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TRACES OF FINITE SETS: EXTREMAL PROBLEMS AND GEOMETRIC APPLICATIONS
, 1992
"... Given a hypergraph H and a subset S of its vertices, the trace of H on S is defined as HS = {E ∩ S: E ∈ H}. The Vapnik–Chervonenkis dimension (VCdimension) of H is the size of the largest subset S for which HS has 2 S edges. Hypergraphs of small VCdimension play a central role in many areas o ..."
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Given a hypergraph H and a subset S of its vertices, the trace of H on S is defined as HS = {E ∩ S: E ∈ H}. The Vapnik–Chervonenkis dimension (VCdimension) of H is the size of the largest subset S for which HS has 2 S edges. Hypergraphs of small VCdimension play a central role in many areas of statistics, discrete and computational geometry, and learning theory. We survey some of the most important results related to this concept with special emphasis on (a) hypergraph theoretic methods and (b) geometric applications.
Tight Lower Bounds for the Size of EpsilonNets [Extended Abstract] ABSTRACT
"... 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. The only known range spaces of small VCdimension, in which the ranges are geometric objects in some Euclidean space and the size of the smallest εnets is superlinear in 1 ε, were found by Alon (2010). In his examples, every εnet is of size Ω ( 1 1 g( ε ε)), where g is an extremely slowly growing function, related to the inverse Ackermann function. We show that there exist geometrically defined range spaces, already of VCdimension 2, in which the size of the smallest εnets is Ω () 1 1 log. We also construct range spaces inε ε duced by axisparallel rectangles in the plane, in which the size of the smallest εnets is Ω () 1 1 log log. By a theorem ε ε of Aronov, Ezra, and Sharir (2010), this bound is tight.
. Using probabilistic
"... 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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According to a well known theorem of Haussler and Welzl (1987), any range space of bounded VCdimension admits an εnet of size O ()