A NON-RECURSIVE ALGORITHM FOR POLYGON TRIANGULATION. Predrag S. STANIMIROVI], Predrag V. KRTOLICA, Rade STANOJEVI] 1. INTRODUCTION AND PRELIMINARIES

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1 Yugoslav Joural of Operatios Research 13 (2003), Number 1, A NON-RECURSIVE ALGORITHM FOR POLYGON TRIANGULATION Predrag S. STANIMIROVI], Predrag V. KRTOLICA, Rade STANOJEVI] Faculty of Sciece ad Mathematics, Uiversity of Ni{, Ni{, Serbia pecko@pmf.pmf.i.ac.yu, krca@pmf.pmf.i.ac.yu, rrr@euet.yu Abstract: I this paper a algorithm for the covex polygo triagulatio based o the reverse Polish otatio is proposed. The formal grammar method is used as the startig poit i the ivestigatio. This idea is "traslated" to the arithmetic expressio field eablig applicatio of the reverse Polish otatio method. Keywords: Reverse Polish otatio, covex polygo triagulatio, cotex-free grammar. 1. INTRODUCTION AND PRELIMINARIES The triagulatio of the covex polygo is the followig problem. For the give polygo fid the umber of the possible splittig o triagles by its diagoals without gaps ad overlaps of these splittig. This is the classical problem solved so far i a few ways. Oe solutio from [4] uses the cotext-free grammar with productios: S ass, S b, (1) where a ad b are termials, ad S a o-termial symbol. The triagulatio of polygo is based o the followig priciples: (a) The o-termial S represets a orieted topological segmet. This segmet is amed potetial. (b) The chai as S correspods to the orieted topological triagle with oe real edge a ad two potetial edges S ad S. (c) The productio S a SS meas the replacemet of the potetial segmet S by the triagle ass, cosistig of its real edge a defied orietatio ad potetial edges S, beig the edge a with defied

2 62 P.S. Staimirovi}, P.V. Krtolica, R. Staojevi} / A No-Recursive Algorithm orietatio ad potetial edges S, beig the edge of the polygo which is about to be defied. (d) The replacemet S b meas replacig the potetial edge S with the real edge b. If we apply 2 times the first productio i (1) (i other words, if we produce a word with 2 a' s i it) ad the we apply, the correspodig umber of times, the secod productio i (1), we get oe triagulatio of the polygo with edges. I [4] it is show that the umber of all possible triagulatios of the -edges polygo f( ) is give by the recurret formula f( 2) = f( 3) = 1, 1 i= 2 f( ) = f( i) f( i+ 1) Of course, implemetatio of this approach (especially if we wat to list all triagles by their odes) is a additioal problem. Beig ispired by the method described above, we replace the first grammar rule i (1) ad use the followig rules geeratig the arithmetic expressio i the reverse Polish otatio. S SS+ (1.1) S b (1.2) Before we start with the algorithm costructio, let us expose some of the results cocerig the reverse Polish otatio method based o the properties ivestigated i [5]. Note that the expressio i the reverse Polish otatio is stored i the array postfix, where postfix[] i, for each i 0, is a strig which deotes a expressio elemet, i.e. a variable, a costat, or a operator. Defiitio 1.1. The grasp of the elemet postfix[] i is the umber of its precedig elemets which form operad(s) of the elemet postfix[] i. We deote the grasp of the elemet postfix[] i by GR ( postfix[ i ]). Iteger i is called the idex of the elemet postfix[] i. Idex i of the elemet postfix[] i will be alteratively deoted by IND ( postfix[ i ]). Defiitio 1.2. The grasped elemets of the operator postfix[] i are the grasp left precedig elemets i the array postfix which form operad(s) of the operator postfix[] i. The idex of the most left elemet amog them is called the left grasp boud. The left grasp boud of the operator postfix[] i is deoted by LGB ( postfix[ i ]). Defiitio 1.3. The elemet postfix[] i is called the mai elemet or head for the expressio formed by postfix[] i ad its grasped elemets.

3 P.S. Staimirovi}, P.V. Krtolica, R. Staojevi} / A No-Recursive Algorithm 63 I the secod sectio we ivestigate a 1-1 ad oto mappig F: P T from a subset of the expressios, made by cosecutive applicatio of the rule (1.1) 2 times ad the rule (1.2) 1 times, to the set of a -go triagulatios. I the third sectio, we costruct the algorithms for the polygo triagulatio usig the results of the secod sectio. 2. ARITHMETICAL EXPRESSIONS AND TRIANGULATIONS Sice the productio S ass meas the costructio of the triagle ass with the real orieted side a ad two potetial orieted sides S, oe ca assume that the productio S SS+ meas the costructio of the triagle obtaied by replacig the real side a of the triagle ass by the real side +, keepig the orietatio. I this way, we replace the termial a by the sig + which ca be cosidered as the arithmetic operator, with two S 's as its operads. Usig this correspodece, for every 3 we ca cosider the mappig F : R T, whose domai is the set of expressios made by the cosecutive applicatio of the rule (1.1) 2 times ad the rule (1.2) 1 times, ad the rage is the set of a -go triagulatios. But, it is ot difficult to verify that the set R cotais multiple elemets. I the followig lemma we got uique characterizatio for each elemet i R. Usig kow otatios from [6], by V we desigate all strigs of the legth 2 o V, ad by V we deote the closure set{ ε} V V, where ε is the empty strig. Also, by { b, we deote the subset of the closure set { b, +} cosistig of p + } p, q appearaces of the sig that{ b, + } 00, = {} ε. b ad q appearaces of the sig +. It is easy to see Lemma 2.1. A arbitrary elemet r R, correspodig to the particular triagulatio of -agle polygo, is uiquely determied by the followig two coditios: (C1) It possesses the form r = bbα+, α { b, + } 3 3,, (C2) Each iitial part of the expressio r (the substrig of cosecutive characters which start from the first character) must be of the form {, b + } pq, p> q, p= 1,..., 1, q= 1,..., 2. Proof: We use the iductio by. For = 3 the expressio bb + correspods to the triagle, ad satisfies the coditios (C1) ad (C2). A arbitrary ( k + 1) -go triagulatio is derived replacig by triagle a adequate side of a triagle i the correspodig k -go triagulatio. Cosider the expressios

4 64 P.S. Staimirovi}, P.V. Krtolica, R. Staojevi} / A No-Recursive Algorithm bb+, bbαbβ+, pα, q β α pβ, qβ α α β β α {, b + }, {, b + }, p + q = 4, p + q = 3 (2.1) which satisfy coditio (C2) ad correspod to ay k -go triagulatio. Each of these expressio satisfies (C1). The, correspodig ( k + 1) -go triagulatio is determied by oe of the followig expressios bbb ++, bb + b+, bbb + αbβ+, bb + bαbβ+, bbαbb + β+, (2.2) which satisfy (C2) ad α {, b + } p,, {, },,, α q β b + α pβ q p β α + pβ = 4 qα + q β = 3. Each of the expressios from (2.2) is uiquely determied. Sice the followig trasformatios are valid bbb ++= bbγ+, γ = b+ { b, + } bb + b+ = bbδ+, δ =+ b { b, + } 11, 11, pα, qα pβ, qβ { bbb + αbβ+, bb + bαbβ+, bbαbb + β+, α {, b + }, β {, b + }, 2, 2 pα + pβ = 4, qα + qβ = 3} = { bbγ+, γ { b, + } } we coclude that these expressios satisfy the coditio (C1), too. Usig the same trasformatios ad the iductive hypothesis we ca prove that these expressios also satisfy the coditio (C2). By P we deote the set of expressios geerated by the grammar rules (1.1), (1.2) which satisfy coditios (C1) ad (C2). Corollary 2.1. The grasp of the head elemet i the expressio correspodig to the particular triagulatio of -agle polygo is 2 4. The head elemet has 1 sigs b ad 3 sigs + as its grasped elemets, ad the iitial part of the grasped elemets is of the form pq, bb{ b, + }, p > q, p = 1,..., 3, q = 1,..., 3. Proof: From Lemma 2.1 it follows that the legth of the expressio from correspodig to ay triagulatio of the polygo with agles is 2 3, possesses the form bb{, b },, ad each of its iitial parts is of the form P bb{, b + }, p > q, p = 1,..., 3, q= 1,..., 3. The, the proof is obvious from Defiitio 1.1 ad Defiitio 1.3. Lemma 2.2. The mappig F: P T is well defied, oe-to-oe ad oto for ay 3. Proof: Firstly, we shall prove by the iductio that mappigs F, 3, are well defied. The claim will be proved usig iductio by, 3. For = 3, the uique + pq,

5 P.S. Staimirovi}, P.V. Krtolica, R. Staojevi} / A No-Recursive Algorithm 65 expressio bb +, obtaied by oe applicatio of the rule (1.1) ad two applicatios of the rule (1.2), correspods to the uique ad trivial triagulatio of the triagle. Suppose that the claim is valid for = k, k 3. I the case = k+1, cosider a arbitrary expressio (2.2) geerated by rule (1.1) applied k 1 times ad by rule (1.2) applied k times. If i the expressio uder the cosideratio we replace the appearace of the substrig bb + by b (usig the opposite directios i the rules (1.2) ad (1.1)), the we get the expressio of the form (2.1) geerated by applyig the rule (1.1) k 2 times ad rule (1.2) ( k 1) times. Accordig to the iductive hypothesis it correspods to the uique triagulatio of the k -go. But, returig bb + to the previous place, ad trasformig it ito bb +, we get the startig expressio (2.2) makig the additioal triagle o the edge of the k -go which correspods to the trasformed b. I this way we get oe uique triagulatio of ( k + 1) -go. Similarly, we ca prove by iductio that the mappig I order to prove that the mappig is oe-to-oe. procedure to get the expressio from P startig from the give triagulatio. Suppose that we have oe triagulatio with orieted edges ad diagoals. To every diagoal ad to edge AB assig the + sig ad to the rest of edges assig the b sig. Observe two edges which defie oe triagle with edge AB. Deote the icomig edge sig by L ad outgoig edge sig by R. Geerally, the eeded expressio has the form L+ R. If L or R is equal to + sig, it should be surrouded by the parethesis ad the algorithm recursively applied to it. Whe we complete the recursio, trasform the begotte expressio to the reverse Polish form which is obviously the elemet of the set P. F F is oto, we will costruct a effective 1 2 S THE ALGORITHM We should geerate all postfix expressios correspodig to the particular triagulatios, ad the we shall get oe triagulatio for each of these expressios. All expressios from the set R could be geerated usig the result of Lemma 2.1 ad backtrackig method. We start from the highest allowed expressio i the lexicographic sese. This is the expressio I every further step we fid the first allowed expressio lexicographically less tha the previously foud expressio. I this way we could fid all allowed expressios form R. We eed to geerate triagulatios correspodig to every allowed word from R. Let w be a arbitrary word from R. Hece, w is the postfix otatio of the expressio cotaiig 1 operads ( S 's). It is clear that there is bijectio betwee R ad the set of biary trees with ( 1)/ 2 odes ad depth 2. We could observe every ode as a set of cosecutive operads which are descedats of a give ode. If we umerate the vertices of a give polygo by itegers 1,...,, the we could

6 66 P.S. Staimirovi}, P.V. Krtolica, R. Staojevi} / A No-Recursive Algorithm wee every vertex desiged by umber for 1,..., 1 as a operad from w. Heceforth, every tree ode i the tree correspodig to some polygo represets the set of cosecutive polygo vertices. This set is defied by the first ad the last vertex, i.e. by its last vertex ad by the last vertex of the previous ode i the same tree level. So, it is eough to remember i every tree level the biggest iteger umeratig some of the ode descedats. Thus, we coclude that accompayig the ode v with the first vertex p ad the ode u with the last vertex q is the separatio of the vertices p, p+ 1,..., q from the rest of the vertices. I geometrical sese, it correspods to drawig the diagoal ( p, q + 1). We are ready to preset the algorithm for the -go triagulatio correspodig to the word w. 1. Assig to every S from w a ordial umber from left to right. 2. Let the poiter curr be the first elemet idex of word w. 3. If the value of the elemet poited by curr is S, the curr gets the idex value of the ext elemet ad repeat this step. If its value is equal to + go to step 4, or if it is the ed of strig go to step I this momet poiter curr poits to a + from word w. By Lemma 2.1 it is assured that to this + at least two S precede (which represets odes of the biary tree). Our goal is a trasitio to the higher level. We achieve this by joiig two S odes which proceed to +, ad pritig the diagoal ( pred( pred( pred( curr)) ) + 1, pred( curr) + 1) (the fuctio pred desigates the predecessor of its argumet). Further, i this step we elimiate curret + ad precedig ode S. The poiter curr gets the idex value of ( pred( pred( curr) )), while we remember the last vertex of the ode pred( cu rr ) as the last vertex of the ode ( pred( pred( curr ))). Go to step At this poit, all diagoals correspodig to the word w, makig particular triagulatio of the give polygo, are geerated ad algorithm is stopped. I the followig table we preset behavior of this algorithm. Table 3.1. # of differet triagulatios Processig time [s]

7 P.S. Staimirovi}, P.V. Krtolica, R. Staojevi} / A No-Recursive Algorithm CONCLUSION As we saw, startig from the triagulatio strategy based o the formal grammar from [4] ad usig the arithmetic expressios i the reverse Polish otatio, we get the relatively simple algorithm for polygo triagulatio. Similar idea could be foud i [3], where a algorithm, based o the matrix multiplicatio ad correspodig parse trees, is preseted. But, this algorithm eeds to operate with data structures which are more complex tha i the case of our algorithm. The reader should be aware that our itetio was ot to preset revolutioary better algorithms for polygo triagulatio. The problem of covex polygo triagulatio is a old oe ad has bee solved so far i may ways (see, for example, [1, 2]) ad it is kow that it could be doe i liear time. Our mai goal is to show how to successfully apply reverse Polish method for the symbolic computatio i this area. REFERENCES [1] Chazelle, B., "Triagulatio a simple polygo i liear time", Discrete Comput. Geom., 6 (1991) [2] Chazelle, B., ad Palios, L., "Decompositio algorithms i geometry", i: Ch.L. Bajaj (ed.), Algebraic Geometry ad Its Applicatio, Spriger-Verlag, New York, Ic., 1994, [3] Corma, T.H., Leiserso, C.E., ad Rivest, R.L., Itroductio to Algorithms, MIT Press, Cambridge, Massachusetts, Lodo, Eglad, [4] Kross, M., ad Leti, A., Notios sur les grammaries formelles, Gauthier-Villars, [5] Krtolica, P.V., ad Staimirovi}, P.S., "O some properties of reverse Polish otatio", FILOMAT, 13 (1999) [6] Tremblay, J.P., ad Soreso, P.G., The Theory ad Practice of Compiler Writig, McGraw- Hill Book Compay, New York, 1985.