On the Intriguing Search for Good Permutations

Pausinger, Florian

doi:10.2478/udt-2019-0005

Cited by 5 publications

(9 citation statements)

References 49 publications

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“…This question, originally due to van der Corput, was answered by van Aardenne-Ehrenfest who showed that no such result exists; see [15] for details. The problem was finally solved by Schmidt [23] who proved that D N ≥ cN −1 log N for infinitely many N ∈ ℕ ; see [12,15,20,23] and references therein as well as [16,17] for the currently best constants. Sequences with D N (X) ≤ cN −1 log N for a constant c > 0 and all N are called low discrepancy sequences.…”

Section: A Problem In Combinatorics and Number Theorymentioning

confidence: 99%

“…The original definition of the van der Corput sequence is based on writing integers in binary expansion: the n− th element is given by (1) writing the integer in binary, i.e. 22 = 10110 2 (2) inverting the order of the digits 10110 → 01101 (3) writing a comma in front of it and interpreting it as a real number in [0,1] The van der Corput sequence and its various generalisations are known to be very close to optimal with regards to discrepancy; see [11,12,20].…”

Section: A Problem In Combinatorics and Number Theorymentioning

confidence: 99%

“…Consequently, the only sequence generated from an identity permutation that improves the result of the classical sequence in base 2 is S 3 ; for more information, see the recent survey [20].…”

Section: Discrepancy Of Permuted Van Der Corput Sequencesmentioning

confidence: 99%

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Greedy energy minimization can count in binary: point charges and the van der Corput sequence

Pausinger

2020

Annali di Matematica

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This paper establishes a connection between a problem in Potential Theory and Mathematical Physics, arranging points so as to minimize an energy functional, and a problem in Combinatorics and Number Theory, constructing ’well-distributed’ sequences of points on [0, 1). Let $$f:[0,1] \rightarrow {\mathbb {R}}$$ f : [ 0 , 1 ] → R be (1) symmetric $$f(x) = f(1-x)$$ f ( x ) = f ( 1 - x ) , (2) twice differentiable on (0, 1), and (3) such that $$f''(x)>0$$ f ′ ′ ( x ) > 0 for all $$x \in (0,1)$$ x ∈ ( 0 , 1 ) . We study the greedy dynamical system, where, given an initial set $$\{x_0, \ldots , x_{N-1}\} \subset [0,1)$$ { x 0 , … , x N - 1 } ⊂ [ 0 , 1 ) , the point $$x_N$$ x N is obtained as $$\begin{aligned} x_{N} = \arg \min _{x \in [0,1)} \sum _{k=0}^{N-1}{f(|x-x_k|)}. \end{aligned}$$ x N = arg min x ∈ [ 0 , 1 ) ∑ k = 0 N - 1 f ( | x - x k | ) . We prove that if we start this construction with the single element $$x_0=0$$ x 0 = 0 , then all arising constructions are permutations of the van der Corput sequence (counting in binary and reflected about the comma): greedy energy minimization recovers the way we count in binary. This gives a new construction of the classical van der Corput sequence. The special case $$f(x) = 1-\log (2 \sin (\pi x))$$ f ( x ) = 1 - log ( 2 sin ( π x ) ) answers a question of Steinerberger. Interestingly, the point sets we derive are also known in a different context as Leja sequences on the unit disk.

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Section: A Problem In Combinatorics and Number Theorymentioning

confidence: 99%

Section: A Problem In Combinatorics and Number Theorymentioning

confidence: 99%

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Greedy energy minimization can count in binary: point charges and the van der Corput sequence

Pausinger

2020

Annali di Matematica

Self Cite

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“…for every point in the solution space. To calculate what this number is, the number of permutations possible within our paths must be calculated [46], using…”

Section: Path Determinationmentioning

confidence: 99%

On the Traveling Salesman Problem in Nautical Environments: an Evolutionary Computing Approach to Optimization of Tourist Route Paths in Medulin, Croatia

Šegota

Lorencin

Ohkura

et al. 2019

JMTS

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The Traveling salesman problem (TSP) defines the problem of finding the optimal path between multiple points, connected by paths of a certain cost. This paper applies that problem formulation in the maritime environment, specifically a path planning problem for a tour boat visiting popular tourist locations in Medulin, Croatia. The problem is solved using two evolutionary computing methods – the genetic algorithm (GA) and the simulated annealing (SA) - and comparing the results (are compared) by an extensive search of the solution space. The results show that evolutionary computing algorithms provide comparable results to an extensive search in a shorter amount of time, with SA providing better results of the two.

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“…That is, f * is the supremum over all c such that ( * ) holds. Study in areas of the same flavour have appeared recently in the form of asympototic constants of the corresponding notions of (star and extreme) discrepancy [12,13,15,16]. Returning to our motivation, in his 1986 paper Proinov states a lower bound for f * .…”

Section: Introductionmentioning

confidence: 98%

On Proinov’s Lower Bound for the Diaphony

Kirk

2020

Uniform Distribution Theory

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In 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.

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On the Intriguing Search for Good Permutations

Cited by 5 publications

References 49 publications

Greedy energy minimization can count in binary: point charges and the van der Corput sequence

Greedy energy minimization can count in binary: point charges and the van der Corput sequence

On the Traveling Salesman Problem in Nautical Environments: an Evolutionary Computing Approach to Optimization of Tourist Route Paths in Medulin, Croatia

On Proinov’s Lower Bound for the Diaphony

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