Mercurial > repos > shellac > sam_consensus_v3
annotate env/lib/python3.9/site-packages/networkx/algorithms/approximation/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 """ Fast approximation for node connectivity |
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2 """ |
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3 import itertools |
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4 from operator import itemgetter |
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5 |
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6 import networkx as nx |
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7 |
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8 __all__ = [ |
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9 "local_node_connectivity", |
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10 "node_connectivity", |
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11 "all_pairs_node_connectivity", |
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12 ] |
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13 |
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14 INF = float("inf") |
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15 |
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16 |
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17 def local_node_connectivity(G, source, target, cutoff=None): |
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18 """Compute node connectivity between source and target. |
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19 |
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20 Pairwise or local node connectivity between two distinct and nonadjacent |
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21 nodes is the minimum number of nodes that must be removed (minimum |
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22 separating cutset) to disconnect them. By Menger's theorem, this is equal |
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23 to the number of node independent paths (paths that share no nodes other |
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24 than source and target). Which is what we compute in this function. |
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25 |
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26 This algorithm is a fast approximation that gives an strict lower |
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27 bound on the actual number of node independent paths between two nodes [1]_. |
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28 It works for both directed and undirected graphs. |
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29 |
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30 Parameters |
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31 ---------- |
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32 |
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33 G : NetworkX graph |
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34 |
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35 source : node |
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36 Starting node for node connectivity |
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37 |
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38 target : node |
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39 Ending node for node connectivity |
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40 |
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41 cutoff : integer |
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42 Maximum node connectivity to consider. If None, the minimum degree |
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43 of source or target is used as a cutoff. Default value None. |
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44 |
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45 Returns |
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46 ------- |
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47 k: integer |
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48 pairwise node connectivity |
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49 |
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50 Examples |
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51 -------- |
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52 >>> # Platonic octahedral graph has node connectivity 4 |
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53 >>> # for each non adjacent node pair |
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54 >>> from networkx.algorithms import approximation as approx |
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55 >>> G = nx.octahedral_graph() |
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56 >>> approx.local_node_connectivity(G, 0, 5) |
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57 4 |
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58 |
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59 Notes |
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60 ----- |
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61 This algorithm [1]_ finds node independents paths between two nodes by |
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62 computing their shortest path using BFS, marking the nodes of the path |
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63 found as 'used' and then searching other shortest paths excluding the |
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64 nodes marked as used until no more paths exist. It is not exact because |
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65 a shortest path could use nodes that, if the path were longer, may belong |
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66 to two different node independent paths. Thus it only guarantees an |
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67 strict lower bound on node connectivity. |
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68 |
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69 Note that the authors propose a further refinement, losing accuracy and |
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70 gaining speed, which is not implemented yet. |
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71 |
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72 See also |
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73 -------- |
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74 all_pairs_node_connectivity |
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75 node_connectivity |
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76 |
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77 References |
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78 ---------- |
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79 .. [1] White, Douglas R., and Mark Newman. 2001 A Fast Algorithm for |
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80 Node-Independent Paths. Santa Fe Institute Working Paper #01-07-035 |
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81 http://eclectic.ss.uci.edu/~drwhite/working.pdf |
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82 |
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83 """ |
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84 if target == source: |
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85 raise nx.NetworkXError("source and target have to be different nodes.") |
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86 |
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87 # Maximum possible node independent paths |
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88 if G.is_directed(): |
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89 possible = min(G.out_degree(source), G.in_degree(target)) |
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90 else: |
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91 possible = min(G.degree(source), G.degree(target)) |
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92 |
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93 K = 0 |
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94 if not possible: |
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95 return K |
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96 |
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97 if cutoff is None: |
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98 cutoff = INF |
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99 |
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100 exclude = set() |
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101 for i in range(min(possible, cutoff)): |
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102 try: |
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103 path = _bidirectional_shortest_path(G, source, target, exclude) |
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104 exclude.update(set(path)) |
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105 K += 1 |
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106 except nx.NetworkXNoPath: |
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107 break |
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108 |
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109 return K |
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110 |
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111 |
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112 def node_connectivity(G, s=None, t=None): |
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113 r"""Returns an approximation for node connectivity for a graph or digraph G. |
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114 |
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115 Node connectivity is equal to the minimum number of nodes that |
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116 must be removed to disconnect G or render it trivial. By Menger's theorem, |
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117 this is equal to the number of node independent paths (paths that |
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118 share no nodes other than source and target). |
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119 |
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120 If source and target nodes are provided, this function returns the |
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121 local node connectivity: the minimum number of nodes that must be |
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122 removed to break all paths from source to target in G. |
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123 |
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124 This algorithm is based on a fast approximation that gives an strict lower |
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125 bound on the actual number of node independent paths between two nodes [1]_. |
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126 It works for both directed and undirected graphs. |
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127 |
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128 Parameters |
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129 ---------- |
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130 G : NetworkX graph |
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131 Undirected graph |
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132 |
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133 s : node |
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134 Source node. Optional. Default value: None. |
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135 |
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136 t : node |
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137 Target node. Optional. Default value: None. |
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138 |
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139 Returns |
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140 ------- |
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141 K : integer |
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142 Node connectivity of G, or local node connectivity if source |
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143 and target are provided. |
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144 |
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145 Examples |
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146 -------- |
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147 >>> # Platonic octahedral graph is 4-node-connected |
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148 >>> from networkx.algorithms import approximation as approx |
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149 >>> G = nx.octahedral_graph() |
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150 >>> approx.node_connectivity(G) |
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151 4 |
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152 |
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153 Notes |
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154 ----- |
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155 This algorithm [1]_ finds node independents paths between two nodes by |
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156 computing their shortest path using BFS, marking the nodes of the path |
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157 found as 'used' and then searching other shortest paths excluding the |
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158 nodes marked as used until no more paths exist. It is not exact because |
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159 a shortest path could use nodes that, if the path were longer, may belong |
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160 to two different node independent paths. Thus it only guarantees an |
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161 strict lower bound on node connectivity. |
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162 |
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163 See also |
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164 -------- |
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165 all_pairs_node_connectivity |
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166 local_node_connectivity |
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167 |
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168 References |
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169 ---------- |
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170 .. [1] White, Douglas R., and Mark Newman. 2001 A Fast Algorithm for |
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171 Node-Independent Paths. Santa Fe Institute Working Paper #01-07-035 |
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172 http://eclectic.ss.uci.edu/~drwhite/working.pdf |
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173 |
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174 """ |
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175 if (s is not None and t is None) or (s is None and t is not None): |
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176 raise nx.NetworkXError("Both source and target must be specified.") |
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177 |
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178 # Local node connectivity |
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179 if s is not None and t is not None: |
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180 if s not in G: |
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181 raise nx.NetworkXError(f"node {s} not in graph") |
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182 if t not in G: |
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183 raise nx.NetworkXError(f"node {t} not in graph") |
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184 return local_node_connectivity(G, s, t) |
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185 |
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186 # Global node connectivity |
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187 if G.is_directed(): |
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188 connected_func = nx.is_weakly_connected |
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189 iter_func = itertools.permutations |
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190 |
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191 def neighbors(v): |
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192 return itertools.chain(G.predecessors(v), G.successors(v)) |
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193 |
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194 else: |
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195 connected_func = nx.is_connected |
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196 iter_func = itertools.combinations |
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197 neighbors = G.neighbors |
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198 |
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199 if not connected_func(G): |
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200 return 0 |
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201 |
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202 # Choose a node with minimum degree |
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203 v, minimum_degree = min(G.degree(), key=itemgetter(1)) |
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204 # Node connectivity is bounded by minimum degree |
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205 K = minimum_degree |
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206 # compute local node connectivity with all non-neighbors nodes |
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207 # and store the minimum |
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208 for w in set(G) - set(neighbors(v)) - {v}: |
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209 K = min(K, local_node_connectivity(G, v, w, cutoff=K)) |
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210 # Same for non adjacent pairs of neighbors of v |
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211 for x, y in iter_func(neighbors(v), 2): |
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212 if y not in G[x] and x != y: |
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213 K = min(K, local_node_connectivity(G, x, y, cutoff=K)) |
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214 return K |
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215 |
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216 |
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217 def all_pairs_node_connectivity(G, nbunch=None, cutoff=None): |
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218 """ Compute node connectivity between all pairs of nodes. |
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219 |
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220 Pairwise or local node connectivity between two distinct and nonadjacent |
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221 nodes is the minimum number of nodes that must be removed (minimum |
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222 separating cutset) to disconnect them. By Menger's theorem, this is equal |
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223 to the number of node independent paths (paths that share no nodes other |
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224 than source and target). Which is what we compute in this function. |
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225 |
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226 This algorithm is a fast approximation that gives an strict lower |
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227 bound on the actual number of node independent paths between two nodes [1]_. |
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228 It works for both directed and undirected graphs. |
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229 |
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230 |
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231 Parameters |
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232 ---------- |
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233 G : NetworkX graph |
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234 |
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235 nbunch: container |
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236 Container of nodes. If provided node connectivity will be computed |
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237 only over pairs of nodes in nbunch. |
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238 |
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239 cutoff : integer |
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240 Maximum node connectivity to consider. If None, the minimum degree |
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241 of source or target is used as a cutoff in each pair of nodes. |
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242 Default value None. |
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243 |
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244 Returns |
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245 ------- |
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246 K : dictionary |
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247 Dictionary, keyed by source and target, of pairwise node connectivity |
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248 |
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249 See Also |
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250 -------- |
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251 local_node_connectivity |
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252 node_connectivity |
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253 |
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254 References |
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255 ---------- |
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256 .. [1] White, Douglas R., and Mark Newman. 2001 A Fast Algorithm for |
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257 Node-Independent Paths. Santa Fe Institute Working Paper #01-07-035 |
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258 http://eclectic.ss.uci.edu/~drwhite/working.pdf |
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259 """ |
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260 if nbunch is None: |
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261 nbunch = G |
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262 else: |
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263 nbunch = set(nbunch) |
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264 |
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265 directed = G.is_directed() |
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266 if directed: |
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267 iter_func = itertools.permutations |
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268 else: |
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269 iter_func = itertools.combinations |
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270 |
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271 all_pairs = {n: {} for n in nbunch} |
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272 |
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273 for u, v in iter_func(nbunch, 2): |
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274 k = local_node_connectivity(G, u, v, cutoff=cutoff) |
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275 all_pairs[u][v] = k |
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276 if not directed: |
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277 all_pairs[v][u] = k |
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278 |
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279 return all_pairs |
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280 |
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281 |
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282 def _bidirectional_shortest_path(G, source, target, exclude): |
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283 """Returns shortest path between source and target ignoring nodes in the |
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284 container 'exclude'. |
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285 |
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286 Parameters |
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287 ---------- |
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288 |
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289 G : NetworkX graph |
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290 |
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291 source : node |
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292 Starting node for path |
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293 |
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294 target : node |
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295 Ending node for path |
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296 |
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297 exclude: container |
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298 Container for nodes to exclude from the search for shortest paths |
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299 |
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300 Returns |
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301 ------- |
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302 path: list |
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303 Shortest path between source and target ignoring nodes in 'exclude' |
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304 |
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305 Raises |
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306 ------ |
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307 NetworkXNoPath |
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308 If there is no path or if nodes are adjacent and have only one path |
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309 between them |
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310 |
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311 Notes |
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312 ----- |
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313 This function and its helper are originally from |
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314 networkx.algorithms.shortest_paths.unweighted and are modified to |
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315 accept the extra parameter 'exclude', which is a container for nodes |
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316 already used in other paths that should be ignored. |
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317 |
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318 References |
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319 ---------- |
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320 .. [1] White, Douglas R., and Mark Newman. 2001 A Fast Algorithm for |
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321 Node-Independent Paths. Santa Fe Institute Working Paper #01-07-035 |
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322 http://eclectic.ss.uci.edu/~drwhite/working.pdf |
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323 |
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324 """ |
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325 # call helper to do the real work |
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326 results = _bidirectional_pred_succ(G, source, target, exclude) |
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327 pred, succ, w = results |
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328 |
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329 # build path from pred+w+succ |
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330 path = [] |
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331 # from source to w |
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332 while w is not None: |
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333 path.append(w) |
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334 w = pred[w] |
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335 path.reverse() |
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336 # from w to target |
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337 w = succ[path[-1]] |
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338 while w is not None: |
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339 path.append(w) |
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340 w = succ[w] |
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341 |
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342 return path |
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343 |
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344 |
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345 def _bidirectional_pred_succ(G, source, target, exclude): |
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346 # does BFS from both source and target and meets in the middle |
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347 # excludes nodes in the container "exclude" from the search |
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348 if source is None or target is None: |
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349 raise nx.NetworkXException( |
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350 "Bidirectional shortest path called without source or target" |
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351 ) |
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352 if target == source: |
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353 return ({target: None}, {source: None}, source) |
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354 |
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355 # handle either directed or undirected |
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356 if G.is_directed(): |
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357 Gpred = G.predecessors |
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358 Gsucc = G.successors |
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359 else: |
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360 Gpred = G.neighbors |
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361 Gsucc = G.neighbors |
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362 |
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363 # predecesssor and successors in search |
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364 pred = {source: None} |
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365 succ = {target: None} |
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366 |
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367 # initialize fringes, start with forward |
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368 forward_fringe = [source] |
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369 reverse_fringe = [target] |
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370 |
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371 level = 0 |
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372 |
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373 while forward_fringe and reverse_fringe: |
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374 # Make sure that we iterate one step forward and one step backwards |
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375 # thus source and target will only trigger "found path" when they are |
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376 # adjacent and then they can be safely included in the container 'exclude' |
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377 level += 1 |
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378 if not level % 2 == 0: |
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379 this_level = forward_fringe |
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380 forward_fringe = [] |
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381 for v in this_level: |
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382 for w in Gsucc(v): |
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383 if w in exclude: |
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384 continue |
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385 if w not in pred: |
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386 forward_fringe.append(w) |
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387 pred[w] = v |
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388 if w in succ: |
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389 return pred, succ, w # found path |
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390 else: |
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391 this_level = reverse_fringe |
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392 reverse_fringe = [] |
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393 for v in this_level: |
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394 for w in Gpred(v): |
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395 if w in exclude: |
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396 continue |
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397 if w not in succ: |
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398 succ[w] = v |
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399 reverse_fringe.append(w) |
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400 if w in pred: |
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401 return pred, succ, w # found path |
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402 |
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403 raise nx.NetworkXNoPath(f"No path between {source} and {target}.") |