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On approximate halfspace range counting and relative epsilonapproximations
 In Proc. 23rd Annu. ACM Sympos. Comput. Geom
, 2007
"... The paper consists of two major parts. In the first part, we reexamine relative εapproximations, previously studied in [12, 13, 18, 25], and their relation to certain geometric problems, most notably to approximate range counting. We give a simple constructive proof of their existence in general r ..."
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Cited by 17 (7 self)
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The paper consists of two major parts. In the first part, we reexamine relative εapproximations, previously studied in [12, 13, 18, 25], and their relation to certain geometric problems, most notably to approximate range counting. We give a simple constructive proof of their existence in general range spaces with finite VC dimension, and of a sharp bound on their size, close to the best known one. We then give a construction of smallersize relative εapproximations for range spaces that involve points and halfspaces in two and higher dimensions. The planar construction is based on a new structure—spanning trees with small relative crossing number, which we believe to be of independent interest. In the second part, we consider the approximate halfspace rangecounting problem in R d with relative error ε, and show that relative εapproximations, combined with the shallow partitioning data structures of Matouˇsek, yields efficient solutions to this problem. For example, one of our data structures requires linear storage and O(n 1+δ) preprocessing time, for any δ> 0, and answers a query in time O(ε −γ n 1−1/⌊d/2 ⌋ 2 b log ∗ n), for any γ> 2/⌊d/2⌋; the choice of γ and δ affects b and the implied constants. Several variants and extensions are also discussed.
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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Cited by 15 (0 self)
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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.
Relative εApproximations in Geometry
, 2007
"... We reexamine relative εapproximations, previously studied in [Pol86, Hau92, LLS01, CKMS06], and their relation to certain geometric problems. We give a simple constructive proof of their existence in general range spaces with finite VCdimension, and of a sharp bound on their size, close to the be ..."
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Cited by 13 (5 self)
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We reexamine relative εapproximations, previously studied in [Pol86, Hau92, LLS01, CKMS06], and their relation to certain geometric problems. We give a simple constructive proof of their existence in general range spaces with finite VCdimension, and of a sharp bound on their size, close to the best known one. We then give a construction of smallersize relative εapproximations for range spaces that involve points and halfspaces in two and higher dimensions. The planar construction is based on a new structure—spanning trees with small relative crossing number, which we believe to be of independent interest. We also consider applications of the new structures for approximate range counting and related problems.
Approximation of MultiColor Discrepancy
 Randomization, Approximation and Combinatorial Optimization (Proceedings of APPROXRANDOM 1999), volume 1671 of Lecture Notes in Computer Science
, 1999
"... . In this article we introduce (combinatorial) multicolor discrepancy and generalize some classical results from 2color discrepancy theory to c colors. We give a recursive method that constructs ccolorings from approximations to the 2color discrepancy. This method works for a large class of ..."
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Cited by 9 (7 self)
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. In this article we introduce (combinatorial) multicolor discrepancy and generalize some classical results from 2color discrepancy theory to c colors. We give a recursive method that constructs ccolorings from approximations to the 2color discrepancy. This method works for a large class of theorems like the sixstandarddeviation theorem of Spencer, the BeckFiala theorem and the results of Matousek, Welzl and Wernisch for bounded VCdimension. On the other hand there are examples showing that discrepancy in c colors can not be bounded in terms of twocolor discrepancy even if c is a power of 2. For the linear discrepancy version of the BeckFiala theorem the recursive approach also fails. Here we extend the method of floating colors to multicolorings and prove multicolor versions of the the BeckFiala theorem and the BaranyGrunberg theorem. 1 Introduction Combinatorial discrepancy theory deals with the problem of partitioning the vertices of a hypergraph (set...
VapnikChervonenkis dimension and (pseudo)hyperplane arrangements
, 1997
"... An arrangement of oriented pseudohyperplanes in affine dspace defines on its set X of pseudohyperplanes a set system (or range space) (X, R), R ⊆ 2 X of VCdimension d in a natural way: to every cell c in the arrangement assign the subset of pseudohyperplanes having c on their positive side, and let ..."
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Cited by 6 (1 self)
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An arrangement of oriented pseudohyperplanes in affine dspace defines on its set X of pseudohyperplanes a set system (or range space) (X, R), R ⊆ 2 X of VCdimension d in a natural way: to every cell c in the arrangement assign the subset of pseudohyperplanes having c on their positive side, and let R be the collection of all these subsets. We investigate and characterize the range spaces corresponding to simple arrangements of pseudohyperplanes in this way; such range spaces are called pseudogeometric, and they have the property that the cardinality of R is maximum for the given VCdimension. In general, such range spaces are called maximum, and we show that the number of ranges R ∈ R for which also X −R ∈ R, determines whether a maximum range space is pseudogeometric. Two other characterizations go via a simple duality concept and ‘small ’ subspaces. The correspondence to arrangements is obtained indirectly via a new characterization of uniform oriented matroids: a range space (X, R) naturally corresponds to a uniform oriented matroid of rank X  − d if and only if its VCdimension is d, R ∈ R implies X − R ∈ R and R  is maximum under these conditions.
Approximate Halfspace Range Counting
, 2008
"... We present a simple scheme extending the shallow partitioning data structures of Matouˇsek, that supports efficient approximate halfspace rangecounting queries in R d with relative error ε. Specifically, the problem is, given a set P of n points in R d, to preprocess them into a data structure that ..."
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Cited by 5 (3 self)
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We present a simple scheme extending the shallow partitioning data structures of Matouˇsek, that supports efficient approximate halfspace rangecounting queries in R d with relative error ε. Specifically, the problem is, given a set P of n points in R d, to preprocess them into a data structure that returns, for a query halfspace h, a number t so that (1−ε)h∩P  ≤ t ≤ (1+ε)h∩P . One of our data structures requires linear storage and O(n 1+δ) preprocessing time, for any δ> 0, and answers a query in time O ( ε −γ n 1−1/⌊d/2 ⌋ 2 blog ∗ n) , for any γ> 2/⌊d/2⌋; the choice of γ and δ affects b and the implied constants. Several variants and extensions are also discussed. As presented, the construction of the structure is mostly deterministic, except for one critical randomized step. The query efficiency is guaranteed with high probability, for all queries. The construction can also be fully derandomized, at the expense of increasing preprocessing time.
QUASIMONTECARLO METHODS AND THE DISPERSION OF POINT SEQUENCES
, 1996
"... QuasiMonteCarlo methods are wellknown for solving different problems of numerical analysis such as integration, optimization, etc. The error estimates for global optimization depend on the dispersion of the point sequence with respect to balls. In general, the dispersion of a point set with respe ..."
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Cited by 4 (0 self)
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QuasiMonteCarlo methods are wellknown for solving different problems of numerical analysis such as integration, optimization, etc. The error estimates for global optimization depend on the dispersion of the point sequence with respect to balls. In general, the dispersion of a point set with respect to various classes of range spaces, like balls, squares, triangles, axisparallel and arbitrary rectangles, spherical caps and slices, is the area of the largest empty range, and it is a measure for the distribution of the points. The main purpose of our paper is to give a survey about this topic, including some folklore results. Furthermore, we prove several properties of the dispersion, generalizing investigations of Niederreiter and others concerning balls. For several wellknown uniformly distributed point sets we estimate the dispersion with respect to triangles, and we also compare them computationally. For the dispersion with respect to spherical slices we mention an application to the polygonal approximation of curves in space.
Approximating Majority Depth
 in: Proceedings of the 24rd Canadian Conference on Computational Geometry (CCCG’12
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