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# steiner_tree

steiner_tree(G, terminal_nodes, weight='weight', method=None)

  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.