On greedy algorithms for binary de Bruijn sequences

Chang, Zuling; Ezerman, Martianus Frederic; Fahreza, Adamas Aqsa

doi:10.1007/s00200-020-00459-3

Cited by 7 publications

(4 citation statements)

References 17 publications

(15 reference statements)

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“…Inspired by some greedy algorithms in the construction of certain classes of de Bruijn sequences discussed by Chang, Ezerman, and Fahreza in [17], we devise two basic algorithms. Algorithm 1 prefers the complement over the double arcs when moving from the current vertex to the next vertex.…”

Section: Hamiltonian Cycles By Two Greedy Algorithmsmentioning

confidence: 99%

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A New Approach to Determine the Minimal Polynomials of Binary Modified de Bruijn Sequences

Musthofa¹,

Wijayanti²,

Palupi³

et al. 2022

Preprint

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A binary modified de Bruijn sequence is an infinite and periodic binary sequence derived by removing a zero from the longest run of zeros in a binary de Bruijn sequence. The minimal polynomial of the modified sequence is its unique least-degree characteristic polynomial. Leveraging on a recent characterization, we devise a novel general approach to determine the minimal polynomial. We translate the characterization into a problem of identifying a Hamiltonian cycle in a specially constructed graph. Along the way, we demonstrate the usefullness of computational tools from the cycle joining method in the modified setup.

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Section: Hamiltonian Cycles By Two Greedy Algorithmsmentioning

confidence: 99%

“…is the set of all vertices in Γ n . If a(x) ∈ Ω, then the two possible values of b(x) = x a(x) are given in Equations ( 16) and (17). Hence, the set O corresponds to a Hamiltonian cycle in Γ n .…”

Section: The Number Of Distinct Hamiltonian Cycles Inmentioning

confidence: 99%

A New Approach to Determine the Minimal Polynomials of Binary Modified de Bruijn Sequences

Musthofa¹,

Wijayanti²,

Palupi³

et al. 2022

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…Inspired by some greedy algorithms in the construction of certain classes of de Bruijn sequences discussed by Chang, Ezerman and Fahreza in [18], we devise two basic algorithms. Algorithm 1 prefers the complement over the double arcs when moving from the current vertex to the next vertex.…”

Section: Hamiltonian Cycles By Two Greedy Algorithmsmentioning

confidence: 99%

A New Approach to Determine the Minimal Polynomials of Binary Modified de Bruijn Sequences

et al. 2022

Self Cite

View full text Add to dashboard Cite

A binary modified de Bruijn sequence is an infinite and periodic binary sequence derived by removing a zero from the longest run of zeros in a binary de Bruijn sequence. The minimal polynomial of the modified sequence is its unique least-degree characteristic polynomial. Leveraging a recent characterization, we devise a novel general approach to determine the minimal polynomial. We translate the characterization into a problem of identifying a Hamiltonian cycle in a specially constructed graph. The graph is isomorphic to the modified de Bruijn–Good graph. Along the way, we demonstrate the usefulness of some computational tools from the cycle joining method in the modified setup.

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“…The use of preference functions is prevalent in many greedy algorithms for binary de Bruijn sequence. A general greedy algorithm, recently proposed by Chang et al in [6], included a discussion on how such functions fit into a unifying picture.…”

Section: Prior Literaturementioning

confidence: 99%

Binary de Bruijn Sequences via Zech’s Logarithms

et al. 2021

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The focus of this work is to show how to combine Zech's logarithms and each of the cycle joining and cross-join pairing methods to construct binary de Bruijn sequences. Basic implementations are supplied as proofs of concept. The cycles, in the cycle joining method, are typically generated by a linear feedback shift register. We prove a crucial characterization that determining Zech's logarithms is equivalent to identifying conjugate pairs shared by any two distinct cycles. This speeds up the task of building a connected adjacency subgraph that contains all vertices of the complete adjacency graph. Distinct spanning trees in either graph correspond to cyclically inequivalent de Bruijn sequences. As the cycles are being joined, guided by the conjugate pairs, we track the changes in the feedback function. We show how to produce certificates of existence for spanning trees of certain types to conveniently handle large order cases. The characterization of conjugate pairs via Zech's logarithms, as positional markings, is then adapted to identify cross-join pairs. A modified m-sequence is initially used, for ease of generation. The process can be repeated on each of the resulting de Bruijn sequences. Most prior constructions in the literature measure the complexity of the corresponding bit-by-bit algorithms. Our approach is different. We aim first to build a connected adjacency subgraph that is certified to contain all of the cycles as vertices. The ingredients are computed just once and concisely stored. Simple strategies are offered to keep the complexities low as the order grows.

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On greedy algorithms for binary de Bruijn sequences

Cited by 7 publications

References 17 publications

A New Approach to Determine the Minimal Polynomials of Binary Modified de Bruijn Sequences

A New Approach to Determine the Minimal Polynomials of Binary Modified de Bruijn Sequences

A New Approach to Determine the Minimal Polynomials of Binary Modified de Bruijn Sequences

Binary de Bruijn Sequences via Zech’s Logarithms

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