Geodesic Spanners for Points in $\mathbb{R}^3$ amid Axis-parallel Boxes

Mohammad Ali Abam; Mohammad Javad Rezaei Seraji

Home/Archive/arXiv

arXiv · 2004.01599

Geodesic Spanners for Points in $\mathbb{R}^3$ amid Axis-parallel Boxes

Mohammad Ali Abam·Mohammad Javad Rezaei Seraji

View original source ↗Read source document ↗Export metadata ↓

Abstract

Let $P$ be a set of $n$ points in $\mathbb{R}^3$ amid a bounded number of obstacles. When obstacles are axis-parallel boxes, we prove that $P$ admits an $8\sqrt{3}$-spanner with $O(n\log^3 n)$ edges with respect to the geodesic distance.

Explore related subjects

cs.CG

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Mohammad Ali Abam, Mohammad Javad Rezaei Seraji. 2020-08-08. Geodesic Spanners for Points in $\mathbb{R}^3$ amid Axis-parallel Boxes. https://arxiv.org/abs/2004.01599

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

k-Planar and Fan-Crossing Drawings and Transductions of Embeddable Graphs

We introduce, for every surface $Σ$, a two-way connection between definability of a graph class $\mathcal C$ by FO transductions (first-order logical transformations) of the graphs embeddable in $Σ$ and a certain variant of fan-crossing drawings of the graphs from $\mathcal C$ in $Σ$. If $\mathcal C$ is additionally of bounded maximum degree, then the restriction on drawings of the graphs from $\mathcal C$ in $Σ$ is simply to have a bounded number of crossings per edge (such as being $k$-planar for fixed~$k$ if $Σ$ is the plane). For graph classes, this connection allows us to derive non-transducibility results from the nonexistence of the said drawings and, conversely, from the nonexistence of a transduction to derive nonexistence of the said drawings. One example of such reasoning is as follows; since the class of 3D-grids is not transducible from the class of planar graphs, we can conclude that the class of 3D-grids is not $k$-planar for any fixed~$k$. On the other hand, the fact that the class of 3D-grids is not $k$-planar for any fixed~$k$ is known also via other means, and this conversely implies that the class of 3D-grids is not transducible from the class of planar graphs. We hope that this connection will help to draw a path to a possible proof that not all toroidal graphs are transducible from planar graphs. The result is based on a recent characterization of weakly sparse FO transductions of classes of bounded expansion by [Gajarský, Gładkowski, Jedelský, Pilipczuk and Toruńczyk, arXiv:2505.15655].

cs.CG↗

arXiv2026

Low-distortion planar embedding of rod-based structures

Rod-based structures are commonly used in practical applications in science and engineering. However, in many design, analysis, and manufacturing tasks, handling the rod-based structures in three dimensions directly is generally challenging. To simplify the tasks, it is usually more desirable to achieve a two-dimensional representation of the rod-based structures via some suitable geometric mappings. In this work, we develop a novel method for computing a low-distortion planar embedding of rod-based structures. Specifically, we identify geometrical constraints that aim to preserve key length and angle quantities of the 3D rod-based structures and prevent the occurrence of overlapping rods in the planar embedding. Experimental results with a variety of rod-based structures are presented to demonstrate the effectiveness of our approach. Moreover, our method can be naturally extended to the design and mapping of hybrid structures consisting of both rods and surface elements. Altogether, our approach paves a new way for the efficient design and fabrication of novel three-dimensional geometric structures for practical applications.

cs.CG↗

arXiv2026

A new O(n log n) approach for the Euclidean maximum weight matching problem

In a weighted graph $G = (V, E)$, the maximum weight matching problem (MWM) asks for a matching (i.e. pairing) of its vertices, such that each vertex is paired with at most one other vertex and the total sum of weights of all edges connecting paired vertices is maximised. If the vertices of the graph correspond to points in the Euclidean plane and the weights to their pairwise Euclidean distances, we get the Euclidean maximum weight matching problem (Euclidean MWM). The best optimum-solution algorithm for this problem runs in $O(n^{2.5})$. Furthermore, there exists an FPTAS guaranteeing a $(1 - ε)$-approximation ratio, which runs in $O(m ε^{-1} \log ε^{-1})$ time. Heuristics with a subquadratic running time (with respect to the number of vertices $|V|$) are known, but often yield solutions of a modest quality. In this paper, we present a novel algorithm for solving the Euclidean MWM running in $O(n \log n)$ time and providing excellent solution quality, especially for larger instances. In particular, in our computational tests, the algorithm yielded optimum or near-optimum solutions for all test instances; the worst observed optimality gap was less than $1.38\%$. This makes the algorithm highly attractive for practical applications, especially when exact methods become computationally prohibitive due to the size of the instance.

cs.CG↗