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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 ''m''<sup>''n''</sup> 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 set of ''m'' symbols <math>S:=\{s_1,\dots,s_m\}</math> then the set of vertices is:
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| : <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>
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| 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 (aka directed edges) is
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| : <math>E=\{((v_1,v_2,\dots,v_n),(v_2,\dots,v_n,s_i)) : i=1,\dots,m \}.</math>
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| Although De Bruijn graphs are named after [[Nicolaas Govert de Bruijn]], they were discovered independently by both De Bruijn<ref name="Bruijn1946"/> and [[I. J. Good]].<ref name="Good1946"/> Much earlier, Camille Flye Sainte-Marie<ref name="Flye1894"/> implicitly used their properties.
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| == Properties ==
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| * If <math>n=1</math> then the condition for any two vertices forming an edge holds vacuously, and hence all the vertices are connected forming a total of <math>m^2</math> edges.
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| * Each vertex has exactly <math>m</math> incoming and <math>m</math> outgoing edges.
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| * Each <math>n</math>-dimensional De Bruijn graph is the [[line graph|line digraph]] of the <math>(n - 1)</math>-dimensional De Bruijn graph with the same set of symbols.<ref name="Zhang1987"/>
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| * Each De Bruijn graph is [[Euler cycle|Eulerian]] and [[Hamiltonian graph|Hamiltonian]]. The Euler cycles and Hamiltonian cycles of these graphs (equivalent to each other via the line graph construction) are [[De Bruijn sequence]]s.
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| 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 <math>n</math>-dimensional De Bruijn graph corresponds to an edge of the <math>(n - 1)</math>-dimensional De Bruijn graph, and each edge in the <math>n</math>-dimensional De Bruijn graph corresponds to a two-edge path in the <math>(n - 1)</math>-dimensional De Bruijn graph.
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| [[Image:DeBruijn-as-line-digraph.svg|center|600px]]
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| == Dynamical systems ==
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| Binary De Bruijn graphs can be drawn (below, left) in such a way that they resemble objects from the theory of [[dynamical system]]s, such as the [[Lorenz attractor]] (below, right):
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| :::[[Image:DeBruijn-3-2.svg|360px]] [[Image:Lorenz attractor yb.svg|200px]]
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| This analogy can be made rigorous: the ''n''-dimensional ''m''-symbol De Bruijn graph is a model of the [[Bernoulli map]]
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| :<math>x\mapsto mx\ \bmod\ 1</math>
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| The Bernoulli map (also called the [[2x mod 1 map]] for ''m'' = 2) is an [[ergodic]] dynamical system, which can be understood to be a single [[shift operator|shift]] of a [[p-adic|m-adic number]].<ref name="Leroux2002"/> 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 [[Radix|base]]-''m'' representation of ''x''. Equivalently, walks in the De Bruijn graph correspond to trajectories in a one-sided [[subshift of finite type]].
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| == Uses ==
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| * Some [[grid network]] topologies are De Bruijn graphs.
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| * The [[distributed hash table]] protocol [[Koorde]] uses a De Bruijn graph.
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| * In [[bioinformatics]], De Bruijn graphs are used for ''de novo'' assembly of (short) read sequences into a [[genome]].<ref name="Pevzner2001a"/><ref name="Pevzner2001b"/><ref name="zerbino2008"/>
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| == See also ==
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| * [[De Bruijn sequence]]
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| * [[De Bruijn torus]]
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| * [[Kautz graph]]
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| * [[Free monoid]]
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| * [[Semiautomata]]
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| == References ==
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| <references>
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| <ref name="Bruijn1946">
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| {{cite journal
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| | author = de Bruijn, N. G.
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| | authorlink = Nicolaas Govert de Bruijn
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| | title = A Combinatorial Problem
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| | journal = Koninklijke Nederlandse Akademie v. Wetenschappen
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| | volume = 49
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| | pages = 758–764
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| | year = 1946}}
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| </ref>
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| <ref name="Flye1894">
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| {{cite journal
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| | author = Flye Sainte-Marie, C.
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| | title = Question 48
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| | journal = L'Intermédiaire Math.
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| | volume = 1
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| | year = 1894
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| | pages = 107–110}}
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| </ref>
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| <ref name="Good1946">
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| {{cite journal
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| | author = Good, I. J.
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| | authorlink = I. J. Good
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| | title = Normal recurring decimals
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| | journal = Journal of the London Mathematical Society
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| | volume = 21
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| | issue = 3
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| | year = 1946
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| | pages = 167–169
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| | doi = 10.1112/jlms/s1-21.3.167}}
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| </ref>
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| <ref name="Leroux2002">
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| {{citation
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| | last= Leroux | first = Philippe
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| | year = 2002
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| | bibcode = 2002quant.ph..9100P
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| | title = Coassociative grammar, periodic orbits and quantum random walk over Z
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| | arxiv = quant-ph/0209100}}
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| </ref>
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| <ref name="Zhang1987">
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| {{cite journal
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| | author = Zhang, Fu Ji; Lin, Guo Ning
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| | title = On the de Bruijn-Good graphs
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| | journal = Acta Math. Sinica
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| | volume = 30
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| | year = 1987
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| | issue = 2
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| | pages = 195–205}}
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| </ref>
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| <ref name ="Pevzner2001a">
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| {{cite journal
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| | author = Pevzner, Pavel A.; Tang, Haixu; Waterman, Michael S.
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| | title = An Eulerian path approach to DNA fragment assembly
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| | journal = PNAS
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| | volume = 98
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| | issue = 17
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| | year = 2001
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| | pages = 9748–9753
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| | pmid = 11504945
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| | pmc = 55524
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| | doi = 10.1073/pnas.171285098
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| | bibcode = 2001PNAS...98.9748P}}
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| </ref>
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| <ref name ="Pevzner2001b">
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| {{cite journal
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| | author = Pevzner, Pavel A.; Tang, Haixu
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| | title = Fragment Assembly with Double-Barreled Data
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| | journal = Bioinformatics/ISMB
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| | volume = 1
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| | year = 2001
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| | pages = 1–9}}
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| </ref>
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| <ref name="zerbino2008">
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| {{cite journal
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| | author = Zerbino, Daniel R.; Birney, Ewan
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| | title = Velvet: algorithms for de novo short read assembly using de Bruijn graphs
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| | journal = Genome Research
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| | volume = 18
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| | issue = 5
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| | year = 2008
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| | pages = 821–829
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| | doi = 10.1101/gr.074492.107
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| | pmid = 18349386
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| | pmc = 2336801}}
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| </ref>
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| </references>
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| ==External links==
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| * {{MathWorld|title=De Bruijn Graph|id=deBruijnGraph}}
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| * [http://www.homolog.us/Tutorials/index.php?p=2.1&s=1 Tutorial on using De Bruijn Graphs in Bioinformatics] by Homolog.us
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| [[Category:Dynamical systems]]
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| [[Category:Automata theory]]
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| [[Category:Parametric families of graphs]]
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| [[Category:Directed graphs]]
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