# Source code for networkx.algorithms.approximation.steinertree

```from itertools import chain

from networkx.utils import pairwise, not_implemented_for
import networkx as nx

__all__ = ["metric_closure", "steiner_tree"]

[docs]@not_implemented_for("directed")
def metric_closure(G, weight="weight"):
"""Return the metric closure of a graph.

The metric closure of a graph *G* is the complete graph in which each edge
is weighted by the shortest path distance between the nodes in *G* .

Parameters
----------
G : NetworkX graph

Returns
-------
NetworkX graph
Metric closure of the graph `G`.

"""
M = nx.Graph()

Gnodes = set(G)

# check for connected graph while processing first node
all_paths_iter = nx.all_pairs_dijkstra(G, weight=weight)
u, (distance, path) = next(all_paths_iter)
if Gnodes - set(distance):
msg = "G is not a connected graph. metric_closure is not defined."
raise nx.NetworkXError(msg)
Gnodes.remove(u)
for v in Gnodes:

# first node done -- now process the rest
for u, (distance, path) in all_paths_iter:
Gnodes.remove(u)
for v in Gnodes:

return M

[docs]@not_implemented_for("directed")
def steiner_tree(G, terminal_nodes, weight="weight"):
"""Return an approximation to the minimum Steiner tree of a graph.

The minimum Steiner tree of `G` w.r.t a set of `terminal_nodes`
is a tree within `G` that spans those nodes and has minimum size
(sum of edge weights) among all such trees.

The minimum Steiner tree can be approximated by computing the minimum
spanning tree of the subgraph of the metric closure of *G* induced by the
terminal nodes, where the metric closure of *G* is the complete graph in
which each edge is weighted by the shortest path distance between the
nodes in *G* .
This algorithm produces a tree whose weight is within a (2 - (2 / t))
factor of the weight of the optimal Steiner tree where *t* is number of
terminal nodes.

Parameters
----------
G : NetworkX graph

terminal_nodes : list
A list of terminal nodes for which minimum steiner tree is
to be found.

Returns
-------
NetworkX graph
Approximation to the minimum steiner tree of `G` induced by
`terminal_nodes` .

Notes
-----
For multigraphs, the edge between two nodes with minimum weight is the
edge put into the Steiner tree.

References
----------
..  Steiner_tree_problem on Wikipedia.
https://en.wikipedia.org/wiki/Steiner_tree_problem
"""
# H is the subgraph induced by terminal_nodes in the metric closure M of G.
M = metric_closure(G, weight=weight)
H = M.subgraph(terminal_nodes)
# Use the 'distance' attribute of each edge provided by M.
mst_edges = nx.minimum_spanning_edges(H, weight="distance", data=True)
# Create an iterator over each edge in each shortest path; repeats are okay
edges = chain.from_iterable(pairwise(d["path"]) for u, v, d in mst_edges)
# For multigraph we should add the minimal weight edge keys
if G.is_multigraph():
edges = (
(u, v, min(G[u][v], key=lambda k: G[u][v][k][weight])) for u, v in edges
)
T = G.edge_subgraph(edges)
return T
```