Related papers: Degree Four Plane Spanners: Simpler and Better
In a geometric network G = (S, E), the graph distance between two vertices u, v in S is the length of the shortest path in G connecting u to v. The dilation of G is the maximum factor by which the graph distance of a pair of vertices…
$\renewcommand{\Re}{{\rm I\!\hspace{-0.025em} R}} \newcommand{\SetX}{\mathsf{X}} \newcommand{\eps}{\varepsilon} \newcommand{\VorX}[1]{\mathcal{V} \pth{#1}} \newcommand{\Polygon}{\mathsf{P}} \newcommand{\IntRange}[1]{[ #1 ]}…
Let $P$ be a finite point set in $\mathbb{R}^2$ with the set of distance $n$-chains defined as $$ \Delta_n(P)=\{(|p_1-p_2|,|p_2-p_3|,\ldots,|p_n-p_{n+1}|):p_i \in P\}.$$ We show that for $2\leq n=O_{|P|}(1)$ we have $$|\Delta_n(P)|\gtrsim…
In this paper, we give upper and lower bounds on the number of Steiner points required to construct a strictly convex quadrilateral mesh for a planar point set. In particular, we show that $3{\lfloor\frac{n}{2}\rfloor}$ internal Steiner…
Consider a set $P$ of $n$ points in $\mathbb{R}^d$. In the discrete median line segment problem, the objective is to find a line segment bounded by a pair of points in $P$ such that the sum of the Euclidean distances from $P$ to the line…
Refining a classical proof of Whitney, we show that any $4$-connected planar triangulation can be decomposed into a Hamiltonian path and two trees. Therefore, every $4$-connected planar graph decomposes into three forests, one having…
In this paper, we present approximate distance and shortest-path oracles for fault-tolerant Euclidean spanners motivated by the routing problem in real-world road networks. An $f$-fault-tolerant Euclidean $t$-spanner for a set $V$ of $n$…
Floor-planning is a fundamental step in VLSI chip design. Based upon the concept of orderly spanning trees, we present a simple O(n)-time algorithm to construct a floor-plan for any n-node plane triangulation. In comparison with previous…
The slope number of a graph $G$ is the smallest number of slopes needed for the segments representing the edges in any straight-line drawing of $G$. It serves as a measure of the visual complexity of a graph drawing. Several bounds on the…
We study spanners in planar domains, including polygonal domains, polyhedral terrain, and planar metrics. Previous work showed that for any constant $\epsilon\in (0,1)$, one could construct a $(2+\epsilon)$-spanner with $O(n\log(n))$ edges…
Inspired by the seminal works of Khuller et al. (STOC 1994) and Chan (SoCG 2003) we study the bottleneck version of the Euclidean bounded-degree spanning tree problem. A bottleneck spanning tree is a spanning tree whose largest edge-length…
Recently, there has been interest in representing single graphs by multiple drawings; for example, using graph stories, storyplans, or uncrossed collections. In this paper, we apply this idea to orthogonal graph drawing. Due to the…
A finite set $P$ of points in the plane is $n$-universal with respect to a class $\mathcal{C}$ of planar graphs if every $n$-vertex graph in $\mathcal{C}$ admits a crossing-free straight-line drawing with vertices at points of $P$. For the…
For a planar point set $P$, its convex hull is the smallest convex polygon that encloses all points in $P$. The construction of the convex hull from an array $I_P$ containing $P$ is a fundamental problem in computational geometry. By…
Let $P$ be a set of $n$ points in the plane. A crossing-free structure on $P$ is a plane graph with vertex set $P$. Examples of crossing-free structures include triangulations of $P$, spanning cycles of $P$, also known as polygonalizations…
In this work we consider triangulations of point sets in the Euclidean plane, i.e., maximal straight-line crossing-free graphs on a finite set of points. Given a triangulation of a point set, an edge flip is the operation of removing one…
The greedy spanner in a low dimensional Euclidean space is a fundamental geometric construction that has been extensively studied over three decades as it possesses the two most basic properties of a good spanner: constant maximum degree…
In the pairwise weighted spanner problem, the input consists of an $n$-vertex-directed graph, where each edge is assigned a cost and a length. Given $k$ vertex pairs and a distance constraint for each pair, the goal is to find a…
The Hilbert metric is a distance function defined for points lying within the interior of a convex body. It arises in the analysis and processing of convex bodies, machine learning, and quantum information theory. In this paper, we show how…
The $\textit{planar slope number}$ $psn(G)$ of a planar graph $G$ is the minimum number of edge slopes in a planar straight-line drawing of $G$. It is known that $psn(G) \in O(c^\Delta)$ for every planar graph $G$ of maximum degree…