De Bruijn Sequences Research Articles

Locating patterns in the de Bruijn torus

The de Bruijn torus (or grid) problem looks to find an n-by-m binary matrix in which every possible j-by-k submatrix appears exactly once. The existence and construction of these binary matrices were determined in the 70s, with generalizations to d-ary matrices in the 80s and 90s. However, these constructions lacked efficient decoding methods, leading to new constructions in the early 2000s. The new constructions develop cross-shaped patterns (rather than rectangular), and rely on a concept known as a half de Bruijn sequence. In this paper, we further advance this construction beyond cross-shape patterns. Furthermore, we show results for universal cycle grids, based off of the one-dimensional universal cycles introduced by Chung, Diaconis, and Graham, in the 90s. These grids have many applications such as robotic vision, location detection, and projective touch-screen displays.

2D Lyndon Words and Applications

A Lyndon word is a primitive string which is lexicographically smallest among cyclic permutations of its characters. Lyndon words are used for constructing bases in free Lie algebras, constructing de Bruijn sequences, finding the lexicographically smallest or largest substring in a string, and succinct suffix---prefix matching of highly periodic strings. In this paper, we extend the concept of the Lyndon word to two dimensions. We introduce the 2D Lyndon word and use it to capture 2D horizontal periodicity of a matrix in which each row is highly periodic, and to efficiently solve 2D horizontal suffix---prefix matching among a set of patterns. This yields a succinct and efficient algorithm for 2D dictionary matching. We present several algorithms that compute the 2D Lyndon word that represents a matrix. The final algorithm achieves linear time complexity even when the least common multiple of the periods of the rows is exponential in the matrix width.

A surprisingly simple de Bruijn sequence construction

Pick any length n binary string b1b2⋯bn and remove the first bit b1. If b2b3⋯bn1 is a necklace, then append the complement of b1 to the end of the remaining string; otherwise append b1. By repeating this process, eventually all 2n binary strings will be visited cyclically. This shift rule leads to a new de Bruijn sequence construction that can be generated in O(1)-amortized time per bit.

Optimized projection patterns for stereo systems

This paper describes how to generate optimal projection patterns to supplement general stereo camera systems. In contrast to structured light, the active stereo systems utilize the projected patterns only as auxiliary information in correspondence search, whereas the structured light systems have to detect the patterns and decode them to compute depth. The concept of non-recurring De Bruijn sequences is introduced, and a few algorithms based on the non-recurring De Bruijn sequence are designed to build optimized projection patterns for several stereo parameters. When only the search window size of a stereo system is given, we show that a non-recurring De Bruijn sequence with corresponding parameters makes the longest functional pattern, and presents experimental results using real scenes to show the effectiveness of the proposed projection patterns. Additionally if the pattern length is given in the form of maximum disparity search range, the algorithm using branch-and-bound search scheme to find an optimal sub-sequence of a non-recurring De Bruijn sequence is proposed.

An algorithm for generating all CR sequences in the de Bruijn sequences of length 2<I><sup>n</sup></I> where <I>n</I> is any odd number

An algorithm for generating all CR sequences in the de Bruijn sequences of length 2<I><sup>n</sup></I> where <I>n</I> is any odd number

A Class of de Bruijn Sequences

In this paper, a class of linear feedback shift registers (LFSRs) with characteristic polynomial (1 + x <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> )p(x) is discussed, where p(x) is a primitive polynomial of degree n > 2. The cycle structure and adjacency graphs of the LFSRs are determined. A new class of de Bruijn sequences is constructed from these LFSRs, and the number of de Bruijn sequences in the class is also considered. To illustrate the efficiency of constructing de Bruijn sequences from these LFSRs, an algorithm for producing some corresponding maximum-length nonlinear feedback shift registers with time and memory complexity O(n) is also proposed.

Dense 3D Reconstruction from High Frame-Rate Video Using a Static Grid Pattern.

Dense 3D reconstruction of fast moving objects could contribute to various applications such as body structure analysis, accident avoidance, and so on. In this paper, we propose a technique based on a one-shot scanning method, which reconstructs 3D shapes for each frame of a high frame-rate video capturing the scenes projected by a static pattern. To avoid instability of image processing, we restrict the number of colors used in the pattern to less than two. The proposed technique comprises (1) an efficient algorithm to eliminate ambiguity of projected parallel-line patterns by using intersection points, (2) a batch reconstruction algorithm of multiple frames by using spatio-temporal constraints, and (3) an efficient detection method of color-encoded grid pattern based on de Bruijn sequence. In the experiments, the line detection algorithm worked effectively and the dense reconstruction algorithm produces accurate and robust results. We also show the improved results by using temporal constraints. Finally, the dense reconstructions of fast moving objects in a high frame-rate video are presented.

A binary algorithm with low divergence for modular inversion on SIMD architectures

When performing modular arithmetic the most computationally expensive operation is the modular inversion of an integer. Its cost might be a problem for cryptanalytical applications (like the transformation from projective to affine coordinates within a Pollard rho algorithm implementation to solve the discrete logarithm problem on an elliptic curve) where performances constitute a key aspect. Good platforms for such operations are single-instruction multiple-data architectures, like graphic processing units (GPUs) because of their extremely competitive performance/price ratio. Unfortunately, when a single thread computes a single inversion, the whole computation on GPUs can be significantly slowed down in the presence of divergent threads. In this paper we describe a new algorithm to compute modular inversion on GPUs based on Stein’s Binary GCD. By exploiting the De Bruijn sequences and the Montgomery arithmetic, our version of Stein’s algorithm better fits GPUs, since it reduces the divergence among threads of the original algorithm. The paper includes a brief report on tests of our algorithm in six prime fields with characteristics of size ranging from 109 to 359 bits and in the two prime fields associated to the Mersenne primes \(2^{521} - 1\) and \(2^{607} - 1\).

Normality and Finite-State Dimension of Liouville Numbers

Liouville numbers were the first class of real numbers which were proven to be transcendental. It is easy to construct non-normal Liouville numbers. Kano (1993) and Bugeaud (2002) have proved, using analytic techniques, that there are normal Liouville numbers. Here, for a given base k ≥ 2, we give a new construction of a Liouville number which is normal to the base k. This construction is combinatorial, and is based on de Bruijn sequences. A real number in the unit interval is normal if and only if its finite-state dimension is 1. We generalize our construction to prove that for any rational r in the closed unit interval, there is a Liouville number with finite state dimension r. This refines Staiger’s result [19] that the set of Liouville numbers has constructive Hausdorff dimension zero, showing a new quantitative classification of Liouville numbers can be attained using finite-state dimension.

Beyond the interference problem: hierarchical patterns for multiple-projector structured light system.

Three-dimensional reconstruction of a dynamic object based on the structured light (SL) technique has attracted intensive research. Since a single projector only covers a limited area of the scene, multiple projectors should be employed simultaneously to extend the imaging area for a 3D object. However, patterns projected by different projectors superpose each other in a light-intersected area, which makes it difficult to recognize the original patterns and obtain the correct depth maps. To solve such a problem, we propose a method to design hierarchical patterns that can be separated from each other. In the proposed patterns, each pixel in binary patterns based on the de Bruijn sequence is replaced by a different bin with limited size. Then the proposed patterns can be separated by identifying distributions of colors in each bin in superposed patterns, and depth maps are obtained by decoding the separated patterns. To verify the performance of the proposed method, we design two hierarchical patterns and conduct several experiments in different scenes. Experimental results demonstrate that the proposed patterns can be separated in a multiple-projector SL system to obtain accurate depth maps, and they are robust for different conditions.

On Cross-Correlation Values of de Bruijn Sequences

quences of length 2 n , while the total number of the former sequences is much less than 2 n =n, the total number of the celebrated de Bruijn sequences is known to be 2 2 n−1 −n

A construction of all CR sequences in the de Bruijn sequences of length 2&lt;sup&gt;2&lt;I&gt;p&lt;/I&gt;+1&lt;/sup&gt; where &lt;I&gt;p&lt;/I&gt; is a prime number

A construction of all CR sequences in the de Bruijn sequences of length 2&lt;sup&gt;2&lt;I&gt;p&lt;/I&gt;+1&lt;/sup&gt; where &lt;I&gt;p&lt;/I&gt; is a prime number

A New Necessary Condition for Feedback Functions of de Bruijn Sequences

Recently nonlinear feedback shift registers (NFSRs) have frequently been used as basic building blocks for stream ciphers. A major problem concerning NFSRs is to construct NFSRs which generate de Bruijn sequences, namely maximum period sequences. In this paper, we present a new necessary condition for NFSRs to generate de Bruijn sequences. The new condition can not be deduced from the previously proposed necessary conditions. It is shown that the number of NFSRs whose feedback functions satisfy all the previous necessary conditions but not the new one is very large.

De Bruijn binary sequences and spread spectrum applications: A marriage possible?

The evaluations and results provided in the show that a proper selection of De Bruijn binary sequences as spreading codes in multiuser communications allows to obtain improved performance, with respect to more traditional solutions, like those based on the use of classical Gold codes, m sequences, and OVSF codes. Promising results have been obtained also in DS sequence SS radar simulations, where the number of available sequences may be important to cope with specific scenarios, such as anticollision vehicular radar applications. The behavior exhibited by De Bruijn sequences motivates the efforts aimed at providing an efficient generation algorithm to obtain the complete set of codes for even greater values of n and performing an exhaustive evaluation of their properties. This activity is currently ongoing.

Design of shortest double-stranded DNA sequences covering all k-mers with applications to protein-binding microarrays and synthetic enhancers

Motivation: Novel technologies can generate large sets of short double-stranded DNA sequences that can be used to measure their regulatory effects. Microarrays can measure in vitro the binding intensity of a protein to thousands of probes. Synthetic enhancer sequences inserted into an organism’s genome allow us to measure in vivo the effect of such sequences on the phenotype. In both applications, by using sequence probes that cover all k-mers, a comprehensive picture of the effect of all possible short sequences on gene regulation is obtained. The value of k that can be used in practice is, however, severely limited by cost and space considerations. A key challenge is, therefore, to cover all k-mers with a minimal number of probes. The standard way to do this uses the de Bruijn sequence of length . However, as probes are double stranded, when a k-mer is included in a probe, its reverse complement k-mer is accounted for as well.Results: Here, we show how to efficiently create a shortest possible sequence with the property that it contains each k-mer or its reverse complement, but not necessarily both. The length of the resulting sequence approaches half that of the de Bruijn sequence as k increases resulting in a more efficient array, which allows covering more longer sequences; alternatively, additional sequences with redundant k-mers of interest can be added.Availability: The software is freely available from our website http://acgt.cs.tau.ac.il/shortcake/.Contact: rshamir@tau.ac.il

CG methylated microarrays identify a novel methylated sequence bound by the CEBPB|ATF4 heterodimer that is active in vivo

To evaluate the effect of CG methylation on DNA binding of sequence-specific B-ZIP transcription factors (TFs) in a high-throughput manner, we enzymatically methylated the cytosine in the CG dinucleotide on protein binding microarrays. Two Agilent DNA array designs were used. One contained 40,000 features using de Bruijn sequences where each 8-mer occurs 32 times in various positions in the DNA sequence. The second contained 180,000 features with each CG containing 8-mer occurring three times. The first design was better for identification of binding motifs, while the second was better for quantification. Using this novel technology, we show that CG methylation enhanced binding for CEBPA and CEBPB and inhibited binding for CREB, ATF4, JUN, JUND, CEBPD, and CEBPG. The CEBPB|ATF4 heterodimer bound a novel motif CGAT|GCAA 10-fold better when methylated. The electrophoretic mobility shift assay (EMSA) confirmed these results. CEBPB ChIP-seq data using primary female mouse dermal fibroblasts with 50× methylome coverage for each strand indicate that the methylated sequences well-bound on the arrays are also bound in vivo. CEBPB bound 39% of the methylated canonical 10-mers ATTGC|GCAAT in the mouse genome. After ATF4 protein induction by thapsigargin which results in ER stress, CEBPB binds methylated CGAT|GCAA in vivo, recapitulating what was observed on the arrays. This methodology can be used to identify new methylated DNA sequences preferentially bound by TFs, which may be functional in vivo.

Hamiltonicity of large generalized de Bruijn cycles

In this article, we determine when the large generalized de Bruijn cycles are Hamiltonian. These digraphs have been introduced by Gómez, Padró and Pérennes as large interconnection networks with small diameter and they are a family of generalized cycles. They are Kronecker products of generalized de Bruijn digraphs and dicycles.

Generalized Fibonacci recurrences and the lex-least de Bruijn sequence

The skew of a binary string is the difference between the number of zeroes and the number of ones, while the length of the string is the sum of these two numbers. We consider certain suffixes of the lexicographically-least de Bruijn sequence at natural breakpoints of the binary string. We show that the skew and length of these suffixes are enumerated by sequences generalizing the Fibonacci and Lucas numbers, respectively.

Constructive method for synthesis of complete classes of multilevel de Bruijn sequences

Two new presentation forms of multilevel de Bruijn sequences (BS) have been introduced as geometric and algebraic structures. The attractive and practical properties of these structures have been found. This formed the basis for proposing a constructive method for the synthesis of generating and complete classes of BS. It has been shown that the application of the found classes of quaternary BS in encryption ensures a two-fold reduction of the memory space required for storing the cryptographic substitution boxes (S-boxes).

Universal Cycles for Weak Orders

Universal cycles are generalizations of de Bruijn cycles and Gray codes that were introduced originally by Chung, Diaconis, and Graham in 1992. They have been developed by many authors since, for various combinatorial objects such as strings, subsets, permutations, partitions, vector spaces, and designs. One generalization of universal cycles, which require almost complete overlap of consecutive words, is $s$-overlap cycles, which relax such a constraint. In this paper we study weak orders, which are relations that are transitive and complete. We prove the existence of universal and $s$-overlap cycles for weak orders, as well as for fixed height and/or weight weak orders, and apply the results to cycles for ordered partitions.