Covering the Boundary of a Simple Polygon with Geodesic Unit Disks

We consider the problem of covering the boundary of a simple polygon on n vertices using the minimum number of geodesic unit disks. We present an O(n \log^2 n+k) time 2-approximation algorithm for finding the centers of the disks, with k denoting the number centers found by the algorithm

Detecting Weakly Simple Polygons

A closed curve in the plane is weakly simple if it is the limit (in the Fr\'echet metric) of a sequence of simple closed curves. We describe an algorithm to determine whether a closed walk of length n in a simple plane graph is weakly simple in O(n log n) time, improving an earlier O(n^3)-time algorithm of Cortese et al. [Discrete Math. 2009]. As an immediate corollary, we obtain the first efficient algorithm to determine whether an arbitrary n-vertex polygon is weakly simple; our algorithm runs in O(n^2 log n) time. We also describe algorithms that detect weak simplicity in O(n log n) time for two interesting classes of polygons. Finally, we discuss subtle errors in several previously published definitions of weak simplicity.Comment: 25 pages and 13 figures, submitted to SODA 201

Light Euclidean Steiner Spanners in the Plane

Lightness is a fundamental parameter for Euclidean spanners; it is the ratio of the spanner weight to the weight of the minimum spanning tree of a finite set of points in $\mathbb{R}^d$. In a recent breakthrough, Le and Solomon (2019) established the precise dependencies on $\varepsilon>0$ and $d\in \mathbb{N}$ of the minimum lightness of $(1+\varepsilon)$-spanners, and observed that additional Steiner points can substantially improve the lightness. Le and Solomon (2020) constructed Steiner $(1+\varepsilon)$-spanners of lightness $O(\varepsilon^{-1}\log\Delta)$ in the plane, where $\Delta\geq \Omega(\sqrt{n})$ is the \emph{spread} of the point set, defined as the ratio between the maximum and minimum distance between a pair of points. They also constructed spanners of lightness $\tilde{O}(\varepsilon^{-(d+1)/2})$ in dimensions $d\geq 3$. Recently, Bhore and T\'{o}th (2020) established a lower bound of $\Omega(\varepsilon^{-d/2})$ for the lightness of Steiner $(1+\varepsilon)$-spanners in $\mathbb{R}^d$, for $d\ge 2$. The central open problem in this area is to close the gap between the lower and upper bounds in all dimensions $d\geq 2$. In this work, we show that for every finite set of points in the plane and every $\varepsilon>0$, there exists a Euclidean Steiner $(1+\varepsilon)$-spanner of lightness $O(\varepsilon^{-1})$; this matches the lower bound for $d=2$. We generalize the notion of shallow light trees, which may be of independent interest, and use directional spanners and a modified window partitioning scheme to achieve a tight weight analysis.Comment: 29 pages, 14 figures. A 17-page extended abstract will appear in the Proceedings of the 37th International Symposium on Computational Geometr

Geometric Separation and Packing Problems

The first part of this thesis investigates combinatorial and algorithmic aspects of geometric separation problems in the plane. In such a setting one is given a set of points and a set of separators such as lines, line segments or disks. The goal is to select a small subset of those separators such that every path between any two points is intersected by at least one separator. We first look at several problems which arise when one is given a set of points and a set of unit disks embedded in the plane and the goal is to separate the points using a small subset of the given disks. Next, we focus on a separation problem involving only one region: Given a region in the plane, bounded by a piecewise linear closed curve, such as a fence, place few guards inside the fenced region such that wherever an intruder cuts through the fence, the closest guard is at most a distance one away. Restricting the separating objects to be lines, we investigate combinatorial aspects which arise when we use them to pairwise separate a set of points in the plane; hereafter we generalize the notion of separability to arbitrary sets and present several enumeration results. Lastly, we investigate a packing problem with a non-convex shape in ℝ3. We show that ℝ3 can be packed at a density of 0.222 with tori of major radius one and minor radius going to zero. Furthermore, we show that the same torus arrangement yields the asymptotically optimal number of pairwise linked tori

Light Orthogonal Networks with Constant Geometric Dilation

An orthogonal spanner network for a given set of n points in the plane is a plane straight line graph with axis-aligned edges that connects all input points. We show that for any set of n points in the plane, there is an orthogonal spanner network that (i) is short having a total edge length of at most a constant times the length of a Euclidean minimum spanning tree for the point set; (ii) is small having O(n) vertices and edges; and (iii) has constant geometric dilation, which means that for any two points u and v in the network, the shortest path in the network between u and v is at most a constant times longer than the (Euclidean) distance between u and v. Such a network can be constructed in O(n log n) time

Light Orthogonal Networks with Constant Geometric Dilation

Abstract An orthogonal network for a given set of n points in the plane is an axis-aligned planar straightline graph that connects all input points. We show that for any set of n points in the plane, there is anorthogonal network that (i) is short having a total edge length of O(jT j), where jT j denotes the lengthof a minimum Euclidean spanning tree for the point set; (ii) is small having O(n) vertices and edges;and (iii) has constant geometric dilation, which means that for any two points u and v in the network,the shortest path in the network between u and v is at most constant times longer than the (Euclidean)distance between u and v. 1 Introduction A typical problem in the theory of metric embeddings asks for a mapping from one metric space to anotherthat distorts the distances between point pairs as little as possible. In this paper, we address the following problem about geometric dilation: Given a finite set S of points in the plane, find a small plane graph G(S) containing S so that the distortion between the L2 distance and the Euclidean shortest path distancebetween any two points (on edges or at vertices) of G(S) is bounded by a constant.A special case of this problem received frantic attention in the late 80s and early 90s in the context o