arXiv subjects

Manuel Wettstein

Publications and source records attributed to Manuel Wettstein.

7 recordsLinked to original sources

arXiv2023

Chains, Koch Chains, and Point Sets with many Triangulations

We introduce the abstract notion of a chain, which is a sequence of $n$ points in the plane, ordered by $x$-coordinates, so that the edge between any two consecutive points is unavoidable as far as triangulations are concerned. A general theory of the structural properties of chains is developed, alongside a general understanding of their number of triangulations. We also describe an intriguing new and concrete configuration, which we call the Koch chain due to its similarities to the Koch curve. A specific construction based on Koch chains is then shown to have $Ω(9.08^n)$ triangulations. This is a significant improvement over the previous and long-standing lower bound of $Ω(8.65^n)$ for the maximum number of triangulations of planar point sets.

cs.CG↗

arXiv2023

A Note on the Faces of the Dual Koch Arrangement

We analyze the faces of the dual Koch arrangement, which is the arrangement of $2^s + 1$ lines obtained by projective duality from the Koch chain $K_s$. In particular, we show that this line arrangement does not contain any $k$-gons for $k > 5$, and that the number of pentagons is $3 \cdot 2^{s-1} - 3$.

cs.CG↗

arXiv2019

From Crossing-Free Graphs on Wheel Sets to Embracing Simplices and Polytopes with Few Vertices

A set $P = H \cup \{w\}$ of $n+1$ points in general position in the plane is called a wheel set if all points but $w$ are extreme. We show that for the purpose of counting crossing-free geometric graphs on such a set $P$, it suffices to know the frequency vector of $P$. While there are roughly $2^n$ distinct order types that correspond to wheel sets, the number of frequency vectors is only about $2^{n/2}$. We give simple formulas in terms of the frequency vector for the number of crossing-free spanning cycles, matchings, triangulations, and many more. Based on that, the corresponding numbers of graphs can be computed efficiently. In particular, we rediscover an already known formula for $w$-embracing triangles spanned by $H$. Also in higher dimensions, wheel sets turn out to be a suitable model to approach the problem of computing the simplicial depth of a point $w$ in a set $H$, i.e., the number of $w$-embracing simplices. While our previous arguments in the plane do not generalize easily, we show how to use similar ideas in $\mathbb{R}^d$ for any fixed $d$. The result is an $O(n^{d-1})$ time algorithm for computing the simplicial depth of a point $w$ in a set $H$ of $n$ points, improving on the previously best bound of $O(n^d\log n)$. Based on our result about simplicial depth, we can compute the number of facets of the convex hull of $n=d+k$ points in general position in $\mathbb{R}^d$ in time $O(n^{\max\{ω,k-2\}})$ where $ω\approx 2.373$, even though the asymptotic number of facets may be as large as $n^k$.

cs.CG↗

arXiv2016

Arc diagrams, flip distances, and Hamiltonian triangulations

We show that every triangulation (maximal planar graph) on $n\ge 6$ vertices can be flipped into a Hamiltonian triangulation using a sequence of less than $n/2$ combinatorial edge flips. The previously best upper bound uses $4$-connectivity as a means to establish Hamiltonicity. But in general about $3n/5$ flips are necessary to reach a $4$-connected triangulation. Our result improves the upper bound on the diameter of the flip graph of combinatorial triangulations on $n$ vertices from $5.2n-33.6$ to $5n-23$. We also show that for every triangulation on $n$ vertices there is a simultaneous flip of less than $2n/3$ edges to a $4$-connected triangulation. The bound on the number of edges is tight, up to an additive constant. As another application we show that every planar graph on $n$ vertices admits an arc diagram with less than $n/2$ biarcs, that is, after subdividing less than $n/2$ (of potentially $3n-6$) edges the resulting graph admits a $2$-page book embedding.

cs.CG↗

arXiv2016

An Optimal Algorithm for Reconstructing Point Set Order Types from Radial Orderings

Let $P$ be a set of $n$ labeled points in the plane. The radial system of $P$ describes, for each $p\in P$, the order in which a ray that rotates around $p$ encounters the points in $P \setminus \{p\}$. This notion is related to the order type of $P$, which describes the orientation (clockwise or counterclockwise) of every ordered triple in $P$. Given only the order type, the radial system is uniquely determined and can easily be obtained. The converse, however, is not true. Indeed, let $R$ be the radial system of $P$, and let $T(R)$ be the set of all order types with radial system $R$ (we define $T(R) = \emptyset$ for the case that $R$ is not a valid radial system). Aichholzer et al. (Reconstructing Point Set Order Types from Radial Orderings, in ISAAC 2014) show that $T(R)$ may contain up to $n-1$ order types. They also provide polynomial-time algorithms to compute $T(R)$ when only $R$ is given. We describe a new algorithm for finding $T(R)$. The algorithm constructs the convex hulls of all possible point sets with the radial system $R$. After that, orientation queries on point triples can be answered in constant time. A representation of this set of convex hulls can be found in $O(n)$ queries to the radial system, using $O(n)$ additional processing time. This is optimal. Our results also generalize to abstract order types.

cs.CG↗

arXiv2016

Counting and Enumerating Crossing-free Geometric Graphs

We describe a framework for counting and enumerating various types of crossing-free geometric graphs on a planar point set. The framework generalizes ideas of Alvarez and Seidel, who used them to count triangulations in time $O(2^nn^2)$ where $n$ is the number of points. The main idea is to reduce the problem of counting geometric graphs to counting source-sink paths in a directed acyclic graph. The following new results will emerge. The number of all crossing-free geometric graphs can be computed in time $O(c^nn^4)$ for some $c < 2.83929$. The number of crossing-free convex partitions can be computed in time $O(2^nn^4)$. The number of crossing-free perfect matchings can be computed in time $O(2^nn^4)$. The number of convex subdivisions can be computed in time $O(2^nn^4)$. The number of crossing-free spanning trees can be computed in time $O(c^nn^4)$ for some $c < 7.04313$. The number of crossing-free spanning cycles can be computed in time $O(c^nn^4)$ for some $c < 5.61804$. With the same bounds on the running time we can construct data structures which allow fast enumeration of the respective classes. For example, after $O(2^nn^4)$ time of preprocessing we can enumerate the set of all crossing-free perfect matchings using polynomial time per enumerated object. For crossing-free perfect matchings and convex partitions we further obtain enumeration algorithms where the time delay for each (in particular, the first) output is bounded by a polynomial in $n$. All described algorithms are comparatively simple, both in terms of their analysis and implementation.

cs.CG↗

arXiv2016

Trapezoidal Diagrams, Upward Triangulations, and Prime Catalan Numbers

The d-dimensional Catalan numbers form a well-known sequence of numbers which count balanced bracket expressions over an alphabet of size d. In this paper, we introduce and study what we call d-dimensional prime Catalan numbers, a sequence of numbers which count only a very specific subset of indecomposable balanced bracket expressions. We further introduce the notion of a trapezoidal diagram of a crossing-free geometric graph, such as a triangulation or a crossing-free perfect matching. In essence, such a diagram is obtained by augmenting the geometric graph in question with its trapezoidal decomposition, and then forgetting about the precise coordinates of individual vertices while preserving the vertical visibility relations between vertices and segments. We note that trapezoidal diagrams of triangulations are closely related to abstract upward triangulations. We study the numbers of such diagrams in the cases of (i) perfect matchings and (ii) triangulations. We give bijective proofs which establish relations with 3-dimensional (prime) Catalan numbers. This allows us to determine the corresponding exponential growth rates exactly as (i) 5.196^n and (ii) 23.459^n (bases are rounded to 3 decimal places). Finally, we give lower bounds for the maximum number of embeddings of a trapezoidal diagram on any given point set.

cs.CG↗