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# gomory_hu_tree

gomory_hu_tree(G, capacity='capacity', flow_func=None)

  Returns the Gomory-Hu tree of an undirected graph G.

  A Gomory-Hu tree of an undirected graph with capacities is a weighted tree that represents the minimum s-t cuts for all s-t pairs in the graph.

  It only requires `n-1` minimum cut computations instead of the obvious `n(n-1)/2`. The tree represents all s-t cuts as the minimum cut value among any pair of nodes is the minimum edge weight in the shortest path between the two nodes in the Gomory-Hu tree.

  The Gomory-Hu tree also has the property that removing the edge with the minimum weight in the shortest path between any two nodes leaves two connected components that form a partition of the nodes in G that defines the minimum s-t cut.

  See Examples section below for details.

  Parameters:
    G NetworkX graph

      Undirected graph

    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’.

    flow_func function

      Function to perform the underlying flow computations. Default value `edmonds_karp()`. This function performs better in sparse graphs with right tailed degree distributions. `shortest_augmenting_path()` will perform better in denser graphs.

  Returns:
    Tree NetworkX graph

      A NetworkX graph representing the Gomory-Hu tree of the input graph.

  Raises:
    NetworkXNotImplemented

      Raised if the input graph is directed.

    NetworkXError

      Raised if the input graph is an empty Graph.

  See also

  `minimum_cut()`
  `maximum_flow()`

  Notes

  This implementation is based on Gusfield approach [1] to compute Gomory-Hu trees, which does not require node contractions and has the same computational complexity than the original method.

  References

  [1]

  Gusfield D: Very simple methods for all pairs network flow analysis. SIAM J Comput 19(1):143-155, 1990.

  Examples

      >>> G = nx.karate_club_graph()
      >>> nx.set_edge_attributes(G, 1, "capacity")
      >>> T = nx.gomory_hu_tree(G)
      >>> # The value of the minimum cut between any pair
      ... # of nodes in G is the minimum edge weight in the
      ... # shortest path between the two nodes in the
      ... # Gomory-Hu tree.
      ... def minimum_edge_weight_in_shortest_path(T, u, v):
      ...     path = nx.shortest_path(T, u, v, weight="weight")
      ...     return min((T[u][v]["weight"], (u, v)) for (u, v) in zip(path, path[1:]))
      >>> u, v = 0, 33
      >>> cut_value, edge = minimum_edge_weight_in_shortest_path(T, u, v)
      >>> cut_value
      10
      >>> nx.minimum_cut_value(G, u, v)
      10
      >>> # The Gomory-Hu tree also has the property that removing the
      ... # edge with the minimum weight in the shortest path between
      ... # any two nodes leaves two connected components that form
      ... # a partition of the nodes in G that defines the minimum s-t
      ... # cut.
      ... cut_value, edge = minimum_edge_weight_in_shortest_path(T, u, v)
      >>> T.remove_edge(*edge)
      >>> U, V = list(nx.connected_components(T))
      >>> # Thus U and V form a partition that defines a minimum cut
      ... # between u and v in G. You can compute the edge cut set,
      ... # that is, the set of edges that if removed from G will
      ... # disconnect u from v in G, with this information:
      ... cutset = set()
      >>> for x, nbrs in ((n, G[n]) for n in U):
      ...     cutset.update((x, y) for y in nbrs if y in V)
      >>> # Because we have set the capacities of all edges to 1
      ... # the cutset contains ten edges
      ... len(cutset)
      10
      >>> # You can use any maximum flow algorithm for the underlying
      ... # flow computations using the argument flow_func
      ... from networkx.algorithms import flow
      >>> T = nx.gomory_hu_tree(G, flow_func=flow.boykov_kolmogorov)
      >>> cut_value, edge = minimum_edge_weight_in_shortest_path(T, u, v)
      >>> cut_value
      10
      >>> nx.minimum_cut_value(G, u, v, flow_func=flow.boykov_kolmogorov)
      10