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Expansive motions and the polytope of pointed pseudotriangulations
 Discrete and Computational Geometry  The GoodmanPollack Festschrift, Algorithms and Combinatorics
, 2003
"... We introduce the polytope of pointed pseudotriangulations of a point set in the plane, defined as the polytope of infinitesimal expansive motions of the points subject to certain constraints on the increase of their distances. Its 1skeleton is the graph whose vertices are the pointed pseudotriang ..."
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Cited by 45 (15 self)
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We introduce the polytope of pointed pseudotriangulations of a point set in the plane, defined as the polytope of infinitesimal expansive motions of the points subject to certain constraints on the increase of their distances. Its 1skeleton is the graph whose vertices are the pointed pseudotriangulations of the point set and whose edges are flips of interior pseudotriangulation edges. For points in convex position we obtain a new realization of the associahedron, i.e., a geometric representation of the set of triangulations of an ngon, or of the set of binary trees on n vertices, or of many other combinatorial objects that are counted by the Catalan numbers. By considering the 1dimensional version of the polytope of constrained expansive motions we obtain a second distinct realization of the associahedron as a perturbation of the positive cell in a Coxeter arrangement. Our methods produce as a byproduct a new proof that every simple polygon or polygonal arc in the plane has expansive motions, a key step in the proofs of
Tight degree bounds for pseudotriangulations of points
, 2003
"... We show that every set of n points in general position has a minimum pseudotriangulation whose maximum vertex degree is five. In addition, we demonstrate that every point set in general position has a minimum pseudotriangulation whose maximum face degree is four (i.e., each interior face of this p ..."
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Cited by 33 (11 self)
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We show that every set of n points in general position has a minimum pseudotriangulation whose maximum vertex degree is five. In addition, we demonstrate that every point set in general position has a minimum pseudotriangulation whose maximum face degree is four (i.e., each interior face of this pseudotriangulation has at most four vertices). Both degree bounds are tight. Minimum pseudotriangulations realizing these bounds (individually but not jointly) can be constructed in O(n log n) time.
Efficiently approximating polygonal paths in three and higher dimensions
 Algorithmica
, 1998
"... Abstract. We present efficient algorithms for solving polygonalpath approximation problems in three and higher dimensions. Given an nvertex polygonal curve P in R d, d ≥ 3, we approximate P by another polygonal curve P ′ of m ≤ n vertices in R d such that the vertex sequence of P ′ is an ordered s ..."
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Cited by 23 (5 self)
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Abstract. We present efficient algorithms for solving polygonalpath approximation problems in three and higher dimensions. Given an nvertex polygonal curve P in R d, d ≥ 3, we approximate P by another polygonal curve P ′ of m ≤ n vertices in R d such that the vertex sequence of P ′ is an ordered subsequence of the vertices of P. The goal is either to minimize the size m of P ′ for a given error tolerance ε (called the min # problem), or to minimize the deviation error ε between P and P ′ for a given size m of P ′ (called the minε problem). Our techniques enable us to develop efficient nearquadratictime algorithms in three dimensions and subcubictime algorithms in four dimensions for solving the min # and minε problems. We discuss extensions of our solutions to ddimensional space, where d> 4, and for the L1 and L∞ metrics. Key Words. Curve approximation, Parametric searching. 1. Introduction. In
Counting Triangulations and PseudoTriangulations of Wheels
 IN PROC. 13TH CANAD. CONF. COMPUT. GEOM
, 2001
"... Motivated by several open questions on triangulations and pseudotriangulations, we give closed form expressions for the number of triangulations and the number of minimum pseudotriangulations of n points in wheel configurations, that is, with n  1 in convex position. Although the numbers of trian ..."
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Cited by 21 (5 self)
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Motivated by several open questions on triangulations and pseudotriangulations, we give closed form expressions for the number of triangulations and the number of minimum pseudotriangulations of n points in wheel configurations, that is, with n  1 in convex position. Although the numbers of triangulations and pseudotriangulations vary depending on the placement of the interior point, their difference is always the (n2)nd Catalan number. We also prove an inequality #PT # 3 i #T for the numbers of minimum pseudotriangulations and triangulations of any point configuration with i interior points.
The zigzag path of a pseudotriangulation
 In Proc. 8th International Workshop on Algorithms and Data Structures (WADS
, 2003
"... We define the path of a pseudotriangulation, a data structure generalizing the path of a triangulation of a point set. This structure allows us to use divideandconquer type of approaches for suitable (i.e. decomposable) problems on pseudotriangulations. We illustrate this method by presenting a n ..."
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Cited by 16 (5 self)
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We define the path of a pseudotriangulation, a data structure generalizing the path of a triangulation of a point set. This structure allows us to use divideandconquer type of approaches for suitable (i.e. decomposable) problems on pseudotriangulations. We illustrate this method by presenting a novel algorithm that counts the number of pseudotriangulations of a point set. 1
Biased Skip Lists
 Algorithmica
, 2004
"... We design a variation of skip lists that performs well for generally biased access sequences. ..."
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Cited by 13 (1 self)
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We design a variation of skip lists that performs well for generally biased access sequences.
PseudoTriangulations  a Survey
 CONTEMPORARY MATHEMATICS
"... A pseudotriangle is a simple polygon with exactly three convex vertices, and a pseudotriangulation is a facetoface tiling of a planar region into pseudotriangles. Pseudotriangulations appear as data structures in computational geometry, as planar barandjoint frameworks in rigidity theory an ..."
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Cited by 13 (4 self)
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A pseudotriangle is a simple polygon with exactly three convex vertices, and a pseudotriangulation is a facetoface tiling of a planar region into pseudotriangles. Pseudotriangulations appear as data structures in computational geometry, as planar barandjoint frameworks in rigidity theory and as projections of locally convex surfaces. This survey of current literature includes combinatorial properties and counting of special classes, rigidity theoretical results, representations as polytopes, straightline drawings from abstract versions called combinatorial pseudotriangulations, algorithms and applications of pseudotriangulations.
The polytope of noncrossing graphs on a planar point set
, 2003
"... For any finite set A of n points in R 2, we define a (3n − 3)dimensional simple polyhedron whose face poset is isomorphic to the poset of “noncrossing marked graphs” with vertex set A, where a marked graph is defined as a geometric graph together with a subset of its vertices. The poset of noncr ..."
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Cited by 11 (5 self)
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For any finite set A of n points in R 2, we define a (3n − 3)dimensional simple polyhedron whose face poset is isomorphic to the poset of “noncrossing marked graphs” with vertex set A, where a marked graph is defined as a geometric graph together with a subset of its vertices. The poset of noncrossing graphs on A appears as the complement of the star of a face in that polyhedron. The polyhedron has a unique maximal bounded face, of dimension 2ni + n − 3 where ni is the number of points of A in the interior of conv(A). The vertices of this polytope are all the pseudotriangulations of A, and the edges are flips of two types: the traditional diagonal flips (in pseudotriangulations) and the removal or insertion of a single edge. As a byproduct of our construction we prove that all pseudotriangulations are infinitesimally rigid graphs.
A Sum of Squares Theorem for Visibility Complexes and Applications
, 2001
"... We present a new method to implement in constant amortized time the ip operation of the socalled Greedy Flip Algorithm, an optimal algorithm to compute the visibility graph or the visibility complex of a collection of pairwise disjoint bounded convex sets of constant complexity. The method uses sim ..."
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Cited by 11 (1 self)
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We present a new method to implement in constant amortized time the ip operation of the socalled Greedy Flip Algorithm, an optimal algorithm to compute the visibility graph or the visibility complex of a collection of pairwise disjoint bounded convex sets of constant complexity. The method uses simple data structures and only the leftturn or counterclockwise predicate; it relies, among other things, on a sum of squares like theorem for visibility complexes stated and proved in this paper. (The sum of squares theorem for a simple arrangement of lines states that the average value of the square of the number of vertices of a face of the arrangement is a O(1).)