scholarly journals Construction of Minimal Bracketing Covers for Rectangles

10.37236/819 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Michael Gnewuch
Keyword(s):  
Asymptotic Expansion ◽  
Lower Bound ◽  
Unit Cube ◽  
Two Dimensional ◽  
Minimal Cardinality ◽  
Significant Term ◽  
Dimensional Unit ◽  
Axis Parallel ◽  
Geometric Discrepancy

We construct explicit $\delta$-bracketing covers with minimal cardinality for the set system of (anchored) rectangles in the two dimensional unit cube. More precisely, the cardinality of these $\delta$-bracketing covers are bounded from above by $\delta^{-2} + o(\delta^{-2})$. A lower bound for the cardinality of arbitrary $\delta$-bracketing covers for $d$-dimensional anchored boxes from [M. Gnewuch, Bracketing numbers for axis-parallel boxes and applications to geometric discrepancy, J. Complexity 24 (2008) 154-172] implies the lower bound $\delta^{-2}+O(\delta^{-1})$ in dimension $d=2$, showing that our constructed covers are (essentially) optimal. We study also other $\delta$-bracketing covers for the set system of rectangles, deduce the coefficient of the most significant term $\delta^{-2}$ in the asymptotic expansion of their cardinality, and compute their cardinality for explicit values of $\delta$.

scholarly journals Bounds for the Average $L^p$-Extreme and the $L^\infty$-Extreme Discrepancy

10.37236/1951 ◽  
2005 ◽  
Vol 12 (1) ◽  
Author(s):  
Michael Gnewuch
Keyword(s):  
Upper Bounds ◽  
Unit Cube ◽  
Point Sets ◽  
Dimensional Unit ◽  
Axis Parallel ◽  
Geometric Discrepancy

The extreme or unanchored discrepancy is the geometric discrepancy of point sets in the $d$-dimensional unit cube with respect to the set system of axis-parallel boxes. For $2\leq p < \infty$ we provide upper bounds for the average $L^p$-extreme discrepancy. With these bounds we are able to derive upper bounds for the inverse of the $L^\infty$-extreme discrepancy with optimal dependence on the dimension $d$ and explicitly given constants.

scholarly journals On Proinov’s Lower Bound for the Diaphony

2020 ◽  
Vol 15 (2) ◽  
pp. 39-72
Author(s):  
Nathan Kirk
Keyword(s):  
Lower Bound ◽  
Uniform Distribution ◽  
Lower Bounds ◽  
Unit Cube ◽  
Integral Approximation ◽  
Quantitative Theory ◽  
Dimensional Unit ◽  
Explicit Lower Bound ◽  
Irregularities Of Distribution ◽  
Infinite Sequences

AbstractIn 1986, Proinov published an explicit lower bound for the diaphony of finite and infinite sequences of points contained in the d−dimensional unit cube [Proinov, P. D.:On irregularities of distribution, C. R. Acad. Bulgare Sci. 39 (1986), no. 9, 31–34]. However, his widely cited paper does not contain the proof of this result but simply states that this will appear elsewhere. To the best of our knowledge, this proof was so far only available in a monograph of Proinov written in Bulgarian [Proinov, P. D.: Quantitative Theory of Uniform Distribution and Integral Approximation, University of Plovdiv, Bulgaria (2000)]. The first contribution of our paper is to give a self contained version of Proinov’s proof in English. Along the way, we improve the explicit asymptotic constants implementing recent, and corrected results of [Hinrichs, A.—Markhasin, L.: On lower bounds for the ℒ2-discrepancy, J. Complexity 27 (2011), 127–132.] and [Hinrichs, A.—Larcher, G.: An improved lower bound for the ℒ2-discrepancy, J. Complexity 34 (2016), 68–77]. (The corrections are due to a note in [Hinrichs, A.—Larcher, G. An improved lower bound for the ℒ2-discrepancy, J. Complexity 34 (2016), 68–77].) Finally, as a main result, we use the method of Proinov to derive an explicit lower bound for the dyadic diaphony of finite and infinite sequences in a similar fashion.

scholarly journals Encoding Two-Dimensional Range Top-k Queries

Algorithmica ◽  
2021 ◽  
Author(s):  
Seungbum Jo ◽  
Rahul Lingala ◽  
Srinivasa Rao Satti
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Upper Bound ◽  
Total Order ◽  
Two Dimensional ◽  
Information Theoretic ◽  
Cartesian Tree ◽  
Dimensional Range

AbstractWe consider the problem of encoding two-dimensional arrays, whose elements come from a total order, for answering $${\text{Top-}}{k}$$ Top- k queries. The aim is to obtain encodings that use space close to the information-theoretic lower bound, which can be constructed efficiently. For an $$m \times n$$ m × n array, with $$m \le n$$ m ≤ n , we first propose an encoding for answering 1-sided $${\textsf {Top}}{\text {-}}k{}$$ Top - k queries, whose query range is restricted to $$[1 \dots m][1 \dots a]$$ [ 1 ⋯ m ] [ 1 ⋯ a ] , for $$1 \le a \le n$$ 1 ≤ a ≤ n . Next, we propose an encoding for answering for the general (4-sided) $${\textsf {Top}}{\text {-}}k{}$$ Top - k queries that takes $$(m\lg {{(k+1)n \atopwithdelims ()n}}+2nm(m-1)+o(n))$$ ( m lg ( k + 1 ) n n + 2 n m ( m - 1 ) + o ( n ) ) bits, which generalizes the joint Cartesian tree of Golin et al. [TCS 2016]. Compared with trivial $$O(nm\lg {n})$$ O ( n m lg n ) -bit encoding, our encoding takes less space when $$m = o(\lg {n})$$ m = o ( lg n ) . In addition to the upper bound results for the encodings, we also give lower bounds on encodings for answering 1 and 4-sided $${\textsf {Top}}{\text {-}}k{}$$ Top - k queries, which show that our upper bound results are almost optimal.

A new lower bound for the non-oriented two-dimensional bin-packing problem

2007 ◽  
Vol 35 (3) ◽  
pp. 365-373 ◽  
Author(s):  
François Clautiaux ◽  
Antoine Jouglet ◽  
Joseph El Hayek
Keyword(s):  
Lower Bound ◽  
Bin Packing ◽  
Packing Problem ◽  
Two Dimensional ◽  
Bin Packing Problem

scholarly journals The number of lattice rules having given invariants

1992 ◽  
Vol 46 (3) ◽  
pp. 479-495 ◽  
Author(s):  
Stephen Joe ◽  
David C. Hunt
Keyword(s):  
Normal Form ◽  
Quadrature Rule ◽  
Unit Cube ◽  
Numerical Results ◽  
Smith Normal Form ◽  
Lattice Rules ◽  
Dimensional Unit ◽  
Lattice Rule ◽  
Positive Integers ◽  
Theoretical Results

A lattice rule is a quadrature rule used for the approximation of integrals over the s-dimensional unit cube. Every lattice rule may be characterised by an integer r called the rank of the rule and a set of r positive integers called the invariants. By exploiting the group-theoretic structure of lattice rules we determine the number of distinct lattice rules having given invariants. Some numerical results supporting the theoretical results are included. These numerical results are obtained by calculating the Smith normal form of certain integer matrices.

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