Upload folder using huggingface_hub (part 10)

.gitattributes

@@ -230,3 +230,4 @@ saved_model/**/* filter=lfs diff=lfs merge=lfs -text
 .venv/lib/python3.12/site-packages/mpmath/__pycache__/function_docs.cpython-312.pyc filter=lfs diff=lfs merge=lfs -text
 .venv/lib/python3.12/site-packages/mpmath/tests/__pycache__/test_fp.cpython-312.pyc filter=lfs diff=lfs merge=lfs -text
 .venv/lib/python3.12/site-packages/mpmath/tests/__pycache__/test_functions2.cpython-312.pyc filter=lfs diff=lfs merge=lfs -text
+.venv/lib/python3.12/site-packages/networkx/drawing/__pycache__/nx_pylab.cpython-312.pyc filter=lfs diff=lfs merge=lfs -text

.venv/lib/python3.12/site-packages/networkx/algorithms/approximation/__init__.py

@@ -0,0 +1,26 @@
+"""Approximations of graph properties and Heuristic methods for optimization.
+
+The functions in this class are not imported into the top-level ``networkx``
+namespace so the easiest way to use them is with::
+
+    >>> from networkx.algorithms import approximation
+
+Another option is to import the specific function with
+``from networkx.algorithms.approximation import function_name``.
+
+"""
+
+from networkx.algorithms.approximation.clustering_coefficient import *
+from networkx.algorithms.approximation.clique import *
+from networkx.algorithms.approximation.connectivity import *
+from networkx.algorithms.approximation.distance_measures import *
+from networkx.algorithms.approximation.dominating_set import *
+from networkx.algorithms.approximation.kcomponents import *
+from networkx.algorithms.approximation.matching import *
+from networkx.algorithms.approximation.ramsey import *
+from networkx.algorithms.approximation.steinertree import *
+from networkx.algorithms.approximation.traveling_salesman import *
+from networkx.algorithms.approximation.treewidth import *
+from networkx.algorithms.approximation.vertex_cover import *
+from networkx.algorithms.approximation.maxcut import *
+from networkx.algorithms.approximation.density import *

.venv/lib/python3.12/site-packages/networkx/algorithms/approximation/clique.py

@@ -0,0 +1,259 @@
+"""Functions for computing large cliques and maximum independent sets."""
+
+import networkx as nx
+from networkx.algorithms.approximation import ramsey
+from networkx.utils import not_implemented_for
+
+__all__ = [
+    "clique_removal",
+    "max_clique",
+    "large_clique_size",
+    "maximum_independent_set",
+]
+
+
+@not_implemented_for("directed")
+@not_implemented_for("multigraph")
+@nx._dispatchable
+def maximum_independent_set(G):
+    """Returns an approximate maximum independent set.
+
+    Independent set or stable set is a set of vertices in a graph, no two of
+    which are adjacent. That is, it is a set I of vertices such that for every
+    two vertices in I, there is no edge connecting the two. Equivalently, each
+    edge in the graph has at most one endpoint in I. The size of an independent
+    set is the number of vertices it contains [1]_.
+
+    A maximum independent set is a largest independent set for a given graph G
+    and its size is denoted $\\alpha(G)$. The problem of finding such a set is called
+    the maximum independent set problem and is an NP-hard optimization problem.
+    As such, it is unlikely that there exists an efficient algorithm for finding
+    a maximum independent set of a graph.
+
+    The Independent Set algorithm is based on [2]_.
+
+    Parameters
+    ----------
+    G : NetworkX graph
+        Undirected graph
+
+    Returns
+    -------
+    iset : Set
+        The apx-maximum independent set
+
+    Examples
+    --------
+    >>> G = nx.path_graph(10)
+    >>> nx.approximation.maximum_independent_set(G)
+    {0, 2, 4, 6, 9}
+
+    Raises
+    ------
+    NetworkXNotImplemented
+        If the graph is directed or is a multigraph.
+
+    Notes
+    -----
+    Finds the $O(|V|/(log|V|)^2)$ apx of independent set in the worst case.
+
+    References
+    ----------
+    .. [1] `Wikipedia: Independent set
+        <https://en.wikipedia.org/wiki/Independent_set_(graph_theory)>`_
+    .. [2] Boppana, R., & Halldórsson, M. M. (1992).
+       Approximating maximum independent sets by excluding subgraphs.
+       BIT Numerical Mathematics, 32(2), 180–196. Springer.
+    """
+    iset, _ = clique_removal(G)
+    return iset
+
+
+@not_implemented_for("directed")
+@not_implemented_for("multigraph")
+@nx._dispatchable
+def max_clique(G):
+    r"""Find the Maximum Clique
+
+    Finds the $O(|V|/(log|V|)^2)$ apx of maximum clique/independent set
+    in the worst case.
+
+    Parameters
+    ----------
+    G : NetworkX graph
+        Undirected graph
+
+    Returns
+    -------
+    clique : set
+        The apx-maximum clique of the graph
+
+    Examples
+    --------
+    >>> G = nx.path_graph(10)
+    >>> nx.approximation.max_clique(G)
+    {8, 9}
+
+    Raises
+    ------
+    NetworkXNotImplemented
+        If the graph is directed or is a multigraph.
+
+    Notes
+    -----
+    A clique in an undirected graph G = (V, E) is a subset of the vertex set
+    `C \subseteq V` such that for every two vertices in C there exists an edge
+    connecting the two. This is equivalent to saying that the subgraph
+    induced by C is complete (in some cases, the term clique may also refer
+    to the subgraph).
+
+    A maximum clique is a clique of the largest possible size in a given graph.
+    The clique number `\omega(G)` of a graph G is the number of
+    vertices in a maximum clique in G. The intersection number of
+    G is the smallest number of cliques that together cover all edges of G.
+
+    https://en.wikipedia.org/wiki/Maximum_clique
+
+    References
+    ----------
+    .. [1] Boppana, R., & Halldórsson, M. M. (1992).
+        Approximating maximum independent sets by excluding subgraphs.
+        BIT Numerical Mathematics, 32(2), 180–196. Springer.
+        doi:10.1007/BF01994876
+    """
+    # finding the maximum clique in a graph is equivalent to finding
+    # the independent set in the complementary graph
+    cgraph = nx.complement(G)
+    iset, _ = clique_removal(cgraph)
+    return iset
+
+
+@not_implemented_for("directed")
+@not_implemented_for("multigraph")
+@nx._dispatchable
+def clique_removal(G):
+    r"""Repeatedly remove cliques from the graph.
+
+    Results in a $O(|V|/(\log |V|)^2)$ approximation of maximum clique
+    and independent set. Returns the largest independent set found, along
+    with found maximal cliques.
+
+    Parameters
+    ----------
+    G : NetworkX graph
+        Undirected graph
+
+    Returns
+    -------
+    max_ind_cliques : (set, list) tuple
+        2-tuple of Maximal Independent Set and list of maximal cliques (sets).
+
+    Examples
+    --------
+    >>> G = nx.path_graph(10)
+    >>> nx.approximation.clique_removal(G)
+    ({0, 2, 4, 6, 9}, [{0, 1}, {2, 3}, {4, 5}, {6, 7}, {8, 9}])
+
+    Raises
+    ------
+    NetworkXNotImplemented
+        If the graph is directed or is a multigraph.
+
+    References
+    ----------
+    .. [1] Boppana, R., & Halldórsson, M. M. (1992).
+        Approximating maximum independent sets by excluding subgraphs.
+        BIT Numerical Mathematics, 32(2), 180–196. Springer.
+    """
+    graph = G.copy()
+    c_i, i_i = ramsey.ramsey_R2(graph)
+    cliques = [c_i]
+    isets = [i_i]
+    while graph:
+        graph.remove_nodes_from(c_i)
+        c_i, i_i = ramsey.ramsey_R2(graph)
+        if c_i:
+            cliques.append(c_i)
+        if i_i:
+            isets.append(i_i)
+    # Determine the largest independent set as measured by cardinality.
+    maxiset = max(isets, key=len)
+    return maxiset, cliques
+
+
+@not_implemented_for("directed")
+@not_implemented_for("multigraph")
+@nx._dispatchable
+def large_clique_size(G):
+    """Find the size of a large clique in a graph.
+
+    A *clique* is a subset of nodes in which each pair of nodes is
+    adjacent. This function is a heuristic for finding the size of a
+    large clique in the graph.
+
+    Parameters
+    ----------
+    G : NetworkX graph
+
+    Returns
+    -------
+    k: integer
+       The size of a large clique in the graph.
+
+    Examples
+    --------
+    >>> G = nx.path_graph(10)
+    >>> nx.approximation.large_clique_size(G)
+    2
+
+    Raises
+    ------
+    NetworkXNotImplemented
+        If the graph is directed or is a multigraph.
+
+    Notes
+    -----
+    This implementation is from [1]_. Its worst case time complexity is
+    :math:`O(n d^2)`, where *n* is the number of nodes in the graph and
+    *d* is the maximum degree.
+
+    This function is a heuristic, which means it may work well in
+    practice, but there is no rigorous mathematical guarantee on the
+    ratio between the returned number and the actual largest clique size
+    in the graph.
+
+    References
+    ----------
+    .. [1] Pattabiraman, Bharath, et al.
+       "Fast Algorithms for the Maximum Clique Problem on Massive Graphs
+       with Applications to Overlapping Community Detection."
+       *Internet Mathematics* 11.4-5 (2015): 421--448.
+       <https://doi.org/10.1080/15427951.2014.986778>
+
+    See also
+    --------
+
+    :func:`networkx.algorithms.approximation.clique.max_clique`
+        A function that returns an approximate maximum clique with a
+        guarantee on the approximation ratio.
+
+    :mod:`networkx.algorithms.clique`
+        Functions for finding the exact maximum clique in a graph.
+
+    """
+    degrees = G.degree
+
+    def _clique_heuristic(G, U, size, best_size):
+        if not U:
+            return max(best_size, size)
+        u = max(U, key=degrees)
+        U.remove(u)
+        N_prime = {v for v in G[u] if degrees[v] >= best_size}
+        return _clique_heuristic(G, U & N_prime, size + 1, best_size)
+
+    best_size = 0
+    nodes = (u for u in G if degrees[u] >= best_size)
+    for u in nodes:
+        neighbors = {v for v in G[u] if degrees[v] >= best_size}
+        best_size = _clique_heuristic(G, neighbors, 1, best_size)
+    return best_size

.venv/lib/python3.12/site-packages/networkx/algorithms/approximation/clustering_coefficient.py

@@ -0,0 +1,71 @@
+import networkx as nx
+from networkx.utils import not_implemented_for, py_random_state
+
+__all__ = ["average_clustering"]
+
+
+@not_implemented_for("directed")
+@py_random_state(2)
+@nx._dispatchable(name="approximate_average_clustering")
+def average_clustering(G, trials=1000, seed=None):
+    r"""Estimates the average clustering coefficient of G.
+
+    The local clustering of each node in `G` is the fraction of triangles
+    that actually exist over all possible triangles in its neighborhood.
+    The average clustering coefficient of a graph `G` is the mean of
+    local clusterings.
+
+    This function finds an approximate average clustering coefficient
+    for G by repeating `n` times (defined in `trials`) the following
+    experiment: choose a node at random, choose two of its neighbors
+    at random, and check if they are connected. The approximate
+    coefficient is the fraction of triangles found over the number
+    of trials [1]_.
+
+    Parameters
+    ----------
+    G : NetworkX graph
+
+    trials : integer
+        Number of trials to perform (default 1000).
+
+    seed : integer, random_state, or None (default)
+        Indicator of random number generation state.
+        See :ref:`Randomness<randomness>`.
+
+    Returns
+    -------
+    c : float
+        Approximated average clustering coefficient.
+
+    Examples
+    --------
+    >>> from networkx.algorithms import approximation
+    >>> G = nx.erdos_renyi_graph(10, 0.2, seed=10)
+    >>> approximation.average_clustering(G, trials=1000, seed=10)
+    0.214
+
+    Raises
+    ------
+    NetworkXNotImplemented
+        If G is directed.
+
+    References
+    ----------
+    .. [1] Schank, Thomas, and Dorothea Wagner. Approximating clustering
+       coefficient and transitivity. Universität Karlsruhe, Fakultät für
+       Informatik, 2004.
+       https://doi.org/10.5445/IR/1000001239
+
+    """
+    n = len(G)
+    triangles = 0
+    nodes = list(G)
+    for i in [int(seed.random() * n) for i in range(trials)]:
+        nbrs = list(G[nodes[i]])
+        if len(nbrs) < 2:
+            continue
+        u, v = seed.sample(nbrs, 2)
+        if u in G[v]:
+            triangles += 1
+    return triangles / trials

.venv/lib/python3.12/site-packages/networkx/algorithms/approximation/connectivity.py

@@ -0,0 +1,412 @@
+"""Fast approximation for node connectivity"""
+
+import itertools
+from operator import itemgetter
+
+import networkx as nx
+
+__all__ = [
+    "local_node_connectivity",
+    "node_connectivity",
+    "all_pairs_node_connectivity",
+]
+
+
+@nx._dispatchable(name="approximate_local_node_connectivity")
+def local_node_connectivity(G, source, target, cutoff=None):
+    """Compute node connectivity between source and target.
+
+    Pairwise or local node connectivity between two distinct and nonadjacent
+    nodes is the minimum number of nodes that must be removed (minimum
+    separating cutset) to disconnect them. By Menger's theorem, this is equal
+    to the number of node independent paths (paths that share no nodes other
+    than source and target). Which is what we compute in this function.
+
+    This algorithm is a fast approximation that gives an strict lower
+    bound on the actual number of node independent paths between two nodes [1]_.
+    It works for both directed and undirected graphs.
+
+    Parameters
+    ----------
+
+    G : NetworkX graph
+
+    source : node
+        Starting node for node connectivity
+
+    target : node
+        Ending node for node connectivity
+
+    cutoff : integer
+        Maximum node connectivity to consider. If None, the minimum degree
+        of source or target is used as a cutoff. Default value None.
+
+    Returns
+    -------
+    k: integer
+       pairwise node connectivity
+
+    Examples
+    --------
+    >>> # Platonic octahedral graph has node connectivity 4
+    >>> # for each non adjacent node pair
+    >>> from networkx.algorithms import approximation as approx
+    >>> G = nx.octahedral_graph()
+    >>> approx.local_node_connectivity(G, 0, 5)
+    4
+
+    Notes
+    -----
+    This algorithm [1]_ finds node independents paths between two nodes by
+    computing their shortest path using BFS, marking the nodes of the path
+    found as 'used' and then searching other shortest paths excluding the
+    nodes marked as used until no more paths exist. It is not exact because
+    a shortest path could use nodes that, if the path were longer, may belong
+    to two different node independent paths. Thus it only guarantees an
+    strict lower bound on node connectivity.
+
+    Note that the authors propose a further refinement, losing accuracy and
+    gaining speed, which is not implemented yet.
+
+    See also
+    --------
+    all_pairs_node_connectivity
+    node_connectivity
+
+    References
+    ----------
+    .. [1] White, Douglas R., and Mark Newman. 2001 A Fast Algorithm for
+        Node-Independent Paths. Santa Fe Institute Working Paper #01-07-035
+        http://eclectic.ss.uci.edu/~drwhite/working.pdf
+
+    """
+    if target == source:
+        raise nx.NetworkXError("source and target have to be different nodes.")
+
+    # Maximum possible node independent paths
+    if G.is_directed():
+        possible = min(G.out_degree(source), G.in_degree(target))
+    else:
+        possible = min(G.degree(source), G.degree(target))
+
+    K = 0
+    if not possible:
+        return K
+
+    if cutoff is None:
+        cutoff = float("inf")
+
+    exclude = set()
+    for i in range(min(possible, cutoff)):
+        try:
+            path = _bidirectional_shortest_path(G, source, target, exclude)
+            exclude.update(set(path))
+            K += 1
+        except nx.NetworkXNoPath:
+            break
+
+    return K
+
+
+@nx._dispatchable(name="approximate_node_connectivity")
+def node_connectivity(G, s=None, t=None):
+    r"""Returns an approximation for node connectivity for a graph or digraph G.
+
+    Node connectivity is equal to the minimum number of nodes that
+    must be removed to disconnect G or render it trivial. By Menger's theorem,
+    this is equal to the number of node independent paths (paths that
+    share no nodes other than source and target).
+
+    If source and target nodes are provided, this function returns the
+    local node connectivity: the minimum number of nodes that must be
+    removed to break all paths from source to target in G.
+
+    This algorithm is based on a fast approximation that gives an strict lower
+    bound on the actual number of node independent paths between two nodes [1]_.
+    It works for both directed and undirected graphs.
+
+    Parameters
+    ----------
+    G : NetworkX graph
+        Undirected graph
+
+    s : node
+        Source node. Optional. Default value: None.
+
+    t : node
+        Target node. Optional. Default value: None.
+
+    Returns
+    -------
+    K : integer
+        Node connectivity of G, or local node connectivity if source
+        and target are provided.
+
+    Examples
+    --------
+    >>> # Platonic octahedral graph is 4-node-connected
+    >>> from networkx.algorithms import approximation as approx
+    >>> G = nx.octahedral_graph()
+    >>> approx.node_connectivity(G)
+    4
+
+    Notes
+    -----
+    This algorithm [1]_ finds node independents paths between two nodes by
+    computing their shortest path using BFS, marking the nodes of the path
+    found as 'used' and then searching other shortest paths excluding the
+    nodes marked as used until no more paths exist. It is not exact because
+    a shortest path could use nodes that, if the path were longer, may belong
+    to two different node independent paths. Thus it only guarantees an
+    strict lower bound on node connectivity.
+
+    See also
+    --------
+    all_pairs_node_connectivity
+    local_node_connectivity
+
+    References
+    ----------
+    .. [1] White, Douglas R., and Mark Newman. 2001 A Fast Algorithm for
+        Node-Independent Paths. Santa Fe Institute Working Paper #01-07-035
+        http://eclectic.ss.uci.edu/~drwhite/working.pdf
+
+    """
+    if (s is not None and t is None) or (s is None and t is not None):
+        raise nx.NetworkXError("Both source and target must be specified.")
+
+    # Local node connectivity
+    if s is not None and t is not None:
+        if s not in G:
+            raise nx.NetworkXError(f"node {s} not in graph")
+        if t not in G:
+            raise nx.NetworkXError(f"node {t} not in graph")
+        return local_node_connectivity(G, s, t)
+
+    # Global node connectivity
+    if G.is_directed():
+        connected_func = nx.is_weakly_connected
+        iter_func = itertools.permutations
+
+        def neighbors(v):
+            return itertools.chain(G.predecessors(v), G.successors(v))
+
+    else:
+        connected_func = nx.is_connected
+        iter_func = itertools.combinations
+        neighbors = G.neighbors
+
+    if not connected_func(G):
+        return 0
+
+    # Choose a node with minimum degree
+    v, minimum_degree = min(G.degree(), key=itemgetter(1))
+    # Node connectivity is bounded by minimum degree
+    K = minimum_degree
+    # compute local node connectivity with all non-neighbors nodes
+    # and store the minimum
+    for w in set(G) - set(neighbors(v)) - {v}:
+        K = min(K, local_node_connectivity(G, v, w, cutoff=K))
+    # Same for non adjacent pairs of neighbors of v
+    for x, y in iter_func(neighbors(v), 2):
+        if y not in G[x] and x != y:
+            K = min(K, local_node_connectivity(G, x, y, cutoff=K))
+    return K
+
+
+@nx._dispatchable(name="approximate_all_pairs_node_connectivity")
+def all_pairs_node_connectivity(G, nbunch=None, cutoff=None):
+    """Compute node connectivity between all pairs of nodes.
+
+    Pairwise or local node connectivity between two distinct and nonadjacent
+    nodes is the minimum number of nodes that must be removed (minimum
+    separating cutset) to disconnect them. By Menger's theorem, this is equal
+    to the number of node independent paths (paths that share no nodes other
+    than source and target). Which is what we compute in this function.
+
+    This algorithm is a fast approximation that gives an strict lower
+    bound on the actual number of node independent paths between two nodes [1]_.
+    It works for both directed and undirected graphs.
+
+
+    Parameters
+    ----------
+    G : NetworkX graph
+
+    nbunch: container
+        Container of nodes. If provided node connectivity will be computed
+        only over pairs of nodes in nbunch.
+
+    cutoff : integer
+        Maximum node connectivity to consider. If None, the minimum degree
+        of source or target is used as a cutoff in each pair of nodes.
+        Default value None.
+
+    Returns
+    -------
+    K : dictionary
+        Dictionary, keyed by source and target, of pairwise node connectivity
+
+    Examples
+    --------
+    A 3 node cycle with one extra node attached has connectivity 2 between all
+    nodes in the cycle and connectivity 1 between the extra node and the rest:
+
+    >>> G = nx.cycle_graph(3)
+    >>> G.add_edge(2, 3)
+    >>> import pprint  # for nice dictionary formatting
+    >>> pprint.pprint(nx.all_pairs_node_connectivity(G))
+    {0: {1: 2, 2: 2, 3: 1},
+     1: {0: 2, 2: 2, 3: 1},
+     2: {0: 2, 1: 2, 3: 1},
+     3: {0: 1, 1: 1, 2: 1}}
+
+    See Also
+    --------
+    local_node_connectivity
+    node_connectivity
+
+    References
+    ----------
+    .. [1] White, Douglas R., and Mark Newman. 2001 A Fast Algorithm for
+        Node-Independent Paths. Santa Fe Institute Working Paper #01-07-035
+        http://eclectic.ss.uci.edu/~drwhite/working.pdf
+    """
+    if nbunch is None:
+        nbunch = G
+    else:
+        nbunch = set(nbunch)
+
+    directed = G.is_directed()
+    if directed:
+        iter_func = itertools.permutations
+    else:
+        iter_func = itertools.combinations
+
+    all_pairs = {n: {} for n in nbunch}
+
+    for u, v in iter_func(nbunch, 2):
+        k = local_node_connectivity(G, u, v, cutoff=cutoff)
+        all_pairs[u][v] = k
+        if not directed:
+            all_pairs[v][u] = k
+
+    return all_pairs
+
+
+def _bidirectional_shortest_path(G, source, target, exclude):
+    """Returns shortest path between source and target ignoring nodes in the
+    container 'exclude'.
+
+    Parameters
+    ----------
+
+    G : NetworkX graph
+
+    source : node
+        Starting node for path
+
+    target : node
+        Ending node for path
+
+    exclude: container
+        Container for nodes to exclude from the search for shortest paths
+
+    Returns
+    -------
+    path: list
+        Shortest path between source and target ignoring nodes in 'exclude'
+
+    Raises
+    ------
+    NetworkXNoPath
+        If there is no path or if nodes are adjacent and have only one path
+        between them
+
+    Notes
+    -----
+    This function and its helper are originally from
+    networkx.algorithms.shortest_paths.unweighted and are modified to
+    accept the extra parameter 'exclude', which is a container for nodes
+    already used in other paths that should be ignored.
+
+    References
+    ----------
+    .. [1] White, Douglas R., and Mark Newman. 2001 A Fast Algorithm for
+        Node-Independent Paths. Santa Fe Institute Working Paper #01-07-035
+        http://eclectic.ss.uci.edu/~drwhite/working.pdf
+
+    """
+    # call helper to do the real work
+    results = _bidirectional_pred_succ(G, source, target, exclude)
+    pred, succ, w = results
+
+    # build path from pred+w+succ
+    path = []
+    # from source to w
+    while w is not None:
+        path.append(w)
+        w = pred[w]
+    path.reverse()
+    # from w to target
+    w = succ[path[-1]]
+    while w is not None:
+        path.append(w)
+        w = succ[w]
+
+    return path
+
+
+def _bidirectional_pred_succ(G, source, target, exclude):
+    # does BFS from both source and target and meets in the middle
+    # excludes nodes in the container "exclude" from the search
+
+    # handle either directed or undirected
+    if G.is_directed():
+        Gpred = G.predecessors
+        Gsucc = G.successors
+    else:
+        Gpred = G.neighbors
+        Gsucc = G.neighbors
+
+    # predecessor and successors in search
+    pred = {source: None}
+    succ = {target: None}
+
+    # initialize fringes, start with forward
+    forward_fringe = [source]
+    reverse_fringe = [target]
+
+    level = 0
+
+    while forward_fringe and reverse_fringe:
+        # Make sure that we iterate one step forward and one step backwards
+        # thus source and target will only trigger "found path" when they are
+        # adjacent and then they can be safely included in the container 'exclude'
+        level += 1
+        if level % 2 != 0:
+            this_level = forward_fringe
+            forward_fringe = []
+            for v in this_level:
+                for w in Gsucc(v):
+                    if w in exclude:
+                        continue
+                    if w not in pred:
+                        forward_fringe.append(w)
+                        pred[w] = v
+                    if w in succ:
+                        return pred, succ, w  # found path
+        else:
+            this_level = reverse_fringe
+            reverse_fringe = []
+            for v in this_level:
+                for w in Gpred(v):
+                    if w in exclude:
+                        continue
+                    if w not in succ:
+                        succ[w] = v
+                        reverse_fringe.append(w)
+                    if w in pred:
+                        return pred, succ, w  # found path
+
+    raise nx.NetworkXNoPath(f"No path between {source} and {target}.")

.venv/lib/python3.12/site-packages/networkx/algorithms/approximation/density.py

@@ -0,0 +1,396 @@
+"""Fast algorithms for the densest subgraph problem"""
+
+import math
+
+import networkx as nx
+
+__all__ = ["densest_subgraph"]
+
+
+def _greedy_plus_plus(G, iterations):
+    if G.number_of_edges() == 0:
+        return 0.0, set()
+    if iterations < 1:
+        raise ValueError(
+            f"The number of iterations must be an integer >= 1. Provided: {iterations}"
+        )
+
+    loads = {node: 0 for node in G.nodes}  # Load vector for Greedy++.
+    best_density = 0.0  # Highest density encountered.
+    best_subgraph = set()  # Nodes of the best subgraph found.
+
+    for _ in range(iterations):
+        # Initialize heap for fast access to minimum weighted degree.
+        heap = nx.utils.BinaryHeap()
+
+        # Compute initial weighted degrees and add nodes to the heap.
+        for node, degree in G.degree:
+            heap.insert(node, loads[node] + degree)
+        # Set up tracking for current graph state.
+        remaining_nodes = set(G.nodes)
+        num_edges = G.number_of_edges()
+        current_degrees = dict(G.degree)
+
+        while remaining_nodes:
+            num_nodes = len(remaining_nodes)
+
+            # Current density of the (implicit) graph
+            current_density = num_edges / num_nodes
+
+            # Update the best density.
+            if current_density > best_density:
+                best_density = current_density
+                best_subgraph = set(remaining_nodes)
+
+            # Pop the node with the smallest weighted degree.
+            node, _ = heap.pop()
+            if node not in remaining_nodes:
+                continue  # Skip nodes already removed.
+
+            # Update the load of the popped node.
+            loads[node] += current_degrees[node]
+
+            # Update neighbors' degrees and the heap.
+            for neighbor in G.neighbors(node):
+                if neighbor in remaining_nodes:
+                    current_degrees[neighbor] -= 1
+                    num_edges -= 1
+                    heap.insert(neighbor, loads[neighbor] + current_degrees[neighbor])
+
+            # Remove the node from the remaining nodes.
+            remaining_nodes.remove(node)
+
+    return best_density, best_subgraph
+
+
+def _fractional_peeling(G, b, x, node_to_idx, edge_to_idx):
+    """
+    Optimized fractional peeling using NumPy arrays.
+
+    Parameters
+    ----------
+    G : networkx.Graph
+        The input graph.
+    b : numpy.ndarray
+        Induced load vector.
+    x : numpy.ndarray
+        Fractional edge values.
+    node_to_idx : dict
+        Mapping from node to index.
+    edge_to_idx : dict
+        Mapping from edge to index.
+
+    Returns
+    -------
+    best_density : float
+        The best density found.
+    best_subgraph : set
+        The subset of nodes defining the densest subgraph.
+    """
+    heap = nx.utils.BinaryHeap()
+
+    remaining_nodes = set(G.nodes)
+
+    # Initialize heap with b values
+    for idx, node in enumerate(G):
+        heap.insert(node, b[idx])
+
+    num_edges = G.number_of_edges()
+
+    best_density = 0.0
+    best_subgraph = set()
+
+    while remaining_nodes:
+        num_nodes = len(remaining_nodes)
+        current_density = num_edges / num_nodes
+
+        if current_density > best_density:
+            best_density = current_density
+            best_subgraph = set(remaining_nodes)
+
+        # Pop the node with the smallest b
+        node, _ = heap.pop()
+        while node not in remaining_nodes:
+            node, _ = heap.pop()  # Clean the heap from stale values
+
+        # Update neighbors b values by subtracting fractional x value
+        for neighbor in G.neighbors(node):
+            if neighbor in remaining_nodes:
+                neighbor_idx = node_to_idx[neighbor]
+                # Take off fractional value
+                b[neighbor_idx] -= x[edge_to_idx[(neighbor, node)]]
+                num_edges -= 1
+                heap.insert(neighbor, b[neighbor_idx])
+
+        remaining_nodes.remove(node)  # peel off node
+
+    return best_density, best_subgraph
+
+
+def _fista(G, iterations):
+    if G.number_of_edges() == 0:
+        return 0.0, set()
+    if iterations < 1:
+        raise ValueError(
+            f"The number of iterations must be an integer >= 1. Provided: {iterations}"
+        )
+    import numpy as np
+
+    # 1. Node Mapping: Assign a unique index to each node and edge
+    node_to_idx = {node: idx for idx, node in enumerate(G)}
+    num_nodes = G.number_of_nodes()
+    num_undirected_edges = G.number_of_edges()
+
+    # 2. Edge Mapping: Assign a unique index to each bidirectional edge
+    bidirectional_edges = [(u, v) for u, v in G.edges] + [(v, u) for u, v in G.edges]
+    edge_to_idx = {edge: idx for idx, edge in enumerate(bidirectional_edges)}
+
+    num_edges = len(bidirectional_edges)
+
+    # 3. Reverse Edge Mapping: Map each (bidirectional) edge to its reverse edge index
+    reverse_edge_idx = np.empty(num_edges, dtype=np.int32)
+    for idx in range(num_undirected_edges):
+        reverse_edge_idx[idx] = num_undirected_edges + idx
+    for idx in range(num_undirected_edges, 2 * num_undirected_edges):
+        reverse_edge_idx[idx] = idx - num_undirected_edges
+
+    # 4. Initialize Variables as NumPy Arrays
+    x = np.full(num_edges, 0.5, dtype=np.float32)
+    y = x.copy()
+    z = np.zeros(num_edges, dtype=np.float32)
+    b = np.zeros(num_nodes, dtype=np.float32)  # Induced load vector
+    tk = 1.0  # Momentum term
+
+    # 5. Precompute Edge Source Indices
+    edge_src_indices = np.array(
+        [node_to_idx[u] for u, _ in bidirectional_edges], dtype=np.int32
+    )
+
+    # 6. Compute Learning Rate
+    max_degree = max(deg for _, deg in G.degree)
+    # 0.9 for floating point errs when max_degree is very large
+    learning_rate = 0.9 / max_degree
+
+    # 7. Iterative Updates
+    for _ in range(iterations):
+        # 7a. Update b: sum y over outgoing edges for each node
+        b[:] = 0.0  # Reset b to zero
+        np.add.at(b, edge_src_indices, y)  # b_u = \sum_{v : (u,v) \in E(G)} y_{uv}
+
+        # 7b. Compute z, z_{uv} = y_{uv} - 2 * learning_rate * b_u
+        z = y - 2.0 * learning_rate * b[edge_src_indices]
+
+        # 7c. Update Momentum Term
+        tknew = (1.0 + math.sqrt(1 + 4.0 * tk**2)) / 2.0
+
+        # 7d. Update x in a vectorized manner, x_{uv} = (z_{uv} - z_{vu} + 1.0) / 2.0
+        new_xuv = (z - z[reverse_edge_idx] + 1.0) / 2.0
+        clamped_x = np.clip(new_xuv, 0.0, 1.0)  # Clamp x_{uv} between 0 and 1
+
+        # Update y using the FISTA update formula (similar to gradient descent)
+        y = (
+            clamped_x
+            + ((tk - 1.0) / tknew) * (clamped_x - x)
+            + (tk / tknew) * (clamped_x - y)
+        )
+
+        # Update x
+        x = clamped_x
+
+        # Update tk, the momemntum term
+        tk = tknew
+
+    # Rebalance the b values! Otherwise performance is a bit suboptimal.
+    b[:] = 0.0
+    np.add.at(b, edge_src_indices, x)  # b_u = \sum_{v : (u,v) \in E(G)} x_{uv}
+
+    # Extract the actual (approximate) dense subgraph.
+    return _fractional_peeling(G, b, x, node_to_idx, edge_to_idx)
+
+
+ALGORITHMS = {"greedy++": _greedy_plus_plus, "fista": _fista}
+
+
+@nx.utils.not_implemented_for("directed")
+@nx.utils.not_implemented_for("multigraph")
+@nx._dispatchable
+def densest_subgraph(G, iterations=1, *, method="fista"):
+    r"""Returns an approximate densest subgraph for a graph `G`.
+
+    This function runs an iterative algorithm to find the densest subgraph,
+    and returns both the density and the subgraph. For a discussion on the
+    notion of density used and the different algorithms available on
+    networkx, please see the Notes section below.
+
+    Parameters
+    ----------
+    G : NetworkX graph
+        Undirected graph.
+
+    iterations : int, optional (default=1)
+        Number of iterations to use for the iterative algorithm. Can be
+        specified positionally or as a keyword argument.
+
+    method : string, optional (default='fista')
+        The algorithm to use to approximate the densest subgraph. Supported
+        options: 'greedy++' by Boob et al. [2]_ and 'fista' by Harb et al. [3]_.
+        Must be specified as a keyword argument. Other inputs produce a
+        ValueError.
+
+    Returns
+    -------
+    d : float
+        The density of the approximate subgraph found.
+
+    S : set
+        The subset of nodes defining the approximate densest subgraph.
+
+    Examples
+    --------
+    >>> G = nx.star_graph(4)
+    >>> nx.approximation.densest_subgraph(G, iterations=1)
+    (0.8, {0, 1, 2, 3, 4})
+
+    Notes
+    -----
+    **Problem Definition:**
+    The densest subgraph problem (DSG) asks to find the subgraph
+    $S \subseteq V(G)$ with maximum density. For a subset of the nodes of
+    $G$, $S \subseteq V(G)$, define $E(S) = \{ (u,v) : (u,v)\in E(G),
+    u\in S, v\in S \}$ as the set of edges with both endpoints in $S$.
+    The density of $S$ is defined as $|E(S)|/|S|$, the ratio between the
+    edges in the subgraph $G[S]$ and the number of nodes in that subgraph.
+    Note that this is different from the standard graph theoretic definition
+    of density, defined as $\frac{2|E(S)|}{|S|(|S|-1)}$, for historical
+    reasons.
+
+    **Exact Algorithms:**
+    The densest subgraph problem is polynomial time solvable using maximum
+    flow, commonly referred to as Goldberg's algorithm. However, the
+    algorithm is quite involved. It first binary searches on the optimal
+    density, $d^\ast$. For a guess of the density $d$, it sets up a flow
+    network $G'$ with size $O(m)$. The maximum flow solution either
+    informs the algorithm that no subgraph with density $d$ exists, or it
+    provides a subgraph with density at least $d$. However, this is
+    inherently bottlenecked by the maximum flow algorithm. For example, [2]_
+    notes that Goldberg’s algorithm was not feasible on many large graphs
+    even though they used a highly optimized maximum flow library.
+
+    **Charikar's Greedy Peeling:**
+    While exact solution algorithms are quite involved, there are several
+    known approximation algorithms for the densest subgraph problem.
+
+    Charikar [1]_ described a very simple 1/2-approximation algorithm for DSG
+    known as the greedy "peeling" algorithm. The algorithm creates an
+    ordering of the nodes as follows. The first node $v_1$ is the one with
+    the smallest degree in $G$ (ties broken arbitrarily). It selects
+    $v_2$ to be the smallest degree node in $G \setminus v_1$. Letting
+    $G_i$ be the graph after removing $v_1, ..., v_i$ (with $G_0=G$),
+    the algorithm returns the graph among $G_0, ..., G_n$ with the highest
+    density.
+
+    **Greedy++:**
+    Boob et al. [2]_ generalized this algorithm into Greedy++, an iterative
+    algorithm that runs several rounds of "peeling". In fact, Greedy++ with 1
+    iteration is precisely Charikar's algorithm. The algorithm converges to a
+    $(1-\epsilon)$ approximate densest subgraph in $O(\Delta(G)\log
+    n/\epsilon^2)$ iterations, where $\Delta(G)$ is the maximum degree,
+    and $n$ is the number of nodes in $G$. The algorithm also has other
+    desirable properties as shown by [4]_ and [5]_.
+
+    **FISTA Algorithm:**
+    Harb et al. [3]_ gave a faster and more scalable algorithm using ideas
+    from quadratic programming for the densest subgraph, which is based on a
+    fast iterative shrinkage-thresholding algorithm (FISTA) algorithm. It is
+    known that computing the densest subgraph can be formulated as the
+    following convex optimization problem:
+
+    Minimize $\sum_{u \in V(G)} b_u^2$
+
+    Subject to:
+
+    $b_u = \sum_{v: \{u,v\} \in E(G)} x_{uv}$ for all $u \in V(G)$
+
+    $x_{uv} + x_{vu} = 1.0$ for all $\{u,v\} \in E(G)$
+
+    $x_{uv} \geq 0, x_{vu} \geq 0$ for all $\{u,v\} \in E(G)$
+
+    Here, $x_{uv}$ represents the fraction of edge $\{u,v\}$ assigned to
+    $u$, and $x_{vu}$ to $v$.
+
+    The FISTA algorithm efficiently solves this convex program using gradient
+    descent with projections. For a learning rate $\alpha$, the algorithm
+    does:
+
+    1. **Initialization**: Set $x^{(0)}_{uv} = x^{(0)}_{vu} = 0.5$ for all
+    edges as a feasible solution.
+
+    2. **Gradient Update**: For iteration $k\geq 1$, set
+    $x^{(k+1)}_{uv} = x^{(k)}_{uv} - 2 \alpha \sum_{v: \{u,v\} \in E(G)}
+    x^{(k)}_{uv}$. However, now $x^{(k+1)}_{uv}$ might be infeasible!
+    To ensure feasibility, we project $x^{(k+1)}_{uv}$.
+
+    3. **Projection to the Feasible Set**: Compute
+    $b^{(k+1)}_u = \sum_{v: \{u,v\} \in E(G)} x^{(k)}_{uv}$ for all
+    nodes $u$. Define $z^{(k+1)}_{uv} = x^{(k+1)}_{uv} - 2 \alpha
+    b^{(k+1)}_u$. Update $x^{(k+1)}_{uv} =
+    CLAMP((z^{(k+1)}_{uv} - z^{(k+1)}_{vu} + 1.0) / 2.0)$, where
+    $CLAMP(x) = \max(0, \min(1, x))$.
+
+    With a learning rate of $\alpha=1/\Delta(G)$, where $\Delta(G)$ is
+    the maximum degree, the algorithm converges to the optimum solution of
+    the convex program.
+
+    **Fractional Peeling:**
+    To obtain a **discrete** subgraph, we use fractional peeling, an
+    adaptation of the standard peeling algorithm which peels the minimum
+    degree vertex in each iteration, and returns the densest subgraph found
+    along the way. Here, we instead peel the vertex with the smallest
+    induced load $b_u$:
+
+    1. Compute $b_u$ and $x_{uv}$.
+
+    2. Iteratively remove the vertex with the smallest $b_u$, updating its
+    neighbors' load by $x_{vu}$.
+
+    Fractional peeling transforms the approximately optimal fractional
+    values $b_u, x_{uv}$ into a discrete subgraph. Unlike traditional
+    peeling, which removes the lowest-degree node, this method accounts for
+    fractional edge contributions from the convex program.
+
+    This approach is both scalable and theoretically sound, ensuring a quick
+    approximation of the densest subgraph while leveraging fractional load
+    balancing.
+
+    References
+    ----------
+    .. [1] Charikar, Moses. "Greedy approximation algorithms for finding dense
+       components in a graph." In International workshop on approximation
+       algorithms for combinatorial optimization, pp. 84-95. Berlin, Heidelberg:
+       Springer Berlin Heidelberg, 2000.
+
+    .. [2] Boob, Digvijay, Yu Gao, Richard Peng, Saurabh Sawlani, Charalampos
+       Tsourakakis, Di Wang, and Junxing Wang. "Flowless: Extracting densest
+       subgraphs without flow computations." In Proceedings of The Web Conference
+       2020, pp. 573-583. 2020.
+
+    .. [3] Harb, Elfarouk, Kent Quanrud, and Chandra Chekuri. "Faster and scalable
+       algorithms for densest subgraph and decomposition." Advances in Neural
+       Information Processing Systems 35 (2022): 26966-26979.
+
+    .. [4] Harb, Elfarouk, Kent Quanrud, and Chandra Chekuri. "Convergence to
+       lexicographically optimal base in a (contra) polymatroid and applications
+       to densest subgraph and tree packing." arXiv preprint arXiv:2305.02987
+       (2023).
+
+    .. [5] Chekuri, Chandra, Kent Quanrud, and Manuel R. Torres. "Densest
+       subgraph: Supermodularity, iterative peeling, and flow." In Proceedings of
+       the 2022 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp.
+       1531-1555. Society for Industrial and Applied Mathematics, 2022.
+    """
+    try:
+        algo = ALGORITHMS[method]
+    except KeyError as e:
+        raise ValueError(f"{method} is not a valid choice for an algorithm.") from e
+
+    return algo(G, iterations)

.venv/lib/python3.12/site-packages/networkx/algorithms/approximation/distance_measures.py

@@ -0,0 +1,150 @@
+"""Distance measures approximated metrics."""
+
+import networkx as nx
+from networkx.utils.decorators import py_random_state
+
+__all__ = ["diameter"]
+
+
+@py_random_state(1)
+@nx._dispatchable(name="approximate_diameter")
+def diameter(G, seed=None):
+    """Returns a lower bound on the diameter of the graph G.
+
+    The function computes a lower bound on the diameter (i.e., the maximum eccentricity)
+    of a directed or undirected graph G. The procedure used varies depending on the graph
+    being directed or not.
+
+    If G is an `undirected` graph, then the function uses the `2-sweep` algorithm [1]_.
+    The main idea is to pick the farthest node from a random node and return its eccentricity.
+
+    Otherwise, if G is a `directed` graph, the function uses the `2-dSweep` algorithm [2]_,
+    The procedure starts by selecting a random source node $s$ from which it performs a
+    forward and a backward BFS. Let $a_1$ and $a_2$ be the farthest nodes in the forward and
+    backward cases, respectively. Then, it computes the backward eccentricity of $a_1$ using
+    a backward BFS and the forward eccentricity of $a_2$ using a forward BFS.
+    Finally, it returns the best lower bound between the two.
+
+    In both cases, the time complexity is linear with respect to the size of G.
+
+    Parameters
+    ----------
+    G : NetworkX graph
+
+    seed : integer, random_state, or None (default)
+        Indicator of random number generation state.
+        See :ref:`Randomness<randomness>`.
+
+    Returns
+    -------
+    d : integer
+       Lower Bound on the Diameter of G
+
+    Examples
+    --------
+    >>> G = nx.path_graph(10)  # undirected graph
+    >>> nx.diameter(G)
+    9
+    >>> G = nx.cycle_graph(3, create_using=nx.DiGraph)  # directed graph
+    >>> nx.diameter(G)
+    2
+
+    Raises
+    ------
+    NetworkXError
+        If the graph is empty or
+        If the graph is undirected and not connected or
+        If the graph is directed and not strongly connected.
+
+    See Also
+    --------
+    networkx.algorithms.distance_measures.diameter
+
+    References
+    ----------
+    .. [1] Magnien, Clémence, Matthieu Latapy, and Michel Habib.
+       *Fast computation of empirically tight bounds for the diameter of massive graphs.*
+       Journal of Experimental Algorithmics (JEA), 2009.
+       https://arxiv.org/pdf/0904.2728.pdf
+    .. [2] Crescenzi, Pierluigi, Roberto Grossi, Leonardo Lanzi, and Andrea Marino.
+       *On computing the diameter of real-world directed (weighted) graphs.*
+       International Symposium on Experimental Algorithms. Springer, Berlin, Heidelberg, 2012.
+       https://courses.cs.ut.ee/MTAT.03.238/2014_fall/uploads/Main/diameter.pdf
+    """
+    # if G is empty
+    if not G:
+        raise nx.NetworkXError("Expected non-empty NetworkX graph!")
+    # if there's only a node
+    if G.number_of_nodes() == 1:
+        return 0
+    # if G is directed
+    if G.is_directed():
+        return _two_sweep_directed(G, seed)
+    # else if G is undirected
+    return _two_sweep_undirected(G, seed)
+
+
+def _two_sweep_undirected(G, seed):
+    """Helper function for finding a lower bound on the diameter
+        for undirected Graphs.
+
+        The idea is to pick the farthest node from a random node
+        and return its eccentricity.
+
+        ``G`` is a NetworkX undirected graph.
+
+    .. note::
+
+        ``seed`` is a random.Random or numpy.random.RandomState instance
+    """
+    # select a random source node
+    source = seed.choice(list(G))
+    # get the distances to the other nodes
+    distances = nx.shortest_path_length(G, source)
+    # if some nodes have not been visited, then the graph is not connected
+    if len(distances) != len(G):
+        raise nx.NetworkXError("Graph not connected.")
+    # take a node that is (one of) the farthest nodes from the source
+    *_, node = distances
+    # return the eccentricity of the node
+    return nx.eccentricity(G, node)
+
+
+def _two_sweep_directed(G, seed):
+    """Helper function for finding a lower bound on the diameter
+        for directed Graphs.
+
+        It implements 2-dSweep, the directed version of the 2-sweep algorithm.
+        The algorithm follows the following steps.
+        1. Select a source node $s$ at random.
+        2. Perform a forward BFS from $s$ to select a node $a_1$ at the maximum
+        distance from the source, and compute $LB_1$, the backward eccentricity of $a_1$.
+        3. Perform a backward BFS from $s$ to select a node $a_2$ at the maximum
+        distance from the source, and compute $LB_2$, the forward eccentricity of $a_2$.
+        4. Return the maximum between $LB_1$ and $LB_2$.
+
+        ``G`` is a NetworkX directed graph.
+
+    .. note::
+
+        ``seed`` is a random.Random or numpy.random.RandomState instance
+    """
+    # get a new digraph G' with the edges reversed in the opposite direction
+    G_reversed = G.reverse()
+    # select a random source node
+    source = seed.choice(list(G))
+    # compute forward distances from source
+    forward_distances = nx.shortest_path_length(G, source)
+    # compute backward distances  from source
+    backward_distances = nx.shortest_path_length(G_reversed, source)
+    # if either the source can't reach every node or not every node
+    # can reach the source, then the graph is not strongly connected
+    n = len(G)
+    if len(forward_distances) != n or len(backward_distances) != n:
+        raise nx.NetworkXError("DiGraph not strongly connected.")
+    # take a node a_1 at the maximum distance from the source in G
+    *_, a_1 = forward_distances
+    # take a node a_2 at the maximum distance from the source in G_reversed
+    *_, a_2 = backward_distances
+    # return the max between the backward eccentricity of a_1 and the forward eccentricity of a_2
+    return max(nx.eccentricity(G_reversed, a_1), nx.eccentricity(G, a_2))

.venv/lib/python3.12/site-packages/networkx/algorithms/approximation/dominating_set.py

@@ -0,0 +1,149 @@
+"""Functions for finding node and edge dominating sets.
+
+A `dominating set`_ for an undirected graph *G* with vertex set *V*
+and edge set *E* is a subset *D* of *V* such that every vertex not in
+*D* is adjacent to at least one member of *D*. An `edge dominating set`_
+is a subset *F* of *E* such that every edge not in *F* is
+incident to an endpoint of at least one edge in *F*.
+
+.. _dominating set: https://en.wikipedia.org/wiki/Dominating_set
+.. _edge dominating set: https://en.wikipedia.org/wiki/Edge_dominating_set
+
+"""
+
+import networkx as nx
+
+from ...utils import not_implemented_for
+from ..matching import maximal_matching
+
+__all__ = ["min_weighted_dominating_set", "min_edge_dominating_set"]
+
+
+# TODO Why doesn't this algorithm work for directed graphs?
+@not_implemented_for("directed")
+@nx._dispatchable(node_attrs="weight")
+def min_weighted_dominating_set(G, weight=None):
+    r"""Returns a dominating set that approximates the minimum weight node
+    dominating set.
+
+    Parameters
+    ----------
+    G : NetworkX graph
+        Undirected graph.
+
+    weight : string
+        The node attribute storing the weight of an node. If provided,
+        the node attribute with this key must be a number for each
+        node. If not provided, each node is assumed to have weight one.
+
+    Returns
+    -------
+    min_weight_dominating_set : set
+        A set of nodes, the sum of whose weights is no more than `(\log
+        w(V)) w(V^*)`, where `w(V)` denotes the sum of the weights of
+        each node in the graph and `w(V^*)` denotes the sum of the
+        weights of each node in the minimum weight dominating set.
+
+    Examples
+    --------
+    >>> G = nx.Graph([(0, 1), (0, 4), (1, 4), (1, 2), (2, 3), (3, 4), (2, 5)])
+    >>> nx.approximation.min_weighted_dominating_set(G)
+    {1, 2, 4}
+
+    Raises
+    ------
+    NetworkXNotImplemented
+        If G is directed.
+
+    Notes
+    -----
+    This algorithm computes an approximate minimum weighted dominating
+    set for the graph `G`. The returned solution has weight `(\log
+    w(V)) w(V^*)`, where `w(V)` denotes the sum of the weights of each
+    node in the graph and `w(V^*)` denotes the sum of the weights of
+    each node in the minimum weight dominating set for the graph.
+
+    This implementation of the algorithm runs in $O(m)$ time, where $m$
+    is the number of edges in the graph.
+
+    References
+    ----------
+    .. [1] Vazirani, Vijay V.
+           *Approximation Algorithms*.
+           Springer Science & Business Media, 2001.
+
+    """
+    # The unique dominating set for the null graph is the empty set.
+    if len(G) == 0:
+        return set()
+
+    # This is the dominating set that will eventually be returned.
+    dom_set = set()
+
+    def _cost(node_and_neighborhood):
+        """Returns the cost-effectiveness of greedily choosing the given
+        node.
+
+        `node_and_neighborhood` is a two-tuple comprising a node and its
+        closed neighborhood.
+
+        """
+        v, neighborhood = node_and_neighborhood
+        return G.nodes[v].get(weight, 1) / len(neighborhood - dom_set)
+
+    # This is a set of all vertices not already covered by the
+    # dominating set.
+    vertices = set(G)
+    # This is a dictionary mapping each node to the closed neighborhood
+    # of that node.
+    neighborhoods = {v: {v} | set(G[v]) for v in G}
+
+    # Continue until all vertices are adjacent to some node in the
+    # dominating set.
+    while vertices:
+        # Find the most cost-effective node to add, along with its
+        # closed neighborhood.
+        dom_node, min_set = min(neighborhoods.items(), key=_cost)
+        # Add the node to the dominating set and reduce the remaining
+        # set of nodes to cover.
+        dom_set.add(dom_node)
+        del neighborhoods[dom_node]
+        vertices -= min_set
+
+    return dom_set
+
+
+@nx._dispatchable
+def min_edge_dominating_set(G):
+    r"""Returns minimum cardinality edge dominating set.
+
+    Parameters
+    ----------
+    G : NetworkX graph
+      Undirected graph
+
+    Returns
+    -------
+    min_edge_dominating_set : set
+      Returns a set of dominating edges whose size is no more than 2 * OPT.
+
+    Examples
+    --------
+    >>> G = nx.petersen_graph()
+    >>> nx.approximation.min_edge_dominating_set(G)
+    {(0, 1), (4, 9), (6, 8), (5, 7), (2, 3)}
+
+    Raises
+    ------
+    ValueError
+        If the input graph `G` is empty.
+
+    Notes
+    -----
+    The algorithm computes an approximate solution to the edge dominating set
+    problem. The result is no more than 2 * OPT in terms of size of the set.
+    Runtime of the algorithm is $O(|E|)$.
+    """
+    if not G:
+        raise ValueError("Expected non-empty NetworkX graph!")
+    return maximal_matching(G)

.venv/lib/python3.12/site-packages/networkx/algorithms/approximation/kcomponents.py

@@ -0,0 +1,369 @@
+"""Fast approximation for k-component structure"""
+
+import itertools
+from collections import defaultdict
+from collections.abc import Mapping
+from functools import cached_property
+
+import networkx as nx
+from networkx.algorithms.approximation import local_node_connectivity
+from networkx.exception import NetworkXError
+from networkx.utils import not_implemented_for
+
+__all__ = ["k_components"]
+
+
+@not_implemented_for("directed")
+@nx._dispatchable(name="approximate_k_components")
+def k_components(G, min_density=0.95):
+    r"""Returns the approximate k-component structure of a graph G.
+
+    A `k`-component is a maximal subgraph of a graph G that has, at least,
+    node connectivity `k`: we need to remove at least `k` nodes to break it
+    into more components. `k`-components have an inherent hierarchical
+    structure because they are nested in terms of connectivity: a connected
+    graph can contain several 2-components, each of which can contain
+    one or more 3-components, and so forth.
+
+    This implementation is based on the fast heuristics to approximate
+    the `k`-component structure of a graph [1]_. Which, in turn, it is based on
+    a fast approximation algorithm for finding good lower bounds of the number
+    of node independent paths between two nodes [2]_.
+
+    Parameters
+    ----------
+    G : NetworkX graph
+        Undirected graph
+
+    min_density : Float
+        Density relaxation threshold. Default value 0.95
+
+    Returns
+    -------
+    k_components : dict
+        Dictionary with connectivity level `k` as key and a list of
+        sets of nodes that form a k-component of level `k` as values.
+
+    Raises
+    ------
+    NetworkXNotImplemented
+        If G is directed.
+
+    Examples
+    --------
+    >>> # Petersen graph has 10 nodes and it is triconnected, thus all
+    >>> # nodes are in a single component on all three connectivity levels
+    >>> from networkx.algorithms import approximation as apxa
+    >>> G = nx.petersen_graph()
+    >>> k_components = apxa.k_components(G)
+
+    Notes
+    -----
+    The logic of the approximation algorithm for computing the `k`-component
+    structure [1]_ is based on repeatedly applying simple and fast algorithms
+    for `k`-cores and biconnected components in order to narrow down the
+    number of pairs of nodes over which we have to compute White and Newman's
+    approximation algorithm for finding node independent paths [2]_. More
+    formally, this algorithm is based on Whitney's theorem, which states
+    an inclusion relation among node connectivity, edge connectivity, and
+    minimum degree for any graph G. This theorem implies that every
+    `k`-component is nested inside a `k`-edge-component, which in turn,
+    is contained in a `k`-core. Thus, this algorithm computes node independent
+    paths among pairs of nodes in each biconnected part of each `k`-core,
+    and repeats this procedure for each `k` from 3 to the maximal core number
+    of a node in the input graph.
+
+    Because, in practice, many nodes of the core of level `k` inside a
+    bicomponent actually are part of a component of level k, the auxiliary
+    graph needed for the algorithm is likely to be very dense. Thus, we use
+    a complement graph data structure (see `AntiGraph`) to save memory.
+    AntiGraph only stores information of the edges that are *not* present
+    in the actual auxiliary graph. When applying algorithms to this
+    complement graph data structure, it behaves as if it were the dense
+    version.
+
+    See also
+    --------
+    k_components
+
+    References
+    ----------
+    .. [1]  Torrents, J. and F. Ferraro (2015) Structural Cohesion:
+            Visualization and Heuristics for Fast Computation.
+            https://arxiv.org/pdf/1503.04476v1
+
+    .. [2]  White, Douglas R., and Mark Newman (2001) A Fast Algorithm for
+            Node-Independent Paths. Santa Fe Institute Working Paper #01-07-035
+            https://www.santafe.edu/research/results/working-papers/fast-approximation-algorithms-for-finding-node-ind
+
+    .. [3]  Moody, J. and D. White (2003). Social cohesion and embeddedness:
+            A hierarchical conception of social groups.
+            American Sociological Review 68(1), 103--28.
+            https://doi.org/10.2307/3088904
+
+    """
+    # Dictionary with connectivity level (k) as keys and a list of
+    # sets of nodes that form a k-component as values
+    k_components = defaultdict(list)
+    # make a few functions local for speed
+    node_connectivity = local_node_connectivity
+    k_core = nx.k_core
+    core_number = nx.core_number
+    biconnected_components = nx.biconnected_components
+    combinations = itertools.combinations
+    # Exact solution for k = {1,2}
+    # There is a linear time algorithm for triconnectivity, if we had an
+    # implementation available we could start from k = 4.
+    for component in nx.connected_components(G):
+        # isolated nodes have connectivity 0
+        comp = set(component)
+        if len(comp) > 1:
+            k_components[1].append(comp)
+    for bicomponent in nx.biconnected_components(G):
+        # avoid considering dyads as bicomponents
+        bicomp = set(bicomponent)
+        if len(bicomp) > 2:
+            k_components[2].append(bicomp)
+    # There is no k-component of k > maximum core number
+    # \kappa(G) <= \lambda(G) <= \delta(G)
+    g_cnumber = core_number(G)
+    max_core = max(g_cnumber.values())
+    for k in range(3, max_core + 1):
+        C = k_core(G, k, core_number=g_cnumber)
+        for nodes in biconnected_components(C):
+            # Build a subgraph SG induced by the nodes that are part of
+            # each biconnected component of the k-core subgraph C.
+            if len(nodes) < k:
+                continue
+            SG = G.subgraph(nodes)
+            # Build auxiliary graph
+            H = _AntiGraph()
+            H.add_nodes_from(SG.nodes())
+            for u, v in combinations(SG, 2):
+                K = node_connectivity(SG, u, v, cutoff=k)
+                if k > K:
+                    H.add_edge(u, v)
+            for h_nodes in biconnected_components(H):
+                if len(h_nodes) <= k:
+                    continue
+                SH = H.subgraph(h_nodes)
+                for Gc in _cliques_heuristic(SG, SH, k, min_density):
+                    for k_nodes in biconnected_components(Gc):
+                        Gk = nx.k_core(SG.subgraph(k_nodes), k)
+                        if len(Gk) <= k:
+                            continue
+                        k_components[k].append(set(Gk))
+    return k_components
+
+
+def _cliques_heuristic(G, H, k, min_density):
+    h_cnumber = nx.core_number(H)
+    for i, c_value in enumerate(sorted(set(h_cnumber.values()), reverse=True)):
+        cands = {n for n, c in h_cnumber.items() if c == c_value}
+        # Skip checking for overlap for the highest core value
+        if i == 0:
+            overlap = False
+        else:
+            overlap = set.intersection(
+                *[{x for x in H[n] if x not in cands} for n in cands]
+            )
+        if overlap and len(overlap) < k:
+            SH = H.subgraph(cands | overlap)
+        else:
+            SH = H.subgraph(cands)
+        sh_cnumber = nx.core_number(SH)
+        SG = nx.k_core(G.subgraph(SH), k)
+        while not (_same(sh_cnumber) and nx.density(SH) >= min_density):
+            # This subgraph must be writable => .copy()
+            SH = H.subgraph(SG).copy()
+            if len(SH) <= k:
+                break
+            sh_cnumber = nx.core_number(SH)
+            sh_deg = dict(SH.degree())
+            min_deg = min(sh_deg.values())
+            SH.remove_nodes_from(n for n, d in sh_deg.items() if d == min_deg)
+            SG = nx.k_core(G.subgraph(SH), k)
+        else:
+            yield SG
+
+
+def _same(measure, tol=0):
+    vals = set(measure.values())
+    if (max(vals) - min(vals)) <= tol:
+        return True
+    return False
+
+
+class _AntiGraph(nx.Graph):
+    """
+    Class for complement graphs.
+
+    The main goal is to be able to work with big and dense graphs with
+    a low memory footprint.
+
+    In this class you add the edges that *do not exist* in the dense graph,
+    the report methods of the class return the neighbors, the edges and
+    the degree as if it was the dense graph. Thus it's possible to use
+    an instance of this class with some of NetworkX functions. In this
+    case we only use k-core, connected_components, and biconnected_components.
+    """
+
+    all_edge_dict = {"weight": 1}
+
+    def single_edge_dict(self):
+        return self.all_edge_dict
+
+    edge_attr_dict_factory = single_edge_dict  # type: ignore[assignment]
+
+    def __getitem__(self, n):
+        """Returns a dict of neighbors of node n in the dense graph.
+
+        Parameters
+        ----------
+        n : node
+           A node in the graph.
+
+        Returns
+        -------
+        adj_dict : dictionary
+           The adjacency dictionary for nodes connected to n.
+
+        """
+        all_edge_dict = self.all_edge_dict
+        return {
+            node: all_edge_dict for node in set(self._adj) - set(self._adj[n]) - {n}
+        }
+
+    def neighbors(self, n):
+        """Returns an iterator over all neighbors of node n in the
+        dense graph.
+        """
+        try:
+            return iter(set(self._adj) - set(self._adj[n]) - {n})
+        except KeyError as err:
+            raise NetworkXError(f"The node {n} is not in the graph.") from err
+
+    class AntiAtlasView(Mapping):
+        """An adjacency inner dict for AntiGraph"""
+
+        def __init__(self, graph, node):
+            self._graph = graph
+            self._atlas = graph._adj[node]
+            self._node = node
+
+        def __len__(self):
+            return len(self._graph) - len(self._atlas) - 1
+
+        def __iter__(self):
+            return (n for n in self._graph if n not in self._atlas and n != self._node)
+
+        def __getitem__(self, nbr):
+            nbrs = set(self._graph._adj) - set(self._atlas) - {self._node}
+            if nbr in nbrs:
+                return self._graph.all_edge_dict
+            raise KeyError(nbr)
+
+    class AntiAdjacencyView(AntiAtlasView):
+        """An adjacency outer dict for AntiGraph"""
+
+        def __init__(self, graph):
+            self._graph = graph
+            self._atlas = graph._adj
+
+        def __len__(self):
+            return len(self._atlas)
+
+        def __iter__(self):
+            return iter(self._graph)
+
+        def __getitem__(self, node):
+            if node not in self._graph:
+                raise KeyError(node)
+            return self._graph.AntiAtlasView(self._graph, node)
+
+    @cached_property
+    def adj(self):
+        return self.AntiAdjacencyView(self)
+
+    def subgraph(self, nodes):
+        """This subgraph method returns a full AntiGraph. Not a View"""
+        nodes = set(nodes)
+        G = _AntiGraph()
+        G.add_nodes_from(nodes)
+        for n in G:
+            Gnbrs = G.adjlist_inner_dict_factory()
+            G._adj[n] = Gnbrs
+            for nbr, d in self._adj[n].items():
+                if nbr in G._adj:
+                    Gnbrs[nbr] = d
+                    G._adj[nbr][n] = d
+        G.graph = self.graph
+        return G
+
+    class AntiDegreeView(nx.reportviews.DegreeView):
+        def __iter__(self):
+            all_nodes = set(self._succ)
+            for n in self._nodes:
+                nbrs = all_nodes - set(self._succ[n]) - {n}
+                yield (n, len(nbrs))
+
+        def __getitem__(self, n):
+            nbrs = set(self._succ) - set(self._succ[n]) - {n}
+            # AntiGraph is a ThinGraph so all edges have weight 1
+            return len(nbrs) + (n in nbrs)
+
+    @cached_property
+    def degree(self):
+        """Returns an iterator for (node, degree) and degree for single node.
+
+        The node degree is the number of edges adjacent to the node.
+
+        Parameters
+        ----------
+        nbunch : iterable container, optional (default=all nodes)
+            A container of nodes.  The container will be iterated
+            through once.
+
+        weight : string or None, optional (default=None)
+           The edge attribute that holds the numerical value used
+           as a weight.  If None, then each edge has weight 1.
+           The degree is the sum of the edge weights adjacent to the node.
+
+        Returns
+        -------
+        deg:
+            Degree of the node, if a single node is passed as argument.
+        nd_iter : an iterator
+            The iterator returns two-tuples of (node, degree).
+
+        See Also
+        --------
+        degree
+
+        Examples
+        --------
+        >>> G = nx.path_graph(4)
+        >>> G.degree(0)  # node 0 with degree 1
+        1
+        >>> list(G.degree([0, 1]))
+        [(0, 1), (1, 2)]
+
+        """
+        return self.AntiDegreeView(self)
+
+    def adjacency(self):
+        """Returns an iterator of (node, adjacency set) tuples for all nodes
+           in the dense graph.
+
+        This is the fastest way to look at every edge.
+        For directed graphs, only outgoing adjacencies are included.
+
+        Returns
+        -------
+        adj_iter : iterator
+           An iterator of (node, adjacency set) for all nodes in
+           the graph.
+
+        """
+        for n in self._adj:
+            yield (n, set(self._adj) - set(self._adj[n]) - {n})

.venv/lib/python3.12/site-packages/networkx/algorithms/approximation/matching.py

@@ -0,0 +1,44 @@
+"""
+**************
+Graph Matching
+**************
+
+Given a graph G = (V,E), a matching M in G is a set of pairwise non-adjacent
+edges; that is, no two edges share a common vertex.
+
+`Wikipedia: Matching <https://en.wikipedia.org/wiki/Matching_(graph_theory)>`_
+"""
+
+import networkx as nx
+
+__all__ = ["min_maximal_matching"]
+
+
+@nx._dispatchable
+def min_maximal_matching(G):
+    r"""Returns the minimum maximal matching of G. That is, out of all maximal
+    matchings of the graph G, the smallest is returned.
+
+    Parameters
+    ----------
+    G : NetworkX graph
+      Undirected graph
+
+    Returns
+    -------
+    min_maximal_matching : set
+      Returns a set of edges such that no two edges share a common endpoint
+      and every edge not in the set shares some common endpoint in the set.
+      Cardinality will be 2*OPT in the worst case.
+
+    Notes
+    -----
+    The algorithm computes an approximate solution for the minimum maximal
+    cardinality matching problem. The solution is no more than 2 * OPT in size.
+    Runtime is $O(|E|)$.
+
+    References
+    ----------
+    .. [1] Vazirani, Vijay Approximation Algorithms (2001)
+    """
+    return nx.maximal_matching(G)

.venv/lib/python3.12/site-packages/networkx/algorithms/approximation/maxcut.py

@@ -0,0 +1,143 @@
+import networkx as nx
+from networkx.utils.decorators import not_implemented_for, py_random_state
+
+__all__ = ["randomized_partitioning", "one_exchange"]
+
+
+@not_implemented_for("directed")
+@not_implemented_for("multigraph")
+@py_random_state(1)
+@nx._dispatchable(edge_attrs="weight")
+def randomized_partitioning(G, seed=None, p=0.5, weight=None):
+    """Compute a random partitioning of the graph nodes and its cut value.
+
+    A partitioning is calculated by observing each node
+    and deciding to add it to the partition with probability `p`,
+    returning a random cut and its corresponding value (the
+    sum of weights of edges connecting different partitions).
+
+    Parameters
+    ----------
+    G : NetworkX graph
+
+    seed : integer, random_state, or None (default)
+        Indicator of random number generation state.
+        See :ref:`Randomness<randomness>`.
+
+    p : scalar
+        Probability for each node to be part of the first partition.
+        Should be in [0,1]
+
+    weight : object
+        Edge attribute key to use as weight. If not specified, edges
+        have weight one.
+
+    Returns
+    -------
+    cut_size : scalar
+        Value of the minimum cut.
+
+    partition : pair of node sets
+        A partitioning of the nodes that defines a minimum cut.
+
+    Examples
+    --------
+    >>> G = nx.complete_graph(5)
+    >>> cut_size, partition = nx.approximation.randomized_partitioning(G, seed=1)
+    >>> cut_size
+    6
+    >>> partition
+    ({0, 3, 4}, {1, 2})
+
+    Raises
+    ------
+    NetworkXNotImplemented
+        If the graph is directed or is a multigraph.
+    """
+    cut = {node for node in G.nodes() if seed.random() < p}
+    cut_size = nx.algorithms.cut_size(G, cut, weight=weight)
+    partition = (cut, G.nodes - cut)
+    return cut_size, partition
+
+
+def _swap_node_partition(cut, node):
+    return cut - {node} if node in cut else cut.union({node})
+
+
+@not_implemented_for("directed")
+@not_implemented_for("multigraph")
+@py_random_state(2)
+@nx._dispatchable(edge_attrs="weight")
+def one_exchange(G, initial_cut=None, seed=None, weight=None):
+    """Compute a partitioning of the graphs nodes and the corresponding cut value.
+
+    Use a greedy one exchange strategy to find a locally maximal cut
+    and its value, it works by finding the best node (one that gives
+    the highest gain to the cut value) to add to the current cut
+    and repeats this process until no improvement can be made.
+
+    Parameters
+    ----------
+    G : networkx Graph
+        Graph to find a maximum cut for.
+
+    initial_cut : set
+        Cut to use as a starting point. If not supplied the algorithm
+        starts with an empty cut.
+
+    seed : integer, random_state, or None (default)
+        Indicator of random number generation state.
+        See :ref:`Randomness<randomness>`.
+
+    weight : object
+        Edge attribute key to use as weight. If not specified, edges
+        have weight one.
+
+    Returns
+    -------
+    cut_value : scalar
+        Value of the maximum cut.
+
+    partition : pair of node sets
+        A partitioning of the nodes that defines a maximum cut.
+
+    Examples
+    --------
+    >>> G = nx.complete_graph(5)
+    >>> curr_cut_size, partition = nx.approximation.one_exchange(G, seed=1)
+    >>> curr_cut_size
+    6
+    >>> partition
+    ({0, 2}, {1, 3, 4})
+
+    Raises
+    ------
+    NetworkXNotImplemented
+        If the graph is directed or is a multigraph.
+    """
+    if initial_cut is None:
+        initial_cut = set()
+    cut = set(initial_cut)
+    current_cut_size = nx.algorithms.cut_size(G, cut, weight=weight)
+    while True:
+        nodes = list(G.nodes())
+        # Shuffling the nodes ensures random tie-breaks in the following call to max
+        seed.shuffle(nodes)
+        best_node_to_swap = max(
+            nodes,
+            key=lambda v: nx.algorithms.cut_size(
+                G, _swap_node_partition(cut, v), weight=weight
+            ),
+            default=None,
+        )
+        potential_cut = _swap_node_partition(cut, best_node_to_swap)
+        potential_cut_size = nx.algorithms.cut_size(G, potential_cut, weight=weight)
+
+        if potential_cut_size > current_cut_size:
+            cut = potential_cut
+            current_cut_size = potential_cut_size
+        else:
+            break
+
+    partition = (cut, G.nodes - cut)
+    return current_cut_size, partition

.venv/lib/python3.12/site-packages/networkx/algorithms/approximation/ramsey.py

@@ -0,0 +1,53 @@
+"""
+Ramsey numbers.
+"""
+
+import networkx as nx
+from networkx.utils import not_implemented_for
+
+from ...utils import arbitrary_element
+
+__all__ = ["ramsey_R2"]
+
+
+@not_implemented_for("directed")
+@not_implemented_for("multigraph")
+@nx._dispatchable
+def ramsey_R2(G):
+    r"""Compute the largest clique and largest independent set in `G`.
+
+    This can be used to estimate bounds for the 2-color
+    Ramsey number `R(2;s,t)` for `G`.
+
+    This is a recursive implementation which could run into trouble
+    for large recursions. Note that self-loop edges are ignored.
+
+    Parameters
+    ----------
+    G : NetworkX graph
+        Undirected graph
+
+    Returns
+    -------
+    max_pair : (set, set) tuple
+        Maximum clique, Maximum independent set.
+
+    Raises
+    ------
+    NetworkXNotImplemented
+        If the graph is directed or is a multigraph.
+    """
+    if not G:
+        return set(), set()
+
+    node = arbitrary_element(G)
+    nbrs = (nbr for nbr in nx.all_neighbors(G, node) if nbr != node)
+    nnbrs = nx.non_neighbors(G, node)
+    c_1, i_1 = ramsey_R2(G.subgraph(nbrs).copy())
+    c_2, i_2 = ramsey_R2(G.subgraph(nnbrs).copy())
+
+    c_1.add(node)
+    i_2.add(node)
+    # Choose the larger of the two cliques and the larger of the two
+    # independent sets, according to cardinality.
+    return max(c_1, c_2, key=len), max(i_1, i_2, key=len)

.venv/lib/python3.12/site-packages/networkx/algorithms/approximation/steinertree.py

@@ -0,0 +1,265 @@
+from itertools import chain
+
+import networkx as nx
+from networkx.utils import not_implemented_for, pairwise
+
+__all__ = ["metric_closure", "steiner_tree"]
+
+
+@not_implemented_for("directed")
+@nx._dispatchable(edge_attrs="weight", returns_graph=True)
+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`.
+
+    Notes
+    -----
+    .. deprecated:: 3.6
+       `metric_closure` is deprecated and will be removed in NetworkX 3.8.
+       Use :func:`networkx.all_pairs_shortest_path_length` instead.
+
+    """
+    import warnings
+
+    warnings.warn(
+        "metric_closure is deprecated and will be removed in NetworkX 3.8.\n"
+        "Use nx.all_pairs_shortest_path_length instead.",
+        category=DeprecationWarning,
+        stacklevel=5,
+    )
+
+    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 len(G) != len(distance):
+        msg = "G is not a connected graph. metric_closure is not defined."
+        raise nx.NetworkXError(msg)
+    Gnodes.remove(u)
+    for v in Gnodes:
+        M.add_edge(u, v, distance=distance[v], path=path[v])
+
+    # first node done -- now process the rest
+    for u, (distance, path) in all_paths_iter:
+        Gnodes.remove(u)
+        for v in Gnodes:
+            M.add_edge(u, v, distance=distance[v], path=path[v])
+
+    return M
+
+
+def _mehlhorn_steiner_tree(G, terminal_nodes, weight):
+    distances, paths = nx.multi_source_dijkstra(G, terminal_nodes, weight=weight)
+
+    d_1 = {}
+    s = {}
+    for v in G.nodes():
+        s[v] = paths[v][0]
+        d_1[(v, s[v])] = distances[v]
+
+    # G1-G4 names match those from the Mehlhorn 1988 paper.
+    G_1_prime = nx.Graph()
+    # iterate over all edges to complete d1
+    for u, v, data in G.edges(data=True):
+        su, sv = s[u], s[v]
+        weight_here = d_1[(u, su)] + data.get(weight, 1) + d_1[(v, sv)]
+        if not G_1_prime.has_edge(su, sv):
+            G_1_prime.add_edge(su, sv, weight_d1=weight_here)
+        else:
+            new_weight = min(weight_here, G_1_prime[su][sv]["weight_d1"])
+            G_1_prime.add_edge(su, sv, weight_d1=new_weight)
+
+    G_2 = nx.minimum_spanning_edges(G_1_prime, data=True, weight="weight_d1")
+
+    G_3 = nx.Graph()
+    for u, v, _ in G_2:
+        path = nx.shortest_path(G, u, v, weight=weight)
+        for n1, n2 in pairwise(path):
+            G_3.add_edge(n1, n2, weight=G[n1][n2].get(weight, 1))
+
+    G_3_mst = list(nx.minimum_spanning_edges(G_3, data=False, weight=weight))
+    if G.is_multigraph():
+        G_3_mst = (
+            (u, v, min(G[u][v], key=lambda k: G[u][v][k].get(weight, 1)))
+            for u, v in G_3_mst
+        )
+    G_4 = G.edge_subgraph(G_3_mst).copy()
+    _remove_nonterminal_leaves(G_4, terminal_nodes)
+    return G_4.edges()
+
+
+def _kou_steiner_tree(G, terminal_nodes, weight):
+    # Compute the metric closure only for terminal nodes
+    # Create a complete graph H from the metric edges
+    H = nx.Graph()
+    unvisited_terminals = set(terminal_nodes)
+
+    # check for connected graph while processing first node
+    u = unvisited_terminals.pop()
+    distances, paths = nx.single_source_dijkstra(G, source=u, weight=weight)
+    if len(G) != len(distances):
+        msg = "G is not a connected graph."
+        raise nx.NetworkXError(msg)
+    for v in unvisited_terminals:
+        H.add_edge(u, v, distance=distances[v], path=paths[v])
+
+    # first node done -- now process the rest
+    for u in unvisited_terminals.copy():
+        distances, paths = nx.single_source_dijkstra(G, source=u, weight=weight)
+        unvisited_terminals.remove(u)
+        for v in unvisited_terminals:
+            H.add_edge(u, v, distance=distances[v], path=paths[v])
+
+    # Use the 'distance' attribute of each edge provided by H.
+    mst_edges = nx.minimum_spanning_edges(H, weight="distance", data=True)
+
+    # Create an iterator over each edge in each shortest path; repeats are okay
+    mst_all_edges = chain.from_iterable(pairwise(d["path"]) for u, v, d in mst_edges)
+    if G.is_multigraph():
+        mst_all_edges = (
+            (u, v, min(G[u][v], key=lambda k: G[u][v][k].get(weight, 1)))
+            for u, v in mst_all_edges
+        )
+
+    # Find the MST again, over this new set of edges
+    G_S = G.edge_subgraph(mst_all_edges)
+    T_S = nx.minimum_spanning_edges(G_S, weight="weight", data=False)
+
+    # Leaf nodes that are not terminal might still remain; remove them here
+    T_H = G.edge_subgraph(T_S).copy()
+    _remove_nonterminal_leaves(T_H, terminal_nodes)
+
+    return T_H.edges()
+
+
+def _remove_nonterminal_leaves(G, terminals):
+    terminal_set = set(terminals)
+    leaves = {n for n in G if len(set(G[n]) - {n}) == 1}
+    nonterminal_leaves = leaves - terminal_set
+
+    while nonterminal_leaves:
+        # Removing a node may create new non-terminal leaves, so we limit
+        # search for candidate non-terminal nodes to neighbors of current
+        # non-terminal nodes
+        candidate_leaves = set.union(*(set(G[n]) for n in nonterminal_leaves))
+        candidate_leaves -= nonterminal_leaves | terminal_set
+        # Remove current set of non-terminal nodes
+        G.remove_nodes_from(nonterminal_leaves)
+        # Find any new non-terminal nodes from the set of candidates
+        leaves = {n for n in candidate_leaves if len(set(G[n]) - {n}) == 1}
+        nonterminal_leaves = leaves - terminal_set
+
+
+ALGORITHMS = {
+    "kou": _kou_steiner_tree,
+    "mehlhorn": _mehlhorn_steiner_tree,
+}
+
+
+@not_implemented_for("directed")
+@nx._dispatchable(preserve_all_attrs=True, returns_graph=True)
+def steiner_tree(G, terminal_nodes, weight="weight", method=None):
+    r"""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` (also *S*)
+    is a tree within `G` that spans those nodes and has minimum size (sum of
+    edge weights) among all such trees.
+
+    The approximation algorithm is specified with the `method` keyword
+    argument. All three available algorithms produce a tree whose weight is
+    within a ``(2 - (2 / l))`` factor of the weight of the optimal Steiner tree,
+    where ``l`` is the minimum number of leaf nodes across all possible Steiner
+    trees.
+
+    * ``"kou"`` [2]_ (runtime $O(|S| |V|^2)$) computes 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*.
+
+    * ``"mehlhorn"`` [3]_ (runtime $O(|E|+|V|\log|V|)$) modifies Kou et al.'s
+      algorithm, beginning by finding the closest terminal node for each
+      non-terminal. This data is used to create a complete graph containing only
+      the terminal nodes, in which edge is weighted with the shortest path
+      distance between them. The algorithm then proceeds in the same way as Kou
+      et al..
+
+    Parameters
+    ----------
+    G : NetworkX graph
+
+    terminal_nodes : list
+         A list of terminal nodes for which minimum steiner tree is
+         to be found.
+
+    weight : string (default = 'weight')
+        Use the edge attribute specified by this string as the edge weight.
+        Any edge attribute not present defaults to 1.
+
+    method : string, optional (default = 'mehlhorn')
+        The algorithm to use to approximate the Steiner tree.
+        Supported options: 'kou', 'mehlhorn'.
+        Other inputs produce a ValueError.
+
+    Returns
+    -------
+    NetworkX graph
+        Approximation to the minimum steiner tree of `G` induced by
+        `terminal_nodes` .
+
+    Raises
+    ------
+    NetworkXNotImplemented
+        If `G` is directed.
+
+    ValueError
+        If the specified `method` is not supported.
+
+    Notes
+    -----
+    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
+    .. [2] Kou, L., G. Markowsky, and L. Berman. 1981.
+           ‘A Fast Algorithm for Steiner Trees’.
+           Acta Informatica 15 (2): 141–45.
+           https://doi.org/10.1007/BF00288961.
+    .. [3] Mehlhorn, Kurt. 1988.
+           ‘A Faster Approximation Algorithm for the Steiner Problem in Graphs’.
+           Information Processing Letters 27 (3): 125–28.
+           https://doi.org/10.1016/0020-0190(88)90066-X.
+    """
+    if method is None:
+        method = "mehlhorn"
+
+    try:
+        algo = ALGORITHMS[method]
+    except KeyError as e:
+        raise ValueError(f"{method} is not a valid choice for an algorithm.") from e
+
+    edges = algo(G, terminal_nodes, weight)
+    # 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