Computing the maximum bichromatic discrepancy is an interesting theoretical problem with important applications in computational learning theory, computational geometry and computer graphics. In this paper we give algorithms to compute the maximum bichromatic discrepancy for simple geometric ranges, including rectangles and halfspaces. In addition, we give extensions to other discrepancy problems. 1. Introduction The main theme of this paper is to present efficient algorithms that solve the problem of computing the maximum bichromatic discrepancy for axis oriented rectangles. This problem arises naturally in different areas of computer science, such as computational 1 The research work of these authors was supported by NSF Grant CCR93-01254 and the Geometry Center. learning theory, computational geometry and computer graphics ([Ma], [DG]), and has applications in all these areas. In computational learning theory, the problem of agnostic PAC-learning with simple geometric hypothese...

We describe efficient constructions of small probability spaces that approximate the independent distribution for general random variables. Previous work on efficient constructions concentrate on approximations of the independent distribution for the special case of uniform boolean-valued random variables. Our results yield efficient constructions of small sets with low discrepancy in high dimensional space and have applications to derandomizing randomized algorithms. 1 Introduction The problem of constructing small sample spaces that "approximate" the independent distribution on n random variables has received considerable attention recently (cf. [6, Chor Goldreich] [8, Karp Wigderson], [11, Luby], [1, Alon Babai Itai], [13, Naor Naor], [2, Alon Goldreich Hastad Peralta], [3, Azar Motwani Naor]). The primary motivation for this line of research is that random variables that are "approximately" independent suffices for the analysis of many interesting randomized algorithm and hence c...

We present two techniques for approximating probability distributions. The first is a simple method for constructing the small-bias probability spaces introduced by Naor & Naor. We show how to efficiently combine this construction with the method of conditional probabilities to yield improved NC algorithms for many problems such as set discrepancy, finding large cuts in graphs, finding large acyclic subgraphs etc. The second is a construction of small probability spaces approximating general independent distributions, which is of smaller size than the constructions of Even, Goldreich, Luby, Nisan & Velickovi'c. Such approximations are useful, e.g., for the derandomization of certain randomized algorithms. Keywords. Derandomization, parallel algorithms, discrepancy, graph coloring, small sample spaces, explicit constructions. 1 Introduction Derandomization, the development of general tools to derive efficient deterministic algorithms from their randomized counterparts, has blossomed ...

Quasi-random (or low discrepancy) sequences are sequences for which the convergence to the uniform distribution on [0; 1) s occurs rapidly. Such sequences are used in quasi-Monte Carlo methods for which the convergence speed, with respect to the N first terms of the sequence, is in O(N \Gamma1 (ln N) s ), where s is the mathematical dimension of the problem considered. The disadvantage of these methods is that error bounds, even if they exist theoretically, are inefficient in practice. Nevertheless, to take advantage of these methods for what concerns their convergence speed, we use them as a variance reduction technique, which lead to great improvements with respect to standard Monte Carlo methods. We consider in this paper two different approaches which combine Monte Carlo and quasi-Monte Carlo methods. The first one can use every low discrepancy sequence and the second one, called Owen's method, uses only Niederreiter sequences. We prove that the first approach has the same...

We investigate the energy of arrangements of N points on the surface of the unit sphere S d in R d+1 that interact through a power law potential V = 1=r s ; where s ? 0 and r is Euclidean distance. With Ed(s; N) denoting the minimal energy for such N-point arrangements we obtain bounds (valid for all N) for E d (s; N) in the cases when 0 ! s ! d and 2 d ! s. For s = d, we determine the precise asymptotic behavior of Ed(d; N) as N !1. As a corollary, lower bounds are given for the separation of any pair of points in an N-point minimal energy configuration, when s d 2. For the unit sphere in R 3 (d = 2), we present two conjectures concerning the asymptotic expansion of E 2 (s; N) that relate to the zeta function iL(s) for a hexagonal lattice in the plane. We prove an asymptotic upper bound that supports the first of these conjectures. Of related interest, we derive an asymptotic formula for the partial sums of iL (s) when 0 ! s ! 2 (the divergent case). 1 Introduction and ...

A common subproblem of DNF approximate counting and derandomizing RL is the discrepancy problem for combinatorial rectangles. We explicitly construct a poly(n)-size sample space that approximates the volume of any combinatorial rectangle in [n] n to within o(1) error (improving on the constructions of [EGLNV92]). The construction extends the techniques of [LLSZ95] for the analogous hitting set problem, most notably via discrepancy preserving reductions. 1 Introduction In a general discrepancy problem, we are given a family of sets and want to construct a small sample space that approximates the volume of an arbitrary set in the family. This problem is closely related to other important issues in combinatorial constructions such as the problem of constructing small sample spaces that approximate the independent distributions on many multivalued random variables [KW84, Lub85, ABI86, CG89, NN90, AGHP90, EGLNV92, Sch92, KM93, KK94], and the problem of constructing pseudorandom generat...

We survey techniques for replacing randomized algorithms in computational geometry by deterministic ones with a similar asymptotic running time. 1 Randomized algorithms and derandomization A rapid growth of knowledge about randomized algorithms stimulates research in derandomization, that is, replacing randomized algorithms by deterministic ones with as small decrease of efficiency as possible. Related to the problem of derandomization is the question of reducing the amount of random bits needed by a randomized algorithm while retaining its efficiency; the derandomization can be viewed as an ultimate case. Randomized algorithms are also related to probabilistic proofs and constructions in combinatorics (which came first historically), whose development has similarly been accompanied by the effort to replace them by explicit, non-random constructions whenever possible. Derandomization of algorithms can be seen as a part of an effort to map the power of randomness and explain its role. ...

Every set S of n points in the plane has a spanning tree such that no line disjoint from S has more than O( p n) intersections with the tree (where the edges are embedded as straight line segments). We review the proof of this result (originally proved by Bernard Chazelle and the author in a more general setting), point at some methods for constructing such a tree, and describe some algorithmic and combinatorial applications. 1 Introduction. Over the recent years there has been considerable progress in the simplex range searching problem. In the planar version of this problem we are required to store a set S of n points such that the number of points in any query triangle can be determined efficiently. One of the combinatorial tools developed for this problem are spanning trees with low crossing numbers. Let S be set of n points in the plane. For a spanning tree on S and a line h, the crossing number of h in the tree is defined as c h = a + b 2 , where a is the number of edges ...

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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.