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  • Parametrization of Neural Networks with Connected Abelian Lie Groups as Data Manifold
    arXiv.cs.CG Pub Date : 2020-04-06
    Luciano Melodia; Richard Lenz

    Neural nets have been used in an elusive number of scientific disciplines. Nevertheless, their parameterization is largely unexplored. Dense nets are the coordinate transformations of a manifold from which the data is sampled. After processing through a layer, the representation of the original manifold may change. This is crucial for the preservation of its topological structure and should therefore

    更新日期:2020-04-08
  • Probing a Set of Trajectories to Maximize Captured Information
    arXiv.cs.CG Pub Date : 2020-04-07
    Sándor P. Fekete; Alexander Hill; Dominik Krupke; Tyler Mayer; Joseph S. B. Mitchell; Ojas Parekh; Cynthia A. Phillips

    We study a trajectory analysis problem we call the Trajectory Capture Problem (TCP), in which, for a given input set ${\cal T}$ of trajectories in the plane, and an integer $k\geq 2$, we seek to compute a set of $k$ points (``portals'') to maximize the total weight of all subtrajectories of ${\cal T}$ between pairs of portals. This problem naturally arises in trajectory analysis and summarization.

    更新日期:2020-04-08
  • Circuminvariants of 3-Periodics in the Elliptic Billiard
    arXiv.cs.CG Pub Date : 2020-04-06
    Dan Reznik; Ronaldo Garcia

    A Circumconic passes through a triangle's vertices; an Inconic is tangent to the sidelines. We study the variable geometry of certain conics derived from the 1d family of 3-periodics in the Elliptic Billiard. Some display intriguing invariances such as aspect ratio and pairwise ratio of focal lengths. We also define the Circumbilliard, a circumellipse to a generic triangle for which the latter is a

    更新日期:2020-04-08
  • Local digital algorithms applied to Boolean models
    arXiv.cs.CG Pub Date : 2015-11-03
    Julia Hörrmann; Anne Marie Svane

    We investigate the estimation of specific intrinsic volumes of stationary Boolean models by local digital algorithms; that is, by weighted sums of $n \times\ldots \times n$ configuration counts. We show that asymptotically unbiased estimators for the specific surface area or integrated mean curvature do not exist if the dimension is at least two or three, respectively. For 3-dimensional stationary

    更新日期:2020-04-08
  • Reconstruction of convex bodies from moments
    arXiv.cs.CG Pub Date : 2016-05-20
    Astrid Kousholt; Julia Schulte

    We investigate how much information about a convex body can be retrieved from a finite number of its geometric moments. We give a sufficient condition for a convex body to be uniquely determined by a finite number of its geometric moments, and we show that among all convex bodies, those which are uniquely determined by a finite number of moments form a dense set. Further, we derive a stability result

    更新日期:2020-04-08
  • Dots & Polygons
    arXiv.cs.CG Pub Date : 2020-04-02
    Kevin Buchin; Mart Hagedoorn; Irina Kostitsyna; Max van Mulken; Jolan Rensen; Leo van Schooten

    We present a new game, Dots & Polygons, played on a planar point set. Players take turns connecting two points, and when a player closes a (simple) polygon, the player scores its area. We show that deciding whether the game can be won from a given state, is NP-hard. We do so by a reduction from vertex-disjoint cycle packing in cubic planar graphs, including a self-contained reduction from planar 3-Satisfiability

    更新日期:2020-04-06
  • Geodesic Spanners for Points in $\mathbb{R}^3$ amid Axis-parallel Boxes
    arXiv.cs.CG Pub Date : 2020-04-03
    Mohammad Ali Abam; Mohammad Javad Rezaei Seraji

    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.

    更新日期:2020-04-06
  • Geometry-Driven Detection, Tracking and Visual Analysis of Viscous and Gravitational Fingers
    arXiv.cs.CG Pub Date : 2019-11-27
    Jiayi Xu; Soumya Dutta; Wenbin He; Joachim Moortgat; Han-Wei Shen

    Viscous and gravitational flow instabilities cause a displacement front to break up into finger-like fluids. The detection and evolutionary analysis of these fingering instabilities are critical in multiple scientific disciplines such as fluid mechanics and hydrogeology. However, previous detection methods of the viscous and gravitational fingers are based on density thresholding, which provides limited

    更新日期:2020-04-06
  • k-Median clustering under discrete Fréchet and Hausdorff distances
    arXiv.cs.CG Pub Date : 2020-04-01
    Abhinandan Nath; Erin Taylor

    We give the first near-linear time $(1+\eps)$-approximation algorithm for $k$-median clustering of polygonal trajectories under the discrete Fr\'{e}chet distance, and the first polynomial time $(1+\eps)$-approximation algorithm for $k$-median clustering of finite point sets under the Hausdorff distance, provided the cluster centers, ambient dimension, and $k$ are bounded by a constant. The main technique

    更新日期:2020-04-03
  • Terrain Visibility Graphs: Persistence is Not Enough
    arXiv.cs.CG Pub Date : 2020-04-02
    Safwa Ameer; Matt Gibson-Lopez; Erik Krohn; Sean Soderman; Qing Wang

    In this paper, we consider the Visibility Graph Recognition and Reconstruction problems in the context of terrains. Here, we are given a graph $G$ with labeled vertices $v_0, v_1, \ldots, v_{n-1}$ such that the labeling corresponds with a Hamiltonian path $H$. $G$ also may contain other edges. We are interested in determining if there is a terrain $T$ with vertices $p_0, p_1, \ldots, p_{n-1}$ such

    更新日期:2020-04-03
  • A Spectral Approach to the Shortest Path Problem
    arXiv.cs.CG Pub Date : 2020-04-02
    Stefan Steinerberger

    Let $G=(V,E)$ be a simple, connected graph. One is often interested in a short path between two vertices $u,v$. We propose a spectral algorithm: construct the function $\phi:V \rightarrow \mathbb{R}_{\geq 0}$ $$ \phi = \arg\min_{f:V \rightarrow \mathbb{R} \atop f(u) = 0, f \not\equiv 0} \frac{\sum_{(w_1, w_2) \in E}{(f(w_1)-f(w_2))^2}}{\sum_{w \in V}{f(w)^2}}.$$ $\phi$ can also be understood as the

    更新日期:2020-04-03
  • Parallel Computation of Alpha Complex for Biomolecules
    arXiv.cs.CG Pub Date : 2019-08-16
    Talha Bin Masood; Tathagata Ray; Vijay Natarajan

    The alpha complex, a subset of the Delaunay triangulation, has been extensively used as the underlying representation for biomolecular structures. We propose a GPU-based parallel algorithm for the computation of the alpha complex, which exploits the knowledge of typical spatial distribution and sizes of atoms in a biomolecule. Unlike existing methods, this algorithm does not require prior construction

    更新日期:2020-04-03
  • SPARE3D: A Dataset for SPAtial REasoning on Three-View Line Drawings
    arXiv.cs.CG Pub Date : 2020-03-31
    Wenyu Han; Siyuan Xiang; Chenhui Liu; Ruoyu Wang; Chen Feng

    Spatial reasoning is an important component of human intelligence. We can imagine the shapes of 3D objects and reason about their spatial relations by merely looking at their three-view line drawings in 2D, with different levels of competence. Can deep networks be trained to perform spatial reasoning tasks? How can we measure their "spatial intelligence"? To answer these questions, we present the SPARE3D

    更新日期:2020-04-01
  • Drawing Halin-graphs with small height
    arXiv.cs.CG Pub Date : 2020-03-31
    Therese Biedl; Milap Sheth

    In this paper, we study how to draw Halin-graphs, i.e., planar graphs that consist of a tree $T$ and a cycle among the leaves of that tree. Based on tree-drawing algorithms and the pathwidth $ pw(T) $, a well-known graph parameter, we find poly-line drawings of height at most $6pw(T)+3\in O(\log n)$. We also give an algorithm for straight-line drawings, and achieve height at most $12pw(T)+1$ for Halin-graphs

    更新日期:2020-04-01
  • Representing Graphs and Hypergraphs by Touching Polygons in 3D
    arXiv.cs.CG Pub Date : 2019-08-22
    William Evans; Paweł Rzążewski; Noushin Saeedi; Chan-Su Shin; Alexander Wolff

    Contact representations of graphs have a long history. Most research has focused on problems in 2D, but 3D contact representations have also been investigated, mostly concerning fully-dimensional geometric objects such as spheres or cubes. In this paper we study contact representations with convex polygons in 3D. We show that every graph admits such a representation. Since our representations use super-polynomial

    更新日期:2020-04-01
  • Synthesizing Structured CAD Models with Equality Saturation and Inverse Transformations
    arXiv.cs.CG Pub Date : 2019-09-26
    Chandrakana Nandi; Max Willsey; Adam Anderson; James R. Wilcox; Eva Darulova; Dan Grossman; Zachary Tatlock

    Recent program synthesis techniques help users customize CAD models(e.g., for 3D printing) by decompiling low-level triangle meshes to Constructive Solid Geometry (CSG) expressions. Without loops or functions, editing CSG can require many coordinated changes, and existing mesh decompilers use heuristics that can obfuscate high-level structure. This paper proposes a second decompilation stage to robustly

    更新日期:2020-04-01
  • Plane-filling trails
    arXiv.cs.CG Pub Date : 2020-03-28
    Herman Haverkort

    The order in which plane-filling curves visit points in the plane can be exploited to design efficient algorithms. Typically, the curves are useful because they preserve locality: points that are close to each other along the curve tend to be close to each other in the plane, and vice versa. However, sketches of plane-filling curves do not show this well: they are hard to read on different levels of

    更新日期:2020-03-31
  • Single-Point Visibility Constraint Minimum Link Paths in Simple Polygons
    arXiv.cs.CG Pub Date : 2020-03-28
    Mohammad Reza Zarrabi; Nasrollah Moghaddam Charkari

    We address the following problem: Given a simple polygon P with n vertices and two points s and t inside it, find a minimum link path between them such that a given target point q is visible from at least one point on the path. The method is based on partitioning a portion of P into a number of faces of equal link distance from a source point. This partitioning is essentially a shortest path map (SPM)

    更新日期:2020-03-31
  • Hardness of Reconfiguring Robot Swarms with Uniform External Control in Limited Directions
    arXiv.cs.CG Pub Date : 2020-03-29
    David Caballero; Angel A. Cantu; Timothy Gomez; Austin Luchsinger; Robert Schweller; Tim Wylie

    Motivated by advances is nanoscale applications and simplistic robot agents, we look at problems based on using a global signal to move all agents when given a limited number of directional signals and immovable geometry. We study a model where unit square particles move within a 2D grid based on uniform external forces. Movement is based on a sequence of uniform commands which cause all particles

    更新日期:2020-03-31
  • How to Find a Point in the Convex Hull Privately
    arXiv.cs.CG Pub Date : 2020-03-30
    Haim Kaplan; Micha Sharir; Uri Stemmer

    We study the question of how to compute a point in the convex hull of an input set $S$ of $n$ points in ${\mathbb R}^d$ in a differentially private manner. This question, which is trivial non-privately, turns out to be quite deep when imposing differential privacy. In particular, it is known that the input points must reside on a fixed finite subset $G\subseteq{\mathbb R}^d$, and furthermore, the size

    更新日期:2020-03-31
  • Optimizing Geometry Compression using Quantum Annealing
    arXiv.cs.CG Pub Date : 2020-03-30
    Sebastian Feld; Markus Friedrich; Claudia Linnhoff-Popien

    The compression of geometry data is an important aspect of bandwidth-efficient data transfer for distributed 3d computer vision applications. We propose a quantum-enabled lossy 3d point cloud compression pipeline based on the constructive solid geometry (CSG) model representation. Key parts of the pipeline are mapped to NP-complete problems for which an efficient Ising formulation suitable for the

    更新日期:2020-03-31
  • Long Alternating Paths Exist
    arXiv.cs.CG Pub Date : 2020-03-30
    Wolfgang Mulzer; Pavel Valtr

    Let $P$ be a set of $2n$ points in convex position, such that $n$ points are colored red and $n$ points are colored blue. A non-crossing alternating path on $P$ of length $\ell$ is a sequence $p_1, \dots, p_\ell$ of $\ell$ points from $P$ so that (i) all points are pairwise distinct; (ii) any two consecutive points $p_i$, $p_{i+1}$ have different colors; and (iii) any two segments $p_i p_{i+1}$ and

    更新日期:2020-03-31
  • Faster Approximation Algorithms for Geometric Set Cover
    arXiv.cs.CG Pub Date : 2020-03-26
    Timothy M. Chan; Qizheng He

    We improve the running times of $O(1)$-approximation algorithms for the set cover problem in geometric settings, specifically, covering points by disks in the plane, or covering points by halfspaces in three dimensions. In the unweighted case, Agarwal and Pan [SoCG 2014] gave a randomized $O(n\log^4 n)$-time, $O(1)$-approximation algorithm, by using variants of the multiplicative weight update (MWU)

    更新日期:2020-03-31
  • Barycentric cuts through a convex body
    arXiv.cs.CG Pub Date : 2020-03-27
    Zuzana Patáková; Martin Tancer; Uli Wagner

    Let $K$ be a convex body in $\mathbb{R}^n$ (i.e., a compact convex set with nonempty interior). Given a point $p$ in the interior of $K$, a hyperplane $h$ passing through $p$ is called barycentric if $p$ is the barycenter of $K \cap h$. In 1961, Gr\"{u}nbaum raised the question whether, for every $K$, there exists an interior point $p$ through which there are at least $n+1$ distinct barycentric hyperplanes

    更新日期:2020-03-31
  • Connectivity of Triangulation Flip Graphs in the Plane
    arXiv.cs.CG Pub Date : 2020-03-30
    Uli Wagner; Emo Welzl

    Given a finite point set $P$ in general position in the plane, a full triangulation is a maximal straight-line embedded plane graph on $P$. A partial triangulation on $P$ is a full triangulation of some subset $P'$ of $P$ containing all extreme points in $P$. A bistellar flip on a partial triangulation either flips an edge, removes a non-extreme point of degree 3, or adds a point in $P \setminus P'$

    更新日期:2020-03-31
  • Combinatorial Properties of Self-Overlapping Curves and Interior Boundaries
    arXiv.cs.CG Pub Date : 2020-03-30
    Parker Evans; Carola Wenk

    We study the interplay between the recently defined concept of minimum homotopy area and the classical topic of self-overlapping curves. The latter are plane curves which are the image of the boundary of an immersed disk. Our first contribution is to prove new sufficient combinatorial conditions for a curve to be self-overlapping. We show that a curve $\gamma$ with Whitney index 1 and without any self-overlapping

    更新日期:2020-03-31
  • PU-GCN: Point Cloud Upsampling using Graph Convolutional Networks
    arXiv.cs.CG Pub Date : 2019-11-30
    Guocheng Qian; Abdulellah Abualshour; Guohao Li; Ali Thabet; Bernard Ghanem

    The effectiveness of learning-based point cloud upsampling pipelines heavily relies on the upsampling modules and feature extractors used therein. We propose three novel point upsampling modules: Multi-branch GCN, Clone GCN, and NodeShuffle. Our modules use Graph Convolutional Networks (GCNs) to better encode local point information from the point neighborhood. These upsampling modules are versatile

    更新日期:2020-03-31
  • LIMP: Learning Latent Shape Representations with Metric Preservation Priors
    arXiv.cs.CG Pub Date : 2020-03-27
    Luca Cosmo; Antonio Norelli; Oshri Halimi; Ron Kimmel; Emanuele Rodolà

    In this paper, we advocate the adoption of metric preservation as a powerful prior for learning latent representations of deformable 3D shapes. Key to our construction is the introduction of a geometric distortion criterion, defined directly on the decoded shapes, translating the preservation of the metric on the decoding to the formation of linear paths in the underlying latent space. Our rationale

    更新日期:2020-03-30
  • Further Results on Colored Range Searching
    arXiv.cs.CG Pub Date : 2020-03-25
    Timothy M. Chan; Qizheng He; Yakov Nekrich

    We present a number of new results about range searching for colored (or "categorical") data: 1. For a set of $n$ colored points in three dimensions, we describe randomized data structures with $O(n\mathop{\rm polylog}n)$ space that can report the distinct colors in any query orthogonal range (axis-aligned box) in $O(k\mathop{\rm polyloglog} n)$ expected time, where $k$ is the number of distinct colors

    更新日期:2020-03-27
  • Geometric Pattern Matching Reduces to k-SUM
    arXiv.cs.CG Pub Date : 2020-03-26
    Boris Aronov; Jean Cardinal

    We prove that some exact geometric pattern matching problems reduce in linear time to $k$-SUM when the pattern has a fixed size $k$. This holds in the real RAM model for searching for a similar copy of a set of $k\geq 3$ points within a set of $n$ points in the plane, and for searching for an affine image of a set of $k\geq d+2$ points within a set of $n$ points in $d$-space. As corollaries, we obtain

    更新日期:2020-03-27
  • Geometric Sparsification of Closeness Relations: Eigenvalue Clustering for Computing Matrix Functions
    arXiv.cs.CG Pub Date : 2020-03-23
    Nir Goren; Dan Halperin; Sivan Toledo

    We show how to efficiently solve a clustering problem that arises in a method to evaluate functions of matrices. The problem requires finding the connected components of a graph whose vertices are eigenvalues of a real or complex matrix and whose edges are pairs of eigenvalues that are at most \delta away from each other. Davies and Higham proposed solving this problem by enumerating the edges of the

    更新日期:2020-03-27
  • A Topological Characterization of Modulo-p Arguments and Implications for Necklace Splitting
    arXiv.cs.CG Pub Date : 2020-03-26
    Aris Filos-Ratsikas; Alexandros Hollender; Katerina Sotiraki; Manolis Zampetakis

    The classes PPA-$p$ have attracted attention lately, because they are the main candidates for capturing the complexity of Necklace Splitting with $p$ thieves, for prime $p$. However, these classes are not known to have complete problems of a topological nature, which impedes any progress towards settling the complexity of the problem. On the contrary, such problems have been pivotal in obtaining completeness

    更新日期:2020-03-27
  • Taming Near Repeat Calculation for Crime Analysis via Cohesive Subgraph Computing
    arXiv.cs.CG Pub Date : 2017-05-18
    Zhaoming Yin; Xuan Shi

    Near repeat (NR) is a well known phenomenon in crime analysis assuming that crime events exhibit correlations within a given time and space frame. Traditional NR calculation generates 2 event pairs if 2 events happened within a given space and time limit. When the number of events is large, however, NR calculation is time consuming and how these pairs are organized are not yet explored. In this paper

    更新日期:2020-03-27
  • Expected dispersion of uniformly distributed points
    arXiv.cs.CG Pub Date : 2019-11-27
    Aicke Hinrichs; David Krieg; Robert J. Kunsch; Daniel Rudolf

    The dispersion of a point set in $[0,1]^d$ is the volume of the largest axis parallel box inside the unit cube that does not intersect with the point set. We study the expected dispersion with respect to a random set of $n$ points determined by an i.i.d. sequence of uniformly distributed random variables. Depending on the number of points $n$ and the dimension $d$ we provide an upper and lower bound

    更新日期:2020-03-27
  • Polymake.jl: A new interface to polymake
    arXiv.cs.CG Pub Date : 2020-03-23
    Marek Kaluba; Benjamin Lorenz; Sascha Timme

    We present the Julia interface Polymake.jl to polymake, a software for research in polyhedral geometry. We describe the technical design and how the integration into Julia makes it possible to combine polymake with state-of-the-art numerical software.

    更新日期:2020-03-26
  • Two almost-circles, and two real ones
    arXiv.cs.CG Pub Date : 2020-03-25
    Zoltán Kovács

    Implicit locus equations in GeoGebra allow the user to do experiments with generalization of the concept of ellipses, namely with $n$-ellipses. By experimenting we obtain a geometric object that is very similar to a set of two circles.

    更新日期:2020-03-26
  • An Implicit Representation of Swept Volumes based on Local Shapes and Movements
    arXiv.cs.CG Pub Date : 2020-03-25
    Clément Laroche

    We introduce a new way to implicitly represent swept volumes in 3D. We first implicitize the base volume and then apply the time-dependent rigid transformation to build an implicit representation of the swept volume. This way, we build a fitting representation, where geometric details generated by the rigid transformation and details of the base volume itself are both taken into account. This representation

    更新日期:2020-03-26
  • Weighted Maximum Independent Set of Geometric Objects in Turnstile Streams
    arXiv.cs.CG Pub Date : 2019-02-27
    Ainesh Bakshi; Nadiia Chepurko; David P. Woodruff

    We study the Maximum Independent Set problem for geometric objects given in the data stream model. A set of geometric objects is said to be independent if the objects are pairwise disjoint. We consider geometric objects in one and two dimensions, i.e., intervals and disks. Let $\alpha$ be the cardinality of the largest independent set. Our goal is to estimate $\alpha$ in a small amount of space, given

    更新日期:2020-03-26
  • Testing Polynomials for Vanishing on Cartesian Products of Planar Point Sets
    arXiv.cs.CG Pub Date : 2020-03-21
    Boris Aronov; Esther Ezra; Micha Sharir

    We present subquadratic algorithms, in the algebraic decision-tree model of computation, for detecting whether there exists a triple of points, belonging to three respective sets $A$, $B$, and $C$ of points in the plane, that satisfy a certain polynomial equation or two equations. The best known instance of such a problem is testing for the existence of a collinear triple of points in $A\times B\times

    更新日期:2020-03-24
  • Volumetric density-equalizing reference map with applications
    arXiv.cs.CG Pub Date : 2020-03-21
    Gary P. T. Choi; Chris H. Rycroft

    The density-equalizing map, a technique developed for cartogram creation, has been widely applied to data visualization but only for 2D applications. In this work, we propose a novel method called the volumetric density-equalizing reference map (VDERM) for computing density-equalizing map for volumetric domains. Given a prescribed density distribution in a volumetric domain in $\mathbb{R}^3$, the proposed

    更新日期:2020-03-24
  • On the snappability and singularity-distance of frameworks with bars and triangular plates
    arXiv.cs.CG Pub Date : 2020-03-22
    Georg Nawratil

    In a recent article the author presented a method to measure the snapping capability -- shortly called snappability -- of bar-joint frameworks based on the total elastic strain energy by computing the deformation of all bars using Hooke's law and the definition of Cauchy/Engineering strain. Within the paper at hand, we extend this approach to frameworks composed of bars and triangular plates by using

    更新日期:2020-03-24
  • 1 x 1 Rush Hour with Fixed Blocks is PSPACE-complete
    arXiv.cs.CG Pub Date : 2020-03-22
    Josh Brunner; Lily Chung; Erik D. Demaine; Dylan Hendrickson; Adam Hesterberg; Adam Suhl; Avi Zeff

    Consider $n^2-1$ unit-square blocks in an $n \times n$ square board, where each block is labeled as movable horizontally (only), movable vertically (only), or immovable -- a variation of Rush Hour with only $1 \times 1$ cars and fixed blocks. We prove that it is PSPACE-complete to decide whether a given block can reach the left edge of the board, by reduction from Nondeterministic Constraint Logic

    更新日期:2020-03-24
  • A Toroidal Maxwell-Cremona-Delaunay Correspondence
    arXiv.cs.CG Pub Date : 2020-03-23
    Jeff Erickson; Patrick Lin

    We consider three classes of geodesic embeddings of graphs on Euclidean flat tori: (1) A torus graph $G$ is equilibrium if it is possible to place positive weights on the edges, such that the weighted edge vectors incident to each vertex of G sum to zero. (2) A torus graph $G$ is reciprocal if there is a geodesic embedding of the dual graph $G^*$ on the same flat torus, where each edge of G is orthogonal

    更新日期:2020-03-24
  • Euclidean TSP in Narrow Strip
    arXiv.cs.CG Pub Date : 2020-03-22
    Henk Alkema; Mark de Berg; Sándor Kisfaludi-Bak

    We investigate how the complexity of Euclidean TSP for point sets $P$ inside the strip $(-\infty,+\infty)\times [0,\delta]$ depends on the strip width $\delta$. We obtain two main results. First, for the case where the points have distinct integer $x$-coordinates, we prove that a shortest bitonic tour (which can be computed in $O(n\log^2 n)$ time using an existing algorithm) is guaranteed to be a shortest

    更新日期:2020-03-24
  • The Stretch Factor of Hexagon-Delaunay Triangulations
    arXiv.cs.CG Pub Date : 2017-10-31
    Michael Dennis; Ljubomir Perković; Duru Türkoğlu

    The problem of computing the exact stretch factor (i.e., the tight bound on the worst case stretch factor) of a Delaunay triangulation is one of the longstanding open problems in computational geometry. Over the years, a series of upper and lower bounds on the exact stretch factor have been obtained but the gap between them is still large. An alternative approach to solving the problem is to develop

    更新日期:2020-03-24
  • On embeddability of unit disk graphs onto straight lines
    arXiv.cs.CG Pub Date : 2018-11-24
    Onur Çağırıcı

    Unit disk graphs are the intersection graphs of unit radius disks in the Euclidean plane. Deciding whether there exists an embedding of a given unit disk graph, i.e. unit disk graph recognition, is an important geometric problem, and has many application areas. In general, this problem is known to be $\exists\mathbb{R}$-complete. In some applications, the objects that correspond to unit disks, have

    更新日期:2020-03-24
  • Approximating the Existential Theory of the Reals
    arXiv.cs.CG Pub Date : 2018-10-02
    Argyrios Deligkas; John Fearnley; Themistoklis Melissourgos; Paul G. Spirakis

    The Existential Theory of the Reals (ETR) consists of existentially quantified Boolean formulas over equalities and inequalities of polynomial functions of variables in $\mathbb{R}$. In this paper we propose and study the approximate existential theory of the reals ($\epsilon$-ETR), in which the constraints only need to be satisfied approximately. We first show that when the domain of the variables

    更新日期:2020-03-24
  • Convex Hulls of Random Order Types
    arXiv.cs.CG Pub Date : 2020-03-18
    Xavier Goaoc; Emo Welzl

    We establish the following two main results on order types of points in general position in the plane (realizable simple planar order types, realizable uniform acyclic oriented matroids of rank~$3$): - The number of extreme points in an $n$-point order type, chosen uniformly at random from all such order types, is on average $4+o(1)$. For labeled order types, this number has average $4-\frac{8}{n^2-n+2}$

    更新日期:2020-03-20
  • Fan-Crossing Free Graphs
    arXiv.cs.CG Pub Date : 2020-03-18
    Franz J. Brandenburg

    A graph is fan-crossing free if it admits a drawing in the plane so that each edge can be crossed by independent edges. Then the crossing edges have distinct vertices. In complement, a graph is fan-crossing if each edge can be crossed by edges of a fan. Then the crossing edges are incident to a common vertex. Graphs are k-planar if each edge is crossed by at most k edges, and k-gap-planar if each crossing

    更新日期:2020-03-20
  • PT2PC: Learning to Generate 3D Point Cloud Shapes from Part Tree Conditions
    arXiv.cs.CG Pub Date : 2020-03-19
    Kaichun Mo; He Wang; Xinchen Yan; Leonidas J. Guibas

    3D generative shape modeling is a fundamental research area in computer vision and interactive computer graphics, with many real-world applications. This paper investigates the novel problem of generating 3D shape point cloud geometry from a symbolic part tree representation. In order to learn such a conditional shape generation procedure in an end-to-end fashion, we propose a conditional GAN "part

    更新日期:2020-03-20
  • Minimum Scan Cover with Angular Transition Costs
    arXiv.cs.CG Pub Date : 2020-03-19
    Sándor P. Fekete; Linda Kleist; Dominik Krupke

    We provide a comprehensive study of a natural geometric optimization problem motivated by questions in the context of satellite communication and astrophysics. In the problem Minimum Scan Cover with Angular Costs (MSC), we are given a graph $G$ that is embedded in Euclidean space. The edges of $G$ need to be scanned, i.e., probed from both of their vertices. In order to scan their edge, two vertices

    更新日期:2020-03-20
  • GPU-Accelerated Computation of Vietoris-Rips Persistence Barcodes
    arXiv.cs.CG Pub Date : 2020-03-17
    Simon Zhang; Mengbai Xiao; Hao Wang

    The computation of Vietoris-Rips persistence barcodes is both execution-intensive and memory-intensive. In this paper, we study the computational structure of Vietoris-Rips persistence barcodes, and identify several unique mathematical properties and algorithmic opportunities with connections to the GPU. Mathematically and empirically, we look into the properties of apparent pairs, which are independently

    更新日期:2020-03-19
  • Site2Vec: a reference frame invariant algorithm for vector embedding of protein-ligand binding sites
    arXiv.cs.CG Pub Date : 2020-03-18
    Arnab Bhadra; Kalidas Y

    Protein-ligand interaction is one of the fundamental molecular interactions of living systems. Proteins are the building blocks of functions in life at the molecular level. Ligands are small molecules that interact with proteins at specific regions on the surface of proteins called binding sites. Understanding the physicochemical properties of ligand-binding sites is very important in the field of

    更新日期:2020-03-19
  • Worst-Case Optimal Covering of Rectangles by Disks
    arXiv.cs.CG Pub Date : 2020-03-18
    Sándor P. Fekete; Utkarsh Gupta; Phillip Keldenich; Christian Scheffer; Sahil Shah

    We provide the solution for a fundamental problem of geometric optimization by giving a complete characterization of worst-case optimal disk coverings of rectangles: For any $\lambda\geq 1$, the critical covering area $A^*(\lambda)$ is the minimum value for which any set of disks with total area at least $A^*(\lambda)$ can cover a rectangle of dimensions $\lambda\times 1$. We show that there is a threshold

    更新日期:2020-03-19
  • Dynamic Distribution-Sensitive Point Location
    arXiv.cs.CG Pub Date : 2020-03-18
    Siu-Wing Cheng; Man-Kit Lau

    We propose a dynamic data structure for the distribution-sensitive point location problem. Suppose that there is a fixed query distribution in $\mathbb{R}^2$, and we are given an oracle that can return in $O(1)$ time the probability of a query point falling into a polygonal region of constant complexity. We can maintain a convex subdivision $\cal S$ with $n$ vertices such that each query is answered

    更新日期:2020-03-19
  • A Generalization of Self-Improving Algorithms
    arXiv.cs.CG Pub Date : 2020-03-18
    Siu-Wing Cheng; Man-Kwun Chiu; Kai Jin; Man Ting Wong

    Ailon et al.~[SICOMP'11] proposed self-improving algorithms for sorting and Delaunay triangulation (DT) when the input instances $x_1,\cdots,x_n$ follow some unknown \emph{product distribution}. That is, $x_i$ comes from a fixed unknown distribution $\mathcal{D}_i$, and the $x_i$'s are drawn independently. After spending $O(n^{1+\varepsilon})$ time in a learning phase, the subsequent expected running

    更新日期:2020-03-19
  • Tatamibari is NP-complete
    arXiv.cs.CG Pub Date : 2020-03-18
    Aviv Adler; Jeffrey Bosboom; Erik D. Demaine; Martin L. Demaine; Quanquan C. Liu; Jayson Lynch

    In the Nikoli pencil-and-paper game Tatamibari, a puzzle consists of an $m \times n$ grid of cells, where each cell possibly contains a clue among +, -, |. The goal is to partition the grid into disjoint rectangles, where every rectangle contains exactly one clue, rectangles containing + are square, rectangles containing - are strictly longer horizontally than vertically, rectangles containing | are

    更新日期:2020-03-19
  • On beta-Plurality Points in Spatial Voting Games
    arXiv.cs.CG Pub Date : 2020-03-17
    Boris Aronov; Mark de Berg; Joachim Gudmundsson; Michael Horton

    Let $V$ be a set of $n$ points in $\mathbb{R}^d$, called voters. A point $p\in \mathbb{R}^d$ is a plurality point for $V$ when the following holds: for every $q\in\mathbb{R}^d$ the number of voters closer to $p$ than to $q$ is at least the number of voters closer to $q$ than to $p$. Thus, in a vote where each $v\in V$ votes for the nearest proposal (and voters for which the proposals are at equal distance

    更新日期:2020-03-18
  • Book Embeddings of Nonplanar Graphs with Small Faces in Few Pages
    arXiv.cs.CG Pub Date : 2020-03-17
    Michael A. Bekos; Giordano Da Lozzo; Svenja Griesbach; Martin Gronemann; Fabrizio Montecchiani; Chrysanthi Raftopoulou

    An embedding of a graph in a book, called book embedding, consists of a linear ordering of its vertices along the spine of the book and an assignment of its edges to the pages of the book, so that no two edges on the same page cross. The book thickness of a graph is the minimum number of pages over all its book embeddings. For planar graphs, a fundamental result is due to Yannakakis, who proposed an

    更新日期:2020-03-18
  • The Parameterized Complexity of Guarding Almost Convex Polygons
    arXiv.cs.CG Pub Date : 2020-03-17
    Akanksha Agrawal; Kristine V. K. Knudsen; Daniel Lokshtanov; Saket Saurabh; Meirav Zehavi

    Art Gallery is a fundamental visibility problem in Computational Geometry. The input consists of a simple polygon P, (possibly infinite) sets G and C of points within P, and an integer k; the task is to decide if at most k guards can be placed on points in G so that every point in C is visible to at least one guard. In the classic formulation of Art Gallery, G and C consist of all the points within

    更新日期:2020-03-18
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