text: # -*- coding: utf-8 -*-
  """
  Capacity scaling minimum cost flow algorithm.
  """
  
  __author__ = """ysitu <ysitu@users.noreply.github.com>"""
  # Copyright (C) 2014 ysitu <ysitu@users.noreply.github.com>
  # All rights reserved.
  # BSD license.
  
  __all__ = ['capacity_scaling']
  
  from itertools import chain
  from math import log
  import networkx as nx
  from ...utils import BinaryHeap
  from ...utils import generate_unique_node
  from ...utils import not_implemented_for
  from ...utils import arbitrary_element
  
  def _detect_unboundedness(R):
      """Detect infinite-capacity negative cycles.
      """
      s = generate_unique_node()
      G = nx.DiGraph()
      G.add_nodes_from(R)
  
      # Value simulating infinity.
      inf = R.graph['inf']
      # True infinity.
      f_inf = float('inf')
      for u in R:
          for v, e in R[u].items():
              # Compute the minimum weight of infinite-capacity (u, v) edges.
              w = f_inf
              for k, e in e.items():
                  if e['capacity'] == inf:
                      w = min(w, e['weight'])
              if w != f_inf:
                  G.add_edge(u, v, weight=w)
  
      if nx.negative_edge_cycle(G):
          raise nx.NetworkXUnbounded(
              'Negative cost cycle of infinite capacity found. '
              'Min cost flow may be unbounded below.')
  
  
  @not_implemented_for('undirected')
  def _build_residual_network(G, demand, capacity, weight):
      """Build a residual network and initialize a zero flow.
      """
      if sum(G.node[u].get(demand, 0) for u in G) != 0:
          raise nx.NetworkXUnfeasible("Sum of the demands should be 0.")
  
      R = nx.MultiDiGraph()
      R.add_nodes_from((u, {'excess': -G.node[u].get(demand, 0),
                            'potential': 0}) for u in G)
  
      inf = float('inf')
      # Detect selfloops with infinite capacities and negative weights.
      for u, v, e in G.selfloop_edges(data=True):
          if e.get(weight, 0) < 0 and e.get(capacity, inf) == inf:
              raise nx.NetworkXUnbounded(
                  'Negative cost cycle of infinite capacity found. '
                  'Min cost flow may be unbounded below.')
  
      # Extract edges with positive capacities. Self loops excluded.
      if G.is_multigraph():
          edge_list = [(u, v, k, e)
                       for u, v, k, e in G.edges_iter(data=True, keys=True)
                       if u != v and e.get(capacity, inf) > 0]
      else:
          edge_list = [(u, v, 0, e) for u, v, e in G.edges_iter(data=True)
                       if u != v and e.get(capacity, inf) > 0]
      # Simulate infinity with the larger of the sum of absolute node imbalances
      # the sum of finite edge capacities or any positive value if both sums are
      # zero. This allows the infinite-capacity edges to be distinguished for
      # unboundedness detection and directly participate in residual capacity
      # calculation.
      inf = max(sum(abs(R.node[u]['excess']) for u in R),
                2 * sum(e[capacity] for u, v, k, e in edge_list
                        if capacity in e and e[capacity] != inf)) or 1
      for u, v, k, e in edge_list:
          r = min(e.get(capacity, inf), inf)
          w = e.get(weight, 0)
          # Add both (u, v) and (v, u) into the residual network marked with the
          # original key. (key[1] == True) indicates the (u, v) is in the
          # original network.
          R.add_edge(u, v, key=(k, True), capacity=r, weight=w, flow=0)
          R.add_edge(v, u, key=(k, False), capacity=0, weight=-w, flow=0)
  
      # Record the value simulating infinity.
      R.graph['inf'] = inf
  
      _detect_unboundedness(R)
  
      return R
  
  
  def _build_flow_dict(G, R, capacity, weight):
      """Build a flow dictionary from a residual network.
      """
      inf = float('inf')
      flow_dict = {}
      if G.is_multigraph():
          for u in G:
              flow_dict[u] = {}
              for v, es in G[u].items():
                  flow_dict[u][v] = dict(
                      # Always saturate negative selfloops.
                      (k, (0 if (u != v or e.get(capacity, inf) <= 0 or
                                 e.get(weight, 0) >= 0) else e[capacity]))
                      for k, e in es.items())
              for v, es in R[u].items():
                  if v in flow_dict[u]:
                      flow_dict[u][v].update((k[0], e['flow'])
                                             for k, e in es.items()
                                             if e['flow'] > 0)
      else:
          for u in G:
              flow_dict[u] = dict(
                  # Always saturate negative selfloops.
                  (v, (0 if (u != v or e.get(capacity, inf) <= 0 or
                             e.get(weight, 0) >= 0) else e[capacity]))
                  for v, e in G[u].items())
              flow_dict[u].update((v, e['flow']) for v, es in R[u].items()
                                  for e in es.values() if e['flow'] > 0)
      return flow_dict
  
  
  def capacity_scaling(G, demand='demand', capacity='capacity', weight='weight',
                       heap=BinaryHeap):
      """Find a minimum cost flow satisfying all demands in digraph G.
  
      This is a capacity scaling successive shortest augmenting path algorithm.
  
      G is a digraph with edge costs and capacities and in which nodes
      have demand, i.e., they want to send or receive some amount of
      flow. A negative demand means that the node wants to send flow, a
      positive demand means that the node want to receive flow. A flow on
      the digraph G satisfies all demand if the net flow into each node
      is equal to the demand of that node.
  
      Parameters
      ----------
      G : NetworkX graph
          DiGraph or MultiDiGraph on which a minimum cost flow satisfying all
          demands is to be found.
  
      demand : string
          Nodes of the graph G are expected to have an attribute demand
          that indicates how much flow a node wants to send (negative
          demand) or receive (positive demand). Note that the sum of the
          demands should be 0 otherwise the problem in not feasible. If
          this attribute is not present, a node is considered to have 0
          demand. Default value: 'demand'.
  
      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'.
  
      weight : string
          Edges of the graph G are expected to have an attribute weight
          that indicates the cost incurred by sending one unit of flow on
          that edge. If not present, the weight is considered to be 0.
          Default value: 'weight'.
  
      heap : class
          Type of heap to be used in the algorithm. It should be a subclass of
          :class:`MinHeap` or implement a compatible interface.
  
          If a stock heap implementation is to be used, :class:`BinaryHeap` is
          recommeded over :class:`PairingHeap` for Python implementations without
          optimized attribute accesses (e.g., CPython) despite a slower
          asymptotic running time. For Python implementations with optimized
          attribute accesses (e.g., PyPy), :class:`PairingHeap` provides better
          performance. Default value: :class:`BinaryHeap`.
  
      Returns
      -------
      flowCost: integer
          Cost of a minimum cost flow satisfying all demands.
  
      flowDict: dictionary
          Dictionary of dictionaries keyed by nodes such that
          flowDict[u][v] is the flow edge (u, v) if G is a digraph.
  
          Dictionary of dictionaries of dictionaries keyed by nodes such that
          flowDict[u][v][key] is the flow edge (u, v, key) if G is a
          multidigraph.
  
      Raises
      ------
      NetworkXError
          This exception is raised if the input graph is not directed,
          not connected.
  
      NetworkXUnfeasible
          This exception is raised in the following situations:
              * The sum of the demands is not zero. Then, there is no
                flow satisfying all demands.
              * There is no flow satisfying all demand.
  
      NetworkXUnbounded
          This exception is raised if the digraph G has a cycle of
          negative cost and infinite capacity. Then, the cost of a flow
          satisfying all demands is unbounded below.
  
      Notes
      -----
      This algorithm does not work if edge weights are floating-point numbers.
  
      See also
      --------
      :meth:`network_simplex`
  
      Examples
      --------
      A simple example of a min cost flow problem.
  
      >>> import networkx as nx
      >>> G = nx.DiGraph()
      >>> G.add_node('a', demand = -5)
      >>> G.add_node('d', demand = 5)
      >>> G.add_edge('a', 'b', weight = 3, capacity = 4)
      >>> G.add_edge('a', 'c', weight = 6, capacity = 10)
      >>> G.add_edge('b', 'd', weight = 1, capacity = 9)
      >>> G.add_edge('c', 'd', weight = 2, capacity = 5)
      >>> flowCost, flowDict = nx.capacity_scaling(G)
      >>> flowCost
      24
      >>> flowDict # doctest: +SKIP
      {'a': {'c': 1, 'b': 4}, 'c': {'d': 1}, 'b': {'d': 4}, 'd': {}}
  
      It is possible to change the name of the attributes used for the
      algorithm.
  
      >>> G = nx.DiGraph()
      >>> G.add_node('p', spam = -4)
      >>> G.add_node('q', spam = 2)
      >>> G.add_node('a', spam = -2)
      >>> G.add_node('d', spam = -1)
      >>> G.add_node('t', spam = 2)
      >>> G.add_node('w', spam = 3)
      >>> G.add_edge('p', 'q', cost = 7, vacancies = 5)
      >>> G.add_edge('p', 'a', cost = 1, vacancies = 4)
      >>> G.add_edge('q', 'd', cost = 2, vacancies = 3)
      >>> G.add_edge('t', 'q', cost = 1, vacancies = 2)
      >>> G.add_edge('a', 't', cost = 2, vacancies = 4)
      >>> G.add_edge('d', 'w', cost = 3, vacancies = 4)
      >>> G.add_edge('t', 'w', cost = 4, vacancies = 1)
      >>> flowCost, flowDict = nx.capacity_scaling(G, demand = 'spam',
      ...                                          capacity = 'vacancies',
      ...                                          weight = 'cost')
      >>> flowCost
      37
      >>> flowDict  # doctest: +SKIP
      {'a': {'t': 4}, 'd': {'w': 2}, 'q': {'d': 1}, 'p': {'q': 2, 'a': 2}, 't': {'q': 1, 'w': 1}, 'w': {}}
      """
      R = _build_residual_network(G, demand, capacity, weight)
  
      inf = float('inf')
      # Account cost of negative selfloops.
      flow_cost = sum(
          0 if e.get(capacity, inf) <= 0 or e.get(weight, 0) >= 0
          else e[capacity] * e[weight]
          for u, v, e in G.selfloop_edges(data=True))
  
      # Determine the maxmimum edge capacity.
      wmax = max(chain([-inf],
                       (e['capacity'] for u, v, e in R.edges_iter(data=True))))
      if wmax == -inf:
          # Residual network has no edges.
          return flow_cost, _build_flow_dict(G, R, capacity, weight)
  
      R_node = R.node
      R_succ = R.succ
  
      delta = 2 ** int(log(wmax, 2))
      while delta >= 1:
          # Saturate Δ-residual edges with negative reduced costs to achieve
          # Δ-optimality.
          for u in R:
              p_u = R_node[u]['potential']
              for v, es in R_succ[u].items():
                  for k, e in es.items():
                      flow = e['capacity'] - e['flow']
                      if e['weight'] - p_u + R_node[v]['potential'] < 0:
                          flow = e['capacity'] - e['flow']
                          if flow >= delta:
                              e['flow'] += flow
                              R_succ[v][u][(k[0], not k[1])]['flow'] -= flow
                              R_node[u]['excess'] -= flow
                              R_node[v]['excess'] += flow
          # Determine the Δ-active nodes.
          S = set()
          T = set()
          S_add = S.add
          S_remove = S.remove
          T_add = T.add
          T_remove = T.remove
          for u in R:
              excess = R_node[u]['excess']
              if excess >= delta:
                  S_add(u)
              elif excess <= -delta:
                  T_add(u)
          # Repeatedly augment flow from S to T along shortest paths until
          # Δ-feasibility is achieved.
          while S and T:
              s = arbitrary_element(S)
              t = None
              # Search for a shortest path in terms of reduce costs from s to
              # any t in T in the Δ-residual network.
              d = {}
              pred = {s: None}
              h = heap()
              h_insert = h.insert
              h_get = h.get
              h_insert(s, 0)
              while h:
                  u, d_u = h.pop()
                  d[u] = d_u
                  if u in T:
                      # Path found.
                      t = u
                      break
                  p_u = R_node[u]['potential']
                  for v, es in R_succ[u].items():
                      if v in d:
                          continue
                      wmin = inf
                      # Find the minimum-weighted (u, v) Δ-residual edge.
                      for k, e in es.items():
                          if e['capacity'] - e['flow'] >= delta:
                              w = e['weight']
                              if w < wmin:
                                  wmin = w
                                  kmin = k
                                  emin = e
                      if wmin == inf:
                          continue
                      # Update the distance label of v.
                      d_v = d_u + wmin - p_u + R_node[v]['potential']
                      if h_insert(v, d_v):
                          pred[v] = (u, kmin, emin)
              if t is not None:
                  # Augment Δ units of flow from s to t.
                  while u != s:
                      v = u
                      u, k, e = pred[v]
                      e['flow'] += delta
                      R_succ[v][u][(k[0], not k[1])]['flow'] -= delta
                  # Account node excess and deficit.
                  R_node[s]['excess'] -= delta
                  R_node[t]['excess'] += delta
                  if R_node[s]['excess'] < delta:
                      S_remove(s)
                  if R_node[t]['excess'] > -delta:
                      T_remove(t)
                  # Update node potentials.
                  d_t = d[t]
                  for u, d_u in d.items():
                      R_node[u]['potential'] -= d_u - d_t
              else:
                  # Path not found.
                  S_remove(s)
          delta //= 2
  
      if any(R.node[u]['excess'] != 0 for u in R):
          raise nx.NetworkXUnfeasible('No flow satisfying all demands.')
  
      # Calculate the flow cost.
      for u in R:
          for v, es in R_succ[u].items():
              for e in es.values():
                  flow = e['flow']
                  if flow > 0:
                      flow_cost += flow * e['weight']
  
      return flow_cost, _build_flow_dict(G, R, capacity, weight)
  