Mercurial > repos > shellac > sam_consensus_v3
annotate env/lib/python3.9/site-packages/networkx/algorithms/connectivity/disjoint_paths.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 """Flow based node and edge disjoint paths.""" |
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2 import networkx as nx |
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3 from networkx.exception import NetworkXNoPath |
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4 |
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5 # Define the default maximum flow function to use for the undelying |
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6 # maximum flow computations |
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7 from networkx.algorithms.flow import edmonds_karp |
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8 from networkx.algorithms.flow import preflow_push |
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9 from networkx.algorithms.flow import shortest_augmenting_path |
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10 |
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11 default_flow_func = edmonds_karp |
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12 # Functions to build auxiliary data structures. |
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13 from .utils import build_auxiliary_node_connectivity |
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14 from .utils import build_auxiliary_edge_connectivity |
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15 |
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16 from itertools import filterfalse as _filterfalse |
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17 |
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18 __all__ = ["edge_disjoint_paths", "node_disjoint_paths"] |
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19 |
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20 |
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21 def edge_disjoint_paths( |
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22 G, s, t, flow_func=None, cutoff=None, auxiliary=None, residual=None |
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23 ): |
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24 """Returns the edges disjoint paths between source and target. |
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25 |
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26 Edge disjoint paths are paths that do not share any edge. The |
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27 number of edge disjoint paths between source and target is equal |
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28 to their edge connectivity. |
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29 |
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30 Parameters |
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31 ---------- |
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32 G : NetworkX graph |
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33 |
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34 s : node |
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35 Source node for the flow. |
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36 |
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37 t : node |
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38 Sink node for the flow. |
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39 |
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40 flow_func : function |
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41 A function for computing the maximum flow among a pair of nodes. |
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42 The function has to accept at least three parameters: a Digraph, |
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43 a source node, and a target node. And return a residual network |
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44 that follows NetworkX conventions (see :meth:`maximum_flow` for |
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45 details). If flow_func is None, the default maximum flow function |
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46 (:meth:`edmonds_karp`) is used. The choice of the default function |
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47 may change from version to version and should not be relied on. |
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48 Default value: None. |
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49 |
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50 cutoff : int |
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51 Maximum number of paths to yield. Some of the maximum flow |
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52 algorithms, such as :meth:`edmonds_karp` (the default) and |
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53 :meth:`shortest_augmenting_path` support the cutoff parameter, |
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54 and will terminate when the flow value reaches or exceeds the |
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55 cutoff. Other algorithms will ignore this parameter. |
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56 Default value: None. |
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57 |
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58 auxiliary : NetworkX DiGraph |
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59 Auxiliary digraph to compute flow based edge connectivity. It has |
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60 to have a graph attribute called mapping with a dictionary mapping |
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61 node names in G and in the auxiliary digraph. If provided |
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62 it will be reused instead of recreated. Default value: None. |
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63 |
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64 residual : NetworkX DiGraph |
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65 Residual network to compute maximum flow. If provided it will be |
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66 reused instead of recreated. Default value: None. |
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67 |
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68 Returns |
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69 ------- |
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70 paths : generator |
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71 A generator of edge independent paths. |
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72 |
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73 Raises |
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74 ------ |
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75 NetworkXNoPath |
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76 If there is no path between source and target. |
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77 |
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78 NetworkXError |
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79 If source or target are not in the graph G. |
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80 |
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81 See also |
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82 -------- |
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83 :meth:`node_disjoint_paths` |
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84 :meth:`edge_connectivity` |
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85 :meth:`maximum_flow` |
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86 :meth:`edmonds_karp` |
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87 :meth:`preflow_push` |
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88 :meth:`shortest_augmenting_path` |
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89 |
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90 Examples |
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91 -------- |
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92 We use in this example the platonic icosahedral graph, which has node |
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93 edge connectivity 5, thus there are 5 edge disjoint paths between any |
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94 pair of nodes. |
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95 |
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96 >>> G = nx.icosahedral_graph() |
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97 >>> len(list(nx.edge_disjoint_paths(G, 0, 6))) |
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98 5 |
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99 |
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100 |
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101 If you need to compute edge disjoint paths on several pairs of |
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102 nodes in the same graph, it is recommended that you reuse the |
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103 data structures that NetworkX uses in the computation: the |
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104 auxiliary digraph for edge connectivity, and the residual |
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105 network for the underlying maximum flow computation. |
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106 |
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107 Example of how to compute edge disjoint paths among all pairs of |
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108 nodes of the platonic icosahedral graph reusing the data |
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109 structures. |
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110 |
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111 >>> import itertools |
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112 >>> # You also have to explicitly import the function for |
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113 >>> # building the auxiliary digraph from the connectivity package |
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114 >>> from networkx.algorithms.connectivity import build_auxiliary_edge_connectivity |
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115 >>> H = build_auxiliary_edge_connectivity(G) |
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116 >>> # And the function for building the residual network from the |
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117 >>> # flow package |
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118 >>> from networkx.algorithms.flow import build_residual_network |
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119 >>> # Note that the auxiliary digraph has an edge attribute named capacity |
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120 >>> R = build_residual_network(H, "capacity") |
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121 >>> result = {n: {} for n in G} |
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122 >>> # Reuse the auxiliary digraph and the residual network by passing them |
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123 >>> # as arguments |
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124 >>> for u, v in itertools.combinations(G, 2): |
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125 ... k = len(list(nx.edge_disjoint_paths(G, u, v, auxiliary=H, residual=R))) |
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126 ... result[u][v] = k |
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127 >>> all(result[u][v] == 5 for u, v in itertools.combinations(G, 2)) |
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128 True |
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129 |
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130 You can also use alternative flow algorithms for computing edge disjoint |
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131 paths. For instance, in dense networks the algorithm |
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132 :meth:`shortest_augmenting_path` will usually perform better than |
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133 the default :meth:`edmonds_karp` which is faster for sparse |
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134 networks with highly skewed degree distributions. Alternative flow |
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135 functions have to be explicitly imported from the flow package. |
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136 |
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137 >>> from networkx.algorithms.flow import shortest_augmenting_path |
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138 >>> len(list(nx.edge_disjoint_paths(G, 0, 6, flow_func=shortest_augmenting_path))) |
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139 5 |
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140 |
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141 Notes |
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142 ----- |
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143 This is a flow based implementation of edge disjoint paths. We compute |
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144 the maximum flow between source and target on an auxiliary directed |
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145 network. The saturated edges in the residual network after running the |
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146 maximum flow algorithm correspond to edge disjoint paths between source |
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147 and target in the original network. This function handles both directed |
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148 and undirected graphs, and can use all flow algorithms from NetworkX flow |
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149 package. |
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150 |
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151 """ |
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152 if s not in G: |
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153 raise nx.NetworkXError(f"node {s} not in graph") |
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154 if t not in G: |
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155 raise nx.NetworkXError(f"node {t} not in graph") |
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156 |
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157 if flow_func is None: |
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158 flow_func = default_flow_func |
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159 |
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160 if auxiliary is None: |
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161 H = build_auxiliary_edge_connectivity(G) |
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162 else: |
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163 H = auxiliary |
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164 |
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165 # Maximum possible edge disjoint paths |
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166 possible = min(H.out_degree(s), H.in_degree(t)) |
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167 if not possible: |
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168 raise NetworkXNoPath |
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169 |
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170 if cutoff is None: |
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171 cutoff = possible |
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172 else: |
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173 cutoff = min(cutoff, possible) |
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174 |
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175 # Compute maximum flow between source and target. Flow functions in |
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176 # NetworkX return a residual network. |
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177 kwargs = dict( |
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178 capacity="capacity", residual=residual, cutoff=cutoff, value_only=True |
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179 ) |
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180 if flow_func is preflow_push: |
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181 del kwargs["cutoff"] |
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182 if flow_func is shortest_augmenting_path: |
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183 kwargs["two_phase"] = True |
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184 R = flow_func(H, s, t, **kwargs) |
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185 |
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186 if R.graph["flow_value"] == 0: |
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187 raise NetworkXNoPath |
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188 |
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189 # Saturated edges in the residual network form the edge disjoint paths |
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190 # between source and target |
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191 cutset = [ |
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192 (u, v) |
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193 for u, v, d in R.edges(data=True) |
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194 if d["capacity"] == d["flow"] and d["flow"] > 0 |
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195 ] |
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196 # This is equivalent of what flow.utils.build_flow_dict returns, but |
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197 # only for the nodes with saturated edges and without reporting 0 flows. |
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198 flow_dict = {n: {} for edge in cutset for n in edge} |
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199 for u, v in cutset: |
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200 flow_dict[u][v] = 1 |
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201 |
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202 # Rebuild the edge disjoint paths from the flow dictionary. |
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203 paths_found = 0 |
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204 for v in list(flow_dict[s]): |
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205 if paths_found >= cutoff: |
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206 # preflow_push does not support cutoff: we have to |
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207 # keep track of the paths founds and stop at cutoff. |
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208 break |
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209 path = [s] |
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210 if v == t: |
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211 path.append(v) |
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212 yield path |
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213 continue |
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214 u = v |
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215 while u != t: |
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216 path.append(u) |
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217 try: |
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218 u, _ = flow_dict[u].popitem() |
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219 except KeyError: |
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220 break |
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221 else: |
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222 path.append(t) |
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223 yield path |
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224 paths_found += 1 |
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225 |
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226 |
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227 def node_disjoint_paths( |
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228 G, s, t, flow_func=None, cutoff=None, auxiliary=None, residual=None |
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229 ): |
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230 r"""Computes node disjoint paths between source and target. |
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231 |
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232 Node disjoint paths are paths that only share their first and last |
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233 nodes. The number of node independent paths between two nodes is |
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234 equal to their local node connectivity. |
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235 |
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236 Parameters |
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237 ---------- |
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238 G : NetworkX graph |
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239 |
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240 s : node |
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241 Source node. |
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242 |
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243 t : node |
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244 Target node. |
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245 |
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246 flow_func : function |
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247 A function for computing the maximum flow among a pair of nodes. |
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248 The function has to accept at least three parameters: a Digraph, |
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249 a source node, and a target node. And return a residual network |
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250 that follows NetworkX conventions (see :meth:`maximum_flow` for |
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251 details). If flow_func is None, the default maximum flow function |
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252 (:meth:`edmonds_karp`) is used. See below for details. The choice |
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253 of the default function may change from version to version and |
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254 should not be relied on. Default value: None. |
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255 |
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256 cutoff : int |
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257 Maximum number of paths to yield. Some of the maximum flow |
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258 algorithms, such as :meth:`edmonds_karp` (the default) and |
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259 :meth:`shortest_augmenting_path` support the cutoff parameter, |
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260 and will terminate when the flow value reaches or exceeds the |
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261 cutoff. Other algorithms will ignore this parameter. |
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262 Default value: None. |
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263 |
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264 auxiliary : NetworkX DiGraph |
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265 Auxiliary digraph to compute flow based node connectivity. It has |
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266 to have a graph attribute called mapping with a dictionary mapping |
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267 node names in G and in the auxiliary digraph. If provided |
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268 it will be reused instead of recreated. Default value: None. |
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269 |
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270 residual : NetworkX DiGraph |
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271 Residual network to compute maximum flow. If provided it will be |
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272 reused instead of recreated. Default value: None. |
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273 |
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274 Returns |
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275 ------- |
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276 paths : generator |
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277 Generator of node disjoint paths. |
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278 |
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279 Raises |
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280 ------ |
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281 NetworkXNoPath |
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282 If there is no path between source and target. |
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283 |
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284 NetworkXError |
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285 If source or target are not in the graph G. |
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286 |
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287 Examples |
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288 -------- |
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289 We use in this example the platonic icosahedral graph, which has node |
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290 node connectivity 5, thus there are 5 node disjoint paths between any |
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291 pair of non neighbor nodes. |
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292 |
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293 >>> G = nx.icosahedral_graph() |
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294 >>> len(list(nx.node_disjoint_paths(G, 0, 6))) |
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295 5 |
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296 |
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297 If you need to compute node disjoint paths between several pairs of |
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298 nodes in the same graph, it is recommended that you reuse the |
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299 data structures that NetworkX uses in the computation: the |
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300 auxiliary digraph for node connectivity and node cuts, and the |
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301 residual network for the underlying maximum flow computation. |
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302 |
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303 Example of how to compute node disjoint paths reusing the data |
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304 structures: |
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305 |
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306 >>> # You also have to explicitly import the function for |
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307 >>> # building the auxiliary digraph from the connectivity package |
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308 >>> from networkx.algorithms.connectivity import build_auxiliary_node_connectivity |
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309 >>> H = build_auxiliary_node_connectivity(G) |
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310 >>> # And the function for building the residual network from the |
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311 >>> # flow package |
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312 >>> from networkx.algorithms.flow import build_residual_network |
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313 >>> # Note that the auxiliary digraph has an edge attribute named capacity |
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314 >>> R = build_residual_network(H, "capacity") |
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315 >>> # Reuse the auxiliary digraph and the residual network by passing them |
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316 >>> # as arguments |
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317 >>> len(list(nx.node_disjoint_paths(G, 0, 6, auxiliary=H, residual=R))) |
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318 5 |
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319 |
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320 You can also use alternative flow algorithms for computing node disjoint |
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321 paths. For instance, in dense networks the algorithm |
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322 :meth:`shortest_augmenting_path` will usually perform better than |
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323 the default :meth:`edmonds_karp` which is faster for sparse |
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324 networks with highly skewed degree distributions. Alternative flow |
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325 functions have to be explicitly imported from the flow package. |
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326 |
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327 >>> from networkx.algorithms.flow import shortest_augmenting_path |
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328 >>> len(list(nx.node_disjoint_paths(G, 0, 6, flow_func=shortest_augmenting_path))) |
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329 5 |
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330 |
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331 Notes |
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332 ----- |
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333 This is a flow based implementation of node disjoint paths. We compute |
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334 the maximum flow between source and target on an auxiliary directed |
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335 network. The saturated edges in the residual network after running the |
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336 maximum flow algorithm correspond to node disjoint paths between source |
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337 and target in the original network. This function handles both directed |
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338 and undirected graphs, and can use all flow algorithms from NetworkX flow |
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339 package. |
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340 |
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341 See also |
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342 -------- |
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343 :meth:`edge_disjoint_paths` |
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344 :meth:`node_connectivity` |
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345 :meth:`maximum_flow` |
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346 :meth:`edmonds_karp` |
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347 :meth:`preflow_push` |
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348 :meth:`shortest_augmenting_path` |
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349 |
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350 """ |
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351 if s not in G: |
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352 raise nx.NetworkXError(f"node {s} not in graph") |
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353 if t not in G: |
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354 raise nx.NetworkXError(f"node {t} not in graph") |
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355 |
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356 if auxiliary is None: |
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357 H = build_auxiliary_node_connectivity(G) |
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358 else: |
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359 H = auxiliary |
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360 |
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361 mapping = H.graph.get("mapping", None) |
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362 if mapping is None: |
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363 raise nx.NetworkXError("Invalid auxiliary digraph.") |
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364 |
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365 # Maximum possible edge disjoint paths |
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366 possible = min(H.out_degree(f"{mapping[s]}B"), H.in_degree(f"{mapping[t]}A")) |
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367 if not possible: |
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368 raise NetworkXNoPath |
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369 |
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370 if cutoff is None: |
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371 cutoff = possible |
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372 else: |
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373 cutoff = min(cutoff, possible) |
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374 |
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375 kwargs = dict(flow_func=flow_func, residual=residual, auxiliary=H, cutoff=cutoff) |
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376 |
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377 # The edge disjoint paths in the auxiliary digraph correspond to the node |
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378 # disjoint paths in the original graph. |
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379 paths_edges = edge_disjoint_paths(H, f"{mapping[s]}B", f"{mapping[t]}A", **kwargs) |
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380 for path in paths_edges: |
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381 # Each node in the original graph maps to two nodes in auxiliary graph |
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382 yield list(_unique_everseen(H.nodes[node]["id"] for node in path)) |
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383 |
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384 |
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385 def _unique_everseen(iterable): |
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386 # Adapted from https://docs.python.org/3/library/itertools.html examples |
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387 "List unique elements, preserving order. Remember all elements ever seen." |
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388 # unique_everseen('AAAABBBCCDAABBB') --> A B C D |
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389 seen = set() |
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390 seen_add = seen.add |
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391 for element in _filterfalse(seen.__contains__, iterable): |
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392 seen_add(element) |
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393 yield element |