Evaluation of pairwise distances among points forming a regular orthogonal grid in a hypercube

Cartesian grid is a basic arrangement of points that form a regular orthogonal grid (ROG). In some applications, it is needed to evaluate all pairwise distances among ROG points. This paper focuses on ROG discretization of a unit hypercube of arbitrary dimension. A method for the fast enumeration o...

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Main Authors: Václav Sadílek, Miroslav Vořechovský
Format: Article
Language:English
Published: Vilnius Gediminas Technical University 2018-09-01
Series:Journal of Civil Engineering and Management
Subjects:
Online Access:https://www.tede.vgtu.lt/index.php/JCEM/article/view/5189
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spelling doaj-125fbae7d69346c2b15b29d41c641f002021-07-02T12:33:19ZengVilnius Gediminas Technical UniversityJournal of Civil Engineering and Management1392-37301822-36052018-09-0124510.3846/jcem.2018.5189Evaluation of pairwise distances among points forming a regular orthogonal grid in a hypercubeVáclav Sadílek0Miroslav Vořechovský1Institute of Structural Mechanics, Faculty of Civil Engineering, Brno University of TechnologyInstitute of Structural Mechanics, Faculty of Civil Engineering, Brno University of Technology, Czech Republic Cartesian grid is a basic arrangement of points that form a regular orthogonal grid (ROG). In some applications, it is needed to evaluate all pairwise distances among ROG points. This paper focuses on ROG discretization of a unit hypercube of arbitrary dimension. A method for the fast enumeration of all pairwise distances and their counts for a high number of points arranged into high-dimensional ROG is presented. The proposed method exploits the regular and collapsible pattern of ROG to reduce the number of evaluated distances. The number of unique distances is identified and frequencies are determined using combinatorial rules. The measured computational speed-up compared to a naïve approach corresponds to the presented theoretical analysis. The proposed method and algorithm may find applications in various fields. The paper shows application focused on the behaviour of various distance measures with the motivation to find the lower bounds on the criteria of point distribution uniformity in Monte Carlo integration. https://www.tede.vgtu.lt/index.php/JCEM/article/view/5189full factorial designdesign of experimentspairwise distancesAudze-Eglãjs criterionoptimizationperiodic space
collection DOAJ
language English
format Article
sources DOAJ
author Václav Sadílek
Miroslav Vořechovský
spellingShingle Václav Sadílek
Miroslav Vořechovský
Evaluation of pairwise distances among points forming a regular orthogonal grid in a hypercube
Journal of Civil Engineering and Management
full factorial design
design of experiments
pairwise distances
Audze-Eglãjs criterion
optimization
periodic space
author_facet Václav Sadílek
Miroslav Vořechovský
author_sort Václav Sadílek
title Evaluation of pairwise distances among points forming a regular orthogonal grid in a hypercube
title_short Evaluation of pairwise distances among points forming a regular orthogonal grid in a hypercube
title_full Evaluation of pairwise distances among points forming a regular orthogonal grid in a hypercube
title_fullStr Evaluation of pairwise distances among points forming a regular orthogonal grid in a hypercube
title_full_unstemmed Evaluation of pairwise distances among points forming a regular orthogonal grid in a hypercube
title_sort evaluation of pairwise distances among points forming a regular orthogonal grid in a hypercube
publisher Vilnius Gediminas Technical University
series Journal of Civil Engineering and Management
issn 1392-3730
1822-3605
publishDate 2018-09-01
description Cartesian grid is a basic arrangement of points that form a regular orthogonal grid (ROG). In some applications, it is needed to evaluate all pairwise distances among ROG points. This paper focuses on ROG discretization of a unit hypercube of arbitrary dimension. A method for the fast enumeration of all pairwise distances and their counts for a high number of points arranged into high-dimensional ROG is presented. The proposed method exploits the regular and collapsible pattern of ROG to reduce the number of evaluated distances. The number of unique distances is identified and frequencies are determined using combinatorial rules. The measured computational speed-up compared to a naïve approach corresponds to the presented theoretical analysis. The proposed method and algorithm may find applications in various fields. The paper shows application focused on the behaviour of various distance measures with the motivation to find the lower bounds on the criteria of point distribution uniformity in Monte Carlo integration.
topic full factorial design
design of experiments
pairwise distances
Audze-Eglãjs criterion
optimization
periodic space
url https://www.tede.vgtu.lt/index.php/JCEM/article/view/5189
work_keys_str_mv AT vaclavsadilek evaluationofpairwisedistancesamongpointsformingaregularorthogonalgridinahypercube
AT miroslavvorechovsky evaluationofpairwisedistancesamongpointsformingaregularorthogonalgridinahypercube
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