Construction of Low-Discrepancy Point Sets of Small Size by Bracketing Covers and Dependent Randomized Rounding

Doerr, Benjamin; Gnewuch, Michael

doi:10.1007/978-3-540-74496-2_17

Cited by 18 publications

(26 citation statements)

References 19 publications

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“…This establishes (2). Let us identify the functions 1 [0,x) in C d with the corresponding points x ∈ [0, 1] d and the functions 1 [x,y) …”

mentioning

confidence: 94%

Bracketing numbers for axis-parallel boxes and applications to geometric discrepancy

Gnewuch

2008

Journal of Complexity

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In the first part of this paper we derive lower bounds and constructive upper bounds for the bracketing numbers of anchored and unanchored axis-parallel boxes in the d-dimensional unit cube.In the second part we apply these results to geometric discrepancy. We derive upper bounds for the inverse of the star and the extreme discrepancy with explicitly given small constants and an optimal dependence on the dimension d, and provide corresponding bounds for the star and the extreme discrepancy itself. These bounds improve known results from [B. Doerr, M. Gnewuch, A. Srivastav, Bounds and constructions for the star-discrepancy via -covers, J. Complexity 21 (2005) 691-709], [M. Gnewuch, Bounds for the average L p -extreme and the L ∞ -extreme discrepancy, Electron. J. Combin. 12 (2005) Research Paper 54] and [H. N. Mhaskar, On the tractability of multivariate integration and approximation by neural networks, J. Complexity 20 (2004) 561-590].We also discuss an algorithm from [E. Thiémard, An algorithm to compute bounds for the star discrepancy, J. Complexity 17 (2001) 850-880] to approximate the star-discrepancy of a given n-point set. Our lower bound on the bracketing number of anchored boxes, e.g., leads directly to a lower bound of the running time of Thiémard's algorithm. Furthermore, we show how one can use our results to modify the algorithm to approximate the extreme discrepancy of a given set.

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“…This establishes (2). Let us identify the functions 1 [0,x) in C d with the corresponding points x ∈ [0, 1] d and the functions 1 [x,y) …”

mentioning

confidence: 94%

Bracketing numbers for axis-parallel boxes and applications to geometric discrepancy

Gnewuch

2008

Journal of Complexity

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“…This bound is obviously weaker than the bounds in (1.1) and (1.2), but the run-time of our derandomized algorithm improves considerably on the run-times of the algorithms in [4,5] generating point sets satisfying (1.2). In Section 5 we compare the run-times of the algorithms to each other and relate the problem of constructing low-discrepancy samples of small size to the problem of approximating the star discrepancy of a given point set.…”

Section: Our Resultsmentioning

confidence: 76%

“…In the light of these results it is not too surprising that the run-times of the derandomized algorithms in this paper and in [4,5] are exponentially in s, since we cannot expect to do the (deterministic) derandomized construction with less effort than the (probabilistic) semi-construction.…”

Section: Resultsmentioning

confidence: 78%

“…The first is the concept of so-called δ-covers, as used in [4,5,7]. A δ-cover is defined as follows.…”

Section: Infinite Dimensional Infinite Sequencesmentioning

confidence: 99%

“…An algorithm using a different derandomization technique was presented in the recent paper [4]. It also generates an N -point set in dimension s satisfying the bound (1.2), but its run-time improves considerably over the algorithm from [5].…”

Section: Introductionmentioning

confidence: 99%

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Component-by-component construction of low-discrepancy point sets of small size

Doerr

Gnewuch²,

Kritzer³

et al. 2008

McMa

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Abstract. We investigate the problem of constructing small point sets with low star discrepancy in the s-dimensional unit cube. The size of the point set shall always be polynomial in the dimension s. Our particular focus is on extending the dimension of a given low-discrepancy point set.This results in a deterministic algorithm that constructs N -point sets with small discrepancy in a component-by-component fashion. The algorithm also provides the exact star discrepancy of the output set. Its run-time considerably improves on the run-times of the currently known deterministic algorithms that generate low-discrepancy point sets of comparable quality.We also study infinite sequences of points with infinitely many components such that all initial subsegments projected down to all finite dimensions have low discrepancy. To this end, we introduce the inverse of the star discrepancy of such a sequence, and derive upper bounds for it as well as for the star discrepancy of the projections of finite subsequences with explicitly given constants. In particular, we establish the existence of sequences whose inverse of the star discrepancy depends linearly on the dimension.

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