python-networkx 1.8.1-0ubuntu3 / usr / lib / python2.7 / dist-packages / networkx / algorithms / flow / maxflow.py

# -*- coding: utf-8 -*-
"""
Maximum flow (and minimum cut) algorithms on capacitated graphs.
"""

__author__ = """Loïc Séguin-C. <loicseguin@gmail.com>"""
# Copyright (C) 2010 Loïc Séguin-C. <loicseguin@gmail.com>
# All rights reserved.
# BSD license.

import networkx as nx

__all__ = ['ford_fulkerson',
           'ford_fulkerson_flow',
           'ford_fulkerson_flow_and_auxiliary',
           'max_flow',
           'min_cut']

def ford_fulkerson_flow_and_auxiliary(G, s, t, capacity='capacity'):
    """Find a maximum single-commodity flow using the Ford-Fulkerson
    algorithm.
    
    This function returns both the value of the maximum flow and the 
    auxiliary network resulting after finding the maximum flow, which
    is also named residual network in the literature. The 
    auxiliary network has edges with capacity equal to the capacity 
    of the edge in the original network minus the flow that went 
    throught that edge. Notice that it can happen that a flow 
    from v to u is allowed in the auxiliary network, though disallowed 
    in the original network. A dictionary with infinite capacity edges 
    can be found as an attribute of the auxiliary network.

    Parameters
    ----------
    G : NetworkX graph
        Edges of the graph are expected to have an attribute called
        'capacity'. If this attribute is not present, the edge is
        considered to have infinite capacity.

    s : node
        Source node for the flow.

    t : node
        Sink node for the flow.

    capacity: string
        Edges of the graph G are expected to have an attribute capacity
        that indicates how much flow the edge can support. If this
        attribute is not present, the edge is considered to have
        infinite capacity. Default value: 'capacity'.

    Returns
    -------
    flow_value : integer, float
        Value of the maximum flow, i.e., net outflow from the source.

    auxiliary : DiGraph
        Residual/auxiliary network after finding the maximum flow. 
        A dictionary with infinite capacity edges can be found as 
        an attribute of this network: auxiliary.graph['inf_capacity_flows']

    Raises
    ------
    NetworkXError
        The algorithm does not support MultiGraph and MultiDiGraph. If
        the input graph is an instance of one of these two classes, a
        NetworkXError is raised.

    NetworkXUnbounded
        If the graph has a path of infinite capacity, the value of a 
        feasible flow on the graph is unbounded above and the function
        raises a NetworkXUnbounded.

    Notes
    -----
    This algorithm uses Edmonds-Karp-Dinitz path selection rule which
    guarantees a running time of `O(nm^2)` for `n` nodes and `m` edges.

    Examples
    --------
    >>> import networkx as nx
    >>> G = nx.DiGraph()
    >>> G.add_edge('x','a', capacity=3.0)
    >>> G.add_edge('x','b', capacity=1.0)
    >>> G.add_edge('a','c', capacity=3.0)
    >>> G.add_edge('b','c', capacity=5.0)
    >>> G.add_edge('b','d', capacity=4.0)
    >>> G.add_edge('d','e', capacity=2.0)
    >>> G.add_edge('c','y', capacity=2.0)
    >>> G.add_edge('e','y', capacity=3.0)
    >>> flow, auxiliary = nx.ford_fulkerson_flow_and_auxiliary(G, 'x', 'y')
    >>> flow
    3.0
    >>> # A dictionary with infinite capacity flows can be found as an
    >>> # attribute of the auxiliary network
    >>> inf_capacity_flows = auxiliary.graph['inf_capacity_flows']

    """
    if G.is_multigraph():
        raise nx.NetworkXError(
                'MultiGraph and MultiDiGraph not supported (yet).')

    if s not in G:
        raise nx.NetworkXError('node %s not in graph' % str(s))
    if t not in G:
        raise nx.NetworkXError('node %s not in graph' % str(t))

    auxiliary = _create_auxiliary_digraph(G, capacity=capacity)
    inf_capacity_flows = auxiliary.graph['inf_capacity_flows']

    flow_value = 0   # Initial feasible flow.

    # As long as there is an (s, t)-path in the auxiliary digraph, find
    # the shortest (with respect to the number of arcs) such path and
    # augment the flow on this path.
    while True:
        try:
            path_nodes = nx.bidirectional_shortest_path(auxiliary, s, t)
        except nx.NetworkXNoPath:
            break

        # Get the list of edges in the shortest path.
        path_edges = list(zip(path_nodes[:-1], path_nodes[1:]))

        # Find the minimum capacity of an edge in the path.
        try:
            path_capacity = min([auxiliary[u][v][capacity]
                                for u, v in path_edges
                                if capacity in auxiliary[u][v]])
        except ValueError: 
            # path of infinite capacity implies no max flow
            raise nx.NetworkXUnbounded(
                    "Infinite capacity path, flow unbounded above.")
        
        flow_value += path_capacity

        # Augment the flow along the path.
        for u, v in path_edges:
            edge_attr = auxiliary[u][v]
            if capacity in edge_attr:
                edge_attr[capacity] -= path_capacity
                if edge_attr[capacity] == 0:
                    auxiliary.remove_edge(u, v)
            else:
                inf_capacity_flows[(u, v)] += path_capacity

            if auxiliary.has_edge(v, u):
                if capacity in auxiliary[v][u]:
                    auxiliary[v][u][capacity] += path_capacity
            else:
                auxiliary.add_edge(v, u, {capacity: path_capacity})
    
    auxiliary.graph['inf_capacity_flows'] = inf_capacity_flows
    return flow_value, auxiliary

def _create_auxiliary_digraph(G, capacity='capacity'):
    """Initialize an auxiliary digraph and dict of infinite capacity
    edges for a given graph G.
    Ignore edges with capacity <= 0.
    """
    auxiliary = nx.DiGraph()
    auxiliary.add_nodes_from(G)
    inf_capacity_flows = {}
    if nx.is_directed(G):
        for edge in G.edges(data = True):
            if capacity in edge[2]:
                if edge[2][capacity] > 0:
                    auxiliary.add_edge(*edge)
            else:
                auxiliary.add_edge(*edge)
                inf_capacity_flows[(edge[0], edge[1])] = 0
    else:
        for edge in G.edges(data = True):
            if capacity in edge[2]:
                if edge[2][capacity] > 0:
                    auxiliary.add_edge(*edge)
                    auxiliary.add_edge(edge[1], edge[0], edge[2])
            else:
                auxiliary.add_edge(*edge)
                auxiliary.add_edge(edge[1], edge[0], edge[2])
                inf_capacity_flows[(edge[0], edge[1])] = 0
                inf_capacity_flows[(edge[1], edge[0])] = 0

    auxiliary.graph['inf_capacity_flows'] = inf_capacity_flows
    return auxiliary


def _create_flow_dict(G, H, capacity='capacity'):
    """Creates the flow dict of dicts on G corresponding to the
    auxiliary digraph H and infinite capacity edges flows
    inf_capacity_flows.
    """
    inf_capacity_flows = H.graph['inf_capacity_flows']
    flow = dict([(u, {}) for u in G])

    if G.is_directed():
        for u, v in G.edges_iter():
            if H.has_edge(u, v):
                if capacity in G[u][v]:
                    flow[u][v] = max(0, G[u][v][capacity] - H[u][v][capacity])
                elif G.has_edge(v, u) and not capacity in G[v][u]:
                    flow[u][v] = max(0, inf_capacity_flows[(u, v)] -
                                     inf_capacity_flows[(v, u)])
                else:
                    flow[u][v] = max(0, H[v].get(u, {}).get(capacity, 0) -
                                     G[v].get(u, {}).get(capacity, 0))
            else:
                flow[u][v] = G[u][v][capacity]

    else: # undirected
        for u, v in G.edges_iter():
            if H.has_edge(u, v):
                if capacity in G[u][v]:
                    flow[u][v] = abs(G[u][v][capacity] - H[u][v][capacity])
                else:
                    flow[u][v] = abs(inf_capacity_flows[(u, v)] -
                                     inf_capacity_flows[(v, u)])
            else:
                flow[u][v] = G[u][v][capacity]
            flow[v][u] = flow[u][v]

    return flow

def ford_fulkerson(G, s, t, capacity='capacity'):
    """Find a maximum single-commodity flow using the Ford-Fulkerson
    algorithm.
    
    This algorithm uses Edmonds-Karp-Dinitz path selection rule which
    guarantees a running time of `O(nm^2)` for `n` nodes and `m` edges.


    Parameters
    ----------
    G : NetworkX graph
        Edges of the graph are expected to have an attribute called
        'capacity'. If this attribute is not present, the edge is
        considered to have infinite capacity.

    s : node
        Source node for the flow.

    t : node
        Sink node for the flow.

    capacity: string
        Edges of the graph G are expected to have an attribute capacity
        that indicates how much flow the edge can support. If this
        attribute is not present, the edge is considered to have
        infinite capacity. Default value: 'capacity'.

    Returns
    -------
    flow_value : integer, float
        Value of the maximum flow, i.e., net outflow from the source.

    flow_dict : dictionary
        Dictionary of dictionaries keyed by nodes such that
        flow_dict[u][v] is the flow edge (u, v).

    Raises
    ------
    NetworkXError
        The algorithm does not support MultiGraph and MultiDiGraph. If
        the input graph is an instance of one of these two classes, a
        NetworkXError is raised.

    NetworkXUnbounded
        If the graph has a path of infinite capacity, the value of a 
        feasible flow on the graph is unbounded above and the function
        raises a NetworkXUnbounded.

    Examples
    --------
    >>> import networkx as nx
    >>> G = nx.DiGraph()
    >>> G.add_edge('x','a', capacity=3.0)
    >>> G.add_edge('x','b', capacity=1.0)
    >>> G.add_edge('a','c', capacity=3.0)
    >>> G.add_edge('b','c', capacity=5.0)
    >>> G.add_edge('b','d', capacity=4.0)
    >>> G.add_edge('d','e', capacity=2.0)
    >>> G.add_edge('c','y', capacity=2.0)
    >>> G.add_edge('e','y', capacity=3.0)
    >>> flow, F = nx.ford_fulkerson(G, 'x', 'y')
    >>> flow
    3.0
    """
    flow_value, auxiliary = ford_fulkerson_flow_and_auxiliary(G, 
                                s, t, capacity=capacity)
    flow_dict = _create_flow_dict(G, auxiliary, capacity=capacity)
    return flow_value, flow_dict

def ford_fulkerson_flow(G, s, t, capacity='capacity'):
    """Return a maximum flow for a single-commodity flow problem.

    Parameters
    ----------
    G : NetworkX graph
        Edges of the graph are expected to have an attribute called
        'capacity'. If this attribute is not present, the edge is
        considered to have infinite capacity.

    s : node
        Source node for the flow.

    t : node
        Sink node for the flow.

    capacity: string
        Edges of the graph G are expected to have an attribute capacity
        that indicates how much flow the edge can support. If this
        attribute is not present, the edge is considered to have
        infinite capacity. Default value: 'capacity'.

    Returns
    -------
    flow_dict : dictionary
        Dictionary of dictionaries keyed by nodes such that
        flow_dict[u][v] is the flow edge (u, v).

    Raises
    ------
    NetworkXError
        The algorithm does not support MultiGraph and MultiDiGraph. If
        the input graph is an instance of one of these two classes, a
        NetworkXError is raised.

    NetworkXUnbounded
        If the graph has a path of infinite capacity, the value of a 
        feasible flow on the graph is unbounded above and the function
        raises a NetworkXUnbounded.

    Examples
    --------
    >>> import networkx as nx
    >>> G = nx.DiGraph()
    >>> G.add_edge('x','a', capacity=3.0)
    >>> G.add_edge('x','b', capacity=1.0)
    >>> G.add_edge('a','c', capacity=3.0)
    >>> G.add_edge('b','c', capacity=5.0)
    >>> G.add_edge('b','d', capacity=4.0)
    >>> G.add_edge('d','e', capacity=2.0)
    >>> G.add_edge('c','y', capacity=2.0)
    >>> G.add_edge('e','y', capacity=3.0)
    >>> F = nx.ford_fulkerson_flow(G, 'x', 'y')
    >>> for u, v in sorted(G.edges_iter()):
    ...     print('(%s, %s) %.2f' % (u, v, F[u][v]))
    ... 
    (a, c) 2.00
    (b, c) 0.00
    (b, d) 1.00
    (c, y) 2.00
    (d, e) 1.00
    (e, y) 1.00
    (x, a) 2.00
    (x, b) 1.00
    """
    flow_value, auxiliary = ford_fulkerson_flow_and_auxiliary(G,
                                s, t, capacity=capacity)
    return _create_flow_dict(G, auxiliary, capacity=capacity)

def max_flow(G, s, t, capacity='capacity'):
    """Find the value of a maximum single-commodity flow.
    
    Parameters
    ----------
    G : NetworkX graph
        Edges of the graph are expected to have an attribute called
        'capacity'. If this attribute is not present, the edge is
        considered to have infinite capacity.

    s : node
        Source node for the flow.

    t : node
        Sink node for the flow.

    capacity: string
        Edges of the graph G are expected to have an attribute capacity
        that indicates how much flow the edge can support. If this
        attribute is not present, the edge is considered to have
        infinite capacity. Default value: 'capacity'.

    Returns
    -------
    flow_value : integer, float
        Value of the maximum flow, i.e., net outflow from the source.

    Raises
    ------
    NetworkXError
        The algorithm does not support MultiGraph and MultiDiGraph. If
        the input graph is an instance of one of these two classes, a
        NetworkXError is raised.

    NetworkXUnbounded
        If the graph has a path of infinite capacity, the value of a 
        feasible flow on the graph is unbounded above and the function
        raises a NetworkXUnbounded.

    Examples
    --------
    >>> import networkx as nx
    >>> G = nx.DiGraph()
    >>> G.add_edge('x','a', capacity=3.0)
    >>> G.add_edge('x','b', capacity=1.0)
    >>> G.add_edge('a','c', capacity=3.0)
    >>> G.add_edge('b','c', capacity=5.0)
    >>> G.add_edge('b','d', capacity=4.0)
    >>> G.add_edge('d','e', capacity=2.0)
    >>> G.add_edge('c','y', capacity=2.0)
    >>> G.add_edge('e','y', capacity=3.0)
    >>> flow = nx.max_flow(G, 'x', 'y')
    >>> flow
    3.0
    """
    return ford_fulkerson_flow_and_auxiliary(G, s, t, capacity=capacity)[0]


def min_cut(G, s, t, capacity='capacity'):
    """Compute the value of a minimum (s, t)-cut.

    Use the max-flow min-cut theorem, i.e., the capacity of a minimum
    capacity cut is equal to the flow value of a maximum flow.

    Parameters
    ----------
    G : NetworkX graph
        Edges of the graph are expected to have an attribute called
        'capacity'. If this attribute is not present, the edge is
        considered to have infinite capacity.

    s : node
        Source node for the flow.

    t : node
        Sink node for the flow.

    capacity: string
        Edges of the graph G are expected to have an attribute capacity
        that indicates how much flow the edge can support. If this
        attribute is not present, the edge is considered to have
        infinite capacity. Default value: 'capacity'.

    Returns
    -------
    cutValue : integer, float
        Value of the minimum cut.
    
    Raises
    ------
    NetworkXUnbounded
        If the graph has a path of infinite capacity, all cuts have
        infinite capacity and the function raises a NetworkXError.
    
    Examples
    --------
    >>> import networkx as nx
    >>> G = nx.DiGraph()
    >>> G.add_edge('x','a', capacity = 3.0)
    >>> G.add_edge('x','b', capacity = 1.0)
    >>> G.add_edge('a','c', capacity = 3.0)
    >>> G.add_edge('b','c', capacity = 5.0)
    >>> G.add_edge('b','d', capacity = 4.0)
    >>> G.add_edge('d','e', capacity = 2.0)
    >>> G.add_edge('c','y', capacity = 2.0)
    >>> G.add_edge('e','y', capacity = 3.0)
    >>> nx.min_cut(G, 'x', 'y')
    3.0
    """

    try:
        return ford_fulkerson_flow_and_auxiliary(G, s, t, capacity=capacity)[0]
    except nx.NetworkXUnbounded:
        raise nx.NetworkXUnbounded(
                "Infinite capacity path, no minimum cut.")