Английская Википедия:De Bruijn graph

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In graph theory, an Шаблон:Mvar-dimensional De Bruijn graph of Шаблон:Mvar symbols is a directed graph representing overlaps between sequences of symbols. It has Шаблон:Mvar vertices, consisting of all possible Шаблон:Nowrap sequences of the given symbols; the same symbol may appear multiple times in a sequence. For a set of Шаблон:Mvar symbols Шаблон:Math the set of vertices is:

<math>V=S^n=\{(s_1,\dots,s_1,s_1),(s_1,\dots,s_1,s_2),\dots,(s_1,\dots,s_1,s_m),(s_1,\dots,s_2,s_1),\dots,(s_m,\dots,s_m,s_m)\}.</math>

If one of the vertices can be expressed as another vertex by shifting all its symbols by one place to the left and adding a new symbol at the end of this vertex, then the latter has a directed edge to the former vertex. Thus the set of arcs (that is, directed edges) is

<math>E=\{((t_1,t_2,\dots,t_n),(t_2,\dots,t_n,s_j)) : t_i \in S, 1\le i\le n, 1\le j\le m \}.</math>

Although De Bruijn graphs are named after Nicolaas Govert de Bruijn, they were invented independently by both de Bruijn and I. J. Good. Much earlier, Camille Flye Sainte-Marie implicitly used their properties.

Properties

• If Шаблон:Math, then the condition for any two vertices forming an edge holds vacuously, and hence all the vertices are connected, forming a total of Шаблон:Math edges.
• Each vertex has exactly Шаблон:Mvar incoming and Шаблон:Mvar outgoing edges.
• Each Шаблон:Mvar-dimensional De Bruijn graph is the line digraph of the Шаблон:Nowrap De Bruijn graph with the same set of symbols.
• Each De Bruijn graph is Eulerian and Hamiltonian. The Euler cycles and Hamiltonian cycles of these graphs (equivalent to each other via the line graph construction) are De Bruijn sequences.

The line graph construction of the three smallest binary De Bruijn graphs is depicted below. As can be seen in the illustration, each vertex of the Шаблон:Mvar-dimensional De Bruijn graph corresponds to an edge of the Шаблон:Nowrap De Bruijn graph, and each edge in the Шаблон:Mvar-dimensional De Bruijn graph corresponds to a two-edge path in the Шаблон:Nowrap De Bruijn graph.

Dynamical systems

Binary De Bruijn graphs can be drawn in such a way that they resemble objects from the theory of dynamical systems, such as the Lorenz attractor: Шаблон:Multiple image This analogy can be made rigorous: the Шаблон:Mvar-dimensional Шаблон:Mvar-symbol De Bruijn graph is a model of the Bernoulli map

<math>x\mapsto mx\ \bmod\ 1.</math>

The Bernoulli map (also called the Шаблон:Math map for Шаблон:Math) is an ergodic dynamical system, which can be understood to be a single shift of a [[p-adic|Шаблон:Mvar-adic number]]. The trajectories of this dynamical system correspond to walks in the De Bruijn graph, where the correspondence is given by mapping each real Шаблон:Mvar in the interval Шаблон:Math to the vertex corresponding to the first Шаблон:Mvar digits in the base-Шаблон:Mvar representation of Шаблон:Mvar. Equivalently, walks in the De Bruijn graph correspond to trajectories in a one-sided subshift of finite type.

Файл:De Bruijn binary graph.svg

Directed graphs of two Шаблон:Math de Bruijn sequences and a Шаблон:Math sequence. In Шаблон:Math, each vertex is visited once, whereas in Шаблон:Math, each edge is traversed once.

Embeddings resembling this one can be used to show that the binary De Bruijn graphs have queue number 2 and that they have book thickness at most 5.

Uses

• Some grid network topologies are De Bruijn graphs.
• The distributed hash table protocol Koorde uses a De Bruijn graph.
• In bioinformatics, De Bruijn graphs are used for de novo assembly of sequencing reads into a genome.

See also

• De Bruijn torus
• Hamming graph
• Kautz graph

External links

• Шаблон:MathWorld
• Tutorial on using De Bruijn Graphs in Bioinformatics by Homolog.us