prompt: We are currently solving the following issue within our repository. Here is the issue text:
  --- BEGIN ISSUE ---
  steiner_tree should accept MultiGraph
  I'm using `steiner_tree` on a road network which may have multiple edges between nodes. 
  
  It looks like `steiner_tree` will fail if passed a `MultiGraph`:
  - as a next-to-last step, edges are generated as `(u, v)` tuples pairwise [here](https://github.com/networkx/networkx/blob/master/networkx/algorithms/approximation/steinertree.py#L87)
  - before being passed to `G.edge_subgraph` which raises a `ValueError` from `nx.filter.show_multiedges`
  
  This should reproduce the issue:
  
  ```python
  import networkx as nx
  import networkx.algorithms.approximation as nxa
  
  def test_simple_steiner_tree():
      G = nx.Graph()
      G.add_weighted_edges_from([
          (1, 2, 1),
          (2, 3, 1),
          (3, 4, 1),
          (3, 5, 1)
      ])
      terminal_nodes = [2, 4, 5]
      expected_edges = [
          (2, 3),
          (3, 4),
          (3, 5)
      ]
      T = nxa.steiner_tree(G, terminal_nodes)
      assert list(T.edges) == expected_edges
  
  def test_multi_steiner_tree():
      G = nx.MultiGraph()
      G.add_weighted_edges_from([
          (1, 2, 1),
          (2, 3, 1),
          (2, 3, 999),
          (3, 4, 1),
          (3, 5, 1)
      ])
      terminal_nodes = [2, 4, 5]
      expected_edges = [
          (2, 3, 0),
          (3, 4, 0),  # first edge has weight one
          (3, 5, 0)
      ]
      T = nxa.steiner_tree(G, terminal_nodes)
  
  test_simple_steiner_tree()  # passes
  test_multi_steiner_tree()  # throws ValueError 
  ```
  
  The quick fix might be to add `@not_implemented_for('multigraph')`. 
  
  For my current purposes, the following does the trick to handle the `MultiGraph` case:
  
  ```python
  # get unique links pairwise (u, v)
  links = set(chain.from_iterable(
      pairwise(d['path'])
      for u, v, d in mst_edges
  ))
  
  # for each link in the chain
  multi_edges = []
  for u, v in links:    
      # consider each edge  between the pair of nodes,
      # keeping track of the one with the minimum weight
      # (there may be a better way - convenience functions/accessors?)
      num_edges = G.number_of_edges(u, v)
      min_k = 0
      min_weight = None
      for k in range(num_edges):
          curr_weight = G.edges[u, v, k][weight]
          if min_weight is None:
              min_weight = curr_weight
          elif curr_weight < min_weight:
              min_weight = curr_weight
              min_k = k
      multi_edges.append((u, v, min_k))
  
  # create subgraph from multi edges - list of (u, v, k)
  T = G.edge_subgraph(multi_edges)
  ```
  
  
  --- END ISSUE ---
  
  Below are some code segments, each from a relevant file. One or more of these files may contain bugs.
  --- BEGIN FILES ---
  Path: `networkx/algorithms/approximation/steinertree.py`
  Content:
  ```
  1 from itertools import chain
  2 
  3 from networkx.utils import pairwise, not_implemented_for
  4 import networkx as nx
  5 
  6 __all__ = ["metric_closure", "steiner_tree"]
  7 
  8 
  9 @not_implemented_for("directed")
  10 def metric_closure(G, weight="weight"):
  11     """  Return the metric closure of a graph.
  12 
  13     The metric closure of a graph *G* is the complete graph in which each edge
  14     is weighted by the shortest path distance between the nodes in *G* .
  15 
  16     Parameters
  17     ----------
  18     G : NetworkX graph
  19 
  20     Returns
  21     -------
  22     NetworkX graph
  23         Metric closure of the graph `G`.
  24 
  25     """
  26     M = nx.Graph()
  27 
  28     Gnodes = set(G)
  29 
  30     # check for connected graph while processing first node
  31     all_paths_iter = nx.all_pairs_dijkstra(G, weight=weight)
  32     u, (distance, path) = next(all_paths_iter)
  33     if Gnodes - set(distance):
  34         msg = "G is not a connected graph. metric_closure is not defined."
  35         raise nx.NetworkXError(msg)
  36     Gnodes.remove(u)
  37     for v in Gnodes:
  38         M.add_edge(u, v, distance=distance[v], path=path[v])
  39 
  40     # first node done -- now process the rest
  41     for u, (distance, path) in all_paths_iter:
  42         Gnodes.remove(u)
  43         for v in Gnodes:
  44             M.add_edge(u, v, distance=distance[v], path=path[v])
  45 
  46     return M
  47 
  48 
  49 @not_implemented_for("multigraph")
  50 @not_implemented_for("directed")
  51 def steiner_tree(G, terminal_nodes, weight="weight"):
  52     """ Return an approximation to the minimum Steiner tree of a graph.
  53 
  54     Parameters
  55     ----------
  56     G : NetworkX graph
  57 
  58     terminal_nodes : list
  59          A list of terminal nodes for which minimum steiner tree is
  60          to be found.
  61 
  62     Returns
  63     -------
  64     NetworkX graph
  65         Approximation to the minimum steiner tree of `G` induced by
  66         `terminal_nodes` .
  67 
  68     Notes
  69     -----
  70     Steiner tree can be approximated by computing the minimum spanning
  71     tree of the subgraph of the metric closure of the graph induced by the
  72     terminal nodes, where the metric closure of *G* is the complete graph in
  73     which each edge is weighted by the shortest path distance between the
  74     nodes in *G* .
  75     This algorithm produces a tree whose weight is within a (2 - (2 / t))
  76     factor of the weight of the optimal Steiner tree where *t* is number of
  77     terminal nodes.
  78 
  79     """
  80     # M is the subgraph of the metric closure induced by the terminal nodes of
  81     # G.
  82     M = metric_closure(G, weight=weight)
  83     # Use the 'distance' attribute of each edge provided by the metric closure
  84     # graph.
  85     H = M.subgraph(terminal_nodes)
  86     mst_edges = nx.minimum_spanning_edges(H, weight="distance", data=True)
  87     # Create an iterator over each edge in each shortest path; repeats are okay
  88     edges = chain.from_iterable(pairwise(d["path"]) for u, v, d in mst_edges)
  89     T = G.edge_subgraph(edges)
  90     return T
  91
  ```
  --- END FILES ---
  
  Please first localize the bug based on the issue statement, and then generate a patch according to the `git diff` format fenced by three backticks.
  
  Here is an example:
  
  ```diff
  diff --git a/examples/server_async.py b/examples/server_async.py
  --- a/examples/server_async.py
  +++ b/examples/server_async.py
  @@ -313,4 +313,4 @@
   
   
   if __name__ == "__main__":
  -    asyncio.run(run_async_server("."), debug=True)
  +    asyncio.run(run_async_server(), debug=True)
  diff --git a/examples/server_sync.py b/examples/server_sync.py
  --- a/examples/server_sync.py
  +++ b/examples/server_sync.py
  @@ -313,5 +313,5 @@
   
   
   if __name__ == "__main__":
  -    server = run_sync_server(".")
  +    server = run_sync_server()
       server.shutdown()
  
  ```
  
golden_diff: diff --git a/networkx/algorithms/approximation/steinertree.py b/networkx/algorithms/approximation/steinertree.py
  --- a/networkx/algorithms/approximation/steinertree.py
  +++ b/networkx/algorithms/approximation/steinertree.py
  @@ -46,11 +46,23 @@
       return M
   
   
  -@not_implemented_for("multigraph")
   @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
  @@ -67,24 +79,26 @@
   
       Notes
       -----
  -    Steiner tree can be approximated by computing the minimum spanning
  -    tree of the subgraph of the metric closure of the graph 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.
  +    For multigraphs, the edge between two nodes with minimum weight is the
  +    edge put into the Steiner tree.
   
  +
  +    References
  +    ----------
  +    .. [1] Steiner_tree_problem on Wikipedia.
  +       https://en.wikipedia.org/wiki/Steiner_tree_problem
       """
  -    # M is the subgraph of the metric closure induced by the terminal nodes of
  -    # G.
  +    # H is the subgraph induced by terminal_nodes in the metric closure M of G.
       M = metric_closure(G, weight=weight)
  -    # Use the 'distance' attribute of each edge provided by the metric closure
  -    # graph.
       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
  
verification_info: {"golden_diff": "diff --git a/networkx/algorithms/approximation/steinertree.py b/networkx/algorithms/approximation/steinertree.py\n--- a/networkx/algorithms/approximation/steinertree.py\n+++ b/networkx/algorithms/approximation/steinertree.py\n@@ -46,11 +46,23 @@\n     return M\n \n \n-@not_implemented_for(\"multigraph\")\n @not_implemented_for(\"directed\")\n def steiner_tree(G, terminal_nodes, weight=\"weight\"):\n     \"\"\" Return an approximation to the minimum Steiner tree of a graph.\n \n+    The minimum Steiner tree of `G` w.r.t a set of `terminal_nodes`\n+    is a tree within `G` that spans those nodes and has minimum size\n+    (sum of edge weights) among all such trees.\n+\n+    The minimum Steiner tree can be approximated by computing the minimum\n+    spanning tree of the subgraph of the metric closure of *G* induced by the\n+    terminal nodes, where the metric closure of *G* is the complete graph in\n+    which each edge is weighted by the shortest path distance between the\n+    nodes in *G* .\n+    This algorithm produces a tree whose weight is within a (2 - (2 / t))\n+    factor of the weight of the optimal Steiner tree where *t* is number of\n+    terminal nodes.\n+\n     Parameters\n     ----------\n     G : NetworkX graph\n@@ -67,24 +79,26 @@\n \n     Notes\n     -----\n-    Steiner tree can be approximated by computing the minimum spanning\n-    tree of the subgraph of the metric closure of the graph induced by the\n-    terminal nodes, where the metric closure of *G* is the complete graph in\n-    which each edge is weighted by the shortest path distance between the\n-    nodes in *G* .\n-    This algorithm produces a tree whose weight is within a (2 - (2 / t))\n-    factor of the weight of the optimal Steiner tree where *t* is number of\n-    terminal nodes.\n+    For multigraphs, the edge between two nodes with minimum weight is the\n+    edge put into the Steiner tree.\n \n+\n+    References\n+    ----------\n+    .. [1] Steiner_tree_problem on Wikipedia.\n+       https://en.wikipedia.org/wiki/Steiner_tree_problem\n     \"\"\"\n-    # M is the subgraph of the metric closure induced by the terminal nodes of\n-    # G.\n+    # H is the subgraph induced by terminal_nodes in the metric closure M of G.\n     M = metric_closure(G, weight=weight)\n-    # Use the 'distance' attribute of each edge provided by the metric closure\n-    # graph.\n     H = M.subgraph(terminal_nodes)\n+    # Use the 'distance' attribute of each edge provided by M.\n     mst_edges = nx.minimum_spanning_edges(H, weight=\"distance\", data=True)\n     # Create an iterator over each edge in each shortest path; repeats are okay\n     edges = chain.from_iterable(pairwise(d[\"path\"]) for u, v, d in mst_edges)\n+    # For multigraph we should add the minimal weight edge keys\n+    if G.is_multigraph():\n+        edges = (\n+            (u, v, min(G[u][v], key=lambda k: G[u][v][k][weight])) for u, v in edges\n+        )\n     T = G.edge_subgraph(edges)\n     return T\n", "issue": "steiner_tree should accept MultiGraph\nI'm using `steiner_tree` on a road network which may have multiple edges between nodes. \r\n\r\nIt looks like `steiner_tree` will fail if passed a `MultiGraph`:\r\n- as a next-to-last step, edges are generated as `(u, v)` tuples pairwise [here](https://github.com/networkx/networkx/blob/master/networkx/algorithms/approximation/steinertree.py#L87)\r\n- before being passed to `G.edge_subgraph` which raises a `ValueError` from `nx.filter.show_multiedges`\r\n\r\nThis should reproduce the issue:\r\n\r\n```python\r\nimport networkx as nx\r\nimport networkx.algorithms.approximation as nxa\r\n\r\ndef test_simple_steiner_tree():\r\n    G = nx.Graph()\r\n    G.add_weighted_edges_from([\r\n        (1, 2, 1),\r\n        (2, 3, 1),\r\n        (3, 4, 1),\r\n        (3, 5, 1)\r\n    ])\r\n    terminal_nodes = [2, 4, 5]\r\n    expected_edges = [\r\n        (2, 3),\r\n        (3, 4),\r\n        (3, 5)\r\n    ]\r\n    T = nxa.steiner_tree(G, terminal_nodes)\r\n    assert list(T.edges) == expected_edges\r\n\r\ndef test_multi_steiner_tree():\r\n    G = nx.MultiGraph()\r\n    G.add_weighted_edges_from([\r\n        (1, 2, 1),\r\n        (2, 3, 1),\r\n        (2, 3, 999),\r\n        (3, 4, 1),\r\n        (3, 5, 1)\r\n    ])\r\n    terminal_nodes = [2, 4, 5]\r\n    expected_edges = [\r\n        (2, 3, 0),\r\n        (3, 4, 0),  # first edge has weight one\r\n        (3, 5, 0)\r\n    ]\r\n    T = nxa.steiner_tree(G, terminal_nodes)\r\n\r\ntest_simple_steiner_tree()  # passes\r\ntest_multi_steiner_tree()  # throws ValueError \r\n```\r\n\r\nThe quick fix might be to add `@not_implemented_for('multigraph')`. \r\n\r\nFor my current purposes, the following does the trick to handle the `MultiGraph` case:\r\n\r\n```python\r\n# get unique links pairwise (u, v)\r\nlinks = set(chain.from_iterable(\r\n    pairwise(d['path'])\r\n    for u, v, d in mst_edges\r\n))\r\n\r\n# for each link in the chain\r\nmulti_edges = []\r\nfor u, v in links:    \r\n    # consider each edge  between the pair of nodes,\r\n    # keeping track of the one with the minimum weight\r\n    # (there may be a better way - convenience functions/accessors?)\r\n    num_edges = G.number_of_edges(u, v)\r\n    min_k = 0\r\n    min_weight = None\r\n    for k in range(num_edges):\r\n        curr_weight = G.edges[u, v, k][weight]\r\n        if min_weight is None:\r\n            min_weight = curr_weight\r\n        elif curr_weight < min_weight:\r\n            min_weight = curr_weight\r\n            min_k = k\r\n    multi_edges.append((u, v, min_k))\r\n\r\n# create subgraph from multi edges - list of (u, v, k)\r\nT = G.edge_subgraph(multi_edges)\r\n```\r\n\n", "before_files": [{"content": "from itertools import chain\n\nfrom networkx.utils import pairwise, not_implemented_for\nimport networkx as nx\n\n__all__ = [\"metric_closure\", \"steiner_tree\"]\n\n\n@not_implemented_for(\"directed\")\ndef metric_closure(G, weight=\"weight\"):\n    \"\"\"  Return the metric closure of a graph.\n\n    The metric closure of a graph *G* is the complete graph in which each edge\n    is weighted by the shortest path distance between the nodes in *G* .\n\n    Parameters\n    ----------\n    G : NetworkX graph\n\n    Returns\n    -------\n    NetworkX graph\n        Metric closure of the graph `G`.\n\n    \"\"\"\n    M = nx.Graph()\n\n    Gnodes = set(G)\n\n    # check for connected graph while processing first node\n    all_paths_iter = nx.all_pairs_dijkstra(G, weight=weight)\n    u, (distance, path) = next(all_paths_iter)\n    if Gnodes - set(distance):\n        msg = \"G is not a connected graph. metric_closure is not defined.\"\n        raise nx.NetworkXError(msg)\n    Gnodes.remove(u)\n    for v in Gnodes:\n        M.add_edge(u, v, distance=distance[v], path=path[v])\n\n    # first node done -- now process the rest\n    for u, (distance, path) in all_paths_iter:\n        Gnodes.remove(u)\n        for v in Gnodes:\n            M.add_edge(u, v, distance=distance[v], path=path[v])\n\n    return M\n\n\n@not_implemented_for(\"multigraph\")\n@not_implemented_for(\"directed\")\ndef steiner_tree(G, terminal_nodes, weight=\"weight\"):\n    \"\"\" Return an approximation to the minimum Steiner tree of a graph.\n\n    Parameters\n    ----------\n    G : NetworkX graph\n\n    terminal_nodes : list\n         A list of terminal nodes for which minimum steiner tree is\n         to be found.\n\n    Returns\n    -------\n    NetworkX graph\n        Approximation to the minimum steiner tree of `G` induced by\n        `terminal_nodes` .\n\n    Notes\n    -----\n    Steiner tree can be approximated by computing the minimum spanning\n    tree of the subgraph of the metric closure of the graph induced by the\n    terminal nodes, where the metric closure of *G* is the complete graph in\n    which each edge is weighted by the shortest path distance between the\n    nodes in *G* .\n    This algorithm produces a tree whose weight is within a (2 - (2 / t))\n    factor of the weight of the optimal Steiner tree where *t* is number of\n    terminal nodes.\n\n    \"\"\"\n    # M is the subgraph of the metric closure induced by the terminal nodes of\n    # G.\n    M = metric_closure(G, weight=weight)\n    # Use the 'distance' attribute of each edge provided by the metric closure\n    # graph.\n    H = M.subgraph(terminal_nodes)\n    mst_edges = nx.minimum_spanning_edges(H, weight=\"distance\", data=True)\n    # Create an iterator over each edge in each shortest path; repeats are okay\n    edges = chain.from_iterable(pairwise(d[\"path\"]) for u, v, d in mst_edges)\n    T = G.edge_subgraph(edges)\n    return T\n", "path": "networkx/algorithms/approximation/steinertree.py"}], "after_files": [{"content": "from itertools import chain\n\nfrom networkx.utils import pairwise, not_implemented_for\nimport networkx as nx\n\n__all__ = [\"metric_closure\", \"steiner_tree\"]\n\n\n@not_implemented_for(\"directed\")\ndef metric_closure(G, weight=\"weight\"):\n    \"\"\"  Return the metric closure of a graph.\n\n    The metric closure of a graph *G* is the complete graph in which each edge\n    is weighted by the shortest path distance between the nodes in *G* .\n\n    Parameters\n    ----------\n    G : NetworkX graph\n\n    Returns\n    -------\n    NetworkX graph\n        Metric closure of the graph `G`.\n\n    \"\"\"\n    M = nx.Graph()\n\n    Gnodes = set(G)\n\n    # check for connected graph while processing first node\n    all_paths_iter = nx.all_pairs_dijkstra(G, weight=weight)\n    u, (distance, path) = next(all_paths_iter)\n    if Gnodes - set(distance):\n        msg = \"G is not a connected graph. metric_closure is not defined.\"\n        raise nx.NetworkXError(msg)\n    Gnodes.remove(u)\n    for v in Gnodes:\n        M.add_edge(u, v, distance=distance[v], path=path[v])\n\n    # first node done -- now process the rest\n    for u, (distance, path) in all_paths_iter:\n        Gnodes.remove(u)\n        for v in Gnodes:\n            M.add_edge(u, v, distance=distance[v], path=path[v])\n\n    return M\n\n\n@not_implemented_for(\"directed\")\ndef steiner_tree(G, terminal_nodes, weight=\"weight\"):\n    \"\"\" Return an approximation to the minimum Steiner tree of a graph.\n\n    The minimum Steiner tree of `G` w.r.t a set of `terminal_nodes`\n    is a tree within `G` that spans those nodes and has minimum size\n    (sum of edge weights) among all such trees.\n\n    The minimum Steiner tree can be approximated by computing the minimum\n    spanning tree of the subgraph of the metric closure of *G* induced by the\n    terminal nodes, where the metric closure of *G* is the complete graph in\n    which each edge is weighted by the shortest path distance between the\n    nodes in *G* .\n    This algorithm produces a tree whose weight is within a (2 - (2 / t))\n    factor of the weight of the optimal Steiner tree where *t* is number of\n    terminal nodes.\n\n    Parameters\n    ----------\n    G : NetworkX graph\n\n    terminal_nodes : list\n         A list of terminal nodes for which minimum steiner tree is\n         to be found.\n\n    Returns\n    -------\n    NetworkX graph\n        Approximation to the minimum steiner tree of `G` induced by\n        `terminal_nodes` .\n\n    Notes\n    -----\n    For multigraphs, the edge between two nodes with minimum weight is the\n    edge put into the Steiner tree.\n\n\n    References\n    ----------\n    .. [1] Steiner_tree_problem on Wikipedia.\n       https://en.wikipedia.org/wiki/Steiner_tree_problem\n    \"\"\"\n    # H is the subgraph induced by terminal_nodes in the metric closure M of G.\n    M = metric_closure(G, weight=weight)\n    H = M.subgraph(terminal_nodes)\n    # Use the 'distance' attribute of each edge provided by M.\n    mst_edges = nx.minimum_spanning_edges(H, weight=\"distance\", data=True)\n    # Create an iterator over each edge in each shortest path; repeats are okay\n    edges = chain.from_iterable(pairwise(d[\"path\"]) for u, v, d in mst_edges)\n    # For multigraph we should add the minimal weight edge keys\n    if G.is_multigraph():\n        edges = (\n            (u, v, min(G[u][v], key=lambda k: G[u][v][k][weight])) for u, v in edges\n        )\n    T = G.edge_subgraph(edges)\n    return T\n", "path": "networkx/algorithms/approximation/steinertree.py"}]}