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Open AccessFeature PaperArticle

Gray Codes Generation Algorithm and Theoretical Evaluation of Random Walks in N-Cubes

Laboratoire Lorrain de Recherche en Informatique et ses Applications (LORIA), UMR 7503, Université de Lorraine, F-54506 Vandoeuvre-lès-Nancy, France
FEMTO-ST Institute, UMR 6174, Université Bourgogne Franche-Comté, 19 Av. du Maréchal Juin, F-90000 Belfort, France
Author to whom correspondence should be addressed.
Mathematics 2018, 6(6), 98;
Received: 1 February 2018 / Revised: 24 May 2018 / Accepted: 29 May 2018 / Published: 8 June 2018
In previous works, some of the authors have proposed a canonical form of Gray Codes (GCs) in N-cubes (hypercubes of dimension N). This form allowed them to draw an algorithm that theoretically provides exactly all the GCs for a given dimension N. In another work, we first have shown that any of these GC can be used to build the transition function of a Pseudorandom Number Generator (PRNG). Also, we have found a theoretical quadratic upper bound of the mixing time, i.e., the number of iterations that are required to provide a PRNG whose output is uniform. This article, extends these two previous works both practically and theoretically. On the one hand, another algorithm for generating GCs is proposed that provides an efficient generation of subsets of the entire set of GCs related to a given dimension N. This offers a large choice of GC to be used in the construction of Choatic Iterations based PRNGs (CI-PRNGs), leading to a large class of possible PRNGs. On the other hand, the mixing time has been theoretically shown to be in Nlog(N), which was anticipated in the previous article, but not proven. View Full-Text
Keywords: Gray Codes; Hamiltonian cycles; N-cube; mixing time; PRNG Gray Codes; Hamiltonian cycles; N-cube; mixing time; PRNG
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Contassot-Vivier, S.; Couchot, J.-F.; Héam, P.-C. Gray Codes Generation Algorithm and Theoretical Evaluation of Random Walks in N-Cubes. Mathematics 2018, 6, 98.

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