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Placing two disks in a convex polygon

Sung Kwon Kim, Chan-Su Shin, Tae-Cheon Yang
2000 Information Processing Letters
Theorem 3 Given a convex polygon P of n vertices, we can nd the maximum value of the minimum radius of two disks of P in O(n log 2 n) time if the separating lines are r estricted t o o n e s t h r ough  ...  two vertices of P.  ...  Besides, there is a related problem of nding two largest disks whose union covers (the boundary of) a convex polygon, instead of inscribing two disks in a convex polygon.  ...

Packing and Covering a Polygon with Geodesic Disks [article]

Ivo Vigan
2013 arXiv   pre-print
Given a polygon P, for two points s and t contained in the polygon, their geodesic distance is the length of the shortest st-path within P.  ...  Lastly, we present a polynomial time exact algorithm which covers a polygon with two geodesic disks of minimum maximal radius.  ...  For polygons, in [37] a O(n 2 log 3 n) time algorithm for covering a convex polygon with two Euclidean disks of minimum maximal radius is presented. This paper is organized as follows.  ...

Max-Min k-Dispersion on a Convex Polygon [article]

Vishwanath R. Singireddy, Manjanna Basappa
2022 arXiv   pre-print
Given a set S of n points placed in the plane in a convex position, and an integer k (0<k<n), the objective is to compute a subset S'⊂ S such that |S'|=k and the minimum distance between a pair of points  ...  We then present an O(logn)-time 1/2√(2)-approximation algorithm for the problem when k=3 if the points are given in convex position order.  ...  Now, we can model this as the k-dispersion problem on a convex polygon to maximize the distance between any two squads.  ...

Approximating the Maximum Overlap of Polygons under Translation [article]

Sariel Har-Peled, Subhro Roy
2014 arXiv   pre-print
Let P and Q be two simple polygons in the plane of total complexity n, each of which can be decomposed into at most k convex parts.  ...  This suggest that for polygons that are "close" to being convex, the problem can be solved (approximately), in near linear time.  ...  In particular, the improved construction of Lemma 4.4 was suggested by an anonymous referee.  ...

Once Punctured Disks, Non-Convex Polygons, and Pointihedra

Hugo Parlier, Lionel Pournin
2018 Annals of Combinatorics
In particular, we consider the topological flip-graphs of once-punctured polygons which, in turn, contain all possible geometric flip-graphs of polygons with a marked point as embedded sub-graphs.  ...  In particular, we show that the embeddings between them are strongly convex (or, said otherwise, totally geodesic).  ...  We are grateful for conversations with Delphine Milenovic who studied a particular variation of the geometric punctured flip-graph in her  ...

Convex Transversals [chapter]

Esther M. Arkin, Claudia Dieckmann, Christian Knauer, Joseph S. B. Mitchell, Valentin Polishchuk, Lena Schlipf, Shang Yang
2011 Lecture Notes in Computer Science
We also consider stabbing with vertices of a regular polygon -a problem closely related to approximate symmetry detection: Given a set of disks in the plane, is it possible to find a point per disk so  ...  On the positive side, we give a polynomial-time algorithm to find a convex transversal of a maximum number of pairwise-disjoint segments (or convex polygons) in 2D if the vertices of the transversal are  ...  Acknowledgments We thank Joe Blitzstein (Harvard University) for pointers to work in convex regression, and the anonymous reviewer for helpful comments. E. Arkin and J.  ...

Convex transversals

Esther M. Arkin, Claudia Dieckmann, Christian Knauer, Joseph S.B. Mitchell, Valentin Polishchuk, Lena Schlipf, Shang Yang
2014 Computational geometry
We also consider stabbing with vertices of a regular polygon -a problem closely related to approximate symmetry detection: Given a set of disks in the plane, is it possible to find a point per disk so  ...  On the positive side, we give a polynomial-time algorithm to find a convex transversal of a maximum number of pairwise-disjoint segments (or convex polygons) in 2D if the vertices of the transversal are  ...  Acknowledgments We thank Joe Blitzstein (Harvard University) for pointers to work in convex regression, and the anonymous reviewer for helpful comments. E. Arkin and J.  ...

Existence and hardness of conveyor belts [article]

Molly Baird, Sara C. Billey, Erik D. Demaine, Martin L. Demaine, David Eppstein, Sándor Fekete, Graham Gordon, Sean Griffin, Joseph S. B. Mitchell, Joshua P. Swanson
2019 arXiv   pre-print
An open problem of Manuel Abellanas asks whether every set of disjoint closed unit disks in the plane can be connected by a conveyor belt, which means a tight simple closed curve that touches the boundary  ...  First, for unit disks whose centers are both x-monotone and y-monotone, or whose centers have x-coordinates that differ by at least two units, a conveyor belt always exists and can be found efficiently  ...  This work was continued at the 32nd Bellairs Winter Workshop on Computational Geometry, held in 2017 at the Bellairs Research Institute, Barbados.  ...

Optimal Base Station Placement for Wireless Sensor Networks with Successive Interference Cancellation

Lei Shi, Jianjun Zhang, Yi Shi, Xu Ding, Zhenchun Wei
2015 Sensors
We build a mathematical model for SIC. Although this model cannot be solved directly, it enables us to identify a necessary condition for SIC on distances from sensor nodes to the base station.  ...  Simulation results show that this algorithm can achieve about 25% improvement compared with the case that the base station is placed at the center of the network coverage area when using SIC.  ...  Suppose that we reduce an m-edge convex polygon to an k-edge convex polygon, k < m, by Steps A and B, and we know the maximum inside disk of the k-edge polygon.  ...

Page 515 of Mathematical Reviews Vol. , Issue 2001A [page]

2001 Mathematical Reviews
in a convex polygon.  ...  We want to find a folding line / and two largest equiradial nonoverlapping disks, one in P;(/) and the other in P>(/), so that if we fold P along /, then the disks coincide in the folded polygon.  ...

Cutting Polygons into Small Pieces with Chords: Laser-Based Localization

Esther M. Arkin, Rathish Das, Jie Gao, Mayank Goswami, Joseph S. B. Mitchell, Valentin Polishchuk, Csaba D. Tóth, Peter Sanders, Fabrizio Grandoni, Grzegorz Herman
2020 European Symposium on Algorithms
In particular, we consider the area, the diameter, and the radius of the largest inscribed circle as a measure of the size of a piece.  ...  Motivated by indoor localization by tripwire lasers, we study the problem of cutting a polygon into small-size pieces, using the chords of the polygon.  ...  The algorithm has two steps. In the first step, a collection P of convex polygons is created such that the vertex set of the complement P \ ( A∈P A) is L.  ...

On the modality of convex polygons

Karl Abrahamson
1990 Discrete & Computational Geometry
Under two definitions of random convex polygons, the expected modality of a random convex polygon grows without bound as the number of vertices grows.  ...  This refutes a conjecture of Aggarwal and Melville.  ...  The two chosen definitions of a random convex polygon are (1) the convex hull of n points drawn uniformly from a disk in the plane, and (2) the convex hull of n points drawn from a two-dimensional normal  ...

Range Assignment of Base-Stations Maximizing Coverage Area without Interference [article]

Ankush Acharyya, Minati De, Subhas C. Nandy, Bodhayan Roy
2017 arXiv   pre-print
If the points are placed on a straight-line and the objects are disks, then the problem is solvable in polynomial time.  ...  All these approximation results hold for the area maximization problem by regular convex polygons with even number of edges centered at the given points.  ...  length × height This property does not hold for odd regular convex polygon.  ...

Discretization of Planar Geometric Cover Problems [article]

Dae-Sung Jang, Han-Lim Choi
2014 arXiv   pre-print
We propose a reduced finite solution space that consists of distinct canonical translates and present polynomial algorithms to find the reduce solution space for disks, convex/non-convex polygons (including  ...  We consider discretization of the 'geometric cover problem' in the plane: Given a set P of n points in the plane and a compact planar object T_0, find a minimum cardinality collection of planar translates  ...  However, for non-convex polygons, the perimeters of two translates intersect in the same order as two arbitrary (non-translate) polygons intersect: the following lemma is for the algorithm analysis in  ...

Minimum Weight Convex Steiner Partitions

Adrian Dumitrescu, Csaba D. Tóth
2009 Algorithmica
We observed that some of the faces of such networks are often non-convex. Theorem 3 shows that non-convex faces are in general unavoidable in a spanner network with properties (i)-(iii). Motivation.  ...  Specifically, we show that the minimum length of a convex Steiner partition for n points in the plane is at most O(log n/ log log n) times longer than a Euclidean minimum spanning tree (EMST), and this  ...  Such a partition can be computed in O(n log n) time. Figure 5 : 5 place a half-disk of radius |ab|/(32k i ) ≤ ℓ/(32k) centered at v and bounded by the tangent line of the circular arc at v.  ...
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