"""
Gomory-Hu tree of undirected Graphs.
"""
import networkx as nx
from networkx.utils import not_implemented_for
from .edmondskarp import edmonds_karp
from .utils import build_residual_network
default_flow_func = edmonds_karp
__all__ = ["gomory_hu_tree"]
[docs]@not_implemented_for("directed")
def gomory_hu_tree(G, capacity="capacity", flow_func=None):
r"""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
:func:`edmonds_karp`. This function performs better in sparse graphs
with right tailed degree distributions.
:func:`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.
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 Comory-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
Notes
-----
This implementation is based on Gusfield approach [1]_ to compute
Comory-Hu trees, which does not require node contractions and has
the same computational complexity than the original method.
See also
--------
:func:`minimum_cut`
:func:`maximum_flow`
References
----------
.. [1] Gusfield D: Very simple methods for all pairs network flow analysis.
SIAM J Comput 19(1):143-155, 1990.
"""
if flow_func is None:
flow_func = default_flow_func
if len(G) == 0: # empty graph
msg = "Empty Graph does not have a Gomory-Hu tree representation"
raise nx.NetworkXError(msg)
# Start the tree as a star graph with an arbitrary node at the center
tree = {}
labels = {}
iter_nodes = iter(G)
root = next(iter_nodes)
for n in iter_nodes:
tree[n] = root
# Reuse residual network
R = build_residual_network(G, capacity)
# For all the leaves in the star graph tree (that is n-1 nodes).
for source in tree:
# Find neighbor in the tree
target = tree[source]
# compute minimum cut
cut_value, partition = nx.minimum_cut(
G, source, target, capacity=capacity, flow_func=flow_func, residual=R
)
labels[(source, target)] = cut_value
# Update the tree
# Source will always be in partition[0] and target in partition[1]
for node in partition[0]:
if node != source and node in tree and tree[node] == target:
tree[node] = source
labels[node, source] = labels.get((node, target), cut_value)
#
if target != root and tree[target] in partition[0]:
labels[source, tree[target]] = labels[target, tree[target]]
labels[target, source] = cut_value
tree[source] = tree[target]
tree[target] = source
# Build the tree
T = nx.Graph()
T.add_nodes_from(G)
T.add_weighted_edges_from(((u, v, labels[u, v]) for u, v in tree.items()))
return T