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

Chaya Keller, Rom Pinchasi
2019 arXiv   pre-print
Let P 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

Micha Sharir, Emo Welzl
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 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

Malte Milatz, Marc Herbstritt
2018 International Symposium on Computational Geometry
We show 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]

Malte Milatz
2018 arXiv   pre-print
We show 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

John Cobb
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

MICHAEL SEGAL
1999 International journal of computational geometry and applications
We are given a collection 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]

Michael Segal
1997 Lecture Notes in Computer Science
We are given a collection 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

Chanderjit Bajaj, Ming Li
1989 Discrete & Computational Geometry
Completeness 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

B. V. Subramanya Bharadwaj, Chintan Rao, Pradeesha Ashok, Sathish Govindarajan
2012 Canadian Conference on Computational Geometry
This is the first 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

Katz, Nielsen, Segal
2003 Algorithmica
It is not difficult to see though 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]

Gergely Ambrus, Imre Barany, Peter Frankl, Daniel Varga
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

Narcís Coll, Marta Fort, Joan Antoni Sellarès
2005 European Workshop on Computational Geometry
Our approach is based 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]

Natan Rubin
2021 arXiv   pre-print
Given a finite 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

Oswin Aichholzer, Thomas Hackl, Clemens Huemer, Ferran Hurtado, Birgit Vogtenhuber
2010 SIAM Journal on Discrete Mathematics
We show 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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