We introduce the polytope of pointed pseudo-triangulations 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 1-skeleton is the graph whose vertices are the pointed pseudo-triangulations of the point set and whose edges are flips of interior pseudo-triangulation 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 n-gon, 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 1-dimensional 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 by-product a new proof that every simple polygon or polygonal arc in the plane has expansive motions, a key step in the proofs of

We show that every set of n points in general position has a minimum pseudo-triangulation whose maximum vertex degree is five. In addition, we demonstrate that every point set in general position has a minimum pseudo-triangulation whose maximum face degree is four (i.e., each interior face of this pseudo-triangulation has at most four vertices). Both degree bounds are tight. Minimum pseudo-triangulations realizing these bounds (individually but not jointly) can be constructed in O(n log n) time.

Abstract. We present efficient algorithms for solving polygonal-path approximation problems in three and higher dimensions. Given an n-vertex 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 min-ε problem). Our techniques enable us to develop efficient near-quadratic-time algorithms in three dimensions and subcubic-time algorithms in four dimensions for solving the min- # and min-ε problems. We discuss extensions of our solutions to d-dimensional space, where d> 4, and for the L1 and L∞ metrics. Key Words. Curve approximation, Parametric searching. 1. Introduction. In

Motivated by several open questions on triangulations and pseudo-triangulations, we give closed form expressions for the number of triangulations and the number of minimum pseudo-triangulations of n points in wheel configurations, that is, with n - 1 in convex position. Although the numbers of triangulations and pseudo-triangulations vary depending on the placement of the interior point, their difference is always the (n-2)nd Catalan number. We also prove an inequality #PT # 3 i #T for the numbers of minimum pseudo-triangulations and triangulations of any point configuration with i interior points.

We define the path of a pseudo-triangulation, a data structure generalizing the path of a triangulation of a point set. This structure allows us to use divide-andconquer type of approaches for suitable (i.e. decomposable) problems on pseudo-triangulations. We illustrate this method by presenting a novel algorithm that counts the number of pseudo-triangulations of a point set. 1

Abstract not found

Abstract. 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 “non-crossing 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 non-crossing 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 pseudo-triangulations of A, and the edges are flips of two types: the traditional diagonal flips (in pseudo-triangulations) and the removal or insertion of a single edge. As a by-product of our construction we prove that all pseudo-triangulations are infinitesimally rigid graphs. 1.

We design a variation of skip lists that performs well for generally biased access sequences.

We present a new method to implement in constant amortized time the ip operation of the so-called 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 left-turn 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).)

this paper, the "carpenter's ruler conjecture" of Lenhart-Whitesidesand Mitchell has been proved ([2, 5]): Any polygonal linkage with fixed length links and hinged joints, can be straightened while maintaining simplicity (without the linkage crossing it- A fuller version of this paper draft is available at http:// www.ams.sunysb.edu/jsbm/paperclips.ps.gz. Research supported by HRL Labs, NSF, NASA, Northrop-Grumman, ONR, Sandia, Sun Microsystems