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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.
Geometric Set Systems
, 1998
"... Let X be a finite point set in the plane. We consider the set system on X whose sets are all intersections of X with a halfplane. Similarly one can investigate set systems defined on point sets in higherdimensional spaces by other classes of simple geometric figures (simplices, balls, ellipsoids, e ..."
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Let X be a finite point set in the plane. We consider the set system on X whose sets are all intersections of X with a halfplane. Similarly one can investigate set systems defined on point sets in higherdimensional spaces by other classes of simple geometric figures (simplices, balls, ellipsoids, etc.). It turns out that simple combinatorial properties of such set systems (most notably the VapnikChervonenkis dimension and related concepts of shatter functions) play an important role in several areas of mathematics and theoretical computer science. Here we concentrate on applications in discrepancy theory, in combinatorial geometry, in derandomization of geometric algorithms, and in geometric range searching. We believe that the described tools might be useful in other areas of mathematics too. 1 Introduction For a set system S ` 2 X on an arbitrary ground set X and for A ` X, we write Sj A = fS " A; S 2 Sg for the set system induced by S on A (or the trace of S on A). Let H den...
Discrepancy of Point Sequences on Fractal Sets
"... We consider asymptotic bounds for the discrepancy of point sets on a class of fractal sets. By a method of R. Alexander, we prove that for a wide class of fractals, the L 2 discrepancy (and consequently also the worstcase discrepancy) of an Npoint set with respect to halfspaces is at least of the ..."
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We consider asymptotic bounds for the discrepancy of point sets on a class of fractal sets. By a method of R. Alexander, we prove that for a wide class of fractals, the L 2 discrepancy (and consequently also the worstcase discrepancy) of an Npoint set with respect to halfspaces is at least of the order N^(1/21/(2s)) , where s is the Hausdorff dimension of the fractal. We also show that for many fractals, this bound is tight for the L 2 discrepancy. Determining the correct order of magnitude of the worstcase discrepancy remains a challenging open problem.
A SizeSensitive Discrepancy Bound for Set Systems of Bounded Primal Shatter Dimension ∗
"... Let (X,S) be a set system on an npoint set X. The discrepancy of S is defined as the minimum of the largest deviation from an even split, over all subsets of S ∈ S and twocolorings χ on X. We consider the scenario where, for any subset X ′ ⊆ X of size m ≤ n and for any parameter 1 ≤ k ≤ m, the nu ..."
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Let (X,S) be a set system on an npoint set X. The discrepancy of S is defined as the minimum of the largest deviation from an even split, over all subsets of S ∈ S and twocolorings χ on X. We consider the scenario where, for any subset X ′ ⊆ X of size m ≤ n and for any parameter 1 ≤ k ≤ m, the number of restrictions of the sets of S to X ′ of size at most k is only O(m d1 k d−d1), for fixed integers d> 0 and 1 ≤ d1 ≤ d (this generalizes the standard notion of bounded primal shatter dimension when d1 = d). In this case we show that there exists a coloring χ with discrepancy bound O ∗ (S  1/2−d1/(2d) n (d1−1)/(2d)), for each S ∈ S, where O ∗ (·) hides a polylogarithmic factor in n. This bound is tight up to a polylogarithmic factor [25, 27] and the corresponding coloring χ can be computed in expected polynomial time using the very recent machinery of Lovett and Meka for constructive discrepancy minimization [24]. Our bound improves and generalizes the bounds obtained from the machinery of HarPeled and Sharir [19] (and the followup work in [32]) for points and halfspaces in dspace for d ≥ 3. Last but not least, we show that our bound yields improved bounds for the size of relative (ε,δ)approximations for set systems of the above kind.