Complexity and algorithms for computing Voronoi cells of lattices

by Mathieu Dutour Sikiric and Achill Schürmann and Frank Vallentin


Preprint series: 08-19, Preprints

03D15 Complexity of computation, See also {68Q15}
11H56 Automorphism groups of lattices
11H06 Lattices and convex bodies, See also {11P21, 52C05, 52C07}
52B55 Computational aspects related to convexity, {For computational geometry and algorithms, See 68Q20, 68Q25, 68U05; for numerical algorithms, See 65Yxx}
52B12 Special polytopes (linear programming, centrally symmetric, etc.)


Abstract: In this paper we are concerned with finding the vertices of the Voronoi cell of a Euclidean lattice. Given a basis of a lattice, we prove that computing the number of vertices is a \sharpp-hard problem. On the other hand we describe an algorithm for this problem which is especially suited for low dimensional (say dimensions at most $12$) and for highly-symmetric lattices. We use our implementation, which drastically outperforms those of current computer algebra systems, to find the vertices of Voronoi cells and quantizer constants of some prominent lattices.

Keywords: lattice, Voronoi cell, Delone cell, covering radius, quantizer constant, lattice isomorphism problem, zonotope

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Letzte Änderung: 01.03.2018 - Ansprechpartner:

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