arXiv subjects

Gabriela Majewska

Publications and source records attributed to Gabriela Majewska.

3 recordsLinked to original sources

arXiv2015

$β$-skeletons for a set of line segments in $R^2$

$β$-skeletons are well-known neighborhood graphs for a set of points. We extend this notion to sets of line segments in the Euclidean plane and present algorithms computing such skeletons for the entire range of $β$ values. The main reason of such extension is the possibility to study $β$-skeletons for points moving along given line segments. We show that relations between $β$-skeletons for $β> 1$, $1$-skeleton (Gabriel Graph), and the Delaunay triangulation for sets of points hold also for sets of segments. We present algorithms for computing circle-based and lune-based $β$-skeletons. We describe an algorithm that for $β\geq 1$ computes the $β$-skeleton for a set $S$ of $n$ segments in the Euclidean plane in $O(n^2 α(n) \log n)$ time in the circle-based case and in $O(n^2 λ_4(n))$ in the lune-based one, where the construction relies on the Delaunay triangulation for $S$, $α$ is a functional inverse of Ackermann function and $λ_4(n)$ denotes the maximum possible length of a $(n,4)$ Davenport-Schinzel sequence. When $0 < β< 1$, the $β$-skeleton can be constructed in a $O(n^3 λ_4(n))$ time. In the special case of $β= 1$, which is a generalization of Gabriel Graph, the construction can be carried out in a $O(n \log n)$ time.

cs.CG↗

arXiv2014

Generalized $β$-skeletons

$β$-skeletons, a prominent member of the neighborhood graph family, have interesting geometric properties and various applications ranging from geographic networks to archeology. This paper focuses on developing a new, more general than the present one, definition of $β$-skeletons based only on the distance criterion. It allows us to consider them in many different cases, e.g. for weighted graphs or objects other than points. Two types of $β$-skeletons are especially well-known: the Gabriel Graph (for $β= 1$) and the Relative Neighborhood Graph (for $β= 2$). The new definition retains relations between those graphs and the other well-known ones (minimum spanning tree and Delaunay triangulation). We also show several new algorithms finding $β$-skeletons.

cs.CG↗

arXiv2014

Multidimensional $β$-skeletons in $L_1$ and $L_{\infty}$ metric

The $β$-skeleton $\{G_β(V)\}$ for a point set V is a family of geometric graphs, defined by the notion of neighborhoods parameterized by real number $0 < β< \infty$. By using the distance-based version definition of $β$-skeletons we study those graphs for a set of points in $\mathbb{R}^d$ space with $l_1$ and $l_{\infty}$ metrics. We present algorithms for the entire spectrum of $β$ values and we discuss properties of lens-based and circle-based $β$-skeletons in those metrics. Let $V \in \mathbb{R}^d$ in $L_{\infty}$ metric be a set of $n$ points in general position. Then, for $β<2$ lens-based $β$-skeleton $G_β(V)$ can be computed in $O(n^2 \log^d n)$ time. For $β\geq 2$ there exists an $O(n \log^{d-1} n)$ time algorithm that constructs $β$-skeleton for the set $V$. We show that in $\mathbb{R}^d$ with $L_{\infty}$ metric, for $β<2$ $β$-skeleton $G_β(V)$ for $n$ points can be computed in $O(n^2 \log^d n)$ time. For $β\geq 2$ there exists an $O(n \log^{d-1} n)$ time algorithm. In $\mathbb{R}^d$ with $L_1$ metric for a set of $n$ points in arbitrary position $β$-skeleton $G_β(V)$ can be computed in $O(n^2 \log^{d+2} n)$ time.

cs.CG↗