Computational Complexity of a Pop-up Book

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1 omputtionl omplexity of Pop-up ook Ryuhei Uehr Shio Termoto strt Origmi is the enturies-ol rt of foling pper, n reently, it is investigte s omputer siene: Given n origmi ith reses, the prolem to etermine if it n e flt fter foling ll reses is NP-hr. nother hunres-ol rt of foling pper is pop-up ook. moel for the pop-up ook esign prolem is given, n its omputtionl omplexity is investigte. We sho tht oth of the opening ook prolem n the losing ook prolem re NP-hr. Keyors: omputtionl omplexity, Origmi, Pper foling, Pop-up ook. 1 Introution Origmi is the enturies-ol rt of foling pper. Reently, some mthemtiins n omputer sientists hve strte to stuy origmi. For exmple, geometri pproh to origmi esign hs tken, n one of useful tehniques is knon s TreeMker progrm y Lng [8]. On the other hn, glol flt folility of n origmi is onsiere. The prolem to fin pproprite overlp orer to fol given origmi flt is NP-hr [1]. The pper foling prolem n e generlize. For exmple, foling mp seems to e similr to the prolem of origmi. The reer n fin omprehensive survey of the omplexity of foling n origmi n relte results ue to Demine & Demine [3] n Demine & O Rourke [4]. nother hunres-ol rt of foling pper is pop-up ook. ontemporry pop-up ook rtists invent mny sulpture of gret euty n intriy (see, e.g., [9]). pop-up ook hs to mjor ifferenes ompring to origmi. First, it hs to surfe overs ith hinge, n the essentil movement epens on them. Hene the movement is strongly restrite (see, e.g., [7, 2] for possile movements). Seon, ook is not only lose (or fole) ut lso opene (or unfole). For pop-up ook esigner, the prolem is to esign sulptures y pper eteen to overs, n mke the ook e le to e opene n lose. Moreover, to see pge of the ook, e usully open or lose the pge one. Tht is, e o not repet the movements open n lose to see pge in the ook. From the viepoint of the omputtion of the movement, this point lso strongly restrits ourselves. In this pper, e first give moel for the pop-up ook esign prolem. Next, e sho tht oth of the opening ook prolem n the losing ook prolem re NP-hr. We note tht our results o not use the overlp orer tehnique use in [1] to sho the NP-hrness of the folility prolem of n origmi. 2 Definitions n input of the prolems is pper sulpture eteen ook struture. Tht is, ook onsists of to (surfe) overs hih re joine y hinge, n some pper ojets re fixe eteen the overs. pper ojet eteen the overs hs some fes n reses. In our moel, reses re given s prt of input, n e re not lloe to mke ne rese. rese n e fole in oth ys, n it is lloe to not e fole (unless mking ne rese). Given input is the (possile) esign of pop-up ook. Tht onsists of to surfe overs ith fixe egree, sy θ 0, n our ojetive is opening or losing the ook. More preisely, for given egree θ 1, e im to mke the egree of the ook from θ 0 to θ 1 ithout mking ne rese. No, e enote y POP(θ 0, θ 1 ) the prolem to eie if given pop-up ook ith to overs of egree θ 0 n e opene or lose to egree θ 1 ithout mking ne rese. The size of n input (or pop-up ook) is efine y the summtion of the numer of lines (or eges of ppers), the numer of (preefine) reses, n the numer of orners. In this pper, ll This is tenttive/unpulishe rft for 4OSME t the liforni Institute of Tehnology Psen, liforni, US, 2006/9/8-10. prt of this pper s presente t G [10]. The PDF file of this rft n e foun t uehr/pf/popup3.pf Shool of Informtion Siene, JIST, Ishik , Jpn. {uehr,s-termo}@jist..jp 1

2 2 Reverse Stopper Reverse Stopper (2) (3) (1) Figure 1: REVSTOP gget Figure 2: LUSE gget orers (n reses) of pper onsist of stright lines. Tht is, e o not el ith the se tht the orer of pper mkes urve. 3 losing pop-up ook In this setion, e sho NP-hrness of the losing pop-up ook. More preisely, min theorem in this setion is the folloing: Theorem 1 The prolem POP(θ 0, θ 1 ) is NP-hr for ny θ 0 > θ 1 0. We reue from ell knon NP-omplete prolem, NE3ST efine s follos [5, LO3]. Input: formul F onsists of m luses 1, 2,..., m of 3 literls ith n vriles x 1, x 2,..., x n. Output: Yes if there is truth ssignment suh tht eh luse hs t lest one true literl n t lest one flse literl. To reue the prolem, e mke three kins of ggets lle REVSTOP, LUSE, n VRILE y pper. The REVSTOP is esrie in Figure 1; for the fe, the fe n e flippe from egree 0 to egree 180 entere t the line. The LUSE is esrie in Figure 2. LUSE onsists of three prts (Figure 2(1)). On the ppers n, the right upper prts form REVSTOP. To see esily, they re omitte in Figure 2(2) n (3). Figure 2 (3) is the finl form of the LUSE (ith REVSTOP). The VRILE is esrie in Figure 3; to ottom lines ill e glue to to surfe overs, respetively. The neutrl position is epite in Figure 3(0). Sine the ottom lines hve the sme height, e hve four possile ses to fol the VRILE flt shon in Figure 3(1)-(4). mong them, the ses (3) n (4) ill e inhiite y other ggets. Hene e ill represent the true n flse ssignments y the forms (1) n (2), respetively. We ll to lines lele y n in the gget riges. When to folings (1) n (2) re exhnge, the heights of to riges (ex)hnge 2. No, e onstrut pper sulpture, or esign of pop-up ook, from formul F (Figure 4). For eh i = 1, 2,..., n, the VRILE X i for x i re glue to to overs t the ottom lines. Initilly, eh VRILE is in neutrl position; to riges re t the sme height. For luse j = (l i1, l i2, l i3 ) ith l i = x i or l i = x i, the LUSE j is onnete to VRILE X i1, X i2, n X i3 s follos: If l i1 = x i1, the ottom line of in Figure 2 is onnete to the right rige of the VRILE X i1. If l i2 = x i2, the ottom line of in Figure 2 is onnete to the left rige of the VRILE X i2. The ottom line of in Figure 2 is onnete to the rige of the VRILE X i3 similrly. The onnetions re one in nturl y; see Figure 4 for the luse j = (x 1, x 2, x n ). In Figure 4, the riges imply x 1 is true, x 2 is flse, n x n is true. We note tht eh VRILE is in neutrl position, n ll riges hve the sme height. Thus, eh LUSE is lso in neutrl position s Figure 2(3). We o not glue the ggets to the overs exept the ottom lines of VRILE s. fter onneting LUSE s n VRILE s, eh VRILE nnot e fole in the form in Figure 3(3) n (4) ithout mking ne rese. The reution n e one in polynomil time of the size of F. No e re rey to sho the key lemm: 2

3 (1) True (2) Flse (3) Illegl (4) Illegl /2 non-fole 3/2 (0) Neutrl uttom lines vlley mountin Figure 3: VRILE gget X 1 X 2 X n... lo lo Glue to uttom lines to the overs over Mke luse gget here. high (1) high (2) R (3) Hinge Figure 5: Folle n unfolle ses Figure 4: onstrution from F 3

4 D G h D E F G H I E F H I Figure 6: VRILE o gget Lemm 2 The pop-up ook onstrute ove n e lose ompletely if n only if there is truth ssignment of F suh tht eh luse hs t lest one true literl n t lest one flse literl. Proof. Eh rige of VRILE n e high hen it is on the top of the mountin, n lo hen it is on the ottom of the vlley. To fol eh VRILE flt, one of to riges is high n the other rige is lo. Hene the prts,, of LUSE n tke only to sttes, sy, high n lo. We first sho fesile ses for LUSE. When n orrespon to the sme height, n orrespons to the ifferent height, is let ome ner to, n then n e move up or on 2 height to fol them flt (Figure 5(1)). On the other hn, hen n orrespon to the sme height n tkes the ifferent height, n re let go frther to oth sies, n then n e move up or on 2 height to fol them flt (Figure 5(2)). Using the symmetri y, LUSE n e fol flt hen one of,, n is high n one of them is lo. The other ys to fol them flt n e lssifie in to ses. The first se is three ifferent heights; from the form in Figure 5(2), e n fol,, n flt ith three ifferent heights in this orer or vie vers. Hoever, this se is impossile sine three prts n tke either high or lo from the restrition y the VRILE s. The lst se is the se tht,, n hve the sme height. This foling n e one if n re fole symmetrilly s shon in Figure 5(3) here the fe, hih forms symmetri shpe of, is omitte to see the se esier. Hoever, this se is lso impossile. In the se, to symmetri fes, mrke y R in Figure 5(3), of n hve to mke 360 egree. Hoever, the reverse movement is inhiite y the REVSTOP in Figure 2(1). Therefore, the LUSE j n e fole flt if n only if one vrile tkes the ifferent vlue from the other to vriles. Hene the pop-up ook n e lose if n only if F is yes instne of NE3ST. No e prove the min theorem in this setion. In Lemm 2, mking the ggets smll enough, e n prove the theorem if θ 0 is smll enough n θ 1 = 0. When θ 1 > 0 n θ 0 is lose enough to θ 1, e mke the ggets eteen to inner overs, n put some stle stns eteen the inner overs n surfe overs. On the other hn, hen θ 1 is lrge, e join the inner overs n surfe overs y long pper rion ith one rese. It is esy to just the length of them to fit for given θ 1 n θ 0. This ompletes the proof of Theorem 1. 4 Opening pop-up ook In this setion, e sho NP-hrness of the opening pop-up ook. More preisely, min theorem in this setion is the folloing: Theorem 3 The prolem POP(θ 0, θ 1 ) is NP-hr for ny θ 1 > θ 0 0. We reue from the 3ST, ell knon NP-omplete prolem [5]. Let F e n instne of 3ST, hih onsists of m luses 1, 2,..., m of 3 literls ith n vriles x 1, x 2,..., x n. To reue F, e mke to kins of ggets lle VRILE o n LUSE o y pper. The VRILE o is esrie in Figure 6; tht onsists of three thik retngles n six thin retngles. To eges of the sme lel re glue s in Figure 6. We note tht the resultnt gget is ompletely flt. Let h e the ommon height of the retngles. Next, to hnles re glue to the VRILE o t height h/2 s in Figure 7(1). (To hnles n e fol flt t the enter reses.) Then, there re only to ys to mke to hnles 2h prt shon in Figure 7(2) n (3). (It hs the sme struture to n ol sin ooen toy hih onsists of severl ors ne like Figure 6, n they n e ontinuously flippe y tisting hnle.) We ll the se 4

5 Left hnle h/2 h/2 h Right hnle (1) (2) True se (3) Flse se Figure 7: Hnles ith VRILE o gget Left Right rm 3 rm 3 Vrile -h/2 h Joints for justment h h h h h h x y R Vlley Mountin z R ottom lines h/2 h/4 x y P Q Glue to hinge P Q Figure 8: gget rms for VRILE o Figure 9: LUSE o gget (2) True n (2) Flse. No, e tth to kins of rms in Figure 8 to the VRILE o. (The numer of rms ill e esrie lter.) The lele eges re glue to the orresponing eges in Figure 6. (Preisely, the left rm is eteen n F t 3, n the right rm is eteen n I t 3.) The joints for justment re fole flt s in Figure 8. No, from the ompletely lose VRILE o, hen e mke to hnles 2h prt s in Figure 7(2), the left rm n go on t most h/2 sine it is free exept t the ege 3, ut the right rm hs to go up h/2 sine it is ught y n, n pulle up. Hene, the ottom line of the left rm n go on h ith unfoling the joint, n the ottom line of the right rm nnot go on from the initil position. We note tht, in the se, the left rm n hoose to sty t the initil position ith using the joint. Similrly, hen e mke to hnles 2h prt s in Figure 7(3), the right rm n go on h, n the left rm nnot go on t ll. The LUSE o is esrie in Figure 9. LUSE o onsists of three rions P, Q, n R. The rion R hs length 6h, n oth sies re glue to the overs t istne from the hinge. The rion P joins the hinge n one of the vlley on R, n the rion Q joins the hinge n nother vlley on R. No, e onstrut pper sulpture, or esign of pop-up ook, from formul F. For eh i = 1, 2,..., n, the VRILE o X i for x i re glue to to overs y to hnles t istne 2 from the hinge. For luse j = (l i1, l i2, l i3 ) ith l i = x i or l i = x i, the LUSE o j is onnete to VRILE o X i1, X i2, n X i3 s follos: If l i1 = x i1, one of three mountins on the rion R of j is onnete to the ottom line of the left rm of X i1. If l i2 = x i2, nother mountin on R is onnete to the ottom line of the right rm of X i2. The lst mountin of R is onnete to X i3 similrly. Hene, X i hs l i left rms n r i right rms, here l i n r i re the numer of ourrenes of x i n x i in F, respetively. We note tht, ith suitle hoie of h n, ll ggets n e fole flt, n the resultnt pop-up ook n e lose ompletely. The reution n e one in polynomil time of the size of F. No e re rey to sho the key lemm: Lemm 4 The pop-up ook onstrute ove n e opene if n only if there is truth ssignment of F suh tht eh luse hs t lest one true literl. 5

6 Proof. We try to open the ook ith the ssignment for eh VRILE o. For eh luse j, if t lest one of three literls is true, the orresponing rm omes on to j, n hene it n e opene to θ ith sin θ = h. Hoever, if none of them re true, no rms ome lose to j, n hene it nnot e opene. Hene F is stisfile if n only if the pop-up ook n e opene to θ. No e prove the min theorem. In Lemm 4, letting h, e hve the theorem for POP(0, θ 1 ) for smll θ 1 > 0. We use the sme trik in Setion 3 for the other ses. This ompletes the proof of Theorem 3. 5 onluing remrks For the prolems for n origmi n pop-up ook, e i not sho tht they re in NP. In ft, the prolems might e PSPE-hr in some moel; they seem to e similr to the movement prolems for 2-imensionl linkges, hih is PSPE-hr ue to Hoproft, Joseph, n Whitesies [6]. Referenes [1] M. ern n. Hyes. The omplexity of Flt Origmi. In Pro. 7th nn. M-SIM Symp. on Disrete lgorithms, pges , [2] D.. rter n J. Diz. Elements of Pop Up : Pop Up ook for spiring Pper Engineers. Little Simon, [3] E. D. Demine n M. L. Demine. Reent Results in omputtionl Origmi. In Proeeings of the 3r Interntionl Meeting of Origmi Siene, Mth, n Eution (OSME 2001), pges 3 16, [4] E. D. Demine n J. O Rourke. Survey of Foling n Unfoling in omputtionl Geometry In omintoril n omputtionl Geometry, Volume 52 of Mthemtil Sienes Reserh Institute Pulitions, pges mrige University Press, [5] M. R. Grey n D. S. Johnson. omputers n Intrtility Guie to the Theory of NP-ompleteness. Freemn, [6] J. E. Hoproft, D.. Joseph, n S. H. Whitesies. Movement Prolems for 2-Dimensionl Linkges. SIM J. omput., 13: , [7] P. Jkson. The Pop-up ook. Ol ooks, [8] R. J. Lng. Origmi Design Serets. K Peters LTD, [9] R. Su. Winter s Tle: n Originl Pop-up Journey. Little Simon, [10] R. Uehr n S. Termoto. The omplexity of Pop-up ook. In Pro. 18th nin onferene on omputtionl Geometry, pges 3 6, uehr/pf/popup2.pf 6