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Placing two disks in a convex polygon

2000
*
Information Processing Letters
*

Theorem 3 Given

doi:10.1016/s0020-0190(99)00154-4
fatcat:gz4m3ahmqfb6hoqzhvwhikc42i
*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*. ...##
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Packing and Covering a Polygon with Geodesic Disks
[article]

2013
*
arXiv
*
pre-print

Given

arXiv:1311.6033v1
fatcat:dkzsuhyw6nd6vbxv24sj7up7uq
*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. ...##
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Max-Min k-Dispersion on a Convex Polygon
[article]

2022
*
arXiv
*
pre-print

Given

arXiv:2205.02021v1
fatcat:lsjsjfmqmndnflbfzpeikk6uuu
*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. ...##
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Approximating the Maximum Overlap of Polygons under Translation
[article]

2014
*
arXiv
*
pre-print

Let P and Q be

arXiv:1406.5778v1
fatcat:in5hsgcwezgpngo74mjljkzoly
*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. ...##
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Once Punctured Disks, Non-Convex Polygons, and Pointihedra

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 ...

##
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Convex Transversals
[chapter]

2011
*
Lecture Notes in Computer Science
*

We also consider stabbing with vertices of

doi:10.1007/978-3-642-22300-6_5
fatcat:umuhgqo4ereznaojqn77p5taqe
*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. ...##
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Convex transversals

2014
*
Computational geometry
*

We also consider stabbing with vertices of

doi:10.1016/j.comgeo.2012.10.009
fatcat:f4ch64gemvamlkb5mrpnagni7m
*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]

2019
*
arXiv
*
pre-print

An open problem of Manuel Abellanas asks whether every set of disjoint closed unit

arXiv:1908.07668v1
fatcat:axpitwnxqvhmjo3m5oufawr55i
*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. ...##
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Optimal Base Station Placement for Wireless Sensor Networks with Successive Interference Cancellation

2015
*
Sensors
*

We build

doi:10.3390/s150101676
pmid:25594600
pmcid:PMC4327097
fatcat:ghugp7gl4ze3rcramfdt3getle
*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*. ...##
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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*. ...

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

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. ...

##
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On the modality of convex polygons

1990
*
Discrete & Computational Geometry
*

Under

doi:10.1007/bf02187802
fatcat:sqkddt3ftjeo3b3tvxlmjkh4qe
*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 ...##
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Range Assignment of Base-Stations Maximizing Coverage Area without Interference
[article]

2017
*
arXiv
*
pre-print

If the points are

arXiv:1705.09346v3
fatcat:mi6y3mqyabaaldgeb7qwyv6vji
*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*. ...##
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Discretization of Planar Geometric Cover Problems
[article]

2014
*
arXiv
*
pre-print

We propose

arXiv:1411.6810v1
fatcat:cj2ega664ves3dyqwm7ge7f4na
*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*...##
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Minimum Weight Convex Steiner Partitions

2009
*
Algorithmica
*

We observed that some of the faces of such networks are often non-

doi:10.1007/s00453-009-9329-9
fatcat:6rh6xmedofgwpb5r724lp23yyu
*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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