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Ipelets for the Convex Polygonal Geometry arXiv.cs.CG Pub Date : 2024-03-15 Nithin Parepally, Ainesh Chatterjee, Auguste Gezalyan, Hongyang Du, Sukrit Mangla, Kenny Wu, Sarah Hwang, David Mount
There are many structures, both classical and modern, involving convex polygonal geometries whose deeper understanding would be facilitated through interactive visualizations. The Ipe extensible drawing editor, developed by Otfried Cheong, is a widely used software system for generating geometric figures. One of its features is the capability to extend its functionality through programs called Ipelets
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Solution-Hashing Search Based on Layout-Graph Transformation for Unequal Circle Packing arXiv.cs.CG Pub Date : 2024-03-10 Jianrong Zhou, Jiyao He, Kun He
The problem of packing unequal circles into a circular container stands as a classic and challenging optimization problem in computational geometry. This study introduces a suite of innovative and efficient methods to tackle this problem. Firstly, we present a novel layout-graph transformation method that represents configurations as graphs, together with an inexact hash method facilitating fast comparison
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A Clique-Based Separator for Intersection Graphs of Geodesic Disks in $\mathbb{R}^2$ arXiv.cs.CG Pub Date : 2024-03-07 Boris Aronov, Mark de Berg, Leonidas Theocharous
Let $d$ be a (well-behaved) shortest-path metric defined on a path-connected subset of $\mathbb{R}^2$ and let $\mathcal{D}=\{D_1,\ldots,D_n\}$ be a set of geodesic disks with respect to the metric $d$. We prove that $\mathcal{G}^{\times}(\mathcal{D})$, the intersection graph of the disks in $\mathcal{D}$, has a clique-based separator consisting of $O(n^{3/4+\varepsilon})$ cliques. This significantly
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A Coreset for Approximate Furthest-Neighbor Queries in a Simple Polygon arXiv.cs.CG Pub Date : 2024-03-07 Mark de Berg, Leonidas Theocharous
Let $\mathcal{P}$ be a simple polygon with $m$ vertices and let $P$ be a set of $n$ points inside $\mathcal{P}$. We prove that there exists, for any $\varepsilon>0$, a set $\mathcal{C} \subset P$ of size $O(1/\varepsilon^2)$ such that the following holds: for any query point $q$ inside the polygon $\mathcal{P}$, the geodesic distance from $q$ to its furthest neighbor in $\mathcal{C}$ is at least $1-\varepsilon$
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Fine-Grained Complexity of Earth Mover's Distance under Translation arXiv.cs.CG Pub Date : 2024-03-07 Karl Bringmann, Frank Staals, Karol Węgrzycki, Geert van Wordragen
The Earth Mover's Distance is a popular similarity measure in several branches of computer science. It measures the minimum total edge length of a perfect matching between two point sets. The Earth Mover's Distance under Translation ($\mathrm{EMDuT}$) is a translation-invariant version thereof. It minimizes the Earth Mover's Distance over all translations of one point set. For $\mathrm{EMDuT}$ in $\mathbb{R}^1$
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An Overview of Minimum Convex Cover and Maximum Hidden Set arXiv.cs.CG Pub Date : 2024-03-03 Reilly Browne
We give a review of results on the minimum convex cover and maximum hidden set problems. In addition, we give some new results. First we show that it is NP-hard to determine whether a polygon has the same convex cover number as its hidden set number. We then give some important examples in which these quantities don't always coincide. Finally, We present some consequences of insights from Browne, Kasthurirangan
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On the Parameterized Complexity of Motion Planning for Rectangular Robots arXiv.cs.CG Pub Date : 2024-02-27 Iyad Kanj, Salman Parsa
We study computationally-hard fundamental motion planning problems where the goal is to translate $k$ axis-aligned rectangular robots from their initial positions to their final positions without collision, and with the minimum number of translation moves. Our aim is to understand the interplay between the number of robots and the geometric complexity of the input instance measured by the input size
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Practical Software for Triangulating and Simplifying 4-Manifolds arXiv.cs.CG Pub Date : 2024-02-23 Rhuaidi Antonio Burke
Dimension 4 is the first dimension in which exotic smooth manifold pairs appear -- manifolds which are topologically the same but for which there is no smooth deformation of one into the other. Whilst smooth and triangulated 4-manifolds do coincide, comparatively little work has been done towards gaining an understanding of smooth 4-manifolds from the discrete and algorithmic perspective. In this paper
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Embeddings and near-neighbor searching with constant additive error for hyperbolic spaces arXiv.cs.CG Pub Date : 2024-02-22 Eunku Park, Antoine Vigneron
We give an embedding of the Poincar\'e halfspace $H^D$ into a discrete metric space based on a binary tiling of $H^D$, with additive distortion $O(\log D)$. It yields the following results. We show that any subset $P$ of $n$ points in $H^D$ can be embedded into a graph-metric with $2^{O(D)}n$ vertices and edges, and with additive distortion $O(\log D)$. We also show how to construct, for any $k$, an
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On $k$-Plane Insertion into Plane Drawings arXiv.cs.CG Pub Date : 2024-02-22 Julia Katheder, Philipp Kindermann, Fabian Klute, Irene Parada, Ignaz Rutter
We introduce the $k$-Plane Insertion into Plane drawing ($k$-PIP) problem: given a plane drawing of a planar graph $G$ and a set of edges $F$, insert the edges in $F$ into the drawing such that the resulting drawing is $k$-plane. In this paper, we focus on the $1$-PIP scenario. We present a linear-time algorithm for the case that $G$ is a triangulation, while proving NP-completeness for the case that
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Generalized Red-Blue Circular Annulus Cover Problem arXiv.cs.CG Pub Date : 2024-02-21 Sukanya Maji, Supantha Pandit, Sanjib Sadhu
We study the Generalized Red-Blue Annulus Cover problem for two sets of points, red ($R$) and blue ($B$), where each point $p \in R\cup B$ is associated with a positive penalty ${\cal P}(p)$. The red points have non-covering penalties, and the blue points have covering penalties. The objective is to compute a circular annulus ${\cal A}$ such that the value of the function ${\cal P}({R}^{out})$ + ${\cal
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Ordering Topological Descriptors arXiv.cs.CG Pub Date : 2024-02-21 Brittany Terese Fasy, David L. Millman, Anna Schenfisch
Recent developments in shape reconstruction and comparison call for the use of many different types of topological descriptors (persistence diagrams, Euler characteristic functions, etc.). We establish a framework that allows for quantitative comparisons of topological descriptor types and therefore may be used as a tool in more rigorously justifying choices made in applications. We then use this framework
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Greedy Monochromatic Island Partitions arXiv.cs.CG Pub Date : 2024-02-20 Steven van den Broek, Wouter Meulemans, Bettina Speckmann
Constructing partitions of colored points is a well-studied problem in discrete and computational geometry. We study the problem of creating a minimum-cardinality partition into monochromatic islands. Our input is a set $S$ of $n$ points in the plane where each point has one of $k \geq 2$ colors. A set of points is monochromatic if it contains points of only one color. An island $I$ is a subset of
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A Geometric Algorithm for Tubular Shape Reconstruction from Skeletal Representation arXiv.cs.CG Pub Date : 2024-02-20 Guoqing Zhang, Songzi Cat, Juzi Cat
We introduce a novel approach for the reconstruction of tubular shapes from skeletal representations. Our method processes all skeletal points as a whole, eliminating the need for splitting input structure into multiple segments. We represent the tubular shape as a truncated signed distance function (TSDF) in a voxel hashing manner, in which the signed distance between a voxel center and the object
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Barking dogs: A Fréchet distance variant for detour detection arXiv.cs.CG Pub Date : 2024-02-20 Ivor van der Hoog, Fabian Klute, Irene Parada, Patrick Schnider
Imagine you are a dog behind a fence $Q$ and a hiker is passing by at constant speed along the hiking path $P$. In order to fulfil your duties as a watchdog, you desire to bark as long as possible at the human. However, your barks can only be heard in a fixed radius $\rho$ and, as a dog, you have bounded speed $s$. Can you optimize your route along the fence $Q$ in order to maximize the barking time
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Faster and Deterministic Subtrajectory Clustering arXiv.cs.CG Pub Date : 2024-02-20 Ivor van der Hoog, Thijs van der Horst, Tim Ophelders
Given a trajectory $T$ and a distance $\Delta$, we wish to find a set $C$ of curves of complexity at most $\ell$, such that we can cover $T$ with subcurves that each are within Fr\'echet distance $\Delta$ to at least one curve in $C$. We call $C$ an $(\ell,\Delta)$-clustering and aim to find an $(\ell,\Delta)$-clustering of minimum cardinality. This problem was introduced by Akitaya $et$ $al.$ (2021)
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Computing Enclosing Depth arXiv.cs.CG Pub Date : 2024-02-19 Bernd Gärtner, Fatime Rasiti, Patrick Schnider
Enclosing depth is a recently introduced depth measure which gives a lower bound to many depth measures studied in the literature. So far, enclosing depth has only been studied from a combinatorial perspective. In this work, we give the first algorithms to compute the enclosing depth of a query point with respect to a data point set in any dimension. In the plane we are able to optimize the algorithm
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Flip Graphs of Pseudo-Triangulations With Face Degree at Most 4 arXiv.cs.CG Pub Date : 2024-02-19 Maarten Löffler, Tamara Mchedlidze, David Orden, Josef Tkadlec, Jules Wulms
A pseudo-triangle is a simple polygon with exactly three convex vertices, and all other vertices (if any) are distributed on three concave chains. A pseudo-triangulation~$\mathcal{T}$ of a point set~$P$ in~$\mathbb{R}^2$ is a partitioning of the convex hull of~$P$ into pseudo-triangles, such that the union of the vertices of the pseudo-triangles is exactly~$P$. We call a size-4 pseudo-triangle a dart
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Capturing the Shape of a Point Set with a Line-Segment arXiv.cs.CG Pub Date : 2024-02-19 Nathan van Beusekom, Marc van Kreveld, Max van Mulken, Marcel Roeloffzen, Bettina Speckmann, Jules Wulms
Detecting location-correlated groups in point sets is an important task in a wide variety of applications areas. In addition to merely detecting such groups, the group's shape carries meaning as well. In this paper, we represent a group's shape using a simple geometric object, a line-segment. Specifically, given a radius $r$, we say a line-segment is shape-representing of a point set $P$ if it is within
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Constrained Boundary Labeling arXiv.cs.CG Pub Date : 2024-02-19 Thomas Depian, Martin Nöllenburg, Soeren Terziadis, Markus Wallinger
Boundary labeling is a technique used to label dense sets of feature points in an illustration. It involves placing labels along a rectangular boundary box and connecting each label with its corresponding feature using non-crossing leader lines. Although boundary labeling is well-studied, semantic constraints on the labels have not been investigated thoroughly. In this paper, we consider grouping and
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Exact solutions to the Weighted Region Problem arXiv.cs.CG Pub Date : 2024-02-19 Sarita de Berg, Guillermo Esteban, Rodrigo I. Silveira, Frank Staals
In this paper, we consider the Weighted Region Problem. In the Weighted Region Problem, the length of a path is defined as the sum of the weights of the subpaths within each region, where the weight of a subpath is its Euclidean length multiplied by a weight $ \alpha \geq 0 $ depending on the region. We study a restricted version of the problem of determining shortest paths through a single weighted
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Reeb Complements for Exploring Inclusions Between Isosurfaces From Two Scalar Fields arXiv.cs.CG Pub Date : 2024-02-15 Akito Fujii, Osamu Saeki, Daisuke Sakurai
This article proposes to integrate two Reeb graphs with the information of their isosurfaces' inclusion relation. As computing power evolves, there arise numerical data that have small-scale physics inside larger ones -- for example, small clouds in a simulation can be contained inside an atmospheric layer, which is further contained in an enormous hurricane. Extracting such inclusions between isosurfaces
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Fully Dynamic Geometric Vertex Cover and Matching arXiv.cs.CG Pub Date : 2024-02-12 Sujoy Bhore, Timothy M. Chan
In this work, we study two fundamental graph optimization problems, minimum vertex cover (MVC) and maximum-cardinality matching (MCM), for intersection graphs of geometric objects, e.g., disks, rectangles, hypercubes, etc., in $d$-dimensional Euclidean space. We consider the problems in fully dynamic settings, allowing insertions and deletions of objects. We develop a general framework for dynamic
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Optimizing Visibility-based Search in Polygonal Domains arXiv.cs.CG Pub Date : 2024-02-08 Kien C. Huynh, Joseph S. B. Mitchell, Linh Nguyen, Valentin Polishchuk
Given a geometric domain $P$, visibility-based search problems seek routes for one or more mobile agents (``watchmen'') to move within $P$ in order to be able to see a portion (or all) of $P$, while optimizing objectives, such as the length(s) of the route(s), the size (e.g., area or volume) of the portion seen, the probability of detecting a target distributed within $P$ according to a prior distribution
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Dynamic Geometric Connectivity in the Plane with Constant Query Time arXiv.cs.CG Pub Date : 2024-02-08 Timothy M. Chan, Zhengcheng Huang
We present the first fully dynamic connectivity data structures for geometric intersection graphs achieving constant query time and sublinear amortized update time for most types of geometric objects in 2D. Our data structures can answer connectivity queries between two objects, as well as "global" connectivity queries (e.g., deciding whether the entire graph is connected). Previously, the data structure
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Making Multicurves Cross Minimally on Surfaces arXiv.cs.CG Pub Date : 2024-02-07 Loïc Dubois
On an orientable surface $S$, consider a collection $\Gamma$ of closed curves. The (geometric) intersection number $i_S(\Gamma)$ is the minimum number of self-intersections that a collection $\Gamma'$ can have, where $\Gamma'$ results from a continuous deformation (homotopy) of $\Gamma$. We provide algorithms that compute $i_S(\Gamma)$ and such a $\Gamma'$, assuming that $\Gamma$ is given by a collection
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Bezier surfaces with prescribed diagonals arXiv.cs.CG Pub Date : 2024-02-06 A. Arnal, J. Monterde
The affine space of all tensor product B\'ezier patches of degree nxn with prescribed main diagonal curves is determined. First, the pair of B\'ezier curves which can be diagonals of a B\'ezier patch is characterized. Besides prescribing the diagonal curves, other related problems are considered, those where boundary curves or tangent planes along boundary curves are also prescribed.
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On Approximation Schemes for Stabbing Rectilinear Polygons arXiv.cs.CG Pub Date : 2024-02-04 Arindam Khan, Aditya Subramanian, Tobias Widmann, Andreas Wiese
We study the problem of stabbing rectilinear polygons, where we are given $n$ rectilinear polygons in the plane that we want to stab, i.e., we want to select horizontal line segments such that for each given rectilinear polygon there is a line segment that intersects two opposite (parallel) edges of it. Our goal is to find a set of line segments of minimum total length such that all polygons are stabbed
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Approximating the Smallest $k$-Enclosing Geodesic Disc in a Simple Polygon arXiv.cs.CG Pub Date : 2024-02-01 Prosenjit Bose, Anthony D'Angelo, Stephane Durocher
We consider the problem of finding a geodesic disc of smallest radius containing at least $k$ points from a set of $n$ points in a simple polygon that has $m$ vertices, $r$ of which are reflex vertices. We refer to such a disc as a SKEG disc. We present an algorithm to compute a SKEG disc using higher-order geodesic Voronoi diagrams with worst-case time $O(k^{2} n + k^{2} r + \min(kr, r(n-k)) + m)$
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Integrable Frame Fields using Odeco Tensors arXiv.cs.CG Pub Date : 2024-01-30 Mattéo Couplet, Alexandre Chemin, Jean-François Remacle
We propose a method for computing integrable orthogonal frame fields on planar surfaces. Frames and their symmetries are implicitly represented using orthogonally decomposable (odeco) tensors. To formulate an integrability criterion, we express the frame field's Lie bracket solely in terms of the tensor representation; this is made possible by studying the sensitivity of the frame with respect to perturbations
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Some fast algorithms for curves in surfaces arXiv.cs.CG Pub Date : 2024-01-29 Marc Lackenby
We present some algorithms that provide useful topological information about curves in surfaces. One of the main algorithms computes the geometric intersection number of two properly embedded 1-manifolds $C_1$ and $C_2$ in a compact orientable surface $S$. The surface $S$ is presented via a triangulation or a handle structure, and the 1-manifolds are given in normal form via their normal coordinates
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A simple and complete discrete exterior calculus on general polygonal meshes arXiv.cs.CG Pub Date : 2024-01-27 Lenka Ptackova, Luiz Velho
Discrete exterior calculus (DEC) offers a coordinate-free discretization of exterior calculus especially suited for computations on curved spaces. In this work, we present an extended version of DEC on surface meshes formed by general polygons that bypasses the need for combinatorial subdivision and does not involve any dual mesh. At its core, our approach introduces a new polygonal wedge product that
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Faster Fréchet Distance Approximation through Truncated Smoothing arXiv.cs.CG Pub Date : 2024-01-26 Thijs van der Horst, Tim Ophelders
The Fr\'echet distance is a popular distance measure for curves. Computing the Fr\'echet distance between two polygonal curves of $n$ vertices takes roughly quadratic time, and conditional lower bounds suggest that even approximating to within a factor $3$ cannot be done in strongly-subquadratic time, even in one dimension. The current best approximation algorithms present trade-offs between approximation
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On Sparse Covers of Minor Free Graphs, Low Dimensional Metric Embeddings, and other applications arXiv.cs.CG Pub Date : 2024-01-25 Arnold Filtser
Given a metric space $(X,d_X)$, a $(\beta,s,\Delta)$-sparse cover is a collection of clusters $\mathcal{C}\subseteq P(X)$ with diameter at most $\Delta$, such that for every point $x\in X$, the ball $B_X(x,\frac\Delta\beta)$ is fully contained in some cluster $C\in \mathcal{C}$, and $x$ belongs to at most $s$ clusters in $\mathcal{C}$. Our main contribution is to show that the shortest path metric
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EMP: Effective Multidimensional Persistence for Graph Representation Learning arXiv.cs.CG Pub Date : 2024-01-24 Ignacio Segovia-Dominguez, Yuzhou Chen, Cuneyt G. Akcora, Zhiwei Zhen, Murat Kantarcioglu, Yulia R. Gel, Baris Coskunuzer
Topological data analysis (TDA) is gaining prominence across a wide spectrum of machine learning tasks that spans from manifold learning to graph classification. A pivotal technique within TDA is persistent homology (PH), which furnishes an exclusive topological imprint of data by tracing the evolution of latent structures as a scale parameter changes. Present PH tools are confined to analyzing data
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Using Java Geometry Expert as Guide in the Preparations for Math Contests arXiv.cs.CG Pub Date : 2024-01-22 Ines GanglmayrThe Private University College of Education of the Diocese of Linz, Austria, Zoltán KovácsThe Private University College of Education of the Diocese of Linz, Austria
We give an insight into Java Geometry Expert (JGEX) in use in a school context, focusing on the Austrian school system. JGEX can offer great support in some classroom situations, especially for solving mathematical competition tasks. Also, we discuss some limitations of the program.
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Solving Some Geometry Problems of the Náboj 2023 Contest with Automated Deduction in GeoGebra Discovery arXiv.cs.CG Pub Date : 2024-01-22 Amela HotaThe Private University College of Education of the Diocese of Linz, Austria, Zoltán KovácsThe Private University College of Education of the Diocese of Linz, Austria, Alexander VujicThe Private University College of Education of the Diocese of Linz, Austria
In this article, we solve some of the geometry problems of the N\'aboj 2023 competition with the help of a computer, using examples that the software tool GeoGebra Discovery can calculate. In each case, the calculation requires symbolic computations. We analyze the difficulty of feeding the problem into the machine and set further goals to make the problems of this type of contests even more tractable
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Improving Angular Speed Uniformity by Piecewise Radical Reparameterization arXiv.cs.CG Pub Date : 2024-01-22 Hoon HongNorth Carolina State University, Dongming WangBeihang University, Jing YangGuangxi Minzu University
For a rational parameterization of a curve, it is desirable that its angular speed is as uniform as possible. Hence, given a rational parameterization, one wants to find re-parameterization with better uniformity. One natural way is to use piecewise rational reparameterization. However, it turns out that the piecewise rational reparameterization does not help when the angular speed of the given rational
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Accelerating Material Property Prediction using Generically Complete Isometry Invariants arXiv.cs.CG Pub Date : 2024-01-22 Jonathan Balasingham, Viktor Zamaraev, Vitaliy Kurlin
Material or crystal property prediction using machine learning has grown popular in recent years as it provides a computationally efficient replacement to classical simulation methods. A crucial first step for any of these algorithms is the representation used for a periodic crystal. While similar objects like molecules and proteins have a finite number of atoms and their representation can be built
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Reliability-based G1 Continuous Arc Spline Approximation arXiv.cs.CG Pub Date : 2024-01-18 Jinhwan Jeon, Yoonjin Hwang, Seibum B. Choi
In this paper, we present an algorithm to approximate a set of data points with G1 continuous arcs, using points' covariance data. To the best of our knowledge, previous arc spline approximation approaches assumed that all data points contribute equally (i.e. have the same weights) during the approximation process. However, this assumption may cause serious instability in the algorithm, if the collected
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Barcodes for the topological analysis of gradient-like vector fields arXiv.cs.CG Pub Date : 2024-01-16 Clemens Bannwart, Claudia Landi
Intending to introduce a method for the topological analysis of fields, we present a pipeline that takes as an input a weighted and based chain complex, produces a tame epimorphic parametrized chain complex, and encodes it as a barcode of tagged intervals. We show how to apply this pipeline to the weighted and based chain complex of a gradient-like Morse-Smale vector field on a compact Riemannian manifold
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Measure Theoretic Reeb Graphs and Reeb Spaces arXiv.cs.CG Pub Date : 2024-01-12 Qingsong Wang, Guanquan Ma, Raghavendra Sridharamurthy, Bei Wang
A Reeb graph is a graphical representation of a scalar function $f: X \to \mathbb{R}$ on a topological space $X$ that encodes the topology of the level sets. A Reeb space is a generalization of the Reeb graph to a multivariate function $f: X \to \mathbb{R}^d$. In this paper, we propose novel constructions of Reeb graphs and Reeb spaces that incorporate the use of a measure. Specifically, we introduce
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On the number of iterations of the DBA algorithm arXiv.cs.CG Pub Date : 2024-01-11 Frederik Brüning, Anne Driemel, Alperen Ergür, Heiko Röglin
The DTW Barycenter Averaging (DBA) algorithm is a widely used algorithm for estimating the mean of a given set of point sequences. In this context, the mean is defined as a point sequence that minimises the sum of dynamic time warping distances (DTW). The algorithm is similar to the $k$-means algorithm in the sense that it alternately repeats two steps: (1) computing an optimal assignment to the points
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Intersection properties of finite disk collections arXiv.cs.CG Pub Date : 2024-01-11 Jesús F. Espinoza, Cynthia G. Esquer-Pérez
In this work we study the intersection properties of a finite disk system in the euclidean space. We accomplish this by utilizing subsets of spheres with varying dimensions and analyze specific points within them, referred to as poles. Additionally, we introduce two applications: estimating the common scale factor for the radii that makes the re-scaled disks intersects in a single point, this is the
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Range Reporting for Time Series via Rectangle Stabbing arXiv.cs.CG Pub Date : 2024-01-08 Lotte Blank, Anne Driemel
We study the Fr\'echet queries problem. It is a data structure problem, where we are given a set $S$ of $n$ polygonal curves and a distance threshold $\rho$. The data structure should support queries with a polygonal curve $q$ for the elements of $S$, for which the continuous Fr\'echet distance to $q$ is at most $\rho$. Afshani and Driemel in 2018 studied this problem for two-dimensional polygonal
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Revisiting the Fréchet distance between piecewise smooth curves arXiv.cs.CG Pub Date : 2024-01-07 Jacobus Conradi, Anne Driemel, Benedikt Kolbe
Since its introduction to computational geometry by Alt and Godau in 1992, the Fr\'echet distance has been a mainstay of algorithmic research on curve similarity computations. The focus of the research has been on comparing polygonal curves, with the notable exception of an algorithm for the decision problem for planar piecewise smooth curves due to Rote (2007). We present an algorithm for the decision
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Colored Points Traveling Salesman Problem arXiv.cs.CG Pub Date : 2024-01-06 Saeed Asaeedi
The Colored Points Traveling Salesman Problem (Colored Points TSP) is introduced in this work as a novel variation of the traditional Traveling Salesman Problem (TSP) in which the set of points is partitioned into multiple classes, each of which is represented by a distinct color (or label). The goal is to find a minimum cost cycle $C$ that visits all the colors and only makes each one appears once
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On 1-bend Upward Point-set Embeddings of $st$-digraphs arXiv.cs.CG Pub Date : 2024-01-06 Emilio Di Giacomo, Henry Förster, Daria Kokhovich, Tamara Mchedlidze, Fabrizio Montecchiani, Antonios Symvonis, Anaïs Villedieu
We study the upward point-set embeddability of digraphs on one-sided convex point sets with at most 1 bend per edge. We provide an algorithm to compute a 1-bend upward point-set embedding of outerplanar $st$-digraphs on arbitrary one-sided convex point sets. We complement this result by proving that for every $n \geq 18$ there exists a $2$-outerplanar $st$-digraph $G$ with $n$ vertices and a one-sided
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Robust Bichromatic Classification using Two Lines arXiv.cs.CG Pub Date : 2024-01-05 Erwin Glazenburg, Thijs van der Horst, Tom Peters, Bettina Speckmann, Frank Staals
Given two sets $\mathit{R}$ and $\mathit{B}$ of at most $\mathit{n}$ points in the plane, we present efficient algorithms to find a two-line linear classifier that best separates the ``red'' points in $\mathit{R}$ from the ``blue'' points in $B$ and is robust to outliers. More precisely, we find a region $\mathit{W}_\mathit{B}$ bounded by two lines, so either a halfplane, strip, wedge, or double wedge
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Recognition of Unit Segment and Polyline Graphs is $\exists\mathbb{R}$-Complete arXiv.cs.CG Pub Date : 2024-01-04 Michael Hoffmann, Tillmann Miltzow, Simon Weber, Lasse Wulf
Given a set of objects O in the plane, the corresponding intersection graph is defined as follows. A vertex is created for each object and an edge joins two vertices whenever the corresponding objects intersect. We study here the case of unit segments and polylines with exactly k bends. In the recognition problem, we are given a graph and want to decide whether the graph can be represented as the intersection
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Competitive Searching over Terrains arXiv.cs.CG Pub Date : 2024-01-02 Sarita de Berg, Nathan van Beusekom, Max van Mulken, Kevin Verbeek, Jules Wulms
We study a variant of the searching problem where the environment consists of a known terrain and the goal is to obtain visibility of an unknown target point on the surface of the terrain. The searcher starts on the surface of the terrain and is allowed to fly above the terrain. The goal is to devise a searching strategy that minimizes the competitive ratio, that is, the worst-case ratio between the
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Polygon Detection from a Set of Lines arXiv.cs.CG Pub Date : 2023-12-27 Alfredo Ferreira Jr., Manuel J. Fonseca, Joaquim A. Jorge
Detecting polygons defined by a set of line segments in a plane is an important step in analyzing vector drawings. This paper presents an approach combining several algorithms to detect basic polygons from arbitrary line segments. The resulting algorithm runs in polynomial time and space, with complexities of $O\bigl((N + M)^4\bigr)$ and $O\bigl((N + M)^2\bigr)$, where $N$ is the number of line segments
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Routing on Heavy Path WSPD Spanners arXiv.cs.CG Pub Date : 2023-12-23 Prosenjit Bose, Tyler Tuttle
In this article, we present a construction of a spanner on a set of $n$ points in $\mathbf{R}^d$ that we call a heavy path WSPD spanner. The construction is parameterized by a constant $s > 2$ called the separation ratio. The size of the graph is $O(s^dn)$ and the spanning ratio is at most $1 + 2/s + 2/(s - 1)$. We also show that this graph has a hop spanning ratio of at most $2\lg n + 1$. We present
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The Exact Spanning Ratio of the Parallelogram Delaunay Graph arXiv.cs.CG Pub Date : 2023-12-21 Prosenjit Bose, Jean-Lou De Carufel, Sandrine Njoo
Finding the exact spanning ratio of a Delaunay graph has been one of the longstanding open problems in Computational Geometry. Currently there are only four convex shapes for which the exact spanning ratio of their Delaunay graph is known: the equilateral triangle, the square, the regular hexagon and the rectangle. In this paper, we show the exact spanning ratio of the parallelogram Delaunay graph
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Point Location in Constant Time arXiv.cs.CG Pub Date : 2023-12-21 Sairam Chaganti, Yijie Han
We preprocess the input subdivision with $n$ points on the plane in $O(n\sqrt{\log n})$ time to facilitate point location in constant time. Previously the preprocessing time is $O(n\log n)$ and point location takes $O(\log n)$ time.
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Fast Approximations and Coresets for (k, l)-Median under Dynamic Time Warping arXiv.cs.CG Pub Date : 2023-12-15 Jacobus Conradi, Benedikt Kolbe, Ioannis Psarros, Dennis Rohde
We present algorithms for the computation of $\varepsilon$-coresets for $k$-median clustering of point sequences in $\mathbb{R}^d$ under the $p$-dynamic time warping (DTW) distance. Coresets under DTW have not been investigated before, and the analysis is not directly accessible to existing methods as DTW is not a metric. The three main ingredients that allow our construction of coresets are the adaptation
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Clustering with Few Disks to Minimize the Sum of Radii arXiv.cs.CG Pub Date : 2023-12-14 Mikkel Abrahamsen, Sarita de Berg, Lucas Meijer, André Nusser, Leonidas Theocharous
Given a set of $n$ points in the Euclidean plane, the $k$-MinSumRadius problem asks to cover this point set using $k$ disks with the objective of minimizing the sum of the radii of the disks. After a long line of research on related problems, it was finally discovered that this problem admits a polynomial time algorithm [GKKPV~'12]; however, the running time of this algorithm is $O(n^{881})$, and its
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Delaunay Triangulations in the Hilbert Metric arXiv.cs.CG Pub Date : 2023-12-10 Auguste Gezalyan, Soo Kim, Carlos Lopez, Daniel Skora, Zofia Stefankovic, David M. Mount
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 to adapt the Euclidean Delaunay triangulation to the Hilbert geometry defined by a convex polygon in the plane. We analyze the geometric properties of the Hilbert
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Geometric Thickness of Multigraphs is $\exists \mathbb{R}$-complete arXiv.cs.CG Pub Date : 2023-12-08 Henry Förster, Philipp Kindermann, Tillmann Miltzow, Irene Parada, Soeren Terziadis, Birgit Vogtenhuber
We say that a (multi)graph $G = (V,E)$ has geometric thickness $t$ if there exists a straight-line drawing $\varphi : V \rightarrow \mathbb{R}^2$ and a $t$-coloring of its edges where no two edges sharing a point in their relative interior have the same color. The Geometric Thickness problem asks whether a given multigraph has geometric thickness at most $t$. This problem was shown to be NP-hard for
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Constraint Model for the Satellite Image Mosaic Selection Problem arXiv.cs.CG Pub Date : 2023-12-07 Manuel Combarro Simón, Pierre Talbot, Grégoire Danoy, Jedrzej Musial, Mohammed Alswaitti, Pascal Bouvry
Satellite imagery solutions are widely used to study and monitor different regions of the Earth. However, a single satellite image can cover only a limited area. In cases where a larger area of interest is studied, several images must be stitched together to create a single larger image, called a mosaic, that can cover the area. Today, with the increasing number of satellite images available for commercial