm he dela-unfolding algorihm, o ugge ha he reuling urface rip are ill narrow bu wider han hoe produced by epilon-unfolding. Our cenral new echnique in dela-unfolding i he concep of heavy and ligh node from heavy-pah decompoiion [12]. Inereingly, heavy-pah decompoiion i a common echnique for balancing ree in he field of daa rucure, bu no o well known in compuaional geomery.

4 The Dela-Unfolding Algorihm 3 Even wih hi echnique in hand, however, dela-unfolding require a careful modificaion and engineering of he echnique ued by epilon-unfolding. Thu, Secion 2 and 3 ar wih reviewing he main echnique of epilonunfolding; hen Secion 4 modifie hoe echnique; and finally Secion 5 pu hee echnique ogeher o obain our main reul. 2 Overview of Epilon-Unfolding We begin wih a review he epilon-unfolding algorihm [6], aring in hi ecion wih a high-level overview, and hen in Secion 3 deailing hoe apec of he algorihm ha we modify o achieve quadraic refinemen. Throughou hi paper, P denoe a genu-zero orhogonal polyhedron whoe edge are parallel o he coordinae axe and whoe urface i a 2- manifold. We ake he z-axi o define he verical direcion, he x-axi o deermine lef and righ, and he y-axi o deermine fron and back. We conienly ake he viewpoin from y =. The face of P are diinguihed by heir ouward normal: forward i y; rearward i +y; lef i x; righ i +x; boom i z; op i +z. 1 The epilon-unfolding algorihm pariion P ino lab by licing i wih y-perpendicular plane hrough each verex. Le Y 0, Y 1, Y 2,... be he licing plane ored by y coordinae. A lab i a conneced componen of P locaed beween wo conecuive plane Y i and Y i+1. Each lab i a imple orhogonal polygon exruded in he y-direcion. The cycle of {lef, righ, op, boom} face urrounding i called a band, and he band edge in Y i (and imilarly in Y i+1 ) form a cycle called a rim. A z-beam i a narrow verical rip on a forward or rearward face of P connecing he rim of wo band. The order in which he band unfold i deermined by an unfolding ree T U whoe node are band, and whoe arc correpond o z-beam, each of which connec a paren band o a child band in T U. The unfolding ree T U will be furher decribed in Secion 3.1 below. The unfolding of a band b i deermined by a hin urface piral, denoed ξ, ha ar on one of b rim, cycle around b while diplacing oward he oher rim, where i urn around and reurn o he poin i ared. A he piral pae by a z-beam connecing b o one of i children, i ener hrough he z-beam o he child rim, and hen recurively vii he ubree rooed a he child. Once he complee piral i deermined, i can be hickened in he ±y direcion o ha i enirely cover all band face. The hickened piral i uch ha i can be laid fla in he plane o form a monoonic aircae rip. The forward and rearward face of P can hen be laid fla wihou overlap by aaching hem in rip above and below he aircae. 1 The ±y face are given he awkward name forward and rearward o avoid confuion wih oher ue of fron and back inroduced laer.

5 4 Mirela Damian e al. 3 Epilon-Unfolding Exruion Almo all algorihmic iue in epilon-unfolding are preen in unfolding polyhedra ha are z-exruion of imple orhogonal polygon in he xy plane. Therefore, we follow [6] in decribing he algorihm for hi imple hape cla, before exending he idea o all orhogonal polyhedra. All modificaion needed for dela-unfolding are alo preen in unfolding orhogonal exruion, and o we decribe hem in erm of hi imple hape cla. We herefore review in deail he epilon-unfolding algorihm for orhogonal exruion. 3.1 Unfolding Tree Le P be a polyhedron ha i he verical exruion of a imple orhogonal polygon, uch a ha illuraed in Figure 1a. The algorihm begin by licing P ino lab, which in hi pecial cae are all block (cuboid), uing y-perpendicular plane hrough each verex. The dual graph i a ree, T U, having a node for each band and an edge beween each pair of adjacen band. In hi pecial cae, all z-beam are degenerae, i.e., of zero z-heigh. The roo i eleced arbirarily from among all band wih a rim of minimum y coordinae. For example, he polyhedron in Figure 1a i liced ino nine block, wih b 1 a he roo and i unfolding ree a hown in Figure 1b. b9 b 1 b 7 b 4 b8 b 2 b 6 b 3 b 2 b 1 b 3 b 4 b5 (a) y z x b 6 b 5 b 8 b 7 (b) b 9 Fig. 1 (a) Exruion of an orhogonal polygon, pariioned by y perpendicular plane. (b) Unfolding ree. Back children are repreened by haded node. The rim of he roo band wih he maller y coordinae i i fron rim, and he oher rim i i back rim. For any oher band, he rim adjacen o i paren in T U i i fron rim, and i oher rim i i back rim. Children aached along he fron rim of heir paren are fron children; children aached along he back rim of heir paren are back children. Noe ha fron and back modifier for rim and children derive from he rucure of T U, and are no relaed o he forward and rearward ±y direcion. For example, b 9 i a fron-child of b 8, alhough i i aached o he rearward face of b 8, and he fron rim of b 5 lie on he rearward face of b 5.

6 The Dela-Unfolding Algorihm Recurive Unfolding The key o he epilon-unfolding mehod i he exience of a hin, non-croing piral ξ ha cycle around each band a lea once, and unfold o a aircae when flaened ino he plane. A aircae i an orhogonal pah in he plane whoe urn alernae beween 90 lef and 90 righ, and o i a monoone pah. The pah ha ξ follow i deermined recurively. We review hi piral ξ, aring wih he bae cae Single Band Bae Cae Figure 2a how he pah followed by ξ for a ingle band correponding o a leaf of T U. I ar a an enering poin on he op edge of he fron rim and piral in a clockwie direcion around he op, righ, boom, and lef band face oward he back rim. We call hi piral piece up o he poin i reache he back rim he enering piral. When i reache he back rim, ξ croe he rearward face upward oward he op face. From here, i rerace he enering piral in he oppoie (counerclockwie) direcion oward an exiing poin lying nex o on he fron rim. When ξ i cu ou, unfolded, and laid horizonally in he plane, i form a monoonic aircae rip, a hown in Figure 3a, becaue he urn alernae beween lef and righ, 90 each. Oberve ha he x, z-parallel egmen of ξ, correponding o he cycling clockwie and counerclockwie around he band, form he air read ; he y-parallel egmen of ξ and he z-parallel rip from he rearward face form he air rier. z y y x (a) x (b) Fig. 2 (a) R block piral, wih mirror view of face ha canno be een direcly. (b) Abrac 2D repreenaion. Three-dimenional illuraion of ξ like ha in Figure 2a are impracical for more complex orhogonal hape. To eaily illurae more complex unfolding, we ue he 2D repreenaion depiced in Figure 2b. Noe ha he 2D repreenaion capure he direcion of he enering piral and he relaive poiion of and. The arc connecing he enrance o he exi ymbolize he reveral of he unfolding direcion uing a rearward face rip.

7 6 Mirela Damian e al. T L B K T R B (a) T R B L (b) Fig. 3 (a) Spiral from Figure 2a unfolded. (b) Thickened piral wih he back face hung underneah. Doed line delineae urface piece from differen block face and are labeled T (op), R (righ), B (boom), L (lef), and K (back). Eigh variaion of he bae cae piral are illuraed in Figure 4. They differ in he manner in which ξ ener and exi he band b o be unfolded. The four variaion labeled L, L, R, R in he op row are ued when he y-coordinae of b fron rim i maller han he y-coordinae of i back rim. R i imilar o R, bu wih and, and he clockwie/counerclockwie cycling direcion revered; L and L are (repecively) mirror of R and R in an x-perpendicular plane. Noe ha he R and L label indicae he piral cycling direcion when i ener he band: R i clockwie, L i counerclockwie. The piral exi he band cycling in he oppoie direcion. The four variaion in he boom row are labeled L +, L +, R +, R +, and hey are ued when he y-coordinae of b fron rim i greaer han he y-coordinae of i back rim. They are exac reflecion of L, L, R, and R, repecively, in a y-perpendicular plane. The mirror ymmerie imply ha he 3D piral correponding o each 2D abrac repreenaion can be eaily derived from R configuraion, illuraed in Figure 2. y L L R R x L + L + R + R + Fig. 4 Abrac 2D repreenaion of he eigh pah ype viiing one lab Recurive Pah For a node b in T U ha i no a leaf, we decribe he pah ha he piral ξ recurively follow when viiing b. We aume ha b ha one of he eigh

8 The Dela-Unfolding Algorihm 7 configuraion label hown in Figure 4. A in he bae cae, he label idenifie he relaive order of poin and on b fron rim, he piral direcion when enering b, and b rim of lower y coordinae. Wihou lo of generaliy, we aume ha b label i R ; he oher even label are equivalen by ymmery. The inducive aumpion i ha, for any ubree horer han he ubree of T U rooed a b, and for any configuraion label aigned o he roo band of he horer ubree, here i a (non-croing) pah ξ conien wih ha label ha cycle around each band in he maller ubree a lea once, and unfold in he plane a a aircae rip. Afer ξ ener b a poin, i vii each of b fron children, aring wih he fron child, call i b 1, fir encounered a i cycle clockwie along he fron rim of b. (See Figure 5). For reaon oon o be explained, child b 1 i aigned he label R + wih wo poin 1 and 1 idenified on he op edge of i fron rim, wih 1 righ of 1. The piral ξ ener b 1 a poin 1 and recurively vii i (and he ubree i roo). By he inducive hypohei, ξ exi b 1 a poin 1 cycling counerclockwie. The label R + i aigned o b 1 becaue ξ i cycling in he direcion R (o he righ, or clockwie) on b ju before i ener b 1, and o i ener b 1 wih ha ame direcion; he + upercrip i neceary becaue he y-coordinae of he fron rim of b 1 i higher han ha of i back rim; and he ordering i neceary o preven ξ from being rapped beneah he porion of ξ beween and 1 upon reurning o b, hu cuing ielf off from reaching b oher children (becaue i canno cro ielf). b (R ) y b 2 (L ) x b 1 (R ) b 3 (R ) b 4 (L ) Fig. 5 Need inide-ou alernaing pah vii he fron children. Doed line how ξ where i cycle underneah on he boom face. Afer recurively viiing b 1, ξ cycle counerclockwie on b o he fir unviied child i pae when on b op face. Thi child, call i b 2, i aigned he label L + wih idenified poin 2 and 2 on i fron rim, conien wih i label, and i i recurively viied. In hi need manner, ξ vii he children clockwie and counerclockwie from from he inide ou, alernaely aigning he label R + and L +. Figure 5 illurae he pah ξ ake when b ha four fron children. (To keep he example imple, only one level of recurion i illuraed, wih all children leave of T U.) Afer viiing he fron children, ξ make a complee cycle around b and hen begin viiing he back children. Aume for concreene ha afer viiing he fron children, ξ i cycling clockwie on b, a hown in Figure 6. I hen ravel clockwie o he back child farhe o he righ along he op

9 8 Mirela Damian e al. b 6 (L ) b 8 (L ) b 7 (R ) b 5 (R ) y 6 x b (_ ) Fig. 6 Need ouide-in alernaing pah vii he back children. face. (See child b 5 in Figure 6). When i reurn from recurively viiing hi back child, i will be cycling counerclockwie (by he inducive hypohei). I i hu imporan ha he child exi poin be o he lef of i enering poin o ha ξ i no blocked from viiing oher back children. (See poin 5 and 5 in Figure 6.) Thu hi fir back child i aigned he configuraion R and i recurively viied. The piral hen move o he unviied child farhe o he lef (ee child b 6 ) and vii i in a imilar way, aigning i he label L. Thu he neing of ξ alernaing pah i ouide-in for back children, wih he label R and L being alernaely aigned. The la back child viied, b k, however, i an excepion when i come o i label aignmen, for he following reaon. When he piral exi b k (ee b 8 in Figure 6), i will rerace i pah (in revere direcion) back o he fron rim of b and hen exi a poin. For band b, define i enering piral, ξ e (b), o be he porion of ξ ha begin a and end a he exiing poin k of b k ( 8 in Figure 6). I exiing piral, ξ x (b), i he porion of ξ ha begin a k and end a on he fron rim of b. The exiing piral ξ x (b) imply parallel alongide he enering piral ξ e (b), reracing he porion of ξ e (b) from o k bu in he oppoie direcion. Since b ha a label, he exiing piral mu leave b wih he enering piral on i lef, from he poin of view of one walking on b along he pah aken by ξ x (b). Thu b k mu alo be aigned he label (conien wih b label), o ha from he beginning of he rerace and hroughou, ξ x (b) ha he enering piral o i lef. We call hi a lef rerace; when ξ x (b) keep he enering piral on i righ during a rerace, we call i a righ rerace. We noe ha if b ha no back children, hen he piral revere direcion uing a rip from b rearward face, a in he bae cae. 3.3 Compleing he Unfolding We have focued on ξ recurive pah becaue ha i where he modificaion for dela-unfolding occur. Bu for compleene, we briefly ummarize he remainder of he epilon-unfolding algorihm for exruion, and refer he reader o [6] for addiional deail. To complee he unfolding of P, ξ i hicken in he +y and y direcion (a viewed in he 3D coordinae yem of Figure 2a) o ha i compleely cover each band. Thi reul in a hicker unfolded aircae rip. Then he forward and rearward face of P are pariioned by imagining he band op rim edge illuminaing downward ligh ray in hee face.

10 The Dela-Unfolding Algorihm 9 The illuminaed piece are hen hung above and below he hickened aircae, along he correponding illuminaing rim egmen which lie along he horizonal edge of he aircae. See Figure 3b. 3.4 Level of Refinemen In [6] i wa hown ha he unfolding echnique dicued o far can make an exponenial number of cu on he family of polyhedra depiced in Figure 7. Each polyhedron coni of n = 2k + 1, k 1, block arranged a hown for k = 1 in Figure 7a, and for k = 2 in Figure 7b. For analyi purpoe, we formally define a vii o a band o begin when he piral croe i fron rim o ener he band (eiher he fir ime, or in a rerace) and end when i croe he fron rim o exi he band. In Figure 7a, b 3 vii begin when ξ ener i a poin 3 cycling counerclockwie. The piral vii back child b 1 and hen b 2. The vii of b 2 rigger a rerace which involve a econd vii of b 1, and hen back hrough b 3 o exiing poin 3, which complee b 3 vii. We b 1 b 2 b 1 b b b b y x 5 5 b 5 (a) (b) Fig. 7 Family of polyhedra requiring exponenial refinemen. Block b 1 i viied wo ime in (a), four ime in (b), and in general 2 n/2 ime for an n-block objec. can wrie hi vii order uing he ring Q 3 = ( 3 ( 1 1 ) ( 2 2 ) ( 1 1 ) 3 ), where an open parenhei followed by a aring poin mark he ar of a vii and an exiing poin followed by a cloing parenhei mark he end. The ubcrip on Q i he number of block in he polyhedron. Oberve ha block b 1 i viied wice. For he five block polyhedron in Figure 7b, ξ ar a poin 5 on b 5, recurively vii block b 3 in he manner ju decribed, hen vii b 4 which rigger a rerace hrough b 3. Afer reviiing b 3, ξ reurn o b 5 and exi a poin 5. The correponding vii ring i Q 5 = ( 5 Q 3 ( 4 4 ) Q 3 5 ). The number of vii o b 1 double o 4. In general, an n block polyhedron in hi family give rie o 2 n/2 vii o b 1, reuling in an exponenial number of cu on b 1.

11 10 Mirela Damian e al. 4 Dela-Unfolding Exruion To achieve quadraic refinemen, we modify he order in which children are viied baed on he heavy/ligh claificaion of node ued in heavy-pah decompoiion [12]. In heavy-pah decompoiion, each ree node v i aigned a weigh n(v), which i he number of decendan in i ubree, including ielf. An edge from paren p o child c i heavy if n(c) > 1 2n(p), and ligh oherwie. We ay a child c i heavy (ligh) if he edge beween c and i paren i heavy (ligh). Oberve ha a node can have a mo one heavy child. If a node b in T U ha a heavy child, hen we modify he pah of he enering piral ξ e (b) o ha i vii he heavy child la, o preven he need for reviiing he heavy child; we will how ha hi raegy quadraically bound he number of vii ξ make o each child. For example, conider he polyhedron in Figure 7b, and oberve ha b 3 i a heavy child. Wih epilonunfolding, ξ e (b) viied child b 3 before b 4. The vii o b 4 riggered a complee rerace of he ubree rooed a b 3, hu leading o he vii ring Q 5 = ( 5 Q 3 ( 4 4 ) Q 3 5 ), and a oal of four vii o b 1. Bu if we revere he vii order o ha ξ e (b) vii b 3 afer b 4, hen he vii ring become Q 5 = ( 5 ( 4 4 ) Q 3 ( 4 4 ) 5 ), and no block i viied more han wice. Since any fron or back child could be heavy, we focu fir on he challenge of finding a roue for he enering piral o ha i vii any pecified child la. If we can achieve hi, hen we can organize he vii o minimize reracing. We hen formally preen he algorihm and analyze he reuling level of refinemen. 4.1 Fron Child Viied La We ar wih he cae when we deire o vii a fron child, call i b l, la. The idea i o vii all he fron children excluding b l, and all he back children in exacly he manner decribed in Secion 3.2.2, a if b l were no preen. Figure 8 how he enering piral ξ e (b) viiing all bu b l. (Noe ha he complee cycle ha ξ e (b) make beween viiing he fron and back children i no fully depiced in he 2D repreenaion.) Afer viiing he la back child (b 9 in he figure), ξ rerace i pah in revere. I i during hi rerace ep ha child b l i viied. We explain he modificaion neceary o accomplih hi for a paren block b wih a label of ype R or L ; label of ype R + and L + are handled ymmerically. Oberve fir ha ince ξ e (b) alernaely vii all he fron children excep for b l and hen make a complee cycle around b, ome coniguou ecion of i, call i f, run alongide he op edge of b l fron rim. Specifically, f i he ecion of ξ e (b) hi by y-parallel ray ho from b l op fron rim edge oward he back rim of b. See Figure 8 where f i marked. Noe ha ince ξ e (b) i cycling oward he back rim of b, f repreen he fir ime ξ e (b) pae by b l op edge. All ubequen pae are behind f.

12 The Dela-Unfolding Algorihm 11 b 6 ( L ) b 8 ( L ) b 9 ( R ) b 7 ( R ) b 5 ( R ) b (R ) f y b 2 (L + ) b 1 (R + ) b 3 (R + ) b 4 (L + ) x Fig. 8 Enering piral vii fron and back children, wih he excepion of fron child b l, which ge viied la (ee Figure 9). bl b 6 ( L ) b 8 ( L ) b 9 ( R ) b 7 ( R ) b 5 ( R ) b (R ) l l y b + 2 (L ) b (R ) bl (R ) + + b 3 (R ) b 4 (L ) x Fig. 9 Enering and reurn piral. The reurn piral pae by b l o ha b l can be viied. During he rerace ep, ξ need o run in fron of f, o ha i ha unobruced acce o b l. If he enering piral i cycling clockwie in ecion f, hen he reracing piral (which run alongide f in he oppoie direcion) need o righ-rerace, becaue ha will keep he enering piral o i righ and poiion i in fron of f. (Recall ha clockwie i o he righ and counerclockwie i o he lef.) To rigger a righ-rerace, we aign he la viied back child he label. If he enering piral i cycling counerclockwie in ecion f, hen he reracing piral need o lef-rerace, hu keeping he enering piral on i lef. To rigger a lef-rerace, we aign he la back child

13 12 Mirela Damian e al. he label. So inead of maching he or label of he la viied back child o ha of b (a in epilon-unfolding), we inead aign i o ha he reracing piral pae alongide b l. When he reracing piral reache ecion f, i upend he rerace, ener b l a poin l, and vii i. We call he porion of ξ from he exiing poin of he la viied back child o l he reurn piral and label i ξ r (b). See Figure 9 which how ξ r (b) in gray exending from 9 o l. Upon exiing b l a poin l, he piral rerace i pah in he revere direcion, bringing i o he exi poin on b. Specifically, i follow he enire pah from o l in revere. Thi econd rerace i b exiing piral, ξ x (b). (In Figure 9, ξ x (b) i no illuraed, bu i begin a l and follow he gray and hen he black pah o, keeping hem o i lef.) The label or aigned o b l mu be conien wih b label in he following way. If b ha he label, a lef rerace aring from l i needed o ha he piral exi a on he correc ide of conien wih b label. Thu, b l i aigned he label, he oppoie of b label. If, however, b ha he label, a righ rerace i needed, and o b l i aigned he label. Becaue he label aigned o he la back child viied depend on he direcion of ξ e (b) in he f-ecion of he pah, we how here ha deermining ha direcion i raighforward. We dicu he cae in which ξ e (b) ener b cycling clockwie; he cae when i i cycling counerclockwie i ymmeric. We alo aume ha here are a lea wo fron children (no including b l ) and hey are labeled b 1, b 2, b 3,..., in he order in which hey are viied along he alernaing pah (a in Figure 8). Oberve ha, if b l i locaed beween and b 1 (a viewed from above), hen ξ e (b) fir pae by b l op edge cycling clockwie, and he ame i rue if i i locaed beween b i and b i+2, for i odd (i {1, 3, 5,...}). Thu in hee cae, f i ravered clockwie. Figure 8 illurae he cae when b l i beween b 1 and b 3. If b l i locaed beween and b 2 or beween b i and b i+2, for i {2, 4, 6,...}, hen ξ e (b) fir pae by he op edge of b l cycling counerclockwie. Thu in hee cae f i ravered counerclockwie. If he op edge of b l i o he righ of he la odd numbered child or o he lef of he la even numbered child, hen ξ e (b) fir pae over b l during i complee cycle around b. During hi cycle, ξ e (b) i heading clockwie if he la viied child wa even and counerclockwie if he la viied child wa odd. Cae when here are fewer han wo fron children are eaily handled: if b l i he only fron child, or if i i locaed beween and b 1, hen f i ravered clockwie; oherwie, f i ravered counerclockwie. 4.2 Back Child Viied La In hi ecion we dicu he iuaion in which we deire o vii a paricular back child b l la. In hi cae, ξ e (b) vii he fron children a decribed in Secion I hen vii he back children a decribed in Secion bu wih an alered viiing order. We conider he cae when b ha a L or R ype

14 The Dela-Unfolding Algorihm 13 configuraion label and he enering piral ξ e (b) i cycling counerclockwie afer viiing he fron children; he oher cae are ymmeric. Le m 0 be he number of back children of b no including b l, and le b 1, b 2,... b j be he fron children, for j 0. Conider he back children of b in he cyclic clockwie order in which heir op edge occur around b back rim. When m i odd, we label he m back children (b j+1, b j+3,..., b m 2, b m, b l, b m 1,..., b j+4, b j+2 ), according o heir poiion relaive o b l in hi cyclic ordering. When m i even, he labeling i (b j+1, b j+3,..., b m 1, b l, b m,..., b j+4, b j+2 ), a depiced in Figure 10. The piral ξ e (b) vii he back children from he ouide-in, following he vii order b j+1, b j+2,..., b m 1, b m, b l. I i alway poible o vii b j+1 fir, wih a full cycle of he piral around b (if neceary) o ge he piral o he op edge of b j+1. Thi i illuraed in Figure 10 for five back children (and no fron children). b 2 ( R ) 2 2 y x b 1 ( L ) 1 1 b 3 ( L ) 3 3 b l l ( L ) l b 4 ( R ) 4 4 b (L ) Fig. 10 Label and pah followed by piral when viiing back children, when he la child o be viied i back child b l ; dahed line depic piral piece on he boom of he paren block. The aignmen of L and R label o he back children of b i he ame a decribed in Secion Specifically, he label for he children alernae beween L and R wih repec o he viiing order. The piral ξ e (b) i cycling counerclockwie (o he lef) when i reache b j+1, which mache b j+1 L label. The recurive unfolding of b j+1 revere he direcion of he piral, o ha i ener b j+2 cycling clockwie (o he righ), hu maching b j+2 R label, and imilarly for he oher back children. The alernaing, label of he children enure an ouide-in neing of ξ, which enable i o reach each back child. A in Secion 3.2.2, he one excepion o he alernaing label i he la viied child b l, whoe or label need o mach ha of i paren b. Afer viiing b l, he exiing piral follow he enering piral in revere o a in Secion 3.2.2, hu compleing he vii of b.

15 14 Mirela Damian e al. 4.3 The Dela-Unfolding Algorihm for Exruion Wha we call he dela-unfolding algorihm i a modified verion of he epilonunfolding algorihm, which require ha a each node b in T U wih a heavy child, he piral ξ vii he heavy child la. Specifically, if he heavy child i a fron child, hen ξ follow he pah decribed in Secion 4.1; if he heavy child i a back child, hen ξ follow he pah decribed in Secion 4.2. If b ha no heavy child, hen i children are viied in he epilon-unfolding order (Secion 3.2.2). All remaining ep of he dela-unfolding algorihm for exruion he hickening of ξ, he unfolding of ξ a a aircae in he plane, and he pariioning and hanging of he fronward and rearward face from he flaened aircae are he ame a for epilon-unfolding. 4.4 Refinemen Analyi We now urn o analyzing he refinemen for exruion. The pah aken by ξ on a band i compoed of a erie of axi-parallel egmen. We deermine an aympoic upper bound on he number of uch egmen on any band face, becaue hi i an aympoic upper bound on he oal number of cu on a grid face in he unfolding. We compue hi by bounding he number of egmen on any op face, a he number of egmen on all four face of a band i aympoically bounded by he number of egmen on i op face. Define he fir vii of ξ o a band b o begin when ξ fir ener b a poin, include he recurive viiing of b children, and end when i exi b a poin. Band b and he band in i ubree may be reviied by ξ many ime during ubequen reracing, bu each of hee reracing merely follow he pah raced during he fir vii o b. Le R(n(b)) be an aympoic upper bound on he number of egmen ha ξ fir vii o b induce on a op face of any band in he unfolding ubree rooed a b. Then a bound on he number of egmen on any op face in b ubree induced by ξ (in i enirey) i R(n(b)) muliplied by he oal number of ime ξ vii b. We now eablih hree properie of ξ fir vii o b: (i) ξ induce a mo O(n(b)) egmen on b op face; (ii) he ligh children of b are each viied a mo four ime; and (iii) if b ha a heavy child, he heavy child i viied only once. For (i), he wor cae occur when b ha O(n(b)) children and a heavy fron child b l. In hi cae, he alernaing pah of b enering piral ξ e (b) ha have i vii each fron child (excluding b l ) may induce O(n(b)) egmen on b op face, and imilarly for he alernaing pah o each back child. Then b reurn piral ξ r (b) rerace hee alernaing pah up o he poin ha i reache b l, which a mo double he number of egmen. Afer viiing b l, he exiing piral ξ x (b) rerace he pah ξ r (b) and hen he pah ξ e (b) in revere back o poin on b, which again a mo double he number of egmen on b. Thu he oal number of egmen i O(n(b)).

16 The Dela-Unfolding Algorihm 15 For (ii), he maximum vii o ligh children occur when b ha a heavy fron child. In hi cae, ξ e (b) vii each ligh child once. Then ξ r (b) vii each ligh child a mo once on i way o he heavy fron child. Afer viiing he heavy fron child, ξ x (b) rerace ξ r (b) and hen rerace ξ e (b) o he enering poin of b, hu viiing each ligh child a mo wice more. Therefore, each ligh child i viied a mo four ime. For (iii), if b ha a heavy fron child, hen he pah ravered by ξ (deailed in Secion 4.1) immediaely eablihe ha he heavy fron child i viied only once. Similarly, if b ha a heavy back child, he pah deailed in Secion 4.2 eablihe ha he heavy back child i viied exacly once. Properie (i), (ii) and (iii) eablihed above imply ha R(n(b)) i deermined by he larger of hree quaniie: (a) he number of egmen on b op face induced during ξ fir vii o b; (b) 4 max i=1...k R(n(b i )), where b 1, b 2,... b k are b ligh children; (c) R(n(b l )), where b l i b heavy child, if i ha one. A muliplier of four i neceary in cae (b) becaue ligh children may be viied up o four ime during b fir vii; no muliplier i neceary for he heavy child (c) becaue i i viied only once. For he bae cae, R(1) = c, for ome conan c > 1, becaue he fir vii of ξ o a leaf node band (a decribed in Secion 3.2.1) induce a conan number of egmen. And in general, { R(n(b)) = max max } R(n(b i)), R(n(b l )) i=1...k O(n(b)), 4 max { O(n(b)), 4 max R ( 1 i=1...k 2 n(b)), R(n(b) 1) = max { O(n(b)), 4R ( 1 2 n(b)), R(n(b) 1) } } noing ha he ligh children ubree conain a mo 1 2 n(b) node, and he heavy child ubree conain a mo n(b) 1 node. I i raighforward o verify by inducion ha R(n(b)) = O(n(b) 2 ). Applying hi o he roo r of T U wih n = n(r) node and noing ha ξ vii r only once in he dela-unfolding algorihm, yield a maximum of O(n 2 ) parallel egmen on any op face. Thi alo bound he number of cu on any grid face in he unfolding. Specifically, in he hickening ep ξ expand in he +y and y direcion o a o cover he enire band, bu hi doe no aympoically increae i number of edge. Afer he hickening, dijoin ecion of ξ run along he enirey of boh band rim. In he pariioning ep, he dijoin ecion along he op rim edge induce he diviion of he fronward and rearward face ino rip; i.e., each dijoin ecion delimi he verical rip beneah i. Becaue O(n 2 ) bound he number of dijoin ecion along he op edge, i alo bound he number of rip a fronward/rearward face i pariioned ino.

17 16 Mirela Damian e al A wor cae refinemen example. A imple example eablihe ha he bound O(n 2 ) i igh: a polyhedron wih n = 2 h+1 1 block, whoe unfolding ree T U i a perfec binary ree of heigh h (i.e., each inernal node ha wo children, and all leave are a he ame level). There are no heavy node in T U, and he number of cu in a vii of he roo i given by he recurrence relaion becaue R(n) = 4R((n 1)/2) = 4 h R(1) = (n + 1) 2 R(1)/4, 4 h = 4 log 2 (n+1) 1 = 2 log 2 (n+1)2 /4 = (n + 1) 2 /4. And ince R(1) = c, for ome conan c, i follow ha R(n) = O(n 2 ), eablihing our claim. 5 Dela-Unfolding of Genu-Zero Orhogonal Polyhedra The dela-unfolding algorihm and i refinemen analyi generalize o all genu-zero orhogonal polyhedra in he ame way he epilon-unfolding algorihm doe, o we ummarize he idea here and refer he reader o [6] for deail. Inead of pariioning P ino block, he general algorihm pariion P ino lab a defined in Secion 2. I hen creae an unfolding ree, T U, where each node correpond o a band urrounding a lab. Each paren-child arc in T U correpond o a z-beam, which i a verical rip from a fronward or rearward face connecing he paren rim o he child rim. For a paren band b, i fron (back) children are hoe whoe z-beam connec o b fron (back) rim. The piral ξ ener and exi b a poin and locaed a he inerecion beween b fron rim and he z-beam connecing b o i paren. Oberve ha here i a naural cyclic ordering of b fron (back) children ha i deermined by heir z-beam connecion around b fron (back) rim. Uing hi cyclic ordering, i i raighforward o generalize he pah ha ξ follow o reach he fron and back children, decribed in Secion 4.1 and 4.2. See for example Figure 11 ha how a band wih i face flaened in he plane (he ligher color mark op/boom face, and he darker color mark righ/lef face). Alo depiced are he z-beam connecion (flaened ino he plane) and he pah ξ e (b) follow o vii he children, auming b l i a heavy child. Oberve ha he pah i he ame a in Figure 8, excep ha i exend acro muliple band face. When ξ vii a child, i move from b o he connecing z-beam and ravel verically (in 3D) along he z-beam o reach he child; when i exi he child i ravel along he z-beam back o b. In he unfolded aircae, he porion of ξ on he z-beam correpond o a verical rier. Thickening ξ i done a in he cae of exruion. The pariioning of he forward and rearward face i alo done a in he cae of exruion, bu in addiion o hooing illuminaing ray down from op rim edge, boom rim edge ha are no

18 The Dela-Unfolding Algorihm 17 hi by hee ray mu hemelve hoo ray upward o illuminae porion of face no illuminaed by he op edge. The face piece reuling from hi pariioning mehod are hung from he aircae a decribed in Secion 3.3. b 6 b 8 b 9 b 7 b 5 b (R ) b 2 b 1 b b l b 3 4 Fig. 11 Band b of a lab, cu and laid fla wih op/boom face ligh gray and righ/lef face dark gray. z-beam connecion o b paren, children b 1... b 9, and child b l are marked along he fron and back rim. The pah ha ξ e(b) follow when b l i heavy i depiced. The O(n 2 ) upper bound on he level of grid refinemen for exruion alo applie o general orhogonal polyhedra by he following argumen. In he cae of exruion, for any block b wih children, ξ make urn only on b op face, becaue all acce o he children i from he op face; i make no urn on he oher hree face of b. Therefore, we analyze he number of egmen (each correponding o a urn) on he op face. For a band b of an arbirary orhogonal polyhedra, ξ vii b children in he ame manner a for an exruion, excep ha he urn made o acce he children are made on whaever op or boom face ha he connecing z-beam, a in Figure 11. In paricular, for a band b wih a given number of fron and back children, he ame number of urn are made, wheher b urround a block of an exruion or a lab of a arbirary orhogonal polyhedron. In erm of maximum refinemen, he wor cae occur when all he urn are concenraed on a ingle face, which i exacly he iuaion handled by our upper bound analyi in he cae of exruion. 6 Concluion We preen modificaion o he epilon-unfolding algorihm from [6] ha reduce he level of grid refinemen neceary o grid-unfold any genu-zero orhogonal polyhedron from exponenial o quadraic. The nex naural ep i o eek a refined grid edge-unfolding of all genu-zero orhogonal polyhedra ha require ubquadraic refinemen of he grid face, o dae only achieved for highly rericed clae of orhogonal polyhedra [4, 7, 8]. I i unlikely ha he echnique ued in hi paper could be exended o produce uch an unfolding, due o he backracking naure of our recurive unfolding algorihm. However, our preliminary inveigaion embolden u o conjecure ha a conan refinemen of he verex grid uffice o grid-unfold all orhogonal polyhedra.

19 18 Mirela Damian e al. Acknowledgemen. The auhor would like o hank Joeph O Rourke for hi careful reading and helpful uggeion. The econd auhor hank Greg Price for helpful early dicuion on hi problem. Reference 1. Zachary Abel and Erik D. Demaine. Edge-unfolding orhogonal polyhedra i rongly np-complee. In Proceeding of he 23rd Canadian Conference on Compuaional Geomery, Augu Greg Aloupi, Proenji K. Boe, Sebaien Collee, Erik D. Demaine, Marin L. Demaine, Karim Douieb, Vida Dujmović, John Iacono, Sefan Langerman, and Pa Morin. Common unfolding of polyominoe and polycube. In Revied Paper from he China- Japan Join Conference on Compuaional Geomery, Graph and Applicaion, volume 7033 of Lecure Noe in Compuer Science, page 44 54, November Marhall Bern, Erik D. Demaine, David Eppein, Eric Kuo, Andrea Manler, and Jack Snoeyink. Ununfoldable polyhedra wih convex face. Compu. Geom.: Theory and Appl., 24(2):51 62, February Theree Biedl, Erik Demaine, Marin Demaine, Anna Lubiw, Mark Overmar, Joeph O Rourke, Seve Robbin, and Sue Whieide. Unfolding ome clae of orhogonal polyhedra. In Proceeding of he 10h Canadian Conference on Compuaional Geomery, Augu Mirela Damian, Robin Flaland, Henk Meijer, and Joeph O Rourke. Unfolding welleparaed orhoree. In Abrac from he 15h Annual Fall Workhop on Compuaional Geomery, November Mirela Damian, Robin Flaland, and Joeph O Rourke. Epilon-unfolding orhogonal polyhedra. Graph and Comb., 23(1): , Mirela Damian, Robin Flaland, and Joeph O Rourke. Unfolding Manhaan ower. Compu. Geom.: Theory and Appl., 40: , Mirela Damian and Henk Meijer. Edge-unfolding orhoack wih orhogonally convex lab. In Abrac from he 14h Annual Fall Workhop on Compuaional Geomery, page 20 21, November hp://cgw2004.cail.mi.edu/alk/34.p. 9. Erik D. Demaine and Joeph O Rourke. A urvey of folding and unfolding in compuaional geomery. In Jacob E. Goodman, Jáno Pach, and Emo Welzl, edior, Dicree and Compuaional Geomery, Mahemaical Science Reearch Iniue Publicaion, page Cambridge Univeriy Pre, Erik D. Demaine and Joeph O Rourke. Geomeric Folding Algorihm: Linkage, Origami, Polyhedra. Cambridge Univeriy Pre, July Joeph O Rourke. Unfolding orhogonal polyhedra. In J. E. Goodman, J. Pach, and R. Pollack, edior, Survey on Dicree and Compuaional Geomery: Tweny Year Laer, page American Mahemaical Sociey, Daniel D. Sleaor and Rober Endre Tarjan. A daa rucure for dynamic ree. J. of Compu. and Sy. Sci., 24(3): , June 1983.