Counting Triangulations and other Crossing-Free Structures Approximately

We consider the problem of counting straight-edge triangulations of a given set $P$ of $n$ points in the plane. Until very recently it was not known whether the exact number of triangulations of $P$ can be computed asymptotically faster than by enumerating all triangulations. We now know that the number of triangulations of $P$ can be computed in $O^{*}(2^{n})$ time, which is less than the lower bound of $\Omega(2.43^{n})$ on the number of triangulations of any point set. In this paper we address the question of whether one can approximately count triangulations in sub-exponential time. We present an algorithm with sub-exponential running time and sub-exponential approximation ratio, that is, denoting by $\Lambda$ the output of our algorithm, and by $c^{n}$ the exact number of triangulations of $P$, for some positive constant $c$, we prove that $c^{n}\leq\Lambda\leq c^{n}\cdot 2^{o(n)}$. This is the first algorithm that in sub-exponential time computes a $(1+o(1))$-approximation of the base of the number of triangulations, more precisely, $c\leq\Lambda^{\frac{1}{n}}\leq(1 + o(1))c$. Our algorithm can be adapted to approximately count other crossing-free structures on $P$, keeping the quality of approximation and running time intact. In this paper we show how to do this for matchings and spanning trees.Comment: 19 pages, 2 figures. A preliminary version appeared at CCCG 201

Multitriangulations, pseudotriangulations and primitive sorting networks

We study the set of all pseudoline arrangements with contact points which cover a given support. We define a natural notion of flip between these arrangements and study the graph of these flips. In particular, we provide an enumeration algorithm for arrangements with a given support, based on the properties of certain greedy pseudoline arrangements and on their connection with sorting networks. Both the running time per arrangement and the working space of our algorithm are polynomial. As the motivation for this work, we provide in this paper a new interpretation of both pseudotriangulations and multitriangulations in terms of pseudoline arrangements on specific supports. This interpretation explains their common properties and leads to a natural definition of multipseudotriangulations, which generalizes both. We study elementary properties of multipseudotriangulations and compare them to iterations of pseudotriangulations.Comment: 60 pages, 40 figures; minor corrections and improvements of presentatio

The Zigzag Path of a Pseudo-Triangulation

We define the zigzag path of a pseudo-triangulation, a concept generalizing the path of a triangulation of a point set. The pseudotriangulation zigzag path allows us to use divide-and-conquer type of approaches for suitable (i.e., decomposable) problems on pseudo-triangulations. For this we provide an algorithm that enumerates all pseudotriangulation zigzag paths (of all pseudo-triangulations of a given point set with respect to a given line) in O(n2) time per path and O(n2) space, where n is the number of points. We illustrate applications of our scheme which include a novel algorithm to count the number of pseudotriangulations of a point set. © Springer-Verlag Berlin Heidelberg 2003

Selected topics in algorithmic geometry

Let P be a set of n points on the plane with no three points on a line. A crossing-free structure on P is a straight-edge plane graph whose vertex set is P. In this thesis we consider problems of two different topics in the area of algorithmic geometry: Geometry using Steiner points, and counting algorithms. These topics have certain crossing-free structures on P as our primary objects of study. Our results can roughly be described as follows: i) Given a k-coloring of P, with k >= 3 colors, we will show how to construct a set of Steiner points S = S(P) such that a k-colored quadrangulation can always be constructed on (P U S). The bound we show of |S| significantly improves on previously known results. ii) We also show how to construct a se S = S(P) of Steiner points such that a triangulation of (P U S) having all its vertices of even (odd) degree can always be constructed. We show that |S| <= n/3 + c, where c is a constant. We also look at other variants of this problem. iii) With respect to counting algorithms, we show new algorithms for counting triangulations, pseudo-triangulations, crossing-free matchings and crossing-free spanning cycles on P. Our algorithms are simple and allow good analysis of their running times. These algorithms significantly improve over previously known results. We also show an algorithm that counts triangulations approximately, and a hardness result of a particular instance of the problem of counting triangulations exactly. iv) We show experiments comparing our algorithms for counting triangulations with another well-known algorithm that is supposed to be very fast in practice.Sei P eine Menge von n Punkte in der Ebene, so dass keine drei Punkten auf einer Geraden liegen. Eine kreuzungsfreie Struktur von P ist ein geradliniger ebener Graph, der P als Knotenmenge hat. In dieser Dissertation behandeln wir zwei verschiedene Problemkreise auf dem Gebiet der algorithmischen Geometrie: Geometrie mit Steinerpunkten und Anzahl bestimmende Algorithmen auf P und auf gewissen kreuzungsfreien Strukturen von P. Unsere Resultate können wie folgt beschrieben werden: i) Gegeben sei eine k-Färbung von P, mit k >= 3 Farben. Es wird gezeigt, wie eine Menge S = S(P) von Steiner Punkten konstruiert werden kann, die die Konstruktion einer k-gefärbten Quadrangulierung von (P U S) ermöglicht. Die von uns gezeigte Schranke für |S| verbessert die bisher bekannte Schranke. ii) Gezeigt wird auch die Konstruktion einer Menge S = S(P) von Steiner Punkten, so dass eine Triangulierung von (P U S) konstruiert werden kann, bei der der Grad aller Knoten gerade (ungerade) ist. Wir zeigen, dass |S| <= n/3 + c möglich ist, wobei c eine Konstante ist. Wir betrachten auch andere Varianten dieses Problems. iii) Was die Anzahl bestimmenden Algorithmen betrifft, zeigen wir neue Algorithmen, um Triangulierungen, Pseudotriangulierungen, kreuzungsfreie Matchings und kreuzungsfreie aufspannende Zyklen von P zu zählen. Unsere Algorithmen sind einfach und lassen eine gute Analyse der Laufzeiten zu. Diese neuen Algorithmen verbessern wesentlich die bisherigen Ergebnisse. Weiter zeigen wir einen Algorithmus, der Triangulierungen approximativ zählt, und bestimmen die Komplexitätsklasse einer bestimmten Variante des Problems des exakten Zählens von Triangulierungen. iv) Wir zeigen Experimente, die unsere triangulierungszählenden Algorithmen mit einem anderen bekannten Algorithmus vergleichen, der in der Praxis als besonders schnell bekannt ist

Non-crossing frameworks with non-crossing reciprocals

We study non-crossing frameworks in the plane for which the classical reciprocal on the dual graph is also non-crossing. We give a complete description of the self-stresses on non-crossing frameworks whose reciprocals are non-crossing, in terms of: the types of faces (only pseudo-triangles and pseudo-quadrangles are allowed); the sign patterns in the self-stress; and a geometric condition on the stress vectors at some of the vertices. As in other recent papers where the interplay of non-crossingness and rigidity of straight-line plane graphs is studied, pseudo-triangulations show up as objects of special interest. For example, it is known that all planar Laman circuits can be embedded as a pseudo-triangulation with one non-pointed vertex. We show that if such an embedding is sufficiently generic, then the reciprocal is non-crossing and again a pseudo-triangulation embedding of a planar Laman circuit. For a singular (i.e., non-generic) pseudo-triangulation embedding of a planar Laman circuit, the reciprocal is still non-crossing and a pseudo-triangulation, but its underlying graph may not be a Laman circuit. Moreover, all the pseudo-triangulations which admit a non-crossing reciprocal arise as the reciprocals of such, possibly singular, stresses on pseudo-triangulation embeddings of Laman circuits. All self-stresses on a planar graph correspond to liftings to piece-wise linear surfaces in 3-space. We prove characteristic geometric properties of the lifts of such non-crossing reciprocal pairs.Comment: 32 pages, 23 figure

Pseudo-Triangulations, Rigidity and Motion Planning

This paper proposes a combinatorial approach to planning non-colliding trajectories for a polygonal bar-and-joint framework with n vertices. It is based on a new class of simple motions induced by expansive one-degree-of-freedom mechanisms, which guarantee noncollisions by moving all points away from each other. Their combinatorial structure is captured by pointed pseudo-triangulations, a class of embedded planar graphs for which we give several equivalent characterizations and exhibit rich rigidity theoretic properties. The main application is an efficient algorithm for the Carpenter\u27s Rule Problem: convexify a simple bar-and-joint planar polygonal linkage using only non-self-intersecting planar motions. A step of the algorithm consists in moving a pseudo-triangulation-based mechanism along its unique trajectory in configuration space until two adjacent edges align. At the alignment event, a local alteration restores the pseudo-triangulation. The motion continues for O(n3) steps until all the points are in convex position. © 2005 Springer Science+Business Media, Inc

Exploring Euclidean Dynamical Triangulations with a Non-trivial Measure Term

We investigate a nonperturbative formulation of quantum gravity defined via Euclidean dynamical triangulations (EDT) with a non-trivial measure term in the path integral. We are motivated to revisit this older formulation of dynamical triangulations by hints from renormalization group approaches that gravity may be asymptotically safe and by the emergence of a semiclassical phase in causal dynamical triangulations (CDT). We study the phase diagram of this model and identify the two phases that are well known from previous work: the branched polymer phase and the collapsed phase. We verify that the order of the phase transition dividing the branched polymer phase from the collapsed phase is almost certainly first-order. The nontrivial measure term enlarges the phase diagram, allowing us to explore a region of the phase diagram that has been dubbed the crinkled region. Although the collapsed and branched polymer phases have been studied extensively in the literature, the crinkled region has not received the same scrutiny. We find that the crinkled region is likely a part of the collapsed phase with particularly large finite-size effects. Intriguingly, the behavior of the spectral dimension in the crinkled region at small volumes is similar to that of CDT, as first reported in arXiv:1104.5505, but for sufficiently large volumes the crinkled region does not appear to have 4-dimensional semiclassical features. Thus, we find that the crinkled region of the EDT formulation does not share the good features of the extended phase of CDT, as we first suggested in arXiv:1104.5505. This agrees with the recent results of arXiv:1307.2270, in which the authors used a somewhat different discretization of EDT from the one presented here.Comment: 36 pages, 27 figures. Typos corrected, improved analysis of phase transition, and clarifications added. Conclusions unchanged. Conforms with version published in JHE

The greedy flip tree of a subword complex

We describe a canonical spanning tree of the ridge graph of a subword complex on a finite Coxeter group. It is based on properties of greedy facets in subword complexes, defined and studied in this paper. Searching this tree yields an enumeration scheme for the facets of the subword complex. This algorithm extends the greedy flip algorithm for pointed pseudotriangulations of points or convex bodies in the plane.Comment: 14 pages, 10 figures; various corrections (in particular deletion of Section 4 which contained a serious mistake pointed out by an anonymous referee). This paper is subsumed by our joint results with Christian Stump on "EL-labelings and canonical spanning trees for subword complexes" (http://arxiv.org/abs/1210.1435) and will therefore not be publishe