De Bruijn
Graphs &
Sequences
Brian Breitsch
Hamilton Path/Cycle
- a Hamilton path visits each vertex exactly once
- a Hamilton cycle is a Hamilton path that loops back to the first vertex
Eulerean Path/Cycle
- an Eulerean path visits each edge exactly once
- an Eulerean cycle is an Eulerean path that loops back to the first edge
Line Digraph
- vertices are the edges of original graph
- when
wx
wx
vw \to xy
vw→xy
w = x
w=x
Example
Nicolaas Govert de Bruijn
- pronounced 'du bruyn'
- 9 July 1918 – 17 February 2012
- analysis, number theory, combinatorics, logic
- wrote AUTOMATH language in 1960s
De Bruijn Graph
- alphabet of size
- words of length
- possible words
- a directed edge exists between and when a left shift in the word corresponding to matches the first letters of the second word
A
A
|A| = k
∣A∣=k
n
n
v_1
v1
v_2
v2
v_2
v2
k-1
k−1
k^n
kn
0123 \to
0123→
1230
1230
1231
1231
1232
1232
1233
1233
A = \{0, 1, 2, 3\}
A={0,1,2,3}
V =
V=
|V| =
∣V∣=
GdB(k,n)
GdB(k,n)
Another Interpretation
- digits of size
- numbers with digits
- vertices are subset of numbers
- a directed edge exists between and when for some
S
S
|S| = s
∣S∣=s
k
k
v_1
v1
v_2
v2
v_2 = sv_1 + a
v2=sv1+a
a \in S
a∈S
0123 \times 10 + a\to
0123×10+a→
1230, a = 0
1230,a=0
1231, a = 1
1231,a=1
1232, a = 2
1232,a=2
1233, a = 3
1233,a=3
A = \{0, 1, 2, 3\}
A={0,1,2,3}
Trivial Examples
ary De Bruijn graph with is a complete graph
k-
k−
n = 1
n=1
trivial loop graph
|A| = 1 \rightarrow |V| = 1
∣A∣=1→∣V∣=1
diameter
Properties
n
n
\partial GdB(k,n)
∂GdB(k,n)
GdB(k,n)
GdB(k,n)
is balanced (i.e. regular directed graph)
is strongly connected
is
GdB(k,n)
GdB(k,n)
De Bruijn Sequences
A ary De Bruijn sequence of order is a cyclic sequence in which every possible word of length from an alphabet appears exactly once.
NOTE: sometimes people restrict the definition to a binary alphabet
k-
k−
n
n
|A| = k
∣A∣=k
A = \{0, 1\}
A={0,1}
n = 3
n=3
00010111
00010111
is a De Bruijn sequence
n
n
- every De Bruijn graph has Euler and Hamliton cycles
Properties
Traversing an Euler/Hamilton cycle yields a De Bruijn Sequence
- start with complete directed graph
- take line graph
- repeat times
n
n
FACT: is the line graph of
Properties
GdB(k,n)
GdB(k,n)
GdB(k,n-1)
GdB(k,n−1)
Binary De Bruijn Graphs & Sequences
- #edges
- some can be generated using finite cell automata [9] (maybe?)
- e.g.
- the lexigraphically-least sequence is called "grand-daddy"
- e.g.
|V| = 2^n
∣V∣=2n
= 2^{n+1}
=2n+1
00001000110010101111
00001000110010101111
How many binary De Bruijn sequences are there for given n?
GdB(2,n-1)
GdB(2,n−1)
GdB(k,n)
GdB(k,n)
Eulerean tours exist because is balanced
This is equivalent to counting the number of Eulerean cycles in
Consider vertices
v,w
v,w
v = (v_1v_2...v_{n-1})
v=(v1v2...vn−1)
w = (w_1w_2...w_{n-1})
w=(w1w2...wn−1)
Then the path from is
v\to w
v→w
(v_1v_2...v_{n-1})(w_1w_2...w_{n-1})
(v1v2...vn−1)(w1w2...wn−1)
v_1)(v_2...v_{n-1}w_1)(w_2...w_{n-1}
v1)(v2...vn−1w1)(w2...wn−1
v_1v_2)(...v_{n-1}w_1w_2)(...w_{n-1}
v1v2)(...vn−1w1w2)(...wn−1
How many binary De Bruijn sequences are there for given n?
Thus we have exactly one path of length for any pair of vertices
v,w
v,w
n-1
n−1
This means
A^{n-1} = \left[ 1 \right]
An−1=[1]
(all 1 matrix)
is the
2^{n-1} \times 2^{n-1}
2n−1×2n−1
adjacency matrix
A
A
The eigenvalues are
\lambda = 0
λ=0
\lambda = 2
λ=2
with multiplicity
with multiplicity
2^{n-1}
2n−1
1
1
How many binary De Bruijn sequences are there for given n?
Matrix-tree theory:
# of Eulerean cycles
= \sum_{e\in E} \kappa (GdB(2,n), e)
=∑e∈Eκ(GdB(2,n),e)
= \sum_{e\in E} \frac{\lambda_1 \lambda_2 ... \lambda_{2^{n-1}-1}}{2^{n-1}}
=∑e∈E2n−1λ1λ2...λ2n−1−1
= \sum_{e\in E} \frac{2^{2^{n-1}-1}}{2^{n-1}}
=∑e∈E2n−122n−1−1
= 2^n \frac{2^{2^{n-1}-1}}{2^{n-1}}
=2n2n−122n−1−1
= 2^{2^{n-1}}
=22n−1
where
\kappa (GdB(2,n), e)
κ(GdB(2,n),e)
is the vertex connectivity
\kappa (GdB(2,n), e)
κ(GdB(2,n),e)
Applications
- genomic data storage and processing
- a De Bruijn graph data structure for genomics compressed a 340GB (naive approach) to under 5GB [1]
\{A, G, T, C\}
{A,G,T,C}
B(4,2)
B(4,2)
Applications
- fastest way to brute-force attack a number or pad lock that allows uninterrupted input of numbers
...1552537748599...
...1552537748599...
Laundry Machine Dials
- find the sequence of connections to put on dial for maximum number of consecutive unambiguous options [4]
Indexing 1 in Computer (old)
- find the location of first 1 in a word using de Bruijn sequences and an efficient hash table lookup [6].
00100000
00100000
multiply by de Bruijn sequence
0000010111000000
0000010111000000
00010111
00010111
mask first byte and shift by 8
00000011
00000011
this is a bijection (because of the property of de Bruijn sequences) and 1 index can thus be looked up in table
Some Open Problems?
Let be a subset of a general de Bruijn sequence which contains all sequences of weight between and
(these are essentially subgraphs of the de Bruijn graph)
GdB_v^w(k,n)
GdBvw(k,n)
v
v
w
w
- can these sequences be constructed efficiently
- what is the first fixed-weight de Bruijn sequence and can it be constructed without "back-tracking"
- what is the diameter for
GdB_w(k,n)
GdBw(k,n)
GdB_v^w(k,n)
GdBvw(k,n)
?
References
- "Tutorials." Tutorials. Homolog.us – Bioinformatics, 2015. Web. 23 Feb. 2015.
- "De Bruijn Graph." Wikipedia. Wikimedia Foundation, n.d. Web. 23 Feb. 2015.
- "De Bruijn Sequence." Wikipedia. Wikimedia Foundation, n.d. Web. 23 Feb. 2015.
- Higgins, Peter M. Nets, Puzzles, and Postmen: An Exploration of Mathematical Connections. Oxford: Oxford UP, 2009. Print.
- Levine, Lionel. "Lecture 21." Algebraic Combinatorics. 28 Apr. 2011. Web.
- Leiserson, Charles E., Harold Prokop, and Keith H. Randall. "Using De Bruijn Sequences to Index a 1 in a Computer Word." (1998): n. pag. Web.
- Ruskey, Frank, Joe Sawada, and Aaron Williams. "De Bruijn Sequences for Fixed-Weight Binary Strings." SIAM Journal on Discrete Mathematics 26.2 (2012): 605-17. Web.
- Shibata, Y., and Y. Gonda. "Extension of De Bruijn Graph and Kautz Graph." Computers & Mathematics with Applications 30.9 (1995): 51-61. Web.
- Sutner, Klaus. "De Bruijn Graphs and Linear Cellular Automata." Complex Systems 5 (1991): 19-30. Web.
De Bruijn Graphs & Sequences Brian Breitsch
De Bruijn Graphs & Sequences
By Brian Breitsch
De Bruijn Graphs & Sequences
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