text: # dinitz.py - Dinitz' algorithm for maximum flow problems.
  #
  # Copyright 2016-2017 NetworkX developers.
  #
  # This file is part of NetworkX.
  #
  # NetworkX is distributed under a BSD license; see LICENSE.txt for more
  # information.
  #
  # Author: Jordi Torrents <jordi.t21@gmail.com>
  """
  Dinitz' algorithm for maximum flow problems.
  """
  from collections import deque
  
  import networkx as nx
  from networkx.algorithms.flow.utils import build_residual_network
  from networkx.utils import pairwise
  
  __all__ = ['dinitz']
  
  
  def dinitz(G, s, t, capacity='capacity', residual=None, value_only=False, cutoff=None):
      """Find a maximum single-commodity flow using Dinitz' algorithm.
  
      This function returns the residual network resulting after computing
      the maximum flow. See below for details about the conventions
      NetworkX uses for defining residual networks.
  
      This algorithm has a running time of $O(n^2 m)$ for $n$ nodes and $m$
      edges [1]_.
  
  
      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'.
  
      residual : NetworkX graph
          Residual network on which the algorithm is to be executed. If None, a
          new residual network is created. Default value: None.
  
      value_only : bool
          If True compute only the value of the maximum flow. This parameter
          will be ignored by this algorithm because it is not applicable.
  
      cutoff : integer, float
          If specified, the algorithm will terminate when the flow value reaches
          or exceeds the cutoff. In this case, it may be unable to immediately
          determine a minimum cut. Default value: None.
  
      Returns
      -------
      R : NetworkX DiGraph
          Residual network after computing the maximum flow.
  
      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.
  
      See also
      --------
      :meth:`maximum_flow`
      :meth:`minimum_cut`
      :meth:`preflow_push`
      :meth:`shortest_augmenting_path`
  
      Notes
      -----
      The residual network :samp:`R` from an input graph :samp:`G` has the
      same nodes as :samp:`G`. :samp:`R` is a DiGraph that contains a pair
      of edges :samp:`(u, v)` and :samp:`(v, u)` iff :samp:`(u, v)` is not a
      self-loop, and at least one of :samp:`(u, v)` and :samp:`(v, u)` exists
      in :samp:`G`.
  
      For each edge :samp:`(u, v)` in :samp:`R`, :samp:`R[u][v]['capacity']`
      is equal to the capacity of :samp:`(u, v)` in :samp:`G` if it exists
      in :samp:`G` or zero otherwise. If the capacity is infinite,
      :samp:`R[u][v]['capacity']` will have a high arbitrary finite value
      that does not affect the solution of the problem. This value is stored in
      :samp:`R.graph['inf']`. For each edge :samp:`(u, v)` in :samp:`R`,
      :samp:`R[u][v]['flow']` represents the flow function of :samp:`(u, v)` and
      satisfies :samp:`R[u][v]['flow'] == -R[v][u]['flow']`.
  
      The flow value, defined as the total flow into :samp:`t`, the sink, is
      stored in :samp:`R.graph['flow_value']`. If :samp:`cutoff` is not
      specified, reachability to :samp:`t` using only edges :samp:`(u, v)` such
      that :samp:`R[u][v]['flow'] < R[u][v]['capacity']` induces a minimum
      :samp:`s`-:samp:`t` cut.
  
      Examples
      --------
      >>> import networkx as nx
      >>> from networkx.algorithms.flow import dinitz
  
      The functions that implement flow algorithms and output a residual
      network, such as this one, are not imported to the base NetworkX
      namespace, so you have to explicitly import them from the flow package.
  
      >>> 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)
      >>> R = dinitz(G, 'x', 'y')
      >>> flow_value = nx.maximum_flow_value(G, 'x', 'y')
      >>> flow_value
      3.0
      >>> flow_value == R.graph['flow_value']
      True
  
      References
      ----------
      .. [1] Dinitz' Algorithm: The Original Version and Even's Version.
             2006. Yefim Dinitz. In Theoretical Computer Science. Lecture
             Notes in Computer Science. Volume 3895. pp 218-240.
             http://www.cs.bgu.ac.il/~dinitz/Papers/Dinitz_alg.pdf
  
      """
      R = dinitz_impl(G, s, t, capacity, residual, cutoff)
      R.graph['algorithm'] = 'dinitz'
      return R
  
  
  def dinitz_impl(G, s, t, capacity, residual, cutoff):
      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))
      if s == t:
          raise nx.NetworkXError('source and sink are the same node')
  
      if residual is None:
          R = build_residual_network(G, capacity)
      else:
          R = residual
  
      # Initialize/reset the residual network.
      for u in R:
          for e in R[u].values():
              e['flow'] = 0
  
      # Use an arbitrary high value as infinite. It is computed
      # when building the residual network.
      INF = R.graph['inf']
  
      if cutoff is None:
          cutoff = INF
  
      R_succ = R.succ
      R_pred = R.pred
  
      def breath_first_search():
          parents = {}
          queue = deque([s])
          while queue:
              if t in parents:
                  break
              u = queue.popleft()
              for v in R_succ[u]:
                  attr = R_succ[u][v]
                  if v not in parents and attr['capacity'] - attr['flow'] > 0:
                      parents[v] = u
                      queue.append(v)
          return parents
  
      def depth_first_search(parents):
          """Build a path using DFS starting from the sink"""
          path = []
          u = t
          flow = INF
          while u != s:
              path.append(u)
              v = parents[u]
              flow = min(flow, R_pred[u][v]['capacity'] - R_pred[u][v]['flow'])
              u = v
          path.append(s)
          # Augment the flow along the path found
          if flow > 0:
              for u, v in pairwise(path):
                  R_pred[u][v]['flow'] += flow
                  R_pred[v][u]['flow'] -= flow
          return flow
  
      flow_value = 0
      while flow_value < cutoff:
          parents = breath_first_search()
          if t not in parents:
              break
          this_flow = depth_first_search(parents)
          if this_flow * 2 > INF:
              raise nx.NetworkXUnbounded(
                  'Infinite capacity path, flow unbounded above.')
          flow_value += this_flow
  
      R.graph['flow_value'] = flow_value
      return R
  