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

Adolescents’ motivations to perpetrate hate speech and links with social norms

Hate speech has become a widespread phenomenon, however, it remains largely unclear why adolescents engage in it and which factors are associated with their motivations for perpetrating hate speech. To this end, we developed the multidimensional “Motivations for Hate Speech Perpetration Scale” (MHATE) and evaluated the psychometric properties. We also explored the associations between social norms and adolescents’ motivations for hate speech perpetration. The sample consisted of 346 adolescents from Switzerland (54.6% boys; Mage=14; SD=0.96) who reported engagement in hate speech as perpetrators. The analyses revealed good psychometric properties for the MHATE, including good internal consistency. The most frequently endorsed subscale was revenge, followed by ideology, group conformity, status enhancement, exhilaration, and power. The results also showed that descriptive norms and peer pressure were related to a wide range of different motivations for perpetrating hate speech. Injunctive norms, however, were only associated with power. In conclusion, findings indicate that hate speech fulfills various functions. We argue that knowing the specific motivations that underlie hate speech could help us derive individually tailored prevention strategies (e.g., anger management, promoting an inclusive classroom climate). Furthermore, we suggest that practitioners working in the field of hate speech prevention give special attention to social norms surrounding adolescents

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

Abstract. Given a set P of n labeled points in the plane, the radial system of P describes, for each p ∈ P , the radial ordering of the other points around p. This notion is related to the order type of P , which describes the orientation (clockwise or counterclockwise) of every ordered triple of P . Given only the order type of P , it is easy to reconstruct the radial system of P , but the converse is not true. Aichholzer et al. (Reconstructing Point Set Order Types from Radial Orderings, in Proc. ISAAC 2014) defined T (R) to be the set of order types with radial system R and showed that sometimes |T (R)| = n − 1. They give polynomial-time algorithms to compute T (R) when only given R. We describe an optimal O(n 2 ) time algorithm for computing T (R). The algorithm constructs the convex hulls of all possible point sets with the given radial system, after which sidedness queries on point triples can be answered in constant time. This set of convex hulls can be found in O(n) time. Our results generalize to abstract order types

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 ?(9.08?) triangulations. This is a significant improvement over the previous and long-standing lower bound of ?(8.65?) for the maximum number of triangulations of planar point sets

Counting and enumerating crossing-free geometric graphs

We describe a framework for constructing data structures which allow fast counting and enumeration of 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 represent geometric graphs as source-sink paths in a directed acyclic graph.The following 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 $c < 5.61804$.Moreover, after a preprocessing phase with the same time bounds as above, we can enumerate the respective classes efficiently. 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

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. These numbers were encountered during the investigation of what we call trapezoidal diagrams of geometric graphs, such as triangulations or crossing-free perfect matchings. 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) 5.196ⁿ and (ii) 23.459ⁿ (bases are rounded to three decimal places). Finally, we give exponential lower bounds for the maximum number of embeddings that a trapezoidal diagram can have on any given point set.ISSN:0179-5376ISSN:1432-044

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 $\Omega(9.08^n)$ triangulations. This is a significant improvement over the previous and long-standing lower bound of $\Omega(8.65^n)$ for the maximum number of triangulations of planar point sets