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Lower bounds on stabbing lines in 3-space
1993
Computational geometry
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., Lower bounds on stabbing lines in 3-space, Computational Geometry: Theory and Applications 3 (1993) 53-58. ...
Agarwal for many useful discussions. Comments of two anonymous referees have been of great help in improving the overall quality of the paper. ...
Lower bound for extremal stabbing lines Theorem 3. There exists a set 6%' of polyhedra in R3 of total complexity n, with Q(n") extremal stabbing lines. Proof. ...
doi:10.1016/0925-7721(93)90030-a
fatcat:udu4afudfndkpnmbqeuu4lshwi
Development of Algorithm for Identification of Area for Maximum Coverage and Interference
2017
International Journal of Computer Applications
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For a point set P in convex position, derive a lower bound on the size of the stabbing set axis-parallel rectangles induced by each pair of points a,b∈P as the diagonal of the rectangles. ...
For a point set P, where no two points have the same x or y coordinates, derive an upper bound on the size of the stabbing set of axis-parallel rectangles induced by each pair of points a,b ∈ P as the ...
Now we will improve the lower bound of stabbing set, for the points in general position. ...
doi:10.5120/ijca2017915323
fatcat:xl7t2a6z7nfava2hou2mwls5qu
Line Transversals of Convex Polyhedra in ^3
[article]
2008
arXiv
pre-print
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We establish a bound of O(n^2k^1+), for any >0, on the combinatorial complexity of the set of line transversals of a collection of k convex polyhedra in ^3 with a total of n facets, and present a randomized ...
To obtain the above result, we study the set of line transversals which emanate from a fixed line ℓ_0, establish an almost tight bound of O(nk^1+) on the complexity of , and provide a randomized algorithm ...
For the sake of simplicity, we derive the above bound only for extremal stabbing lines defined by pairs of polyhedra. ...
arXiv:0807.1221v1
fatcat:jsu4fcl2xngbbauvxwn4kf5mbi
Finding stabbing lines in 3-space
1992
Discrete & Computational Geometry
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Within the same time bound it is possible to determine if a stabbing line exists and to nd one. ...
A line intersecting all polyhedra in a set B is called a \stabber" for the set B. This paper addresses some combinatorial and algorithmic questions about the set S(B) of all lines stabbing B. ...
Acknowledgments We wish to thank Micha Sharir for proposing the problem and Richard Pollack, Boris Aronov, Pankaj K. Agarwal and Janos Pach for many useful discussions. ...
doi:10.1007/bf02293043
fatcat:xo2rvmyj45flxc2e3546zmniky
Catching a Polygonal Fish with a Minimum Net
[article]
2021
arXiv
pre-print
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2020 pre-print arXiv:2008.06337v2 fatcat:nzdme6ggknfcrdnocuno24uv6iOther Versions
of lines is minimized. ...
We prove the solution is always a regular grid or a set of equidistant parallel lines, whose distance depends on P. ...
Such a net must stab any subset of copies of a shape, and the minimum number of lines for stabbing all copies is a lower bound on the minimum number of lines in the minimum net.
Catching the Fish! ...
arXiv:2008.06337v3
fatcat:7k3bengatrhsfjnwe4jehmshx4
Line Transversals of Convex Polyhedra in $\mathbb{R}^3$
2010
SIAM journal on computing (Print)
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A line is a transversal of P if it intersects every member of P. The set of all line transversals of P is called the transversal space (or stabbing region) of P and is denoted by T (P). ...
We establish a bound of O(n 2 k 1+ε ), for any ε > 0, on the combinatorial complexity of the set T of line transversals of a collection P of k convex polyhedra in R 3 with a total of n facets, and we present ...
We thank the anonymous referees for valuable suggestions that helped us to improve the presentation. ...
doi:10.1137/080744694
fatcat:4ei42ypusjd33cq7z6kywkwv3y
STABBING SIMPLICES OF POINT SETS WITH k-FLATS
2014
International journal of computational geometry and applications
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In this paper we give lower and upper bounds on the size of minimum m k -stabbers of point sets in R d . We study mainly m k -stabbers in the plane and in R 3 . ...
In our previous terminology, determine lower and upper bounds for f 1 1 (n) and f 2 1 (n) for point sets on the plane. ...
The lower bound follows from the fact that r lines, no three of which intersect at a point, divide the plane into 2 + 2 + 3 + · · · + r convex regions, and if a set of r lines stabs all of the triangles ...
doi:10.1142/s021819591460005x
fatcat:h3hb7uvdtrbqphf7xnzxfpc7y4
Minimizing the Stabbing Number of Matchings, Trees, and Triangulations
2008
Discrete & Computational Geometry
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The answer we provide is negative for a number of minimum stabbing problems by showing them NP-hard by means of a general proof technique. It implies non-trivial lower bounds on the approximability. ...
The (axis-parallel) stabbing number of a given set of line segments is the maximum number of segments that can be intersected by any one (axis-parallel) line. ...
We also thank Kamal Jain for some discussions on iterated rounding. ...
doi:10.1007/s00454-008-9114-6
fatcat:lztesrasybbsnnk2yo5s7bgaq4
Stabbing Planes
[article]
2022
arXiv
pre-print
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Finally, we prove linear lower bounds on the rank of Stabbing Planes refutations by adapting lower bounds in communication complexity and use these bounds in order to show that Stabbing Planes proofs cannot ...
We develop a new semi-algebraic proof system called Stabbing Planes which formalizes modern branch-and-cut algorithms for integer programming and is in the style of DPLL-based modern SAT solvers. ...
n) lower bound for both formulas. ...
arXiv:1710.03219v2
fatcat:4dedtl7iordyvcmquixcpwmd6y
Stabbers of line segments in the plane
2011
Computational geometry
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We provide efficient algorithms for the following problems: computing the stabbing wedges for S, finding a stabbing wedge for a set of parallel segments with equal length, and computing other stabbers ...
The problem of computing a representation of the stabbing lines of a set S of segments in the plane was solved by Edelsbrunner et al. ...
Hurtado for their help in the first study of the stabbing problems which was part of her PhD thesis. E. Arkin and J. S. B. Mitchell deserve our thanks for their useful comments. B. ...
doi:10.1016/j.comgeo.2010.12.004
fatcat:ogmalirgnjhv3hfb5v7privnbu
On Rectilinear Partitions with Minimum Stabbing Number
[chapter]
2011
Lecture Notes in Computer Science
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-There are point sets such that any partition with disjoint bounding boxes has stabbing number Ω(r 1−1/d ), while the optimal partition has stabbing number 2. ...
A rectilinear r-partition for S is a collection Ψ (S) := {(S1, b1), . . . , (St, bt)}, such that the sets Si form a partition of S, each bi is the bounding box of Si, and n/2r |Si| 2n/r for all 1 i t. ...
The arcs from the boxes to the sink also have (besides the upper bound of 2n/r on the flow) a lower bound of n/2r on the flow. ...
doi:10.1007/978-3-642-22300-6_26
fatcat:tysdeipp5fcvthhhxev4vkakmq
The Minimum Stabbing Triangulation Problem: IP Models and Computational Evaluation
[chapter]
2012
Lecture Notes in Computer Science
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This paper presents integer programming (ip) formulations for the mstr, that allow us to solve it exactly through ip branch-and-bound (b&b) algorithms. ...
The minimum stabbing triangulation of a set of points in the plane (mstr) was previously investigated in the literature. ...
Lübbecke for making available the grid instances. ...
doi:10.1007/978-3-642-32147-4_5
fatcat:syy6jyqnyneqfoqkcpdnv5d3mm
Lines Tangent to Four Triangles in Three-Dimensional Space
2007
Discrete & Computational Geometry
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We investigate the lines tangent to four triangles in R 3 . By a construction, there can be as many as 62 tangents. ...
In this paper we consider the case of four triangles in R 3 , and establish lower and upper bounds on the number of tangent lines. ...
[7] showed an (n 3 ) lower bound on the complexity of the set of free lines (and thus free segments) among n disjoint homothetic convex polyhedra. Recently, Agarwal et al. ...
doi:10.1007/s00454-006-1278-3
fatcat:vc5udpwucvbg5gu67siwj4bnc4
Triangle-Free Geometric Intersection Graphs with Large Chromatic Number
2013
Discrete & Computational Geometry
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Additionally, we reveal a surprising connection between coloring geometric objects in the plane and on-line coloring of intervals on the line. ...
This provides a negative answer to a question of Gyarfas and Lehel for L-shapes. ...
of line segments is bounded by an absolute constant. ...
doi:10.1007/s00454-013-9534-9
fatcat:sccdix27orhd5p2ucvcdfchhvq
The Approximability and Integrality Gap of Interval Stabbing and Independence Problems
2012
Canadian Conference on Computational Geometry
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Motivated by problems such as rectangle stabbing in the plane, we study the minimum hitting set and maximum independent set problems for families of d-intervals and d-union-intervals. ...
for the hitting set problem on d-intervals; (3) a proof that the approximation ratios for independent set on families of 2-intervals and 2-union-intervals can be improved to match tight duality gap lower ...
The table below summarizes the known integrality and duality gap bounds for large d: d-Interval Lower Bound Upper Bound Duality Gap Ω( d 2 log d ) [13] d 2 − d + 1 [10] Max-IS Integ. ...
dblp:conf/cccg/Ben-DavidGMS12
fatcat:bqo6cp7ol5ebzm7nlpjbbwz2j4
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