D. D. Sleator, R. E. Tarjan, and W. P. Thurston. Rotation distance, triangulations, and hyperbolic geometry. J. Amer. Math. Soc. 1(3):647--681, 1988.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....such a triangulation by the unordered triple ijk of vertices it contains, and similarly for edges ij and vertices i. Let An and Mn denote the set of Catalan triangulations of the disk and Mobius band , respectively. The present paper was motivated by two very well known facts about (see, e.g. [STT]) 1) The cardinality an of An is the Catalan number 2n Gamma4 , which has generating function A(x) 2x 1 1 Gamma 4x (here we are using the convention that a 2 = 1) 2) Any two triangulations in An may be connected by a sequence of operations which we will call ....

.... polytope called the associahedron or Stasheff polytope (see [Lee] The last fact implies that the graph on An has edge connectivity and vertexconnectivity n Gamma 3 by Balinski s Theorem (see [Ba] and the diameter of the graph on An was proven to be 2n Gamma 10 for n sufficiently large in [STT]. What are the edge connectivity, vertex connectivity, and diameter of the graph on Mn Computer calculations show that M 5 ; M 6 ; M 7 , and M 8 have diameters 0; 5; 10, and 16 respectively. 2) In light of Theorem 2, and also the result of Pachner [Pa, Theorem 6.3] which says that any two ....

D. Sleator, R. Tarjan, and W. Thurston, Rotation distance, triangulations, and hyperbolic geometry, J. Amer. Math. Soc. 1 (1988), 647-681.

.... of characterizing the f vectors of triangulations of balls and polytopes (see open problems in [8] In fact, the study of minimal trian gulations of topological balls also received attention due to its connections to data structures, in the calculation of rotation distance of binary trees [25]. Institut f Jr Theoretische Informatik, ETH Z Jrich (belowif.ethz. ch) tDept. of Mathematics, Univ. of California Davis (deloeramath.ucdavis.edu) Institut ffir Theoretische Informatik, ETH Zfirich (richterinf.ethz. The computational geometry literature has several papers interested in ....

....number of tetrahedra ranges from n 3 to ( 2n 3. Both bounds are known to be tight for three dimensions [13] It is also known that the size of a minimal triangulation of a convex 3 polytope must lie between n 3 and 2n 10, when n 12 [13] That the upper bound is tight was proved in [25] using hyperbolic geometry. It is worth noticing at this point that the size of the constant K we constructed in the previous section satisfies in fact n 3 K 2n. Now we discuss an interesting reason why the lower bound is strict: We say that a convex polytope is stacked if it has a ....

Sleator, D.D., Tarjan R.E., and Thurston W.P. Rotation distance, triangulations, and hyperbolic geometry. J. Amer. Math. Soc., 1, 1988, 647- 681.

....the size, and of generating a random element are real problems. The hope is that by studying the walk on T n we could learn about the other two sets of triangulations. Unfortunately little is known about the triangulations walk. This is surprising because G n is so familiar and well studied [7] [12] ( also [4] 5] 6] 8] 10] which are relevant because G n is also the graph of rooted binary trees with n 2 internal nodes, adjacency de ned by the rotation operation) In the present paper we will apply the method of conductance to show that the walk mixes rapidly. This amounts to showing ....

D. Sleator, R. Tarjan, and W. Thurston. Rotation Distance, Triangulations, and Hyperbolic Geometry. J. Amer. Math. Soc. 1, 647-681, 1988. 9

....face if and only if the corresponding trees di#er by a rotation move. In Sleator et al. 1992) it is shown that the maximal rotation distance between two trees on n leaves is O(n log n) while the maximal rotation distance between two trees contained in the same associahedron is exactly 2n 6 (see Sleator et al. 1988)) These results give an indication of the size of our space of trees. 4. GEOMETRY OF THE SPACE OF TREES By the geometry of the space we mean its metric, as opposed to combinatorial, properties. The space of trees comes equipped with a natural distance function, due to the fact that it is made ....

D. D. Sleator, R. E. Tarjan, and W. P. Thurston. Rotation distance, triangulations, and hyperbolic geometry. J. Amer. Math. Soc., 1(3): 647--681, 1988.

....quadrilateral. Note that it is Whitney duality which relates the triangulations and trees (faces correspond to vertices in the dual while edges cross dual edges) hence, the notion of flipping has a meaning for trees as well. The underlying graph Tn of Dn has been called the rotation graph [32] and is the 1 skeleton of the associahedron (or Staheff polytope, also discovered by Mac Lane, see, e.g. Ziegler [38] We are interested in the set Pn of all directed paths in Dn joining T to T 0 ; this is known to be nonempty for any two vertices T; T 0 . In [18] we give two isomorphisms c ....

....or right most parenthesizations, resp. If these two trees, say Ln and Rn , are considered in dual form, the result of the fusion process becomes the cube of a path, P 3 n , and we remarked on its hyperbolic nature in [17] Further, hyperbolic geometry was used by Sleator, Tarjan and Thurston [32] to majorize the diameter of Tn . The cube P 3 n of the path [15] is the graph with nodes f1; ng in which nodes are adjacent if and only if they differ by at most 3. For any given scale s, 0 s 1, there is a linear embedding of Pn into the plane which extends to a linear embedding of ....

D. D. Sleator, R. E. Tarjan and W. P. Thurston, Rotation distance, triangulations, and hyperbolic geometry, J. American Math. Soc. 1 (1988) No. 3, 647-681.

.... Tarjan formulated a dynamic optimality conjecture concerning the performance of splaying [3, 6] It raises the interesting question: Given two binary trees on n (internal) nodes, what is the maximum distance between the two trees via rotation In a beautiful paper, Sleator, Tarjan and Thurston [4][5] solved this problem with a tight bound of 2n Gamma 6. In that paper they first transformed the problem on binary trees to a problem on triangulations on the disk, and then applied techniques of topology and geometry to obtain their bounds. Most remarkable is their use of volume estimate in ....

....of a planar region, and take its dual graph, each triangular face will correspond to a vertex of degree 3. In particular, if we start with a triangulation of the disk, with all its vertices on the boundary circle, and take the dual graph, we obtain precisely the special case of binary trees as in [5] 1 . As observed in [5] a rotation in the dual graph (the binary tree) corresponds to the following operation called an edge flip on the triangulation of the disk: Take any quadrilateral formed by two adjacent triangles sharing an edge e, remove e and replace it with the other diagonal of the ....

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D. D. Sleator, R. E. Tarjan and W. P. Thurston, Rotation distance, triangulations, and hyperbolic geometry, J. AMS, vol 1, Num 3, 1988, 647--681.

.... and Tarjan formulated a dynamic optimality conjecture concerning the performance of splaying [3, 6] It raises the interesting question: Given two binary trees on n (internal) nodes, what is the maximum distance between the two trees via rotation In a beautiful paper, Sleator, Tarjan and Thurston [4][5] solved this problem with a tight bound of 2n Gamma 6. In that paper they first transformed the problem on binary trees to a problem on triangulations on the disk, and then applied techniques of topology and geometry to obtain their bounds. Most remarkable is their use of volume estimate in ....

D. D. Sleator, R. E. Tarjan and W. P. Thurston, Rotation distance, triangulations, and hyperbolic geometry, Proc. ACM STOC, 1986, 122--135.

....leaves Date: September 27, 1998. 1991 Mathematics Subject Classification. Primary: 32G15, 57R20, 81Q30. Secondary: 14H15, 30E15, 30E20, 30F30. 1 2 M. MULASE AND M. PENKAVA of a measured foliation defined on a Riemann surface by a meromorphic quadratic differential called a Strebel differential [2, 11]. When the Riemann surface is defined over Q , it coincides with the same surface that is given by the Child s Drawing correspondence between ribbon graphs and algebraic curves defined over Q . In this paper we give constructive proofs of these facts using canonical coordinate systems arising from ....

....find coordinate patches that represent the complex structure In this section we give a canonical coordinate system on a Riemann surface once a finite number of points on the surface and the same number of positive real numbers are chosen. The key technique is the theory of Strebel differentials [11]. Using Strebel differentials, we can encode the holomorphic structure of a Riemann surface in the combinatorial data of ribbon graphs. Let C be a compact Riemann surface. We choose a finite set of labeled points fp 1 ; p 2 ; Delta Delta Delta ; pn g on C, and call them marked points on the ....

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Daniel D. Sleator, Robert E. Tarjan, and William P. Thurston, Rotation distance, triangulations, and hyperbolic geometry, Journal of the American Mathematical Society 1 (1988), 647--681.

.... by items in the right subtree of T 2 in preorder) Then the total access time is O(n) Sleator and Tarjan state that the Dynamic Optimality Conjecture implies the other two conjectures (the proof is non trivial) There have been several works on, or related to, the optimality of splay trees [STT86], W86,T85,Su89,Luc88a,Luc88b] STT86] shows that the rotation distance between any two binary search trees is at most 2n Gamma 6 and that this bound is tight; they also relate this to distinct triangulations of polygons; although connected to the splay tree conjectures, this result has no ....

.... 2 in preorder) Then the total access time is O(n) Sleator and Tarjan state that the Dynamic Optimality Conjecture implies the other two conjectures (the proof is non trivial) There have been several works on, or related to, the optimality of splay trees [STT86] W86,T85,Su89,Luc88a,Luc88b] [STT86] shows that the rotation distance between any two binary search trees is at most 2n Gamma 6 and that this bound is tight; they also relate this to distinct triangulations of polygons; although connected to the splay tree conjectures, this result has no immediate application to them. W86] ....

D.D. Sleator, R.E. Tarjan, W.P. Thurston. Rotation distance, triangulations, and hyperbolic geometry. In Proceedings Eighteenth Symposium on Theory of Computing, 1986, 122--135.

.... by items in the right subtree of T 2 in preorder) Then the total access time is O(n) Sleator and Tarjan state that the Dynamic Optimality Conjecture implies the other two conjectures (the proof is non trivial) There have been several works on, or related to, the optimality of splay trees [STT86], W86,T85,Su89,Luc88a,Luc88b] STT86] showed that the rotation distance between any two binary search trees is at most 2n Gamma 6 and that this bound is tight; they also related this to distinct triangulations of polygons; although connected to the splay tree conjectures, this result has no ....

.... 2 in preorder) Then the total access time is O(n) Sleator and Tarjan state that the Dynamic Optimality Conjecture implies the other two conjectures (the proof is non trivial) There have been several works on, or related to, the optimality of splay trees [STT86] W86,T85,Su89,Luc88a,Luc88b] [STT86] showed that the rotation distance between any two binary search trees is at most 2n Gamma 6 and that this bound is tight; they also related this to distinct triangulations of polygons; although connected to the splay tree conjectures, this result has no immediate application to them. W86] ....

D.D. Sleator, R.E. Tarjan, W.P. Thurston. Rotation distance, triangulations, and hyperbolic geometry. In Proceedings Eighteenth Symposium on Theory of Computing, 1986, 122--135. 42

.... in dimension two are known to be connected since the early days of computational geometry [25] For the vertex set of a convex polygon, the graph is a classical object in combinatorics, first studied by Stasheff and Tamari [46, 42] and related to associativity structures and to binary trees [26, 41]. It is disturbing that in dimension three, and even assuming convex position, we do not know whether the graph is always connected or not. The graph of triangulations of A contains as an induced subgraph the 1 skeleton of the secondary polytope of A introduced by Gel fand et al. 20] This is a ....

D. Sleator, R. Tarjan and W. Thurston, Rotation distance, triangulations, and hyperbolic geometry, J. Am. Math. Soc. 1 (1988), 647--681.

....ratio (on weighted or unweighted phylogenies) It seems that to obtain a ratio of o(log n) we have to be able to prove nontrivial lower bounds for sorting sequences on trees with nni moves. The nni operation is similar to and slightly more powerful than the rotation operation discussed in [3, 32]. Is it NP complete to compute the rotation distance Can we approximate the rotation distance better than the trivial ratio 2 This question turns out to be subtler than it appears to be. Partial results in this direction can be found in [27] Acknowledgments: We wish to thank J. Felsenstein for ....

D. Sleator, R. Tarjan and W. Thurston, Rotation distance, triangulations, and hyperbolic geometry, J. Amer. Math. Soc., 1(1988), 647-681.

....This has also been ob Minimal Simplicial Dissections and Triangulations of Convex 3 Polytopes 47 Fig. 6. Unique triangulated planar graph with n 5 = 6 and n 4 = 3. served for topological triangulations that extend a triangulated sphere into a triangulated 3 ball (see Section 3. 5 of [20]) Our example for Theorem 1.3 (see Fig. 4) shows that this is not always the case. ....

Sleator, D. D., Tarjan, R. E., and Thurston, W.P. Rotation distance, triangulations, and hyperbolic geometry, J. Amer. Math. Soc. 1(3) (1988), 647--681.

.... families of combinatorial objects, the size is counted by the Catalan numbers, defined for n 0 by C n = 1 n 1 2n n : These include binary trees on n vertices [SW86] well formed sequences of 2n parentheses [SW86] and triangulations of a labeled convex polygon with n 2 vertices [STT88]. Since bijections are known between most members of the Catalan family, a Gray code for one member of the family gives implicitly a listing scheme for every other member of the family. However, the resulting lists may not look like Gray codes, since bijections need not preserve minimal changes ....

....[Luc87] giving a cyclic Gray code. 22 It so happens that under a particular bijection between binary trees with n nodes and the set of all triangulations of a labeled convex polygon with n 2 vertices, rotation in a binary tree corresponds to the flip of a diagonal in the triangulation [STT88]. So, the results of [Luc87, LRR93] also give a listing of all triangulations of a polygon so that successive triangulations differ only by the flip of a single diagonal. 8 Necklaces and Variations An n bead, k color necklace is an equivalence class of k ary n tuples under rotation. Figure 10 ....

D. D. Sleator, R. E. Tarjan, and W. P. Thurston. Rotation distance, triangulations, and hyperbolic geometry. Journal of the American Mathematical Society, 1(3):647--681, 1988.

....The interchange graphs for the three representations are defined as follows (see Fig. 3) ffl Triangulations: The set of triangulations forms the vertex set of the interchange graph, and two triangulations are adjacent if one can be obtained from the other by a diagonal flip, as described in [35]. Every diagonal in a triangulation of a convex polygon defines a quadrilateral. A diagonal flip replaces that diagonal with the other diagonal of the same quadrilateral. Sleator et al. 35] show this move connects the state space and they obtained tight upper and lower bounds (of 2n Gamma6) on ....

....two triangulations are adjacent if one can be obtained from the other by a diagonal flip, as described in [35] Every diagonal in a triangulation of a convex polygon defines a quadrilateral. A diagonal flip replaces that diagonal with the other diagonal of the same quadrilateral. Sleator et al. [35] show this move connects the state space and they obtained tight upper and lower bounds (of 2n Gamma6) on the diameter of this interchange graph and other results on triangulations of the sphere (see [20] for a simpler proof) ffl Binary trees: Two binary trees with n internal nodes are adjacent ....

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Sleator, D., Tarjan, R. and Thurston, W. Rotation distance, triangulations, and hyperbolic geometry. J.AMS. 1, 1988.

....connected since the early days of computational geometry [22] and extensively used in that area. For the vertex set of a convex polygon, the graph is a classical object in combinatorics, first studied by Stasheff and Tamari [38, 35] and related to associativity structures and to binary trees (see [23, 34]) It is disturbing that in dimension three, and even assuming convex position, we do not know whether the graph is always connected or not. The graph of triangulations of A contains as an induced subgraph the 1skeleton of the secondary polytope of A introduced by Gel fand et al. 18] This is a ....

D. Sleator, R. Tarjan and W. Thurston, Rotation distance, triangulations, and hyperbolic geometry, J. Amer. Math. Soc. 1 (1988), 647--681.

....approximate nni with a better ratio (on weighted or unweighted phylogenies) It seems that to obtain a ratio better than log n, we have to be able to prove superlinear lower bounds for sorting sequences on trees with nni moves. 2. Nni is similar to and slightly more powerful than rotation distance [4, 29]. Is rotation distance NP complete Can we approximate the rotation distance better than (the trivial ratio) 2 This question turns out to be subtler than it appears to be. 8 Acknowledgements We thank J. Felsenstein and J. Hein for explaining to us the biological motivations for comparing ....

D. Sleator, R. Tarjan, W. Thurston, Rotation distance, triangulations, and hyperbolic geometry, J. Amer. Math. Soc., 1(1988), 647-681.

....B n , the set of rooted binary trees with n Gamma 2 internal nodes, and also to P n , the set of non negative lattice paths that start at 0, make 2n Gamma 4 steps X i of size Sigma1, and end at X 1 Delta Delta Delta X 2n Gamma4 = 0. It is natural (see e.g. 1] 2] 3] 4] 5] 6] [7]) to ask about the expectation of certain properties of the elements of these sets when we sample them at random, for example the height of a uniformly distributed binary tree in B n . Here we study two functions that measure very natural geometric features of a triangulation 2 T n . These ....

D. Sleator, R. Tarjan, and W. Thurston. Rotation Distance, Triangulations, and Hyperbolic Geometry. J. Amer. Math. Soc. 1, 647-681, 1988.

....approximate nni with a better ratio (on weighted or unweighted phylogenies) It seems that to obtain a ratio better than log n, we have to be able to prove superlinear lower bounds for sorting sequences on trees with nni moves. 2. Nni is similar to and slightly more powerful than rotation distance [4, 28]. Is rotation distance NP complete Can we approximate the rotation distance better than (the trivial ratio) 2 This question turns out to be subtler than it appears to be. 8 7 Acknowledgments We thank J. Felsenstein and J. Hein for explaining to us the biological motivations for comparing ....

D. Sleator, R. Tarjan, W. Thurston, Rotation distance, triangulations, and hyperbolic geometry, J. Amer. Math. Soc., 1(1988), 647-681.

....rotations is n Gamma 1. To convert T 1 to T 2 , convert T 1 to a right path then compute the rotations needed to convert T 2 to a right path, and apply them in reverse to the right spine into which T 1 has been converted. At most 2n Gamma 2 rotations are necessary. Sleator, Tarjan and Thurston [66] improved the upper bound to 2n Gamma 6 using a relation between binary trees and triangulations of polyhedra. They also demonstrated the existence of an infinite family in which 2n Gamma 6 rotations were required. Makinen [52] subsequently showed that a weaker upper bound of 2n Gamma 5 can ....

D.D. Sleator, R.E. Tarjan, and W.P. Thurston. Rotation distance, triangulations, and hyperbolic geometry. In Proc. 18th Symp. on Theory of Computing (STOC), pages 122--135, 1986.

....the task of computing the size, and of generating a random element are open problems. By studying the walk on T n we may learn how to study other sets of triangulations. Unfortunately little is known about the triangulation walk. This is surprising because G n is so familiar and well studied [7] [12] ( also [4] 5] 6] 8] 10] which are relevant because G n is also the graph of rooted binary trees with n Gamma 2 internal nodes, adjacency defined by the rotation operation) In the present paper we will apply the method of conductance to show that the walk mixes rapidly. Write S for T n ....

D. Sleator, R. Tarjan, and W. Thurston. Rotation Distance, Triangulations, and Hyperbolic Geometry. J. Amer. Math. Soc. 1, 647-681, 1988.

....ratio (on weighted or unweighted phylogenies) It seems that to obtain a ratio of o(log n) we have to be able to prove nontrivial lower bounds for sorting sequences on trees with nni moves. ffl The nni operation is similar to and slightly more powerful than the rotation operation discussed in [3, 32]. Is it NP complete to compute the rotation distance Can we approximate the rotation distance better than the trivial ratio 2 This question turns out to be subtler than it appears to be. Partial results in this direction can be found in [27] Acknowledgments: We wish to thank J. Felsenstein ....

D. Sleator, R. Tarjan and W. Thurston, Rotation distance, triangulations, and hyperbolic geometry, J. Amer. Math. Soc., 1(1988), 647-681.

....In the remainder of this paper, we will focus exclusively on the tree morphing problem, ignoring any relationship to the polygon morphing problem. 1.2 Our Results The problem of transforming one tree to another without any weight constraint has been studied before. Sleator, Tarjan, and Thurston [7] show that an n node tree can be made isomorphic to any other n node tree by at most 2n Gamma 6 rotations, slightly improving an earlier result by Culik and Wood [2] Our tree morphing problem, however, is more difficult because of the weight constraint on the nodes, which restricts the choice of ....

D. Sleator, R. E. Tarjan, and W. Thurston. Rotation distance, triangulations, and hyperbolic geometry. J. AMS, 1:647--682, 1988.

....of n leaves to any other tree with the same set of leaves. D. Sleator, R. Tarjan, and W. Thurston [42] proved an Omega Gamma n log n) lower bound for most pair of trees. A restricted version of the nni operation, known as the tree rotation operation (discussed in Section 6) was considered in [41] and a trivial approximation algorithm with approximation ratio of 2 was given. But given two individual pair of trees, computing the nni distance between them (either for labeled or unlabeled trees) has been a long standing open question until recently when this problem was settled (for both ....

....any pair of such triangulations by fd(n) Obviously, rd(n) fd(n 2) Culik and Wood showed that rd(n) 2n Gamma 2( 6] Sleator, Tarjan and Thurston improved this bound to 2n Gamma 6 and showed that the bound is tight for all sufficiently large n using hyperbolic geometry. Theorem 18 ([41]) rd(n) fd(n 2) 2n Gamma 6 for all n 10. Furthermore, the equality holds for all sufficiently large n. The exact values of rd(n) for n 16 are listed below[41] n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 rd(n) 0 1 2 4 5 7 9 11 12 15 16 18 20 22 24 26 However, little is known about the lower ....

[Article contains additional citation context not shown here]

D. Sleator, R. Tarjan, W. Thurston, Rotation distance, triangulations, and hyperbolic geometry, J. Amer. Math. Soc., 1, pp. 647-681, 1988.

.... [26] 59] Particularly relevant are notions of dimension that emerge from such considerations, see, e.g. chapter 5 in [7] The possibility of embedding graphs in spaces other than Euclidean and spheric geometry is very appealing, and hardly anything has been done in this direction (but see [56]) We have also said nothing about modeling geometric objects with graphs, which is a related vast area. To initiate our technical discussion, recall that a norm k Delta k associates a real number kxk with every point x in real d space, where (i) kxk 0, with equality only if x = 0, ii) kxk = jj ....

D. D. Sleator, R. E. Tarjan, and W. P. Thurston, Rotation distance, triangulations, and hyperbolic geometry, J. AMS 1 (1988), 647-681.

....is a variant of the nni distance for rooted, ordered trees. A rotation is an operation that changes one rooted binary tree into another with the same size. Figure 4 shows the general rotation rule. An easy approximation algorithm for computing distance with a performance ratio of 2 is given in [68]. However, it is not known if computing this distance is NP hard or not. A B C u v A B C u v rotation at u rotation at v Figure 4: Left and right rotation operations on a rooted binary tree. 2.6 Distances on Weighted Phylogenies Comparison of weighted evolutionary trees has recently been studied ....

D. Sleator, R. Tarjan, W. Thurston. Rotation distance, triangulations, and hyperbolic geometry, J. Amer. Math. Soc., 1 (1988), pp. 647-681.

....moves up to the line y = i and then moves right for j i steps. It is easy to see that this procedure gives a path in P n and that every such path comes from a distinct triangulation. These bijections are frequently exploited when studying the combinatorics in one of these sets (see especially [14]) and also for the task of randomly generating elements from one of the sets (e,g, 2] 5] 13] To understand what 1 n says about trees, imagine the diagonal v i v j in as directed from the smaller numbered vertex of K to the larger one. Take 0 i n 0 1 and move counter clockwise ....

D. Sleator, R. Tarjan, and W. Thurston. Rotation Distance, Triangulations, and Hyperbolic Geometry. J. Amer. Math. Soc. 1, 647-681, 1988.

.... vertex of any face f is incident to more than O(log r) triangles, and the intersection of any line with a face f meets no more than O(log r) triangles (in both cases, the triangles form a path in the dual tree) For a discussion on the relation between triangulations and binary trees, see, e.g. [16]. See also Figure 4 for an illustration of a tree corresponding to a balanced triangulation. Note that a balanced binary tree with n nodes corresponds to a triangulation of a convex polygon with n 2 vertices, having therefore n triangles. We choose an appropriate edge of the polygon to be the root ....

D.D. Sleator, R.E. Tarjan and W.P. Thurston, Rotation distance, triangulations, and hyperbolic geometry, J. Amer. Math. Soc. 1 (1988), pp. 647--681.

....induction. Lemma 2.1. A minimal triangulation T of an n cycle C consists of n Gamma 3 chords. It partitions C into n Gamma 2 triangles. Any two of these triangles are either disjoint or share a chord. Every chord in T is shared by exactly two triangles. The following lemma is well known (cf. [34] and the proof of Lemma 4.3, which is similar) Lemma 2.2. There is a 1 1 correspondence between the minimal triangulations of a cycle with l vertices and the binary trees with l Gamma 2 internal nodes. Denote by c l the l th Catalan number, i.e. c l = Gamma 2l Delta l 1 . Note that c ....

D. D. Sleator, R. E. Tarjan, and W. P. Thurston, Rotation distance, triangulations, and hyperbolic geometry, Journal of the AMS, 1 (1988), pp. 647--681.

....moves up to the line y = i and then moves right for j i steps. It is easy to see that this procedure gives a path in P n and that every such path comes from a distinct triangulation. These bijections are frequently exploited when studying the combinatorics in one of these sets (see especially [15]) and also for the task of randomly generating elements from one of the sets (e,g, 2] 5] 13] To understand what # n says about trees, imagine the diagonal v i v j in # as directed from the smaller numbered vertex of K to the larger one. Take 0 i n 1 and move counterclockwise along ....

D. Sleator, R. Tarjan, and W. Thurston. Rotation Distance, Triangulations, and Hyperbolic Geometry. J. Amer. Math. Soc. 1, 647-681, 1988. 12

.... by the MCS algorithm described in [31, 32] Using the algorithm described in [30] one can generate all minimal triangulations in O(jCj) time for each, observing that there exists a 1 1 correspondence between minimal triangulations of an l cycle and binary trees with l Gamma 2 internal nodes (cf. [28]) The nodes of this search tree which are actually traversed correspond to supergraphs of G with no more than k additional edges. If one such node is a leaf then we have found a k triangulation. Otherwise, no such triangulation exists. Theorem 2.2 All minimal k triangulations of a graph G can ....

D. D. Sleator, R. E. Tarjan, and W. P. Thurston. Rotation distance, triangulations, and hyperbolic geometry. Journal of the AMS, 1(3):647--681, 1988.

....by induction. Lemma 2.1 A minimal triangulation T of an n cycle C consists of n Gamma 3 chords. It partitions C into n Gamma 2 triangles. Any two of these triangles are either disjoint or share a chord. Every chord in T is shared by exactly two triangles. The following lemma is well known (cf. [33] and the proof of Lemma 4.3, which is similar) Lemma 2.2 There is a 1 1 correspondence between the minimal triangulations of a cycle with l vertices and the binary trees with l Gamma 2 internal nodes. Denote by c l the l th Catalan number, i.e. c l = Gamma 2l l Delta 1 l 1 . Note that ....

D. D. Sleator, R. E. Tarjan, and W. P. Thurston. Rotation distance, triangulations, and hyperbolic geometry. Journal of the AMS, 1(3):647--681, 1988.

No context found.

D. Sleator, R. Tarjan, W. Thurston, Rotation distance, triangulations, and hyperbolic geometry, J. Amer. Math. Soc., 1(1988), 647-681.

....this strategy crosses at most O(log jT j) geodesic triangles in a balanced geodesic triangulation. Our approach is to maintain a geodesic triangulation of polygon P so that its dual tree T is a balanced binary tree in particular, a red black tree [7, 13, 20, 25] Sleator, Tarjan, and Thurston [24] observe that, given a triangulation of a convex polygon P , then any two adjacent triangles 4uvw and 4wzu in this triangulation can be replaced by the triangles 4vwz and 4zuv, and such a diagonal swap corresponds to a rotation in the tree dual to this triangulation. We extend this result to ....

D. D. Sleator, R. E. Tarjan, and W. P. Thurston. Rotation distance, triangulations, and hyperbolic geometry. J. Amer. Math. Soc., 1:647--682, 1988.

No context found.

D. D. Sleator, R. E. Tarjan, and W. P. Thurston. Rotation distance, triangulations, and hyperbolic geometry. J. Amer. Math. Soc. 1(3):647--681, 1988.

No context found.

DD. Sleator, R.E. Tarjan, W.P. Thurston. Rotation Distance, triangulations, and hyperbolic geometry. Journal of AMS, 1,3,(1988), 647--681

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