Polynomial-time algorithms for continuous metrics on atomic clouds of unordered points
Abstract
The most fundamental model of a molecule is a cloud of unordered atoms, even without chemical bonds that can depend on thresholds for distances and angles. The strongest equivalence between clouds of atoms is rigid motion, which is a composition of translations and rotations. The existing datasets of experimental and simulated molecules require a continuous quantification of similarity in terms of a distance metric. While clouds of m ordered points were continuously classified by Lagrange's quadratic forms (distance matrices or Gram matrices), their extensions to m unordered points are impractical due to the exponential number of m! permutations. We propose new metrics that are continuous in general position and are computable in a polynomial time in the number m of unordered points in any Euclidean space of a fixed dimension n.
Cite
@article{arxiv.2207.08502,
title = {Polynomial-time algorithms for continuous metrics on atomic clouds of unordered points},
author = {Vitaliy Kurlin},
journal= {arXiv preprint arXiv:2207.08502},
year = {2023}
}
Comments
30 pages, 2 figures. The paper was published (with grayscale images) in the journal MATCH Communications in Mathematical and in Computer Chemistry, v.91 (2024), p.79-108, https://doi.org/10.46793/match.91-1.079K. This version includes color images. The latest file is at http://kurlin.org/projects/cloud-isometry-spaces/metrics-atomic-clouds.pdf