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.

Quantum collision-resistance of non-uniformly distributed functions: upper and lower bounds

2018 ◽  
Vol 18 (15&16) ◽  
pp. 1332-1349
Author(s):  
Ehsan Ebrahimi ◽  
Dominique Unruh
Keyword(s):  
Lower Bound ◽  
Uniform Distribution ◽  
Lower Bounds ◽  
Upper Bounds ◽  
Query Complexity ◽  
Upper And Lower Bounds ◽  
Collision Resistance ◽  
Quantum Query Complexity ◽  
Quantum Query

We study the quantum query complexity of finding a collision for a function f whose outputs are chosen according to a non-uniform distribution D. We derive some upper bounds and lower bounds depending on the min-entropy and the collision-entropy of D. In particular, we improve the previous lower bound by Ebrahimi Targhi et al. from \Omega(2^{k/9}) to \Omega(2^{k/5}) where k is the min-entropy of D.

Multivariate Uniformity Test Based on Incomplete Samples

2017 ◽  
Vol 24 (05) ◽  
pp. 1750025
Author(s):  
Zhenmin Chen
Keyword(s):  
Order Statistics ◽  
Uniform Distribution ◽  
Population Distribution ◽  
Test Procedure ◽  
Unit Cube ◽  
Complete Sample ◽  
Research Papers ◽  
Uniformity Test ◽  
Dimensional Unit ◽  
Sample Case

The purpose of this paper is to propose an approach for testing multivariate uniformity. The proposed test procedure works for both complete sample case and incomplete sample case. What multivariate uniformity test does is to check whether the population distribution, from which a random sample is drawn is a multivariate uniform distribution. Particularly, it is desired to check whether the population distribution is a uniform distribution on a [Formula: see text]-dimensional unit cube. There are some existing research papers in the literature. The proposed method works when only some arbitrarily selected order statistics are available for the test.

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 $\delta$.

MASS PROBLEMS AND INITIAL SEGMENT COMPLEXITY

2014 ◽  
Vol 79 (01) ◽  
pp. 20-44 ◽  
Author(s):  
W. M. PHILLIP HUDELSON
Keyword(s):  
Asymptotic Behavior ◽  
Lower Bound ◽  
Lower Bounds ◽  
Kolmogorov Complexity ◽  
Initial Segment ◽  
Infinite Sequence ◽  
Finite Sequence ◽  
Randomness Extraction ◽  
The Given ◽  
Infinite Sequences

Abstract By the complexity of a finite sequence of 0’s and 1’s we mean the Kolmogorov complexity, that is the length of the shortest input to a universal recursive function which returns the given sequence as output. By initial segment complexity of an infinite sequence of 0’s and 1’s we mean the asymptotic behavior of the complexity of its finite initial segments. In this paper, we construct infinite sequences of 0’s and 1’s with given recursive lower bounds on initial segment complexity which do not compute any infinite sequences of 0’s and 1’s with a significantly larger recursive lower bound on initial segment complexity. This improves several known results about randomness extraction and separates many natural degrees in the lattice of Muchnik degrees.

A Lower Bound for Schur Numbers and Multicolor Ramsey Numbers

10.37236/1188 ◽  
1994 ◽  
Vol 1 (1) ◽  
Author(s):  
Geoffrey Exoo
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Ramsey Numbers

For $k \geq 5$, we establish new lower bounds on the Schur numbers $S(k)$ and on the k-color Ramsey numbers of $K_3$.

scholarly journals A Lower Bound for the Query Phase of Contraction Hierarchies and Hub Labels and a Provably Optimal Instance-Based Schema

Algorithms ◽  
2021 ◽  
Vol 14 (6) ◽  
pp. 164
Author(s):  
Tobias Rupp ◽  
Stefan Funke
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Real World ◽  
Road Network ◽  
Upper And Lower Bounds ◽  
Bounded Degree ◽  
Shortest Path Routing ◽  
Graph Classes ◽  
Speed Up ◽  
To Come

We prove a Ω(n) lower bound on the query time for contraction hierarchies (CH) as well as hub labels, two popular speed-up techniques for shortest path routing. Our construction is based on a graph family not too far from subgraphs that occur in real-world road networks, in particular, it is planar and has a bounded degree. Additionally, we borrow ideas from our lower bound proof to come up with instance-based lower bounds for concrete road network instances of moderate size, reaching up to 96% of an upper bound given by a constructed CH. For a variant of our instance-based schema applied to some special graph classes, we can even show matching upper and lower bounds.

scholarly journals On Computing Multilinear Polynomials Using Multi- r -ic Depth Four Circuits

2021 ◽  
Vol 13 (3) ◽  
pp. 1-21
Author(s):  
Suryajith Chillara
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Complexity Measure ◽  
Natural Question ◽  
Multilinear Polynomial ◽  
Arithmetic Circuit ◽  
Theory Of Computing ◽  
Small Constant ◽  
Multilinear Polynomials ◽  
The Individual

In this article, we are interested in understanding the complexity of computing multilinear polynomials using depth four circuits in which the polynomial computed at every node has a bound on the individual degree of r ≥ 1 with respect to all its variables (referred to as multi- r -ic circuits). The goal of this study is to make progress towards proving superpolynomial lower bounds for general depth four circuits computing multilinear polynomials, by proving better bounds as the value of r increases. Recently, Kayal, Saha and Tavenas (Theory of Computing, 2018) showed that any depth four arithmetic circuit of bounded individual degree r computing an explicit multilinear polynomial on n O (1) variables and degree d must have size at least ( n / r 1.1 ) Ω(√ d / r ) . This bound, however, deteriorates as the value of r increases. It is a natural question to ask if we can prove a bound that does not deteriorate as the value of r increases, or a bound that holds for a larger regime of r . In this article, we prove a lower bound that does not deteriorate with increasing values of r , albeit for a specific instance of d = d ( n ) but for a wider range of r . Formally, for all large enough integers n and a small constant η, we show that there exists an explicit polynomial on n O (1) variables and degree Θ (log 2 n ) such that any depth four circuit of bounded individual degree r ≤ n η must have size at least exp(Ω(log 2 n )). This improvement is obtained by suitably adapting the complexity measure of Kayal et al. (Theory of Computing, 2018). This adaptation of the measure is inspired by the complexity measure used by Kayal et al. (SIAM J. Computing, 2017).

scholarly journals Lower bounds for the state complexity of probabilistic languages and the language of prime numbers

2020 ◽  
Vol 30 (1) ◽  
pp. 175-192
Author(s):  
NathanaËl Fijalkow
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Prime Numbers ◽  
Regular Languages ◽  
Automata Theory ◽  
Seminal Paper ◽  
Probabilistic Automata ◽  
State Complexity ◽  
Probabilistic Languages ◽  
Alternating Automata

Abstract This paper studies the complexity of languages of finite words using automata theory. To go beyond the class of regular languages, we consider infinite automata and the notion of state complexity defined by Karp. Motivated by the seminal paper of Rabin from 1963 introducing probabilistic automata, we study the (deterministic) state complexity of probabilistic languages and prove that probabilistic languages can have arbitrarily high deterministic state complexity. We then look at alternating automata as introduced by Chandra, Kozen and Stockmeyer: such machines run independent computations on the word and gather their answers through boolean combinations. We devise a lower bound technique relying on boundedly generated lattices of languages, and give two applications of this technique. The first is a hierarchy theorem, stating that there are languages of arbitrarily high polynomial alternating state complexity, and the second is a linear lower bound on the alternating state complexity of the prime numbers written in binary. This second result strengthens a result of Hartmanis and Shank from 1968, which implies an exponentially worse lower bound for the same model.

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.

Lower Bounds to the Eigenvalues in One-Dimensional Problems by a Shift in the Weight Function

1970 ◽  
Vol 37 (2) ◽  
pp. 267-270 ◽  
Author(s):  
D. Pnueli
Keyword(s):  
Lower Bound ◽  
Weight Function ◽  
Lower Bounds ◽  
Variational Formulation ◽  
Ritz Method ◽  
The Other ◽  
One Dimensional ◽  
Systematic Shift ◽  
The One ◽  
Detailed Computation

A method is presented to obtain both upper and lower bound to eigenvalues when a variational formulation of the problem exists. The method consists of a systematic shift in the weight function. A detailed procedure is offered for one-dimensional problems, which makes improvement of the bounds possible, and which involves the same order of detailed computation as the Rayleigh-Ritz method. The main contribution of this method is that it yields the “other bound;” i.e., the one which cannot be obtained by the Rayleigh-Ritz method.

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