De Bruijn sequences

De Bruijn sequences

A De Bruijn sequence is defined as the shortest cyclic sequence that incorporates all substrings of a certain length of an alphabet.

For instance, the \(2^3=8\) binary strings of length 3 are all included in the following string:

sage: DeBruijnSequences(2,3).an_element()
[0, 0, 0, 1, 0, 1, 1, 1]

They can be obtained as a subsequence of the cyclic De Bruijn sequence of parameters \(k=2\) and \(n=3\):

sage: seq = DeBruijnSequences(2,3).an_element()
sage: print Word(seq).string_rep()
00010111
sage: shift = lambda i: [(i+j)%2**3 for j in range(3)]
sage: for i in range(2**3):
...      print (Word(map(lambda (j,b): b if j in shift(i) else '*',
...                                       enumerate(seq))).string_rep())
000*****
*001****
**010***
***101**
****011*
*****111
0*****11
00*****1

This sequence is of length \(k^n\), which is best possible as it is the number of \(k\)-ary strings of length \(n\). One can equivalently define a De Bruijn sequence of parameters \(k\) and \(n\) as a cyclic sequence of length \(k^n\) in which all substring of length \(n\) are different.

See also the Wikipedia article on De Bruijn sequences.

TESTS:

Checking the sequences generated are indeed valid:

sage: for n in range(1, 7):
...      for k in range(1, 7):
...         D = DeBruijnSequences(k, n)
...         if not D.an_element() in D:
...             print "Something's dead wrong (n=%s, k=%s)!" %(n,k)
...             break

AUTHOR:

• Eviatar Bach (2011): initial version
• Nathann Cohen (2011): Some work on the documentation and defined the __contain__ method

class sage.combinat.debruijn_sequence.DeBruijnSequences(k, n)

Bases: sage.structure.unique_representation.UniqueRepresentation, sage.structure.parent.Parent

Represents the De Bruijn sequences of given parameters \(k\) and \(n\).

A De Bruijn sequence of parameters \(k\) and \(n\) is defined as the shortest cyclic sequence that incorporates all substrings of length \(n\) a \(k\)-ary alphabet.

This class can be used to generate the lexicographically smallest De Bruijn sequence, to count the number of existing De Bruijn sequences or to test whether a given sequence is De Bruijn.

INPUT:

• k – A natural number to define arity. The letters used are the integers \(0..k-1\).
• n – A natural number that defines the length of the substring.

EXAMPLES:

Obtaining a De Bruijn sequence:

sage: seq = DeBruijnSequences(2, 3).an_element()
sage: print seq
[0, 0, 0, 1, 0, 1, 1, 1]

Testing whether it is indeed one:

sage: seq in DeBruijnSequences(2, 3)
True

The total number for these parameters:

sage: DeBruijnSequences(2, 3).cardinality()
2

Note

This function only generates one De Bruijn sequence (the smallest lexicographically). Support for generating all possible ones may be added in the future.

TESTS:

Setting k to 1 will return 0:

sage: DeBruijnSequences(1, 3).an_element()
[0]

Setting n to 1 will return the alphabet:

sage: DeBruijnSequences(3, 1).an_element()
[0, 1, 2]

The test suite:

sage: d=DeBruijnSequences(2, 3)
sage: TestSuite(d).run()

an_element()

Returns the lexicographically smallest De Bruijn sequence with the given parameters.

ALGORITHM:

The algorithm is described in the book “Combinatorial Generation” by Frank Ruskey. This program is based on a Ruby implementation by Jonas Elfström, which is based on the C program by Joe Sadawa.

EXAMPLE:

sage: DeBruijnSequences(2, 3).an_element()
[0, 0, 0, 1, 0, 1, 1, 1]

cardinality()

Returns the number of distinct De Bruijn sequences for the object’s parameters.

EXAMPLE:

sage: DeBruijnSequences(2, 5).cardinality()
2048

ALGORITHM:

The formula for cardinality is \(k!^{k^{n-1}}/k^n\) [1].

REFERENCES:

[1]	Rosenfeld, Vladimir Raphael, 2002: Enumerating De Bruijn Sequences. Communications in Math. and in Computer Chem.

sage.combinat.debruijn_sequence.debruijn_sequence(k, n)

The generating function for De Bruijn sequences. This avoids the object creation, so is significantly faster than accessing from DeBruijnSequence. For more information, see the documentation there. The algorithm used is from Frank Ruskey’s “Combinatorial Generation”.

INPUT:

• k – Arity. Must be an integer.
• n – Substring length. Must be an integer.

EXAMPLES:

sage: from sage.combinat.debruijn_sequence import debruijn_sequence
sage: debruijn_sequence(3, 1)
[0, 1, 2]

sage.combinat.debruijn_sequence.is_debruijn_sequence(seq, k, n)

Given a sequence of integer elements in \(0..k-1\), tests whether it corresponds to a De Bruijn sequence of parameters \(k\) and \(n\).

INPUT:

• seq – Sequence of elements in \(0..k-1\).
• n,k – Integers.

EXAMPLE:

sage: from sage.combinat.debruijn_sequence import is_debruijn_sequence
sage: s = DeBruijnSequences(2, 3).an_element()
sage: is_debruijn_sequence(s, 2, 3)
True
sage: is_debruijn_sequence(s + [0], 2, 3)
False
sage: is_debruijn_sequence([1] + s[1:], 2, 3)
False