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Some new bounds for EpsilonNets
"... Abstract. Given any natural nmnber d, 0 < e < 1, let fd(e) denote the smallest integer f such that; every range space of VapnikChervonenkis dimension d has an enet of size at most f. We solve a problem of Hanssler and Welzl by showing that if d> 2, then fd(e)> ~log~ which is not far fr ..."
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Abstract. Given any natural nmnber d, 0 < e < 1, let fd(e) denote the smallest integer f such that; every range space of VapnikChervonenkis dimension d has an enet of size at most f. We solve a problem of Hanssler and Welzl by showing that if d> 2, then fd(e)> ~log~ which is not far
EpsilonNets and Simplex Range Queries
, 1986
"... We present a new technique for halfspace and simplex range query using O(n) space and O(n a) query time, where a < if(al) +7 for all dimensions d ~2 a(al) + 1 and 7> 0. These bounds are better than those previously published for all d ~ 2. The technique uses random sampling to build a part ..."
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Cited by 290 (7 self)
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We present a new technique for halfspace and simplex range query using O(n) space and O(n a) query time, where a < if(al) +7 for all dimensions d ~2 a(al) + 1 and 7> 0. These bounds are better than those previously published for all d ~ 2. The technique uses random sampling to build a
Tight Lower Bounds for the Size of EpsilonNets
"... According to a well known theorem of Haussler and Welzl (1987), any range space of bounded VCdimension admits an εnet of size O () ..."
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Cited by 22 (1 self)
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According to a well known theorem of Haussler and Welzl (1987), any range space of bounded VCdimension admits an εnet of size O ()
Lower bounds for weak epsilonnets and stairconvexity
 IN: PROC. 25TH ACM SYMPOS. COMPUT. GEOM. (SOCG 2009
, 2009
"... A set N ⊂ Rd is called a weak εnet (with respect to convex sets) for a finite X ⊂ Rd if N intersects every convex set C with X ∩ C  ≥ εX. For every fixed d ≥ 2 and every r ≥ 1 we construct sets X ⊂ Rd for which every weak 1 rnet has at least Ω(r logd−1 r) points; this is the first superlinear ..."
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Cited by 13 (5 self)
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superlinear lower bound for weak εnets in a fixed dimension. The construction is a stretched grid, i.e., the Cartesian product of d suitable fastgrowing finite sequences, and convexity in this grid can be analyzed using stairconvexity, a new variant of the usual notion of convexity. We also consider weak εnets
New Constructions of Weak EpsilonNets
 In Proc. 19th Annual ACM Symposium on Computational Geometry
, 2003
"... A nite set N R is a weak "net for an npoint set X R (with respect to convex sets) if N intersects every convex set K with jK \ X j "n. We give an alternative, and arguably simpler, proof of the fact, rst shown by Chazelle et al. [8], that every point set X in R admits a ..."
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Cited by 5 (0 self)
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weak "net of cardinality O(" polylog(1=")). Moreover, for a number of special point sets (e.g., for points on the moment curve), our method gives substantially better bounds. The construction yields an algorithm to construct such weak "nets in time O(n ln(1=")).
New constructions of weak epsilonnets
 In Proc. 19th Annual ACM Symposium on Computational Geometry
, 2003
"... A finite set N ⊂ R d is a weak εnet for an npoint set X ⊂ R d (with respect to convex sets) if N intersects every convex set K with K ∩ X  ≥ εn. We give an alternative, and arguably simpler, proof of the fact, first shown by Chazelle et al. [8], that every point set X in R d admits a weak εnet ..."
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Cited by 3 (0 self)
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of cardinality O(ε −d polylog(1/ε)). Moreover, for a number of special point sets (e.g., for points on the moment curve), our method gives substantially better bounds. The construction yields an algorithm to construct such weak εnets in time O(n ln(1/ε)). 1
Tight Lower Bounds for the Size of EpsilonNets (Extended Abstract)
 SCG '11
, 2011
"... According to a well known theorem of Haussler and Welzl (1987), any range space of bounded VCdimension admits an εnet of size O () 1 1 log. Using probabilistic techniques, ε ε Pach and Woeginger (1990) showed that there exist range spaces of VCdimension 2, for which the above bound is sharp. The ..."
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According to a well known theorem of Haussler and Welzl (1987), any range space of bounded VCdimension admits an εnet of size O () 1 1 log. Using probabilistic techniques, ε ε Pach and Woeginger (1990) showed that there exist range spaces of VCdimension 2, for which the above bound is sharp
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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Cited by 5 (0 self)
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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
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
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