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## Filters

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On sets of n points in general position that determine lines that can be pierced by n points
[article]

2019
*
arXiv
*
pre-print

Let P

arXiv:1908.06390v1
fatcat:rnz3gccs6vdbpinehfebnqk5mu
*be*a*set**of**n**points**in**general**position**in*the plane. ... Let R*be*a*set**of**n**points*disjoint from P such*that*for every x,y ∈ P the*line*through x and y contains a*point**in*R outside*of*the segment delimited*by*x and y. ... As we have seen, there are constructions*of**sets*P*of**n**points**in**general**position*and*sets*R*of**n**points*not*in*P , such*that*every*line**determined**by*P passes through a*point**in*R. ...##
###
Rectilinear and polygonal p-piercing and p-center problems

1996
*
Proceedings of the twelfth annual symposium on Computational geometry - SCG '96
*

*In*the p-

*piercing*problem, we are given a collection

*of*regions, and wish to

*determine*whether there exists a

*set*

*of*p

*points*

*that*intersects each

*of*the given regions. ...

*n*convex c-oriented polygons

*can*

*be*decided

*in*time O(c 2

*n*log

*n*), and the 2-center problem for a convex c-gon

*can*

*be*solved

*in*O(c 5

*n*logn) time. ... We wish to thank Jirka Matou sek, Otfried Schwarzkopf, G unter Rote, Arik Tamir and Eyal Assa for helpful discussions and suggestions

*on*the problems studied

*in*this paper. ...

##
###
Lattice and Non-lattice Piercing of Axis-Parallel Rectangles: Exact Algorithms and a Separation Result
[article]

2022
*
arXiv
*
pre-print

Finally, we sharpen our separation result

arXiv:2204.10385v1
fatcat:updlkgf6l5dwborzq6kcl5zbvi
*by*running the first algorithm*on*a suitable instance, and show*that*the best lattice*can**be*sometimes worse*by*20% than the optimal*piercing**set*. ... For a given family*of*shapes ℱ*in*the plane, we study what is the lowest possible density*of*a*point**set*P*that**pierces*("intersects", "hits") all translates*of*each shape*in*ℱ. ... Let R*be*an arbitrary maximal empty rectangle amidst the*points**in*Λ*that*is*determined**by*two adjacent u-*lines*1 and 2 (i.e., R is incident to two*points**on*each*of*the two*lines*). ...##
###
Random Walks on Polytopes of Constant Corank

2018
*
International Symposium on Computational Geometry
*

We show

doi:10.4230/lipics.socg.2018.60
dblp:conf/compgeom/Milatz18
fatcat:4nkyrebhajhjhnxnzkdsnjorxu
*that*the pivoting process associated with*one**line*and*n**points**in*r-dimensional space may need Ω(log r*n*) steps*in*expectation as*n*→ ∞. ... Our lower bound is also valid for the expected number*of*pivoting steps*in*the following applications: (1) The Random-Edge simplex algorithm*on*linear programs with*n*constraints*in*d =*n*− r variables ... Acknowledgements I would like to thank Bernd Gärtner, Ahad*N*. Zehmakan, Jerri Nummenpalo and Alexander Pilz for many useful suggestions and discussions. ...##
###
Random Walks on Polytopes of Constant Corank
[article]

2018
*
arXiv
*
pre-print

We show

arXiv:1802.09324v1
fatcat:zro7vcbgu5fz5mxlynzbtnasbi
*that*the pivoting process associated with*one**line*and*n**points**in*r-dimensional space may need Ω(^r*n*) steps*in*expectation as*n*→∞. ... Our lower bound is also valid for the expected number*of*pivoting steps*in*the following applications: (1) The Random-Edge simplex algorithm*on*linear programs with*n*constraints*in*d =*n*- r variables ... I would like to thank Bernd Gärtner, Ahad*N*. Zehmakan, Jerri Nummenpalo and Alexander Pilz for many useful suggestions and discussions. ...##
###
Dimensions of strong n-point sets

2006
*
Topology and its Applications
*

*n*-

*point*

*sets*(plane

*sets*which hit each

*line*

*in*

*n*

*points*) and strong

*n*-

*point*

*sets*(

*in*addition hit each circle

*in*

*n*-

*points*) exist (for

*n*2,

*n*3 respectively)

*by*transfinite induction, but their properties ... Recently for

*n*-

*point*

*sets*the question

*of*their possible dimensions has been settled: 2-and 3-

*point*

*sets*are always zero-dimensional, while for

*n*4,

*one*-dimensional

*n*-

*point*

*sets*exist. ... If A is an arc

*that*is not a

*line*segment, then some circle C is

*pierced*

*by*A

*in*at least three

*points*. Proof. ...

##
###
ON PIERCING SETS OF AXIS-PARALLEL RECTANGLES AND RINGS

1999
*
International journal of computational geometry and applications
*

We are given a collection

doi:10.1142/s0218195999000157
fatcat:l6mmvq3javdlzaojfxtoqqmdfy
*of*axis-parallel rectangles*in*the plane, and wish to*determine*whether there exists a*set**of*p*points*whose union intersects all the given rectangles. ... We present e cient algorithms for nding a*piercing**set*(i.e., a*set**of*p*points*as above) for values*of*p = 1; 2; 3; 4; 5. ... Introduction Let R*be*a*set**of**n*axis-parallel rectangles*in*the plane, and let p*be*a*positive*integer. ...##
###
On piercing sets of axis-parallel rectangles and rings
[chapter]

1997
*
Lecture Notes in Computer Science
*

We are given a collection

doi:10.1007/3-540-63397-9_33
fatcat:7mlwjhatqjcflltnl3f42bnnsm
*of*axis-parallel rectangles*in*the plane, and wish to*determine*whether there exists a*set**of*p*points*whose union intersects all the given rectangles. ... We present e cient algorithms for nding a*piercing**set*(i.e., a*set**of*p*points*as above) for values*of*p = 1; 2; 3; 4; 5. ... Introduction Let R*be*a*set**of**n*axis-parallel rectangles*in*the plane, and let p*be*a*positive*integer. ...##
###
Geometric optimization andD P -completeness

1989
*
Discrete & Computational Geometry
*

Completeness

doi:10.1007/bf02187711
fatcat:ncrhsoyb3fanzl3ks4giqyiloq
*in*D p is exhibited under many-*one*and*positive*reductions. Further an OptP(O(log*n*)) result is also obtained for some*of*these optimization problems. ...*In*this paper we show a number*of*natural geometric optimization problems*in*the plane to*be*complete for a class D p. The class D p contains both NP and Co-NP and is contained*in*A p = pNP. ... any instance*of*the problems,*that*is parameter R and the coordinate*points**in**set*T, are restricted to*be**of*size bounded*by*a polynomial*in**n*. ...##
###
On Piercing (Pseudo)Lines and Boxes

2012
*
Canadian Conference on Computational Geometry
*

This is the first

dblp:conf/cccg/BharadwajRAG12
fatcat:caydqpqosrbshomp3rvtymoeky
*positive*result for arbitrary k for a*general*family*of*objects. ...*In*this paper we investigate the existence*of*g(k, d) such*that*if any family*of*objects Danzer and Grünbaum showed*that*g(k, d) is infinite for families*of*boxes and translates*of*centrally symmetric ... . , x d ∈ H d , such*that*x i*pierces*r i . Thus (x 1 , . . . , x d ) ∈ H*pierces*r. Hence every r ∈ C*can**be**pierced**by**one**of*the (at most k d ) grid*points**in*H. ...##
###
Maintenance of a Piercing Set for Intervals with Applications

2003
*
Algorithmica
*

It is not difficult to see though

doi:10.1007/s00453-002-1006-1
fatcat:ecod26gjkffyllk7ziwojdcgli
*that*b(S)*piercing**points*are also sufficient*in*order to*pierce*all intervals*in*S, thus c(S) = b(S), and a minimum*piercing**set*for S*can**be*found*in*time O(*n*log c( ... Let b(S)*be*the maximum size*of*an independent subset*of*S. A*piercing**set*for S is a*set*P*of**points**on*the real*line*, such*that*, for each interval s i ∈ S, s i ∩ P = ∅. ... Actually, the*general*problem*can**be*solved*by*applying the solution for a shooter*on*a*line*to the O(*n*2 )*lines*defined*by*the endpoints*of*the segments, thus obtaining an alternative O(*n*5 )-time solution ...##
###
Piercing the chessboard
[article]

2021
*
arXiv
*
pre-print

*Determining*these values

*can*also

*be*interpreted as a strengthening

*of*the classical plank problem for integer

*points*. Using the symmetric plank theorem

*of*K. ... We consider the minimum number

*of*

*lines*h_n and p_n needed to intersect or

*pierce*, respectively, all the cells

*of*the

*n*×

*n*chessboard. ... More

*generally*, all the cells

*of*Q

*n*

*can*

*be*

*pierced*

*by*

*n*parallel

*lines*

*in*any given direction which are at distance 2

*n*from each other

*in*the 1 distance and do not go through any grid

*points*. ...

##
###
Approximate multi-visibility map computation

2005
*
European Workshop on Computational Geometry
*

Our approach is based

dblp:conf/ewcg/CollFS05
fatcat:etcb2t6l6bes3p4yp6u53lzu6a
*on*an algorithm*that*reconstructs an approximation*of*an unknown planar subdivision from information gathered from linear probes*of*the subdivision. ... A multi-visibility map is the subdivision*of*the domain*of*a terrain into different regions*that*, according to different criteria, encode the visibility with respect to a*set**of*view elements. ...*of*T v,s (3) two edges exchange their order*in*tvc, what means*that*the*pierce**points*they*determine*exchange their angular*position*. ...##
###
Stronger Bounds for Weak Epsilon-Nets in Higher Dimensions
[article]

2021
*
arXiv
*
pre-print

Given a finite

arXiv:2104.12654v1
fatcat:x6tsbztlizhvxpgurtrzx5kljm
*point**set*P*in*ℝ^d, and ϵ>0 we say*that**N*⊆ℝ^d is a weak ϵ-net if it*pierces*every convex*set*K with |K∩ P|≥ϵ |P|. Let d≥ 3. ... This is the first improvement*of*the bound*of*O^*(1/ϵ^d)*that*was obtained*in*1994*by*Chazelle, Edelsbrunner, Grigni, Guibas, Sharir, and Welzl for*general**point**sets**in*dimension d≥ 3. ... Let P*be*an*n*-*point**set**in*a*general**position**in*R d . ...##
###
Large Bichromatic Point Sets Admit Empty Monochromatic 4-Gons

2010
*
SIAM Journal on Discrete Mathematics
*

We show

doi:10.1137/090767947
fatcat:32ie4lyj4feqngz66ckbtlwxe4
*that*any bichromatic*set**of**n*≥ 5044*points**in*Ê 2*in**general**position**determines*at least*one*empty, monochromatic quadrilateral (and thus linearly many). ... We consider a variation*of*a problem stated*by*Erdős and Szekeres*in*1935 about the existence*of*a number f ES (k) such*that*any*set*S*of*at least f ES (k)*points**in**general**position**in*the plane has a ... As already mentioned, throughout this paper we assume S to*be*a*set**of**n**points**in*the plane*in**general**position*,*that*is, no three*points**of*S lie*on*a common*line*. ...
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