Evaluation of Pairwise Distances Among Points Forming a Regular Orthogonal Grid in a Hypercube

Abstract

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.

Periodic version of the minimax distance criterion for Monte Carlo integration

“…Euclidean distance is isotropic, i.e., the distance between two points does not depend on the orientation of the coordinate axes. However, its usage in a higher dimensional hypercube is problematic due to the concentration of distances [39]. The search for D mM is also known as the facility location problem or the set covering location problem [43].…”

“…Based on trial-error analyses, the reasonable N 0 was estimated as N 0 = 3n char , where the characteristic length char = n −1/s represents the shortest distance in the regular orthogonal grid [39]. The procedure usually terminates in the second step, but sometimes more steps are needed.…”

Integrated Sensing and Communications for 3D Object Imaging via Bilinear Inference

“…(3) can be derived, either by studying the N V 3 constituent angles within the highly subdivided geometry of the grid, or via a stochastic geometry-based grid analysis[59]. To the best of the authors' knowledge, however, solutions to this problem exist only for vertex-to-vertex distance distributions[60], with the case of vertex angles never addressed before. Since the focus of this article is to develop ISAC estimators, we leave this matter for a future contribution and meanwhile consider the proposed highly-accurate approximate model, as illustrated in Fig.4.This article has been accepted for publication in IEEE Transactions on Wireless Communications.…”