De Bruijn graph

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In graph theory, an n-dimensional de Bruijn graph of m symbols is a directed graph representing overlaps between sequences of symbols. It has mn vertices, consisting of all possible length-n sequences of the given symbols; the same symbol may appear multiple times in a sequence. If we have the m symbols s_1,\dots,s_m then the set of vertices is:

V=\{(s_1,\dots,s_1,s_1),(s_1,\dots,s_1,s_2),\dots,(s_m,\dots,s_m,s_m)\}

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

E=\{((v_1,v_2,\dots,v_n),(w_1,w_2,\dots,w_n)) : v_2=w_1,v_3=w_2,\dots,v_n=w_{n-1}\}.

Although de Bruijn graphs are named after Nicolaas Govert de Bruijn, they were discovered independently both by de Bruijn (1946) and I. J. Good (1946). Much earlier, Flye Sainte-Marie (1894) implicitly used their properties.

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[edit] Properties

  • If n = 1 then the condition for any two vertices forming an edge holds vacuously, and hence all the vertices are connected forming a total of m2 edges.
  • Each vertex has exactly m incoming and m outgoing edges.
  • Each n-dimensional de Bruijn graph is the line digraph of the (n-1)-dimensional de Bruijn graph with the same set of symbols (Zhang and Lin 1987).
  • 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 n-dimensional de Bruijn graph corresponds to an edge of the (n-1)-dimensional de Bruijn graph, and each edge in the n-dimensional de Bruijn graph corresponds to a two-edge path in the (n-1)-dimensional de Bruijn graph.

[edit] Dynamical systems

Binary de Bruijn graphs can be drawn (below, left) in such a way that they resemble objects from the theory of dynamical systems, such as the Lorenz attractor (below, right):

       

This analogy can be made rigorous: the n-dimensional m-symbol de Bruijn graph forms a discrete model of the chaotic dynamical system

x\mapsto mx \bmod 1

(Philippe 2002). The trajectories of this dynamical system correspond to walks in the de Bruijn graph, where the correspondence is given by mapping each real x in the interval [0,1) to the vertex corresponding to the first n digits in the base-m representation of x. Equivalently, walks in the de Bruijn graph correspond by a similar mapping to trajectories in a one-sided subshift of finite type.

[edit] References

  • de Bruijn, N. G. (1946). "A Combinatorial Problem". Koninklijke Nederlandse Akademie v. Wetenschappen 49: 758–764. 
  • Flye Sainte-Marie, C. (1894). "Question 48". L'Intermédiaire Math. 1: 107–110. 
  • Good, I. J. (1946). "Normal recurring decimals". Journal of the London Mathematical Society 21: 167–169. 
  • Philippe, Leroux (2002). "Coassociative grammar, periodic orbits and quantum random walk over Z". arXiv:quant-ph/0209100. 
  • Zhang, Fu Ji; Lin, Guo Ning (1987). "On the de Bruijn-Good graphs". Acta Math. Sinica 30 (2): 195–205. 

[edit] See also

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