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