code: def preflow_push(
      G,
      s,
      t,
      capacity="capacity",
      residual=None,
      global_relabel_freq=1,
      value_only=False,
  ):
      r"""Find a maximum single-commodity flow using the highest-label
      preflow-push 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 $O(n^2 \sqrt{m})$ 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'.
  
      residual : NetworkX graph
          Residual network on which the algorithm is to be executed. If None, a
          new residual network is created. Default value: None.
  
      global_relabel_freq : integer, float
          Relative frequency of applying the global relabeling heuristic to speed
          up the algorithm. If it is None, the heuristic is disabled. Default
          value: 1.
  
      value_only : bool
          If False, compute a maximum flow; otherwise, compute a maximum preflow
          which is enough for computing the maximum flow value. Default value:
          False.
  
      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:`edmonds_karp`
      :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 node :samp:`u` in :samp:`R`,
      :samp:`R.nodes[u]['excess']` represents the difference between flow into
      :samp:`u` and flow out of :samp:`u`.
  
      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']`. 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
      --------
      >>> from networkx.algorithms.flow import preflow_push
  
      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 = preflow_push(G, "x", "y")
      >>> flow_value = nx.maximum_flow_value(G, "x", "y")
      >>> flow_value == R.graph["flow_value"]
      True
      >>> # preflow_push also stores the maximum flow value
      >>> # in the excess attribute of the sink node t
      >>> flow_value == R.nodes["y"]["excess"]
      True
      >>> # For some problems, you might only want to compute a
      >>> # maximum preflow.
      >>> R = preflow_push(G, "x", "y", value_only=True)
      >>> flow_value == R.graph["flow_value"]
      True
      >>> flow_value == R.nodes["y"]["excess"]
      True
  
      """
      R = preflow_push_impl(
          G, s, t, capacity, residual, global_relabel_freq, value_only
      )
      R.graph["algorithm"] = "preflow_push"
      return R
  
imports: from collections import deque
  from itertools import islice
  import networkx
  from ...utils import arbitrary_element
  from .utils import CurrentEdge, GlobalRelabelThreshold, Level, build_residual_network, detect_unboundedness
label_desc: Use image node_id 2 for calling a global function with example usage: preflow_push(G, s, t, capacity, residual, global_relabel_freq, value_only) and returns: R