Papers › Arbitrary-length analogs to de Bruijn sequences
Arbitrary-length analogs to de Bruijn sequences
Abhinav Nellore, Rachel Ward
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Let α be a length-L cyclic sequence of characters from a size-K alphabet 𝒜 such that the number of occurrences of any length-m string on 𝒜 as a substring of α is ⌊L / Kᵐ ⌋ or ⌈L / Kᵐ ⌉. When L = Kᴺ for any positive integer N, α is a de Bruijn sequence of order N, and when L ≠Kᴺ, α shares many properties with de Bruijn sequences. We describe an algorithm that outputs some α for any combination of K ≥2 and L ≥1 in O(L) time using O(L logK) space. This algorithm extends Lempel's recursive construction of a binary de Bruijn sequence. An implementation written in Python is available at https://github.com/nelloreward/pkl.
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