Mercurial > repos > shellac > sam_consensus_v3
annotate env/lib/python3.9/site-packages/networkx/algorithms/approximation/steinertree.py @ 0:4f3585e2f14b draft default tip
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author | shellac |
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date | Mon, 22 Mar 2021 18:12:50 +0000 |
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1 from itertools import chain |
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2 |
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3 from networkx.utils import pairwise, not_implemented_for |
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4 import networkx as nx |
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5 |
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6 __all__ = ["metric_closure", "steiner_tree"] |
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7 |
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8 |
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9 @not_implemented_for("directed") |
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10 def metric_closure(G, weight="weight"): |
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11 """ Return the metric closure of a graph. |
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12 |
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13 The metric closure of a graph *G* is the complete graph in which each edge |
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14 is weighted by the shortest path distance between the nodes in *G* . |
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15 |
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16 Parameters |
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17 ---------- |
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18 G : NetworkX graph |
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19 |
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20 Returns |
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21 ------- |
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22 NetworkX graph |
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23 Metric closure of the graph `G`. |
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24 |
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25 """ |
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26 M = nx.Graph() |
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27 |
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28 Gnodes = set(G) |
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29 |
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30 # check for connected graph while processing first node |
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31 all_paths_iter = nx.all_pairs_dijkstra(G, weight=weight) |
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32 u, (distance, path) = next(all_paths_iter) |
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33 if Gnodes - set(distance): |
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34 msg = "G is not a connected graph. metric_closure is not defined." |
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35 raise nx.NetworkXError(msg) |
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36 Gnodes.remove(u) |
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37 for v in Gnodes: |
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38 M.add_edge(u, v, distance=distance[v], path=path[v]) |
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39 |
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40 # first node done -- now process the rest |
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41 for u, (distance, path) in all_paths_iter: |
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42 Gnodes.remove(u) |
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43 for v in Gnodes: |
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44 M.add_edge(u, v, distance=distance[v], path=path[v]) |
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45 |
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46 return M |
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47 |
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48 |
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49 @not_implemented_for("directed") |
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50 def steiner_tree(G, terminal_nodes, weight="weight"): |
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51 """ Return an approximation to the minimum Steiner tree of a graph. |
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52 |
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53 The minimum Steiner tree of `G` w.r.t a set of `terminal_nodes` |
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54 is a tree within `G` that spans those nodes and has minimum size |
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55 (sum of edge weights) among all such trees. |
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56 |
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57 The minimum Steiner tree can be approximated by computing the minimum |
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58 spanning tree of the subgraph of the metric closure of *G* induced by the |
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59 terminal nodes, where the metric closure of *G* is the complete graph in |
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60 which each edge is weighted by the shortest path distance between the |
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61 nodes in *G* . |
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62 This algorithm produces a tree whose weight is within a (2 - (2 / t)) |
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63 factor of the weight of the optimal Steiner tree where *t* is number of |
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64 terminal nodes. |
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65 |
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66 Parameters |
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67 ---------- |
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68 G : NetworkX graph |
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69 |
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70 terminal_nodes : list |
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71 A list of terminal nodes for which minimum steiner tree is |
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72 to be found. |
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73 |
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74 Returns |
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75 ------- |
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76 NetworkX graph |
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77 Approximation to the minimum steiner tree of `G` induced by |
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78 `terminal_nodes` . |
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79 |
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80 Notes |
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81 ----- |
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82 For multigraphs, the edge between two nodes with minimum weight is the |
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83 edge put into the Steiner tree. |
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84 |
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85 |
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86 References |
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87 ---------- |
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88 .. [1] Steiner_tree_problem on Wikipedia. |
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89 https://en.wikipedia.org/wiki/Steiner_tree_problem |
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90 """ |
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91 # H is the subgraph induced by terminal_nodes in the metric closure M of G. |
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92 M = metric_closure(G, weight=weight) |
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93 H = M.subgraph(terminal_nodes) |
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94 # Use the 'distance' attribute of each edge provided by M. |
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95 mst_edges = nx.minimum_spanning_edges(H, weight="distance", data=True) |
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96 # Create an iterator over each edge in each shortest path; repeats are okay |
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97 edges = chain.from_iterable(pairwise(d["path"]) for u, v, d in mst_edges) |
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98 # For multigraph we should add the minimal weight edge keys |
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99 if G.is_multigraph(): |
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100 edges = ( |
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101 (u, v, min(G[u][v], key=lambda k: G[u][v][k][weight])) for u, v in edges |
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102 ) |
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103 T = G.edge_subgraph(edges) |
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104 return T |