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Small weak epsilonnets in three dimensions
 In Proceedings of the 18th Canadian Conference on Computational Geometry
, 2006
"... We study the problem of finding small weak εnets in three dimensions and provide new upper and lower bounds on the value of ε for which a weak εnet of a given small constant size exists. The range spaces under consideration are the set of all convex sets and the set of all halfspaces in R3. 1 ..."
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We study the problem of finding small weak εnets in three dimensions and provide new upper and lower bounds on the value of ε for which a weak εnet of a given small constant size exists. The range spaces under consideration are the set of all convex sets and the set of all halfspaces in R3. 1
Weak ɛnets and interval chains
, 2007
"... We construct weak ɛnets of almost linear size for certain types of point sets. Specifically, for planar point sets in convex position we construct weak 1 rnets of size O(rα(r)), where α(r) denotes the inverse Ackermann function. For point sets along the moment curve in Rd we construct weak 1 rnet ..."
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Cited by 4 (1 self)
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We construct weak ɛnets of almost linear size for certain types of point sets. Specifically, for planar point sets in convex position we construct weak 1 rnets of size O(rα(r)), where α(r) denotes the inverse Ackermann function. For point sets along the moment curve in Rd we construct weak 1 rnets of size r · 2poly(α(r)) , where the degree of the polynomial in the exponent depends (quadratically) on d. Our constructions result from a reduction to a new problem, which we call stabbing interval chains with jtuples. Given the range of integers N = [1, n], an interval chain of length k is a sequence of k consecutive, disjoint, nonempty intervals contained in N. A jtuple p = (p1,..., pj) is said to stab an interval chain C = I1 · · · Ik if each pi falls on a different interval of C. The problem is to construct a smallsize family Z of jtuples that stabs all kinterval chains in N. Let z (j)
Small Strong Epsilon Nets
"... In this paper, we initiate the study of small strong ϵnets and prove bounds for axisparallel rectangles, half spaces, strips and wedges. We also give some improved bounds for small weak ϵnets. 1 ..."
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In this paper, we initiate the study of small strong ϵnets and prove bounds for axisparallel rectangles, half spaces, strips and wedges. We also give some improved bounds for small weak ϵnets. 1
Hitting and piercing rectangles induced by a point set
"... Abstract. We consider various hitting and piercing problems for the family of axisparallel 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 ..."
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Abstract. We consider various hitting and piercing problems for the family of axisparallel 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 nearquadratic 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 axisparallel rectangles. Finally, we show that the minimum hitting set problem for all induced lines is NPComplete.
An Optimal Generalization . . .
, 2007
"... We prove an optimal generalization of the centerpoint theorem: given a set P of n points in the plane, there exist two points (not necessarily among input points) that hit all convex objects containing more than 4n/7 points of P. We further prove that this bound is tight. We get this bound as part o ..."
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We prove an optimal generalization of the centerpoint theorem: given a set P of n points in the plane, there exist two points (not necessarily among input points) that hit all convex objects containing more than 4n/7 points of P. We further prove that this bound is tight. We get this bound as part of a more general procedure for finding small number of points hitting convex sets over P, yielding several improvements over previous results.
An optimal extension of . . .
 COMPUTATIONAL GEOMETRY
, 2007
"... We prove an optimal extension of the centerpoint theorem: given a set P of n points in the plane, there exist two points (not necessarily among input points) that hit all convex sets containing more than 4 7 n points of P. We further prove that this bound is tight. We get this bound as part of a mor ..."
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We prove an optimal extension of the centerpoint theorem: given a set P of n points in the plane, there exist two points (not necessarily among input points) that hit all convex sets containing more than 4 7 n points of P. We further prove that this bound is tight. We get this bound as part of a more general procedure for finding small number of points hitting convex sets over P, yielding several improvements over previous results.
Nonorthogonal Range Searching: A Review
, 2008
"... Range searching is a common type of database query where the user enquires about the elements that lie within a certain geometric shape. Typically, a query shape is defined by a simple geometric object, such as a rectangle, a disk, a triangle, a parametric curve, or a combination of them. In particu ..."
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Range searching is a common type of database query where the user enquires about the elements that lie within a certain geometric shape. Typically, a query shape is defined by a simple geometric object, such as a rectangle, a disk, a triangle, a parametric curve, or a combination of them. In particular, nonorthogonal range searching covers the most general cases of range searching since the query shape is not constrained to be a rectangle parallel to the axis coordinates. We review the long history of nonorthogonal range searching from its beginning to the present and show the latest advances in this area. Most traditional scenarios of nonorthogonal range searching have been well studied and the available algorithms are nearoptimal. Latest results include heuristic algorithms, small improvements for higher dimensions, and new problems on range searching with a variety of applications.
Small Weak EpsilonNets in Three Dimensions
"... We study the problem of finding small weak εnets in three dimensions and provide new upper and lower bounds on the value of ε for which a weak εnet of a given small constant size exists. The range spaces under consideration are the set of all convex sets and the set of all halfspaces in R 3. 1 ..."
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We study the problem of finding small weak εnets in three dimensions and provide new upper and lower bounds on the value of ε for which a weak εnet of a given small constant size exists. The range spaces under consideration are the set of all convex sets and the set of all halfspaces in R 3. 1