Quadratic-time, linear-space algorithms for generating orthogonal polygons with a given number of vertices

Quad a ic-Time Linea -Space Algo i hms o Gene a ing O hogonal Polygons wi h a Gi en Numbe o Ve ices ⋆ Ana Paula Tom´as ∗,aand An ´onio Leslie Bajuelos b aDCC-FC & LIACC, Uni e si y o Po o, Po ugal bDepa men o Ma hema ics & CEOC - Cen e o Resea ch in Op imiza ion and Con ol, Uni e si y o A ei o, Po ugal Key wo ds: O hogonal Polygons, Gene a ion, Decomposi ion, Dual G aphs, Mou hs. 1. In oduc ion This wo k ocus on simple polygons wi hou holes. The e o e, we call hem jus polygons. Mo e- o e , “polygon” will some imes mean a polygon oge he wi h i s in e io . Pdeno es a polygon and he numbe o i s e lex e ices. A polygon is o hogonal (o ec ilinea ) i i s edges mee a igh angles. O’Rou ke [4] has shown ha n= 2 + 4 o e e y n- e ex o hogonal polygon (n-ogon, o sho ). Gene ic n-ogons may be ob ained om a pa icula kind o n-ogons, ha we called g id o hogonal polygons, as illus a ed in Fig. 1. Fig. 1. Th ee 12-ogons mapped o he same g id 12-ogon. De ini ion 1 An n-ogon Pis in gene al posi ion i e e y ho izon al and e ical line con ains a mos one edge o P, i.e., i Phas no collinea edges. We call “g id n-ogon” each n-ogon in gene al posi ion de ined in a n 2×n 2squa e g id. We assume ha he g id is de ined by ho i- zon al lines y= 1,...,y =n 2and e ical lines x= 1,...,x=n 2and ha i s no hwes co ne has ⋆(Ex ended abs ac ) Pa ially unded by LIACC h ough P og ama de Financiamen o Plu ianual, Funda¸c˜ao pa a a Ciˆencia e Tecnologia (FCT) and P og ama POSI, and by CEOC (Uni . o A ei o) h ough P og ama POCTI, FCT, co- inanced by EC und FEDER. ∗Co esponding au ho Email add esses: ap @ncc.up.p (Ana Paula Tom´as ), leslie@ma .ua.p (An ´onio Leslie Bajuelos ). coo dina es (1,1). Each g id n-ogon has exac ly one edge in e e y line o he g id. Each n-ogon no in gene al posi ion may be mapped o an n-ogon in gene al posi ion by ǫ-pe u ba ions, o a su icien ly small cons an ǫ > 0. Hence, we es ic gene a ion o n-ogons in gene al posi ion. Each n-ogon in gene al posi- ion is mapped o a unique g id n-ogon h ough op- o-bo om and le - o- igh sweeping. And, ecip ocally, gi en a g id n-ogon we may c ea e an n-ogon ha is an ins ance o i s class by andomly spacing he g id lines in such a way ha hei ela i e o de is kep . 1.1. The pape ’s con ibu ion We p opose wo me hods ha gene a e g id n-ogons in polynomial ime – In la e-Cu and In la e-Pas e. The o me was published in [7], whe e we also ga e implemen a ion de ails, show- ing ha i equi es linea space in nand uns in quad a ic ime in a e age. Two p og ams o gene a ing andom o hogonal polygons, by O’Rou ke (de eloped o he e alua ion o [5]) and by Filguei as 1a e men ioned he e. The main idea o O’Rou ke is o cons uc such a polygon ia g ow h om a seed cell (i.e., uni squa e) in a boa d, gluing oge he a gi en numbe o cells ha a e selec ed andomly using some heu is ics. Filguei as’ me hod sha es a simila idea hough i glues ec angles o la ge a eas and allows hem o o e lap. Nei he o hese me hods allows o con- ol he inal numbe o e ices o he polygon. A majo idea in In la e-Pas e is also o glue 1pe sonal communica ion, DCC-LIACC, 2003. 20 h EWCG Se ille, Spain (2004) 20 h Eu opean Wo kshop on Compu a ional Geome y ec angles. Ne e heless, i es ic s he posi ions whe e ec angles may be glued, which ende s he algo i hm simple and p o ides con ol on he i- nal numbe o e ices. Fo he la e pu pose, he In la e ans o ma ion is c ucial. I is possible o implemen In la e-Pas e so ha i equi es quad a ic- ime in he wo s -case and linea -space. Ou me hods may be also adap ed o gene a e simple o hogonal polygons wi h holes. Indeed, each hole is an o hogonal polygon wi hou holes. 2. In la e, Cu and Pas e ans o ma ions Le i= (xi, yi), o i= 1,...,n, be he e ices o a g id n-ogon P, in CCW o de . In la e akes a g id n-ogon Pand a pai o in ege s (p, q) wi h p, q ∈[0,n 2], and yields a new n- e ex o hogonal polygon ˜ Pwi h e ices ˜ i= (˜xi,˜yi) gi en by ˜xi=xii xi≤pand ˜xi=xi+ 1 i xi> p, and ˜yi=yii yi≤qand ˜yi=yi+ 1 i yi> q, o i= 1,...,n. Thus, i augmen s he g id, c ea ing wo ee lines, x=p+ 1 and y=q+ 1. In la e-Cu : Le Cbe a uni cell in he in e io o P, wi h cen e cand no hwes e ex (p, q). When we apply In la e o Pusing (p, q), cis mapped o ˜c= (p+ 1, q + 1), ha is he cen e o in la ed C. The goal o Cu is o in oduce ˜cas e lex e ex o he polygon. To do ha , i cu s one ec angle (de ined by ˜cand a e ex ˜ mbelonging o one o he ou edges sho by he ho izon al and e ical ays ha emana e om ˜c). We allow such a ec angle o be cu i i con ains no e ex o ˜ P excep ˜ m. I no ec angle may be cu , we say ha Cu ails o C. So, suppose ha ˜sis he poin whe e one o hese ays i s in e sec s he bounda y o ˜ P, ha ˜ m is one o he wo e ices on he edge o ˜ P ha con ains ˜sand ha he ec angle de ined by ˜cand ˜ mmay be cu . Cu cu s his ec angle om ˜ P eplacing ˜ mby ˜s, ˜c, ˜s′i his sequence is in CCW o de (o ˜s′, ˜c, ˜s, o he wise), wi h ˜s′= ˜c+(˜ m−˜s). We may conclude ha ˜s, ˜c, ˜s′is in CCW o de i ˜s belongs o he edge ˜ m−1˜ mand in CW o de i i belongs o ˜ m˜ m+1.Cu always emo es a single e ex o he g id ogon and in oduces h ee new ones. Fig. 2 illus a es his echnique. Because Cu ne e ails i Chas an edge ha is pa o an edge o P,In la e-Cu may be always applied o P. 4 In la e Cell Cu 1 Cu 2 Cu 3 Cu 4 3 21 CC Fig. 2. The wo ec angles de ined by he cen e o C and he e ices o he le mos e ical edge ((1,1),(1,7)) canno be cu . The e emain he ou possibili ies shown. In la e-Pas e: We i s imagine he g id n-ogon me ged in a (n 2+ 2) ×(n 2+ 2) squa e g id, wi h he op, bo om, le mos and igh mos g id lines ee. The op line is x= 0 and he le mos one y= 0, so ha (0,0) is now he no hwes co - ne o his ex ended g id. Le eH( i) ep esen he ho izon al edge o P o which ibelongs. De ini ion 2 Gi en a g id n-ogon Pme ged in o a(n 2+ 2) ×(n 2+ 2) squa e g id, and a con ex e - ex io P, he ee s ai case neighbou hood o i, deno ed by FSN( i), is he la ges s ai case polygon in his g id ha has ias e ex, does no in e sec he in e io o Pand i s base edge con ains eH( i). An example is gi en in Fig. 3. 10 11 12 14 1 2 4 6 3 Fig. 3. A g id n-ogon me ged in o a (n 2+ 2) ×(n 2+ 2) squa e g id and he ee s ai case neighbou hood o each o i s con ex e ices, wi h n= 14. Now, o ans o m Pby In la e-Pas e we i s ake a con ex e ex io P, selec a cell C in FSN( i), and apply In la e o Pusing he no wes co ne (p, q) o C. As be o e, he cen e o cell Cis mapped o ˜c= (p+ 1, q + 1), which will now be a con ex e ex o he new polygon. Pas e glues he ec angle de ined by ˜ iand ˜c o ˜ P, inc easing he numbe o e ices by wo. I eH( i)≡ i i+1 hen Pas e emo es ˜ i= (˜xi,˜yi) and inse s he chain (˜xi, q + 1), ˜c, (p+ 1,˜yi) in i s place. I eH( i)≡ i−1 i,Pas e eplaces ˜ iby he sequence (p+ 1,˜yi), ˜c, (˜xi, q + 1). Fig. 4 illus a es his ans o ma ion. Clea ly, Pas e ne e ails, in con as o Cu . Ma ch 25-26, 2004 Se ille (Spain) 10 Fig. 4. The ou g id 14-ogons ha we may cons uc i we apply In la e-Pas e o he gi en 12-ogon, o ex end he e ical edge ha ends in e ex 10. 3. In la e-Cu and In la e-Pas e Me hods We showed in [7] ha e e y g id n-ogon may be c ea ed om a uni squa e (i.e., he g id 4-ogon) by applying In la e-Cu ans o ma ions. Now, we may show he same esul o In la e-Pas e. A i e a ion k, bo h me hods cons uc a g id (2k+ 4)-ogon om he g id (2(k−1) + 4)-ogon ob ained in he p e ious i e a ion, o 1 ≤k≤ . The In la e-Cu me hod yields a andom g id n-ogon, i cells and ec angles a e chosen a an- dom. This is also ue o In la e-Pas e, hough now o he selec ions o iand o Cin FSN( i). I is no di icul o see ha bo h In la e-Cu and In la e-Pas e yield g id ogons. In con as , he p oo o hei comple eness is no immedia e, as sugges ed by he example gi en in Fig. 5. Fig. 5. The igh mos polygon is he unique g id 16-ogon ha gi es ise o his 18-ogon, i we apply In la e-Cu . Be o e we go h ough he p oo , we need o in- oduce some de ini ions and esul s. De ini ion 3 Gi en a simple o hogonal polygon Pwi hou holes, le ΠH(P)be he ho izon al decom- posi ion o Pin o ec angles ob ained by ex ending he ho izon al edges inciden o e lex e ices o- wa ds he in e io o Pun il hey hi i s bounda y. Each cho d (i.e., edge ex ension) sepa a es exac ly wo adjacen pieces ( aces), since i makes an ho i- zon al cu (see e.g. [9]). The dual g aph o ΠH(P) cap u es he adjacency ela ion be ween pieces o ΠH(P). I s nodes a e he pieces o ΠH(P) and i s non-o ien ed edges connec adjacen pieces. Lemma 4 The dual g aph o ΠH(P)is a ee o all simple o hogonal polygons Pwi hou holes. PROOF. This esul ollows om he well-known Jo dan Cu e Theo em. Suppose he g aph con- ains a simple cycle F0, F1,...,Fd, F0, wi h d≥2. Le γ= (γ0,1γ1,2. . . γd,0) be a simple closed cu e in he in e io o P ha links he cen oids o he aces F0, F1,...,Fd. Deno e by he e lex e ex ha de ines he cho d s ha sepa a es F0 om F1. He e, s is he poin whe e his edge’s ex en- sion in e sec s he bounda y o P. Ei he o s would be in he in e io o γ, because γneeds o c oss he ho izon al line suppo ing s a leas wice and jus γ0,1c osses s . Bu he in e io o γis con ained in he in e io o P, and he e exis poin s in he ex e io o Pin he neighbou hood o and o s , so ha we achie e a con adic ion. ✷ I is wo h no ing ha he e ical decomposi- ion ΠV(P) o Pwould ha e iden ical p ope ies. We shall now p o e P oposi ion 5 ha asse s he comple eness o In la e-Pas e. P oposi ion 5 Fo each g id (n+ 2)-ogon, wi h n≥4, he e is a g id n-ogon ha yields i by In la e-Pas e. PROOF. Gi en a g id (n+ 2)-ogon P, we use Lemma 4 o conclude ha he dual g aph o ΠH(P) is a ee. Each lea o his ee co esponds o a ec angle ha could ha e been glued by Pas e o yield P. Indeed, suppose ha s is he cho d ha sepa a es a lea F om he es o P. Because g id ogons a e in gene al posi ion, s is no a e ex o P. I belongs o he ela i e in e io o an edge o P. The e ex o ec angle F ha is no adjacen o s would be ˜cin In la e-Pas e. I we cu F, we would ob ain an in la ed n-ogon, ha we may de la e o ge a g id n-ogon ha yields P. The wo g id lines y=y˜cand x=x˜ca e ee. Clea ly s is he e ex we called iin he desc ip ion o In la e-Pas e (mo e accu a ely, s is ˜ i) and c= (x˜c−1, y˜c−1) ∈F SN( i). ✷ Fo his pape o be sel -con ained, we ecall now a p oo o he comple eness o In la e-Cu , al- eady ske ched in [7]. I was inspi ed by wo k abou con exi ica ion o simple polygons [2,6,8], in pa i- cula , by a ecen pape by O. Aichholze e al. [1]. I also sha es ideas o a p oo o Meis e s’ Two- Ea s Theo em [3] by O’Rou ke, hough we we e no awa e o his when we w o e i . Fig. 6 illus a es he undamen al ideas. 20 h Eu opean Wo kshop on Compu a ional Geome y pocke pocke A Fig. 6. The wo le mos g ids show a g id 18-ogon and i s pocke s. The shaded ec angle A is a lea o he ee associ- a ed o he e ical pa i ioning o he la ges pocke . The igh mos polygon is an in la ed g id 16-ogon ha yields he ep esen ed g id 18-ogon, i Cu emo es ec angle A. We need some addi ional de ini ions and esul s. De ini ion 6 Apocke o a noncon ex polygon P is a maximal sequence o edges o Pdisjoin om i s con ex hull excep a he endpoin s. The lid is he line segmen joining i s wo endpoin s. Any noncon ex polygon Phas a leas one pocke . Each pocke o an n-ogon, oge he wi h i s lid, de ines a simple polygon wi hou holes, ha is almos o hogonal excep o an edge (lid). I is possible o sligh ly ans o m i o ob ain an o hogonal polygon, as illus a ed in Fig. 6. We shall e e o his polygon as an o hogonalized pocke . Fo e e y o hogonalized pocke Q, i is easy o see ha he pocke ’s lid is con ained in a single ec angle o ei he ΠH(Q) o ΠV(Q). Le Π(Q) ep esen he one whe e he lid is con ained in a single piece. P oposi ion 7 Fo each g id (n+ 2)-ogon, he e is a g id n-ogon ha yields i by In la e-Cu . PROOF. Gi en a g id (n+ 2)-ogon P, le Qbe an o hogonalized pocke o P. Necessa ily, Qis in gene al posi ion. By Lemma 4 he dual g aph o Π(Q) is a ee. We claim ha a leas one o i s lea es con ains o is i sel a ec angle ha migh ha e been emo ed by Cu o yield P. Indeed, he lea es a e o he wo ollowing o ms. c ~ ~ m c ~ ~ m The shaded ec angles a e he ones ha migh ha e been cu . We ha e also ep esen ed he poin s ha would be ˜ mand ˜cin In la e-Cu . He e, we mus be ca e ul abou he lea ha has he pocke ’s lid. Only i he ee consis s o a single node (c. . he smalles pocke in Fig. 6), may his lea be illed. Bu , e e y non-degene a ed ee has a leas wo lea es. Then, in his case he ee has a lea o he han he one ha con ains he lid. ✷ The concep o mou h [8] was c ucial o each he cu en o mula ion o Cu . Ac ually, In la e- Cu is somehow doing he e e se o an algo i hm gi en by Toussain in [8] ha compu es he con ex hull o a polygon globbing-up mou hs o succes- si ely emo e i s conca i ies. Fo o hogonal poly- gons, we would a he de ine ec angula mou hs. De ini ion 8 A e lex e ex io an ogon Pis a ec angula mou h o Pi he in e io o he ec - angle de ined by i−1and i+1 is con ained in he ex e io o Pand nei he his ec angle no i s in e- io con ain e ices o P, excep i−1, iand i+1. To jus i y he co ec ion o ou echnique, we obse e ha when we apply Cu o ob ain a g id (n+ 2)-ogon, he e ex ˜cis always a ec angula mou h o he esul ing (n+ 2)-ogon. In sum, he p oo o P oposi ion 7 gi en abo e jus i ies Co ol- la y 9, which eph ases he One-Mou h Theo em by Toussain . Co olla y 9 Each g id n-ogon has a leas one ec angula mou h, o n≥6. Re e ences [1] Aichholze , O., Co ´es, C., Demaine, E. D., Dujmo ic, V., E ickson, J., Meije , H., O e ma s, M., Palop, B., Ramaswawi, S., Toussain , G. T.: Flip u ning polygons. Disc e e & Compu . Geome y 28 (2002), 231–253. [2] E d¨os, P.: P oblem numbe 3763. Ame ican Ma hema ical Mon hly 42 (1935) 627. [3] Meis e s, G. H.: Polygons ha e ea s. Ame ican Ma hema ical Mon hly 82 (1975) 648-651. [4] O’Rou ke, J.: An al e na e p oo o he ec ilinea a galle y heo em. J. o Geome y 21 (1983) 118–130. [5] O’Rou ke, J., Pashchenko, I., Tewa i, G.: Pa i ioning o hogonal polygons in o a ec angles. In P oc. 13 h Canadian Con e ence on Compu a ional Geome y (CCCG’01) (2001) 133-136. [6] Sz.-Nagy, B.: Solu ion o p oblem 3763. Ame ican Ma hema ical Mon hly 46 (1939) 176–177. [7] Tom´as, A. P., Bajuelos, A. L.: Gene a ing Random O hogonal Polygons. To appea in Pos -con e ence P oceedings o CAEPIA-TTIA’2003, LNAI, Sp inge - Ve lag (2004). [8] Toussain , G. T.: Polygons a e an h opomo phic. 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