def ImaseItoh(self, n, d):
r"""
Returns the digraph of Imase and Itoh of order `n` and degree `d`.
The digraph of Imase and Itoh has been defined in [II83]_. It has vertex
set `V=\{0, 1,..., n-1\}` and there is an arc from vertex `u \in V` to
all vertices `v \in V` such that `v \equiv (-u*d-a-1) \mod{n}` with
`0 \leq a < d`.
When `n = d^{D}`, the digraph of Imase and Itoh is isomorphic to the de
Bruijn digraph of degree `d` and diameter `D`. When `n = d^{D-1}(d+1)`,
the digraph of Imase and Itoh is isomorphic to the Kautz digraph
[Kautz68]_ of degree `d` and diameter `D`.
INPUTS:
- ``n`` -- is the number of vertices of the digraph
- ``d`` -- is the degree of the digraph
EXAMPLES::
sage: II = digraphs.ImaseItoh(8, 2)
sage: II.is_isomorphic(digraphs.DeBruijn(2, 3), certify = True)
(True, {0: '010', 1: '011', 2: '000', 3: '001', 4: '110', 5: '111', 6: '100', 7: '101'})
sage: II = digraphs.ImaseItoh(12, 2)
sage: II.is_isomorphic(digraphs.Kautz(2, 3), certify = True)
(True, {0: '010', 1: '012', 2: '021', 3: '020', 4: '202', 5: '201', 6: '210', 7: '212', 8: '121', 9: '120', 10: '102', 11: '101'})
TESTS:
An exception is raised when the degree is less than one::
sage: G = digraphs.ImaseItoh(2, 0)
Traceback (most recent call last):
...
ValueError: The digraph of Imase and Itoh is defined for degree at least one.
An exception is raised when the order of the graph is less than two::
sage: G = digraphs.ImaseItoh(1, 2)
Traceback (most recent call last):
...
ValueError: The digraph of Imase and Itoh is defined for at least two vertices.
REFERENCE:
.. [II83] M. Imase and M. Itoh. A design for directed graphs with
minimum diameter, *IEEE Trans. Comput.*, vol. C-32, pp. 782-784, 1983.
"""
if n < 2:
raise ValueError(
"The digraph of Imase and Itoh is defined for at least two vertices."
)
if d < 1:
raise ValueError(
"The digraph of Imase and Itoh is defined for degree at least one."
)
II = DiGraph(loops=True)
II.allow_multiple_edges(True)
for u in xrange(n):
for a in xrange(-u * d - d, -u * d):
II.add_edge(u, a % n)
II.name("Imase and Itoh digraph (n=%s, d=%s)" % (n, d))
return II
def DeBruijn(self, k, n, vertices='strings'):
r"""
Returns the De Bruijn digraph with parameters `k,n`.
The De Bruijn digraph with parameters `k,n` is built upon a set of
vertices equal to the set of words of length `n` from a dictionary of
`k` letters.
In this digraph, there is an arc `w_1w_2` if `w_2` can be obtained from
`w_1` by removing the leftmost letter and adding a new letter at its
right end. For more information, see the
:wikipedia:`Wikipedia article on De Bruijn graph <De_Bruijn_graph>`.
INPUT:
- ``k`` -- Two possibilities for this parameter :
- An integer equal to the cardinality of the alphabet to use, that
is the degree of the digraph to be produced.
- An iterable object to be used as the set of letters. The degree
of the resulting digraph is the cardinality of the set of
letters.
- ``n`` -- An integer equal to the length of words in the De Bruijn
digraph when ``vertices == 'strings'``, and also to the diameter of
the digraph.
- ``vertices`` -- 'strings' (default) or 'integers', specifying whether
the vertices are words build upon an alphabet or integers.
EXAMPLES::
sage: db=digraphs.DeBruijn(2,2); db
De Bruijn digraph (k=2, n=2): Looped digraph on 4 vertices
sage: db.order()
4
sage: db.size()
8
TESTS::
sage: digraphs.DeBruijn(5,0)
De Bruijn digraph (k=5, n=0): Looped multi-digraph on 1 vertex
sage: digraphs.DeBruijn(0,0)
De Bruijn digraph (k=0, n=0): Looped multi-digraph on 0 vertices
"""
from sage.combinat.words.words import Words
from sage.rings.integer import Integer
W = Words(range(k) if isinstance(k, Integer) else k, n)
A = Words(range(k) if isinstance(k, Integer) else k, 1)
g = DiGraph(loops=True)
if vertices == 'strings':
if n == 0:
g.allow_multiple_edges(True)
v = W[0]
for a in A:
g.add_edge(v.string_rep(), v.string_rep(), a.string_rep())
else:
for w in W:
ww = w[1:]
for a in A:
g.add_edge(w.string_rep(), (ww * a).string_rep(),
a.string_rep())
else:
d = W.size_of_alphabet()
g = digraphs.GeneralizedDeBruijn(d**n, d)
g.name("De Bruijn digraph (k=%s, n=%s)" % (k, n))
return g
def GeneralizedDeBruijn(self, n, d):
r"""
Returns the generalized de Bruijn digraph of order `n` and degree `d`.
The generalized de Bruijn digraph has been defined in [RPK80]_
[RPK83]_. It has vertex set `V=\{0, 1,..., n-1\}` and there is an arc
from vertex `u \in V` to all vertices `v \in V` such that
`v \equiv (u*d + a) \mod{n}` with `0 \leq a < d`.
When `n = d^{D}`, the generalized de Bruijn digraph is isomorphic to the
de Bruijn digraph of degree `d` and diameter `D`.
INPUTS:
- ``n`` -- is the number of vertices of the digraph
- ``d`` -- is the degree of the digraph
.. SEEALSO::
* :meth:`sage.graphs.generic_graph.GenericGraph.is_circulant` --
checks whether a (di)graph is circulant, and/or returns all
possible sets of parameters.
EXAMPLE::
sage: GB = digraphs.GeneralizedDeBruijn(8, 2)
sage: GB.is_isomorphic(digraphs.DeBruijn(2, 3), certify = True)
(True, {0: '000', 1: '001', 2: '010', 3: '011', 4: '100', 5: '101', 6: '110', 7: '111'})
TESTS:
An exception is raised when the degree is less than one::
sage: G = digraphs.GeneralizedDeBruijn(2, 0)
Traceback (most recent call last):
...
ValueError: The generalized de Bruijn digraph is defined for degree at least one.
An exception is raised when the order of the graph is less than one::
sage: G = digraphs.GeneralizedDeBruijn(0, 2)
Traceback (most recent call last):
...
ValueError: The generalized de Bruijn digraph is defined for at least one vertex.
REFERENCES:
.. [RPK80] S. M. Reddy, D. K. Pradhan, and J. Kuhl. Directed graphs with
minimal diameter and maximal connectivity, School Eng., Oakland Univ.,
Rochester MI, Tech. Rep., July 1980.
.. [RPK83] S. Reddy, P. Raghavan, and J. Kuhl. A Class of Graphs for
Processor Interconnection. *IEEE International Conference on Parallel
Processing*, pages 154-157, Los Alamitos, Ca., USA, August 1983.
"""
if n < 1:
raise ValueError(
"The generalized de Bruijn digraph is defined for at least one vertex."
)
if d < 1:
raise ValueError(
"The generalized de Bruijn digraph is defined for degree at least one."
)
GB = DiGraph(loops=True)
GB.allow_multiple_edges(True)
for u in xrange(n):
for a in xrange(u * d, u * d + d):
GB.add_edge(u, a % n)
GB.name("Generalized de Bruijn digraph (n=%s, d=%s)" % (n, d))
return GB
def ImaseItoh(self, n, d):
r"""
Returns the digraph of Imase and Itoh of order `n` and degree `d`.
The digraph of Imase and Itoh has been defined in [II83]_. It has vertex
set `V=\{0, 1,..., n-1\}` and there is an arc from vertex `u \in V` to
all vertices `v \in V` such that `v \equiv (-u*d-a-1) \mod{n}` with
`0 \leq a < d`.
When `n = d^{D}`, the digraph of Imase and Itoh is isomorphic to the de
Bruijn digraph of degree `d` and diameter `D`. When `n = d^{D-1}(d+1)`,
the digraph of Imase and Itoh is isomorphic to the Kautz digraph
[Kautz68]_ of degree `d` and diameter `D`.
INPUTS:
- ``n`` -- is the number of vertices of the digraph
- ``d`` -- is the degree of the digraph
EXAMPLES::
sage: II = digraphs.ImaseItoh(8, 2)
sage: II.is_isomorphic(digraphs.DeBruijn(2, 3), certify = True)
(True, {0: '010', 1: '011', 2: '000', 3: '001', 4: '110', 5: '111', 6: '100', 7: '101'})
sage: II = digraphs.ImaseItoh(12, 2)
sage: II.is_isomorphic(digraphs.Kautz(2, 3), certify = True)
(True, {0: '010', 1: '012', 2: '021', 3: '020', 4: '202', 5: '201', 6: '210', 7: '212', 8: '121', 9: '120', 10: '102', 11: '101'})
TESTS:
An exception is raised when the degree is less than one::
sage: G = digraphs.ImaseItoh(2, 0)
Traceback (most recent call last):
...
ValueError: The digraph of Imase and Itoh is defined for degree at least one.
An exception is raised when the order of the graph is less than two::
sage: G = digraphs.ImaseItoh(1, 2)
Traceback (most recent call last):
...
ValueError: The digraph of Imase and Itoh is defined for at least two vertices.
REFERENCE:
.. [II83] M. Imase and M. Itoh. A design for directed graphs with
minimum diameter, *IEEE Trans. Comput.*, vol. C-32, pp. 782-784, 1983.
"""
if n < 2:
raise ValueError("The digraph of Imase and Itoh is defined for at least two vertices.")
if d < 1:
raise ValueError("The digraph of Imase and Itoh is defined for degree at least one.")
II = DiGraph(loops = True)
II.allow_multiple_edges(True)
for u in xrange(n):
for a in xrange(-u*d-d, -u*d):
II.add_edge(u, a % n)
II.name( "Imase and Itoh digraph (n=%s, d=%s)"%(n,d) )
return II
def GeneralizedDeBruijn(self, n, d):
r"""
Returns the generalized de Bruijn digraph of order `n` and degree `d`.
The generalized de Bruijn digraph has been defined in [RPK80]_
[RPK83]_. It has vertex set `V=\{0, 1,..., n-1\}` and there is an arc
from vertex `u \in V` to all vertices `v \in V` such that
`v \equiv (u*d + a) \mod{n}` with `0 \leq a < d`.
When `n = d^{D}`, the generalized de Bruijn digraph is isomorphic to the
de Bruijn digraph of degree `d` and diameter `D`.
INPUTS:
- ``n`` -- is the number of vertices of the digraph
- ``d`` -- is the degree of the digraph
.. SEEALSO::
* :meth:`sage.graphs.generic_graph.GenericGraph.is_circulant` --
checks whether a (di)graph is circulant, and/or returns all
possible sets of parameters.
EXAMPLE::
sage: GB = digraphs.GeneralizedDeBruijn(8, 2)
sage: GB.is_isomorphic(digraphs.DeBruijn(2, 3), certify = True)
(True, {0: '000', 1: '001', 2: '010', 3: '011', 4: '100', 5: '101', 6: '110', 7: '111'})
TESTS:
An exception is raised when the degree is less than one::
sage: G = digraphs.GeneralizedDeBruijn(2, 0)
Traceback (most recent call last):
...
ValueError: The generalized de Bruijn digraph is defined for degree at least one.
An exception is raised when the order of the graph is less than one::
sage: G = digraphs.GeneralizedDeBruijn(0, 2)
Traceback (most recent call last):
...
ValueError: The generalized de Bruijn digraph is defined for at least one vertex.
REFERENCES:
.. [RPK80] S. M. Reddy, D. K. Pradhan, and J. Kuhl. Directed graphs with
minimal diameter and maximal connectivity, School Eng., Oakland Univ.,
Rochester MI, Tech. Rep., July 1980.
.. [RPK83] S. Reddy, P. Raghavan, and J. Kuhl. A Class of Graphs for
Processor Interconnection. *IEEE International Conference on Parallel
Processing*, pages 154-157, Los Alamitos, Ca., USA, August 1983.
"""
if n < 1:
raise ValueError("The generalized de Bruijn digraph is defined for at least one vertex.")
if d < 1:
raise ValueError("The generalized de Bruijn digraph is defined for degree at least one.")
GB = DiGraph(loops = True)
GB.allow_multiple_edges(True)
for u in xrange(n):
for a in xrange(u*d, u*d+d):
GB.add_edge(u, a%n)
GB.name( "Generalized de Bruijn digraph (n=%s, d=%s)"%(n,d) )
return GB
def DeBruijn(self, k, n, vertices = 'strings'):
r"""
Returns the De Bruijn digraph with parameters `k,n`.
The De Bruijn digraph with parameters `k,n` is built upon a set of
vertices equal to the set of words of length `n` from a dictionary of
`k` letters.
In this digraph, there is an arc `w_1w_2` if `w_2` can be obtained from
`w_1` by removing the leftmost letter and adding a new letter at its
right end. For more information, see the
:wikipedia:`Wikipedia article on De Bruijn graph <De_Bruijn_graph>`.
INPUT:
- ``k`` -- Two possibilities for this parameter :
- An integer equal to the cardinality of the alphabet to use, that
is the degree of the digraph to be produced.
- An iterable object to be used as the set of letters. The degree
of the resulting digraph is the cardinality of the set of
letters.
- ``n`` -- An integer equal to the length of words in the De Bruijn
digraph when ``vertices == 'strings'``, and also to the diameter of
the digraph.
- ``vertices`` -- 'strings' (default) or 'integers', specifying whether
the vertices are words build upon an alphabet or integers.
EXAMPLES::
sage: db=digraphs.DeBruijn(2,2); db
De Bruijn digraph (k=2, n=2): Looped digraph on 4 vertices
sage: db.order()
4
sage: db.size()
8
TESTS::
sage: digraphs.DeBruijn(5,0)
De Bruijn digraph (k=5, n=0): Looped multi-digraph on 1 vertex
sage: digraphs.DeBruijn(0,0)
De Bruijn digraph (k=0, n=0): Looped multi-digraph on 0 vertices
"""
from sage.combinat.words.words import Words
from sage.rings.integer import Integer
W = Words(range(k) if isinstance(k, Integer) else k, n)
A = Words(range(k) if isinstance(k, Integer) else k, 1)
g = DiGraph(loops=True)
if vertices == 'strings':
if n == 0 :
g.allow_multiple_edges(True)
v = W[0]
for a in A:
g.add_edge(v.string_rep(), v.string_rep(), a.string_rep())
else:
for w in W:
ww = w[1:]
for a in A:
g.add_edge(w.string_rep(), (ww*a).string_rep(), a.string_rep())
else:
d = W.size_of_alphabet()
g = digraphs.GeneralizedDeBruijn(d**n, d)
g.name( "De Bruijn digraph (k=%s, n=%s)"%(k,n) )
return g
def DeBruijn(self,k,n):
r"""
Returns the De Bruijn diraph with parameters `k,n`.
The De Bruijn digraph with parameters `k,n` is built
upon a set of vertices equal to the set of words of
length `n` from a dictionary of `k` letters.
In this digraph, there is an arc `w_1w_2` if `w_2`
can be obtained from `w_1` by removing the leftmost
letter and adding a new letter at its right end.
For more information, see the
`Wikipedia article on De Bruijn graph
<http://en.wikipedia.org/wiki/De_Bruijn_graph>`_.
INPUT:
- ``k`` -- Two possibilities for this parameter :
- an integer equal to the cardinality of the
alphabet to use.
- An iterable object to be used as the set
of letters
- ``n`` -- An integer equal to the length of words in
the De Bruijn digraph.
EXAMPLES::
sage: db=digraphs.DeBruijn(2,2); db
De Bruijn digraph (k=2, n=2): Looped digraph on 4 vertices
sage: db.order()
4
sage: db.size()
8
TESTS::
sage: digraphs.DeBruijn(5,0)
De Bruijn digraph (k=5, n=0): Looped multi-digraph on 1 vertex
sage: digraphs.DeBruijn(0,0)
De Bruijn digraph (k=0, n=0): Looped multi-digraph on 0 vertices
"""
from sage.combinat.words.words import Words
from sage.rings.integer import Integer
W = Words(range(k) if isinstance(k, Integer) else k, n)
A = Words(range(k) if isinstance(k, Integer) else k, 1)
g = DiGraph(loops=True)
if n == 0 :
g.allow_multiple_edges(True)
v = W[0]
for a in A:
g.add_edge(v.string_rep(), v.string_rep(), a.string_rep())
else:
for w in W:
ww = w[1:]
for a in A:
g.add_edge(w.string_rep(), (ww*a).string_rep(), a.string_rep())
g.name( "De Bruijn digraph (k=%s, n=%s)"%(k,n) )
return g
def reduced_rauzy_graph(self, n):
r"""
Returns the reduced Rauzy graph of order `n` of self.
INPUT:
- ``n`` - non negative integer. Every vertex of a reduced
Rauzy graph of order `n` is a factor of length `n` of self.
OUTPUT:
Looped multi-digraph
DEFINITION:
For infinite periodic words (resp. for finite words of type `u^i
u[0:j]`), the reduced Rauzy graph of order `n` (resp. for `n`
smaller or equal to `(i-1)|u|+j`) is the directed graph whose
unique vertex is the prefix `p` of length `n` of self and which has
an only edge which is a loop on `p` labelled by `w[n+1:|w|] p`
where `w` is the unique return word to `p`.
In other cases, it is the directed graph defined as followed. Let
`G_n` be the Rauzy graph of order `n` of self. The vertices are the
vertices of `G_n` that are either special or not prolongable to the
right or to the left. For each couple (`u`, `v`) of such vertices
and each directed path in `G_n` from `u` to `v` that contains no
other vertices that are special, there is an edge from `u` to `v`
in the reduced Rauzy graph of order `n` whose label is the label of
the path in `G_n`.
.. NOTE::
In the case of infinite recurrent non periodic words, this
definition correspond to the following one that can be found in
[1] and [2] where a simple path is a path that begins with a
special factor, ends with a special factor and contains no
other vertices that are special:
The reduced Rauzy graph of factors of length `n` is obtained
from `G_n` by replacing each simple path `P=v_1 v_2 ...
v_{\ell}` with an edge `v_1 v_{\ell}` whose label is the
concatenation of the labels of the edges of `P`.
EXAMPLES::
sage: w = Word(range(10)); w
word: 0123456789
sage: g = w.reduced_rauzy_graph(3); g
Looped multi-digraph on 2 vertices
sage: g.vertices()
[word: 012, word: 789]
sage: g.edges()
[(word: 012, word: 789, word: 3456789)]
For the Fibonacci word::
sage: f = words.FibonacciWord()[:100]
sage: g = f.reduced_rauzy_graph(8);g
Looped multi-digraph on 2 vertices
sage: g.vertices()
[word: 01001010, word: 01010010]
sage: g.edges()
[(word: 01001010, word: 01010010, word: 010), (word: 01010010, word: 01001010, word: 01010), (word: 01010010, word: 01001010, word: 10)]
For periodic words::
sage: from itertools import cycle
sage: w = Word(cycle('abcd'))[:100]
sage: g = w.reduced_rauzy_graph(3)
sage: g.edges()
[(word: abc, word: abc, word: dabc)]
::
sage: w = Word('111')
sage: for i in range(5) : w.reduced_rauzy_graph(i)
Looped digraph on 1 vertex
Looped digraph on 1 vertex
Looped digraph on 1 vertex
Looped multi-digraph on 1 vertex
Looped multi-digraph on 0 vertices
For ultimately periodic words::
sage: sigma = WordMorphism('a->abcd,b->cd,c->cd,d->cd')
sage: w = sigma.fixed_point('a')[:100]; w
word: abcdcdcdcdcdcdcdcdcdcdcdcdcdcdcdcdcdcdcd...
sage: g = w.reduced_rauzy_graph(5)
sage: g.vertices()
[word: abcdc, word: cdcdc]
sage: g.edges()
[(word: abcdc, word: cdcdc, word: dc), (word: cdcdc, word: cdcdc, word: dc)]
AUTHOR:
Julien Leroy (March 2010): initial version
REFERENCES:
- [1] M. Bucci et al. A. De Luca, A. Glen, L. Q. Zamboni, A
connection between palindromic and factor complexity using
return words," Advances in Applied Mathematics 42 (2009) 60-74.
- [2] L'ubomira Balkova, Edita Pelantova, and Wolfgang Steiner.
Sequences with constant number of return words. Monatsh. Math,
155 (2008) 251-263.
"""
from sage.graphs.all import DiGraph
from copy import copy
g = copy(self.rauzy_graph(n))
# Otherwise it changes the rauzy_graph function.
l = [v for v in g if g.in_degree(v)==1 and g.out_degree(v)==1]
if g.num_verts() !=0 and len(l)==g.num_verts():
# In this case, the Rauzy graph is simply a cycle.
g = DiGraph()
g.allow_loops(True)
g.add_vertex(self[:n])
g.add_edge(self[:n],self[:n],self[n:n+len(l)])
else:
g.allow_loops(True)
g.allow_multiple_edges(True)
for v in l:
[i] = g.neighbors_in(v)
[o] = g.neighbors_out(v)
g.add_edge(i,o,g.edge_label(i,v)[0]*g.edge_label(v,o)[0])
g.delete_vertex(v)
return g
