Generalized De Bruijn Research Articles

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.

Криптографические слабости алгоритмов типа «гиперкуб»

The class of block cryptographic XSLP-algorithms called "hypercube" is considered. For algorithms of this class, we obtain estimates for the dispersion index of a linear environment for any number of iterations. It is shown that when choosing a transformation P using generalized de Bruijn graphs for the algorithms under consideration, the avalanche effect may not occur, as a result of which the encryption key can be determined with laboriousness, which is significantly less than the laboriousness of total key testing.

Maximal state complexity and generalized de Bruijn words

We compute the exact maximum state complexity for the language consisting of m words of length N, and characterize languages achieving the maximum. We also consider a special case, namely languages C(w) consisting of the conjugates of a single word w. The words for which the maximum state complexity of C(w) is achieved turn out to be a natural generalization of de Bruijn words. We show that generalized de Bruijn words exist for each length and consider the number of them.

ENTROPY FLOW AND DE BRUIJN'S IDENTITY FOR A CLASS OF STOCHASTIC DIFFERENTIAL EQUATIONS DRIVEN BY FRACTIONAL BROWNIAN MOTION

AbstractMotivated by the classical De Bruijn's identity for the additive Gaussian noise channel, in this paper we consider a generalized setting where the channel is modelled via stochastic differential equations driven by fractional Brownian motion with Hurst parameter H ∈ (0, 1). We derive generalized De Bruijn's identity for Shannon entropy and Kullback–Leibler divergence by means of Itô's formula, and present two applications. In the first application we demonstrate its equivalence with Stein's identity for Gaussian distributions, while in the second application, we show that for H ∈ (0, 1/2], the entropy power is concave in time while for H ∈ (1/2, 1) it is convex in time when the initial distribution is Gaussian. Compared with the classical case of H = 1/2, the time parameter plays an interesting and significant role in the analysis of these quantities.

A method for constructing artificial DNA libraries based on generalized de Bruijn sequences

This work presents an algorithm for constructing artificial DNA libraries that consist of chains having minimal tendency to hybridize with each other. Such libraries can be applied in DNA computing (Adleman, 1994) for encoding problem instances. The algorithm is based on an extension of de Bruijn graphs and de Bruijn sequences. In general, the algorithm can be used to build a chain over any alphabet of even size with defined binary symmetric irreflexive relation of complementarity — the obtained chain is a modified de Bruijn sequence with shifted windows (there are no repetitive k-long subchains that start on positions divisible by a given parameter) where half of these windows are removed to eliminate relation of complementarity among them.

Random Regular Graph and Generalized De Bruijn Graph with $k$ -Shortest Path Routing

The Random regular graph (RRG) has recently been proposed as an interconnect topology for future large scale data centers and HPC clusters. An RRG is a special case of directed regular graph (DRG) where each link is unidirectional and all nodes have the same number of incoming and outgoing links. In this work, we establish bounds for DRGs on diameter, average $k$ -shortest path length, and a load balancing property with $k$ -shortest path routing, and use these bounds to evaluate RRGs. The results indicate that an RRG with $k$ -shortest path routing is not ideal in terms of diameter and load balancing. We further consider the Generalized De Bruijn Graph (GDBG), a deterministic DRG, and prove that for most network configurations, a GDBG is near optimal in terms of diameter, average $k$ -shortest path length, and load balancing with a $k$ -shortest path routing scheme. Finally, we use modeling and simulation to exploit the strengths and weaknesses of RRGs for different traffic conditions by comparing RRGs with GDBGs.

New bounds on the decycling number of generalized de Bruijn digraphs

A set of vertices of a graph whose removal leaves an acyclic graph is referred as a decycling set, or a feedback vertex set, of the graph. The minimum cardinality of a decycling set of a graph [Formula: see text] is referred to as the decycling number of [Formula: see text]. For [Formula: see text], the generalized de Bruijn digraph [Formula: see text] is defined by congruence equations as follows: [Formula: see text] and [Formula: see text]. In this paper, we give a systematic method to find a decycling set of [Formula: see text] and give a new upper bound that improve the best known results. By counting the number of vertex-disjoint cycles with the idea of constrained necklaces, we obtain new lower bounds on the decycling number of generalized de Bruijn digraphs.

V-SUPER VERTEX OUT-MAGIC TOTAL LABELINGS OF DIGRAPHS

Let D be a directed graph with p vertices and q arcs. A vertex out-magic total labeling is a bijection f from <TEX>$V(D){\cup}A(D){\rightarrow}\{1,2,{\ldots},p+q\}$</TEX> with the property that for every <TEX>$v{\in}V(D)$</TEX>, <TEX>$f(v)+\sum_{u{\in}O(v)}f((v,u))=k$</TEX>, for some constant k. Such a labeling is called a V-super vertex out-magic total labeling (V-SVOMT labeling) if <TEX>$f(V(D))=\{1,2,3,{\ldots},p\}$</TEX>. A digraph D is called a V-super vertex out-magic total digraph (V-SVOMT digraph) if D admits a V-SVOMT labeling. In this paper, we provide a method to find the most vital nodes in a network by introducing the above labeling and we study the basic properties of such labelings for digraphs. In particular, we completely solve the problem of finding V-SVOMT labeling of generalized de Bruijn digraphs which are used in the interconnection network topologies.

Efficient absorbants in generalized de Bruijn digraphs

Let D=(V,A) be a digraph with vertex set V and arc set A. An efficient absorbant (respectively, dominating set) of a digraph D=(V,A) is a set S⊆V such that, for every v∈V∖S, there exists exactly one out-neighbor (respectively, in-neighbor) of v in S and there is no arc in the induced digraph of S, where an out-neighbor (respectively, in-neighbor) of v in S is a vertex u with an arc (v,u) (respectively, (u,v)) in A with u∈S. The efficient absorbant conjecture is as follows: there is an efficient absorbant in generalized de Bruijn digraph GB(n,d) if and only if nd+1 is a multiple of gcd(n,d−1), where gcd(x,y) denotes the greatest common divisor of integers x and y. The sufficient condition of this conjecture was proved in DOI:10.1080/00207160.2016.1154949. In this paper, we settle the conjecture. Moreover, we also show that, the subclasses in GB(n,d) having efficient absorbants and efficient dominating sets are equivalent.

Distance domination of generalized de Bruijn and Kautz digraphs

Let $G=(V,A)$ be a digraph and $k\ge 1$ an integer. For $u,v\in V$, we say that the vertex $u$ distance $k$-dominate $v$ if the distance from $u$ to $v$ at most $k$. A set $D$ of vertices in $G$ is a distance $k$-dominating set if for each vertex of $V\setminus D$ is distance $k$-dominated by some vertex of $D$. The {\em distance $k$-domination number} of $G$, denoted by $\gamma_{k}(G)$, is the minimum cardinality of a distance $k$-dominating set of $G$. Generalized de Bruijn digraphs $G_B(n,d)$ and generalized Kautz digraphs $G_K(n,d)$ are good candidates for interconnection networks. Tian and Xu showed that $\big \lceil n\big/\sum_{j=0}^kd^j\big\rceil\le \gamma_{k}(G_B(n,d))\le \big\lceil n/d^{k}\big\rceil$ and $\big \lceil n \big/\sum_{j=0}^kd^j\big\rceil\le \gamma_{k}(G_K(n,d))\le \big\lceil n/d^{k}\big\rceil$. In this paper we prove that every generalized de Bruijn digraph $G_B(n,d)$ has the distance $k$-domination number $\big\lceil n\big/\sum_{j=0}^kd^j\big\rceil$ or $\big\lceil n\big/\sum_{j=0}^kd^j\big\rceil+1$, and the distance $k$-domination number of every generalized Kautz digraph $G_K(n,d)$ bounded above by $\big\lceil n\big/(d^{k-1}+d^{k})\big\rceil$. Additionally, we present various sufficient conditions for $\gamma_{k}(G_B(n,d))=\big\lceil n\big/\sum_{j=0}^kd^j\big\rceil$ and $\gamma_{k}(G_K(n,d))=\big\lceil n\big/\sum_{j=0}^kd^j\big\rceil$.

PanTools: representation, storage and exploration of pan-genomic data.

Next-generation sequencing technology is generating a wealth of highly similar genome sequences for many species, paving the way for a transition from single-genome to pan-genome analyses. Accordingly, genomics research is going to switch from reference-centric to pan-genomic approaches. We define the pan-genome as a comprehensive representation of multiple annotated genomes, facilitating analyses on the similarity and divergence of the constituent genomes at the nucleotide, gene and genome structure level. Current pan-genomic approaches do not thoroughly address scalability, functionality and usability. We introduce a generalized De Bruijn graph as a pan-genome representation, as well as an online algorithm to construct it. This representation is stored in a Neo4j graph database, which makes our approach scalable to large eukaryotic genomes. Besides the construction algorithm, our software package, called PanTools, currently provides functionality for annotating pan-genomes, adding sequences, grouping genes, retrieving gene sequences or genomic regions, reconstructing genomes and comparing and querying pan-genomes. We demonstrate the performance of the tool using datasets of 62 E. coli genomes, 93 yeast genomes and 19 Arabidopsis thaliana genomes. The Java implementation of PanTools is publicly available at http://www.bif.wur.nl sandra.smit@wur.nl.

On efficient absorbant conjecture in generalized De Bruijn digraphs

ABSTRACTAn absorbant of a digraph D is a set such that, for every , there exists an arc with . An absorbant S is efficient if no two vertices in S have a common in-neighbour and the subdigraph induced by S has no arc. The efficient absorbant conjecture in generalized De Bruijn digraphs is as follows: There exists an efficient absorbant in generalized De Bruijn digraph with if and only if c is a multiple of . In this paper, we show that the sufficient condition of the efficient absorbant conjecture in generalized De Bruijn digraphs is affirmative.

Vertex Antimagic Total Labeling of Digraphs

In this paper we investigate the properties of (a, d)-vertex antimagic total labeling of a digraph D = (V, A). In this labeling, we assign to the vertices and arcs the consecutive integers from 1 to |V|+|A| and calculate the sum of labels at each vertex, i.e., the vertex label added to the labels on its out arcs. These sums form an arithmetical progression with initial term a and common difference d. We show the existence and non-existence of (a, d)-vertex antimagic total labeling for several class of digraphs, and show how to construct labelings for generalized de Bruijn digraphs. We conclude this paper with an open problem suitable for further research.

Generalized de Bruijn words for primitive words and powers

We show that for every n≥1 and over any finite alphabet, there is a word whose circular factors of length n have a one-to-one correspondence with the set of primitive words. In particular, we prove that such a word can be obtained by a greedy algorithm, or by concatenating all Lyndon words of length n in increasing lexicographic order. We also look into connections between de Bruijn graphs of primitive words and Lyndon graphs.Finally, we also show that the shortest word that contains every p-power of length pn over a k-letter alphabet has length between pkn and roughly (p+1k)kn, for all integers p≥1. An algorithm that generates a word which achieves the upper bound is provided.

Constructing the minimum dominating sets of generalized de Bruijn digraphs

In a digraph G=(V,A), a vertex u is said to dominate its out-neighbors v, that is, there is an arc (u,v)∈A. A set D⊆V is a dominating set of G if each vertex of V∖D is dominated by some vertex of D. The domination number of G, denoted by γ(G), is the minimum cardinality of a dominating set of G. Generalized de Bruijn digraphs GB(n,d) are good candidates for interconnection networks. Kikuchi and Shibata show that ⌈nd+1⌉≤γ(GB(n,d))≤⌈nd⌉. In this paper, by constructing a (consecutive) dominating set of a generalized de Bruijn digraph, we prove that its domination number has exactly two values, that is, γ(GB(n,d))=⌈nd+1⌉ or ⌈nd+1⌉+1. Additionally, using a related result in number theory, we provide a complete characterization of generalized de Bruijn digraphs with the domination number equal to ⌈nd+1⌉ or ⌈nd+1⌉+1. As a consequence of the above results, we prove a conjecture posed by Kikuchi and Shibata. The conjecture says that each generalized de Bruijn digraph has a minimum dominating set that is consecutive.

Efficient twin domination in generalized De Bruijn digraphs

A twin dominating set in a digraph D=(V,A) is a set S⊆V such that, for every w∈V∖S, there exist arcs (u,w),(w,v)∈A with u,v∈S. A twin dominating set S is efficient if there is no arc in the subdigraph induced by S and, for every w∈V∖S, there exist exactly one vertex u∈S and exactly one vertex v∈S such that arcs (u,w),(w,v)∈A. This paper resolves a conjecture regarding an efficient twin domination set in generalized De Bruijn digraphs. The conjecture says that the generalized De Bruijn digraphs GB(n,d) with n a multiple of d+1 have an efficient twin dominating set if and only if d is even and relatively prime to n. This paper affirms this conjecture.

Sandpile groups of generalized de Bruijn and Kautz graphs and circulant matrices over finite fields

A maximal minor M of the Laplacian of an n-vertex Eulerian digraph Γ gives rise to a finite group Zn−1/Zn−1M known as the sandpile (or critical) groupS(Γ) of Γ. We determine S(Γ) of the generalized de Bruijn graphsΓ=DB(n,d) with vertices 0,…,n−1 and arcs (i,di+k) for 0≤i≤n−1 and 0≤k≤d−1, and closely related generalized Kautz graphs, extending and completing earlier results for the classical de Bruijn and Kautz graphs.Moreover, for a prime p and an n-cycle permutation matrix X∈GLn(p) we show that S(DB(n,p)) is isomorphic to the quotient by 〈X〉 of the centralizer of X in PGLn(p). This offers an explanation for the coincidence of numerical data in sequences A027362 and A003473 of the OEIS, and allows one to speculate upon a possibility to construct normal bases in the finite field Fpn from spanning trees in DB(n,p).

EULERIAN AND HAMILTONIAN DICYCLES IN DIRECTED HYPERGRAPHS

In this paper, we generalize the concepts of Eulerian and Hamiltonian digraphs to directed hypergraphs. A dihypergraphH is a pair (𝒱(H), ℰ(H)), where 𝒱(H) is a non-empty set of elements, called vertices, and ℰ(H) is a collection of ordered pairs of subsets of 𝒱(H), called hyperarcs. It is Eulerian (resp. Hamiltonian) if there is a dicycle containing each hyperarc (resp. each vertex) exactly once. We first present some properties of Eulerian and Hamiltonian dihypergraphs. For example, we show that deciding whether a dihypergraph is Eulerian is an NP-complete problem. We also study when iterated line dihypergraphs are Eulerian and Hamiltonian. Finally, we study when the generalized de Bruijn dihypergraphs are Eulerian and Hamiltonian. In particular, we determine when they contain a complete Berge dicycle, i.e., an Eulerian and Hamiltonian dicycle.

On m-restricted edge connectivity of undirected generalized De Bruijn graphs

An m-restricted edge cut is an edge cut of a connected graph whose removal results in components of order at least m, the minimum cardinality over all m-restricted edge cuts of a graph is its m-restricted edge connectivity. It is known that telecommunication networks with topology having larger m-restricted edge connectivity are locally more reliable for all m≤3. This work shows that if n≥7, then undirected generalized binary De Bruijn graph UBG(2, n) is maximally m-restricted edge connected for all m≤3, where a graph G is maximally m-restricted edge connected if its m-restricted edge connectivity is equal to the minimum number of edges from any connected subgraphs S to G−S.

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.