Python all.DiGraph类代码示例

def _matrix_to_digraph( M ):
    """
    Returns the digraph obtained from the matrix ``M``.
    In order to generate a quiver, we assume that ``M`` is skew-symmetrizable.

    EXAMPLES::

        sage: from sage.combinat.cluster_algebra_quiver.mutation_class import _matrix_to_digraph
        sage: _matrix_to_digraph(matrix(3,[0,1,0,-1,0,-1,0,1,0]))
        Digraph on 3 vertices
    """
    n = M.ncols()

    dg = DiGraph(sparse=True)
    for i,j in M.nonzero_positions():
        if i >= n: a,b = M[i,j],-M[i,j]
        else: a,b = M[i,j],M[j,i]
        if a > 0:
            dg._backend.add_edge(i,j,(a,b),True)
        elif i >= n:
            dg._backend.add_edge(j,i,(-a,-b),True)
    if dg.order() < M.nrows():
        for i in [ index for index in xrange(M.nrows()) if index not in dg ]:
            dg.add_vertex(i)
    return dg

def _dig6_to_digraph( dig6 ):
    """
    Returns the digraph obtained from the dig6 and edge data.

    INPUT:

    - ``dig6`` -- a pair ``(dig6, edges)`` where ``dig6`` is a string encoding a digraph and ``edges`` is a dict or tuple encoding edges

    EXAMPLES::

        sage: from sage.combinat.cluster_algebra_quiver.mutation_class import _digraph_to_dig6
        sage: from sage.combinat.cluster_algebra_quiver.mutation_class import _dig6_to_digraph
        sage: dg = ClusterQuiver(['A',4]).digraph()
        sage: data = _digraph_to_dig6(dg)
        sage: _dig6_to_digraph(data)
        Digraph on 4 vertices
        sage: _dig6_to_digraph(data).edges()
        [(0, 1, (1, -1)), (2, 1, (1, -1)), (2, 3, (1, -1))]
    """
    dig6, edges = dig6
    dg = DiGraph( dig6 )
    if not type(edges) == dict:
        edges = dict( edges )
    for edge in dg._backend.iterator_in_edges(dg,False):
        if edge in edges:
            dg.set_edge_label( edge[0],edge[1],edges[edge] )
        else:
            dg.set_edge_label( edge[0],edge[1], (1,-1) )
    return dg

def _digraph_mutate( dg, k, n, m ):
    """
    Returns a digraph obtained from dg with n+m vertices by mutating at vertex k.

    INPUT:

    - ``dg`` -- a digraph with integral edge labels with ``n+m`` vertices
    - ``k`` -- the vertex at which ``dg`` is mutated

    EXAMPLES::

        sage: from sage.combinat.cluster_algebra_quiver.mutation_class import _digraph_mutate
        sage: dg = ClusterQuiver(['A',4]).digraph()
        sage: dg.edges()
        [(0, 1, (1, -1)), (2, 1, (1, -1)), (2, 3, (1, -1))]
        sage: _digraph_mutate(dg,2,4,0).edges()
        [(0, 1, (1, -1)), (1, 2, (1, -1)), (3, 2, (1, -1))]
    """
    edges = dict( ((v1,v2),label) for v1,v2,label in dg._backend.iterator_in_edges(dg,True) )
    in_edges = [ (v1,v2,edges[(v1,v2)]) for (v1,v2) in edges if v2 == k ]
    out_edges = [ (v1,v2,edges[(v1,v2)]) for (v1,v2) in edges if v1 == k ]
    in_edges_new = [ (v2,v1,(-label[1],-label[0])) for (v1,v2,label) in in_edges ]
    out_edges_new = [ (v2,v1,(-label[1],-label[0])) for (v1,v2,label) in out_edges ]
    diag_edges_new = []
    diag_edges_del = []

    for (v1,v2,label1) in in_edges:
        for (w1,w2,label2) in out_edges:
            l11,l12 = label1
            l21,l22 = label2
            if (v1,w2) in edges:
                diag_edges_del.append( (v1,w2,edges[(v1,w2)]) )
                a,b = edges[(v1,w2)]
                a,b = a+l11*l21, b-l12*l22
                diag_edges_new.append( (v1,w2,(a,b)) )
            elif (w2,v1) in edges:
                diag_edges_del.append( (w2,v1,edges[(w2,v1)]) )
                a,b = edges[(w2,v1)]
                a,b = b+l11*l21, a-l12*l22
                if a<0:
                    diag_edges_new.append( (w2,v1,(b,a)) )
                elif a>0:
                    diag_edges_new.append( (v1,w2,(a,b)) )
            else:
                a,b = l11*l21,-l12*l22
                diag_edges_new.append( (v1,w2,(a,b)) )

    del_edges = in_edges + out_edges + diag_edges_del
    new_edges = in_edges_new + out_edges_new + diag_edges_new
    new_edges += [ (v1,v2,edges[(v1,v2)]) for (v1,v2) in edges if not (v1,v2,edges[(v1,v2)]) in del_edges ]

    dg_new = DiGraph()
    for v1,v2,label in new_edges:
        dg_new._backend.add_edge(v1,v2,label,True)
    if dg_new.order() < n+m:
        dg_new_vertices = [ v for v in dg_new ]
        for i in [ v for v in dg if v not in dg_new_vertices ]:
            dg_new.add_vertex(i)

    return dg_new

    def plot(self, label_elements=True, element_labels=None,
            label_font_size=12,label_font_color='black', layout = "acyclic", **kwds):
        """
        Returns a Graphics object corresponding to the Hasse diagram.
        
        EXAMPLES::
        
            sage: uc = [[2,3], [], [1], [1], [1], [3,4]]
            sage: elm_lbls = Permutations(3).list()
            sage: P = Poset(uc,elm_lbls)
            sage: H = P._hasse_diagram
            sage: levels = H.level_sets()
            sage: heights = dict([[i, levels[i]] for i in range(len(levels))])
            sage: type(H.plot(label_elements=True))
            <class 'sage.plot.graphics.Graphics'>
        
        ::
        
            sage: P = Posets.SymmetricGroupBruhatIntervalPoset([0,1,2,3], [2,3,0,1])
            sage: P._hasse_diagram.plot()
        """
        # Set element_labels to default to the vertex set.
        if element_labels is None:
            element_labels = range(self.num_verts())

        # Create the underlying graph.
        graph = DiGraph(self)
        graph.relabel(element_labels)

        return graph.plot(layout = layout, **kwds)

 def add_edge(self, i, j, label=1):
     """
     EXAMPLES::
     
         sage: from sage.combinat.root_system.dynkin_diagram import DynkinDiagram_class
         sage: d = DynkinDiagram_class(CartanType(['A',3]))
         sage: list(sorted(d.edges()))
         []
         sage: d.add_edge(2, 3)
         sage: list(sorted(d.edges()))
         [(2, 3, 1), (3, 2, 1)]
     """
     DiGraph.add_edge(self, i, j, label)
     if not self.has_edge(j, i):
         self.add_edge(j, i, 1)

 def __getitem__(self, i):
     r"""
     With a tuple (i,j) as argument, returns the scalar product
     `\langle
             \alpha^\vee_i, \alpha_j\rangle`.
     
     Otherwise, behaves as the usual DiGraph.__getitem__
     
     EXAMPLES: We use the `C_4` dynkin diagram as a cartan
     matrix::
     
         sage: g = DynkinDiagram(['C',4])
         sage: matrix([[g[i,j] for j in range(1,5)] for i in range(1,5)])
         [ 2 -1  0  0]
         [-1  2 -1  0]
         [ 0 -1  2 -2]
         [ 0  0 -1  2]
     
     The neighbors of a node can still be obtained in the usual way::
     
         sage: [g[i] for i in range(1,5)]
         [[2], [1, 3], [2, 4], [3]]
     """
     if not isinstance(i, tuple):
         return DiGraph.__getitem__(self, i)
     [i, j] = i
     if i == j:
         return 2
     elif self.has_edge(j, i):
         return -self.edge_label(j, i)
     else:
         return 0

        def digraph(self):
            """
            Returns the DiGraph associated to self.

            EXAMPLES::

                sage: C = Crystals().example(5)
                sage: C.digraph()
                Digraph on 6 vertices

            The edges of the crystal graph are by default colored using blue for edge 1, red for edge 2,
            and green for edge 3::

                sage: C = Crystals().example(3)
                sage: G = C.digraph()
                sage: view(G, pdflatex=True, tightpage=True)  #optional - dot2tex graphviz

            One may also overwrite the colors::

                sage: C = Crystals().example(3)
                sage: G = C.digraph()
                sage: G.set_latex_options(color_by_label = {1:"red", 2:"purple", 3:"blue"})
                sage: view(G, pdflatex=True, tightpage=True)  #optional - dot2tex graphviz

            Or one may add colors to yet unspecified edges::

                sage: C = Crystals().example(4)
                sage: G = C.digraph()
                sage: C.cartan_type()._index_set_coloring[4]="purple"
                sage: view(G, pdflatex=True, tightpage=True)  #optional - dot2tex graphviz

            TODO: add more tests
            """
            from sage.graphs.all import DiGraph
            d = {}
            for x in self:
                d[x] = {}
                for i in self.index_set():
                    child = x.f(i)
                    if child is None:
                        continue
                    d[x][child]=i
            G = DiGraph(d)
            if have_dot2tex():
                G.set_latex_options(format="dot2tex", edge_labels = True, color_by_label = self.cartan_type()._index_set_coloring,
                                    edge_options = lambda (u,v,label): ({"backward":label ==0}))
            return G

    def __init__(self, t=None):
        """
        INPUT:

         - ``t`` - a Cartan type or None

        EXAMPLES::
        
            sage: d = DynkinDiagram(["A", 3])
            sage: d == loads(dumps(d))
            True

        Implementation note: if a Cartan type is given, then the nodes
        are initialized from the index set of this Cartan type.
        """
        DiGraph.__init__(self)
        self._cartan_type = t
        if t is not None:
            self.add_vertices(t.index_set())

    def IntegerPartitions(n):
        """
        Returns the poset of integer partitions on the integer ``n``.

        A partition of a positive integer `n` is a non-increasing list
        of positive integers that sum to `n`. If `p` and `q` are
        integer partitions of `n`, then `p` covers `q` if and only
        if `q` is obtained from `p` by joining two parts of `p`
        (and sorting, if necessary).

        EXAMPLES::
        
            sage: P = Posets.IntegerPartitions(7); P
            Finite poset containing 15 elements
            sage: len(P.cover_relations())
            28
        """
        def lower_covers(partition):
            r"""
            Nested function for computing the lower covers
            of elements in the poset of integer partitions.
            """
            lc = []
            for i in range(0,len(partition)-1):
                for j in range(i+1,len(partition)):
                    new_partition = partition[:]
                    del new_partition[j]
                    del new_partition[i]
                    new_partition.append(partition[i]+partition[j])
                    new_partition.sort(reverse=True)
                    tup = tuple(new_partition)
                    if tup not in lc: 
                        lc.append(tup)
            return lc
        from sage.combinat.partition import partitions_list
        H = DiGraph(dict([[tuple(p),lower_covers(p)] for p in
            partitions_list(n)]))
        return Poset(H.reverse())

    def __init__(self):
        """
        EXAMPLES::

            sage: C = sage.categories.examples.crystals.NaiveCrystal()
            sage: C == Crystals().example(choice='naive')
            True
        """
        Parent.__init__(self, category = ClassicalCrystals())
        self.n = 2
        self._cartan_type = CartanType(['A',2])
        self.G = DiGraph(5)
        self.G.add_edges([ [0,1,1], [1,2,1], [2,3,1], [3,5,1],  [0,4,2], [4,5,2] ])
        self.module_generators = [ self(0) ]

    def _digraph(self):
        r"""
        Constructs the underlying digraph and stores the result as an
        attribute.

        EXAMPLES::
        
            sage: from sage.combinat.yang_baxter_graph import SwapIncreasingOperator
            sage: ops = [SwapIncreasingOperator(i) for i in range(2)]
            sage: Y = YangBaxterGraph(root=(1,2,3), operators=ops)
            sage: Y._digraph
            Digraph on 6 vertices
        """
        digraph = DiGraph()
        digraph.add_vertex(self._root)
        queue = [self._root]
        while queue:
            u = queue.pop()
            for (v, l) in self._succesors(u):
                if v not in digraph:
                    queue.append(v)
                digraph.add_edge(u, v, l)
        return digraph

    def RestrictedIntegerPartitions(n):
        """
        Returns the poset of integer partitions on the integer `n`
        ordered by restricted refinement. That is, if `p` and `q`
        are integer partitions of `n`, then `p` covers `q` if and
        only if `q` is obtained from `p` by joining two distinct
        parts of `p` (and sorting, if necessary).

        EXAMPLES::
        
            sage: P = Posets.RestrictedIntegerPartitions(7); P
            Finite poset containing 15 elements
            sage: len(P.cover_relations())
            17
        """
        def lower_covers(partition):
            r"""
            Nested function for computing the lower covers of elements in the
            restricted poset of integer partitions.
            """
            lc = []
            for i in range(0,len(partition)-1):
                for j in range(i+1,len(partition)):
                    if partition[i] != partition[j]:
                        new_partition = partition[:]
                        del new_partition[j]
                        del new_partition[i]
                        new_partition.append(partition[i]+partition[j])
                        new_partition.sort(reverse=True)
                        tup = tuple(new_partition)
                        if tup not in lc: 
                            lc.append(tup)
            return lc
        from sage.combinat.partition import Partitions
        H = DiGraph(dict([[tuple(p),lower_covers(p)] for p in
            Partitions(n)]))
        return Poset(H.reverse())

    def to_dag(self):
        """
        Returns a directed acyclic graph corresponding to the skew
        partition.
        
        EXAMPLES::
        
            sage: dag = SkewPartition([[3, 2, 1], [1, 1]]).to_dag()
            sage: dag.edges()
            [('0,1', '0,2', None), ('0,1', '1,1', None)]
            sage: dag.vertices()
            ['0,1', '0,2', '1,1', '2,0']
        """
        i = 0

        #Make the skew tableau from the shape
        skew = [[1]*row_length for row_length in self.outer()]
        inner = self.inner()
        for i in range(len(inner)):
            for j in range(inner[i]):
                skew[i][j] = None

        G = DiGraph()
        for row in range(len(skew)):
            for column in range(len(skew[row])):
                if skew[row][column] is not None:
                    string = "%d,%d" % (row, column)
                    G.add_vertex(string)
                    #Check to see if there is a node to the right
                    if column != len(skew[row]) - 1:
                        newstring = "%d,%d" % (row, column+1)
                        G.add_edge(string, newstring)

                    #Check to see if there is anything below
                    if row != len(skew) - 1:
                        if len(skew[row+1]) > column:
                            if skew[row+1][column] is not None:
                                newstring = "%d,%d" % (row+1, column)
                                G.add_edge(string, newstring)
        return G

def _is_valid_digraph_edge_set( edges, frozen=0 ):
    """
    Returns True if the input data is the edge set of a digraph for a quiver (no loops, no 2-cycles, edge-labels of the specified format), and returns False otherwise.

    INPUT:

    - ``frozen`` -- (integer; default:0) The number of frozen vertices.

    EXAMPLES::

        sage: from sage.combinat.cluster_algebra_quiver.mutation_class import _is_valid_digraph_edge_set
        sage: _is_valid_digraph_edge_set( [[0,1,'a'],[2,3,(1,-1)]] )
        The given digraph has edge labels which are not integral or integral 2-tuples.
        False
        sage: _is_valid_digraph_edge_set( [[0,1],[2,3,(1,-1)]] )
        True
        sage: _is_valid_digraph_edge_set( [[0,1,'a'],[2,3,(1,-1)],[3,2,(1,-1)]] )
        The given digraph or edge list contains oriented 2-cycles.
        False
    """
    try:
        dg = DiGraph()
        dg.allow_multiple_edges(True)
        dg.add_edges( edges )

        # checks if the digraph contains loops
        if dg.has_loops():
            print "The given digraph or edge list contains loops."
            return False

        # checks if the digraph contains oriented 2-cycles
        if _has_two_cycles( dg ):
            print "The given digraph or edge list contains oriented 2-cycles."
            return False

        # checks if all edge labels are 'None', positive integers or tuples of positive integers
        if not all( i == None or ( i in ZZ and i > 0 ) or ( type(i) == tuple and len(i) == 2 and i[0] in ZZ and i[1] in ZZ ) for i in dg.edge_labels() ):
            print "The given digraph has edge labels which are not integral or integral 2-tuples."
            return False

        # checks if all edge labels for multiple edges are 'None' or positive integers
        if dg.has_multiple_edges():
            for e in set( dg.multiple_edges(labels=False) ):
                if not all( i == None or ( i in ZZ and i > 0 ) for i in dg.edge_label( e[0], e[1] ) ):
                    print "The given digraph or edge list contains multiple edges with non-integral labels."
                    return False

        n = dg.order() - frozen
        if n < 0:
            print "The number of frozen variables is larger than the number of vertices."
            return False

        if [ e for e in dg.edges(labels=False) if e[0] >= n] <> []:
            print "The given digraph or edge list contains edges within the frozen vertices."
            return False

        return True
    except StandardError:
        print "Could not even build a digraph from the input data."
        return False

    def RandomPoset(n,p):
        r"""
        Generate a random poset on ``n`` vertices according to a
        probability ``p``.

        INPUT:

        - ``n`` - number of vertices, a non-negative integer

        - ``p`` - a probability, a real number between 0 and 1 (inclusive)

        OUTPUT:

        A poset on ``n`` vertices.  The construction decides to make an
        ordered pair of vertices comparable in the poset with probability
        ``p``, however a pair is not made comparable if it would violate
        the defining properties of a poset, such as transitivity.

        So in practice, once the probability exceeds a small number the
        generated posets may be very similar to a chain.  So to create
        interesting examples, keep the probability small, perhaps on the
        order of `1/n`.

        EXAMPLES::

            sage: Posets.RandomPoset(17,.15)
            Finite poset containing 17 elements

        TESTS::

            sage: Posets.RandomPoset('junk', 0.5)
            Traceback (most recent call last):
            ...
            TypeError: number of elements must be an integer, not junk

            sage: Posets.RandomPoset(-6, 0.5)
            Traceback (most recent call last):
            ...
            ValueError: number of elements must be non-negative, not -6

            sage: Posets.RandomPoset(6, 'garbage')
            Traceback (most recent call last):
            ...
            TypeError: probability must be a real number, not garbage

            sage: Posets.RandomPoset(6, -0.5)
            Traceback (most recent call last):
            ...
            ValueError: probability must be between 0 and 1, not -0.5
        """
        try:
            n = Integer(n)
        except:
            raise TypeError("number of elements must be an integer, not {0}".format(n))
        if n < 0:
            raise ValueError("number of elements must be non-negative, not {0}".format(n))
        try:
            p = float(p)
        except:
            raise TypeError("probability must be a real number, not {0}".format(p))
        if p < 0 or p> 1:
            raise ValueError("probability must be between 0 and 1, not {0}".format(p))
        
        D = DiGraph(loops=False,multiedges=False)
        D.add_vertices(range(n))
        for i in range(n):
            for j in range(n):
                if random.random() < p:
                    D.add_edge(i,j)
                    if not D.is_directed_acyclic():
                        D.delete_edge(i,j)
        return Poset(D,cover_relations=False)