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Hierarchical Categories in Colored Searching Comput. Geom. (IF 0.6) Pub Date : 2024-03-04 Peyman Afshani, Rasmus Killmann, Kasper G. Larsen
In colored range counting (CRC), the input is a set of points where each point is assigned a “color” (or a “category”) and the goal is to store them in a data structure such that the number of distinct categories inside a given query range can be counted efficiently. CRC has strong motivations as it allows data structure to deal with categorical data.
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Accelerating Iterated Persistent Homology Computations with Warm Starts Comput. Geom. (IF 0.6) Pub Date : 2024-03-01 Yuan Luo, Bradley J. Nelson
Persistent homology is a topological feature used in a variety of applications such as generating features for data analysis and penalizing optimization problems. We develop an approach to accelerate persistent homology computations performed on many similar filtered topological spaces which is based on updating associated matrix factorizations. Our approach improves the update scheme of Cohen-Steiner
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Topological regularization via persistence-sensitive optimization Comput. Geom. (IF 0.6) Pub Date : 2024-02-28 Arnur Nigmetov, Aditi Krishnapriyan, Nicole Sanderson, Dmitriy Morozov
Optimization, a key tool in machine learning and statistics, relies on regularization to reduce overfitting. Traditional regularization methods control a norm of the solution to ensure its smoothness. Recently, topological methods have emerged as a way to provide a more precise and expressive control over the solution, relying on persistent homology to quantify and reduce its roughness. All such existing
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Piercing families of convex sets in the plane that avoid a certain subfamily with lines Comput. Geom. (IF 0.6) Pub Date : 2024-02-27 Daniel McGinnis
We define a to be a family of sets such that when (indices are taken modulo ). We show that if is a family of compact, convex sets that does not contain a , then there are lines that pierce . Additionally, we give an example of a family of compact, convex sets that contains no and cannot be pierced by lines.
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Erratum to: “Densest Lattice Packings of 3–Polytopes” [Computational Geometry 16 (2000) 157–186] Comput. Geom. (IF 0.6) Pub Date : 2023-12-20 Martin Henk
Abstract not available
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Bounds on soft rectangle packing ratios Comput. Geom. (IF 0.6) Pub Date : 2023-12-22 Judith Brecklinghaus, Ulrich Brenner, Oliver Kiss
We examine rectangle packing problems where only the areas a1,…,an of the rectangles to be packed are given while their aspect ratios may be chosen from a given interval [1γ,γ]. In particular, we ask for the smallest possible size of a rectangle R such that, under these constraints, any collection a1,…,an of rectangle areas of total size 1 can be packed into R. As for standard square packing problems
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Global strong convexity and characterization of critical points of time-of-arrival-based source localization Comput. Geom. (IF 0.6) Pub Date : 2023-12-19 Yuen-Man Pun, Anthony Man-Cho So
In this work, we study a least-squares formulation of the source localization problem given time-of-arrival measurements. We show that the formulation, albeit non-convex in general, is globally strongly convex under certain condition on the geometric configuration of the anchors and the source and on the measurement noise. Next, we derive a characterization of the critical points of the least-squares
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Rational tensegrities through the lens of toric geometry Comput. Geom. (IF 0.6) Pub Date : 2023-11-30 Fatemeh Mohammadi, Xian Wu
A classical tensegrity model consists of an embedded graph in a vector space with rigid bars representing edges, and an assignment of a stress to every edge such that at every vertex of the graph the stresses sum up to zero. The tensegrity frameworks have been recently extended from the two dimensional graph case to the multidimensional setting. We study the multidimensional tensegrities using tools
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Generalized class cover problem with axis-parallel strips Comput. Geom. (IF 0.6) Pub Date : 2023-11-17 Apurva Mudgal, Supantha Pandit
We initiate the study of a generalization of the class cover problem [Cannon and Cowen [1], Bereg et al. [2]] the generalized class cover problem, where we are allowed to misclassify some points provided we pay an associated positive penalty for every misclassified point. Two versions: single coverage and multiple coverage, of the generalized class cover problem are investigated. We study five different
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Enumerating combinatorial resultant trees Comput. Geom. (IF 0.6) Pub Date : 2023-10-31 Goran Malić, Ileana Streinu
A 2D rigidity circuit is a minimal graph G=(V,E) supporting a non-trivial stress in any generic placement of its vertices in the Euclidean plane. All 2D rigidity circuits can be constructed from K4 graphs using combinatorial resultant (CR) operations. A combinatorial resultant tree (CR-tree) is a rooted binary tree capturing the structure of such a construction. The CR operation has a specific algebraic
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Geometric triangulations and discrete Laplacians on manifolds: An update Comput. Geom. (IF 0.6) Pub Date : 2023-10-29 David Glickenstein
This paper uses the technology of weighted triangulations to study discrete versions of the Laplacian on piecewise Euclidean manifolds. Given a collection of Euclidean simplices glued together along their boundary, a geometric structure on the Poincaré dual may be constructed by considering weights at the vertices. We show that this is equivalent to specifying sphere radii at vertices and generalized
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Flexibility and rigidity of frameworks consisting of triangles and parallelograms Comput. Geom. (IF 0.6) Pub Date : 2023-10-05 Georg Grasegger, Jan Legerský
A framework, which is a (possibly infinite) graph with a realization of its vertices in the plane, is called flexible if it can be continuously deformed while preserving the edge lengths. We focus on flexibility of frameworks in which 4-cycles form parallelograms. For the class of frameworks considered in this paper (allowing triangles), we prove that the following are equivalent: flexibility, infinitesimal
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Distance measures for geometric graphs Comput. Geom. (IF 0.6) Pub Date : 2023-10-06 Sushovan Majhi, Carola Wenk
A geometric graph is a combinatorial graph, endowed with a geometry that is inherited from its embedding in a Euclidean space. Formulation of a meaningful measure of (dis-)similarity in both the combinatorial and geometric structures of two such geometric graphs is a challenging problem in pattern recognition. We study two notions of distance measures for geometric graphs, called the geometric edit
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The dispersive art gallery problem Comput. Geom. (IF 0.6) Pub Date : 2023-10-02 Christian Rieck, Christian Scheffer
We introduce a new variant of the art gallery problem that comes from safety issues. In this variant we are not interested in guard sets of smallest cardinality, but in guard sets with largest possible distances between these guards. To the best of our knowledge, this variant has not been considered before. We call it the Dispersive Art Gallery Problem. In particular, in the dispersive art gallery
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On reverse shortest paths in geometric proximity graphs Comput. Geom. (IF 0.6) Pub Date : 2023-09-11 Pankaj K. Agarwal, Matthew J. Katz, Micha Sharir
Let S be a set of n geometric objects of constant complexity (e.g., points, line segments, disks, ellipses) in R2, and let ϱ:S×S→R≥0 be a distance function on S. For a parameter r≥0, we define the proximity graph G(r)=(S,E) where E={(e1,e2)∈S×S|e1≠e2,ϱ(e1,e2)≤r}. Given S, s,t∈S, and an integer k≥1, the reverse-shortest-path (RSP) problem asks for computing the smallest value r⁎≥0 such that G(r⁎) contains
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Clustering with faulty centers Comput. Geom. (IF 0.6) Pub Date : 2023-08-21 Emily Fox, Hongyao Huang, Benjamin Raichel
In this paper we introduce and formally study the problem of k-clustering with faulty centers. Specifically, we study the faulty versions of k-center, k-median, and k-means clustering, where centers have some probability of not existing, as opposed to prior work where clients had some probability of not existing. For all three problems we provide fixed parameter tractable algorithms, in the parameters
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On algorithmic complexity of imprecise spanners Comput. Geom. (IF 0.6) Pub Date : 2023-08-16 Abolfazl Poureidi, Mohammad Farshi
Let t>1 be a real number. A geometric t-spanner is a geometric graph for a point set in Rd with straight line segments between vertices such that the ratio of the shortest-path distance between every pair of vertices in the graph (with Euclidean edge lengths) to their actual Euclidean distance is at most t. An imprecise point set is modeled by a set R of regions in Rd. If one chooses a point inside
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Straight-line drawings of 1-planar graphs Comput. Geom. (IF 0.6) Pub Date : 2023-07-04 Franz J. Brandenburg
A graph is 1-planar if it can be drawn in the plane such that each edge is crossed at most once. However, there are 1-planar graphs that do not admit a straight-line 1-planar drawing. We show that every 1-planar graph has a straight-line drawing with a two-coloring of the edges such that edges of the same color do not cross. Thus 1-planar graphs have geometric thickness two. In addition, the drawing
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Guest editorial: Special issue on the 33rd Canadian Conference on Computational Geometry (CCCG) Comput. Geom. (IF 0.6) Pub Date : 2023-06-28 Meng He, Don Sheehy
Abstract not available
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Edge-unfolding nested prismatoids Comput. Geom. (IF 0.6) Pub Date : 2023-06-28 Manuel Radons
A 3-prismatoid is the convex hull of two convex polygons A and B which lie in parallel planes HA,HB⊂R3. Let A˜ be the orthogonal projection of A onto HB. A 3-prismatoid is called nested if A˜ is properly contained in B, or vice versa. We show that every nested 3-prismatoid has an edge-unfolding to a non-overlapping polygon in the plane.
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Approximating Gromov-Hausdorff distance in Euclidean space Comput. Geom. (IF 0.6) Pub Date : 2023-06-24 Sushovan Majhi, Jeffrey Vitter, Carola Wenk
The Gromov-Hausdorff distance (dGH) proves to be a useful distance measure between shapes. In order to approximate dGH for X,Y⊂Rd, we look into its relationship with dH,iso, the infimum Hausdorff distance under Euclidean isometries. As already known for dimension d≥2, dH,iso cannot be bounded above by a constant factor times dGH. For d=1, however, we prove that dH,iso≤54dGH. We also show that the bound
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From trees to barcodes and back again II: Combinatorial and probabilistic aspects of a topological inverse problem Comput. Geom. (IF 0.6) Pub Date : 2023-06-22 Justin Curry, Jordan DeSha, Adélie Garin, Kathryn Hess, Lida Kanari, Brendan Mallery
In this paper we consider two aspects of the inverse problem of how to construct merge trees realizing a given barcode. Much of our investigation exploits a recently discovered connection between the symmetric group and barcodes in general position, based on the simple observation that death order is a permutation of birth order. We show how to lift this combinatorial characterization of barcodes to
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Density of triangulated ternary disc packings Comput. Geom. (IF 0.6) Pub Date : 2023-06-20 Thomas Fernique, Daria Pchelina
We consider ternary disc packings of the plane, i.e. the packings using discs of three different radii. Packings in which each “hole” is bounded by three pairwise tangent discs are called triangulated. There are 164 pairs (r,s), 1>r>s, allowing triangulated packings by discs of radii 1, r and s. In this paper, we enhance existing methods of dealing with maximal-density packings in order to find ternary
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Range updates and range sum queries on multidimensional points with monoid weights Comput. Geom. (IF 0.6) Pub Date : 2023-06-15 Shangqi Lu, Yufei Tao
Let P be a set of n points in Rd where each point p∈P carries a weight drawn from a commutative monoid (M,+,0). Given a d-rectangle rupd (i.e., an orthogonal rectangle in Rd) and a value Δ∈M, a range update adds Δ to the weight of every point p∈P∩rupd; given a d-rectangle rqry, a range sum query returns the total weight of the points in P∩rqry. The goal is to store P in a structure to support updates
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Half-plane point retrieval queries with independent and dependent geometric uncertainties Comput. Geom. (IF 0.6) Pub Date : 2023-06-12 Rivka Gitik, Leo Joskowicz
This paper addresses a family of geometric half-plane retrieval queries of points in the plane in the presence of geometric uncertainty. The problems include exact and uncertain point sets and half-plane queries defined by an exact or uncertain line whose location uncertainties are independent or dependent and are defined by k real-valued parameters. Point coordinate uncertainties are modeled with
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Colouring bottomless rectangles and arborescences Comput. Geom. (IF 0.6) Pub Date : 2023-06-01 Jean Cardinal, Kolja Knauer, Piotr Micek, Dömötör Pálvölgyi, Torsten Ueckerdt, Narmada Varadarajan
We study problems related to colouring families of bottomless rectangles in the plane, in an attempt to improve the polychromatic k-colouring number mk⁎. This number is the smallest m such that any collection of bottomless rectangles can be k-coloured so that any m-fold covered point is covered by all k colours. We show that for many families of bottomless rectangles, such as unit-width bottomless
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Multi-robot motion planning for unit discs with revolving areas Comput. Geom. (IF 0.6) Pub Date : 2023-05-26 Pankaj K. Agarwal, Tzvika Geft, Dan Halperin, Erin Taylor
We study the problem of motion planning for a collection of n labeled unit disc robots in a polygonal environment. We assume that the robots have revolving areas around their start and final positions: that each start and each final is contained in a radius 2 disc lying in the free space, not necessarily concentric with the start or final position, which is free from other start or final positions
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Algorithms for radius-optimally augmenting trees in a metric space Comput. Geom. (IF 0.6) Pub Date : 2023-05-24 Joachim Gudmundsson, Yuan Sha
Let T be a tree with n vertices in a metric space. We consider the problem of adding one shortcut edge to T to minimize the radius of the resulting graph. For the continuous version of the problem where a center may be a point in the interior of an edge of the graph we give a linear time algorithm. In the case when the center is restricted to lie on a vertex, the discrete version, we give an O(nlogn)
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Simple linear time algorithms for piercing pairwise intersecting disks Comput. Geom. (IF 0.6) Pub Date : 2023-05-10 Ahmad Biniaz, Prosenjit Bose, Yunkai Wang
A set D of disks in the plane is said to be pierced by a point set P if each disk in D contains a point of P. Any set of pairwise intersecting unit disks can be pierced by 3 points (Hadwiger and Debrunner (1955) [7]). Stachó and independently Danzer established that any set of pairwise intersecting arbitrary disks can be pierced by 4 points (Stachó (1981–1984) [16]. Danzer (1986) [4]). Existing linear-time
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The constant of point–line incidence constructions Comput. Geom. (IF 0.6) Pub Date : 2023-04-25 Martin Balko, Adam Sheffer, Ruiwen Tang
We study a lower bound for the constant of the Szemerédi–Trotter theorem. In particular, we show that a recent infinite family of point-line configurations satisfies I(P,L)≥(c+o(1))|P|2/3|L|2/3, with c≈1.27. Our technique is based on studying a variety of properties of Euler's totient function. We also improve the current best constant for Elekes's construction from 1 to about 1.27. From an expository
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Cut locus realizations on convex polyhedra Comput. Geom. (IF 0.6) Pub Date : 2023-04-18 Joseph O'Rourke, Costin Vîlcu
We prove that every positively weighted tree T can be realized as the cut locus C(x) of a point x on a convex polyhedron P, with T edge weights matching C(x) edge lengths. If T has n leaves, P has (in general) n+1 vertices. We show there is in fact a continuum of polyhedra P each realizing T for some x∈P. Three main tools in the proof are properties of the star unfolding of P, Alexandrov's gluing theorem
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Geometric dominating-set and set-cover via local-search Comput. Geom. (IF 0.6) Pub Date : 2023-03-17 Minati De, Abhiruk Lahiri
In this paper, we study two classic optimization problems: minimum geometric dominating set and set cover. In the dominating-set problem, for a given set of objects in the plane as input, the objective is to choose a minimum number of input objects such that every input object is dominated by the chosen set of objects. Here, we say that one object is dominated by another if their intersection is nonempty
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Keep your distance: Land division with separation Comput. Geom. (IF 0.6) Pub Date : 2023-03-14 Edith Elkind, Erel Segal-Halevi, Warut Suksompong
This paper is part of an ongoing endeavor to bring the theory of fair division closer to practice by handling requirements from real-life applications. We focus on two requirements originating from the division of land estates: (1) each agent should receive a plot of a usable geometric shape, and (2) plots of different agents must be physically separated. With these requirements, the classic fairness
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Augmenting graphs to minimize the radius Comput. Geom. (IF 0.6) Pub Date : 2023-03-09 Joachim Gudmundsson, Yuan Sha
We study the problem of augmenting a metric graph by adding k edges while minimizing the radius of the augmented graph. We give a simple 3-approximation algorithm and show that there is no polynomial-time (5/3−ϵ)-approximation algorithm, for any ϵ>0, unless P=NP. We also give two exact algorithms for the special case when the input graph is a tree, one of which is generalized to handle metric graphs
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Any platonic solid can transform to another by O(1) refoldings Comput. Geom. (IF 0.6) Pub Date : 2023-03-01 Erik D. Demaine, Martin L. Demaine, Yevhenii Diomidov, Tonan Kamata, Ryuhei Uehara, Hanyu Alice Zhang
We show that several classes of polyhedra are joined by a sequence of O(1) refolding steps, where each refolding step unfolds the current polyhedron (allowing cuts anywhere on the surface and allowing overlap) and folds that unfolding into exactly the next polyhedron; in other words, a polyhedron is refoldable into another polyhedron if they share a common unfolding. Specifically, assuming equal surface
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Bottleneck matching in the plane Comput. Geom. (IF 0.6) Pub Date : 2023-02-13 Matthew J. Katz, Micha Sharir
We present a randomized algorithm that with high probability finds a bottleneck matching in a set of n=2ℓ points in the plane. The algorithm's running time is O(nω/2logn), where ω>2 is a constant such that any two n×n matrices can be multiplied in time O(nω). The state of the art in fast matrix multiplication allows us to set ω=2.3728596.
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On the geometric priority set cover problem Comput. Geom. (IF 0.6) Pub Date : 2023-02-13 Aritra Banik, Rajiv Raman, Saurabh Ray
We study the priority set cover problem for simple geometric set systems in the plane. For pseudo-halfspaces in the plane we obtain a PTAS via local search by showing that the corresponding set system admits a planar support. We show that the problem is APX-hard even for unit disks in the plane and argue that in this case the standard local search algorithm can output a solution that is arbitrarily
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Partial matchings induced by morphisms between persistence modules Comput. Geom. (IF 0.6) Pub Date : 2023-02-03 R. Gonzalez-Diaz, M. Soriano-Trigueros, A. Torras-Casas
We study how to obtain partial matchings using the block function Mf, induced by a morphism f between persistence modules. Mf is defined algebraically and is linear with respect to direct sums of morphisms. We study some interesting properties of Mf, and provide a way of obtaining Mf using matrix operations.
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Linear-time approximation scheme for k-means clustering of axis-parallel affine subspaces Comput. Geom. (IF 0.6) Pub Date : 2023-01-24 Kyungjin Cho, Eunjin Oh
In this paper, we present a linear-time approximation scheme for k-means clustering of incomplete data points in d-dimensional Euclidean space. An incomplete data point with Δ>0 unspecified entries is represented as an axis-parallel affine subspace of dimension Δ. The distance between two incomplete data points is defined as the Euclidean distance between two closest points in the axis-parallel affine
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Shortcut hulls: Vertex-restricted outer simplifications of polygons Comput. Geom. (IF 0.6) Pub Date : 2023-01-19 Annika Bonerath, Jan-Henrik Haunert, Joseph S.B. Mitchell, Benjamin Niedermann
Let P be a polygon and C a set of shortcuts, where each shortcut is a directed straight-line segment connecting two vertices of P. A shortcut hull of P is another polygon that encloses P and whose oriented boundary is composed of elements from C. We require P and the output shortcut hull to be weakly simple polygons, which we define as a generalization of simple polygons. Shortcut hulls find their
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Angles of arc-polygons and Lombardi drawings of cacti Comput. Geom. (IF 0.6) Pub Date : 2023-01-14 David Eppstein, Daniel Frishberg, Martha C. Osegueda
We characterize the triples of interior angles that are possible in non-self-crossing triangles with circular-arc sides, and we prove that a given cyclic sequence of angles can be realized by a non-self-crossing polygon with circular-arc sides whenever all angles are ≤π. As a consequence of these results, we prove that every cactus has a planar Lombardi drawing (a drawing with edges depicted as circular
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Editorial Comput. Geom. (IF 0.6) Pub Date : 2022-12-22 Emilio Di Giacomo, Fabrizio Montecchiani
Abstract not available
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How close is a quad mesh to a polycube? Comput. Geom. (IF 0.6) Pub Date : 2022-12-22 Markus Baumeister, Leif Kobbelt
We compute the shortest sequence of local connectivity modifications that transform a genus 0 quad mesh to a polycube. The modification operations are (dual) loop preserving and thus, we are restricted to quad meshes where loops don't self-intersect and two loops intersect at most twice. The intersection patterns of the loops are encoded in a simplicial complex, which we call loop complex. To formulate
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An algorithmic framework for the single source shortest path problem with applications to disk graphs Comput. Geom. (IF 0.6) Pub Date : 2022-12-09 Katharina Klost
Shortest path problems are among the fundamental problems in graph theory. It is folklore that the unweighted single source shortest path (SSSP) problem in general graphs can be solved optimally with breadth first search (BFS) in O(n+m) time. In this paper, we develop an algorithmic framework that generalizes a batched BFS approach to give efficient SSSP algorithms for several graph classes. The running
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Untangling circular drawings: Algorithms and complexity Comput. Geom. (IF 0.6) Pub Date : 2022-12-05 Sujoy Bhore, Guangping Li, Martin Nöllenburg, Ignaz Rutter, Hsiang-Yun Wu
We consider the problem of untangling a given (non-planar) straight-line circular drawing δG of an outerplanar graph G=(V,E) into a planar straight-line circular drawing of G by shifting a minimum number of vertices to a new position on the circle. For an outerplanar graph G, it is obvious that such a crossing-free circular drawing always exists and we define the circular shifting number shift∘(δG)
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Dynamic data structures for k-nearest neighbor queries Comput. Geom. (IF 0.6) Pub Date : 2022-12-05 Sarita de Berg, Frank Staals
Our aim is to develop dynamic data structures that support k-nearest neighbors (k-NN) queries for a set of n point sites in the plane in O(f(n)+k) time, where f(n) is some polylogarithmic function of n. The key component is a general query algorithm that allows us to find the k-NN spread over t substructures simultaneously, thus reducing an O(tk) term in the query time to O(k). Combining this technique
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Unfoldings and nets of regular polytopes Comput. Geom. (IF 0.6) Pub Date : 2022-12-05 Satyan L. Devadoss, Matthew Harvey
Over a decade ago, it was shown that every edge unfolding of the Platonic solids was without self-overlap, yielding a valid net. We consider this property for their higher-dimensional analogs, the regular polytopes. Three classes of regular polytopes exist for all dimensions (n-simplex, n-cube, n-orthoplex) and three additional regular polytopes appear only in four-dimensions (24-cell, 120-cell, 600-cell)
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Complexity results on untangling red-blue matchings Comput. Geom. (IF 0.6) Pub Date : 2022-12-02 Arun Kumar Das, Sandip Das, Guilherme D. da Fonseca, Yan Gerard, Bastien Rivier
Given a matching between n red points and n blue points by line segments in the plane, we consider the problem of obtaining a crossing-free matching through flip operations that replace two crossing segments by two non-crossing ones. We first show that (i) it is NP-hard to α-approximate the shortest flip sequence, for any constant α. Second, we show that when the red points are collinear, (ii) given
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Rectangle stabbing and orthogonal range reporting lower bounds in moderate dimensions Comput. Geom. (IF 0.6) Pub Date : 2022-11-23 Peyman Afshani, Rasmus Killmann
We study the orthogonal range reporting and rectangle stabbing problems in moderate dimensions, i.e., when the dimension is clog(n) for some constant c. In orthogonal range reporting, the input is a set of n points in d dimensions, and the goal is to store these n points in a data structure such that given a query rectangle, we can report all the input points contained in the rectangle. The rectangle
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Discrete Fréchet distance for closed curves Comput. Geom. (IF 0.6) Pub Date : 2022-11-14 Evgeniy Vodolazskiy
The paper presents a discrete variation of the Fréchet distance between closed curves, which can be seen as an approximation of the continuous measure. A rather straightforward approach to compute the discrete Fréchet distance between two closed sequences of m and n points using binary search takes O(mnlogmn) time. We present an algorithm that takes O(mnlog⁎mn) time, where log⁎ is the iterated logarithm
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Intersecting Disks using Two Congruent Disks Comput. Geom. (IF 0.6) Pub Date : 2022-11-15 Byeonguk Kang, Jongmin Choi, Hee-Kap Ahn
We consider the following Euclidean 2-center problem. Given n disks in the plane, find two smallest congruent disks such that every input disk intersects at least one of the two congruent disks. We present a deterministic algorithm for the problem that returns an optimal pair of congruent disks in O(n2log3n/loglogn) time. We also present a randomized algorithm with O(n2log2n/loglogn) expected
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Rectangular Partitions of a Rectilinear Polygon Comput. Geom. (IF 0.6) Pub Date : 2022-11-15 Hwi Kim, Jaegun Lee, Hee-Kap Ahn
We investigate the problem of partitioning a rectilinear polygon P with n vertices and no holes into rectangles using disjoint line segments drawn inside P under two optimality criteria. In the minimum ink partition, the total length of the line segments drawn inside P is minimized. We present an O(n3)-time algorithm using O(n2) space that returns a minimum ink partition of P. In the thick partition
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Time and space efficient collinearity indexing Comput. Geom. (IF 0.6) Pub Date : 2022-11-08 Boris Aronov, Esther Ezra, Micha Sharir, Guy Zigdon
The collinearity testing problem is a basic problem in computational geometry, in which, given three sets A, B, C in the plane, of n points each, the task is to detect a collinear triple of points in A×B×C or report there is no such triple. In this paper we consider a preprocessing variant of this question, namely, the collinearity indexing problem, in which we are given two sets A and B, each of n
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Drawing outerplanar graphs using thirteen edge lengths Comput. Geom. (IF 0.6) Pub Date : 2022-11-08 Ziv Bakhajian, Ohad Noy Feldheim
We show that every outerplanar graph can be linearly embedded in the plane such that the number of distinct distances between pairs of adjacent vertices is at most thirteen and there is no intersection between the image of a vertex and that of an edge not containing it. This extends the work of Alon and the second author, where only overlap between vertices was disallowed, thus settling a problem posed
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Editorial Comput. Geom. (IF 0.6) Pub Date : 2022-11-03 Tamara Mchedlidze, Elena Arseneva
Abstract not available