Voronoi Cells of Lattices with Respect to Arbitrary Norms
| Autor: | Johannes Blömer, Kathlén Kohn |
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| Rok vydání: | 2018 |
| Předmět: |
Computational Geometry (cs.CG)
FOS: Computer and information sciences 0102 computer and information sciences Computer Science::Computational Geometry 01 natural sciences Combinatorics Mathematics - Metric Geometry Vector problem Lattice (order) Euclidean geometry FOS: Mathematics Mathematics - Combinatorics Mathematics::Metric Geometry 0101 mathematics Mathematics Algebra and Number Theory Applied Mathematics Lattice problem 010102 general mathematics Metric Geometry (math.MG) Exponential function Euclidean distance 010201 computation theory & mathematics Computer Science - Computational Geometry Combinatorics (math.CO) Geometry and Topology Convex function Voronoi diagram |
| Zdroj: | SIAM Journal on Applied Algebra and Geometry. 2:314-338 |
| ISSN: | 2470-6566 |
| DOI: | 10.1137/17m1132045 |
| Popis: | Motivated by the deterministic single exponential time algorithm of Micciancio and Voulgaris for solving the shortest and closest vector problem for the Euclidean norm, we study the geometry and complexity of Voronoi cells of lattices with respect to arbitrary norms.On the positive side, we show that for strictly convex and smooth norms the geometry of Voronoi cells of lattices in any dimension is similar to the Euclidean case, i.e., the Voronoi cells are defined by the so-called Voronoi-relevant vectors and the facets of a Voronoi cell are in one-to-one correspondence with these vectors. On the negative side, we show that combinatorially Voronoi cells for arbitrary strictly convex and smooth norms are much more complicated than in the Euclidean case.In particular, we construct a family of three-dimensional lattices whose number of Voronoi-relevant vectors with respect to the l_3-norm is unbounded.Since the algorithm of Micciancio and Voulgaris and its run time analysis crucially dependonthefactthatfortheEuclidean normthenumber of Voronoi-relevant vectors is single exponential in the lattice dimension, this indicates that the techniques of Micciancio and Voulgaris cannot be extended to achieve deterministic single exponential time algorithms for lattice problems with respect to arbitrary l_p-norms. |
| Databáze: | OpenAIRE |
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