Flip Graph of Pseudo Triangulation with a Fixed ... - Semantic Scholar

Flip Graph of Pseudo Triangulation with a Fixed Set of Pointed Vertices Vivek Kumar Singh & Shashank K Mehta ∗ Department of Computer Science and Engineering Indian Institute of Technology, Kanpur India 208016 {viks}{skmehta}@cse.iitk.ac.in

Abstract If P is a set of points in a plane and S is a subset of non-hull points, then TSP denotes the set of all pseudotriangulations in which S vertices are non-pointed. It is shown that the flip graph on TSP pseudo-triangulations is connected and no flip edge exists between pseudotriangulations of TSP and TSP0 when S 6= S0 . As a consequence of this result, Sergei Bespamyatnikh’s algorithm to enumerate members of T0/P (minimum pseudo-triangulations) can also enumerate all pseudo-triangulations in O(log |P|) time per pseudotriangulation.

1

Introduction

Pseudo-triangulations are a generalization of triangulations, which find application in visibility, ray shooting, kinetic collision detection, and guarding. Although notion of pseudo-triangulations was introduced long ago, only recently have they received much attention. Although several geometric properties of pseudotriangulations have been established, some combinatorial questions are still open. In [6], Jack Snoeyink conjectured that for a planar point set, the number of minimum pseudo-triangulations is never less than the number of triangulations. The conjecture has been verified for all topologies with at most 10 points [9]. In [5] Randall, Rote, and Snoeyink verified the conjecture for a set of points with exactly one interior (non-hull) point by deriving closed form expressions for the numbers of triangulations and of minimum pseudo-triangulations. They

further proved a coarse upper bound on the number of minimum pseudo-triangulations on a set of points in general position. Aichholzer et al. [1] established that the number of minimum pseudo-triangulations on n points is minimized when the points are in convex position. Ketter et al. [10] showed that minimum pseudo-triangulations, having pseudo-triangles with at most one non-convex vertex, can be constructed in O(n log n) time. Recently Speckmann and Eindhoven [7] conjectured a stronger claim that if a non-pointed vertex p becomes pointed then the number of possible pseudoP triangulations does not decrease, i.e., |TSP | ≤ |TS−{p} |. It is believed that better understanding of the flip graph of the pseudo-triangulation might help establish the conjecture. It is well known that the flip graph of the triangulations is connected, i.e., one can construct any triangulation by a series of flip operation starting from any other triangulation. In [8], Rote and Streinu introduced the polytope of minimum pseudo-triangulations of a point set in the plane whose 1-skeleton is the flip graph of pseudotriangulations of the point set. Bespamyatnikh [3] had devised an algorithm to enumerate all triangulation using the connectivity of the flip graph of the triangulations. In [2, 4] he showed that the flip graph of minimum pseudo-triangulations is connected and has a diameter of at most (n − 1)(n − 4) where n is the number of points. Further, he gave an algorithm to enumerate all the minimum pseudotriangulations in O(log n) time per pseudo-triangulation using linear space. In this work, we show that the flip graph of the collection of pseudo-triangulations with a fixed set of pointed vertices is connected, using some of the techniques of Sergei Bespamyatnikh. This result enables his algorithm to enumerate all the pseudo-triangulations in O(log n) time per pseudo-triangulation using linear space. The proofs have been omitted in this extended abstract and can be found in the full paper.

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Figure 1. A pseudo-triangulation

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Figure 2. Geodesic between p and q

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Terminology and Basic Definitions

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Figure 3. Schematics of ways two pseudo-triangles can share an edge

hull edge is shared by two pseudo-triangles. Figure 3 shows schematic diagrams of the possible ways two triangles can share an edge. Let t1 and t2 be two pseudotriangles sharing a non-hull edge e. We denote the region formed by the union of t1 and t2 after deletion of e by re , see Figure 4. Under the assumption that no collinear triplet exists, re is a simple polygon. From the figure it is easy to verify that re can have at most four sides, therefore re must be either a pseudo-triangle or a pseudoquadrilateral as pseudo 2-gon is impossible.

A simple polygon is referred as m-sided pseudo-polygon if the internal angle of m vertices is less than 180 degrees. These vertices will be called convex vertices. The sequence of edges between two successive convex vertices will be called a reflex chain or a concave chain or a side. Let P be a set of planar points. A spatial subdivision by line segments between points of P in which all regions, except the outermost, are pseudo-triangles is called a pseudo-triangulation of P, see Figure 1 . In any pseudotriangulation of a point set all edges of the convex-hull are present. In this article we will assume that the underlying point-set P has no three collinear points.

Let u be the convex vertex of t1 opposite the side containing e. Similarly let v be the convex vertex of t2 opposite the side containing e. If the geodesic between u and v in re already exists then this region must be a pseudotriangle and e is said to have no dual, Figure 4(c,d). In other case when the geodesic is not fully present, then we know that one edge, e0 , is missing. This edge is called the dual of e, see Figure 4(a,b). Replacement of e by e0 is called flipping e. The arrangement resulting from a flip is also a pseudo-triangulation. It is easy to show that if the dual of an edge exists, then dual of the dual also exists and it is the original edge.

A non-hull edge e of a pseudo-triangulation is said to be deletable if its deletion results in a pseudo-triangulation. Similarly, if insertion of an edge in a pseudo-triangulation results again in a pseudo-triangulation then that edge is called insertable.

If the dual of e does not exist, then e will be called unflippable. Under the assumption that there are no collinear triplets, two pseudo-triangles cannot share more than one edge. Thus each non-hull edge has a unique pair of pseudo-triangles that share it.

Inside a simple polygon the shortest path between any two vertices, p and q, which does not intersect any edge (i.e., does not exit the polygon) is called the geodesic between them. It is unique, see Figure 2. In a polygon with at most 4 convex vertices, the geodesic between a given pair of vertices is either already present in the polygon or one edge of the geodesic is absent.

In a pseudo-triangulation the set of edges incident upon a point p will be denoted by E p . If any two adjacent edges in E p form an angle greater than 180-degrees between them, then p is called pointed. Trivially, every point on the convex-hull of the base points-set is pointed in every pseudo-triangulation. If S is a subset of interior (non-hull) points of the base-set P, then TSP denotes the set of all pseudo-triangulations over P in which S vertices are non-pointed and all other vertices

A pseudo-triangulation is a spatial division so each non-

u=x

between c1 and c2 , then sum of angles contained in re at p is greater than 180 degrees. Hence, re is a pseudotriangle.

u x

re

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(only-if part) By definition, e splits re into t1 and t2 . If re is a pseudo-triangle, then c1 and c2 must be convex vertices of re . So there exists a convex chain between c1 and c2 which implies that the geodesic between c1 and c2 already exists. Hence, e is unflippable.

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L EMMA 3.2. No edge in E p is flippable in a triangulation iff c p is a pseudo-triangle and if p is a non-hull point, then edges of E p = {(p, q0 ), (p, q1 ), (p, q2 )} are first edges of the respective geodesics from p to the three convex vertices of c p .

re

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Figure 4. In (a) and (b) dual of e exists: e0 = (x, y)

P ROOF. From observation 3.1 we know that edge e = (p, q), shared by pseudo-triangles t1 and t2 , is unflippable iff re is a pseudo-triangle. If re has p as a convex vertex, we call it of type-I otherwise if re has q as a convex vertex, we call it of type-II.

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Figure 5. Cell of p

are pointed. Pseudo-triangulation of Figure 1 belongs to P . The pseudo-triangulations T P are known as miniT12,14 0/ mum pseudo-triangulations. T P will denote the set of all pseudo-triangulations on point set P. Let T be a pseudo-triangulation of P and p ∈ P. The polygon formed by the union of pseudo-triangles sharing p after all non-hull edges of E p are deleted, will be called the cell of p, c p . See the figure 5.

3

Preliminary Observations

O BSERVATION 3.1. A non-hull edge e is unflippable iff re , the region formed by the union of adjacent triangles on removal of e, is a pseudo-triangle. P ROOF. Let pseudo-triangles that share e be t1 and t2 . Let the convex vertices opposite to e in t1 and t2 be c1 and c2 respectively. (if part) If e = (p, q) is unflippable, then we know at least one of p and q must be a convex vertex of both t1 and t2 . Wlog lets assume that p is a convex vertex of both t1 and t2 . This implies that the geodesics from p to c1 and p to c2 exist. Since e is unflippable so the geodesic between c1 and c2 already exists. If chain c1 − p − c2 is the geodesic

cp

(if-part) By definition E p = {(p, qi )}. Now if p lies on convex hull and c p is pseudo-triangle then all qi must lie on the convex chain of c p opposite to p. Hence the geodesic contained in c p between any qi and q j already exists. This implies no edge in E p is flippable. If p is inside the convex hull, with edges of ∈ E p = {(p, q0 ), (p, q1 ), (p, q2 )} as the first edge of the respective geodesics from p to a convex vertices of c p , then trivially re of each e ∈ E p must be of type-II pseudo-triangle and hence unflippable. (only if-part) If p is a point on the convex hull and each edge e ∈ E p is unflippable then re for all e ∈ E p must be a pseudo-triangle of type-I. Since all the re have a common convex vertex p hence c p which is the union of re over all e ∈ E p will also be a pseudo triangle. If p lies inside the convex hull then if then re of each edge must be of type-II only. If for any edge e ∈ E p , re is of type − I then for the edges e1 = (p, q1 ), e2 = (p, q2 ) on re to be unflippable re1 and re2 must be both of type-II which is not possible. Furthermore, if re of each edge in E p of type-II only, then |E p | must be exactly 3. This is evident from the fact that the sum of the reflex angles at p for each re must be bounded by twice the sum the of all angles around p (= 360). C OROLLARY 3.3. If p is a non-hull point and no edge in E p is flippable, then p is non-pointed. P ROOF. If p is a non-hull point and no edge in E p is flippable then we know |E p | = 3. Lets say E p = {(p, q0 ), (p, q1 ), (p, q2 )}. Further we know that for each edge e ∈ E p , re must be a pseudo-triagle of type-II. Hence, no two edge in E p can have angle more than 180 between them, which implies p is non-pointed. L EMMA 3.4. Flipping preserves pointedness of all points.

P ROOF. In flipping, only 4 points are involved - the endpoints of e and its dual ed . Observe that removing an edge from a pointed vertex cannot make it non-pointed. Similarly adding an edge to a non-pointed vertex cannot make it pointed. Since flipping can be realized as deletion of e and insertion of ed , so the only relevant cases are (Case 1) When end-point of ed is pointed: Realize that dual of an edge is essentially a bitangent between two sides of a pseudo-4-gon (see figure 6). So end points of ed are points of tangency and hence pointed. So if the end points of ed were initially pointed, then they will remain pointed after addition of ed . (Case 2) When end-point of e, say p, is non-pointed: Lets assume that flipping e makes p non-pointed to pointed. Clearly p must be a convex vertex of both the pseudotriangles that shares e. Hence if our assumption is correct then from observation 3.1, re must be a pseudo-triangle. But then e cannot be flippable. Hence, flipping e cannot make a non-pointed vertex pointed.

e e

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Figure 6. Dual of an edge is bitangent between two sides of a pseudo-4-gon

reflex angle at p into 2 non-reflex angles. Now it trivially follows that deleting e would make p from non-pointed to pointed. (only if part) From observation ??, we know that flipping cannot make a non-pointed vertex pointed. Trivially, insertion also cannot make a non-pointed vertex pointed. So the only way we can make a non-pointed vertex pointed is by deleting an edge. Above results give the following equivalence. T HEOREM 3.8. Following statements are equivalent for a non-hull edge e = (p, q) (i) e is unflippable (ii) re is a pseudo-triangle (iii) e is deletable (iv) One of p and q, but not both, transforms from nonpointed to pointed on deletion of e. C OROLLARY 3.9. Addition of an insertable edge makes one pointed vertex non-pointed. The well known result on the number of edges in a minimum pseudo-triangulations is a direct consequence of the above corollary. C OROLLARY 3.10. [11] Any minimum triangulation must have 2.|P| − 3 edges.

pseudo-

C OROLLARY 3.5. TSP is closed under the flip operation, for all S.

More generally, every pseudo-triangulation of TSP has 2.|P| + |S| − 3 edges.

O BSERVATION 3.6. A non-hull edge is unflippable iff it is deletable.

O BSERVATION 3.11. Let p be a non-hull vertex. If the degree of p cannot be altered by flipping of the edges in E p (i.e., |E p | remains constant), then either p is already pointed or all edges in E p are deletable, thus the deletion of any E p edge makes p pointed.

P ROOF. Let pseudo-triangles that share e be t1 and t2 and the convex vertices opposite to e in t1 and t2 be c1 and c2 respectively. (if part) If e is unflippable, then from observation 3.1, re is a pseudo-triangle. Hence, on removing e, the resulting diagram would still remain a pseudo-triangulation. (only if part) If deletion of e is valid, then deleting e yields a pseudo-triangulation. Hence, t1 ∪ t2 (=re ) is a pseudo-triangle. Hence from observation 3.1, e is unflippable. O BSERVATION 3.7. A non-hull edge e = (p, q) is deletable iff deletion converts either p or q, but not both, from non-pointed to pointed. P ROOF. Let the pseudo-triangles that share e be t1 and t2 . (if part) If e is deletable then from observation 3.1 re is a pseudo-triangle. So clearly at least one of the vertices p and q must be convex to both t1 and t2 . Suppose p is convex to both t1 and t2 . Then clearly p is non-pointed because p is a non-convex vertex of re and e splits the

P ROOF. If edges in E p are not flippable then from theorem 3.8 we know all edges in E p are deletable and on deletion will make p pointed. If flipping edges in E p does not alter the degree of p for each edge e ∈ E p , e and its dual ed must share p as a common endpoint. From observation 3.1 we know that re of a flippable edge must be a pseudo-quadilateral. Now for e and ed to share p, p must be the point of intersection of the geodesics between the opposite convex vertices of pseudo-quadilateral re . This implies p must lie on the convex chain of re and hence must be pointed. We conclude this section with an important lemma. This result is used in the proof of Theorem 4 of [4]. O BSERVATION 3.12. [4] Let P0 be a subset of the point set P. Let e = (p, q) be an edge in a triangulation T of P such that p be outside the convex-hull conv(P0 ), and q be in the interior of conv(P0 ). Let t1 and t2 be the two pseudo-triangles sharing e. Suppose these pseudotriangles also contain edges (p, q1 ) and (p, q2 ) respectively where q1 , q2 are on the convex-hull of P0 and are

convex vertices of the respective pseudo-triangles. Then e is flippable. P ROOF. Consider re . Clearly re (= t1 ∪t2 ) is contained in conv(P0 ). Also q1 and q2 are convex vertices of re . Since p lies outside the conv(P0 ) hence any pseudo-triangle having vertices in p ∪ P0 must have p as its convex vertex. So p must be a convex vertex of re (= t1 ∪ t2 ). Now from observation 3.1 if e is unflippable then re must be a pseudo-triangle with q1 , q2 and p as its convex vertices. Now since q1 and q2 lies on the conv(P0 ) thus re can be a pseudo-triangle iff q lies on the conv(P0 ). But q lies inside the conv(P0 ). Hence e must be a flippable edge.

4

half-plane, obtained by the line passing through pl (pr ) and p, that contains c. Let region R be the intersection of the hl and hr . Trivially, R contains c. (refer figure 7) Let q be the point such that (p, q) completes the geodesic from p to c. R also contains q otherwise the sides of the pseudo-triangle will no longer remain reflex. Notice that angle pl pq cannot be reflex otherwise the chain pl − p − q no longer remain reflex. Hence edge (p, q) splits the (reflex) angle between pl , p and p, pr into two non-reflex angles. Hence, addition of unique edge (p, q) makes p non-pointed. Clearly, edges which kills the pointedness of points that belong to the same pseudo-triangle do not intersect as they all lies on the geodesic to the opposite vertex.

Flip-Graph of TSP

The flip-graph, G(T P ), of pseudo-triangulations of a planar set of points P is defined by taking pseudotriangulations as nodes and defining an undirected edge between two nodes T and T 0 iff some flip operation transforms T into T 0 . Duality of the flipping operation ensures that this graph is well-defined. From Corollary 3.5 in the previous section we know that G(T P ) is not connected. In this section we will show that the induced subgraph G(TSP ), for any subset S of interior vertices, is indeed connected. We define a vector representation for each pseudotriangulation which was originally proposed in [3]. Let the vertices of P be ordered according to convex ordering from outside in: p0 , p1 , p2 , . . .. Let Pi denote the set of points {pi , pi+1 , . . . , pn }. The points in P has the property that each pi is on the convex-hull of Pi .Let us denote the edges of a pseudo-triangulation T as ordered pairs (a, b) where a < b under this ordering. Let (a1 , b1 ), (a2 , b2 ), . . . , (am , bm ) be the list of all edges of the pseudo-triangulation in lexicographically increasing order (comparing pairs from left to right). Then vector vT = (a1 , b1 , a2 , b2 , . . . , am , bm ) is a representation of T . It is clear that each pseudo-triangulation has a unique representation and since lexicographic ordering is a total ordering, it linearly orders all pseudo-triangulations. In the sequel, comparison between the pseudo-triangulations will always be done based on this ordering, starting from the minimum edge (left-end).

c q

p

l

R

p pr

Figure 7. Shaded region contains the geodesic from p to the opposite convex vertex In Henneberg’s Type-1 construction, minimum pseudotriangulation is obtained by drawing tangents from the pi to the conv(Pi ), starting from pn−3 . We extend Henneberg’s construction as follows. First construct the minimum pseudo-triangulation from Henneberg’s Type-1 construction. Then for a given subset of interior vertices, S, insert ε p for each p ∈ S. We denote the result by TSH . It is obvious that this pseudo-triangulation belongs to TSP . O BSERVATION 4.2. All pseudo-triangles in TSH have at least two sides constituted of single edge each.

O BSERVATION 4.1. In any pseudo-triangulation every pointed interior vertex p can be converted into nonpointed by insertion of a unique edge ε p . Further, these edges do not intersect.

P ROOF. Clearly every pseudo-triangle in Henneberg’s Type-1 construction has two sides of single edge each and third side as a convex chain. Our algorithm adds edges from pointed vertex to the opposite convex vertex which is shared by the two single edge sides. Hence the edge added clearly splits the pseudo-triangle in two pseudotriangles, both of which have two sides as complete edges From observation 4.1, we know that the edges added by our algorithm do not intersect. Hence final arrangement of edges is independent of order in which the edges were inserted.

P ROOF. Let p be an interior vertex and c be the opposite convex vertex of the pseudo-triangle in which p is a nonconvex vertex. Let pl and pr be the points on the left and right to p on the convex chain. Let hl (hr ) denote the

O BSERVATION 4.3. Let T be an arbitrary pseudotriangulation of TSP and let e = (p, q) be the smallest edge of vT where mismatch occurs with vT H . Then S E p (TSH ) ⊂ E p (T ).

Henneberg’s Type-1 construction is shown to generate a minimum pseudo-triangulation [11]. It is easy to show that this pseudo-triangulation is the maximum among all pseudo-triangulations under the ordering.

P ROOF. Since e = (p, q) is the smallest edge where the mismatch occurs all the edges outside the conv(Pp ) are common in T and TSH . We simply have to prove that every edge in E p (TSH ) belongs to E p (T ). Realize that T must contain the edges from p to the conv(Pp ). If p lies on the conv(P), then it is trivially satisfied. If p lies in the interior and edges from p to the conv(Pp ) are absent, then realize that the polygon containing p as a non-convex vertex will cease to be pseudo-triangle. Hence edge e must lie in the interior of conv(Pp ). Now consider any edge e0 ∈ E p (TSH ) such that e0 = (p, q0 ) doesnot lie on the conv(Pp ). Since e0 lies in E p (TSH ) so our algorithm must have added this edge to kill the pointedness of q0 such that q0 lies on conv(Pp ). If T doesnot contain e0 then the vertex q0 will remain pointed in T because all the remaining edges from q0 in T being lexically larger than (p, q0 ) are contained in conv(Pq0 ). Hence, every edge in E p (TSH ) belongs to E p (T ). O BSERVATION 4.4. TSH is the largest triangulation in TSP w.r.t. the lexical order.

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P ROOF. Since lexical order of edges of a pseudotriangulation is unique for each, so it imposes a total ordering over the set the pseudo-triangulation. Hence their must be unique largest pseudo-triangulation. Now compare the TSH and any other pseudo-triangulation T . From 4.3, we know that the lexically smallest mismatch occurs at edge e = (p, q) ∈ T then E p (TSH ) ⊂ E p (T ). Trivially then TSH is the lexically larger than T .

lexically larger pseudo-triangulation on flipping. C OROLLARY 4.6. The flip-graph of TSP is connected. P ROOF. From observation 4.4 we know that for any PS there exists a unique pseudo-triangulation TSH ∈ TSP which has the lexico-largest vector. Now from above theorem we know TSH can be obtained by a series of flip operations to any pseudo-triangulation T ∈ TSP . Hence there exists a path between any two pseudo-triangulations in TSP . We define insertion graph where each triangulation TSH is a node. Two nodes are connected with an edge if a single insertion/deletion transforms one into the other. From the definition, we know that TSH and TSH0 are adjacent if S and S0 differ by one vertex. Figure 8 show this graph where the super nodes are flip-graphs of TSP .

5 Bespamyatnikh’s Enumeration Algorithm Sergei Bespamyatnikh’s [3] enumeration technique is based on imposing a rooted spanning tree on the flipgraph of triangulations as follows. TSH is the root and the parent of any other pseudo-triangulation, T , is the maximum pseudo-triangulation reachable from T by one flip. Enumeration requires an efficient test to decide flipping which edge leads to the parent or a child node. In [3] these tests are based on two lemmas which are also valid for general pseudo-triangulations. We reproduce these lemmas. Enumeration is performed by doing depth-first traversal on this tree.

T HEOREM 4.5. Every pseudo-triangulation T of TSP , other than TSH , has a flippable edge which, on flipping, results in a larger pseudo-triangulation.

L EMMA 5.1. [3] Flipping lexico-smallest edge among T −TSH gives the parent pseudo-triangulation for any T ∈ TSP .

P ROOF. Let e = (p, q)inT denote the lexically smallest mismatch edge between vT and vT H . From observation S 4.3, we know e must be an interior edge of conv(Pp ).

L EMMA 5.2. [3] Let em = (a, b) be the edge in T ∈ TSP that takes T to its parent. Let (c, d) be the dual of any flippable edge, e, in T other than em and flipping e transforms T into T 0 . Then T 0 is a child of T iff either (c, d) < (a, b) or if (a, b) is not flippable in T 0 then (c, d) is the smallest edge in T 0 − TSH .

(Case 1) If q lies strictly inside the conv(Pp+1 ): From observation 3.12, we know that e = (p, q) is flippable. (Case 2) If q lies on the conv(Pp+1 ): Note that since e does not exist in TSH so q must be pointed. Now if e is unflippable then ce must be a pseudo-triangle with p as a convex vertex. But in that case q must be non-pointed which is not possible. Hence, e must be flippable. Now we claim that the ed = (p0 , q0 ), dual of e, is lexically larger than e i.e lies inside that conv(Pp ). Realize that p cannot be p0 . Suppose p0 is lexically smaller than p, then in that case E p0 (TSH ) and E p0 (T ) must be same. But notice that with fixed set of pointed vertices and E p0 (TSH ) equal to E p0 (T ), the non-hull edge e cannot have a lexically larger dual edge. Hence, e is flippable and gives a

T HEOREM 5.3. All pseudo-triangulations of TSP can be enumerated using Bespamyatnikh’s algorithm in O(log |P|) time per pseudo-triangulation. We adopt a similar strategy to enumerate all pseudotriangulation by imposing a rooted spanning tree on insertion-graph. T0/H is the root. Further, TSH is the parent H of TS∪{p} if p is an internal point with index greater than all points in S. To enumerate all pseudo-triangulation, perform a depth-first traversal on this tree and on visiting each TSH enumerate all pseudo-triangulations of TSP . This enables enumeration of all pseudo-triangulations of a set P in O(log |P|) per pseudo-triangulation in linear space.

H

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H Q − {p1 } H

H

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T{p } k

H {p } k

Figure 8. Flip-insertion graph: only insertion/deletion edges shown

6

Conclusion

We adopted Sergei Bespamyatnikh’s approach for showing the connectivity of flip-graph of minimum pseudotriangulations and proved the similar result for the flipgraph of pseudo-triangulations with a fixed set of nonpointed vertices, TSP . In addition we have shown that nodes of TSP and TSP0 are not connected in flip graph if S 6= S0 . The graph is shown to be connected after adding insertion edges. Consequently Bespamyatnikh’s enumeration algorithm for minimum pseudo-triangulations can also be adopted to enumerate all pseudo-triangulations. Every pseudo-triangulation can be enumerated with time cost of (log |P|) per pseudo-triangulation using linear space.

7

References

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