Small strong epsilon nets

Ashok, Pradeesha; Azmi, Umair; Govindarajan, Sathish

doi:10.1016/j.comgeo.2014.05.002

Cited by 7 publications

(9 citation statements)

References 33 publications

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“…Similarly, y med is present in at least n 2 4 intervals, obtained by projecting the Proof. Let p ∈ P be the strong centerpoint for axis-parallel rectangles [4]. Note that this is also a strong centerpoint for axis-parallel slabs i.e., any axis-parallel slab that contains more than 3n 4 points from P contains p. We claim that p is contained in at least 3n 2 8 induced axis-parallel slabs.…”

Section: Axis-parallel Slabsmentioning

confidence: 97%

“…Let p be the strong centerpoint of P w.r.t axis-parallel rectangles. Then any axis-parallel rectangle that contains more than 3n 4 points from P contains p [4]. We claim that p is contained in at least n 2 16 rectangles from R. Let p partition P into four quadrants as shown in figure 3.…”

Section: Strong Variantmentioning

confidence: 99%

“…The centerpoint question has also been studied for special classes of convex objects like axis-parallel rectangles, halfplanes and disks [2]. Another variant of the centerpoint called strong centerpoint, where the centerpoint is required to be an input point, has also been studied [4].…”

Section: Introductionmentioning

confidence: 99%

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Selection Lemmas for Various Geometric Objects

Ashok

Govindarajan

Rajgopal

2016

Int. J. Comput. Geom. Appl.

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Selection lemmas are classical results in discrete geometry that have been well studied and have applications in many geometric problems like weak epsilon nets and slimming Delaunay triangulations. Selection lemma type results typically show that there exists a point that is contained in many objects that are induced (spanned) by an underlying point set.In the first selection lemma, we consider the set of all the objects induced (spanned) by a point set P . This question has been widely explored for simplices in R d , with tight bounds in R 2 . In our paper, we prove first selection lemma for other classes of geometric objects. We also consider the strong variant of this problem where we add the constraint that the piercing point comes from P . We prove an exact result on the strong and the weak variant of the first selection lemma for axis-parallel rectangles, special subclasses of axis-parallel rectangles like quadrants and slabs, disks (for centrally symmetric point sets). We also show non-trivial bounds on the first selection lemma for axis-parallel boxes and hyperspheres in R d . In the second selection lemma, we consider an arbitrary m sized subset of the set of all objects induced by P . We study this problem for axisparallel rectangles and show that there exists an point in the plane that is contained in m 3 24n 4 rectangles. This is an improvement over the previous bound by Smorodinsky and Sharir [20] when m is almost quadratic.

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Section: Axis-parallel Slabsmentioning

confidence: 97%

Section: Strong Variantmentioning

confidence: 99%

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Selection Lemmas for Various Geometric Objects

Ashok

Govindarajan

Rajgopal

2016

Int. J. Comput. Geom. Appl.

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“…Let p be the strong centerpoint of P w.r.t axis-parallel rectangles. Then any axisparallel rectangle that contains more than 3n 4 points from P contains p [6]. We claim that p is contained in at least …”

Section: Strong Variantmentioning

confidence: 99%

“…To our knowledge, there has been no previous work on the first selection lemma for axis-parallel rectangles. Interestingly, we use the weak and strong centerpoint for rectangles [1,6] to prove this result. -We show bounds on f (m, n) (second selection lemma) for axis-parallel rectangles.…”

Section: Introductionmentioning

confidence: 99%

Hitting and Piercing Rectangles Induced by a Point Set

Rajgopal

Ashok

Govindarajan

et al. 2013

Lecture Notes in Computer Science

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Abstract. We consider various hitting and piercing problems for the family of axis-parallel rectangles induced by a point set. Selection Lemmas on induced objects are classical results in discrete geometry that have been well studied and have applications in many geometric problems like weak epsilon nets and slimming Delaunay triangulations. Selection Lemma type results typically bound the maximum number of induced objects that are hit/pierced by a single point. First, we prove an exact result on the strong and the weak variant of the First Selection Lemma for rectangles. We also show bounds for the Second Selection Lemma which improve upon previous bounds when there are near-quadratic number of induced rectangles. Next, we consider the hitting set problem for induced rectangles. This is a special case of the geometric hitting set problem which has been extensively studied. We give efficient algorithms and show exact combinatorial bounds on the hitting set problem for two special classes of induced axis-parallel rectangles. Finally, we show that the minimum hitting set problem for all induced lines is NP-Complete.

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