text: """
  Boykov-Kolmogorov algorithm for maximum flow problems.
  """
  from collections import deque
  from operator import itemgetter
  
  import networkx as nx
  from networkx.algorithms.flow.utils import build_residual_network
  
  __all__ = ["boykov_kolmogorov"]
  
  
  def boykov_kolmogorov(
      G, s, t, capacity="capacity", residual=None, value_only=False, cutoff=None
  ):
      r"""Find a maximum single-commodity flow using Boykov-Kolmogorov 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 worse case complexity $O(n^2 m |C|)$ for $n$ nodes, $m$
      edges, and $|C|$ the cost of the minimum cut [1]_. This implementation
      uses the marking heuristic defined in [2]_ which improves its running
      time in many practical problems.
  
      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 boykov_kolmogorov
  
      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 = boykov_kolmogorov(G, 'x', 'y')
      >>> flow_value = nx.maximum_flow_value(G, 'x', 'y')
      >>> flow_value
      3.0
      >>> flow_value == R.graph['flow_value']
      True
  
      A nice feature of the Boykov-Kolmogorov algorithm is that a partition
      of the nodes that defines a minimum cut can be easily computed based
      on the search trees used during the algorithm. These trees are stored
      in the graph attribute `trees` of the residual network.
  
      >>> source_tree, target_tree = R.graph['trees']
      >>> partition = (set(source_tree), set(G) - set(source_tree))
  
      Or equivalently:
  
      >>> partition = (set(G) - set(target_tree), set(target_tree))
  
      References
      ----------
      .. [1] Boykov, Y., & Kolmogorov, V. (2004). An experimental comparison
             of min-cut/max-flow algorithms for energy minimization in vision.
             Pattern Analysis and Machine Intelligence, IEEE Transactions on,
             26(9), 1124-1137.
             http://www.csd.uwo.ca/~yuri/Papers/pami04.pdf
  
      .. [2] Vladimir Kolmogorov. Graph-based Algorithms for Multi-camera
             Reconstruction Problem. PhD thesis, Cornell University, CS Department,
             2003. pp. 109-114.
             https://pub.ist.ac.at/~vnk/papers/thesis.pdf
  
      """
      R = boykov_kolmogorov_impl(G, s, t, capacity, residual, cutoff)
      R.graph["algorithm"] = "boykov_kolmogorov"
      return R
  
  
  def boykov_kolmogorov_impl(G, s, t, capacity, residual, cutoff):
      if s not in G:
          raise nx.NetworkXError(f"node {str(s)} not in graph")
      if t not in G:
          raise nx.NetworkXError(f"node {str(t)} not in graph")
      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.
      # This is way too slow
      # nx.set_edge_attributes(R, 0, 'flow')
      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 grow():
          """Bidirectional breadth-first search for the growth stage.
  
             Returns a connecting edge, that is and edge that connects
             a node from the source search tree with a node from the
             target search tree.
             The first node in the connecting edge is always from the
             source tree and the last node from the target tree.
          """
          while active:
              u = active[0]
              if u in source_tree:
                  this_tree = source_tree
                  other_tree = target_tree
                  neighbors = R_succ
              else:
                  this_tree = target_tree
                  other_tree = source_tree
                  neighbors = R_pred
              for v, attr in neighbors[u].items():
                  if attr["capacity"] - attr["flow"] > 0:
                      if v not in this_tree:
                          if v in other_tree:
                              return (u, v) if this_tree is source_tree else (v, u)
                          this_tree[v] = u
                          dist[v] = dist[u] + 1
                          timestamp[v] = timestamp[u]
                          active.append(v)
                      elif v in this_tree and _is_closer(u, v):
                          this_tree[v] = u
                          dist[v] = dist[u] + 1
                          timestamp[v] = timestamp[u]
              _ = active.popleft()
          return None, None
  
      def augment(u, v):
          """Augmentation stage.
  
             Reconstruct path and determine its residual capacity.
             We start from a connecting edge, which links a node
             from the source tree to a node from the target tree.
             The connecting edge is the output of the grow function
             and the input of this function.
          """
          attr = R_succ[u][v]
          flow = min(INF, attr["capacity"] - attr["flow"])
          path = [u]
          # Trace a path from u to s in source_tree.
          w = u
          while w != s:
              n = w
              w = source_tree[n]
              attr = R_pred[n][w]
              flow = min(flow, attr["capacity"] - attr["flow"])
              path.append(w)
          path.reverse()
          # Trace a path from v to t in target_tree.
          path.append(v)
          w = v
          while w != t:
              n = w
              w = target_tree[n]
              attr = R_succ[n][w]
              flow = min(flow, attr["capacity"] - attr["flow"])
              path.append(w)
          # Augment flow along the path and check for saturated edges.
          it = iter(path)
          u = next(it)
          these_orphans = []
          for v in it:
              R_succ[u][v]["flow"] += flow
              R_succ[v][u]["flow"] -= flow
              if R_succ[u][v]["flow"] == R_succ[u][v]["capacity"]:
                  if v in source_tree:
                      source_tree[v] = None
                      these_orphans.append(v)
                  if u in target_tree:
                      target_tree[u] = None
                      these_orphans.append(u)
              u = v
          orphans.extend(sorted(these_orphans, key=dist.get))
          return flow
  
      def adopt():
          """Adoption stage.
  
             Reconstruct search trees by adopting or discarding orphans.
             During augmentation stage some edges got saturated and thus
             the source and target search trees broke down to forests, with
             orphans as roots of some of its trees. We have to reconstruct
             the search trees rooted to source and target before we can grow
             them again.
          """
          while orphans:
              u = orphans.popleft()
              if u in source_tree:
                  tree = source_tree
                  neighbors = R_pred
              else:
                  tree = target_tree
                  neighbors = R_succ
              nbrs = ((n, attr, dist[n]) for n, attr in neighbors[u].items() if n in tree)
              for v, attr, d in sorted(nbrs, key=itemgetter(2)):
                  if attr["capacity"] - attr["flow"] > 0:
                      if _has_valid_root(v, tree):
                          tree[u] = v
                          dist[u] = dist[v] + 1
                          timestamp[u] = time
                          break
              else:
                  nbrs = (
                      (n, attr, dist[n]) for n, attr in neighbors[u].items() if n in tree
                  )
                  for v, attr, d in sorted(nbrs, key=itemgetter(2)):
                      if attr["capacity"] - attr["flow"] > 0:
                          if v not in active:
                              active.append(v)
                      if tree[v] == u:
                          tree[v] = None
                          orphans.appendleft(v)
                  if u in active:
                      active.remove(u)
                  del tree[u]
  
      def _has_valid_root(n, tree):
          path = []
          v = n
          while v is not None:
              path.append(v)
              if v == s or v == t:
                  base_dist = 0
                  break
              elif timestamp[v] == time:
                  base_dist = dist[v]
                  break
              v = tree[v]
          else:
              return False
          length = len(path)
          for i, u in enumerate(path, 1):
              dist[u] = base_dist + length - i
              timestamp[u] = time
          return True
  
      def _is_closer(u, v):
          return timestamp[v] <= timestamp[u] and dist[v] > dist[u] + 1
  
      source_tree = {s: None}
      target_tree = {t: None}
      active = deque([s, t])
      orphans = deque()
      flow_value = 0
      # data structures for the marking heuristic
      time = 1
      timestamp = {s: time, t: time}
      dist = {s: 0, t: 0}
      while flow_value < cutoff:
          # Growth stage
          u, v = grow()
          if u is None:
              break
          time += 1
          # Augmentation stage
          flow_value += augment(u, v)
          # Adoption stage
          adopt()
  
      if flow_value * 2 > INF:
          raise nx.NetworkXUnbounded("Infinite capacity path, flow unbounded above.")
  
      # Add source and target tree in a graph attribute.
      # A partition that defines a minimum cut can be directly
      # computed from the search trees as explained in the docstrings.
      R.graph["trees"] = (source_tree, target_tree)
      # Add the standard flow_value graph attribute.
      R.graph["flow_value"] = flow_value
      return R
  