Mercurial > repos > shellac > sam_consensus_v3
annotate env/lib/python3.9/site-packages/networkx/algorithms/flow/gomory_hu.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 """ |
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2 Gomory-Hu tree of undirected Graphs. |
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3 """ |
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4 import networkx as nx |
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5 from networkx.utils import not_implemented_for |
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6 |
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7 from .edmondskarp import edmonds_karp |
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8 from .utils import build_residual_network |
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9 |
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10 default_flow_func = edmonds_karp |
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11 |
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12 __all__ = ["gomory_hu_tree"] |
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13 |
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14 |
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15 @not_implemented_for("directed") |
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16 def gomory_hu_tree(G, capacity="capacity", flow_func=None): |
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17 r"""Returns the Gomory-Hu tree of an undirected graph G. |
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18 |
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19 A Gomory-Hu tree of an undirected graph with capacities is a |
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20 weighted tree that represents the minimum s-t cuts for all s-t |
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21 pairs in the graph. |
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22 |
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23 It only requires `n-1` minimum cut computations instead of the |
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24 obvious `n(n-1)/2`. The tree represents all s-t cuts as the |
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25 minimum cut value among any pair of nodes is the minimum edge |
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26 weight in the shortest path between the two nodes in the |
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27 Gomory-Hu tree. |
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28 |
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29 The Gomory-Hu tree also has the property that removing the |
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30 edge with the minimum weight in the shortest path between |
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31 any two nodes leaves two connected components that form |
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32 a partition of the nodes in G that defines the minimum s-t |
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33 cut. |
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34 |
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35 See Examples section below for details. |
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36 |
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37 Parameters |
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38 ---------- |
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39 G : NetworkX graph |
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40 Undirected graph |
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41 |
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42 capacity : string |
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43 Edges of the graph G are expected to have an attribute capacity |
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44 that indicates how much flow the edge can support. If this |
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45 attribute is not present, the edge is considered to have |
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46 infinite capacity. Default value: 'capacity'. |
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47 |
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48 flow_func : function |
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49 Function to perform the underlying flow computations. Default value |
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50 :func:`edmonds_karp`. This function performs better in sparse graphs |
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51 with right tailed degree distributions. |
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52 :func:`shortest_augmenting_path` will perform better in denser |
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53 graphs. |
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54 |
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55 Returns |
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56 ------- |
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57 Tree : NetworkX graph |
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58 A NetworkX graph representing the Gomory-Hu tree of the input graph. |
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59 |
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60 Raises |
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61 ------ |
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62 NetworkXNotImplemented |
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63 Raised if the input graph is directed. |
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64 |
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65 NetworkXError |
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66 Raised if the input graph is an empty Graph. |
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67 |
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68 Examples |
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69 -------- |
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70 >>> G = nx.karate_club_graph() |
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71 >>> nx.set_edge_attributes(G, 1, "capacity") |
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72 >>> T = nx.gomory_hu_tree(G) |
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73 >>> # The value of the minimum cut between any pair |
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74 ... # of nodes in G is the minimum edge weight in the |
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75 ... # shortest path between the two nodes in the |
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76 ... # Gomory-Hu tree. |
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77 ... def minimum_edge_weight_in_shortest_path(T, u, v): |
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78 ... path = nx.shortest_path(T, u, v, weight="weight") |
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79 ... return min((T[u][v]["weight"], (u, v)) for (u, v) in zip(path, path[1:])) |
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80 >>> u, v = 0, 33 |
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81 >>> cut_value, edge = minimum_edge_weight_in_shortest_path(T, u, v) |
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82 >>> cut_value |
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83 10 |
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84 >>> nx.minimum_cut_value(G, u, v) |
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85 10 |
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86 >>> # The Comory-Hu tree also has the property that removing the |
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87 ... # edge with the minimum weight in the shortest path between |
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88 ... # any two nodes leaves two connected components that form |
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89 ... # a partition of the nodes in G that defines the minimum s-t |
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90 ... # cut. |
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91 ... cut_value, edge = minimum_edge_weight_in_shortest_path(T, u, v) |
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92 >>> T.remove_edge(*edge) |
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93 >>> U, V = list(nx.connected_components(T)) |
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94 >>> # Thus U and V form a partition that defines a minimum cut |
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95 ... # between u and v in G. You can compute the edge cut set, |
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96 ... # that is, the set of edges that if removed from G will |
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97 ... # disconnect u from v in G, with this information: |
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98 ... cutset = set() |
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99 >>> for x, nbrs in ((n, G[n]) for n in U): |
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100 ... cutset.update((x, y) for y in nbrs if y in V) |
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101 >>> # Because we have set the capacities of all edges to 1 |
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102 ... # the cutset contains ten edges |
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103 ... len(cutset) |
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104 10 |
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105 >>> # You can use any maximum flow algorithm for the underlying |
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106 ... # flow computations using the argument flow_func |
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107 ... from networkx.algorithms import flow |
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108 >>> T = nx.gomory_hu_tree(G, flow_func=flow.boykov_kolmogorov) |
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109 >>> cut_value, edge = minimum_edge_weight_in_shortest_path(T, u, v) |
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110 >>> cut_value |
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111 10 |
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112 >>> nx.minimum_cut_value(G, u, v, flow_func=flow.boykov_kolmogorov) |
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113 10 |
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114 |
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115 Notes |
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116 ----- |
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117 This implementation is based on Gusfield approach [1]_ to compute |
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118 Comory-Hu trees, which does not require node contractions and has |
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119 the same computational complexity than the original method. |
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120 |
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121 See also |
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122 -------- |
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123 :func:`minimum_cut` |
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124 :func:`maximum_flow` |
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125 |
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126 References |
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127 ---------- |
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128 .. [1] Gusfield D: Very simple methods for all pairs network flow analysis. |
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129 SIAM J Comput 19(1):143-155, 1990. |
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130 |
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131 """ |
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132 if flow_func is None: |
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133 flow_func = default_flow_func |
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134 |
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135 if len(G) == 0: # empty graph |
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136 msg = "Empty Graph does not have a Gomory-Hu tree representation" |
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137 raise nx.NetworkXError(msg) |
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138 |
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139 # Start the tree as a star graph with an arbitrary node at the center |
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140 tree = {} |
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141 labels = {} |
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142 iter_nodes = iter(G) |
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143 root = next(iter_nodes) |
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144 for n in iter_nodes: |
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145 tree[n] = root |
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146 |
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147 # Reuse residual network |
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148 R = build_residual_network(G, capacity) |
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149 |
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150 # For all the leaves in the star graph tree (that is n-1 nodes). |
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151 for source in tree: |
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152 # Find neighbor in the tree |
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153 target = tree[source] |
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154 # compute minimum cut |
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155 cut_value, partition = nx.minimum_cut( |
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156 G, source, target, capacity=capacity, flow_func=flow_func, residual=R |
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157 ) |
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158 labels[(source, target)] = cut_value |
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159 # Update the tree |
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160 # Source will always be in partition[0] and target in partition[1] |
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161 for node in partition[0]: |
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162 if node != source and node in tree and tree[node] == target: |
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163 tree[node] = source |
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164 labels[node, source] = labels.get((node, target), cut_value) |
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165 # |
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166 if target != root and tree[target] in partition[0]: |
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167 labels[source, tree[target]] = labels[target, tree[target]] |
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168 labels[target, source] = cut_value |
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169 tree[source] = tree[target] |
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170 tree[target] = source |
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171 |
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172 # Build the tree |
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173 T = nx.Graph() |
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174 T.add_nodes_from(G) |
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175 T.add_weighted_edges_from(((u, v, labels[u, v]) for u, v in tree.items())) |
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176 return T |