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"]

[docs] @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`. """ 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
def _mehlhorn_steiner_tree(G, terminal_nodes, weight): paths = nx.multi_source_dijkstra_path(G, terminal_nodes) d_1 = {} s = {} for v in G.nodes(): s[v] = paths[v][0] d_1[(v, s[v])] = len(paths[v]) - 1 # G1-G4 names match those from the Mehlhorn 1988 paper. G_1_prime = nx.Graph() 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=weight_here) else: new_weight = min(weight_here, G_1_prime[su][sv]["weight"]) G_1_prime.add_edge(su, sv, weight=new_weight) G_2 = nx.minimum_spanning_edges(G_1_prime, data=True) G_3 = nx.Graph() for u, v, d in G_2: path = nx.shortest_path(G, u, v, weight) for n1, n2 in pairwise(path): G_3.add_edge(n1, n2) G_3_mst = list(nx.minimum_spanning_edges(G_3, data=False)) if G.is_multigraph(): G_3_mst = ( (u, v, min(G[u][v], key=lambda k: G[u][v][k][weight])) 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): # 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 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][weight])) 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): terminals_set = set(terminals) for n in list(G.nodes): if n not in terminals_set and == 1: G.remove_node(n) ALGORITHMS = { "kou": _kou_steiner_tree, "mehlhorn": _mehlhorn_steiner_tree, }
[docs] @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. .. [2] Kou, L., G. Markowsky, and L. Berman. 1981. ‘A Fast Algorithm for Steiner Trees’. Acta Informatica 15 (2): 141–45. .. [3] Mehlhorn, Kurt. 1988. ‘A Faster Approximation Algorithm for the Steiner Problem in Graphs’. Information Processing Letters 27 (3): 125–28. """ 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