De Bruijn Sequences Research Articles

Efficient Constructions of the Prefer-Same and Prefer-Opposite de Bruijn Sequences

The greedy Prefer-same de Bruijn sequence construction was first presented by Eldert, Gray, Gurk, and Rubinoff in 1958. As a greedy algorithm, it has one major downside: it requires an exponential amount of space to store the length \(2^{n}\) de Bruijn sequence. Though de Bruijn sequences have been heavily studied over the last 60 years, finding an efficient construction for the Prefer-same de Bruijn sequence has remained a tantalizing open problem. In this article, we unveil the underlying structure of the Prefer-same de Bruijn sequence and solve the open problem by presenting an efficient algorithm to construct it using \(O(n)\) time per bit and only \(O(n)\) space. Following a similar approach, we also present an efficient algorithm to construct the Prefer-opposite de Bruijn sequence.

Generation of Absolute Patterns for Magnetic Encoders of Arbitrary Size With De Bruijn Sequences

Generation of Absolute Patterns for Magnetic Encoders of Arbitrary Size With De Bruijn Sequences

Cut-down de Bruijn sequences

A cut-down de Bruijn sequence is a cyclic string of length L, where 1≤L≤kn, such that every substring of length n appears at most once. Etzion [Theor. Comp. Sci 44 (1986)] introduced an algorithm to construct binary cut-down de Bruijn sequences requiring o(n) simple n-bit operations per symbol generated. In this paper, we simplify the algorithm and improve the running time to O(n) time per symbol generated using O(n) space. Additionally, we develop the first successor-rule approach for constructing a binary cut-down de Bruijn sequence by leveraging recent ranking/unranking algorithms for fixed-density Lyndon words. Finally, we develop an algorithm to generate cut-down de Bruijn sequences for k>2 that runs in O(n) time per symbol using O(n) space after some initialization.

A 3D Reconstruction Method Based on Homogeneous De Bruijn-Encoded Structured Light

Structured light three-dimensional reconstruction is one of the important methods for non-contact acquisition of sparse texture object surfaces. Variations in ambient illumination and disparities in object surface reflectance can significantly impact the fidelity of three-dimensional reconstruction, introducing considerable inaccuracies. We introduce a robust method for color speckle structured light encoding, which is based on a variant of the De Bruijn sequence, termed the Homogeneous De Bruijn Sequence. This innovative approach enhances the reliability and accuracy of structured light techniques for three-dimensional reconstruction by utilizing the distinctive characteristics of Homogeneous De Bruijn Sequences. Through a pruning process applied to the De Bruijn sequence, a structured light pattern with seven distinct color patches is generated. This approach ensures a more equitable distribution of speckle information.

A hardware friendly and meaningful image encryption strategy based on a binary measurement matrix and de Bruijn sequence

A hardware friendly and meaningful image encryption strategy based on a binary measurement matrix and de Bruijn sequence

Structured Light Encoding and Decoding Algorithm Based on Adjacency-Hopping de Bruijn Sequences

Structured Light Encoding and Decoding Algorithm Based on Adjacency-Hopping de Bruijn Sequences

Proofs of Conjectures on Extremal Weight De Bruijn Sequences

De Bruijn sequences can be categorized by the weight of the truth tables of the generating functions. Among all the weight classes, the numbers of de Bruijn sequences with minimum and maximum weights draw much attention. Fredricksen and Mayhew proposed six conjectures about them, some of which have remained unsolved for about forty years. In this paper, we give the exact formulas of de Bruijn sequences with extremal weights and then prove all the conjectures in the affirmative.

On the existence of balanced generalized de Bruijn sequences

A balanced generalized de Bruijn sequence with parameters (n,l,k) is a cyclic sequence of n bits such that (a) the number of 0's equals the number of 1's, and (b) each substring of length l occurs at most k times. We determine necessary and sufficient conditions on n,l, and k for the existence of such a sequence.

On Hierarchies of Balanced Sequences

Balanced sequences and balanced codes have attracted a lot of research in the last seventy years due to their diverse applications in information theory as well as other areas of computer science and engineering. There have been some methods to classify balanced sequences. This work suggests two new different hierarchies to classify these sequences. The first one is based on the largest <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\ell $ </tex-math></inline-formula> for which each <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\ell $ </tex-math></inline-formula> -tuple is contained the same amount of times in the sequence. This property is a generalization for the property required for de Bruijn sequences. The second hierarchy is based on the number of balanced derivatives of the sequence. Enumeration for each such family of sequences and efficient encoding and decoding algorithms are provided in this paper.

Single-shot dense depth sensing with a tricolor RGB fringe pattern

A single-shot fringe pattern composed of red (R), green (G), and blue (B) is proposed for dynamic scene depth acquisition. A tricolor binary fringe is adopted and filtered in the frequency domain to obtain sinusoidal fringes. A Gabor filter is employed to estimate the wrapped phase. To overcome the phase ambiguity, we use the color-encoded De Bruijn sequence to denote the period of the phase information and assign the sequence to two channels of the pattern, while the third channel is equal to the inverse of the composition of the two channels. Benefiting from this coding strategy, two disparity maps with different phase shifts can be acquired in only one shot, and a disparity consistency check is used to eliminate errors. Quantitative and qualitative experiments are conducted to verify the performance of the proposed method and prove that it can achieve a high accuracy and dense depth map.

Graph universal cycles: Compression and connections to universal cycles

Universal cycles, such as De Bruijn cycles, are cyclic sequences of symbols that represent every combinatorial object from some family exactly once as a consecutive subsequence. Graph universal cycles are a graph analogue of universal cycles introduced in 2010. We introduce graph universal partial cycles, a more compact representation of graph classes, which use “do not know” edges. We show how to construct graph universal partial cycles for labeled graphs, threshold graphs, and permutation graphs. For threshold graphs and permutation graphs, we demonstrate that the graph universal cycles and graph universal partial cycles are closely related to universal cycles and compressed universal cycles, respectively. Using the same connection, for permutation graphs, we define and prove the existence of an s-overlap form of graph universal cycles. We also prove the existence of a generalized form of graph universal cycles for unlabeled graphs.

On the generation of necklaces and bracelets in R

This note introduces a code snippet in R aiming to generate necklaces as well as bracelets. Among various uses, necklaces are useful tools to manage traces of products of random matrices. Functionality for necklaces and bracelets is provided with some examples of applications such as Lyndon words and de Bruijn sequences. The routines are collected in the Necklaces package available from the Comprehensive R Archive Network.

Multi de Bruijn Sequences and the Cross-Join Method

We show a method to construct binary multi de Bruijn sequences using the cross-join method. We extend the proof given by Alhakim for ordinary de Bruijn sequences to the case of multi de Bruijn sequences. In particular, we establish that all multi de Bruijn sequences can be obtained by cross-joining an ordinary de Bruijn sequence concatenated with itself an appropriate number of times. We implemented the generation of all multi de Bruijn sequences of type C(2,2,2) and C(3,2,2). We experimentally confirm that some multi de Bruijn sequences can be generated by Galois Nonlinear Feedback Shift Registers (NLFSRs). It is supposed that all multi de Bruijn sequences can be generated using Galois NLFSRs.

Enumeration of Both-Ends-Fixed &lt;i&gt;k&lt;/i&gt;-Ary Necklaces and Its Applications

We consider both-ends-fixed k-ary necklaces and enumerate all such necklaces of length n from the viewpoints of symbolic dynamics and β-expansions, where n and k(≥ 2) are natural numbers and β(> 1) is a real number. Recently, Sawada et al. proposed an efficient construction of k-ary de Bruijn sequence of length kn, which for each n ≥ 1, requires O(n) space but generates a single k-ary de Bruijn sequence of length kn in O(1)-amortized time per bit. Based on the enumeration of both-ends-fixed k-ary necklaces of length n, we evaluate auto-correlation values of the k-ary de Bruijn sequences of length kn constructed by Sawada et al. We also estimate the asymptotic behaviour of the obtained auto-correlation values as n tends to infinity.

3D RNA-scaffolded wireframe origami

Hybrid RNA:DNA origami, in which a long RNA scaffold strand folds into a target nanostructure via thermal annealing with complementary DNA oligos, has only been explored to a limited extent despite its unique potential for biomedical delivery of mRNA, tertiary structure characterization of long RNAs, and fabrication of artificial ribozymes. Here, we investigate design principles of three-dimensional wireframe RNA-scaffolded origami rendered as polyhedra composed of dual-duplex edges. We computationally design, fabricate, and characterize tetrahedra folded from an EGFP-encoding messenger RNA and de Bruijn sequences, an octahedron folded with M13 transcript RNA, and an octahedron and pentagonal bipyramids folded with 23S ribosomal RNA, demonstrating the ability to make diverse polyhedral shapes with distinct structural and functional RNA scaffolds. We characterize secondary and tertiary structures using dimethyl sulfate mutational profiling and cryo-electron microscopy, revealing insight into both global and local, base-level structures of origami. Our top-down sequence design strategy enables the use of long RNAs as functional scaffolds for complex wireframe origami.

Partial Cycle Structure of FSRs and Its Applications in Searching De Bruijn Sequences

We propose the concept of partial cycle structure of feedback shift registers, and study its applications in searching the characteristic functions of de Bruijn sequences. We show that, if a function generates de Bruijn sequences then its partial cycle structure does not contain cycles, and conversely, if the partial cycle structure of a function does not contain cycles then it can be extended into a function that generates de Bruijn sequences. By using this property, we analyze the low degree terms in the characteristic functions of de Bruijn sequences, and in particular give a full description of the linear terms in them. We also design an algorithm to search the characteristic functions of de Bruijn sequences which should perform better than the random search algorithm.

Some necessary conditions for feedback functions of de Bruijn sequences

Abstract De Bruijn sequence is an important kind of nonlinear shift register sequence, which has a very wide range of applications in the fields of communication and cryptography. From the perspective of the relationship between linear terms, quadratic terms and cubic terms in the feedback functions of de Bruijn sequences, some new necessary conditions for feedback functions of de Bruijn sequences are obtained. Some examples show that the known necessary conditions cannot deduce the proposed necessary conditions.

Fast Format-Aware Fuzzing for Structured Input Applications

Fuzzing is one of the most successful software testing techniques used to discover vulnerabilities in programs. Without seeds that fit the input format, existing runtime dependency recognition strategies are limited by incompleteness and high overhead. In this paper, for structured input applications, we propose a fast format-aware fuzzing approach to recognize dependencies from the specified input to the corresponding comparison instruction. We divided the dependencies into Input-to-State (I2S) and indirect dependencies. Our approach has the following advantages compared to existing works: (1) recognizing I2S dependencies more completely and swiftly using the input based on the de Bruijn sequence and its mapping structure; (2) obtaining indirect dependencies with a light dependency existence analysis on the input fragments. We implemented a fast format-aware fuzzing prototype, FFAFuzz, based on our method and evaluated FFAFuzz in real-world structured input applications. The evaluation results showed that FFAFuzz reduced the average time overhead by 76.49% while identifying more completely compared with Redqueen and by 89.10% compared with WEIZZ. FFAFuzz also achieved higher code coverage by 14.53% on average compared to WEIZZ.

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

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.

Designing preference functions for de Bruijn sequences with forbidden words

Designing preference functions for de Bruijn sequences with forbidden words