Source code for networkx.algorithms.approximation.steinertree

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")
def metric_closure(G, weight="weight"):
    """Return the metric closure of a graph.

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

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

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

    """
    M = nx.Graph()

    Gnodes = set(G)

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


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

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

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

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

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

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

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


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