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  • Euclidean minimum spanning trees with independent and dependent geometric uncertainties
    Comput. Geom. (IF 0.476) Pub Date : 2021-01-06
    Rivka Gitik; Or Bartal; Leo Joskowicz

    We address the problems of constructing the Euclidean Minimum Spanning Tree (EMST) of points in the plane with mutually dependent location uncertainties, testing its stability, and computing its total weight. Point coordinate uncertainties are modeled with the Linear Parametric Geometric Uncertainty Model (LPGUM), an expressive and computationally efficient worst-case, first order linear approximation

    更新日期:2021-01-13
  • Coloring Delaunay-edges and their generalizations
    Comput. Geom. (IF 0.476) Pub Date : 2021-01-11
    Eyal Ackerman; Balázs Keszegh; Dömötör Pálvölgyi

    We consider geometric hypergraphs whose vertex set is a finite set of points (e.g., in the plane), and whose hyperedges are the intersections of this set with a family of geometric regions (e.g., axis-parallel rectangles). A typical coloring problem for such geometric hypergraphs asks, given an integer k, for the existence of an integer m=m(k), such that every set of points can be k-colored such that

    更新日期:2021-01-11
  • Distance Measures for Embedded Graphs
    Comput. Geom. (IF 0.476) Pub Date : 2021-01-05
    Hugo A. Akitaya; Maike Buchin; Bernhard Kilgus; Stef Sijben; Carola Wenk

    We introduce new distance measures for comparing straight-line embedded graphs based on the Fréchet distance and the weak Fréchet distance. These graph distances are defined using continuous mappings and thus take the combinatorial structure as well as the geometric embeddings of the graphs into account. We present a general algorithmic approach for computing these graph distances. Although we show

    更新日期:2021-01-05
  • An algorithm for the construction of the tight span of finite subsets of the Manhattan plane
    Comput. Geom. (IF 0.476) Pub Date : 2020-12-18
    Mehmet Kılıç; Şahin Koçak; Yunus Özdemir

    We give a simple algorithm to obtain the tight span of a finite subset of the Manhattan plane.

    更新日期:2020-12-28
  • Largest and smallest area triangles on imprecise points
    Comput. Geom. (IF 0.476) Pub Date : 2020-12-07
    Vahideh Keikha; Maarten Löffler; Ali Mohades

    Assume we are given a set of parallel line segments in the plane, and we wish to place a point on each line segment such that the resulting point set maximizes or minimizes the area of the largest or smallest triangle in the set. We analyze the complexity of the four resulting computational problems, and we show that three of them admit polynomial-time algorithms, while the fourth is NP-hard. Specifically

    更新日期:2020-12-23
  • Guarantees on nearest-neighbor condensation heuristics
    Comput. Geom. (IF 0.476) Pub Date : 2020-11-19
    Alejandro Flores-Velazco; David Mount

    The problem of nearest-neighbor condensation aims to reduce the size of a training set of a nearest-neighbor classifier while maintaining its classification accuracy. Although many condensation techniques have been proposed, few bounds have been proved on the amount of reduction achieved. In this paper, we present one of the first theoretical results for practical nearest-neighbor condensation algorithms

    更新日期:2020-12-08
  • On the number of order types in integer grids of small size
    Comput. Geom. (IF 0.476) Pub Date : 2020-11-19
    Luis E. Caraballo; José-Miguel Díaz-Báñez; Ruy Fabila-Monroy; Carlos Hidalgo-Toscano; Jesús Leaños; Amanda Montejano

    Let {p1,…,pn} and {q1,…,qn} be two sets of n labeled points in general position in the plane. We say that these two point sets have the same order type if for every triple of indices (i,j,k), pk is above the directed line from pi to pj if and only if qk is above the directed line from qi to qj. In this paper we give the first non-trivial lower bounds on the number of different order types of n points

    更新日期:2020-11-27
  • Empty rainbow triangles in k-colored point sets
    Comput. Geom. (IF 0.476) Pub Date : 2020-11-20
    Ruy Fabila-Monroy; Daniel Perz; Ana Laura Trujillo-Negrete

    Let S be a set of n points in general position in the plane. Suppose that each point of S has been assigned one of k≥3 possible colors and that there is the same number, m, of points of each color class. This means n=km. A polygon with vertices on S is empty if it does not contain points of S in its interior; and it is rainbow if all its vertices have different colors. Let f(k,m) be the minimum number

    更新日期:2020-11-26
  • Geometric firefighting in the half-plane
    Comput. Geom. (IF 0.476) Pub Date : 2020-11-18
    Sang-Sub Kim; Rolf Klein; David Kübel; Elmar Langetepe; Barbara Schwarzwald

    In 2006, Alberto Bressan [1] suggested the following problem. Suppose a circular fire spreads in the Euclidean plane at unit speed. The task is to build, in real time, barrier curves to contain the fire. At each time t the total length of all barriers built so far must not exceed t⋅v, where v is a speed constant. How large a speed v is needed? He proved that speed v>2 is sufficient, and that v>1 is

    更新日期:2020-11-23
  • Flipping in spirals
    Comput. Geom. (IF 0.476) Pub Date : 2020-11-18
    Sander Verdonschot

    We study the number of edge flips required to transform any triangulation of an n-vertex spiral polygon into any other. We improve the upper bound from 4n−6 to 3n−9 flips and show a lower bound of 2n−8 flips. Instead of using a single canonical triangulation as the intermediate point in the transformation between two triangulations, we use a family of closely-connected triangulations.

    更新日期:2020-11-21
  • Minimum cuts in geometric intersection graphs
    Comput. Geom. (IF 0.476) Pub Date : 2020-10-29
    Sergio Cabello; Wolfgang Mulzer

    Let D be a set of n disks in the plane. The disk graph GD for D is the undirected graph with vertex set D in which two disks are joined by an edge if and only if they intersect. The directed transmission graph GD→ for D is the directed graph with vertex set D in which there is an edge from a disk D1∈D to a disk D2∈D if and only if D1 contains the center of D2. Given D and two non-intersecting disks

    更新日期:2020-11-06
  • Minimum ply covering of points with disks and squares
    Comput. Geom. (IF 0.476) Pub Date : 2020-10-14
    Therese Biedl; Ahmad Biniaz; Anna Lubiw

    Following the seminal work of Erlebach and van Leeuwen in SODA 2008, we introduce the minimum ply covering problem. Given a set P of points and a set S of geometric objects, both in the plane, our goal is to find a subset S′ of S that covers all points of P while minimizing the maximum number of objects covering any point in the plane (not only points of P). For objects that are unit squares and unit

    更新日期:2020-10-30
  • Angle-monotonicity of Delaunay triangulation
    Comput. Geom. (IF 0.476) Pub Date : 2020-09-29
    Davood Bakhshesh; Mohammad Farshi

    Given an angle γ>0, a geometric path (v1,…,vk) is called angle-monotone with width γ if, for any two integers 1≤i,j

    更新日期:2020-10-02
  • Maximum-area and maximum-perimeter rectangles in polygons
    Comput. Geom. (IF 0.476) Pub Date : 2020-09-25
    Yujin Choi; Seungjun Lee; Hee-Kap Ahn

    We study the problem of finding maximum-area and maximum-perimeter rectangles that are inscribed in polygons in the plane. There has been a fair amount of work on this problem when the rectangles have to be axis-aligned or when the polygons are convex. We consider this problem in polygons with n vertices that are not necessarily convex, possibly with holes, and with no restriction on the orientation

    更新日期:2020-10-02
  • On the approximation of shortest escape paths
    Comput. Geom. (IF 0.476) Pub Date : 2020-09-02
    David Kübel; Elmar Langetepe

    A hiker is lost in a forest of unknown shape. What is a good path for the hiker to follow in order to escape from the forest within a reasonable amount of time? The hiker's dilemma clearly is: Should one start exploring the area close-by and expand the search radii gradually? Or should one rather pick some direction and run straight on? We employ a competitive analysis to prove that a certain spiral

    更新日期:2020-09-21
  • Folding polyominoes with holes into a cube
    Comput. Geom. (IF 0.476) Pub Date : 2020-08-18
    Oswin Aichholzer, Hugo A. Akitaya, Kenneth C. Cheung, Erik D. Demaine, Martin L. Demaine, Sándor P. Fekete, Linda Kleist, Irina Kostitsyna, Maarten Löffler, Zuzana Masárová, Klara Mundilova, Christiane Schmidt

    When can a polyomino piece of paper be folded into a unit cube? Prior work studied tree-like polyominoes, but polyominoes with holes remain an intriguing open problem. We present sufficient conditions for a polyomino with one or several holes to fold into a cube, and conditions under which cube folding is impossible. In particular, we show that all but five special “basic” holes guarantee foldability

    更新日期:2020-08-18
  • Computation of spatial skyline points
    Comput. Geom. (IF 0.476) Pub Date : 2020-08-11
    Binay Bhattacharya, Arijit Bishnu, Otfried Cheong, Sandip Das, Arindam Karmakar, Jack Snoeyink

    The database skyline query (or non-domination query) has a spatial form: Given a set P with n point sites, and a point set S of m locations of interest, a site p∈P is a skyline point if and only if for each q∈P∖{p}, there exists at least one location s∈S that is closer to p than to q. We reduce the problem of determining skyline points to the problem of finding sites that have non-empty cells in an

    更新日期:2020-08-11
  • Dynamic layers of maxima with applications to dominating queries
    Comput. Geom. (IF 0.476) Pub Date : 2020-08-10
    E. Kipouridis, A. Kosmatopoulos, A.N. Papadopoulos, K. Tsichlas

    Over the past years there has been an enormous increase in the amount of data generated on a daily basis. A critical task in handling the information overload is locating the most interesting objects of a dataset according to a specific configuration or ranking function. Our work is based on the concept of dominance which compares data objects based on maximization preferences on the attribute values

    更新日期:2020-08-10
  • On the minimum-area rectangular and square annulus problem
    Comput. Geom. (IF 0.476) Pub Date : 2020-07-29
    Sang Won Bae

    In this paper, we address the minimum-area rectangular and square annulus problem, which asks a rectangular or square annulus of minimum area, either in a fixed orientation or over all orientations, that encloses a set P of n input points in the plane. To our best knowledge, no nontrivial results on the problem have been discussed in the literature, while its minimum-width variants have been intensively

    更新日期:2020-07-29
  • I/O-efficient 2-d orthogonal range skyline and attrition priority queues
    Comput. Geom. (IF 0.476) Pub Date : 2020-07-24
    Casper Kejlberg-Rasmussen, Yufei Tao, Konstantinos Tsakalidis, Kostas Tsichlas, Jeonghun Yoon

    We present the first static and dynamic external memory data structures for variants of 2-d orthogonal range skyline reporting with worst-case logarithmic query and update I/O-complexity. The results are obtained by using persistent data structures and by extending the attrition priorities queues of Sundar (1989) [26] to also support real-time concatenation, a result of independent interest. We show

    更新日期:2020-07-24
  • Tilings of the regular N-gon with triangles of angles π/N,π/N,(N − 2)π/N for N = 5,8,10 and 12
    Comput. Geom. (IF 0.476) Pub Date : 2020-07-21
    M. Laczkovich

    We say that a triangle T tiles a polygon A, if A can be dissected into finitely many nonoverlapping triangles similar to T. We are concerned with the question whether the triangle of angles π/N,π/N, (N−2)π/N tiles the regular N-gon. It is easy to see that if N=3,4 or 6, then the answer is affirmative. We show the same in the cases N=5,8,10 and 12.

    更新日期:2020-07-21
  • Soft subdivision motion planning for complex planar robots
    Comput. Geom. (IF 0.476) Pub Date : 2020-07-15
    Bo Zhou, Yi-Jen Chiang, Chee Yap

    The design and implementation of theoretically-sound robot motion planning algorithms is challenging. Within the framework of resolution-exact algorithms, it is possible to exploit soft predicates for collision detection. The design of soft predicates is a balancing act between their implementability and their accuracy/effectivity. In this paper, we focus on the class of planar polygonal rigid robots

    更新日期:2020-07-15
  • Rectilinear link diameter and radius in a rectilinear polygonal domain
    Comput. Geom. (IF 0.476) Pub Date : 2020-07-11
    Elena Arseneva, Man-Kwun Chiu, Matias Korman, Aleksandar Markovic, Yoshio Okamoto, Aurélien Ooms, André van Renssen, Marcel Roeloffzen

    We study the computation of the diameter and radius under the rectilinear link distance within a rectilinear polygonal domain of n vertices and h holes. We introduce a graph of oriented distances to encode the distance between pairs of points of the domain. This helps us transform the problem so that we can search through the candidates more efficiently. Our algorithm computes both the diameter and

    更新日期:2020-07-11
  • Smallest universal covers for families of triangles
    Comput. Geom. (IF 0.476) Pub Date : 2020-07-09
    Ji-won Park, Otfried Cheong

    A universal cover for a family T of triangles is a convex set that contains a congruent copy of each triangle T∈T. We conjecture that for any family T of triangles of bounded diameter there is a triangle that forms a universal cover for T of smallest possible area. We prove this conjecture for all families of two triangles, and for the family of triangles that fit in the unit disk.

    更新日期:2020-07-09
  • On pseudo-disk hypergraphs
    Comput. Geom. (IF 0.476) Pub Date : 2020-07-09
    Boris Aronov, Anirudh Donakonda, Esther Ezra, Rom Pinchasi

    Let F be a family of pseudo-disks in the plane, and P be a finite subset of F. Consider the hyper-graph H(P,F) whose vertices are the pseudo-disks in P and the edges are all subsets of P of the form {D∈P|D∩S≠∅}, where S is a pseudo-disk in F. We give an upper bound of O(nk3) for the number of edges in H(P,F) of cardinality at most k. This generalizes a result of Buzaglo et al. [4]. As an application

    更新日期:2020-07-09
  • Improved PTASs for convex barrier coverage
    Comput. Geom. (IF 0.476) Pub Date : 2020-07-09
    Paz Carmi, Matthew J. Katz, Rachel Saban, Yael Stein

    Let R be a connected and closed region in the plane and let S be a set of n points (representing mobile sensors) in the interior of R. We think of R's boundary as a barrier which needs to be monitored. This gives rise to the barrier coverage problem, where one needs to move the sensors to the boundary of R, so that every point on the boundary is covered by one of the sensors. We focus on the variant

    更新日期:2020-07-09
  • An improved construction for spanners of disks
    Comput. Geom. (IF 0.476) Pub Date : 2020-07-01
    Michiel Smid

    Let D be a set of n pairwise disjoint disks in the plane. Consider the metric space in which the distance between any two disks D and D′ in D is the length of the shortest line segment that connects D and D′. For any real number ε>0, we show how to obtain a (1+ε)-spanner for this metric space that has at most (2π/ε)⋅n edges. The previously best known result is by Bose et al. (2011) [1]. Their (1+ε)-spanner

    更新日期:2020-07-01
  • Two theorems on point-flat incidences
    Comput. Geom. (IF 0.476) Pub Date : 2020-06-26
    Ben Lund

    We improve the theorem of Beck giving a lower bound on the number of k-flats spanned by a set of points in real space, and improve the bound of Elekes and Tóth on the number of incidences between points and k-flats in real space.

    更新日期:2020-06-26
  • Covering the plane by a sequence of circular disks with a constraint
    Comput. Geom. (IF 0.476) Pub Date : 2020-06-15
    Amitava Bhattacharya, Anupam Mondal

    We are interested in the following problem of covering the plane by a sequence of congruent circular disks with a constraint on the distance between consecutive disks. Let (Dn)n∈N be a sequence of closed unit circular disks such that ∪n∈NDn=R2 with the condition that for n≥2, the center of the disk Dn lies in Dn−1. What is a “most economical” or an optimal way of placing Dn for all n∈N? We answer this

    更新日期:2020-06-15
  • Range closest-pair search in higher dimensions
    Comput. Geom. (IF 0.476) Pub Date : 2020-06-05
    Timothy M. Chan, Saladi Rahul, Jie Xue

    Range closest-pair (RCP) search is a range-search variant of the classical closest-pair problem, which aims to store a given set S of points into some space-efficient data structure such that when a query range Q is specified, the closest pair in S∩Q can be reported quickly. RCP search has received attention over years, but the primary focus was only on R2. In this paper, we study RCP search in higher

    更新日期:2020-06-05
  • Two disjoint 5-holes in point sets
    Comput. Geom. (IF 0.476) Pub Date : 2020-06-02
    Manfred Scheucher

    Given a set of points S⊆R2, a subset X⊆S with |X|=k is called k-gon if all points of X lie on the boundary of the convex hull of X, and k-hole if, in addition, no point of S∖X lies in the convex hull of X. We use computer assistance to show that every set of 17 points in general position admits two disjoint 5-holes, that is, holes with disjoint respective convex hulls. This answers a question of Hosono

    更新日期:2020-06-02
  • Combinatorics of beacon-based routing in three dimensions
    Comput. Geom. (IF 0.476) Pub Date : 2020-05-28
    Jonas Cleve, Wolfgang Mulzer

    A beacon b∈Rd is a point-shaped object in d-dimensional space that can exert a magnetic pull on any other point-shaped object p∈Rd. This object p then moves greedily towards b. The motion stops when p gets stuck at an obstacle or when p reaches b. By placing beacons inside a d-dimensional polyhedron P, we can implement a scheme to route point-shaped objects between any two locations in P. We can also

    更新日期:2020-05-28
  • Output sensitive algorithms for approximate incidences and their applications
    Comput. Geom. (IF 0.476) Pub Date : 2020-05-22
    Dror Aiger, Haim Kaplan, Micha Sharir

    An ε-approximate incidence between a point and some geometric object (line, circle, plane, sphere) occurs when the point and the object lie at distance at most ε from each other. Given a set of points and a set of objects, computing the approximate incidences between them is a major step in many database and web-based applications in computer vision and graphics, including robust model fitting, approximate

    更新日期:2020-05-22
  • Extending upward planar graph drawings
    Comput. Geom. (IF 0.476) Pub Date : 2020-05-22
    Giordano Da Lozzo, Giuseppe Di Battista, Fabrizio Frati

    In this paper we study the computational complexity of the Upward Planarity Extension problem, which takes as input an upward planar drawing ΓH of a subgraph H of a directed graph G and asks whether ΓH can be extended to an upward planar drawing of G. Our study fits into the line of research on the extensibility of partial representations, which has recently become a mainstream in Graph Drawing. We

    更新日期:2020-05-22
  • Dihedral deformation and rigidity
    Comput. Geom. (IF 0.476) Pub Date : 2020-05-12
    Nina Amenta, Carlos Rojas

    We consider defining the embedding of a triangle mesh into R3, up to translation, rotation, and scale, by its vector of dihedral angles. On the theoretical side, we show that locally the map from realizable vectors of dihedrals to mesh embeddings is one-to-one almost everywhere. On the implementation side, we are interested in using the dihedral parameterization in shape analysis. This demands a way

    更新日期:2020-05-12
  • Packing plane spanning trees into a point set
    Comput. Geom. (IF 0.476) Pub Date : 2020-05-05
    Ahmad Biniaz, Alfredo García

    Let P be a set of n points in the plane in general position. We show that at least ⌊n/3⌋ plane spanning trees can be packed into the complete geometric graph on P. This improves the previous best known lower bound Ω(n). Towards our proof of this lower bound we show that the center of a set of points, in the d-dimensional space in general position, is of dimension either 0 or d.

    更新日期:2020-05-05
  • Trajectory planning for an articulated probe
    Comput. Geom. (IF 0.476) Pub Date : 2020-04-28
    Ka Yaw Teo, Ovidiu Daescu, Kyle Fox

    We consider a new trajectory planning problem involving a simple articulated probe. The probe is modeled as two line segments ab and bc, with a joint at the common point b, where bc is of fixed length r and ab is of arbitrarily large length. Initially, ab and bc are collinear. Given a set of obstacles in the form of n line segments and a target point t, the probe is to first be inserted in straight

    更新日期:2020-04-28
  • Reconstructing embedded graphs from persistence diagrams
    Comput. Geom. (IF 0.476) Pub Date : 2020-04-27
    Robin Lynne Belton, Brittany Terese Fasy, Rostik Mertz, Samuel Micka, David L. Millman, Daniel Salinas, Anna Schenfisch, Jordan Schupbach, Lucia Williams

    The persistence diagram (PD) is an increasingly popular topological descriptor. By encoding the size and prominence of topological features at varying scales, the PD provides important geometric and topological information about a space. Recent work has shown that well-chosen (finite) sets of PDs can differentiate between geometric simplicial complexes, providing a method for representing complex shapes

    更新日期:2020-04-27
  • Finding minimum witness sets in orthogonal polygons
    Comput. Geom. (IF 0.476) Pub Date : 2020-04-27
    I. Aldana-Galván, C. Alegría, J.L. Álvarez-Rebollar, N. Marín, E. Solís-Villarreal, J. Urrutia, C. Velarde

    A witness set W in a polygon P is a subset of P such that any set G⊂P that guards W is guaranteed to guard P. We study the problem of finding a minimum witness set for an orthogonal polygon under three models of orthogonal visibility. It is known that not all simple polygons admit a finite witness set under the traditional line-segment visibility and, if a polygon admits a finite minimal witness set

    更新日期:2020-04-27
  • Approximate range closest-pair queries
    Comput. Geom. (IF 0.476) Pub Date : 2020-04-24
    Jie Xue, Yuan Li, Ravi Janardan

    The range closest-pair (RCP) problem, as a range-search version of the classical closest-pair problem, aims to store a dataset of points in some data structure such that whenever a query range Q is given, the closest-pair inside Q can be reported efficiently. In this paper, we study the approximate version of the RCP problem with two approximation criteria. The first criterion is in terms of the query

    更新日期:2020-04-24
  • Parallel computation of alpha complexes for biomolecules
    Comput. Geom. (IF 0.476) Pub Date : 2020-04-14
    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:2020-04-14
  • Improved approximation bounds for the minimum constraint removal problem
    Comput. Geom. (IF 0.476) Pub Date : 2020-04-02
    Sayan Bandyapadhyay, Neeraj Kumar, Subhash Suri, Kasturi Varadarajan

    In the minimum constraint removal problem, we are given a set of overlapping geometric objects as obstacles in the plane, and we want to find the minimum number of obstacles that must be removed to reach a target point t from the source point s by an obstacle-free path. The problem is known to be intractable and no sub-linear approximations are known even for simple obstacles such as rectangles and

    更新日期:2020-04-02
  • Extending Erdős–Beck's theorem to higher dimensions
    Comput. Geom. (IF 0.476) Pub Date : 2020-04-01
    Thao Do

    Erdős-Beck's theorem states that n points in the plane with at most n−x points collinear define at least cxn lines for some positive constant c. It implies n points in the plane define Θ(n2) lines unless most of the points (i.e. n−o(n) points) are collinear. In this paper, we will present two ways to extend this result to higher dimensions. Given a set S of n points in Rd, we want to estimate a lower

    更新日期:2020-04-01
  • An application of the universality theorem for Tverberg partitions to data depth and hitting convex sets
    Comput. Geom. (IF 0.476) Pub Date : 2020-03-16
    Imre Bárány, Nabil H. Mustafa

    We show that, as a consequence of a new result of Pór on universal Tverberg partitions, any large-enough set P of points in Rd has a (d+2)-sized subset whose Radon point has half-space depth at least cd⋅|P|, where cd∈(0,1) depends only on d. We then give two applications of this result. The first is to computing weak ϵ-nets by random sampling. The second is to show that given any set P of points in

    更新日期:2020-03-16
  • Hamiltonicity for convex shape Delaunay and Gabriel graphs
    Comput. Geom. (IF 0.476) Pub Date : 2020-03-09
    Prosenjit Bose, Pilar Cano, Maria Saumell, Rodrigo I. Silveira

    We study Hamiltonicity for some of the most general variants of Delaunay and Gabriel graphs. Instead of defining these proximity graphs using circles, we use an arbitrary convex shape C. Let S be a point set in the plane. The k-order Delaunay graph of S, denoted k-DGC(S), has vertex set S, and edges defined as follows. Given p,q∈S, pq is an edge of k-DGC(S) provided there exists some homothet of C

    更新日期:2020-03-09
  • Symmetric assembly puzzles are hard, beyond a few pieces
    Comput. Geom. (IF 0.476) Pub Date : 2020-03-09
    Erik D. Demaine, Matias Korman, Jason S. Ku, Joseph S.B. Mitchell, Yota Otachi, André van Renssen, Marcel Roeloffzen, Ryuhei Uehara, Yushi Uno

    We study the complexity of symmetric assembly puzzles: given a collection of simple polygons, can we translate, rotate, and possibly flip them so that their interior-disjoint union is line symmetric? On the negative side, we show that the problem is strongly NP-complete even if the pieces are all polyominos. On the positive side, we show that the problem can be solved in polynomial time if the number

    更新日期:2020-03-09
  • Universal hinge patterns for folding strips efficiently into any grid polyhedron
    Comput. Geom. (IF 0.476) Pub Date : 2020-03-03
    Nadia M. Benbernou, Erik D. Demaine, Martin L. Demaine, Anna Lubiw

    We present two universal hinge patterns that enable a strip of material to fold into any connected surface made up of unit squares on the 3D cube grid—for example, the surface of any polycube. The folding is efficient: for target surfaces topologically equivalent to a sphere, the strip needs to have only twice the target surface area, and the folding stacks at most two layers of material anywhere.

    更新日期:2020-03-03
  • Gathering by repulsion
    Comput. Geom. (IF 0.476) Pub Date : 2020-03-02
    Prosenjit Bose, Thomas C. Shermer

    We consider a repulsion actuator located in an n-sided convex environment full of point particles. When the actuator is activated, all the particles move away from the actuator. We study the problem of gathering all the particles to a point. We give an O(n2) time algorithm to compute all the actuator locations that gather the particles to one point with one activation, and an O(n) time algorithm to

    更新日期:2020-03-02
  • Minimizing the continuous diameter when augmenting a geometric tree with a shortcut
    Comput. Geom. (IF 0.476) Pub Date : 2020-02-28
    Jean-Lou De Carufel, Carsten Grimm, Anil Maheshwari, Stefan Schirra, Michiel Smid

    We augment a tree T with a shortcut pq to minimize the largest distance between any two points along the resulting augmented tree T+pq. We study this problem in a continuous and geometric setting where T is a geometric tree in the Euclidean plane, a shortcut is a line segment connecting any two points along the edges of T, and we consider all points on T+pq (i.e., vertices and points along edges) when

    更新日期:2020-02-28
  • Fast and compact planar embeddings
    Comput. Geom. (IF 0.476) Pub Date : 2020-02-28
    Leo Ferres, José Fuentes-Sepúlveda, Travis Gagie, Meng He, Gonzalo Navarro

    There are many representations of planar graphs, but few are as elegant as Turán's (1984): it is simple and practical, uses only 4 bits per edge, can handle self-loops and multi-edges, and can store any specified embedding. Its main disadvantage has been that “it does not allow efficient searching” (Jacobson, 1989). In this paper we show how to add a sublinear number of bits to Turán's representation

    更新日期:2020-02-28
  • 1-bend upward planar slope number of SP-digraphs
    Comput. Geom. (IF 0.476) Pub Date : 2020-02-26
    Emilio Di Giacomo, Giuseppe Liotta, Fabrizio Montecchiani

    It is proved that every series-parallel digraph whose maximum vertex degree is Δ admits an upward planar drawing with at most one bend per edge such that each edge segment has one of Δ distinct slopes. The construction is worst-case optimal in terms of the number of slopes, and it gives rise to drawings with optimal angular resolution πΔ. A variant of the drawing algorithm is used to show that (non-directed)

    更新日期:2020-02-26
  • Evacuating equilateral triangles and squares in the face-to-face model
    Comput. Geom. (IF 0.476) Pub Date : 2020-02-19
    Huda Chuangpishit, Saeed Mehrabi, Lata Narayanan, Jaroslav Opatrny

    Consider k robots initially located at a point inside a region T. Each robot can move anywhere in T independently of the other robots with maximum speed one. The goal of the robots is to evacuate T through an exit at an unknown location on the boundary of T. The objective is to minimize the evacuation time, which is defined as the time the last robot reaches the exit. We consider the face-to-face communication

    更新日期:2020-02-19
  • Shortest paths and convex hulls in 2D complexes with non-positive curvature
    Comput. Geom. (IF 0.476) Pub Date : 2020-02-14
    Anna Lubiw, Daniela Maftuleac, Megan Owen

    Globally non-positively curved, or CAT(0), polyhedral complexes arise in a number of applications, including evolutionary biology and robotics. These spaces have unique shortest paths and are composed of Euclidean polyhedra, yet many algorithms and properties of shortest paths and convex hulls in Euclidean space fail to transfer over. We give an algorithm, using linear programming, to compute the convex

    更新日期:2020-02-14
  • Computing multiparameter persistent homology through a discrete Morse-based approach
    Comput. Geom. (IF 0.476) Pub Date : 2020-02-12
    Sara Scaramuccia, Federico Iuricich, Leila De Floriani, Claudia Landi

    Persistent homology allows for tracking topological features, like loops, holes and their higher-dimensional analogues, along a single-parameter family of nested shapes. Computing descriptors for complex data characterized by multiple parameters is becoming a major challenging task in several applications, including physics, chemistry, medicine, and geography. Multiparameter persistent homology generalizes

    更新日期:2020-02-12
  • Plane hop spanners for unit disk graphs: Simpler and better
    Comput. Geom. (IF 0.476) Pub Date : 2020-02-12
    Ahmad Biniaz

    The unit disk graph (UDG) is a widely employed model for the study of wireless networks. In this model, wireless nodes are represented by points in the plane and there is an edge between two points if and only if their Euclidean distance is at most one. A hop spanner for the UDG is a spanning subgraph H such that for every edge (p,q) in the UDG the topological shortest path between p and q in H has

    更新日期:2020-02-12
  • Middle curves based on discrete Fréchet distance
    Comput. Geom. (IF 0.476) Pub Date : 2020-02-06
    Hee-Kap Ahn, Helmut Alt, Maike Buchin, Eunjin Oh, Ludmila Scharf, Carola Wenk

    Given a set of polygonal curves, we present algorithms for computing a middle curve that serves as a representative for the entire set of curves. We require that the middle curve consists of vertices of the input curves and that it minimizes the maximum discrete Fréchet distance to all input curves. We consider three different variants of a middle curve depending on in which order vertices of the input

    更新日期:2020-02-06
  • Progressive simplification of polygonal curves
    Comput. Geom. (IF 0.476) Pub Date : 2020-02-06
    Kevin Buchin, Maximilian Konzack, Wim Reddingius

    Simplifying polygonal curves at different levels of detail is an important problem with many applications. Existing geometric optimization algorithms are only capable of minimizing the complexity of a simplified curve for a single level of detail. We present an O(n3m)-time algorithm that takes a polygonal curve of n vertices and produces a set of consistent simplifications for m scales while minimizing

    更新日期:2020-02-06
  • Problems on track runners
    Comput. Geom. (IF 0.476) Pub Date : 2020-02-03
    Adrian Dumitrescu, Csaba D. Tóth

    Consider the circle C of length 1 and a circular arc A of length ℓ∈(0,1). It is shown that there exists k=k(ℓ)∈N, and a schedule for k runners along the circle with k constant but distinct positive speeds so that at any time t≥0, at least one of the k runners is not in A. On the other hand, we show the following. Assume that k runners 1,2,…,k, with constant rationally independent (thus distinct) speeds

    更新日期:2020-02-03
  • On the shortest separating cycle
    Comput. Geom. (IF 0.476) Pub Date : 2020-01-31
    Adrian Dumitrescu

    According to a result of Arkin et al. (2016), given n point pairs in the plane, there exists a simple polygonal cycle that separates the two points in each pair to different sides; moreover, a O(n)-factor approximation with respect to the minimum length can be computed in polynomial time. Here the following results are obtained: (I) We extend the problem to geometric hypergraphs and obtain the following

    更新日期:2020-01-31
  • The most-likely skyline problem for stochastic points
    Comput. Geom. (IF 0.476) Pub Date : 2020-01-30
    Akash Agrawal, Yuan Li, Jie Xue, Ravi Janardan

    For a set O of n points in Rd, the skyline consists of the subset of all points of O where no point is dominated by any other point of O. Suppose that each point oi∈O has an associated probability of existence pi∈(0,1]. The problem of computing the skyline with the maximum probability of occurrence is considered. It is shown that in Rd, d≥3, the problem is NP-hard and that the desired skyline cannot

    更新日期:2020-01-30
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