A Successor Rule Framework for Constructing $k$ -Ary de Bruijn Sequences and Universal Cycles

Gabric, Daniel; Sawada, Joe; Williams, Aaron; Wong, Dennis

doi:10.1109/tit.2019.2928292

Cited by 26 publications

(34 citation statements)

References 22 publications

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“…To extract a cycle joining construction from a shift rule, one needs to identify the words whose successors/predecessors are not rotations of them. This is how a cycle joining construction was proved to generate prefer-max in [25]. A similar approach can extract a prefer-max cycle joining construction from the Lyndon words concatenation, i.e.…”

Section: A Cycle Joining Constructionmentioning

confidence: 98%

“…In [5,25], the shift rule was proved based on the correctness of the FKM theorem. In [6], the other direction is established; [6] provides a self-contained correctness proof for the prefer-max shift rule, and show that the FKM theorem follows from it.…”

Section: A Cycle Joining Constructionmentioning

confidence: 99%

“…Second, some DB sequences are proved to be equal to a concatenating of certain words [14,22]. Third, some DB sequences are constructed by a shift rule; a formula to produce the successor of a given word [5,6,14,24,25,40,41].…”

Section: Introductionmentioning

confidence: 99%

“…This paper employs a fourth common construction technique: a cycle joining (a.k.a. cross-join) construction [3,4,7,10,20,25,27,29,39,46]. Roughly speaking, in this technique, the n-length words are divided into the equivalence classes of the 'rotation of' equivalence relation.…”

Section: Introductionmentioning

confidence: 99%

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A Cycle Joining Construction of the Prefer-Max De Bruijn Sequence

Amram,

Rubin,

Weiss

2021

Preprint

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We propose a novel construction for the well-known prefer-max De Bruijn sequence, based on the cycle joining technique. We further show that the construction implies known results from the literature in a straightforward manner. First, it implies the correctness of the onion theorem, stating that, effectively, the reverse of prefer-max is in fact an infinite De Bruijn sequence. Second, it implies the correctness of recently discovered shift rules for prefer-max, prefer-min, and their reversals. Lastly, it forms an alternative proof for the seminal FKM-theorem.

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Section: A Cycle Joining Constructionmentioning

confidence: 98%

Section: A Cycle Joining Constructionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

A Cycle Joining Construction of the Prefer-Max De Bruijn Sequence

Amram,

Rubin,

Weiss

2021

Preprint

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“…In this paper we consider the universal cycles for k-permutations. Jackson [6] showed that the universal cycle for k-permutations always exists when k < n. There are lots of results about the construction of universal cycles for k-permutations, mainly for the case that k = n − 1 named shorthand permutations [4,5,8,10]. Another interesting problem is to compute the number of distinct universal cycles for k-permutations.…”

mentioning

confidence: 99%