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arXiv · 2207.08502

Polynomial-time algorithms for continuous metrics on atomic clouds of unordered points

Vitaliy Kurlin
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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.

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Vitaliy Kurlin. 2022-07-18. Polynomial-time algorithms for continuous metrics on atomic clouds of unordered points. https://doi.org/10.46793/match.91-1.079k

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