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  • Approximating the packedness of polygonal curves
    arXiv.cs.CG Pub Date : 2020-09-16
    Joachim Gudmundsson; Yuan Sha; Sampson Wong

    In 2012 Driemel et al. \cite{DBLP:journals/dcg/DriemelHW12} introduced the concept of $c$-packed curves as a realistic input model. In the case when $c$ is a constant they gave a near linear time $(1+\varepsilon)$-approximation algorithm for computing the Fr\'echet distance between two $c$-packed polygonal curves. Since then a number of papers have used the model. In this paper we consider the problem

    更新日期:2020-09-17
  • Related by Similarity II: Poncelet 3-Periodics in the Homothetic Pair and the Brocard Porism
    arXiv.cs.CG Pub Date : 2020-09-16
    Dan Reznik; Ronaldo Garcia

    Previously we showed the family of 3-periodics in the elliptic billiard (confocal pair) is the image under a variable similarity transform of poristic triangles (those with non-concentric, fixed incircle and circumcircle). Both families conserve the ratio of inradius to circumradius and therefore also the sum of cosines. This is consisten with the fact that a similarity preserves angles. Here we study

    更新日期:2020-09-17
  • Drawing outer-1-planar graphs revisited
    arXiv.cs.CG Pub Date : 2020-09-15
    Therese Biedl

    In a recent article (Auer et al, Algorithmica 2016) it was claimed that every outer-1-planar graph has a planar visibility representation of area $O(n\log n)$. In this paper, we show that this is wrong: There are outer-1-planar graphs that require $\Omega(n^2)$ area in any planar drawing.

    更新日期:2020-09-16
  • Steady and ranging sets in graph persistence
    arXiv.cs.CG Pub Date : 2020-09-15
    Mattia G. Bergomi; Massimo Ferri; Antonella Tavaglione

    Generalised persistence functions (gp-functions) are defined on $(\mathbb{R}, \le)$-indexed diagrams in a given category. A sufficient condition for stability is also introduced. In the category of graphs, a standard way of producing gp-functions is proposed: steady and ranging sets for a given feature. The example of steady and ranging hubs is studied in depth; their meaning is investigated in three

    更新日期:2020-09-16
  • Bounded-Degree Spanners in the Presence of Polygonal Obstacles
    arXiv.cs.CG Pub Date : 2020-09-15
    André van Renssen; Gladys Wong

    Let $V$ be a finite set of vertices in the plane and $S$ be a finite set of polygonal obstacles. We show how to construct a plane $2$-spanner of the visibility graph of $V$ with respect to $S$. As this graph can have unbounded degree, we modify it in three easy-to-follow steps, in order to bound the degree to $7$ at the cost of slightly increasing the spanning ratio to 6.

    更新日期:2020-09-16
  • Interactive Focus+Context Rendering for Hexahedral Mesh Inspection
    arXiv.cs.CG Pub Date : 2020-09-14
    Christoph Neuhauser; Junpeng Wang; Rüdiger Westermann

    The visual inspection of a hexahedral mesh with respect to element quality is difficult due to clutter and occlusions that are produced when rendering all element faces or their edges simultaneously. Current approaches overcome this problem by using focus on specific elements that are then rendered opaque, and carving away all elements occluding their view. In this work, we make use of advanced GPU

    更新日期:2020-09-15
  • Extracting Optimal Solution Manifolds using Constrained Neural Optimization
    arXiv.cs.CG Pub Date : 2020-09-13
    Gurpreet Singh; Soumyajit Gupta; Matthew Lease

    Constrained Optimization solution algorithms are restricted to point based solutions. In practice, single or multiple objectives must be satisfied, wherein both the objective function and constraints can be non-convex resulting in multiple optimal solutions. Real world scenarios include intersecting surfaces as Implicit Functions, Hyperspectral Unmixing and Pareto Optimal fronts. Local or global convexification

    更新日期:2020-09-15
  • Kunneth Theorems for Vietoris-Rips Homology
    arXiv.cs.CG Pub Date : 2020-09-12
    Antonio Rieser; Alejandra Trujillo

    We prove a Kunneth theorem for the Vietoris-Rips homology and cohomology of a semi-uniform space. We then interpret this result for graphs, where we show that the Kunneth theorem holds for graphs with respect to the strong graph product. We finish by computing the Vietoris-Rips cohomology of the torus endowed with diferent semi-uniform structures.

    更新日期:2020-09-15
  • Empty axis-parallel boxes
    arXiv.cs.CG Pub Date : 2020-09-12
    Boris Bukh; Ting-Wei Chao

    We show that, for every set of $n$ points in $d$-dimensional unit cube, there is an empty axis-parallel box of volume at least $\Omega(d/n)$. In the opposite direction, we give a construction without an empty axis-parallel box of volume $O(d^2\log d/n)$. These improve on the previous best bounds of $\Omega(\log d/n)$ and $O(2^{7d}/n)$ respectively.

    更新日期:2020-09-15
  • 2D Fractional Cascading on Axis-aligned Planar Subdivisions
    arXiv.cs.CG Pub Date : 2020-09-11
    Peyman Afshani; Pingan Cheng

    Fractional cascading is one of the influential techniques in data structures, as it provides a general framework for solving the important iterative search problem. In the problem, the input is a graph $G$ with constant degree and a set of values for every vertex of $G$. The goal is to preprocess $G$ such that when given a query value $q$, and a connected subgraph $\pi$ of $G$, we can find the predecessor

    更新日期:2020-09-14
  • Dune-CurvedSurfaceGrid -- A Dune module for surface parametrization
    arXiv.cs.CG Pub Date : 2020-09-10
    Simon Praetorius; Florian Stenger

    In this paper we introduce and describe an implementation of curved surface geometries within the Dune framework for grid-based discretizations. Therefore, we employ the abstraction of geometries as local-functions bound to a grid element, and the abstraction of a grid as connectivity of elements together with a grid-function that can be localized to the elements to provide element local parametrizations

    更新日期:2020-09-11
  • Deterministic Linear Time Constrained Triangulation using Simplified Earcut
    arXiv.cs.CG Pub Date : 2020-09-09
    Marco Livesu; Gianmarco Cherchi; Riccardo Scateni; Marco Attene

    Triangulation algorithms that conform to a set of non-intersecting input segments typically proceed in an incremental fashion, by inserting points first, and then segments. Inserting a segment amounts to delete all the triangles it intersects, define two polygons that fill the so generated hole and have the segment as shared basis, and then re-triangulate each polygon separately. In this paper we prove

    更新日期:2020-09-10
  • Automatic feature-preserving size field for 3D mesh generation
    arXiv.cs.CG Pub Date : 2020-09-04
    Arthur Bawin; François Henrotte; Jean-François Remacle

    This paper presents a methodology aiming at easing considerably the generation of high-quality meshes for complex 3D domains. We show that the whole mesh generation process can be controlled with only five parameters to generate in one stroke quality meshes for arbitrary geometries. The main idea is to build a meshsize field $h(x)$ taking local features of the geometry, such as curvatures, into account

    更新日期:2020-09-10
  • Length-optimal tool path planning for freeform surfaces with preferred feed directions
    arXiv.cs.CG Pub Date : 2020-09-06
    Qiang Zou; Charlie C. L. Wang; Hsi-Yung Feng

    This paper presents a new method to generate tool paths for machining freeform surfaces represented either as parametric surfaces or as triangular meshes. This method allows for the optimal tradeoff between the preferred feed direction field and the constant scallop height, and yields a minimized overall path length. The optimality is achieved by formulating tool path planning as a Poisson problem

    更新日期:2020-09-08
  • Trimmed Spline Surfaces with Accurate Boundary Control
    arXiv.cs.CG Pub Date : 2020-09-05
    Florian Martin; Ulrich Reif

    We introduce trimmed NURBS surfaces with accurate boundary control, briefly called ABC-surfaces, as a solution to the notorious problem of constructing watertight or smooth ($G^1$ and $G^2)$ multi-patch surfaces within the function range of standard CAD/CAM systems and the associated file exchange formats. Our construction is based on the appropriate blend of a base surface, which traces out the intended

    更新日期:2020-09-08
  • Approximating $(k,\ell)$-Median Clustering for Polygonal Curves
    arXiv.cs.CG Pub Date : 2020-09-03
    Maike Buchin; Anne Driemel; Dennis Rohde

    In 2015, Driemel, Krivo\v{s}ija and Sohler introduced the $(k,\ell)$-median problem for clustering polygonal curves under the Fr\'echet distance. Given a set of input curves, the problem asks to find $k$ median curves of at most $\ell$ vertices each that minimize the sum of Fr\'echet distances over all input curves to their closest median curve. A major shortcoming of their algorithm is that the input

    更新日期:2020-09-05
  • TopoMap: A 0-dimensional Homology Preserving Projection of High-Dimensional Data
    arXiv.cs.CG Pub Date : 2020-09-03
    Harish Doraiswamy; Julien Tierny; Paulo J. S. Silva; Luis Gustavo Nonato; Claudio Silva

    Multidimensional Projection is a fundamental tool for high-dimensional data analytics and visualization. With very few exceptions, projection techniques are designed to map data from a high-dimensional space to a visual space so as to preserve some dissimilarity (similarity) measure, such as the Euclidean distance for example. In fact, although adopting distinct mathematical formulations designed to

    更新日期:2020-09-05
  • Strong Hanani-Tutte for the Torus
    arXiv.cs.CG Pub Date : 2020-09-01
    Radoslav Fulek; Michael J. Pelsmajer; Marcus Schaefer

    If a graph can be drawn on the torus so that every two independent edges cross an even number of times, then the graph can be embedded on the torus.

    更新日期:2020-09-05
  • Isotopic Arrangement of Simple Curves: an Exact Numerical Approach based on Subdivision
    arXiv.cs.CG Pub Date : 2020-09-02
    Jyh-Ming Lien; Vikram Sharma; Gert Vegter; Chee Yap

    This paper presents the first purely numerical (i.e., non-algebraic) subdivision algorithm for the isotopic approximation of a simple arrangement of curves. The arrangement is "simple" in the sense that any three curves have no common intersection, any two curves intersect transversally, and each curve is non-singular. A curve is given as the zero set of an analytic function $f:\mathbb{R}^2\rightarrow

    更新日期:2020-09-03
  • Turning machines
    arXiv.cs.CG Pub Date : 2020-09-02
    Irina Kostitsyna; Cai Wood; Damien Woods

    Molecular robotics is challenging, so it seems best to keep it simple. We consider an abstract molecular robotics model based on simple folding instructions that execute asynchronously. Turning Machines are a simple 1D to 2D folding model, also easily generalisable to 2D to 3D folding. A Turning Machine starts out as a line of connected monomers in the discrete plane, each with an associated turning

    更新日期:2020-09-03
  • On Polyhedral Realization with Isosceles Triangles
    arXiv.cs.CG Pub Date : 2020-08-31
    David Eppstein

    Answering a question posed by Joseph Malkevitch, we prove that there exists a polyhedral graph, with triangular faces, such that every realization of it as the graph of a convex polyhedron includes at least one face that is a scalene triangle. Our construction is based on Kleetopes, and shows that there exists an integer $i$ such that all convex $i$-iterated Kleetopes have a scalene face. However,

    更新日期:2020-09-02
  • Topology and Neural Networks
    arXiv.cs.CG Pub Date : 2020-08-31
    Mustafa Hajij; Kyle Istvan

    We utilize classical facts from topology to show that the classification problem in machine learning is always solvable under very mild conditions. Furthermore, given a training dataset, we show how topological formalism can be used to suggest the appropriate architectural choices for neural networks designed to be trained as classifiers on the data. Finally, we show how the architecture of a neural

    更新日期:2020-09-01
  • A Topological Similarity Measure between Multi-Field Data using Multi-Resolution Reeb Spaces
    arXiv.cs.CG Pub Date : 2020-08-28
    Tripti Agarwal; Yashwanth Ramamurthi; Amit Chattopadhyay

    Searching topological similarity between a pair of shapes or data is an important problem in data analysis and visualization. The problem of computing similarity measures using scalar topology has been studied extensively and proven useful in shape and data matching. Even though multi-field (or multivariate) topology-based techniques reveal richer topological features, research on computing similarity

    更新日期:2020-08-31
  • FastORB-SLAM: a Fast ORB-SLAM Method with Coarse-to-Fine Descriptor Independent Keypoint Matching
    arXiv.cs.CG Pub Date : 2020-08-22
    Qiang Fu; Hongshan Yu; Xiaolong Wang; Zhengeng Yang; Hong Zhang; Ajmal Mian

    Indirect methods for visual SLAM are gaining popularity due to their robustness to varying environments. ORB-SLAM2 is a benchmark method in this domain, however, the computation of descriptors in ORB-SLAM2 is time-consuming and the descriptors cannot be reused unless a frame is selected as a keyframe. To overcome these problems, we present FastORB-SLAM which is light-weight and efficient as it tracks

    更新日期:2020-08-28
  • Flip Paths Between Lattice Triangulations
    arXiv.cs.CG Pub Date : 2020-08-25
    William Sims; Meera Sitharam

    The problem of finding a diagonal flip path between two triangulations has been studied for nearly a century in the combinatorial (topological) setting and for decades in the geometric setting. In the geometric setting, finding a diagonal flip path between two triangulations that minimizes the number of diagonal flips over all such paths is NP-complete. However, when restricted to lattice triangulations

    更新日期:2020-08-27
  • On the Maximum Number of Crossings in Star-Simple Drawings of $K_n$ with No Empty Lens
    arXiv.cs.CG Pub Date : 2020-08-25
    Stefan Felsner; Michael Hoffmann; Kristin Knorr; Irene Parada

    A star-simple drawing of a graph is a drawing in which adjacent edges do not cross. In contrast, there is no restriction on the number of crossings between two independent edges. When allowing empty lenses (a face in the arrangement induced by two edges that is bounded by a 2-cycle), two independent edges may cross arbitrarily many times in a star-simple drawing. We consider star-simple drawings of

    更新日期:2020-08-26
  • Simple Topological Drawings of $k$-Planar Graphs
    arXiv.cs.CG Pub Date : 2020-08-25
    Michael Hoffmann; Chih-Hung Liu; Meghana M. Reddy; Csaba D. Tóth

    Every finite graph admits a \emph{simple (topological) drawing}, that is, a drawing where every pair of edges intersects in at most one point. However, in combination with other restrictions simple drawings do not universally exist. For instance, \emph{$k$-planar graphs} are those graphs that can be drawn so that every edge has at most $k$ crossings (i.e., they admit a \emph{$k$-plane drawing}). It

    更新日期:2020-08-26
  • Limiting crossing numbers for geodesic drawings on the sphere
    arXiv.cs.CG Pub Date : 2020-08-24
    Marthe Bonamy; Bojan Mohar; Alexandra Wesolek

    We introduce a model for random geodesic drawings of the complete bipartite graph $K_{n,n}$ on the unit sphere $\mathbb{S}^2$ in $\mathbb{R}^3$, where we select the vertices in each bipartite class of $K_{n,n}$ with respect to two non-degenerate probability measures on $\mathbb{S}^2$. It has been proved recently that many such measures give drawings whose crossing number approximates the Zarankiewicz

    更新日期:2020-08-25
  • Stochastic Gradient Descent Works Really Well for Stress Minimization
    arXiv.cs.CG Pub Date : 2020-08-24
    Katharina Börsig; Ulrik Brandes; Barna Pasztor

    Stress minimization is among the best studied force-directed graph layout methods because it reliably yields high-quality layouts. It thus comes as a surprise that a novel approach based on stochastic gradient descent (Zheng, Pawar and Goodman, TVCG 2019) is claimed to improve on state-of-the-art approaches based on majorization. We present experimental evidence that the new approach does not actually

    更新日期:2020-08-25
  • An Integer-Linear Program for Bend-Minimization in Ortho-RadialDrawings
    arXiv.cs.CG Pub Date : 2020-08-24
    Benjamin Niedermann; Ignaz Rutter

    An ortho-radial grid is described by concentric circles and straight-line spokes emanating from the circles' center. An ortho-radial drawing is the analog of an orthogonal drawing on an ortho-radial grid. Such a drawing has an unbounded outer face and a central face that contains the origin. Building on the notion of an ortho-radial representation (Barth et al., SoCG, 2017), we describe an integer-linear

    更新日期:2020-08-25
  • Computing the Real Isolated Points of an Algebraic Hypersurface
    arXiv.cs.CG Pub Date : 2020-08-24
    Huu Phuoc Le; Mohab Safey El Din; Timo de Wolff

    Let $\mathbb{R}$ be the field of real numbers. We consider the problem of computing the real isolated points of a real algebraic set in $\mathbb{R}^n$ given as the vanishing set of a polynomial system. This problem plays an important role for studying rigidity properties of mechanism in material designs. In this paper, we design an algorithm which solves this problem. It is based on the computations

    更新日期:2020-08-25
  • Polygons with Prescribed Angles in 2D and 3D
    arXiv.cs.CG Pub Date : 2020-08-24
    Alon Efrat; Radoslav Fulek; Stephen Kobourov; Csaba D. Tóth

    We consider the construction of a polygon $P$ with $n$ vertices whose turning angles at the vertices are given by a sequence $A=(\alpha_0,\ldots, \alpha_{n-1})$, $\alpha_i\in (-\pi,\pi)$, for $i\in\{0,\ldots, n-1\}$. The problem of realizing $A$ by a polygon can be seen as that of constructing a straight-line drawing of a graph with prescribed angles at vertices, and hence, it is a special case of

    更新日期:2020-08-25
  • Geometric and statistical techniques for projective mapping of chocolate chip cookies with a large number of consumers
    arXiv.cs.CG Pub Date : 2020-08-24
    David Orden; Encarnación Fernández-Fernández; Marino Tejedor-Romero; Alejandra Martínez-Moraian

    This work uses projective mapping with n=349 consumers to evaluate nine commercial chocolate chips cookies, including a blind duplicate of a multinational best-selling brand and seven private labels. The data obtained are processed using both a statistical technique, Multiple Factor Analysis, and two geometric techniques, the recently appeared method SensoGraph using Gabriel clustering, and a novel

    更新日期:2020-08-25
  • Extending Partial Orthogonal Drawings
    arXiv.cs.CG Pub Date : 2020-08-24
    Patrizio AngeliniJohn Cabot University, Rome, Italy; Ignaz RutterUniversität Passau, Germany; Sandhya T PUniversität Passau, Germany

    We study the planar orthogonal drawing style within the framework of partial representation extension. Let $(G,H,{\Gamma}_H )$ be a partial orthogonal drawing, i.e., G is a graph, $H\subseteq G$ is a subgraph and ${\Gamma}_H$ is a planar orthogonal drawing of H. We show that the existence of an orthogonal drawing ${\Gamma}_G$ of $G$ that extends ${\Gamma}_H$ can be tested in linear time. If such a

    更新日期:2020-08-25
  • $(2+ε)$-ANN for time series under the Fréchet distance
    arXiv.cs.CG Pub Date : 2020-08-21
    Anne Driemel; Ioannis Psarros

    We give the first ANN-data structure for time series under the continuous Fr\'echet distance, where ANN stands for approximate near neighbor. Given a parameter $\epsilon \in (0,1]$, the data structure can be used to preprocess $n$ curves in Euclidean $\mathbb{R}$ (aka time series), each of complexity $m$, to answer queries with a curve of complexity $k$ by either returning a curve that lies within

    更新日期:2020-08-24
  • $2$-Layer $k$-Planar Graphs: Density, Crossing Lemma, Relationships, and Pathwidth
    arXiv.cs.CG Pub Date : 2020-08-21
    Patrizio Angelini; Giordano Da Lozzo; Henry Förster; Thomas Schneck

    The $2$-layer drawing model is a well-established paradigm to visualize bipartite graphs. Several beyond-planar graph classes have been studied under this model. Surprisingly, however, the fundamental class of $k$-planar graphs has been considered only for $k=1$ in this context. We provide several contributions that address this gap in the literature. First, we show tight density bounds for the classes

    更新日期:2020-08-24
  • On Turn-Regular Orthogonal Representations
    arXiv.cs.CG Pub Date : 2020-08-20
    Michael A. Bekos; Carla Binucci; Giuseppe Di Battista; Walter Didimo; Martin Gronemann; Karsten Klein; Maurizio Patrignani; Ignaz Rutter

    An interesting class of orthogonal representations consists of the so-called turn-regular ones, i.e., those that do not contain any pair of reflex corners that "point to each other" inside a face. For such a representation H it is possible to compute in linear time a minimum-area drawing, i.e., a drawing of minimum area over all possible assignments of vertex and bend coordinates of H. In contrast

    更新日期:2020-08-21
  • Plane Spanning Trees in Edge-Colored Simple Drawings of $K_n$
    arXiv.cs.CG Pub Date : 2020-08-20
    Oswin Aichholzer; Michael Hoffmann; Johannes Obenaus; Rosna Paul; Daniel Perz; Nadja Seiferth; Birgit Vogtenhuber; Alexandra Weinberger

    K\'{a}rolyi, Pach, and T\'{o}th proved that every 2-edge-colored straight-line drawing of the complete graph contains a monochromatic plane spanning tree. It is open if this statement generalizes to other classes of drawings, specifically, to simple drawings of the complete graph. These are drawings where edges are represented by Jordan arcs, any two of which intersect at most once. We present two

    更新日期:2020-08-21
  • On transversality of bent hyperplane arrangements and the topological expressiveness of ReLU neural networks
    arXiv.cs.CG Pub Date : 2020-08-20
    J. Elisenda Grigsby; Kathryn Lindsey

    Let F:R^n -> R be a feedforward ReLU neural network. It is well-known that for any choice of parameters, F is continuous and piecewise (affine) linear. We lay some foundations for a systematic investigation of how the architecture of F impacts the geometry and topology of its possible decision regions for binary classification tasks. Following the classical progression for smooth functions in differential

    更新日期:2020-08-21
  • A Simple Proof of Optimal Approximations
    arXiv.cs.CG Pub Date : 2020-08-20
    Mónika Csikós; Nabil H. Mustafa

    The fundamental result of Li, Long, and Srinivasan on approximations of set systems has become a key tool across several communities such as learning theory, algorithms, combinatorics and data analysis (described as `the pinnacle of a long sequence of papers'). The goal of this paper is to give a simpler, self-contained, modular proof of this result for finite set systems. The only ingredient we assume

    更新日期:2020-08-21
  • Quality tetrahedral mesh generation with HXT
    arXiv.cs.CG Pub Date : 2020-08-19
    Célestin Marot; Jean-François Remacle

    We proposed, in a recent paper (10.1002/nme.5987), a fast 3D parallel Delaunay kernel for tetrahedral mesh generation. This kernel was however incomplete in the sense that it lacked the necessary mesh improvement tools. The present paper builds on that previous work and proposes a fast parallel mesh improvement stage that delivers high-quality tetrahedral meshes compared to alternative open-source

    更新日期:2020-08-20
  • Parameterized Algorithms for Queue Layouts
    arXiv.cs.CG Pub Date : 2020-08-19
    Sujoy Bhore; Robert Ganian; Fabrizio Montecchiani; Martin Nöllenburg

    An $h$-queue layout of a graph $G$ consists of a linear order of its vertices and a partition of its edges into $h$ queues, such that no two independent edges of the same queue nest. The minimum $h$ such that $G$ admits an $h$-queue layout is the queue number of $G$. We present two fixed-parameter tractable algorithms that exploit structural properties of graphs to compute optimal queue layouts. As

    更新日期:2020-08-20
  • Planar L-Drawings of Bimodal Graphs
    arXiv.cs.CG Pub Date : 2020-08-18
    Patrizio Angelini; Steven Chaplick; Sabine Cornelsen; Giordano Da Lozzo

    In a planar L-drawing of a directed graph (digraph) each edge e is represented as a polyline composed of a vertical segment starting at the tail of e and a horizontal segment ending at the head of e. Distinct edges may overlap, but not cross. Our main focus is on bimodal graphs, i.e., digraphs admitting a planar embedding in which the incoming and outgoing edges around each vertex are contiguous. We

    更新日期:2020-08-19
  • Lazy caterer jigsaw puzzles: Models, properties, and a mechanical system-based solver
    arXiv.cs.CG Pub Date : 2020-08-17
    Peleg Harel; Ohad Ben-Shahar

    Jigsaw puzzle solving, the problem of constructing a coherent whole from a set of non-overlapping unordered fragments, is fundamental to numerous applications, and yet most of the literature has focused thus far on less realistic puzzles whose pieces are identical squares. Here we formalize a new type of jigsaw puzzle where the pieces are general convex polygons generated by cutting through a global

    更新日期:2020-08-19
  • Drawing Shortest Paths in Geodetic Graphs
    arXiv.cs.CG Pub Date : 2020-08-17
    Sabine Cornelsen; Maximilian Pfister; Henry Förster; Martin Gronemann; Michael Hoffmann; Stephen Kobourov; Thomas Schneck

    Motivated by the fact that in a space where shortest paths are unique, no two shortest paths meet twice, we study a question posed by Greg Bodwin: Given a geodetic graph $G$, i.e., an unweighted graph in which the shortest path between any pair of vertices is unique, is there a philogeodetic drawing of $G$, i.e., a drawing of $G$ in which the curves of any two shortest paths meet at most once? We answer

    更新日期:2020-08-19
  • When Lipschitz Walks Your Dog: Algorithm Engineering of the Discrete Fréchet Distance under Translation
    arXiv.cs.CG Pub Date : 2020-08-17
    Karl Bringmann; Marvin Künnemann; André Nusser

    Consider the natural question of how to measure the similarity of curves in the plane by a quantity that is invariant under translations of the curves. Such a measure is justified whenever we aim to quantify the similarity of the curves' shapes rather than their positioning in the plane, e.g., to compare the similarity of handwritten characters. Perhaps the most natural such notion is the (discrete)

    更新日期:2020-08-18
  • Geodesic Paths for Image Segmentation with Implicit Region-based Homogeneity Enhancement
    arXiv.cs.CG Pub Date : 2020-08-16
    Da Chen; Jian Zhu; Xinxin Zhang; Minglei Shu; Laurent D. Cohen

    Minimal paths are considered as a powerful and efficient tool for boundary detection and image segmentation due to its global optimality and well-established numerical solutions such as fast marching algorithm. In this paper, we introduce a flexible interactive image segmentation model based on the minimal geodesic framework in conjunction with region-based homogeneity enhancement. A key ingredient

    更新日期:2020-08-18
  • On Efficient Low Distortion Ultrametric Embedding
    arXiv.cs.CG Pub Date : 2020-08-15
    Vincent Cohen-Addad; Karthik C. S.; Guillaume Lagarde

    A classic problem in unsupervised learning and data analysis is to find simpler and easy-to-visualize representations of the data that preserve its essential properties. A widely-used method to preserve the underlying hierarchical structure of the data while reducing its complexity is to find an embedding of the data into a tree or an ultrametric. The most popular algorithms for this task are the classic

    更新日期:2020-08-18
  • Towards a characterization of stretchable aligned graphs
    arXiv.cs.CG Pub Date : 2020-08-14
    Marcel Radermacher; Ignaz Rutter; Peter Stumpf

    We consider the problem of stretching pseudolines in a planar straight-line drawing to straight lines while preserving the straightness and the combinatorial embedding of the drawing. We answer open questions by Mchedlidze et al. by showing that not all instances with two pseudolines are stretchable. On the positive side, for $k\geq 2$ pseudolines intersecting in a single point, we prove that in case

    更新日期:2020-08-17
  • Catching a Polygonal Fish with a Minimum Net
    arXiv.cs.CG Pub Date : 2020-08-11
    Sepideh Aghamolaei

    Given a polygon $P$ in the plane that can be translated, rotated and enlarged arbitrarily inside a unit square, the goal is to find a set of lines such that at least one of them always hits $P$ and the number of lines is minimized. We prove the solution is always a regular grid or a set of equidistant parallel lines, whose distance depends on $P$.

    更新日期:2020-08-17
  • Graph Drawing via Gradient Descent, $(GD)^2$
    arXiv.cs.CG Pub Date : 2020-08-12
    Reyan Ahmed; Felice De Luca; Sabin Devkota; Stephen Kobourov; Mingwei Li

    Readability criteria, such as distance or neighborhood preservation, are often used to optimize node-link representations of graphs to enable the comprehension of the underlying data. With few exceptions, graph drawing algorithms typically optimize one such criterion, usually at the expense of others. We propose a layout approach, Graph Drawing via Gradient Descent, $(GD)^2$, that can handle multiple

    更新日期:2020-08-14
  • A short proof of the non-biplanarity of $K_9$
    arXiv.cs.CG Pub Date : 2020-08-12
    Ahmad Biniaz

    Battle, Harary, and Kodama (1962) and independently Tutte (1963) proved that the complete graph with nine vertices is not biplanar. Aiming towards simplicity and brevity, in this note we provide a short proof of this claim.

    更新日期:2020-08-14
  • Analysis of self-equilibrated networks through cellular modeling
    arXiv.cs.CG Pub Date : 2020-08-11
    Omar Aloui; David Orden; Nizar Bel Hadj Ali; Landolf Rhode-Barbarigos

    Network equilibrium models represent a versatile tool for the analysis of interconnected objects and their relationships. They have been widely employed in both science and engineering to study the behavior of complex systems under various conditions, including external perturbations and damage. In this paper, network equilibrium models are revisited through graph-theory laws and attributes with special

    更新日期:2020-08-12
  • Obnoxious facility location: the case of weighted demand points
    arXiv.cs.CG Pub Date : 2020-08-10
    Pawel Kalczynski; Atsuo Suzuki; Zvi Drezner

    The problem considered in this paper is the weighted obnoxious facility location in the convex hull of demand points. The objective function is to maximize the smallest weighted distance between a facility and a set of demand points. Three new optimal solution approaches are proposed. Two variants of the "Big Triangle Small Triangle" global optimization method, and a procedure based on intersection

    更新日期:2020-08-12
  • Consistent High Dimensional Rounding with Side Information
    arXiv.cs.CG Pub Date : 2020-08-09
    Orr Dunkelman; Zeev Geyzel; Chaya Keller; Nathan Keller; Eyal Ronen; Adi Shamir; Ran J. Tessler

    In standard rounding, we want to map each value $X$ in a large continuous space (e.g., $R$) to a nearby point $P$ from a discrete subset (e.g., $Z$). This process seems to be inherently discontinuous in the sense that two consecutive noisy measurements $X_1$ and $X_2$ of the same value may be extremely close to each other and yet they can be rounded to different points $P_1\ne P_2$, which is undesirable

    更新日期:2020-08-11
  • A Sub-linear Time Algorithm for Approximating k-Nearest-Neighbor with Full Quality Guarantee
    arXiv.cs.CG Pub Date : 2020-08-07
    Hengzhao Ma; Jianzhong Li

    In this paper we propose an algorithm for the approximate k-Nearest-Neighbors problem. According to the existing researches, there are two kinds of approximation criterion. One is the distance criteria, and the other is the recall criteria. All former algorithms suffer the problem that there are no theoretical guarantees for the two approximation criterion. The algorithm proposed in this paper unifies

    更新日期:2020-08-10
  • Some `converses' to intrinsic linking theorems
    arXiv.cs.CG Pub Date : 2020-08-06
    R. Karasev; A. Skopenkov

    A low-dimensional version of our main result is the following `converse' of the Conway-Gordon-Sachs Theorem on intrinsic linking of the graph $K_6$ in 3-space: For any integer $z$ there are 6 points $1,2,3,4,5,6$ in 3-space, of which every two $i,j$ are joint by a polygonal line $ij$, the interior of one polygonal line is disjoint with any other polygonal line, the linking coefficient of any pair disjoint

    更新日期:2020-08-08
  • An Evolver program for weighted Steiner trees
    arXiv.cs.CG Pub Date : 2020-08-05
    Henrique Botelho; Francisco Zampirolli; Valério Ramos Batista

    We present an algorithm to find near-optimal weighted Steiner minimal trees in the plane. The algorithm is implemented in Evolver programming language, which already contains many built-in energy minimisation routines. Some are invoked in the program, which enable it to consist of only 183 lines of source code. Our algorithm reproduces the physical experiment of a soap film detaching from connected

    更新日期:2020-08-08
  • The Expected Size of Random Convex Layers and Convex Shells
    arXiv.cs.CG Pub Date : 2020-08-05
    Zhengyang Guo; Yi Li; Shaoyu Pei

    Given a planar point set $X$, we study the convex shells and the convex layers. We prove that when $X$ consists of points independently and uniformly sampled inside a convex polygon with $k$ vertices, the expected number of vertices on the first $t$ convex shells is $O(kt\log{n})$ for $t=O(\sqrt{n})$, and the expected number of vertices on the first $t$ convex layers is $O\left(kt^{3}\log{\frac{n}{t^2}}\right)$

    更新日期:2020-08-06
  • Persistent Homology in $\ell_{\infty}$ Metric
    arXiv.cs.CG Pub Date : 2020-08-05
    Gabriele Beltramo; Primoz Skraba

    Proximity complexes and filtrations are a central construction in topological data analysis. Built using distance functions or more generally metrics, they are often used to infer connectivity information from point clouds. We investigate proximity complexes and filtrations built over the Chebyshev metric, also known as the maximum metric or $\ell_{\infty}$ metric, rather than the classical Euclidean

    更新日期:2020-08-06
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