This paper describes a new algorithm for constructing the set of free bitangents of a collection of n disjoint convex obstacles of constant complexity. The algorithm runs in time O(n log n + k), where k is the output size, and uses O(n) space. While earlier algorithms achieve the same optimal running time, this is the first optimal algorithm that uses only linear space. The visibility graph or the visibility complex can be computed in the same time and space. The only complicated data structure used by the algorithm is a splittable queue, which can be implemented easily using red--black trees. The algorithm is conceptually very simple, and should therefore be easy to implement and quite fast in practice. The algorithm relies on greedy pseudotriangulations, which are subgraphs of the visibility graph with many nice combinatorial properties. These properties, and thus the correctness of the algorithm, are partially derived from properties of a certain partial order on the faces of th...

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.

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.

Abstract not found

The number of minimum pseudo-triangulations is minimized for point sets in convex position.

In this paper we present an algorithm to build a sensor-based, dynamic data structure useful for robot navigation in an unknown, multiply-connected planar environment. This data structure offers a robust framework for robot navigation, avoiding the need of a complete geometric map or explicit localization, by building a minimal representation based entirely on critical events in online sensor measurements made by the robot. There are two sensing requirements for the robot: it must detect when it is close to the walls, to perform wall-following reliably, and it must be able to detect discontinuities in depth information. It is also assumed that the robot is able to drop, detect and recover a marker. The navigation paths generated are optimal up to the homotopy class to which the paths belong, even though no distance information is measured.

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).)

We show that the segment endpoint visibility graph of any finite set of disjoint line segments in the plane admits a simple Hamiltonian polygon, if not all segments are collinear. This proves a conjecture of Mirzaian.

We pose a monotonicity conjecture on the number of pseudo-triangulations of any planar point set, and check it on two prominent families of point sets, namely the so-called double circle and double chain. The latter has asymptotically 12 n n Θ(1) pointed pseudo-triangulations, which lies significantly above the maximum number of triangulations in a planar point set known so far. ⋆ Parts of this work were done while the authors visited the Departament de

Abstract. We consider the problem of computing a minimum weight pseudo-triangulation of a set S of n points in the plane. We first present an O(n log n)-time algorithm that produces a pseudo-triangulation of weight O(log n·wt(M(S))) which is shown to be asymptotically worst-case optimal, i.e., there exists a point set S for which every pseudo-triangulation has weight Ω(log n · wt(M(S))), where wt(M(S)) is the weight of a minimum spanning tree of S. We also present a constant factor approximation algorithm running in cubic time. In the process we give an algorithm that produces a minimum weight pseudo-triangulation of a simple polygon. 1