De Bruijn graph

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

$V=S^n=\{(s_1,\dots ,s_1,s_1),(s_1,\dots ,s_1,s_2),\dots ,(s_1,\dots ,s_1,s_m),(s_1,\dots ,s_2,s_1),\dots ,(s_m,\dots ,s_m,s_m)\}.$

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

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

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 Template:Math, then the condition for any two vertices forming an edge holds vacuously, and hence all the vertices are connected, forming a total of Template:Math edges.
• Each vertex has exactly Template:Mvar incoming and Template:Mvar outgoing edges.
• Each Template:Mvar-dimensional De Bruijn graph is the line digraph of the Template:Math-dimensional 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 Template:Mvar-dimensional De Bruijn graph corresponds to an edge of the Template:Math-dimensional De Bruijn graph, and each edge in the Template:Mvar-dimensional De Bruijn graph corresponds to a two-edge path in the Template:Math-dimensional 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: Template:Multiple image This analogy can be made rigorous: the Template:Mvar-dimensional Template:Mvar-symbol De Bruijn graph is a model of the Bernoulli map

$x\mapsto mxmod1.$

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

File:De Bruijn binary graph.svg

Directed graphs of two Template:Math de Bruijn sequences and a Template:Math sequence. In Template:Math, each vertex is visited once, whereas in Template: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. Instead of the complete De Bruijn graphs described above that contain all possible k-mers, de novo sequence assemblers make use of De Bruijn subgraphs that contain only the k-mers observed in a sequencing dataset.
• In time series forecasting, De Bruijn graphs have been adapted to encode temporal patterns by mapping discrete subsequences (n-grams) of observations to graph nodes. This enables the modeling of sequential dependencies in symbolic or discretized time series data. Multivariate De Bruijn graphs extend this idea by jointly encoding patterns across multiple correlated variables, allowing for the representation of complex inter-variable temporal dynamics in multivariate time series.

See also

• De Bruijn torus
• Hamming graph
• Kautz graph

External links

• Tutorial on using De Bruijn Graphs in Bioinformatics by Homolog.us
• De Bruijn Graph algorithm tutorial by Kurban Intelligence Lab