1,240 research outputs found

    An Improved Bound for Weak Epsilon-Nets in the Plane

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    We show that for any finite set PP of points in the plane and ϵ>0\epsilon>0 there exist O(1ϵ3/2+γ)\displaystyle O\left(\frac{1}{\epsilon^{3/2+\gamma}}\right) points in R2{\mathbb{R}}^2, for arbitrary small γ>0\gamma>0, that pierce every convex set KK with ∣K∩P∣≥ϵ∣P∣|K\cap P|\geq \epsilon |P|. This is the first improvement of the bound of O(1ϵ2)\displaystyle O\left(\frac{1}{\epsilon^2}\right) that was obtained in 1992 by Alon, B\'{a}r\'{a}ny, F\"{u}redi and Kleitman for general point sets in the plane.Comment: A preliminary version to appear in the proceedings of FOCS 201

    Small Strong Epsilon Nets

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    Let P be a set of n points in Rd\mathbb{R}^d. A point x is said to be a centerpoint of P if x is contained in every convex object that contains more than dnd+1dn\over d+1 points of P. We call a point x a strong centerpoint for a family of objects C\mathcal{C} if x∈Px \in P is contained in every object C∈CC \in \mathcal{C} that contains more than a constant fraction of points of P. A strong centerpoint does not exist even for halfspaces in R2\mathbb{R}^2. We prove that a strong centerpoint exists for axis-parallel boxes in Rd\mathbb{R}^d and give exact bounds. We then extend this to small strong ϵ\epsilon-nets in the plane and prove upper and lower bounds for ϵiS\epsilon_i^\mathcal{S} where S\mathcal{S} is the family of axis-parallel rectangles, halfspaces and disks. Here ϵiS\epsilon_i^\mathcal{S} represents the smallest real number in [0,1][0,1] such that there exists an ϵiS\epsilon_i^\mathcal{S}-net of size i with respect to S\mathcal{S}.Comment: 19 pages, 12 figure

    Polychromatic Coloring for Half-Planes

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    We prove that for every integer kk, every finite set of points in the plane can be kk-colored so that every half-plane that contains at least 2k−12k-1 points, also contains at least one point from every color class. We also show that the bound 2k−12k-1 is best possible. This improves the best previously known lower and upper bounds of 43k\frac{4}{3}k and 4k−14k-1 respectively. We also show that every finite set of half-planes can be kk colored so that if a point pp belongs to a subset HpH_p of at least 3k−23k-2 of the half-planes then HpH_p contains a half-plane from every color class. This improves the best previously known upper bound of 8k−38k-3. Another corollary of our first result is a new proof of the existence of small size \eps-nets for points in the plane with respect to half-planes.Comment: 11 pages, 5 figure

    Selection Lemmas for various geometric objects

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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 PP. This question has been widely explored for simplices in Rd\mathbb{R}^d, with tight bounds in R2\mathbb{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 PP. 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 Rd\mathbb{R}^d. In the second selection lemma, we consider an arbitrary mm sized subset of the set of all objects induced by PP. We study this problem for axis-parallel rectangles and show that there exists an point in the plane that is contained in m324n4\frac{m^3}{24n^4} rectangles. This is an improvement over the previous bound by Smorodinsky and Sharir when mm is almost quadratic
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