Efficient linear programming algorithm to generate the densest lattice sphere packings

Étienne Marcotte and Salvatore Torquato
Phys. Rev. E 87, 063303 – Published 7 June 2013

Abstract

Finding the densest sphere packing in d-dimensional Euclidean space Rd is an outstanding fundamental problem with relevance in many fields, including the ground states of molecular systems, colloidal crystal structures, coding theory, discrete geometry, number theory, and biological systems. Numerically generating the densest sphere packings becomes very challenging in high dimensions due to an exponentially increasing number of possible sphere contacts and sphere configurations, even for the restricted problem of finding the densest lattice sphere packings. In this paper we apply the Torquato-Jiao packing algorithm, which is a method based on solving a sequence of linear programs, to robustly reproduce the densest known lattice sphere packings for dimensions 2 through 19. We show that the TJ algorithm is appreciably more efficient at solving these problems than previously published methods. Indeed, in some dimensions, the former procedure can be as much as three orders of magnitude faster at finding the optimal solutions than earlier ones. We also study the suboptimal local density-maxima solutions (inherent structures or “extreme” lattices) to gain insight about the nature of the topography of the “density” landscape.

  • Figure
  • Received 19 April 2013

DOI:https://doi.org/10.1103/PhysRevE.87.063303

©2013 American Physical Society

Authors & Affiliations

Étienne Marcotte1 and Salvatore Torquato1,2,3,4,*

  • 1Department of Physics, Princeton University, Princeton, New Jersey 08544, USA
  • 2Department of Chemistry, Princeton University, Princeton, New Jersey 08544, USA
  • 3Program in Applied and Computational Mathematics, Princeton University, Princeton, New Jersey 08544, USA
  • 4Princeton Institute of the Science and Technology of Materials, Princeton University, Princeton, New Jersey 08544, USA

  • *torquato@princeton.edu

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Issue

Vol. 87, Iss. 6 — June 2013

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