Mercurial > repos > shellac > sam_consensus_v3
annotate env/lib/python3.9/site-packages/networkx/algorithms/connectivity/cuts.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 cut algorithms |
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3 """ |
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4 import itertools |
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5 import networkx as nx |
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6 |
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7 # Define the default maximum flow function to use in all flow based |
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8 # cut algorithms. |
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9 from networkx.algorithms.flow import edmonds_karp |
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10 from networkx.algorithms.flow import build_residual_network |
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11 |
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12 default_flow_func = edmonds_karp |
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13 |
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14 from .utils import build_auxiliary_node_connectivity, build_auxiliary_edge_connectivity |
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15 |
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16 __all__ = [ |
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17 "minimum_st_node_cut", |
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18 "minimum_node_cut", |
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19 "minimum_st_edge_cut", |
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20 "minimum_edge_cut", |
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21 ] |
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22 |
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23 |
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24 def minimum_st_edge_cut(G, s, t, flow_func=None, auxiliary=None, residual=None): |
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25 """Returns the edges of the cut-set of a minimum (s, t)-cut. |
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26 |
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27 This function returns the set of edges of minimum cardinality that, |
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28 if removed, would destroy all paths among source and target in G. |
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29 Edge weights are not considered. See :meth:`minimum_cut` for |
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30 computing minimum cuts considering edge weights. |
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31 |
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32 Parameters |
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33 ---------- |
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34 G : NetworkX graph |
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35 |
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36 s : node |
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37 Source node for the flow. |
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38 |
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39 t : node |
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40 Sink node for the flow. |
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41 |
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42 auxiliary : NetworkX DiGraph |
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43 Auxiliary digraph to compute flow based node connectivity. It has |
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44 to have a graph attribute called mapping with a dictionary mapping |
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45 node names in G and in the auxiliary digraph. If provided |
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46 it will be reused instead of recreated. Default value: None. |
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47 |
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48 flow_func : function |
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49 A function for computing the maximum flow among a pair of nodes. |
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50 The function has to accept at least three parameters: a Digraph, |
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51 a source node, and a target node. And return a residual network |
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52 that follows NetworkX conventions (see :meth:`maximum_flow` for |
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53 details). If flow_func is None, the default maximum flow function |
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54 (:meth:`edmonds_karp`) is used. See :meth:`node_connectivity` for |
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55 details. The choice of the default function may change from version |
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56 to version and should not be relied on. Default value: None. |
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57 |
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58 residual : NetworkX DiGraph |
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59 Residual network to compute maximum flow. If provided it will be |
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60 reused instead of recreated. Default value: None. |
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61 |
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62 Returns |
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63 ------- |
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64 cutset : set |
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65 Set of edges that, if removed from the graph, will disconnect it. |
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66 |
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67 See also |
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68 -------- |
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69 :meth:`minimum_cut` |
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70 :meth:`minimum_node_cut` |
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71 :meth:`minimum_edge_cut` |
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72 :meth:`stoer_wagner` |
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73 :meth:`node_connectivity` |
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74 :meth:`edge_connectivity` |
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75 :meth:`maximum_flow` |
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76 :meth:`edmonds_karp` |
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77 :meth:`preflow_push` |
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78 :meth:`shortest_augmenting_path` |
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79 |
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80 Examples |
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81 -------- |
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82 This function is not imported in the base NetworkX namespace, so you |
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83 have to explicitly import it from the connectivity package: |
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84 |
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85 >>> from networkx.algorithms.connectivity import minimum_st_edge_cut |
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86 |
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87 We use in this example the platonic icosahedral graph, which has edge |
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88 connectivity 5. |
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89 |
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90 >>> G = nx.icosahedral_graph() |
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91 >>> len(minimum_st_edge_cut(G, 0, 6)) |
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92 5 |
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93 |
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94 If you need to compute local edge cuts on several pairs of |
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95 nodes in the same graph, it is recommended that you reuse the |
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96 data structures that NetworkX uses in the computation: the |
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97 auxiliary digraph for edge connectivity, and the residual |
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98 network for the underlying maximum flow computation. |
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99 |
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100 Example of how to compute local edge cuts among all pairs of |
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101 nodes of the platonic icosahedral graph reusing the data |
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102 structures. |
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103 |
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104 >>> import itertools |
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105 >>> # You also have to explicitly import the function for |
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106 >>> # building the auxiliary digraph from the connectivity package |
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107 >>> from networkx.algorithms.connectivity import build_auxiliary_edge_connectivity |
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108 >>> H = build_auxiliary_edge_connectivity(G) |
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109 >>> # And the function for building the residual network from the |
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110 >>> # flow package |
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111 >>> from networkx.algorithms.flow import build_residual_network |
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112 >>> # Note that the auxiliary digraph has an edge attribute named capacity |
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113 >>> R = build_residual_network(H, "capacity") |
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114 >>> result = dict.fromkeys(G, dict()) |
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115 >>> # Reuse the auxiliary digraph and the residual network by passing them |
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116 >>> # as parameters |
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117 >>> for u, v in itertools.combinations(G, 2): |
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118 ... k = len(minimum_st_edge_cut(G, u, v, auxiliary=H, residual=R)) |
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119 ... result[u][v] = k |
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120 >>> all(result[u][v] == 5 for u, v in itertools.combinations(G, 2)) |
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121 True |
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122 |
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123 You can also use alternative flow algorithms for computing edge |
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124 cuts. For instance, in dense networks the algorithm |
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125 :meth:`shortest_augmenting_path` will usually perform better than |
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126 the default :meth:`edmonds_karp` which is faster for sparse |
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127 networks with highly skewed degree distributions. Alternative flow |
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128 functions have to be explicitly imported from the flow package. |
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129 |
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130 >>> from networkx.algorithms.flow import shortest_augmenting_path |
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131 >>> len(minimum_st_edge_cut(G, 0, 6, flow_func=shortest_augmenting_path)) |
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132 5 |
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133 |
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134 """ |
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135 if flow_func is None: |
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136 flow_func = default_flow_func |
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137 |
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138 if auxiliary is None: |
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139 H = build_auxiliary_edge_connectivity(G) |
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140 else: |
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141 H = auxiliary |
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142 |
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143 kwargs = dict(capacity="capacity", flow_func=flow_func, residual=residual) |
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144 |
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145 cut_value, partition = nx.minimum_cut(H, s, t, **kwargs) |
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146 reachable, non_reachable = partition |
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147 # Any edge in the original graph linking the two sets in the |
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148 # partition is part of the edge cutset |
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149 cutset = set() |
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150 for u, nbrs in ((n, G[n]) for n in reachable): |
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151 cutset.update((u, v) for v in nbrs if v in non_reachable) |
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152 |
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153 return cutset |
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154 |
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155 |
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156 def minimum_st_node_cut(G, s, t, flow_func=None, auxiliary=None, residual=None): |
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157 r"""Returns a set of nodes of minimum cardinality that disconnect source |
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158 from target in G. |
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159 |
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160 This function returns the set of nodes of minimum cardinality that, |
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161 if removed, would destroy all paths among source and target in G. |
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162 |
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163 Parameters |
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164 ---------- |
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165 G : NetworkX graph |
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166 |
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167 s : node |
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168 Source node. |
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169 |
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170 t : node |
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171 Target node. |
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172 |
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173 flow_func : function |
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174 A function for computing the maximum flow among a pair of nodes. |
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175 The function has to accept at least three parameters: a Digraph, |
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176 a source node, and a target node. And return a residual network |
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177 that follows NetworkX conventions (see :meth:`maximum_flow` for |
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178 details). If flow_func is None, the default maximum flow function |
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179 (:meth:`edmonds_karp`) is used. See below for details. The choice |
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180 of the default function may change from version to version and |
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181 should not be relied on. Default value: None. |
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182 |
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183 auxiliary : NetworkX DiGraph |
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184 Auxiliary digraph to compute flow based node connectivity. It has |
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185 to have a graph attribute called mapping with a dictionary mapping |
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186 node names in G and in the auxiliary digraph. If provided |
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187 it will be reused instead of recreated. Default value: None. |
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188 |
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189 residual : NetworkX DiGraph |
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190 Residual network to compute maximum flow. If provided it will be |
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191 reused instead of recreated. Default value: None. |
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192 |
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193 Returns |
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194 ------- |
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195 cutset : set |
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196 Set of nodes that, if removed, would destroy all paths between |
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197 source and target in G. |
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198 |
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199 Examples |
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200 -------- |
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201 This function is not imported in the base NetworkX namespace, so you |
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202 have to explicitly import it from the connectivity package: |
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203 |
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204 >>> from networkx.algorithms.connectivity import minimum_st_node_cut |
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205 |
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206 We use in this example the platonic icosahedral graph, which has node |
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207 connectivity 5. |
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208 |
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209 >>> G = nx.icosahedral_graph() |
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210 >>> len(minimum_st_node_cut(G, 0, 6)) |
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211 5 |
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212 |
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213 If you need to compute local st cuts between several pairs of |
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214 nodes in the same graph, it is recommended that you reuse the |
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215 data structures that NetworkX uses in the computation: the |
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216 auxiliary digraph for node connectivity and node cuts, and the |
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217 residual network for the underlying maximum flow computation. |
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218 |
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219 Example of how to compute local st node cuts reusing the data |
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220 structures: |
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221 |
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222 >>> # You also have to explicitly import the function for |
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223 >>> # building the auxiliary digraph from the connectivity package |
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224 >>> from networkx.algorithms.connectivity import build_auxiliary_node_connectivity |
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225 >>> H = build_auxiliary_node_connectivity(G) |
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226 >>> # And the function for building the residual network from the |
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227 >>> # flow package |
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228 >>> from networkx.algorithms.flow import build_residual_network |
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229 >>> # Note that the auxiliary digraph has an edge attribute named capacity |
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230 >>> R = build_residual_network(H, "capacity") |
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231 >>> # Reuse the auxiliary digraph and the residual network by passing them |
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232 >>> # as parameters |
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233 >>> len(minimum_st_node_cut(G, 0, 6, auxiliary=H, residual=R)) |
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234 5 |
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235 |
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236 You can also use alternative flow algorithms for computing minimum st |
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237 node cuts. For instance, in dense networks the algorithm |
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238 :meth:`shortest_augmenting_path` will usually perform better than |
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239 the default :meth:`edmonds_karp` which is faster for sparse |
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240 networks with highly skewed degree distributions. Alternative flow |
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241 functions have to be explicitly imported from the flow package. |
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242 |
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243 >>> from networkx.algorithms.flow import shortest_augmenting_path |
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244 >>> len(minimum_st_node_cut(G, 0, 6, flow_func=shortest_augmenting_path)) |
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245 5 |
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246 |
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247 Notes |
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248 ----- |
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249 This is a flow based implementation of minimum node cut. The algorithm |
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250 is based in solving a number of maximum flow computations to determine |
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251 the capacity of the minimum cut on an auxiliary directed network that |
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252 corresponds to the minimum node cut of G. It handles both directed |
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253 and undirected graphs. This implementation is based on algorithm 11 |
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254 in [1]_. |
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255 |
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256 See also |
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257 -------- |
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258 :meth:`minimum_node_cut` |
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259 :meth:`minimum_edge_cut` |
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260 :meth:`stoer_wagner` |
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261 :meth:`node_connectivity` |
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262 :meth:`edge_connectivity` |
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263 :meth:`maximum_flow` |
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264 :meth:`edmonds_karp` |
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265 :meth:`preflow_push` |
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266 :meth:`shortest_augmenting_path` |
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267 |
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268 References |
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269 ---------- |
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270 .. [1] Abdol-Hossein Esfahanian. Connectivity Algorithms. |
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271 http://www.cse.msu.edu/~cse835/Papers/Graph_connectivity_revised.pdf |
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272 |
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273 """ |
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274 if auxiliary is None: |
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275 H = build_auxiliary_node_connectivity(G) |
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276 else: |
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277 H = auxiliary |
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278 |
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279 mapping = H.graph.get("mapping", None) |
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280 if mapping is None: |
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281 raise nx.NetworkXError("Invalid auxiliary digraph.") |
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282 if G.has_edge(s, t) or G.has_edge(t, s): |
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283 return {} |
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284 kwargs = dict(flow_func=flow_func, residual=residual, auxiliary=H) |
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285 |
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286 # The edge cut in the auxiliary digraph corresponds to the node cut in the |
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287 # original graph. |
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288 edge_cut = minimum_st_edge_cut(H, f"{mapping[s]}B", f"{mapping[t]}A", **kwargs) |
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289 # Each node in the original graph maps to two nodes of the auxiliary graph |
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290 node_cut = {H.nodes[node]["id"] for edge in edge_cut for node in edge} |
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291 return node_cut - {s, t} |
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292 |
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293 |
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294 def minimum_node_cut(G, s=None, t=None, flow_func=None): |
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295 r"""Returns a set of nodes of minimum cardinality that disconnects G. |
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296 |
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297 If source and target nodes are provided, this function returns the |
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298 set of nodes of minimum cardinality that, if removed, would destroy |
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299 all paths among source and target in G. If not, it returns a set |
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300 of nodes of minimum cardinality that disconnects G. |
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301 |
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302 Parameters |
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303 ---------- |
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304 G : NetworkX graph |
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305 |
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306 s : node |
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307 Source node. Optional. Default value: None. |
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308 |
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309 t : node |
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310 Target node. Optional. Default value: None. |
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311 |
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312 flow_func : function |
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313 A function for computing the maximum flow among a pair of nodes. |
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314 The function has to accept at least three parameters: a Digraph, |
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315 a source node, and a target node. And return a residual network |
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316 that follows NetworkX conventions (see :meth:`maximum_flow` for |
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317 details). If flow_func is None, the default maximum flow function |
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318 (:meth:`edmonds_karp`) is used. See below for details. The |
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319 choice of the default function may change from version |
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320 to version and should not be relied on. Default value: None. |
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321 |
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322 Returns |
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323 ------- |
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324 cutset : set |
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325 Set of nodes that, if removed, would disconnect G. If source |
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326 and target nodes are provided, the set contains the nodes that |
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327 if removed, would destroy all paths between source and target. |
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328 |
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329 Examples |
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330 -------- |
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331 >>> # Platonic icosahedral graph has node connectivity 5 |
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332 >>> G = nx.icosahedral_graph() |
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333 >>> node_cut = nx.minimum_node_cut(G) |
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334 >>> len(node_cut) |
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335 5 |
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336 |
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337 You can use alternative flow algorithms for the underlying maximum |
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338 flow computation. In dense networks the algorithm |
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339 :meth:`shortest_augmenting_path` will usually perform better |
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340 than the default :meth:`edmonds_karp`, which is faster for |
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341 sparse networks with highly skewed degree distributions. Alternative |
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342 flow functions have to be explicitly imported from the flow package. |
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343 |
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344 >>> from networkx.algorithms.flow import shortest_augmenting_path |
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345 >>> node_cut == nx.minimum_node_cut(G, flow_func=shortest_augmenting_path) |
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346 True |
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347 |
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348 If you specify a pair of nodes (source and target) as parameters, |
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349 this function returns a local st node cut. |
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350 |
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351 >>> len(nx.minimum_node_cut(G, 3, 7)) |
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352 5 |
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353 |
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354 If you need to perform several local st cuts among different |
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355 pairs of nodes on the same graph, it is recommended that you reuse |
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356 the data structures used in the maximum flow computations. See |
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357 :meth:`minimum_st_node_cut` for details. |
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358 |
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359 Notes |
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360 ----- |
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361 This is a flow based implementation of minimum node cut. The algorithm |
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362 is based in solving a number of maximum flow computations to determine |
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363 the capacity of the minimum cut on an auxiliary directed network that |
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364 corresponds to the minimum node cut of G. It handles both directed |
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365 and undirected graphs. This implementation is based on algorithm 11 |
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366 in [1]_. |
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367 |
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368 See also |
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369 -------- |
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370 :meth:`minimum_st_node_cut` |
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371 :meth:`minimum_cut` |
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372 :meth:`minimum_edge_cut` |
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373 :meth:`stoer_wagner` |
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374 :meth:`node_connectivity` |
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375 :meth:`edge_connectivity` |
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376 :meth:`maximum_flow` |
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377 :meth:`edmonds_karp` |
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378 :meth:`preflow_push` |
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379 :meth:`shortest_augmenting_path` |
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380 |
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381 References |
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382 ---------- |
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383 .. [1] Abdol-Hossein Esfahanian. Connectivity Algorithms. |
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384 http://www.cse.msu.edu/~cse835/Papers/Graph_connectivity_revised.pdf |
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385 |
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386 """ |
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387 if (s is not None and t is None) or (s is None and t is not None): |
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388 raise nx.NetworkXError("Both source and target must be specified.") |
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389 |
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390 # Local minimum node cut. |
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391 if s is not None and t is not None: |
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392 if s not in G: |
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393 raise nx.NetworkXError(f"node {s} not in graph") |
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394 if t not in G: |
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395 raise nx.NetworkXError(f"node {t} not in graph") |
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396 return minimum_st_node_cut(G, s, t, flow_func=flow_func) |
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397 |
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398 # Global minimum node cut. |
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399 # Analog to the algorithm 11 for global node connectivity in [1]. |
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400 if G.is_directed(): |
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401 if not nx.is_weakly_connected(G): |
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402 raise nx.NetworkXError("Input graph is not connected") |
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403 iter_func = itertools.permutations |
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404 |
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405 def neighbors(v): |
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406 return itertools.chain.from_iterable([G.predecessors(v), G.successors(v)]) |
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407 |
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408 else: |
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409 if not nx.is_connected(G): |
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410 raise nx.NetworkXError("Input graph is not connected") |
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411 iter_func = itertools.combinations |
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412 neighbors = G.neighbors |
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413 |
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414 # Reuse the auxiliary digraph and the residual network. |
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415 H = build_auxiliary_node_connectivity(G) |
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416 R = build_residual_network(H, "capacity") |
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417 kwargs = dict(flow_func=flow_func, auxiliary=H, residual=R) |
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418 |
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419 # Choose a node with minimum degree. |
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420 v = min(G, key=G.degree) |
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421 # Initial node cutset is all neighbors of the node with minimum degree. |
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422 min_cut = set(G[v]) |
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423 # Compute st node cuts between v and all its non-neighbors nodes in G. |
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424 for w in set(G) - set(neighbors(v)) - {v}: |
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425 this_cut = minimum_st_node_cut(G, v, w, **kwargs) |
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426 if len(min_cut) >= len(this_cut): |
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427 min_cut = this_cut |
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428 # Also for non adjacent pairs of neighbors of v. |
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429 for x, y in iter_func(neighbors(v), 2): |
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430 if y in G[x]: |
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431 continue |
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432 this_cut = minimum_st_node_cut(G, x, y, **kwargs) |
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433 if len(min_cut) >= len(this_cut): |
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434 min_cut = this_cut |
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435 |
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436 return min_cut |
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437 |
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438 |
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439 def minimum_edge_cut(G, s=None, t=None, flow_func=None): |
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440 r"""Returns a set of edges of minimum cardinality that disconnects G. |
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441 |
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442 If source and target nodes are provided, this function returns the |
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443 set of edges of minimum cardinality that, if removed, would break |
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444 all paths among source and target in G. If not, it returns a set of |
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445 edges of minimum cardinality that disconnects G. |
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446 |
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447 Parameters |
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448 ---------- |
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449 G : NetworkX graph |
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450 |
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451 s : node |
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452 Source node. Optional. Default value: None. |
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453 |
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454 t : node |
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455 Target node. Optional. Default value: None. |
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456 |
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457 flow_func : function |
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458 A function for computing the maximum flow among a pair of nodes. |
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459 The function has to accept at least three parameters: a Digraph, |
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460 a source node, and a target node. And return a residual network |
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461 that follows NetworkX conventions (see :meth:`maximum_flow` for |
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462 details). If flow_func is None, the default maximum flow function |
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463 (:meth:`edmonds_karp`) is used. See below for details. The |
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464 choice of the default function may change from version |
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465 to version and should not be relied on. Default value: None. |
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466 |
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467 Returns |
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468 ------- |
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469 cutset : set |
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470 Set of edges that, if removed, would disconnect G. If source |
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471 and target nodes are provided, the set contains the edges that |
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472 if removed, would destroy all paths between source and target. |
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473 |
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474 Examples |
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475 -------- |
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476 >>> # Platonic icosahedral graph has edge connectivity 5 |
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477 >>> G = nx.icosahedral_graph() |
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478 >>> len(nx.minimum_edge_cut(G)) |
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479 5 |
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480 |
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481 You can use alternative flow algorithms for the underlying |
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482 maximum flow computation. In dense networks the algorithm |
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483 :meth:`shortest_augmenting_path` will usually perform better |
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484 than the default :meth:`edmonds_karp`, which is faster for |
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485 sparse networks with highly skewed degree distributions. |
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486 Alternative flow functions have to be explicitly imported |
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487 from the flow package. |
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488 |
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489 >>> from networkx.algorithms.flow import shortest_augmenting_path |
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490 >>> len(nx.minimum_edge_cut(G, flow_func=shortest_augmenting_path)) |
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491 5 |
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492 |
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493 If you specify a pair of nodes (source and target) as parameters, |
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494 this function returns the value of local edge connectivity. |
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495 |
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496 >>> nx.edge_connectivity(G, 3, 7) |
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497 5 |
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498 |
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499 If you need to perform several local computations among different |
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500 pairs of nodes on the same graph, it is recommended that you reuse |
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501 the data structures used in the maximum flow computations. See |
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502 :meth:`local_edge_connectivity` for details. |
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503 |
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504 Notes |
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505 ----- |
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506 This is a flow based implementation of minimum edge cut. For |
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507 undirected graphs the algorithm works by finding a 'small' dominating |
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508 set of nodes of G (see algorithm 7 in [1]_) and computing the maximum |
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509 flow between an arbitrary node in the dominating set and the rest of |
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510 nodes in it. This is an implementation of algorithm 6 in [1]_. For |
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511 directed graphs, the algorithm does n calls to the max flow function. |
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512 The function raises an error if the directed graph is not weakly |
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513 connected and returns an empty set if it is weakly connected. |
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514 It is an implementation of algorithm 8 in [1]_. |
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515 |
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516 See also |
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517 -------- |
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518 :meth:`minimum_st_edge_cut` |
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519 :meth:`minimum_node_cut` |
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520 :meth:`stoer_wagner` |
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521 :meth:`node_connectivity` |
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522 :meth:`edge_connectivity` |
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523 :meth:`maximum_flow` |
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524 :meth:`edmonds_karp` |
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525 :meth:`preflow_push` |
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526 :meth:`shortest_augmenting_path` |
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527 |
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528 References |
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529 ---------- |
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530 .. [1] Abdol-Hossein Esfahanian. Connectivity Algorithms. |
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531 http://www.cse.msu.edu/~cse835/Papers/Graph_connectivity_revised.pdf |
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532 |
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533 """ |
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534 if (s is not None and t is None) or (s is None and t is not None): |
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535 raise nx.NetworkXError("Both source and target must be specified.") |
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536 |
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537 # reuse auxiliary digraph and residual network |
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538 H = build_auxiliary_edge_connectivity(G) |
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539 R = build_residual_network(H, "capacity") |
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540 kwargs = dict(flow_func=flow_func, residual=R, auxiliary=H) |
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541 |
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542 # Local minimum edge cut if s and t are not None |
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543 if s is not None and t is not None: |
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544 if s not in G: |
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545 raise nx.NetworkXError(f"node {s} not in graph") |
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546 if t not in G: |
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547 raise nx.NetworkXError(f"node {t} not in graph") |
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548 return minimum_st_edge_cut(H, s, t, **kwargs) |
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549 |
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550 # Global minimum edge cut |
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551 # Analog to the algorithm for global edge connectivity |
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552 if G.is_directed(): |
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553 # Based on algorithm 8 in [1] |
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554 if not nx.is_weakly_connected(G): |
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555 raise nx.NetworkXError("Input graph is not connected") |
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556 |
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557 # Initial cutset is all edges of a node with minimum degree |
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558 node = min(G, key=G.degree) |
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559 min_cut = set(G.edges(node)) |
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560 nodes = list(G) |
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561 n = len(nodes) |
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562 for i in range(n): |
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563 try: |
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564 this_cut = minimum_st_edge_cut(H, nodes[i], nodes[i + 1], **kwargs) |
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565 if len(this_cut) <= len(min_cut): |
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566 min_cut = this_cut |
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567 except IndexError: # Last node! |
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568 this_cut = minimum_st_edge_cut(H, nodes[i], nodes[0], **kwargs) |
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569 if len(this_cut) <= len(min_cut): |
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570 min_cut = this_cut |
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571 |
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572 return min_cut |
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573 |
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574 else: # undirected |
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575 # Based on algorithm 6 in [1] |
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576 if not nx.is_connected(G): |
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577 raise nx.NetworkXError("Input graph is not connected") |
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578 |
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579 # Initial cutset is all edges of a node with minimum degree |
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580 node = min(G, key=G.degree) |
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581 min_cut = set(G.edges(node)) |
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582 # A dominating set is \lambda-covering |
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583 # We need a dominating set with at least two nodes |
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584 for node in G: |
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585 D = nx.dominating_set(G, start_with=node) |
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586 v = D.pop() |
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587 if D: |
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588 break |
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589 else: |
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590 # in complete graphs the dominating set will always be of one node |
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591 # thus we return min_cut, which now contains the edges of a node |
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592 # with minimum degree |
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593 return min_cut |
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594 for w in D: |
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595 this_cut = minimum_st_edge_cut(H, v, w, **kwargs) |
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596 if len(this_cut) <= len(min_cut): |
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597 min_cut = this_cut |
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598 |
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599 return min_cut |