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 \to 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

v_1

v_2

v_2

v_2

v_2

k-1

k-1

k^n

k^n

0123 \to

0123 \to

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

v_1

v_2

v_2

v_2 = sv_1 + a

v_2 = sv_1 + a

a \in S

a \in S

0123 \times 10 + a\to

0123 \times 10 + a\to

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 \rightarrow |V| = 1

diameter

Properties

n

n

\partial GdB(k,n)

\partial 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"

|V| = 2^n

|V| = 2^n

= 2^{n+1}

= 2^{n+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 = (v_1v_2...v_{n-1})

w = (w_1w_2...w_{n-1})

w = (w_1w_2...w_{n-1})

Then the path from is

v\to w

v\to w

(v_1v_2...v_{n-1})(w_1w_2...w_{n-1})

(v_1v_2...v_{n-1})(w_1w_2...w_{n-1})

v_1)(v_2...v_{n-1}w_1)(w_2...w_{n-1}

v_1)(v_2...v_{n-1}w_1)(w_2...w_{n-1}

v_1v_2)(...v_{n-1}w_1w_2)(...w_{n-1}

v_1v_2)(...v_{n-1}w_1w_2)(...w_{n-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]

A^{n-1} = \left[ 1 \right]

(all 1 matrix)

is the

2^{n-1} \times 2^{n-1}

2^{n-1} \times 2^{n-1}

adjacency matrix

A

A

The eigenvalues are

\lambda = 0

\lambda = 0

\lambda = 2

\lambda = 2

with multiplicity

2^{n-1}

2^{n-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)

= \sum_{e\in E} \kappa (GdB(2,n), e)

= \sum_{e\in E} \frac{\lambda_1 \lambda_2 ... \lambda_{2^{n-1}-1}}{2^{n-1}}

= \sum_{e\in E} \frac{\lambda_1 \lambda_2 ... \lambda_{2^{n-1}-1}}{2^{n-1}}

= \sum_{e\in E} \frac{2^{2^{n-1}-1}}{2^{n-1}}

= \sum_{e\in E} \frac{2^{2^{n-1}-1}}{2^{n-1}}

= 2^n \frac{2^{2^{n-1}-1}}{2^{n-1}}

= 2^n \frac{2^{2^{n-1}-1}}{2^{n-1}}

= 2^{2^{n-1}}

= 2^{2^{n-1}}

where

\kappa (GdB(2,n), e)

\kappa (GdB(2,n), e)

is the vertex connectivity

\kappa (GdB(2,n), e)

\kappa (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)

GdB_v^w(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)

GdB_w(k,n)

GdB_v^w(k,n)

GdB_v^w(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.