Mercurial > repos > shellac > sam_consensus_v3
annotate env/lib/python3.9/site-packages/networkx/algorithms/connectivity/connectivity.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 Flow based connectivity algorithms |
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3 """ |
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4 |
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5 import itertools |
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6 from operator import itemgetter |
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7 |
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8 import networkx as nx |
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9 |
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10 # Define the default maximum flow function to use in all flow based |
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11 # connectivity algorithms. |
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12 from networkx.algorithms.flow import boykov_kolmogorov |
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13 from networkx.algorithms.flow import dinitz |
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14 from networkx.algorithms.flow import edmonds_karp |
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15 from networkx.algorithms.flow import shortest_augmenting_path |
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16 from networkx.algorithms.flow import build_residual_network |
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17 |
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18 default_flow_func = edmonds_karp |
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19 |
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20 from .utils import build_auxiliary_node_connectivity, build_auxiliary_edge_connectivity |
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21 |
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22 __all__ = [ |
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23 "average_node_connectivity", |
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24 "local_node_connectivity", |
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25 "node_connectivity", |
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26 "local_edge_connectivity", |
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27 "edge_connectivity", |
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28 "all_pairs_node_connectivity", |
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29 ] |
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30 |
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31 |
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32 def local_node_connectivity( |
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33 G, s, t, flow_func=None, auxiliary=None, residual=None, cutoff=None |
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34 ): |
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35 r"""Computes local node connectivity for nodes s and t. |
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36 |
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37 Local node connectivity for two non adjacent nodes s and t is the |
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38 minimum number of nodes that must be removed (along with their incident |
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39 edges) to disconnect them. |
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40 |
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41 This is a flow based implementation of node connectivity. We compute the |
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42 maximum flow on an auxiliary digraph build from the original input |
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43 graph (see below for details). |
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44 |
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45 Parameters |
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46 ---------- |
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47 G : NetworkX graph |
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48 Undirected graph |
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49 |
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50 s : node |
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51 Source node |
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52 |
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53 t : node |
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54 Target node |
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55 |
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56 flow_func : function |
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57 A function for computing the maximum flow among a pair of nodes. |
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58 The function has to accept at least three parameters: a Digraph, |
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59 a source node, and a target node. And return a residual network |
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60 that follows NetworkX conventions (see :meth:`maximum_flow` for |
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61 details). If flow_func is None, the default maximum flow function |
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62 (:meth:`edmonds_karp`) is used. See below for details. The choice |
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63 of the default function may change from version to version and |
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64 should not be relied on. Default value: None. |
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65 |
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66 auxiliary : NetworkX DiGraph |
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67 Auxiliary digraph to compute flow based node connectivity. It has |
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68 to have a graph attribute called mapping with a dictionary mapping |
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69 node names in G and in the auxiliary digraph. If provided |
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70 it will be reused instead of recreated. Default value: None. |
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71 |
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72 residual : NetworkX DiGraph |
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73 Residual network to compute maximum flow. If provided it will be |
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74 reused instead of recreated. Default value: None. |
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75 |
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76 cutoff : integer, float |
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77 If specified, the maximum flow algorithm will terminate when the |
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78 flow value reaches or exceeds the cutoff. This is only for the |
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79 algorithms that support the cutoff parameter: :meth:`edmonds_karp` |
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80 and :meth:`shortest_augmenting_path`. Other algorithms will ignore |
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81 this parameter. Default value: None. |
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82 |
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83 Returns |
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84 ------- |
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85 K : integer |
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86 local node connectivity for nodes s and t |
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87 |
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88 Examples |
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89 -------- |
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90 This function is not imported in the base NetworkX namespace, so you |
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91 have to explicitly import it from the connectivity package: |
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92 |
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93 >>> from networkx.algorithms.connectivity import local_node_connectivity |
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94 |
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95 We use in this example the platonic icosahedral graph, which has node |
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96 connectivity 5. |
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97 |
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98 >>> G = nx.icosahedral_graph() |
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99 >>> local_node_connectivity(G, 0, 6) |
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100 5 |
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101 |
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102 If you need to compute local connectivity on several pairs of |
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103 nodes in the same graph, it is recommended that you reuse the |
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104 data structures that NetworkX uses in the computation: the |
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105 auxiliary digraph for node connectivity, and the residual |
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106 network for the underlying maximum flow computation. |
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107 |
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108 Example of how to compute local node connectivity among |
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109 all pairs of nodes of the platonic icosahedral graph reusing |
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110 the data structures. |
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111 |
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112 >>> import itertools |
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113 >>> # You also have to explicitly import the function for |
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114 >>> # building the auxiliary digraph from the connectivity package |
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115 >>> from networkx.algorithms.connectivity import build_auxiliary_node_connectivity |
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116 ... |
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117 >>> H = build_auxiliary_node_connectivity(G) |
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118 >>> # And the function for building the residual network from the |
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119 >>> # flow package |
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120 >>> from networkx.algorithms.flow import build_residual_network |
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121 >>> # Note that the auxiliary digraph has an edge attribute named capacity |
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122 >>> R = build_residual_network(H, "capacity") |
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123 >>> result = dict.fromkeys(G, dict()) |
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124 >>> # Reuse the auxiliary digraph and the residual network by passing them |
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125 >>> # as parameters |
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126 >>> for u, v in itertools.combinations(G, 2): |
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127 ... k = local_node_connectivity(G, u, v, auxiliary=H, residual=R) |
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128 ... result[u][v] = k |
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129 ... |
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130 >>> all(result[u][v] == 5 for u, v in itertools.combinations(G, 2)) |
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131 True |
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132 |
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133 You can also use alternative flow algorithms for computing node |
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134 connectivity. For instance, in dense networks the algorithm |
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135 :meth:`shortest_augmenting_path` will usually perform better than |
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136 the default :meth:`edmonds_karp` which is faster for sparse |
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137 networks with highly skewed degree distributions. Alternative flow |
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138 functions have to be explicitly imported from the flow package. |
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139 |
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140 >>> from networkx.algorithms.flow import shortest_augmenting_path |
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141 >>> local_node_connectivity(G, 0, 6, flow_func=shortest_augmenting_path) |
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142 5 |
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143 |
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144 Notes |
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145 ----- |
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146 This is a flow based implementation of node connectivity. We compute the |
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147 maximum flow using, by default, the :meth:`edmonds_karp` algorithm (see: |
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148 :meth:`maximum_flow`) on an auxiliary digraph build from the original |
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149 input graph: |
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150 |
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151 For an undirected graph G having `n` nodes and `m` edges we derive a |
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152 directed graph H with `2n` nodes and `2m+n` arcs by replacing each |
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153 original node `v` with two nodes `v_A`, `v_B` linked by an (internal) |
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154 arc in H. Then for each edge (`u`, `v`) in G we add two arcs |
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155 (`u_B`, `v_A`) and (`v_B`, `u_A`) in H. Finally we set the attribute |
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156 capacity = 1 for each arc in H [1]_ . |
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157 |
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158 For a directed graph G having `n` nodes and `m` arcs we derive a |
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159 directed graph H with `2n` nodes and `m+n` arcs by replacing each |
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160 original node `v` with two nodes `v_A`, `v_B` linked by an (internal) |
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161 arc (`v_A`, `v_B`) in H. Then for each arc (`u`, `v`) in G we add one arc |
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162 (`u_B`, `v_A`) in H. Finally we set the attribute capacity = 1 for |
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163 each arc in H. |
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164 |
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165 This is equal to the local node connectivity because the value of |
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166 a maximum s-t-flow is equal to the capacity of a minimum s-t-cut. |
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167 |
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168 See also |
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169 -------- |
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170 :meth:`local_edge_connectivity` |
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171 :meth:`node_connectivity` |
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172 :meth:`minimum_node_cut` |
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173 :meth:`maximum_flow` |
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174 :meth:`edmonds_karp` |
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175 :meth:`preflow_push` |
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176 :meth:`shortest_augmenting_path` |
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177 |
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178 References |
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179 ---------- |
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180 .. [1] Kammer, Frank and Hanjo Taubig. Graph Connectivity. in Brandes and |
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181 Erlebach, 'Network Analysis: Methodological Foundations', Lecture |
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182 Notes in Computer Science, Volume 3418, Springer-Verlag, 2005. |
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183 http://www.informatik.uni-augsburg.de/thi/personen/kammer/Graph_Connectivity.pdf |
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184 |
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185 """ |
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186 if flow_func is None: |
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187 flow_func = default_flow_func |
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188 |
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189 if auxiliary is None: |
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190 H = build_auxiliary_node_connectivity(G) |
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191 else: |
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192 H = auxiliary |
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193 |
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194 mapping = H.graph.get("mapping", None) |
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195 if mapping is None: |
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196 raise nx.NetworkXError("Invalid auxiliary digraph.") |
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197 |
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198 kwargs = dict(flow_func=flow_func, residual=residual) |
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199 if flow_func is shortest_augmenting_path: |
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200 kwargs["cutoff"] = cutoff |
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201 kwargs["two_phase"] = True |
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202 elif flow_func is edmonds_karp: |
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203 kwargs["cutoff"] = cutoff |
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204 elif flow_func is dinitz: |
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205 kwargs["cutoff"] = cutoff |
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206 elif flow_func is boykov_kolmogorov: |
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207 kwargs["cutoff"] = cutoff |
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208 |
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209 return nx.maximum_flow_value(H, f"{mapping[s]}B", f"{mapping[t]}A", **kwargs) |
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210 |
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211 |
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212 def node_connectivity(G, s=None, t=None, flow_func=None): |
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213 r"""Returns node connectivity for a graph or digraph G. |
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214 |
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215 Node connectivity is equal to the minimum number of nodes that |
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216 must be removed to disconnect G or render it trivial. If source |
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217 and target nodes are provided, this function returns the local node |
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218 connectivity: the minimum number of nodes that must be removed to break |
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219 all paths from source to target in G. |
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220 |
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221 Parameters |
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222 ---------- |
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223 G : NetworkX graph |
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224 Undirected graph |
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225 |
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226 s : node |
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227 Source node. Optional. Default value: None. |
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228 |
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229 t : node |
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230 Target node. Optional. Default value: None. |
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231 |
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232 flow_func : function |
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233 A function for computing the maximum flow among a pair of nodes. |
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234 The function has to accept at least three parameters: a Digraph, |
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235 a source node, and a target node. And return a residual network |
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236 that follows NetworkX conventions (see :meth:`maximum_flow` for |
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237 details). If flow_func is None, the default maximum flow function |
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238 (:meth:`edmonds_karp`) is used. See below for details. The |
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239 choice of the default function may change from version |
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240 to version and should not be relied on. Default value: None. |
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241 |
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242 Returns |
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243 ------- |
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244 K : integer |
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245 Node connectivity of G, or local node connectivity if source |
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246 and target are provided. |
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247 |
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248 Examples |
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249 -------- |
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250 >>> # Platonic icosahedral graph is 5-node-connected |
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251 >>> G = nx.icosahedral_graph() |
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252 >>> nx.node_connectivity(G) |
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253 5 |
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254 |
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255 You can use alternative flow algorithms for the underlying maximum |
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256 flow computation. In dense networks the algorithm |
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257 :meth:`shortest_augmenting_path` will usually perform better |
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258 than the default :meth:`edmonds_karp`, which is faster for |
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259 sparse networks with highly skewed degree distributions. Alternative |
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260 flow functions have to be explicitly imported from the flow package. |
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261 |
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262 >>> from networkx.algorithms.flow import shortest_augmenting_path |
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263 >>> nx.node_connectivity(G, flow_func=shortest_augmenting_path) |
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264 5 |
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265 |
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266 If you specify a pair of nodes (source and target) as parameters, |
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267 this function returns the value of local node connectivity. |
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268 |
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269 >>> nx.node_connectivity(G, 3, 7) |
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270 5 |
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271 |
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272 If you need to perform several local computations among different |
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273 pairs of nodes on the same graph, it is recommended that you reuse |
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274 the data structures used in the maximum flow computations. See |
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275 :meth:`local_node_connectivity` for details. |
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276 |
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277 Notes |
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278 ----- |
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279 This is a flow based implementation of node connectivity. The |
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280 algorithm works by solving $O((n-\delta-1+\delta(\delta-1)/2))$ |
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281 maximum flow problems on an auxiliary digraph. Where $\delta$ |
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282 is the minimum degree of G. For details about the auxiliary |
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283 digraph and the computation of local node connectivity see |
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284 :meth:`local_node_connectivity`. This implementation is based |
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285 on algorithm 11 in [1]_. |
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286 |
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287 See also |
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288 -------- |
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289 :meth:`local_node_connectivity` |
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290 :meth:`edge_connectivity` |
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291 :meth:`maximum_flow` |
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292 :meth:`edmonds_karp` |
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293 :meth:`preflow_push` |
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294 :meth:`shortest_augmenting_path` |
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295 |
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296 References |
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297 ---------- |
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298 .. [1] Abdol-Hossein Esfahanian. Connectivity Algorithms. |
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299 http://www.cse.msu.edu/~cse835/Papers/Graph_connectivity_revised.pdf |
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300 |
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301 """ |
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302 if (s is not None and t is None) or (s is None and t is not None): |
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303 raise nx.NetworkXError("Both source and target must be specified.") |
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304 |
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305 # Local node connectivity |
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306 if s is not None and t is not None: |
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307 if s not in G: |
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308 raise nx.NetworkXError(f"node {s} not in graph") |
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309 if t not in G: |
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310 raise nx.NetworkXError(f"node {t} not in graph") |
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311 return local_node_connectivity(G, s, t, flow_func=flow_func) |
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312 |
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313 # Global node connectivity |
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314 if G.is_directed(): |
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315 if not nx.is_weakly_connected(G): |
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316 return 0 |
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317 iter_func = itertools.permutations |
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318 # It is necessary to consider both predecessors |
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319 # and successors for directed graphs |
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320 |
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321 def neighbors(v): |
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322 return itertools.chain.from_iterable([G.predecessors(v), G.successors(v)]) |
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323 |
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324 else: |
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325 if not nx.is_connected(G): |
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326 return 0 |
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327 iter_func = itertools.combinations |
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328 neighbors = G.neighbors |
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329 |
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330 # Reuse the auxiliary digraph and the residual network |
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331 H = build_auxiliary_node_connectivity(G) |
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332 R = build_residual_network(H, "capacity") |
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333 kwargs = dict(flow_func=flow_func, auxiliary=H, residual=R) |
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334 |
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335 # Pick a node with minimum degree |
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336 # Node connectivity is bounded by degree. |
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337 v, K = min(G.degree(), key=itemgetter(1)) |
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338 # compute local node connectivity with all its non-neighbors nodes |
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339 for w in set(G) - set(neighbors(v)) - {v}: |
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340 kwargs["cutoff"] = K |
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341 K = min(K, local_node_connectivity(G, v, w, **kwargs)) |
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342 # Also for non adjacent pairs of neighbors of v |
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343 for x, y in iter_func(neighbors(v), 2): |
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344 if y in G[x]: |
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345 continue |
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346 kwargs["cutoff"] = K |
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347 K = min(K, local_node_connectivity(G, x, y, **kwargs)) |
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348 |
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349 return K |
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350 |
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351 |
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352 def average_node_connectivity(G, flow_func=None): |
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353 r"""Returns the average connectivity of a graph G. |
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354 |
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355 The average connectivity `\bar{\kappa}` of a graph G is the average |
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356 of local node connectivity over all pairs of nodes of G [1]_ . |
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357 |
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358 .. math:: |
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359 |
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360 \bar{\kappa}(G) = \frac{\sum_{u,v} \kappa_{G}(u,v)}{{n \choose 2}} |
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361 |
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362 Parameters |
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363 ---------- |
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364 |
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365 G : NetworkX graph |
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366 Undirected graph |
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367 |
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368 flow_func : function |
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369 A function for computing the maximum flow among a pair of nodes. |
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370 The function has to accept at least three parameters: a Digraph, |
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371 a source node, and a target node. And return a residual network |
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372 that follows NetworkX conventions (see :meth:`maximum_flow` for |
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373 details). If flow_func is None, the default maximum flow function |
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374 (:meth:`edmonds_karp`) is used. See :meth:`local_node_connectivity` |
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375 for details. The choice of the default function may change from |
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376 version to version and should not be relied on. Default value: None. |
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377 |
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378 Returns |
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379 ------- |
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380 K : float |
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381 Average node connectivity |
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382 |
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383 See also |
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384 -------- |
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385 :meth:`local_node_connectivity` |
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386 :meth:`node_connectivity` |
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387 :meth:`edge_connectivity` |
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388 :meth:`maximum_flow` |
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389 :meth:`edmonds_karp` |
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390 :meth:`preflow_push` |
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391 :meth:`shortest_augmenting_path` |
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392 |
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393 References |
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394 ---------- |
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395 .. [1] Beineke, L., O. Oellermann, and R. Pippert (2002). The average |
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396 connectivity of a graph. Discrete mathematics 252(1-3), 31-45. |
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397 http://www.sciencedirect.com/science/article/pii/S0012365X01001807 |
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398 |
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399 """ |
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400 if G.is_directed(): |
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401 iter_func = itertools.permutations |
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402 else: |
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403 iter_func = itertools.combinations |
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404 |
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405 # Reuse the auxiliary digraph and the residual network |
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406 H = build_auxiliary_node_connectivity(G) |
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407 R = build_residual_network(H, "capacity") |
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408 kwargs = dict(flow_func=flow_func, auxiliary=H, residual=R) |
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409 |
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410 num, den = 0, 0 |
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411 for u, v in iter_func(G, 2): |
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412 num += local_node_connectivity(G, u, v, **kwargs) |
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413 den += 1 |
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414 |
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415 if den == 0: # Null Graph |
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416 return 0 |
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417 return num / den |
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418 |
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419 |
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420 def all_pairs_node_connectivity(G, nbunch=None, flow_func=None): |
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421 """Compute node connectivity between all pairs of nodes of G. |
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422 |
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423 Parameters |
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424 ---------- |
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425 G : NetworkX graph |
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426 Undirected graph |
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427 |
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428 nbunch: container |
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429 Container of nodes. If provided node connectivity will be computed |
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430 only over pairs of nodes in nbunch. |
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431 |
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432 flow_func : function |
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433 A function for computing the maximum flow among a pair of nodes. |
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434 The function has to accept at least three parameters: a Digraph, |
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435 a source node, and a target node. And return a residual network |
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436 that follows NetworkX conventions (see :meth:`maximum_flow` for |
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437 details). If flow_func is None, the default maximum flow function |
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438 (:meth:`edmonds_karp`) is used. See below for details. The |
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439 choice of the default function may change from version |
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440 to version and should not be relied on. Default value: None. |
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441 |
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442 Returns |
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443 ------- |
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444 all_pairs : dict |
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445 A dictionary with node connectivity between all pairs of nodes |
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446 in G, or in nbunch if provided. |
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447 |
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448 See also |
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449 -------- |
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450 :meth:`local_node_connectivity` |
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451 :meth:`edge_connectivity` |
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452 :meth:`local_edge_connectivity` |
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453 :meth:`maximum_flow` |
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454 :meth:`edmonds_karp` |
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455 :meth:`preflow_push` |
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456 :meth:`shortest_augmenting_path` |
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457 |
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458 """ |
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459 if nbunch is None: |
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460 nbunch = G |
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461 else: |
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462 nbunch = set(nbunch) |
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463 |
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464 directed = G.is_directed() |
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465 if directed: |
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466 iter_func = itertools.permutations |
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467 else: |
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468 iter_func = itertools.combinations |
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469 |
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470 all_pairs = {n: {} for n in nbunch} |
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471 |
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472 # Reuse auxiliary digraph and residual network |
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473 H = build_auxiliary_node_connectivity(G) |
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474 mapping = H.graph["mapping"] |
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475 R = build_residual_network(H, "capacity") |
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476 kwargs = dict(flow_func=flow_func, auxiliary=H, residual=R) |
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477 |
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478 for u, v in iter_func(nbunch, 2): |
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479 K = local_node_connectivity(G, u, v, **kwargs) |
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480 all_pairs[u][v] = K |
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481 if not directed: |
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482 all_pairs[v][u] = K |
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483 |
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484 return all_pairs |
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485 |
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486 |
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487 def local_edge_connectivity( |
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488 G, s, t, flow_func=None, auxiliary=None, residual=None, cutoff=None |
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489 ): |
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490 r"""Returns local edge connectivity for nodes s and t in G. |
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491 |
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492 Local edge connectivity for two nodes s and t is the minimum number |
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493 of edges that must be removed to disconnect them. |
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494 |
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495 This is a flow based implementation of edge connectivity. We compute the |
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496 maximum flow on an auxiliary digraph build from the original |
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497 network (see below for details). This is equal to the local edge |
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498 connectivity because the value of a maximum s-t-flow is equal to the |
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499 capacity of a minimum s-t-cut (Ford and Fulkerson theorem) [1]_ . |
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500 |
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501 Parameters |
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502 ---------- |
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503 G : NetworkX graph |
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504 Undirected or directed graph |
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505 |
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506 s : node |
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507 Source node |
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508 |
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509 t : node |
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510 Target node |
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511 |
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512 flow_func : function |
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513 A function for computing the maximum flow among a pair of nodes. |
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514 The function has to accept at least three parameters: a Digraph, |
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515 a source node, and a target node. And return a residual network |
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516 that follows NetworkX conventions (see :meth:`maximum_flow` for |
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517 details). If flow_func is None, the default maximum flow function |
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518 (:meth:`edmonds_karp`) is used. See below for details. The |
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519 choice of the default function may change from version |
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520 to version and should not be relied on. Default value: None. |
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521 |
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522 auxiliary : NetworkX DiGraph |
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523 Auxiliary digraph for computing flow based edge connectivity. If |
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524 provided it will be reused instead of recreated. Default value: None. |
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525 |
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526 residual : NetworkX DiGraph |
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527 Residual network to compute maximum flow. If provided it will be |
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528 reused instead of recreated. Default value: None. |
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529 |
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530 cutoff : integer, float |
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531 If specified, the maximum flow algorithm will terminate when the |
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532 flow value reaches or exceeds the cutoff. This is only for the |
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533 algorithms that support the cutoff parameter: :meth:`edmonds_karp` |
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534 and :meth:`shortest_augmenting_path`. Other algorithms will ignore |
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535 this parameter. Default value: None. |
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536 |
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537 Returns |
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538 ------- |
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539 K : integer |
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540 local edge connectivity for nodes s and t. |
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541 |
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542 Examples |
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543 -------- |
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544 This function is not imported in the base NetworkX namespace, so you |
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545 have to explicitly import it from the connectivity package: |
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546 |
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547 >>> from networkx.algorithms.connectivity import local_edge_connectivity |
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548 |
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549 We use in this example the platonic icosahedral graph, which has edge |
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550 connectivity 5. |
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551 |
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552 >>> G = nx.icosahedral_graph() |
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553 >>> local_edge_connectivity(G, 0, 6) |
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554 5 |
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555 |
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556 If you need to compute local connectivity on several pairs of |
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557 nodes in the same graph, it is recommended that you reuse the |
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558 data structures that NetworkX uses in the computation: the |
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559 auxiliary digraph for edge connectivity, and the residual |
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560 network for the underlying maximum flow computation. |
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561 |
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562 Example of how to compute local edge connectivity among |
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563 all pairs of nodes of the platonic icosahedral graph reusing |
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564 the data structures. |
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565 |
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566 >>> import itertools |
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567 >>> # You also have to explicitly import the function for |
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568 >>> # building the auxiliary digraph from the connectivity package |
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569 >>> from networkx.algorithms.connectivity import build_auxiliary_edge_connectivity |
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570 >>> H = build_auxiliary_edge_connectivity(G) |
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571 >>> # And the function for building the residual network from the |
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572 >>> # flow package |
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573 >>> from networkx.algorithms.flow import build_residual_network |
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574 >>> # Note that the auxiliary digraph has an edge attribute named capacity |
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575 >>> R = build_residual_network(H, "capacity") |
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576 >>> result = dict.fromkeys(G, dict()) |
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577 >>> # Reuse the auxiliary digraph and the residual network by passing them |
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578 >>> # as parameters |
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579 >>> for u, v in itertools.combinations(G, 2): |
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580 ... k = local_edge_connectivity(G, u, v, auxiliary=H, residual=R) |
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581 ... result[u][v] = k |
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582 >>> all(result[u][v] == 5 for u, v in itertools.combinations(G, 2)) |
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583 True |
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584 |
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585 You can also use alternative flow algorithms for computing edge |
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586 connectivity. For instance, in dense networks the algorithm |
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587 :meth:`shortest_augmenting_path` will usually perform better than |
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588 the default :meth:`edmonds_karp` which is faster for sparse |
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589 networks with highly skewed degree distributions. Alternative flow |
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590 functions have to be explicitly imported from the flow package. |
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591 |
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592 >>> from networkx.algorithms.flow import shortest_augmenting_path |
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593 >>> local_edge_connectivity(G, 0, 6, flow_func=shortest_augmenting_path) |
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594 5 |
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595 |
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596 Notes |
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597 ----- |
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598 This is a flow based implementation of edge connectivity. We compute the |
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599 maximum flow using, by default, the :meth:`edmonds_karp` algorithm on an |
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600 auxiliary digraph build from the original input graph: |
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601 |
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602 If the input graph is undirected, we replace each edge (`u`,`v`) with |
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603 two reciprocal arcs (`u`, `v`) and (`v`, `u`) and then we set the attribute |
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604 'capacity' for each arc to 1. If the input graph is directed we simply |
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605 add the 'capacity' attribute. This is an implementation of algorithm 1 |
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606 in [1]_. |
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607 |
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608 The maximum flow in the auxiliary network is equal to the local edge |
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609 connectivity because the value of a maximum s-t-flow is equal to the |
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610 capacity of a minimum s-t-cut (Ford and Fulkerson theorem). |
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611 |
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612 See also |
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613 -------- |
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614 :meth:`edge_connectivity` |
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615 :meth:`local_node_connectivity` |
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616 :meth:`node_connectivity` |
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617 :meth:`maximum_flow` |
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618 :meth:`edmonds_karp` |
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619 :meth:`preflow_push` |
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620 :meth:`shortest_augmenting_path` |
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621 |
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622 References |
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623 ---------- |
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624 .. [1] Abdol-Hossein Esfahanian. Connectivity Algorithms. |
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625 http://www.cse.msu.edu/~cse835/Papers/Graph_connectivity_revised.pdf |
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626 |
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627 """ |
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628 if flow_func is None: |
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629 flow_func = default_flow_func |
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630 |
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631 if auxiliary is None: |
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632 H = build_auxiliary_edge_connectivity(G) |
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633 else: |
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634 H = auxiliary |
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635 |
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636 kwargs = dict(flow_func=flow_func, residual=residual) |
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637 if flow_func is shortest_augmenting_path: |
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638 kwargs["cutoff"] = cutoff |
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639 kwargs["two_phase"] = True |
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640 elif flow_func is edmonds_karp: |
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641 kwargs["cutoff"] = cutoff |
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642 elif flow_func is dinitz: |
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643 kwargs["cutoff"] = cutoff |
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644 elif flow_func is boykov_kolmogorov: |
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645 kwargs["cutoff"] = cutoff |
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646 |
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647 return nx.maximum_flow_value(H, s, t, **kwargs) |
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648 |
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649 |
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650 def edge_connectivity(G, s=None, t=None, flow_func=None, cutoff=None): |
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651 r"""Returns the edge connectivity of the graph or digraph G. |
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652 |
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653 The edge connectivity is equal to the minimum number of edges that |
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654 must be removed to disconnect G or render it trivial. If source |
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655 and target nodes are provided, this function returns the local edge |
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656 connectivity: the minimum number of edges that must be removed to |
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657 break all paths from source to target in G. |
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658 |
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659 Parameters |
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660 ---------- |
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661 G : NetworkX graph |
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662 Undirected or directed graph |
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663 |
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664 s : node |
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665 Source node. Optional. Default value: None. |
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666 |
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667 t : node |
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668 Target node. Optional. Default value: None. |
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669 |
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670 flow_func : function |
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671 A function for computing the maximum flow among a pair of nodes. |
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672 The function has to accept at least three parameters: a Digraph, |
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673 a source node, and a target node. And return a residual network |
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674 that follows NetworkX conventions (see :meth:`maximum_flow` for |
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675 details). If flow_func is None, the default maximum flow function |
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676 (:meth:`edmonds_karp`) is used. See below for details. The |
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677 choice of the default function may change from version |
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678 to version and should not be relied on. Default value: None. |
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679 |
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680 cutoff : integer, float |
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681 If specified, the maximum flow algorithm will terminate when the |
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682 flow value reaches or exceeds the cutoff. This is only for the |
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683 algorithms that support the cutoff parameter: e.g., :meth:`edmonds_karp` |
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684 and :meth:`shortest_augmenting_path`. Other algorithms will ignore |
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685 this parameter. Default value: None. |
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686 |
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687 Returns |
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688 ------- |
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689 K : integer |
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690 Edge connectivity for G, or local edge connectivity if source |
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691 and target were provided |
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692 |
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693 Examples |
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694 -------- |
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695 >>> # Platonic icosahedral graph is 5-edge-connected |
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696 >>> G = nx.icosahedral_graph() |
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697 >>> nx.edge_connectivity(G) |
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698 5 |
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699 |
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700 You can use alternative flow algorithms for the underlying |
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701 maximum flow computation. In dense networks the algorithm |
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702 :meth:`shortest_augmenting_path` will usually perform better |
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703 than the default :meth:`edmonds_karp`, which is faster for |
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704 sparse networks with highly skewed degree distributions. |
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705 Alternative flow functions have to be explicitly imported |
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706 from the flow package. |
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707 |
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708 >>> from networkx.algorithms.flow import shortest_augmenting_path |
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709 >>> nx.edge_connectivity(G, flow_func=shortest_augmenting_path) |
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710 5 |
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711 |
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712 If you specify a pair of nodes (source and target) as parameters, |
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713 this function returns the value of local edge connectivity. |
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714 |
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715 >>> nx.edge_connectivity(G, 3, 7) |
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716 5 |
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717 |
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718 If you need to perform several local computations among different |
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719 pairs of nodes on the same graph, it is recommended that you reuse |
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720 the data structures used in the maximum flow computations. See |
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721 :meth:`local_edge_connectivity` for details. |
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722 |
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723 Notes |
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724 ----- |
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725 This is a flow based implementation of global edge connectivity. |
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726 For undirected graphs the algorithm works by finding a 'small' |
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727 dominating set of nodes of G (see algorithm 7 in [1]_ ) and |
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728 computing local maximum flow (see :meth:`local_edge_connectivity`) |
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729 between an arbitrary node in the dominating set and the rest of |
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730 nodes in it. This is an implementation of algorithm 6 in [1]_ . |
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731 For directed graphs, the algorithm does n calls to the maximum |
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732 flow function. This is an implementation of algorithm 8 in [1]_ . |
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733 |
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734 See also |
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735 -------- |
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736 :meth:`local_edge_connectivity` |
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737 :meth:`local_node_connectivity` |
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738 :meth:`node_connectivity` |
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739 :meth:`maximum_flow` |
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740 :meth:`edmonds_karp` |
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741 :meth:`preflow_push` |
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742 :meth:`shortest_augmenting_path` |
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743 :meth:`k_edge_components` |
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744 :meth:`k_edge_subgraphs` |
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745 |
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746 References |
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747 ---------- |
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748 .. [1] Abdol-Hossein Esfahanian. Connectivity Algorithms. |
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749 http://www.cse.msu.edu/~cse835/Papers/Graph_connectivity_revised.pdf |
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750 |
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751 """ |
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752 if (s is not None and t is None) or (s is None and t is not None): |
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753 raise nx.NetworkXError("Both source and target must be specified.") |
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754 |
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755 # Local edge connectivity |
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756 if s is not None and t is not None: |
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757 if s not in G: |
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758 raise nx.NetworkXError(f"node {s} not in graph") |
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759 if t not in G: |
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760 raise nx.NetworkXError(f"node {t} not in graph") |
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761 return local_edge_connectivity(G, s, t, flow_func=flow_func, cutoff=cutoff) |
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762 |
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763 # Global edge connectivity |
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764 # reuse auxiliary digraph and residual network |
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765 H = build_auxiliary_edge_connectivity(G) |
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766 R = build_residual_network(H, "capacity") |
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767 kwargs = dict(flow_func=flow_func, auxiliary=H, residual=R) |
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768 |
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769 if G.is_directed(): |
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770 # Algorithm 8 in [1] |
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771 if not nx.is_weakly_connected(G): |
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772 return 0 |
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773 |
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774 # initial value for \lambda is minimum degree |
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775 L = min(d for n, d in G.degree()) |
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776 nodes = list(G) |
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777 n = len(nodes) |
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778 |
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779 if cutoff is not None: |
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780 L = min(cutoff, L) |
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781 |
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782 for i in range(n): |
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783 kwargs["cutoff"] = L |
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784 try: |
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785 L = min(L, local_edge_connectivity(G, nodes[i], nodes[i + 1], **kwargs)) |
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786 except IndexError: # last node! |
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787 L = min(L, local_edge_connectivity(G, nodes[i], nodes[0], **kwargs)) |
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788 return L |
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789 else: # undirected |
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790 # Algorithm 6 in [1] |
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791 if not nx.is_connected(G): |
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792 return 0 |
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793 |
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794 # initial value for \lambda is minimum degree |
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795 L = min(d for n, d in G.degree()) |
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796 |
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797 if cutoff is not None: |
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798 L = min(cutoff, L) |
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799 |
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800 # A dominating set is \lambda-covering |
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801 # We need a dominating set with at least two nodes |
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802 for node in G: |
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803 D = nx.dominating_set(G, start_with=node) |
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804 v = D.pop() |
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805 if D: |
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806 break |
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807 else: |
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808 # in complete graphs the dominating sets will always be of one node |
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809 # thus we return min degree |
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810 return L |
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811 |
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812 for w in D: |
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813 kwargs["cutoff"] = L |
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814 L = min(L, local_edge_connectivity(G, v, w, **kwargs)) |
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815 |
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816 return L |