The paper consists of two major parts. In the first part, we re-examine relative ε-approximations, 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 smaller-size relative ε-approximations 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 range-counting problem in R d with relative error ε, and show that relative ε-approximations, 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.

. In this article we introduce (combinatorial) multi--color discrepancy and generalize some classical results from 2--color discrepancy theory to c colors. We give a recursive method that constructs c--colorings from approximations to the 2--color discrepancy. This method works for a large class of theorems like the six--standard--deviation theorem of Spencer, the Beck--Fiala theorem and the results of Matousek, Welzl and Wernisch for bounded VC--dimension. On the other hand there are examples showing that discrepancy in c colors can not be bounded in terms of two--color discrepancy even if c is a power of 2. For the linear discrepancy version of the Beck--Fiala theorem the recursive approach also fails. Here we extend the method of floating colors to multi--colorings and prove multi--color versions of the the Beck--Fiala theorem and the Barany--Grunberg theorem. 1 Introduction Combinatorial discrepancy theory deals with the problem of partitioning the vertices of a hypergraph (set--...

Given a hypergraph H and a subset S of its vertices, the trace of H on S is defined as H|S = {E ∩ S: E ∈ H}. The Vapnik–Chervonenkis dimension (VC-dimension) of H is the size of the largest subset S for which H|S has 2 |S| edges. Hypergraphs of small VC-dimension 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.

An arrangement of oriented pseudohyperplanes in affine d-space 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 VC-dimension. 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 VC-dimension is d, R ∈ R implies X − R ∈ R and |R | is maximum under these conditions.

We present a simple scheme extending the shallow partitioning data structures of Matouˇsek, that supports efficient approximate halfspace range-counting 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.

We re-examine relative ε-approximations, 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 VC-dimension, and of a sharp bound on their size, close to the best known one. We then give a construction of smaller-size relative ε-approximations 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. 1

Abstract. Quasi-Monte-Carlo methods are well-known for solving di erent problems of numerical analysis such asintegration, 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, axis-parallel 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 well-known 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. 1.