Python flow.build_residual_network函数代码示例

    def test_gl1(self):
        G = read_graph("gl1")
        s = 1
        t = len(G)
        R = build_residual_network(G, "capacity")
        kwargs = dict(residual=R)

        for flow_func in flow_funcs:
            validate_flows(G, s, t, 156545, flow_func(G, s, t, **kwargs), flow_func)

    def test_wlm3(self):
        G = read_graph("wlm3")
        s = 1
        t = len(G)
        R = build_residual_network(G, "capacity")
        kwargs = dict(residual=R)

        for flow_func in flow_funcs:
            validate_flows(G, s, t, 11875108, flow_func(G, s, t, **kwargs), flow_func)

    def test_gw1(self):
        G = read_graph('gw1')
        s = 1
        t = len(G)
        R = build_residual_network(G, 'capacity')
        kwargs = dict(residual=R)

        for flow_func in flow_funcs:
            validate_flows(G, s, t, 1202018, flow_func(G, s, t, **kwargs),
                           flow_func)

    def test_pyramid(self):
        N = 10
        # N = 100 # this gives a graph with 5051 nodes
        G = gen_pyramid(N)
        R = build_residual_network(G, "capacity")
        kwargs = dict(residual=R)

        for flow_func in flow_funcs:
            kwargs["flow_func"] = flow_func
            flow_value = nx.maximum_flow_value(G, (0, 0), "t", **kwargs)
            assert_almost_equal(flow_value, 1.0, msg=msg.format(flow_func.__name__))

    def test_complete_graph(self):
        N = 50
        G = nx.complete_graph(N)
        nx.set_edge_attributes(G, "capacity", 5)
        R = build_residual_network(G, "capacity")
        kwargs = dict(residual=R)

        for flow_func in flow_funcs:
            kwargs["flow_func"] = flow_func
            flow_value = nx.maximum_flow_value(G, 1, 2, **kwargs)
            assert_equal(flow_value, 5 * (N - 1), msg=msg.format(flow_func.__name__))

    def test_wlm3(self):
        G = read_graph('wlm3')
        s = 1
        t = len(G)
        R = build_residual_network(G, 'capacity')
        kwargs = dict(residual=R)

        # do one flow_func to save time
        flow_func = flow_funcs[0]
        validate_flows(G, s, t, 11875108, flow_func(G, s, t, **kwargs),
                           flow_func)

 def test_reusing_residual(self):
     G = self.G
     fv = 3.0
     s, t = 'x', 'y'
     R = build_residual_network(G, 'capacity')
     for interface_func in interface_funcs:
         for flow_func in flow_funcs:
             for i in range(3):
                 result = interface_func(G, 'x', 'y', flow_func=flow_func,
                                         residual=R)
                 if interface_func in max_min_funcs:
                     result = result[0]
                 assert_equal(fv, result,
                              msg=msgi.format(flow_func.__name__,
                                              interface_func.__name__))

#.........这里部分代码省略.........
    >>> nx.edge_connectivity(G, flow_func=shortest_augmenting_path)
    5

    If you specify a pair of nodes (source and target) as parameters,
    this function returns the value of local edge connectivity.

    >>> nx.edge_connectivity(G, 3, 7)
    5

    If you need to perform several local computations among different
    pairs of nodes on the same graph, it is recommended that you reuse
    the data structures used in the maximum flow computations. See
    :meth:`local_edge_connectivity` for details.

    Notes
    -----
    This is a flow based implementation of global edge connectivity.
    For undirected graphs the algorithm works by finding a 'small'
    dominating set of nodes of G (see algorithm 7 in [1]_ ) and
    computing local maximum flow (see :meth:`local_edge_connectivity`)
    between an arbitrary node in the dominating set and the rest of
    nodes in it. This is an implementation of algorithm 6 in [1]_ .
    For directed graphs, the algorithm does n calls to the maximum
    flow function. This is an implementation of algorithm 8 in [1]_ .

    See also
    --------
    :meth:`local_edge_connectivity`
    :meth:`local_node_connectivity`
    :meth:`node_connectivity`
    :meth:`maximum_flow`
    :meth:`edmonds_karp`
    :meth:`preflow_push`
    :meth:`shortest_augmenting_path`

    References
    ----------
    .. [1] Abdol-Hossein Esfahanian. Connectivity Algorithms.
        http://www.cse.msu.edu/~cse835/Papers/Graph_connectivity_revised.pdf

    """
    if (s is not None and t is None) or (s is None and t is not None):
        raise nx.NetworkXError('Both source and target must be specified.')

    # Local edge connectivity
    if s is not None and t is not None:
        if s not in G:
            raise nx.NetworkXError('node %s not in graph' % s)
        if t not in G:
            raise nx.NetworkXError('node %s not in graph' % t)
        return local_edge_connectivity(G, s, t, flow_func=flow_func)

    # Global edge connectivity
    # reuse auxiliary digraph and residual network
    H = build_auxiliary_edge_connectivity(G)
    R = build_residual_network(H, 'capacity')
    kwargs = dict(flow_func=flow_func, auxiliary=H, residual=R)

    if G.is_directed():
        # Algorithm 8 in [1]
        if not nx.is_weakly_connected(G):
            return 0

        # initial value for \lambda is minimum degree
        L = min(G.degree().values())
        nodes = G.nodes()
        n = len(nodes)
        for i in range(n):
            kwargs['cutoff'] = L
            try:
                L = min(L, local_edge_connectivity(G, nodes[i], nodes[i+1],
                                                   **kwargs))
            except IndexError: # last node!
                L = min(L, local_edge_connectivity(G, nodes[i], nodes[0],
                                                   **kwargs))
        return L
    else: # undirected
        # Algorithm 6 in [1]
        if not nx.is_connected(G):
            return 0

        # initial value for \lambda is minimum degree
        L = min(G.degree().values())
        # A dominating set is \lambda-covering
        # We need a dominating set with at least two nodes
        for node in G:
            D = nx.dominating_set(G, start_with=node)
            v = D.pop()
            if D:
                break
        else:
            # in complete graphs the dominating sets will always be of one node
            # thus we return min degree
            return L

        for w in D:
            kwargs['cutoff'] = L
            L = min(L, local_edge_connectivity(G, v, w, **kwargs))

        return L

def all_pairs_node_connectivity(G, nbunch=None, flow_func=None):
    """Compute node connectivity between all pairs of nodes of G.

    Parameters
    ----------
    G : NetworkX graph
        Undirected graph

    nbunch: container
        Container of nodes. If provided node connectivity will be computed
        only over pairs of nodes in nbunch.

    flow_func : function
        A function for computing the maximum flow among a pair of nodes.
        The function has to accept at least three parameters: a Digraph,
        a source node, and a target node. And return a residual network
        that follows NetworkX conventions (see :meth:`maximum_flow` for
        details). If flow_func is None, the default maximum flow function
        (:meth:`edmonds_karp`) is used. See below for details. The
        choice of the default function may change from version
        to version and should not be relied on. Default value: None.

    Returns
    -------
    all_pairs : dict
        A dictionary with node connectivity between all pairs of nodes
        in G, or in nbunch if provided.

    See also
    --------
    :meth:`local_node_connectivity`
    :meth:`edge_connectivity`
    :meth:`local_edge_connectivity`
    :meth:`maximum_flow`
    :meth:`edmonds_karp`
    :meth:`preflow_push`
    :meth:`shortest_augmenting_path`

    """
    if nbunch is None:
        nbunch = G
    else:
        nbunch = set(nbunch)

    directed = G.is_directed()
    if directed:
        iter_func = itertools.permutations
    else:
        iter_func = itertools.combinations

    all_pairs = {n: {} for n in nbunch}

    # Reuse auxiliary digraph and residual network
    H = build_auxiliary_node_connectivity(G)
    mapping = H.graph['mapping']
    R = build_residual_network(H, 'capacity')
    kwargs = dict(flow_func=flow_func, auxiliary=H, residual=R)

    for u, v in iter_func(nbunch, 2):
        K = local_node_connectivity(G, u, v, **kwargs)
        all_pairs[u][v] = K
        if not directed:
            all_pairs[v][u] = K

    return all_pairs

def average_node_connectivity(G, flow_func=None):
    r"""Returns the average connectivity of a graph G.

    The average connectivity `\bar{\kappa}` of a graph G is the average
    of local node connectivity over all pairs of nodes of G [1]_ .

    .. math::

        \bar{\kappa}(G) = \frac{\sum_{u,v} \kappa_{G}(u,v)}{{n \choose 2}}

    Parameters
    ----------

    G : NetworkX graph
        Undirected graph

    flow_func : function
        A function for computing the maximum flow among a pair of nodes.
        The function has to accept at least three parameters: a Digraph,
        a source node, and a target node. And return a residual network
        that follows NetworkX conventions (see :meth:`maximum_flow` for
        details). If flow_func is None, the default maximum flow function
        (:meth:`edmonds_karp`) is used. See :meth:`local_node_connectivity`
        for details. The choice of the default function may change from
        version to version and should not be relied on. Default value: None.

    Returns
    -------
    K : float
        Average node connectivity

    See also
    --------
    :meth:`local_node_connectivity`
    :meth:`node_connectivity`
    :meth:`edge_connectivity`
    :meth:`maximum_flow`
    :meth:`edmonds_karp`
    :meth:`preflow_push`
    :meth:`shortest_augmenting_path`

    References
    ----------
    .. [1]  Beineke, L., O. Oellermann, and R. Pippert (2002). The average
            connectivity of a graph. Discrete mathematics 252(1-3), 31-45.
            http://www.sciencedirect.com/science/article/pii/S0012365X01001807

    """
    if G.is_directed():
        iter_func = itertools.permutations
    else:
        iter_func = itertools.combinations

    # Reuse the auxiliary digraph and the residual network
    H = build_auxiliary_node_connectivity(G)
    R = build_residual_network(H, 'capacity')
    kwargs = dict(flow_func=flow_func, auxiliary=H, residual=R)

    num, den = 0, 0
    for u, v in iter_func(G, 2):
        num += local_node_connectivity(G, u, v, **kwargs)
        den += 1

    if den == 0: # Null Graph
        return 0
    return num / den

#.........这里部分代码省略.........
    >>> nx.node_connectivity(G)
    5

    You can use alternative flow algorithms for the underlying maximum
    flow computation. In dense networks the algorithm
    :meth:`shortest_augmenting_path` will usually perform better
    than the default :meth:`edmonds_karp`, which is faster for
    sparse networks with highly skewed degree distributions. Alternative
    flow functions have to be explicitly imported from the flow package.

    >>> from networkx.algorithms.flow import shortest_augmenting_path
    >>> nx.node_connectivity(G, flow_func=shortest_augmenting_path)
    5

    If you specify a pair of nodes (source and target) as parameters,
    this function returns the value of local node connectivity.

    >>> nx.node_connectivity(G, 3, 7)
    5

    If you need to perform several local computations among different
    pairs of nodes on the same graph, it is recommended that you reuse
    the data structures used in the maximum flow computations. See
    :meth:`local_node_connectivity` for details.

    Notes
    -----
    This is a flow based implementation of node connectivity. The
    algorithm works by solving `O((n-\delta-1+\delta(\delta-1)/2))`
    maximum flow problems on an auxiliary digraph. Where `\delta`
    is the minimum degree of G. For details about the auxiliary
    digraph and the computation of local node connectivity see
    :meth:`local_node_connectivity`. This implementation is based
    on algorithm 11 in [1]_.

    See also
    --------
    :meth:`local_node_connectivity`
    :meth:`edge_connectivity`
    :meth:`maximum_flow`
    :meth:`edmonds_karp`
    :meth:`preflow_push`
    :meth:`shortest_augmenting_path`

    References
    ----------
    .. [1] Abdol-Hossein Esfahanian. Connectivity Algorithms.
        http://www.cse.msu.edu/~cse835/Papers/Graph_connectivity_revised.pdf

    """
    if (s is not None and t is None) or (s is None and t is not None):
        raise nx.NetworkXError('Both source and target must be specified.')

    # Local node connectivity
    if s is not None and t is not None:
        if s not in G:
            raise nx.NetworkXError('node %s not in graph' % s)
        if t not in G:
            raise nx.NetworkXError('node %s not in graph' % t)
        return local_node_connectivity(G, s, t, flow_func=flow_func)

    # Global node connectivity
    if G.is_directed():
        if not nx.is_weakly_connected(G):
            return 0
        iter_func = itertools.permutations
        # It is necessary to consider both predecessors
        # and successors for directed graphs
        def neighbors(v):
            return itertools.chain.from_iterable([G.predecessors_iter(v),
                                                  G.successors_iter(v)])
    else:
        if not nx.is_connected(G):
            return 0
        iter_func = itertools.combinations
        neighbors = G.neighbors_iter

    # Reuse the auxiliary digraph and the residual network
    H = build_auxiliary_node_connectivity(G)
    R = build_residual_network(H, 'capacity')
    kwargs = dict(flow_func=flow_func, auxiliary=H, residual=R)

    # Pick a node with minimum degree
    degree = G.degree()
    minimum_degree = min(degree.values())
    v = next(n for n, d in degree.items() if d == minimum_degree)
    # Node connectivity is bounded by degree.
    K = minimum_degree
    # compute local node connectivity with all its non-neighbors nodes
    for w in set(G) - set(neighbors(v)) - set([v]):
        kwargs['cutoff'] = K
        K = min(K, local_node_connectivity(G, v, w, **kwargs))
    # Also for non adjacent pairs of neighbors of v
    for x, y in iter_func(neighbors(v), 2):
        if y in G[x]:
            continue
        kwargs['cutoff'] = K
        K = min(K, local_node_connectivity(G, x, y, **kwargs))

    return K

def all_node_cuts(G, k=None, flow_func=None):
    r"""Returns all minimum k cutsets of an undirected graph G. 

    This implementation is based on Kanevsky's algorithm [1]_ for finding all
    minimum-size node cut-sets of an undirected graph G; ie the set (or sets) 
    of nodes of cardinality equal to the node connectivity of G. Thus if 
    removed, would break G into two or more connected components.

    Parameters
    ----------
    G : NetworkX graph
        Undirected graph

    k : Integer
        Node connectivity of the input graph. If k is None, then it is 
        computed. Default value: None.

    flow_func : function
        Function to perform the underlying flow computations. Default value
        edmonds_karp. This function performs better in sparse graphs with
        right tailed degree distributions. shortest_augmenting_path will
        perform better in denser graphs.


    Returns
    -------
    cuts : a generator of node cutsets
        Each node cutset has cardinality equal to the node connectivity of
        the input graph.

    Examples
    --------
    >>> # A two-dimensional grid graph has 4 cutsets of cardinality 2
    >>> G = nx.grid_2d_graph(5, 5)
    >>> cutsets = list(nx.all_node_cuts(G))
    >>> len(cutsets)
    4
    >>> all(2 == len(cutset) for cutset in cutsets)
    True
    >>> nx.node_connectivity(G)
    2

    Notes
    -----
    This implementation is based on the sequential algorithm for finding all
    minimum-size separating vertex sets in a graph [1]_. The main idea is to
    compute minimum cuts using local maximum flow computations among a set 
    of nodes of highest degree and all other non-adjacent nodes in the Graph.
    Once we find a minimum cut, we add an edge between the high degree
    node and the target node of the local maximum flow computation to make 
    sure that we will not find that minimum cut again.

    See also
    --------
    node_connectivity
    edmonds_karp
    shortest_augmenting_path

    References
    ----------
    .. [1]  Kanevsky, A. (1993). Finding all minimum-size separating vertex 
            sets in a graph. Networks 23(6), 533--541.
            http://onlinelibrary.wiley.com/doi/10.1002/net.3230230604/abstract

    """
    if not nx.is_connected(G):
        raise nx.NetworkXError('Input graph is disconnected.')

    # Address some corner cases first.
    # For cycle graphs
    if G.order() == G.size():
        if all(2 == d for n, d in G.degree()):
            seen = set()
            for u in G:
                for v in nx.non_neighbors(G, u):
                    if (u, v) not in seen and (v, u) not in seen:
                        yield {v, u}
                        seen.add((v, u))
            return
    # For complete Graphs
    if nx.density(G) == 1:
        for cut_set in combinations(G, len(G) - 1):
            yield set(cut_set)
        return
    # Initialize data structures.
    # Keep track of the cuts already computed so we do not repeat them.
    seen = []
    # Even-Tarjan reduction is what we call auxiliary digraph
    # for node connectivity.
    H = build_auxiliary_node_connectivity(G)
    mapping = H.graph['mapping']
    R = build_residual_network(H, 'capacity')
    kwargs = dict(capacity='capacity', residual=R)
    # Define default flow function
    if flow_func is None:
        flow_func = default_flow_func
    if flow_func is shortest_augmenting_path:
        kwargs['two_phase'] = True
    # Begin the actual algorithm
    # step 1: Find node connectivity k of G
#.........这里部分代码省略.........

#.........这里部分代码省略.........
    pairs of nodes on the same graph, it is recommended that you reuse
    the data structures used in the maximum flow computations. See 
    :meth:`local_edge_connectivity` for details.

    Notes
    -----
    This is a flow based implementation of minimum edge cut. For
    undirected graphs the algorithm works by finding a 'small' dominating
    set of nodes of G (see algorithm 7 in [1]_) and computing the maximum
    flow between an arbitrary node in the dominating set and the rest of
    nodes in it. This is an implementation of algorithm 6 in [1]_. For 
    directed graphs, the algorithm does n calls to the max flow function.
    The function raises an error if the directed graph is not weakly
    connected and returns an empty set if it is weakly connected.
    It is an implementation of algorithm 8 in [1]_.

    See also
    --------
    :meth:`minimum_st_edge_cut`
    :meth:`minimum_node_cut`
    :meth:`stoer_wagner`
    :meth:`node_connectivity`
    :meth:`edge_connectivity`
    :meth:`maximum_flow`
    :meth:`edmonds_karp`
    :meth:`preflow_push`
    :meth:`shortest_augmenting_path`

    References
    ----------
    .. [1] Abdol-Hossein Esfahanian. Connectivity Algorithms.
        http://www.cse.msu.edu/~cse835/Papers/Graph_connectivity_revised.pdf

    """
    if (s is not None and t is None) or (s is None and t is not None):
        raise nx.NetworkXError("Both source and target must be specified.")

    # reuse auxiliary digraph and residual network
    H = build_auxiliary_edge_connectivity(G)
    R = build_residual_network(H, "capacity")
    kwargs = dict(flow_func=flow_func, residual=R, auxiliary=H)

    # Local minimum edge cut if s and t are not None
    if s is not None and t is not None:
        if s not in G:
            raise nx.NetworkXError("node %s not in graph" % s)
        if t not in G:
            raise nx.NetworkXError("node %s not in graph" % t)
        return minimum_st_edge_cut(H, s, t, **kwargs)

    # Global minimum edge cut
    # Analog to the algoritm for global edge connectivity
    if G.is_directed():
        # Based on algorithm 8 in [1]
        if not nx.is_weakly_connected(G):
            raise nx.NetworkXError("Input graph is not connected")

        # Initial cutset is all edges of a node with minimum degree
        node = min(G, key=G.degree)
        min_cut = set(G.edges(node))
        nodes = list(G)
        n = len(nodes)
        for i in range(n):
            try:
                this_cut = minimum_st_edge_cut(H, nodes[i], nodes[i + 1], **kwargs)
                if len(this_cut) <= len(min_cut):
                    min_cut = this_cut
            except IndexError:  # Last node!
                this_cut = minimum_st_edge_cut(H, nodes[i], nodes[0], **kwargs)
                if len(this_cut) <= len(min_cut):
                    min_cut = this_cut

        return min_cut

    else:  # undirected
        # Based on algorithm 6 in [1]
        if not nx.is_connected(G):
            raise nx.NetworkXError("Input graph is not connected")

        # Initial cutset is all edges of a node with minimum degree
        node = min(G, key=G.degree)
        min_cut = set(G.edges(node))
        # A dominating set is \lambda-covering
        # We need a dominating set with at least two nodes
        for node in G:
            D = nx.dominating_set(G, start_with=node)
            v = D.pop()
            if D:
                break
        else:
            # in complete graphs the dominating set will always be of one node
            # thus we return min_cut, which now contains the edges of a node
            # with minimum degree
            return min_cut
        for w in D:
            this_cut = minimum_st_edge_cut(H, v, w, **kwargs)
            if len(this_cut) <= len(min_cut):
                min_cut = this_cut

        return min_cut

#.........这里部分代码省略.........
    You can use alternative flow algorithms for the underlying maximum
    flow computation. In dense networks the algorithm
    :meth:`shortest_augmenting_path` will usually perform better
    than the default :meth:`edmonds_karp`, which is faster for
    sparse networks with highly skewed degree distributions. Alternative
    flow functions have to be explicitly imported from the flow package.

    >>> from networkx.algorithms.flow import shortest_augmenting_path
    >>> node_cut == nx.minimum_node_cut(G, flow_func=shortest_augmenting_path)
    True

    If you specify a pair of nodes (source and target) as parameters,
    this function returns a local st node cut.

    >>> len(nx.minimum_node_cut(G, 3, 7))
    5

    If you need to perform several local st cuts among different
    pairs of nodes on the same graph, it is recommended that you reuse
    the data structures used in the maximum flow computations. See 
    :meth:`minimum_st_node_cut` for details.

    Notes
    -----
    This is a flow based implementation of minimum node cut. The algorithm
    is based in solving a number of maximum flow computations to determine
    the capacity of the minimum cut on an auxiliary directed network that
    corresponds to the minimum node cut of G. It handles both directed
    and undirected graphs. This implementation is based on algorithm 11 
    in [1]_.

    See also
    --------
    :meth:`minimum_st_node_cut`
    :meth:`minimum_cut`
    :meth:`minimum_edge_cut`
    :meth:`stoer_wagner`
    :meth:`node_connectivity`
    :meth:`edge_connectivity`
    :meth:`maximum_flow`
    :meth:`edmonds_karp`
    :meth:`preflow_push`
    :meth:`shortest_augmenting_path`

    References
    ----------
    .. [1] Abdol-Hossein Esfahanian. Connectivity Algorithms.
        http://www.cse.msu.edu/~cse835/Papers/Graph_connectivity_revised.pdf

    """
    if (s is not None and t is None) or (s is None and t is not None):
        raise nx.NetworkXError("Both source and target must be specified.")

    # Local minimum node cut.
    if s is not None and t is not None:
        if s not in G:
            raise nx.NetworkXError("node %s not in graph" % s)
        if t not in G:
            raise nx.NetworkXError("node %s not in graph" % t)
        return minimum_st_node_cut(G, s, t, flow_func=flow_func)

    # Global minimum node cut.
    # Analog to the algoritm 11 for global node connectivity in [1].
    if G.is_directed():
        if not nx.is_weakly_connected(G):
            raise nx.NetworkXError("Input graph is not connected")
        iter_func = itertools.permutations

        def neighbors(v):
            return itertools.chain.from_iterable([G.predecessors(v), G.successors(v)])

    else:
        if not nx.is_connected(G):
            raise nx.NetworkXError("Input graph is not connected")
        iter_func = itertools.combinations
        neighbors = G.neighbors

    # Reuse the auxiliary digraph and the residual network.
    H = build_auxiliary_node_connectivity(G)
    R = build_residual_network(H, "capacity")
    kwargs = dict(flow_func=flow_func, auxiliary=H, residual=R)

    # Choose a node with minimum degree.
    v = min(G, key=G.degree)
    # Initial node cutset is all neighbors of the node with minimum degree.
    min_cut = set(G[v])
    # Compute st node cuts between v and all its non-neighbors nodes in G.
    for w in set(G) - set(neighbors(v)) - set([v]):
        this_cut = minimum_st_node_cut(G, v, w, **kwargs)
        if len(min_cut) >= len(this_cut):
            min_cut = this_cut
    # Also for non adjacent pairs of neighbors of v.
    for x, y in iter_func(neighbors(v), 2):
        if y in G[x]:
            continue
        this_cut = minimum_st_node_cut(G, x, y, **kwargs)
        if len(min_cut) >= len(this_cut):
            min_cut = this_cut

    return min_cut