Computing efficiently the lattice width in any dimension

Authors:
Émilie Charrier;Fabien Feschet;Lilian Buzer
Affiliations:
Université de Savoie, Laboratoire de Mathématiques, UMR 5127 CNRS, F-73376 Le Bourget du Lac, France;Clermont Université, Université dAuvergne, Santé-Technologie, 49 Bd F. Mitterrand, BP 32, F-63001 Clermont-Ferrand, France;Université Paris-Est, LABINFO-IGM, UMR CNRS 8049, ESIEE Paris, 2 Bd Blaise Pascal, BP 99, F-93162 Noisy le Grand Cedex, France
Venue:
Theoretical Computer Science
Year:
2011

Citing 13
Cited 0

Theory of linear and integer programming

Theory of linear and integer programming
Computing the Width of a Set

IEEE Transactions on Pattern Analysis and Machine Intelligence
The generalized Gauss reduction algorithm

Journal of Algorithms
A Polynomial-Time Algorithm for the Knapsack Problem with Two Variables

Journal of the ACM (JACM)
A Polynomial Algorithm for the Two-Variable Integer Programming Problem

Journal of the ACM (JACM)
Computing Two-Dimensional Integer Hulls

SIAM Journal on Computing
Fast 2-Variable Integer Programming

Proceedings of the 8th International IPCO Conference on Integer Programming and Combinatorial Optimization
Finding extremal polygons

STOC '82 Proceedings of the fourteenth annual ACM symposium on Theory of computing
Minimal Ellipsoids and Maximal Simplexes in 3D Euclidean Space

Cybernetics and Systems Analysis
A linear algorithm for integer programming in the plane

Mathematical Programming: Series A and B
On a general method for maximizing and minimizing among certain geometric problems

SFCS '79 Proceedings of the 20th Annual Symposium on Foundations of Computer Science
Efficient lattice width computation in arbitrary dimension

DGCI'09 Proceedings of the 15th IAPR international conference on Discrete geometry for computer imagery
The exact lattice width of planar sets and minimal arithmetical thickness

IWCIA'06 Proceedings of the 11th international conference on Combinatorial Image Analysis

Quantified Score

Hi-index	5.23

Visualization

Abstract

We provide an algorithm for the exact computation of the lattice width of a set of points K in Z^2 in linear-time with respect to the size of K. This method consists in computing a particular surrounding polygon. From this polygon, we deduce a set of candidate vectors allowing the computation of the lattice width. Moreover, we describe how this new algorithm can be extended to an arbitrary dimension thanks to a greedy and practical approach to compute a surrounding polytope. Indeed, this last computation is very efficient in practice as it processes only a few linear time iterations whatever the size of the set of points. Hence, it avoids complex geometric processings.