Enumerating pseudo-triangulations in the plane

Transcription

1 Enumerating pseuo-triangulations in the plane Sergey Bereg Astrat A pseuo-triangle is a simple polygon with exatly three onvex verties. A pseuo-triangulation of a finite point set S in the plane is a partition of the onvex hull of S into interior isjoint pseuo-triangles whose verties are points of S. A pointe pseuo-triangulation is one whih has the least numer of pseuo-triangles. We stuy the graph G whose verties represent the pointe pseuo-triangulations an whose eges represent flips. We present an algorithm for enumerating pointe pseuo-triangulations in O(log n) time per pseuo-triangulation. Keywors: pseuo-triangulation, flip, enumeration, reverse searh. 1 Introution Pseuo-triangles an pseuo-triangulations have reeive muh attention reently eause of appliations in visiility [15, 16], ray shooting [15], ollision etetion [13], an rigi motion [7, 20]. There are many open questions relate to pseuo-triangulations [21]. A pseuo-triangle is a simple polygon with exatly three onvex verties. For a set S of n points in the plane, a pseuo-triangulation T is efine as partition of the onvex hull of S into interior isjoint pseuo-triangulations whose verties are points of S (eah point of S is a vertex of T an vie versa). A minimum or pointe pseuo-triangulation introue y Streinu [20] is a pseuotriangulation with the least numer of eges among all pseuo-triangulations of S. Streinu [20] showe that any pointe pseuo-triangulation has 2n 3 eges. Equivalently, we an efine a pointe pseuo-triangulation as a pseuo-triangulation with minimum numer of pseuo-triangles. Any pointe pseuo-triangulation has n 2 pseuo-triangles sine n f = n e n + 1 = n 2 y Euler s formula. Reent results on pseuo-triangulations inlue [1, 2, 12, 18]. Ranall et al. [18] gave lose form expressions for the numer of triangulations an the numer of pointe pseuo-triangulations when S has only one point insie its onvex hull. For a point set in general position, they prove an upper oun for the numer of pseuo-triangulations. Aihholzer et al. [1] investigate the numer of pseuo-triangulations generate y n points in the plane. They prove that the least numer of pseuo-triangulations is attaine when points are in onvex position. We otaine an algorithm [4] for enumeration of triangulations in O(log log n) time per triangulation. The algorithm is ase on flips, small moifiations of a triangulation. Pseuotriangulations amit flips as well, see Fig. 1. It allows us to efine, for a point set S, a graph of pseuo-triangulations whose verties are the pointe pseuo-triangulations of S an whose eges are the flips of interior pseuo-triangulation eges. Reently, Brönnimann et al. [6] esrie an Department of Computer Siene, University of Texas at Dallas, Box , Riharson, TX 75083, USA. esp@utallas.eu 1

2 algorithm for enumerating all pointe pseuo-triangulations using an effiient tehnique of Pohiola an Vegter [17] for fining a flip in O(1) amortize time. Unfortunately the omplexity of the algorithm is unknown 1 ut the require spae is quarati. It follows from the algorithm that the graph of pseuo-triangulations is onnete. (a) a () Figure 1: (a) Flip of triangles. () Flip of pseuo-triangles. Very reently, Rote et al. [19] introue a polytope of pointe pseuo-triangulations of a point set in the plane, efine as the polytope of infinitesimal expansive motions of the points sujet to ertain onstraints on the inrease of their istanes. The polytope possesses useful properties an its 1-skeleton is the graph of pseuo-triangulations. We present an algorithm for enumerating all pointe pseuo-triangulations in O(log n) time per pseuo-triangulation using linear spae. The algorithm is ase on reverse searh tehnique y Avis an Fukua [3]. As y-prout it implies that the graph of pointe pseuo-triangulations is onnete. Our algorithm an e use to list the verties of the polytope of pointe pseuotriangulations [19]. It also generates a spanning tree of the graph of pseuo-triangulations. Aihholzer et al. [2] onsiere the prolem of ounting the numer of pointe pseuo-triangulations of a point set. They showe that every pseuo-triangulation ontains a zigzag path. It is use to ount pointe pseuo-triangulations effiiently y enumerating zigzag paths in O(n 2 ) time per path. 2 Preliminaries Let S = {p 1, p 2,..., p n } e a set of n points in general position in the plane. We efine a onvex orer of the points as follows. Let S i = {p i, p i+1,..., p n } for i = 1,..., n. The orer (p 1, p 2,..., p n ) is onvex if every point p i, i = 1,..., n lies on the ounary of the onvex hull of S i. An example of the onvex orer is the lexiographial orer y (x, y)-oorinates. This orer is use, for example, 1 The authors onjeture that the running time is O(log n) per pseuo-triangulation. 2

3 in the inremental algorithm for onstruting the onvex hull. Let x(p i ) an y(p i ) enote the x- an y-oorinates of a point p i. A vertex of a polygon is onvex if its interior angle is less than π, otherwise the vertex is reflex. A polygon is alle a pseuo-triangle if it has exatly three onvex verties. A pseuo-quarangle has exatly four onvex verties. In general, a pseuo-k-gon has exatly k onvex verties. A sie of a pseuo-k-gon P onsists of verties an eges etween two onvex verties (the internal verties of a sie are reflex verties of P ). Let T 1 (P ), T 2 (P ),..., T k (P ) enote inary searh trees storing the verties of the sies of P in lokwise orer. Note that every onvex vertex of P partiipates in two trees an every reflex vertex is ontaine in one tree. The following lemma generalizes flips in triangulations to flips in pseuo-triangulations, see Fig. 1. Lemma 1 (Flips in a pseuo-quarangle) Every pseuo-quarangle Q has exatly two partitions into two pseuo-triangles whih are proue y two iagonals. The iagonals either interset properly or share a reflex vertex of the pseuo-quarangle. The iagonals an e ompute in O(log n) time if the verties of Q are store in four alane searh trees T i (Q), i = 1, 2, 3, 4 orresponing to the sies of Q. Proof: The existene of two iagonals has een shown y Rote et al. [19]. Let Q e a pseuoquarangle an let a,,, an e the onvex verties of Q in ounterlokwise orer. The partitions an e otaine y utting Q with the shortest paths from a to an from to in Q (see etails in Lemma 2.2 [19] an Lemma 2.1 [22]). Eah shortest path has exatly one ege insie Q, whih is a iagonal that splits Q into two pseuo-triangles, see Fig. 1 (). a a p q p q a p q a p q (a) () () () Figure 2: Four types of the shortest path etween a to. We show that the iagonals annot share a onvex vertex. Clearly, the shortest path etween a an avois the verties an sine they are onvex. Similarly the shortest path etween an avois the verties a an. Thus two shortest paths a an an share reflex verties only. We show that the iagonals interset. Let p e q e the enpoints of the iagonal from the shortest path a. The line pq is ommon tangent to two sies of Q, one is a or a an the other is or, see Fig. 2. Without loss of generality we an assume that the line pq is horizontal. In all four ases epite in Fig. 2 is aove the line pq an is elow the line pq. The shortest path ontains an ege p q suh that p lies aove or on the line pq an q lies elow or on the line pq. It an e verifie that p q is the seon iagonal an the iagonals interset. Computing iagonals. Eah iagonal is a itangent to two sies of Q. A itangent to two onvex polygons an e ompute in O(log n) time [14]. A sie of Q an e omplete to a onvex 3

4 a a a a (a) () () () Figure 3: Fining a iagonal of Q on the shortest path a. polygon y aing the straight line segment etween the sie enpoints. Let P a, P, P an P a enote the polygons orresponing to the sies a,, an a, respetively. We show how the sies of Q an foun in O(1) time. Suppose that we want to fin a iagonal of the geoesi a. Suppose that the point is elow a. Let a 1 e the ege of the sie a. Similarly we enote the eges a 1, 2, 2, see Fig. 3. We onsier the following ases epening on the loation the segment i i, i = 1, 2 relative to the segment a. If a interset oth 1 1 an 2 2, then a is the require iagonal, see Fig. 3 (a). If oth 1 1 an 2 2 lie on the same sie of the line a, say on the left sie (the ase of the right sie is symmetri), then the iagonal is a itangent of P a an P, see Fig. 3 (). Suppose that the line a separates the segments 1 1 an 2 2, say 1 1 lies to the left of a an 2 2 lies to the right of a. Then the iagonal is a itangent of P a an P, see Fig. 3 (). Suppose that the line a intersets only one of the segments 1 1 an 2 2, say 2 2. Suppose that 1 1 lies to the left of the line a (otherwise it is symmetri). Then the iagonal is a itangent of P a an P, see Fig. 3 (). Thus the iagonals an e foun in O(log n) time. The flip of an ege e in a pseuo-triangulation removes e an inserts another ege e. We all e a ual ege of e. A pseuo-triangulation of S is a partitioning of the onvex hull of S into pseuo-triangles with verties in S suh that eah point of S is a vertex of a pseuo-triangle. A pseuo-triangulation T is a minimum pseuo-triangulation or a pointe pseuo-triangulation if it ontains a minimum numer of eges. Streinu [20] prove that a pointe pseuo-triangulation of a n-point set ontains 2n 3 eges. Henneerg onstrution. Streinu [20] prove that the graph of a pointe pseuo-triangulation is minimally rigi. A minimally rigi graph [9, 23] an e onstrute using the Henneerg upates [10]. There are two types of graph upates: (I) attah a new vertex y two eges, an (II) elete an ege an attah a new vertex to the enpoints of the elete ege plus one other vertex. Figure 4 illustrates the Henneerg onstrution of type I. It an e use to generate a pointe pseuotriangulation of S. Let G e the graph of pointe pseuo-triangulations of S. Reall that its verties orrespon to pointe pseuo-triangulations an the eges orrespon to flips. Similarly the graph of triangulations G T is efine [11, 4]. These graphs share some properties though the graphs inue y the same point sets are ifferent in general. Both graphs G T an G are onnete (the onnetivity of G has een shown in [19] an follows from Theorem 3 elow). They oinie if the points of S are in onvex position. 4

5 CH p p (a) () Figure 4: The Henneerg onstrution of type I applie for a point p lying (a) outsie the onvex hull, an () insie a pseuo-triangle. Hurtao et al. [11] showe that there are 2n points in the plane an two triangulations that require more than (n 1) 2 flips to transform one into the other. Brönnimann et al. [6] showe that O(n 2 ) flips suffie to transform a pointe pseuo-triangulation into another one. Interestingly, the flip istane etween pseuo-triangulations is smaller than the one etween triangulations: reently, we improve the oun to O(n log n) [5]. 3 Tree of Pointe Pseuo-Triangulations Let onv(a) enote the onvex hull of a set A. Let (p 1, p 2,..., p n ) e a onvex orer of the points in S. A pointe pseuo-triangulation T ontains all the eges of onv(s) = onv(s 1 ). We efine an inex of T enote y inex(t ) as the largest k n 2 so that T has the eges of onv(s 1 ),..., onv(s k ). Clearly, k 1 for any pointe pseuo-triangulation. The pseuotriangulation T ontains onv(s k ). We efine a egree of T enote y egree(t ) as the numer of pseuo-triangles of T that are ontaine in S k an have p k as a vertex. With eah pointe pseuo-triangulation we assoiate a vetor α(t ) = (k, l) where k = inex(t ) an l = egree(t ), see Fig. 5 (a) for an example. The lexiographial orer of vetors α(t ) inues a partial orer on the set of pseuo-triangulations. Lemma 2 There is a unique pointe pseuo-triangulation with the lexio-largest vetor α. Proof: Let T e the pseuo-triangulation otaine y the Henneerg onstrution of type I applie for the points in the orer (p n, p n 1,..., p 1 ): start with the triangle p n p n 1 p n 2 an insert points p n 3,..., p 1. The pseuo-triangulation T is pointe (it an e easily verifie) y Theorem 3.1 [20]. We show that the vetor α of T is the lexio-largest vetor among all pointe pseuo-triangulations. The inex of T is n 2 sine T ontains the eges of onv(s i ), i = 1, 2,..., n 2. The inex of T has the maximum value. Any pseuo-triangulation T with inex n 2 ontains all the eges of onv(s i ), i = 1, 2,..., n 2 an, thus, is a supergraph of T. By the efinition of pointe pseuotriangulation T = T. The lemma follows. We enote y T max a pseuo-triangulation with the lexio-largest vetor. 5

6 p 2 p 1 p 4 p 5 p 8 p 7 p 3 p 1 p 4 p 6 p 2 p 3 p 6 p 7 p 8 p 5 (a) () Figure 5: (a) A pseuo-triangulation T with inex(t ) = 3 an α(t ) = (4, 3), () a pseuotriangulation with the lexio-largest vetor α(t ) = (6, 1). Theorem 3 For any pointe pseuo-triangulation T T max, there is a flip making a pseuotriangulation with lexio-larger vetor α. Proof: Let α(t ) = (k, l) e the vetor of T. Sine α(t ) is not the lexio-largest vetor, α(t ) (n 2, 1) an k < n 2. The egree of T, l, must e greater than 1; otherwise T has all the eges of the onvex hull of S k+1 an the first inex must e less than k. In other wors, there are at least two pseuo-triangles insie the onvex hull of S k that have the ommon vertex p k. Let 1 an 2 e two suh pseuo-triangles that share a ommon ege (p k, p i ). Let Q e the polygon whih is the union of 1 an 2. We show that Q is a pseuo-quarangle. Note that Q has at least three onvex verties p k, p k1 an p k2 where p k1 an p k2 are the onvex verties of 1 an 2 ifferent from p k an p i, see Fig. 6. On the other han, there are at most 6 aniates to e onvex verties of Q (the onvex verties of 1 an 2 ). Let γ 1 an γ 2 e the angles etween the segments inient to p i in the pseuo-triangle 1 an 2, respetively. The angles satisfy one of the the following ases. Case 1: γ 1 + γ 2 < π. The numer of onvex verties of Q is four sine p k an p i are the onvex verties of oth 1 an 2, see Fig. 6 (a). Case 2: γ 1 + γ 2 > π an γ 1, γ 2 < π. The numer of onvex verties of Q is three, see Fig. 6 (). It ontraits the fat that T is a pointe pseuo-triangulation. Case 3: γ 1 + γ 2 > π an one of the angles γ 1 or γ 2 is greater than π. The numer of onvex verties of Q is four, see Fig. 6 (). By Lemma 1 the polygon Q an e partitione into two pseuo-triangles in two ways an orresponing iagonals o not share the vertex p k. We show how a spanning tree T of G an e onstrute. It is possile to efine the tree using an approah similar to [4] whih is ase on a lexiographial orer of triangulations using ege vetors (note that the pointe pseuo-triangulations have a fixe numer of eges). Unfortunately, it is not lear how to traverse the resulting tree effiiently. We apply a ifferent approah for pseuo-triangulations. We restrit the orer of points to e monotone in some iretion. For simpliity, we assume that the points are sorte y x-oorinate an their x-oorinates are istint (we an slightly rotate the set of the points to ahieve this). The root of T orrespons to the pseuo-triangulation 6

7 p k2 p k1 p k1 p i p i γ 1 γ 2 p k2 γ 1 γ p k p k (a) () p k1 p k2 p k1 p i γ 1 γ 2 p i γ 1 γ 2 1 p k2 2 p k p k () Figure 6: Cases. (a) an () Q is a pseuo-quarangle. () Q is a pseuo-triangle. T max, see Fig. 5 (). Let T e a pseuo-triangulation istint from T max. Theorem 3 guarantees the existene of an ege whose flip inreases the vetor of T. Let v e the noe of G orresponing to T. We efine the parent noe of v as follows. Let e = (p a, p ), a < e an ege of T suh that (i) the flip of e inreases the vetor α(t ), an (ii) a is the least numer among the eges satisfying (i), an (iii) the vetor p a p has the maximum slope among the eges satisfying (i) an (ii). Clearly, e is well efine. Ausing notation, we all e parent ege of T an eges whose flips lea to the hilren of v as hil eges. The following lemma haraterizes the parent eges. Lemma 4 (Parent Ege) An ege e = (p a, p ), a < is the parent ege of a pseuo-triangulation T if an only if a = inex(t ) an e has the largest slope among the eges of T T max. Proof: Let k = inex(t ). First, we show that a k. Suppose to the ontrary that a < k. By the efinition of inex(t ), T inlues the eges of the polygons onv(s a ) an onv(s a+1 ). Their ifferene onv(s a ) onv(s a+1 ) is a pseuo-triangle, see Fig. 7. The interior of oes not ontain eges of T sine T is the pointe pseuo-triangulation. Let p i an p j e two ajaent verties of onv(s a ). Flip of either p a p i or p a p j in T estroys an introues a new ege inient to a vertex outsie onv(s a ). This flip ereases the vetor α(t ). The ontraition implies a k. 7

8 p i p a onv(s a+1 ) onv(s a ) p j Figure 7: Pseuo-triangle. By Theorem 3 there is an ege inient to p k satisfying the aove onition (i). The onition (ii) implies a = k. It follows from the proof of Theorem 3 that the flip of any ege (p a, p ), > a of T inreases α(t ). The lemma follows. The parent ege e of a pointe pseuo-triangulation T is non-vertial segment an is not an ege of the onvex hull of S. Thus it is inient to two pseuo-triangles of T, one ontains a nei. We enote y (T ) the pseuo-triangle of T that is inient to e parent an is aove e, i.e. an internal point p of the segment e enters (T ) y an infenitesimal motion upwar. An example of (T ) is epite in Fig. 9 where (p k, p u ) is the parent ege. Next we haraterize the hil eges in T. Theorem 5 (Chil Ege) Let v e a noe of T an let T e its pseuo-triangulation with α(t ) = (k, l). Let p k p j p m e the onvex verties of (T ) in lokwise orer. Let C e the sie of (T ) etween p j an p m. Let p j = p j1, p j2,..., p m e the verties of the sie C. An ege e = (p a, p ), a < is a hil ege of T if an only if e is not an ege of the onvex hull of S an one of the following onitions hols (i) a < k, or (ii) e is an ege of onv(s k ), or (iii) {a, } = {j r, j r+1 }, m / {a, } an oth p a an p are visile from p k in (T ), or (iv) (a) {a, } = {j r, j r+1 } an p jr is visile from p k in (T ), an () j r+1 = m or p jr+1 () the ual ege of e is inient to p k. is not visile from p k in (T ), an Proof: If). (i) If a < k then e is a hil ege sine its flip ereases the vetor α(t ), see the proof of Lemma 4. (ii) The ege e lies on the ounary of two pseuo-triangles, say 1 an 2. Sine e is an ege of onv(s k ), one of the pseuo-triangles, say 2, lies in onv(s k ) an the other lies outsie onv(s k ), see Fig. 8. Let x e the largest numer suh that 1 lies in onv(s x ). Note that 1 x < k sine 1 is ontaine in onv(s 1 ) ut not in onv(s k ). Also 1 = onv(s x ) onv(s x+1 ) sine T ontains all the eges of onv(s i ), i = 1,..., k. Let p x p y p z e the onvex verties of 1 an let p e the onvex vertex of 2 ifferent from p a an p, see Fig. 8. The ual ege of e is ontaine in the shortest path etween p x an p in 8

9 p x p y 1 e p p a 2 p k p t p z onv(s k ) onv(s x ) p Figure 8: Pseuo-triangle 1 with onvex vertex p z, x(p z ) < x(p k ). the pseuo-quarangle Q = 1 2. The shortest path p x p avois the onvex verties p x an p y of Q. Therefore the ual ege of e is inient to p x (sine p x is ajaent to p y an p z in Q). Let e = (p x, p t ) e the ual ege of e an let T e the pseuo-triangulation otaine from T y flipping e. The ege e is the parent ege of T y Lemma 7. (iii) Consier the thir ase. The pseuo-triangle (T ) ontains p jr p jr+1. Let 1 = p jr p jr+1 p e the pseuo-triangle of T on the other sie of p jr p jr+1, see Fig. 9. The union Q = (T ) 1 is a pseuo-quarangle sine p k, p j, p an p m are the onvex verties of Q. By Lemma 1, e an its ual ege interset either properly or at a vertex of e. Let q e the intersetion point. The ege e is ompletely visile from p k. Clearly, p k q is the part of the shortest path p k p in the pseuo-quarangle Q. Therefore the ual ege of e is inient to p k. Let (p k, p v ) e the ual ege of e. Let (p k, p u ) e the parent ege of T, see Fig. 9. We show that v / {u, j}. If v = u then the verties p u, p jr+1 an p m oinie. This is impossile sine j r+1 m. If v = j then p is aove the line passing through p k an p j. This is impossile sine p is ontaine in onv(s k ) an (p k, p j ) is its ege. Let T e the pseuo-triangulation otaine y flipping e in T. The parent ege of T is p k p v y Lemma 7. Thus e is a hil ege of T. (iv) The forth ase follows from the onition (). Only If). We show that e is not a hil ege of T if none of the onitions (i)-(iv) hols. This assumption implies that e lies in the interior onv(s k ). Let T e the pseuo-triangulation otaine from T y flipping e, let e e the ual ege of e an let (p k, p u ) e the parent ege of T. Suppose that e is not an ege of (T ). Then (T ) is a pseuo-triangle of T an (p k, p u ) is still the parent ege of T. Therefore e is not a hil ege. Suppose that e is an ege of the sie p j p m of (T ). If oth p a an p are not visile from p k, then e is not visile from p k an p k is not inient to e (y Lemma 1, e intersets e). Then e is not a hil ege. If oth p a an p are visile from p k an m / {a, }, then it is the ase (iii) an e is a hil ege. In the remaining ase e is not inient to p k y the onition (iv)(). Therefore e is not a hil ege. Suppose that e is an ege of the sie p k p m of (T ). Clearly, (p k, p u ) is not a hil ege sine its flip inreases the vetor of T. Suppose that e (p k, p u ). The ege e is not on the ounary of onv(s k ) y the onition (ii) (the hain p k p m annot atually ontain an ege of onv(s k ) sine 9

10 p j p jr e 1 p k p u (T ) p v p jr+1 p onv(s k ) p m Figure 9: The ual ege of e = (p jr, p jr+1 ) is (p k, p v ). p j p j p k p p a e e p p m onv(s k ) p k = p e p a e p p m onv(s k ) (a) () Figure 10: The ege e is not a hil ege of T. the internal verties of p k p m are reflex). Let e the pseuo-triangle with e on its ounary an ifferent from (T ). Let p e the onvex vertex of opposite to the sie of e, see Fig. 10. The ege e is a part of the shortest path from p j to p in the pseuo-quarangle Q = (T ). We show that e is not inient to p k (note that p may oinie with p k ). If p p k, then it follows from the fat that the shortest path p j p avois the onvex vertex p k of Q, see Fig. 10 (a). If p = p k, then it follows from e (p k, p u ), see Fig. 10 (). Theorem 6 (Height of T ) Let S e a set of n points in general position in the plane. The tree of pseuo-triangulations T has height at most ( n 2) 2. The oun is tight in the worst ase. Proof: The tree of pseuo-triangulations is onsistent with the partial orer on the set of pseuotriangulations. In orer to show the upper oun on the height of T it suffies to prove that the length of the partial orer on the set of vetors α(t ) is at most ( n 2) 2. This follows from the fat that there are ( n 2) possile vetors α() = (k, l) (note that every flip hanges α()). We ount all possile vetors α(). Let k e any integer from 1,..., n 3 an let T e a pseuotriangulation with inex k. The numer of possile pseuo-triangles of T in onv(s k ) inient to p k ranges from 2 to n k 1. Thus the numer of vetors α() for the fixe k is n k 2. If k = n 2, then the only pseuo-triangulation with inex k is T max. The total numer of vetors 10

11 p 2 p 3 p 4 p n 1 p 1 p n (a) () () () Figure 11: (a) n points on the ar. () Pseuo-triangulation with lexio-smallest α(). () Pseuotriangulation after n 3 flips. () T max. α() is n 2 n n k 2 = 1 + i = 1 + k=1 i=1 ( ) n 2. 2 It remains to prove the lower oun. We plae n points on the unit ar x 2 + y 2 = 1, y 0, see Fig. 11 (a). The pseuo-triangulation with the lexio-smallest vetor α(t ) = (1, n 2) is epite on Fig. 11 (). There are n 3 parent flips efore we otain a pseuo-triangulation with inex two, see Fig. 11 (). In general, there are n k 2 parent flips on pointe pseuo-triangulations of inex k. The theorem follows. An example of a spanning tree of pseuo-triangulations for a set of five points is illustrate in Fig Enumerating Pointe Pseuo-Triangulations We apply the reverse searh tehnique y Avis an Fukua [3]. A possile approah is to use a reursive proeure that proesses a noe of T. By Theorem 6 the spae requirement for this approah is Ω(n 2 ) in the worst ase. We show that the spae size an e reue to linear. We nee some properties of hil eges for an effiient algorithm. We lassify the hil eges aoring to the ases of Theorem 5, for example, the eges of type (i) orrespon to the ase (i). For every vertex p i S, i = 1, 2,..., n 2 we enote y e u i = (p i, p up(i) ) an e l i = (p i, p low(i) ) two eges of onv(s i ) inient to p i (e u i lies on the upper hull of S i an e l i lies on the lower hull of S i). Lemma 7 Let T e a pseuo-triangulation with inex k. (i) Let n 1 e the numer of hil eges of type (i) in T. Then k 2 n 1 2(k 2). (ii) The hil eges of type (ii) in T an e reporte in O(1) time per ege using the inex funtions up() an low(). Proof: (i) Let (p a, p ), a < e a hil ege of type (i). Then 2 a k 1. By the efinition of the inex of T, there are exatly two eges (p a, p 1 ) an (p a, p 2 ) in T satisfying i > a. This implies the upper oun for n 1. The lower oun follows from the fat that at least one of the eges (p a, p i ), i = 1, 2 lies insie the onvex hull of S (sine p 1 lies outsie onv(s a )). (ii) The eges e u k an el k are the eges of onv(s k) inient to p k. Let C up an C low e the upper hull an the lower hull of onv(s k ), respetively. We show how to fin eges of C up. Suppose that p i is the urrent vertex of C up. Note that p i an e inient to many eges, see Fig. 13. It turns out that the next vertex in C up is always p up(i). 11

12 p 1 p 2 p 3 p 5 (p 2, p 3 ) (p 2, p 4 ) p 4 (p 3, p 4 ) (p 3, p 5 ) (p 3, p 4 ) (p 3, p 5 ) (p 3, p 5 ) (p 2, p 3 ) (p 2, p 4 ) (p 3, p 5 ) (p 2, p 3 ) (p 3, p 5 ) Figure 12: A spanning tree of pseuo-triangulations for five points. The laels of tree eges are the parent eges. Let p j e the next vertex of onv(s k ), see Fig. 13. We laim that j = up(i). The point p up(i) lies in onv(s k ) sine p up(i) onv(s i ) onv(s k ). Thus, the slope of the segment p i p j is at least the slope of p i p up(i). Note that x(p j ) > x(p i ) (or j > i). On the other han, the slope of p i p up(i) is the maximum slope among the segments p i p l, l > i. Therefore p j = p up(i). Thus, the eges of C up an e foun using the funtion up(). The eges of C low an e foun similarly. Theorem 8 Let S e a set of n points in the plane. The pointe pseuo-triangulations of S an e reporte in O(log n) time per pseuo-triangulation using linear spae. Proof: The algorithm maintains ata strutures that allow an effiient traversal of T. There are stati ata strutures that store information aout the orer of the points an T max. The eges of the onvex hull onv(s) are store in a alane inary tree T onv in the lexio-graphial orer so that any ege (p i, p j ), i < j an e teste whether it is in onv(s) in O(log n) time. The eges of T max are store in a inary searh tree aoring to the lexio-graphial orer. We store the values up(i) an low(i) using two arrays. We store ynami strutures relate to the urrent pseuo-triangulation T. Let L t e the list of pseuo-triangles of T. With every sie s of a pseuo-triangle of T we assoiate a inary searh tree T s ( ) storing its points in ounterlokwise orer. We also assume that the operations of 12

13 p i p j p up(k) p k p low(k) onv(s k ) Figure 13: Fining the eges of C up. onatenation an split an e performe in O(log n) time using a union-split ata struture [8] for example. Note that a point p S an our in many trees T s ( ) (the numer of trees is atually at most the egree of p in T ). The preeessor an the suessor of a point p T s ( ) provie aess to the eges inient to p along the sie s. Every ege ours in two sets an we store it twie with pointers to eah other. We also store (i) L e, the list of eges in T. With every ege we store pointers to (at most) two inient pseuo-triangles. (ii) L, the list of eges of T T max in the lexiographial orer. For every point p S we store L 1 (p) the list of eges in T T max inient to p in the sorte orer y the slope. We apply the reverse searh tehnique [3] whih an e viewe as a epth-first traversal of the pseuo-triangulation tree T. For a urrent pseuo-triangulation T, the algorithm maintains its inex, the parent ege an the pseuo-triangle (T ). The value of inex(t ) an e ompute y fining the smallest element in L. The parent ege an e foun using L 1 (p k ). The pseuo-triangle (T ) an e foun y heking the pseuo-triangles inient to the parent ege. We traverse T as follows. The flip making a pseuo-triangulation of the parent is efine y the parent ege. The hil eges an e foun using Theorem 5. We esrie how the hil eges of eah type an e foun. Case (i). We show that the hil eges of type (i) an e foun in O(1) time per ege. The algorithm heks the verties p i, i = 2, 3,..., k 1 an the eges of onv(s i ) inient to p i. We use T onv to test whether an ege e is a hil ege. The total time is O(k) an the numer hil eges of type (i) is Ω(k) y Lemma 7 (i). Case (ii). The eges an e foun y Lemma 7 (ii). Cases (iii-iv). The algorithm traverses the eges of the sie p j p m of (T ) in the ounterlokwise orer (from p j to p m ). Let e = (p jr, p jr+1 ) e the urrent ege. We an etet if p jr an p jr+1 are visile from p k in O(log n) time. By Lemma 1 the ual ege of e an e foun in O(log n) time. Thus we an etet in O(log n) time if e has type (iii) or (iv). Note that the eges of types (iii) an (iv) form a ontinuous path p j p jl. We stop the searh if either e is the last ege of p j p m or e oes not satisfy the onitions (iii) an (iv). The linear spae an e ahieve using a non-reursive algorithm. For this, we maintain a oolean variale NewNoe iniating whether the urrent noe of T is visite for the first time. Depening on NewNoe we fin the hil ege y alling FinFirstChil() or FinNextChil(), see Algorithm 1. We store the urrent hil ege in ChilEge an the next hil ege in 13

14 NewChilEge. Both variales store pointers to the enpoints an the type of the ege so that next hil ege an e ompute. Clearly, the spae is linear. Algorithm 1 Enumeration of pointe pseuo-triangulations. 1: Compute T max, onv(s), T onv, up(), low(), the lists L t, L e, L, L 1 (p). {Initialization} 2: NewNoe = true; 3: loop {main loop} 4: if (NewNoe) then {we arrive at a new noe of T } 5: NewChilEge = FinFirstChil(); 6: else {ol noe of T } 7: NewChilEge = FinNextChil(); 8: en if 9: if (NewChilEge <> NULL) then {there is a new hil ege} 10: NewNoe=true; 11: ChilEge=NewChilEge; 12: Output(ChilEge); {new pseuo-triangulation} 13: Flip(ChilEge); 14: else {all hil eges are visite} 15: if (ParentEge = NULL) then {the root of T } 16: return; 17: en if 18: NewNoe=false; 19: ChilEge=Dual(ParentEge); 20: Output(ParentEge); 21: Flip(ParentEge); 22: en if 23: en loop 5 Conlusion We presente an algorithm for enumerating the pointe pseuo-triangulations of a set of n points in the plane. The algorithm uses flips to generate pseuo-triangulations an its running time is O(log n) per pseuo-triangulation. An interesting algorithmi question is to ount the numer of pseuo-triangulations without generating them. This might help to verify the onjeture [6, 18] that the numer of pseuo-triangulations for a finite set of points in the plane is at most the numer of its triangulations. Referenes [1] O. Aihholzer, F. Aurenhammer, H. Krasser, an B. Spekmann. Convexity minimizes pseuotriangulations. In Pro. 14th Cana. Conf. Comput. Geom., pp , [2] O. Aihholzer, G. Rote, B. Spekmann, an I. Streinu. The zigzag path of a pseuotriangulation. In Pro. 10th Workshop Algorithms Data Strut., pp ,

15 [3] D. Avis an K. Fukua. Reverse searh for enumeration. Disrete Appl. Math., 65:21 46, [4] S. Bespamyatnikh. An effiient algorithm for enumeration of triangulations. Comput. Geom. Theory Appl., 23(3): , 2002, [5] S. Bespamyatnikh. Transforming pseuo-triangulations. In Pro. International Conferene on Computational Siene, LNCS 2657, pp , 2003, link.asp?i=5t160n7unryyeku. [6] H. Brönnimann, L. Kettner, M. Pohiola, an J. Snoeyink. Counting an enumerating pseuotriangulations with the greey flip algorithm. In Fall Workshop on Comput. Geometry, [7] R. Connelly, E. D. Demaine, an G. Rote. Straightening polygonal ars an onvexifying polygonal yles. In Pro. 41th Annu. Sympos. on Foun. of Computer Siene, pp , [8] T. H. Cormen, C. E. Leiserson, R. L. Rivest, an C. Stein. Introution to Algorithms. MIT Press, Camrige, MA, 2n eition, [9] J. Graver, B. Servatius, an H. Servatius. Cominatorial Rigiity, volume 2. Amer. Math. So., Grauate Stuies in Mathematis, [10] L. Henneerg. Die graphishe Statik er starren Systeme. Leipzig, [11] F. Hurtao, M. Noy, an J. Urrutia. Flipping eges in triangulations. Disrete Comput. Geom., 22(3): , [12] L. Kettner, D. Kirkpatrik, an B. Spekmann. Tight egree ouns for pseuo-triangulations of points. In Pro. 13th Cana. Conf. Comput. Geom., pp , [13] D. Kirkpatrik, J. Snoeyink, an B. Spekmann. Kineti ollision etetion for simple polygons. In Pro. 16th Annu. ACM Sympos. Comput. Geom., pp , [14] M. H. Overmars an J. van Leeuwen. Maintenane of onfigurations in the plane. J. Comput. Syst. Si., 23: , [15] M. Pohiola an G. Vegter. Pseuo-triangulations: Theory an appliations. In Pro. 12th Annu. ACM Sympos. Comput. Geom., pp , [16] M. Pohiola an G. Vegter. Topologially sweeping visiility omplexes via pseuotriangulations. Disrete Comput. Geom., 16(4): , [17] M. Pohiola an G. Vegter. The visiility omplex. Int. J. Comput. Geom. Appl., 6(3): , [18] D. Ranall, G. Rote, F. Santos, an J. Snoeyink. Counting triangulations an pseuotriangulations of wheels. In Pro. 13th Cana. Conf. Comput. Geom., pp , [19] G. Rote, F. Santos, an I. Streinu. Expansive motions an the polytope of pointe pseuotriangulations. Disr. Comput. Geom. - The Gooman-Pollak Festshrift, pp , 2003, 15

16 [20] I. Streinu. A ominatorial approah to planar non-olliing root arm motion planning. In Pro. 41st Annu. IEEE Sympos. Foun. Comput. Si., pp , 2000, eu/~streinu/papers/motion.ps.gz. [21] I. Streinu. Foling arpenter s rulers, root arms, proteins: a rigiity theoreti approah. Invite talk, 10th Annual Fall Workshop on Computational Geometry, [22] I. Streinu. A ominatorial approah to planar non-olliing root arm motion planning. preprint, [23] W. Whiteley. Matrois from Disrete Geometry. in Matroi Theory, J. Bonin, J. Oxley an B. Servatius (es.), AMS Contemporary Mathematis,