The Vertex-Adjacency Dual of a Triangulated Irregular Network has a Hamiltonian Cycle

Transcription

1 The Verex-Adjacency Dual of a Triangulaed Irregular Nework ha a Hamilonian Cycle John J. Barholdi, III Paul Goldman November 1, 003 Abrac Triangulaed irregular nework (TIN) are common repreenaion of urface in compuaional graphic. We define he dual of a TIN in a pecialway,baedonverex-adjacency,andhowhaihamiloniancycle alway exi and can be found efficienly. Thi reul ha applicaion in ranmiion of large graphic daae. Keyword: Hamilonian cycle, riangulaed irregular nework, riangle meh, riangulaion Correponding auhor. Addre: School of Indurial and Syem Engineering, Georgia Iniue of Technology, 765 Fer Drive, Alana, Georgia USA. john.barholdi@iye.gaech.edu. We appreciae he uppor of he Office of Naval Reearch hrough Gran #N J-1571, he Naional Science Foundaion hrough Gran #DMI , The Logiic Iniue a Georgia Tech and i ier organizaion, The Logiic Iniue Aia Pacific in Singapore. We are alo graeful for he ueful commen of an anonymou referee. 1

2 1 Inroducion Following Peer Alfeld [1]: A riangulaed irregular nework (TIN, or riangle meh, or riangulaion) T i a collecion of n riangle aifying he following: 1. The inerior of he riangle are pairwie dijoin;. Each edge of a riangle in T i eiher a common edge of wo riangle in T or ele i i on he boundary of he union of all he riangle; 3. The union of all he riangle i homeomorphic o a quare. Alfeld oberve ha The fir requiremen ay ha riangle don overlap. The econd requiremen rule ou combinaion of riangle where one ha a verex in he inerior of an edge of anoher riangle, and he range ounding la requiremen rule ou hole, pinchpoin (where ju riangle mee in a ingle poin) and dijoin e of riangle. TIN are commonly ued o repreen urface uch a errain and 3-D compuer graphic model (ee for example [6]). A dual graph of a TIN can be conruced by defining a verex (node) for each riangle, and an edge for each pair of riangle ha hare an edge. A TIN i aid o be Hamilonian if hi dual ha a Hamilonian cycle (or pah). Thu a Hamilonian cycle order he riangular cell of a TIN in a way ha preerve edge-adjacency. The idea of finding equence of edge-adjacen riangle ha been ued o compre large polyhedral urface model [3, 5, 7, 8, 9]. The problem arie becaue each verex in a TIN may belong o muliple riangle and o a naive mehod of ranmiing or proceing a TIN, uch a by ending he riangle individually, would conain much duplicaed daa. If he TIN can be diaembled ino a Hamilonian pah of riangle ha i, if he dual graph i Hamilonian hen he TIN can be ranmied very economically, by a raegy of ending he ingle remaining verex of he nex riangle along he pah. In hi way, each poin would be proceed once. Unforunaely, i i NP-Complee o deermine wheher or no a TIN i Hamilonian []. Conequenly, hi mehod of compreion i generally implemened by rying o find long equence of edge-adjacen riangle (egmen of pah in he dual graph). Evan, Skiena, and Varhney (1996) dicu ome ypical greedy heuriic o find long pah in he dual in hope of achieving greaer compreion []. However, increaing compuaion rik offeing any gain in compreion. We how ha a weaker verion of he dual guaranee he exience of a Hamilonian circui. We ugge conidering he verex-adjacency dual of a TIN, which i conruced by defining a verex for each riangle, and an edge for each pair of riangle ha hare a verex or an edge. A riangle ha more verex-adjacen neighbor han i doe edge-adjacen neighbor; we how ha hi exra conneciviy i ufficien o guaranee he exience of a Hamilonian

3 (i) (ii) (iii) 5 Figure 1: Three approache o ranmiing TIN daa. (i): a individual riangle, (ii): by edge-adjacency, (iii): by verex-adjacency. circui. Moreover, uch a circui can be conruced in linear ime, o ha i eem o offer a pracical bai for daa compreion. Figure 1 illurae he hree approache o ranmiing a TIN ha we have menioned: (i) naive approach, in which each riangle i en individually, reuling in he ranmiion of 15 verice; (ii) riangle rip approach, in which he TIN i divided ino wo edge-adjacen riangle rip, and 9 verice are ranmied; (iii) verex-adjacency approach, in which 8 verice are ranmied. Conrucing a Hamilonian cycle We how how o conruc a Hamilonian cycle hrough he verex-adjacency dual of a TIN T. The cycle of he node in he dual correpond o a cyclic ordering of he riangle in he TIN. For convenience, where here can be no confuion we will peak inerchangeably of cycle of eiher node (in he dual) or riangle (in he TIN). Each riangle ha wo neighbor in a cycle: a predeceor pred() and a ucceor ucc(). Neighboring riangle of T haring a common edge(wo verice) will be called -adj ; neighboring riangle haring exacly one common verex will be called 1-adj. The cycle ha we creae ha he following pecial propery ha i ueful in avoiding an awkward cae in he conrucion of he cycle. Propery 1 No riangle i 1-adj, a he ame verex, o boh of i neighbor in he cycle. Theorem 1 The verex-adjacency dual of a TIN i Hamilonian. Proof (conrucive) 1. Iniializaion. Aume TIN T ha n riangle. The heorem i rivially rue for n =1 or n =,oaumen 3. Chooe any riangle a, a econd riangle b ha i -adj o a, and a hird riangle c ha i -adj o eiher a or b. Aume wihou lo of generaliy ha c i -adj o b. Le C ignify he 3

4 Figure : Cae I: i 1-adj (a a verex on e) o a lea one neighbor. ube of T compoed of riangle a, b, andc. C i ielf a TIN, and ince riangle a i clearly 1-adj or -adj o riangle c, a Hamilonian cycle of he verex-adjacency dual of C i imply: a, b, c, a. Propery1iaified, ince each riangle i -adj o a lea one oher riangle. The procedure coninue by adding riangle o C, one a a ime, and updaing he cycle of i dual a each ep, unil C include all of T.Afer each updae, we mu verify ha he new cycle i ill Hamilonian (by checking adjacencie around he change). We alo verify ha Propery 1remainaified by howing, for each riangle involved in he updae, ha eiher: i neighbor are unchanged; i ha a -adj neighbor; i i 1-adj o boh neighbor a differen verice; or, i ha exchanged one 1-adj neighbor for a differen 1-adj neighbor a he ame verex. The raighforward deail of all hee check are deferred o he Appendix.. Ieraion. If C include all he riangle in T hen he procedure i done. Oherwie, find a riangle, noinc, ha hare edge e wih ome riangle in C. Add o C, updainghecycleaccordingohefollowingcae,which are pariioned according o relaionhip wih i neighbor: Cae I: i 1-adj, one, o (a lea) one of i neighbor (Figure ). Aume, wihou lo of generaliy, ha i 1-adj, one, opred(). Soluion: Add o he cycle a follow: pred(),,,ucc(). (Noe ha pred() now acually become pred(), bu for clariy we coninue o refer o verice by heir iniial label, here and elewhere.) The curve in Figure i hown o originae a he cener of pred(), and erminae a he cener of. Thiimeanoindicaehawe do no reric how pred() iadjacenopred(pred()), or how i adjacen o ucc() becaue all (legal) poibiliie are covered by he oluion hown. For example, and ucc() couldbe-adj a eiher of open edge, or hey could be 1-adj a any of verice (aide from he verex a which i 1-adj o pred()). Cae II: i -adj o boh neighbor (Figure 3). Noe ha in he figure ome curve begin (end) a a verex, indicaing ha he fir riangle i adjacen o i predeceor (ucceor) a he verex indicaed. When a curve begin (end) a a riangle cener,

5 IIa IIb Oherwie, Figure 3: Cae II: i -adj o boh neighbor. Subcae IIa: pred(pred()) i 1-adj o pred() on e, and ucc(ucc()) i 1-adj o ucc() on e. Subcae IIb: Oherwie. he preceding riangle may be adjacen by any of i edge or verice, a long a he adjacency i legal. IIa: pred(pred()) i 1-adj o pred() one, anducc(ucc()) i 1-adj o ucc() one. Soluion: pred(pred()),,,pred(),ucc(),ucc(ucc()). IIb: Oherwie (pred(pred()) i no 1-adj o pred() one, or ucc(ucc()) i no 1-adj o ucc() one, orbohareno). Aume, wihou lo of generaliy, ha pred(pred()) i no 1-adj o pred() one. Soluion: pred(),,,ucc(). Cae III: i -adj o one neighbor, and 1-adj, off e, o he oher neighbor (Figure ). Aume, wihou lo of generaliy, ha i -adj o pred(), and 1-adj, off e, oucc(). IIIa: pred() i1-adj o pred(pred()), on e. Soluion: pred(pred()),,,pred(),ucc(). IIIb: Oherwie (pred(pred()) i no 1-adj o pred() one). Soluion: pred(),,,ucc(). The procedure conume one riangle wih each ieraion, and o erminae afer n 3 ep. To verify ha he cae comprie he complee e of poibiliie, oberve ha: Each riangle in he curren cycle (a each ep) i eiher 1-adj or -adj o each of i wo neighbor pred() and ucc(). Cae I cover he cae where i 1-adj o boh neighbor (becaue a lea one of he common verice mu be on e, by Propery 1); Cae II cover he cae where i -adj o boh neighbor; Cae III cover he cae where i -adj o one neighbor and 5

6 IIIa IIIb Oherwie, Figure : Cae III: i -adj o one neighbor, and 1-adj (a verex oppoie e) o he oher neighbor. Subcae IIIa: pred() i 1-adj (on e) o pred(pred()). Subcae IIIb: Oherwie. 1-adj o he oher (excep where he 1-adjacency i on e, which i covered in Cae I). Cae II and III, are each pariioned by a pair of muually-excluive ubcae. The remaining poibiliy, in which boh of neighbor are 1-adj a he ame verex, i forbidden by Propery 1 and never occur. 3 Concluion By acceping verex-adjacency, raher han iniing on edge-adjacency, we have hown ha, for any TIN, one can conruc, in linear ime, a ingle equence of adjacen riangle ha encompae he enire TIN. Thi allow an alernaive raegy for compreion. Recen effor have focued on finding a mall number of pah egmen ha pan all riangle of he TIN. Each pah egmen can hen be en efficienly bu here i exra work o find he pah egmen and exra work o hif from one pah egmen o he nex. Our mehod quickly find a ingle pah, bu i may no compre a well a he ideal one baed on edge-adjacency (which may no exi and can be hard o find in any cae). Poenial duplicaion arie in our mehod whenever wo neighboring riangle (in he cycle) hare ju one verex, o ha we mu end wo new verice raher han one o pecify he econd riangle. Thi never require more han wo poin per riangle, which i wore han he (perhap unachievable) lower 6

7 bound of one poin per riangle, bu beer han ending each riangle eparaely (3 poin per riangle). More o he poin, we would no expec verex-adjacen riangle o dominae; informal e ugge ha here will likely be run of edge-adjacen riangle inerpered wih occaional verex-adjacen riangle. Comparion of compreion mehod baed on edge-adjacency and mehod baed on verex-adjacency i an empirical queion ha will be explored in ubequen work. Reference [1] Peer Alfeld. Triangulaion. hp:// riangulaion.hml, March [] Eher M. Arkin, Marin Held, Joeph S. B. Michell, and Seven S. Skiena. Hamilonian riangulaion for fa rendering. In Viualizaion 95: Proceding, Oc 9 Nov 3, 1995, Alana, GA, [3] John J. Barholdi, III and Paul Goldman. Mulireoluion indexing of riangulaed irregular nework. IEEE Tranacion on Viualizaion and Compuer Graphic. Toappear. [] Francine Evan, Seven Skiena, and Amiabh Varhney. Opimizing riangle rip for fa rendering. In Proceeding, IEEE Viualizaion 1996, page , [5] Marin Ienburg. Triangle rip compreion. In Proceeding of Graphic Inerface 000, page 197 0, May 000. [6] Rober Laurini and Derek Thompon. Fundamenal of Spaial Informaion Syem. Academic Pre, Ld., San Diego, CA, 199. [7] Jarek Roignac. Edgebreaker: Conneciviy compreion for riangle mehe. IEEE Tranacion on Viualizaion and Compuer Graphic, 5(1), [8] Gabriel Taubin and Jarek Roignac. Geomeric compreion hrough opological urgery. ACM Tranacion on Graphic, 17():8 115, Apr [9] Coa Touma and Craig Goman. Triangle meh compreion. In Proceeding of Graphic Inerface, 1998, page 6 3, Vancouver, B. C.,