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Gray codes generation algorithm and theoretical evaluation of random walks in \(N\)-cubes. (English) Zbl 06994629
Summary: 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 Chaotic 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 \(N\log N\), which was anticipated in the previous article, but not proven.
MSC:
94A05 Communication theory
05C45 Eulerian and Hamiltonian graphs
68W40 Analysis of algorithms
82C41 Dynamics of random walks, random surfaces, lattice animals, etc. in time-dependent statistical mechanics
Software:
CIPRNG; Diehard; TestU01
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References:
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