Loopless Algorithms to Generate Maximum Length Gray Cycles wrt. k-Character Substitution

09/04/2023

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by   Jean Néraud, et al.

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Given a binary word relation τ onto A * and a finite language X ⊆ A * , a τ-Gray cycle over X consists in a permutation w [i] 0≤i≤|X|–1 of X such that each word w [i] is an image under τ of the previous word w [i–1]. We define the complexity measure λA,τ (n), equal to the largest cardinality of a language X having words of length at most n, and st. some τ-Gray cycle over X exists. The present paper is concerned with τ = σ k , the so-called k-character substitution, st. (u, v) ∈ σ k holds if, and only if, the Hamming distance of u and v is k. We present loopless (resp., constant amortized time) algorithms for computing specific maximum length σ k-Gray cycles.