Mercurial > repos > shellac > sam_consensus_v3
annotate env/lib/python3.9/sitepackages/networkx/algorithms/bipartite/matching.py @ 0:4f3585e2f14b draft default tip
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author  shellac 

date  Mon, 22 Mar 2021 18:12:50 +0000 
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1 # This module uses material from the Wikipedia article HopcroftKarp algorithm 
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2 # <https://en.wikipedia.org/wiki/Hopcroft%E2%80%93Karp_algorithm>, accessed on 
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3 # January 3, 2015, which is released under the Creative Commons 
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4 # AttributionShareAlike License 3.0 
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5 # <http://creativecommons.org/licenses/bysa/3.0/>. That article includes 
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6 # pseudocode, which has been translated into the corresponding Python code. 
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7 # 
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8 # Portions of this module use code from David Eppstein's Python Algorithms and 
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9 # Data Structures (PADS) library, which is dedicated to the public domain (for 
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10 # proof, see <http://www.ics.uci.edu/~eppstein/PADS/ABOUTPADS.txt>). 
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11 """Provides functions for computing maximum cardinality matchings and minimum 
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12 weight full matchings in a bipartite graph. 
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13 
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14 If you don't care about the particular implementation of the maximum matching 
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15 algorithm, simply use the :func:`maximum_matching`. If you do care, you can 
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16 import one of the named maximum matching algorithms directly. 
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17 
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18 For example, to find a maximum matching in the complete bipartite graph with 
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19 two vertices on the left and three vertices on the right: 
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20 
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21 >>> G = nx.complete_bipartite_graph(2, 3) 
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22 >>> left, right = nx.bipartite.sets(G) 
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23 >>> list(left) 
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24 [0, 1] 
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25 >>> list(right) 
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26 [2, 3, 4] 
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27 >>> nx.bipartite.maximum_matching(G) 
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28 {0: 2, 1: 3, 2: 0, 3: 1} 
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29 
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30 The dictionary returned by :func:`maximum_matching` includes a mapping for 
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31 vertices in both the left and right vertex sets. 
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32 
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33 Similarly, :func:`minimum_weight_full_matching` produces, for a complete 
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34 weighted bipartite graph, a matching whose cardinality is the cardinality of 
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35 the smaller of the two partitions, and for which the sum of the weights of the 
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36 edges included in the matching is minimal. 
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37 
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38 """ 
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39 import collections 
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40 import itertools 
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41 
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42 from networkx.algorithms.bipartite.matrix import biadjacency_matrix 
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43 from networkx.algorithms.bipartite import sets as bipartite_sets 
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44 import networkx as nx 
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45 
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46 __all__ = [ 
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47 "maximum_matching", 
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48 "hopcroft_karp_matching", 
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49 "eppstein_matching", 
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50 "to_vertex_cover", 
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51 "minimum_weight_full_matching", 
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52 ] 
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53 
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54 INFINITY = float("inf") 
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55 
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56 
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57 def hopcroft_karp_matching(G, top_nodes=None): 
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58 """Returns the maximum cardinality matching of the bipartite graph `G`. 
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59 
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60 A matching is a set of edges that do not share any nodes. A maximum 
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61 cardinality matching is a matching with the most edges possible. It 
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62 is not always unique. Finding a matching in a bipartite graph can be 
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63 treated as a networkx flow problem. 
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64 
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65 The functions ``hopcroft_karp_matching`` and ``maximum_matching`` 
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66 are aliases of the same function. 
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67 
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68 Parameters 
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69  
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70 G : NetworkX graph 
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71 
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72 Undirected bipartite graph 
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73 
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74 top_nodes : container of nodes 
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75 
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76 Container with all nodes in one bipartite node set. If not supplied 
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77 it will be computed. But if more than one solution exists an exception 
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78 will be raised. 
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79 
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80 Returns 
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81  
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82 matches : dictionary 
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83 
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84 The matching is returned as a dictionary, `matches`, such that 
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85 ``matches[v] == w`` if node `v` is matched to node `w`. Unmatched 
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86 nodes do not occur as a key in `matches`. 
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87 
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88 Raises 
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89  
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90 AmbiguousSolution 
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91 Raised if the input bipartite graph is disconnected and no container 
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92 with all nodes in one bipartite set is provided. When determining 
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93 the nodes in each bipartite set more than one valid solution is 
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94 possible if the input graph is disconnected. 
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95 
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96 Notes 
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97  
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98 This function is implemented with the `HopcroftKarp matching algorithm 
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99 <https://en.wikipedia.org/wiki/Hopcroft%E2%80%93Karp_algorithm>`_ for 
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100 bipartite graphs. 
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101 
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102 See :mod:`bipartite documentation <networkx.algorithms.bipartite>` 
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103 for further details on how bipartite graphs are handled in NetworkX. 
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104 
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105 See Also 
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106  
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107 maximum_matching 
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108 hopcroft_karp_matching 
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109 eppstein_matching 
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110 
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111 References 
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112  
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113 .. [1] John E. Hopcroft and Richard M. Karp. "An n^{5 / 2} Algorithm for 
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114 Maximum Matchings in Bipartite Graphs" In: **SIAM Journal of Computing** 
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115 2.4 (1973), pp. 225231. <https://doi.org/10.1137/0202019>. 
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116 
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117 """ 
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118 # First we define some auxiliary search functions. 
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119 # 
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120 # If you are a human reading these auxiliary search functions, the "global" 
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121 # variables `leftmatches`, `rightmatches`, `distances`, etc. are defined 
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122 # below the functions, so that they are initialized close to the initial 
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123 # invocation of the search functions. 
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124 def breadth_first_search(): 
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125 for v in left: 
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126 if leftmatches[v] is None: 
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127 distances[v] = 0 
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128 queue.append(v) 
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129 else: 
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130 distances[v] = INFINITY 
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131 distances[None] = INFINITY 
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132 while queue: 
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133 v = queue.popleft() 
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134 if distances[v] < distances[None]: 
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135 for u in G[v]: 
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136 if distances[rightmatches[u]] is INFINITY: 
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137 distances[rightmatches[u]] = distances[v] + 1 
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138 queue.append(rightmatches[u]) 
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139 return distances[None] is not INFINITY 
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140 
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141 def depth_first_search(v): 
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142 if v is not None: 
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143 for u in G[v]: 
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144 if distances[rightmatches[u]] == distances[v] + 1: 
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145 if depth_first_search(rightmatches[u]): 
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146 rightmatches[u] = v 
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147 leftmatches[v] = u 
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148 return True 
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149 distances[v] = INFINITY 
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150 return False 
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151 return True 
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152 
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153 # Initialize the "global" variables that maintain state during the search. 
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154 left, right = bipartite_sets(G, top_nodes) 
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155 leftmatches = {v: None for v in left} 
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156 rightmatches = {v: None for v in right} 
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157 distances = {} 
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158 queue = collections.deque() 
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159 
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160 # Implementation note: this counter is incremented as pairs are matched but 
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161 # it is currently not used elsewhere in the computation. 
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162 num_matched_pairs = 0 
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163 while breadth_first_search(): 
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164 for v in left: 
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165 if leftmatches[v] is None: 
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166 if depth_first_search(v): 
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167 num_matched_pairs += 1 
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168 
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169 # Strip the entries matched to `None`. 
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170 leftmatches = {k: v for k, v in leftmatches.items() if v is not None} 
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171 rightmatches = {k: v for k, v in rightmatches.items() if v is not None} 
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172 
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173 # At this point, the left matches and the right matches are inverses of one 
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174 # another. In other words, 
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175 # 
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176 # leftmatches == {v, k for k, v in rightmatches.items()} 
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177 # 
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178 # Finally, we combine both the left matches and right matches. 
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179 return dict(itertools.chain(leftmatches.items(), rightmatches.items())) 
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180 
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181 
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182 def eppstein_matching(G, top_nodes=None): 
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183 """Returns the maximum cardinality matching of the bipartite graph `G`. 
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184 
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185 Parameters 
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186  
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187 G : NetworkX graph 
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188 
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189 Undirected bipartite graph 
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190 
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191 top_nodes : container 
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192 
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193 Container with all nodes in one bipartite node set. If not supplied 
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194 it will be computed. But if more than one solution exists an exception 
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195 will be raised. 
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196 
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197 Returns 
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198  
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199 matches : dictionary 
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200 
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201 The matching is returned as a dictionary, `matching`, such that 
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202 ``matching[v] == w`` if node `v` is matched to node `w`. Unmatched 
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203 nodes do not occur as a key in `matching`. 
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204 
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205 Raises 
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206  
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207 AmbiguousSolution 
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208 Raised if the input bipartite graph is disconnected and no container 
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209 with all nodes in one bipartite set is provided. When determining 
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210 the nodes in each bipartite set more than one valid solution is 
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211 possible if the input graph is disconnected. 
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212 
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213 Notes 
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214  
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215 This function is implemented with David Eppstein's version of the algorithm 
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216 HopcroftKarp algorithm (see :func:`hopcroft_karp_matching`), which 
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217 originally appeared in the `Python Algorithms and Data Structures library 
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218 (PADS) <http://www.ics.uci.edu/~eppstein/PADS/ABOUTPADS.txt>`_. 
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219 
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220 See :mod:`bipartite documentation <networkx.algorithms.bipartite>` 
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221 for further details on how bipartite graphs are handled in NetworkX. 
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222 
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223 See Also 
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224  
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225 
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226 hopcroft_karp_matching 
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227 
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228 """ 
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229 # Due to its original implementation, a directed graph is needed 
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230 # so that the two sets of bipartite nodes can be distinguished 
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231 left, right = bipartite_sets(G, top_nodes) 
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232 G = nx.DiGraph(G.edges(left)) 
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233 # initialize greedy matching (redundant, but faster than full search) 
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234 matching = {} 
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235 for u in G: 
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236 for v in G[u]: 
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237 if v not in matching: 
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238 matching[v] = u 
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239 break 
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240 while True: 
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241 # structure residual graph into layers 
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242 # pred[u] gives the neighbor in the previous layer for u in U 
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243 # preds[v] gives a list of neighbors in the previous layer for v in V 
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244 # unmatched gives a list of unmatched vertices in final layer of V, 
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245 # and is also used as a flag value for pred[u] when u is in the first 
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246 # layer 
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247 preds = {} 
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248 unmatched = [] 
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249 pred = {u: unmatched for u in G} 
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250 for v in matching: 
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251 del pred[matching[v]] 
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252 layer = list(pred) 
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253 
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254 # repeatedly extend layering structure by another pair of layers 
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255 while layer and not unmatched: 
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256 newLayer = {} 
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257 for u in layer: 
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258 for v in G[u]: 
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259 if v not in preds: 
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260 newLayer.setdefault(v, []).append(u) 
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261 layer = [] 
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262 for v in newLayer: 
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263 preds[v] = newLayer[v] 
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264 if v in matching: 
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265 layer.append(matching[v]) 
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266 pred[matching[v]] = v 
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267 else: 
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268 unmatched.append(v) 
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269 
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270 # did we finish layering without finding any alternating paths? 
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271 if not unmatched: 
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272 unlayered = {} 
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273 for u in G: 
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274 # TODO Why is extra inner loop necessary? 
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275 for v in G[u]: 
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276 if v not in preds: 
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277 unlayered[v] = None 
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278 # TODO Originally, this function returned a threetuple: 
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279 # 
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280 # return (matching, list(pred), list(unlayered)) 
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281 # 
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282 # For some reason, the documentation for this function 
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283 # indicated that the second and third elements of the returned 
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284 # threetuple would be the vertices in the left and right vertex 
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285 # sets, respectively, that are also in the maximum independent set. 
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286 # However, what I think the author meant was that the second 
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287 # element is the list of vertices that were unmatched and the third 
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288 # element was the list of vertices that were matched. Since that 
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289 # seems to be the case, they don't really need to be returned, 
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290 # since that information can be inferred from the matching 
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291 # dictionary. 
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292 
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293 # All the matched nodes must be a key in the dictionary 
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294 for key in matching.copy(): 
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295 matching[matching[key]] = key 
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296 return matching 
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297 
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298 # recursively search backward through layers to find alternating paths 
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299 # recursion returns true if found path, false otherwise 
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300 def recurse(v): 
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301 if v in preds: 
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302 L = preds.pop(v) 
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303 for u in L: 
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304 if u in pred: 
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305 pu = pred.pop(u) 
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306 if pu is unmatched or recurse(pu): 
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307 matching[v] = u 
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308 return True 
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309 return False 
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310 
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311 for v in unmatched: 
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312 recurse(v) 
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313 
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314 
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315 def _is_connected_by_alternating_path(G, v, matched_edges, unmatched_edges, targets): 
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316 """Returns True if and only if the vertex `v` is connected to one of 
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317 the target vertices by an alternating path in `G`. 
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318 
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319 An *alternating path* is a path in which every other edge is in the 
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320 specified maximum matching (and the remaining edges in the path are not in 
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321 the matching). An alternating path may have matched edges in the even 
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322 positions or in the odd positions, as long as the edges alternate between 
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323 'matched' and 'unmatched'. 
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324 
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325 `G` is an undirected bipartite NetworkX graph. 
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326 
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327 `v` is a vertex in `G`. 
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328 
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329 `matched_edges` is a set of edges present in a maximum matching in `G`. 
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330 
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331 `unmatched_edges` is a set of edges not present in a maximum 
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332 matching in `G`. 
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333 
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334 `targets` is a set of vertices. 
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335 
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336 """ 
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337 
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338 def _alternating_dfs(u, along_matched=True): 
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339 """Returns True if and only if `u` is connected to one of the 
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340 targets by an alternating path. 
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341 
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342 `u` is a vertex in the graph `G`. 
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343 
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344 If `along_matched` is True, this step of the depthfirst search 
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345 will continue only through edges in the given matching. Otherwise, it 
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346 will continue only through edges *not* in the given matching. 
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347 
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348 """ 
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349 if along_matched: 
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350 edges = itertools.cycle([matched_edges, unmatched_edges]) 
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351 else: 
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352 edges = itertools.cycle([unmatched_edges, matched_edges]) 
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353 visited = set() 
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354 stack = [(u, iter(G[u]), next(edges))] 
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355 while stack: 
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356 parent, children, valid_edges = stack[1] 
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357 try: 
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358 child = next(children) 
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359 if child not in visited: 
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360 if (parent, child) in valid_edges or (child, parent) in valid_edges: 
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361 if child in targets: 
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362 return True 
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363 visited.add(child) 
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364 stack.append((child, iter(G[child]), next(edges))) 
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365 except StopIteration: 
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366 stack.pop() 
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367 return False 
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368 
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369 # Check for alternating paths starting with edges in the matching, then 
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370 # check for alternating paths starting with edges not in the 
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371 # matching. 
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372 return _alternating_dfs(v, along_matched=True) or _alternating_dfs( 
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373 v, along_matched=False 
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374 ) 
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375 
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376 
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377 def _connected_by_alternating_paths(G, matching, targets): 
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378 """Returns the set of vertices that are connected to one of the target 
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379 vertices by an alternating path in `G` or are themselves a target. 
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380 
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381 An *alternating path* is a path in which every other edge is in the 
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382 specified maximum matching (and the remaining edges in the path are not in 
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383 the matching). An alternating path may have matched edges in the even 
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384 positions or in the odd positions, as long as the edges alternate between 
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385 'matched' and 'unmatched'. 
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386 
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387 `G` is an undirected bipartite NetworkX graph. 
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388 
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389 `matching` is a dictionary representing a maximum matching in `G`, as 
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390 returned by, for example, :func:`maximum_matching`. 
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391 
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392 `targets` is a set of vertices. 
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393 
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394 """ 
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395 # Get the set of matched edges and the set of unmatched edges. Only include 
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396 # one version of each undirected edge (for example, include edge (1, 2) but 
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397 # not edge (2, 1)). Using frozensets as an intermediary step we do not 
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398 # require nodes to be orderable. 
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399 edge_sets = {frozenset((u, v)) for u, v in matching.items()} 
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400 matched_edges = {tuple(edge) for edge in edge_sets} 
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401 unmatched_edges = { 
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402 (u, v) for (u, v) in G.edges() if frozenset((u, v)) not in edge_sets 
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403 } 
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404 
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405 return { 
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406 v 
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407 for v in G 
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408 if v in targets 
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409 or _is_connected_by_alternating_path( 
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410 G, v, matched_edges, unmatched_edges, targets 
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411 ) 
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412 } 
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413 
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414 
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415 def to_vertex_cover(G, matching, top_nodes=None): 
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416 """Returns the minimum vertex cover corresponding to the given maximum 
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417 matching of the bipartite graph `G`. 
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418 
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419 Parameters 
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420  
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421 G : NetworkX graph 
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422 
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423 Undirected bipartite graph 
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424 
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425 matching : dictionary 
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426 
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427 A dictionary whose keys are vertices in `G` and whose values are the 
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428 distinct neighbors comprising the maximum matching for `G`, as returned 
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429 by, for example, :func:`maximum_matching`. The dictionary *must* 
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430 represent the maximum matching. 
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431 
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432 top_nodes : container 
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433 
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434 Container with all nodes in one bipartite node set. If not supplied 
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435 it will be computed. But if more than one solution exists an exception 
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436 will be raised. 
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437 
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438 Returns 
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439  
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440 vertex_cover : :class:`set` 
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441 
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442 The minimum vertex cover in `G`. 
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443 
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444 Raises 
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445  
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446 AmbiguousSolution 
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447 Raised if the input bipartite graph is disconnected and no container 
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448 with all nodes in one bipartite set is provided. When determining 
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449 the nodes in each bipartite set more than one valid solution is 
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450 possible if the input graph is disconnected. 
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451 
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452 Notes 
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453  
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454 This function is implemented using the procedure guaranteed by `Konig's 
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455 theorem 
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456 <https://en.wikipedia.org/wiki/K%C3%B6nig%27s_theorem_%28graph_theory%29>`_, 
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457 which proves an equivalence between a maximum matching and a minimum vertex 
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458 cover in bipartite graphs. 
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459 
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460 Since a minimum vertex cover is the complement of a maximum independent set 
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461 for any graph, one can compute the maximum independent set of a bipartite 
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462 graph this way: 
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463 
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464 >>> G = nx.complete_bipartite_graph(2, 3) 
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465 >>> matching = nx.bipartite.maximum_matching(G) 
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466 >>> vertex_cover = nx.bipartite.to_vertex_cover(G, matching) 
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467 >>> independent_set = set(G)  vertex_cover 
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468 >>> print(list(independent_set)) 
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469 [2, 3, 4] 
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470 
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471 See :mod:`bipartite documentation <networkx.algorithms.bipartite>` 
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472 for further details on how bipartite graphs are handled in NetworkX. 
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473 
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474 """ 
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475 # This is a Python implementation of the algorithm described at 
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476 # <https://en.wikipedia.org/wiki/K%C3%B6nig%27s_theorem_%28graph_theory%29#Proof>. 
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477 L, R = bipartite_sets(G, top_nodes) 
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478 # Let U be the set of unmatched vertices in the left vertex set. 
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479 unmatched_vertices = set(G)  set(matching) 
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480 U = unmatched_vertices & L 
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481 # Let Z be the set of vertices that are either in U or are connected to U 
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482 # by alternating paths. 
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483 Z = _connected_by_alternating_paths(G, matching, U) 
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484 # At this point, every edge either has a right endpoint in Z or a left 
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485 # endpoint not in Z. This gives us the vertex cover. 
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486 return (L  Z)  (R & Z) 
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487 
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488 
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489 #: Returns the maximum cardinality matching in the given bipartite graph. 
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490 #: 
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491 #: This function is simply an alias for :func:`hopcroft_karp_matching`. 
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492 maximum_matching = hopcroft_karp_matching 
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493 
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494 
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495 def minimum_weight_full_matching(G, top_nodes=None, weight="weight"): 
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496 r"""Returns a minimum weight full matching of the bipartite graph `G`. 
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497 
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498 Let :math:`G = ((U, V), E)` be a weighted bipartite graph with real weights 
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499 :math:`w : E \to \mathbb{R}`. This function then produces a matching 
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500 :math:`M \subseteq E` with cardinality 
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501 
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502 .. math:: 
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503 \lvert M \rvert = \min(\lvert U \rvert, \lvert V \rvert), 
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504 
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505 which minimizes the sum of the weights of the edges included in the 
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506 matching, :math:`\sum_{e \in M} w(e)`, or raises an error if no such 
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507 matching exists. 
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508 
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509 When :math:`\lvert U \rvert = \lvert V \rvert`, this is commonly 
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510 referred to as a perfect matching; here, since we allow 
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511 :math:`\lvert U \rvert` and :math:`\lvert V \rvert` to differ, we 
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512 follow Karp [1]_ and refer to the matching as *full*. 
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513 
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514 Parameters 
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515  
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516 G : NetworkX graph 
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517 
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518 Undirected bipartite graph 
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519 
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520 top_nodes : container 
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521 
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522 Container with all nodes in one bipartite node set. If not supplied 
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523 it will be computed. 
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524 
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525 weight : string, optional (default='weight') 
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526 
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527 The edge data key used to provide each value in the matrix. 
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528 
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529 Returns 
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530  
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531 matches : dictionary 
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532 
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533 The matching is returned as a dictionary, `matches`, such that 
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534 ``matches[v] == w`` if node `v` is matched to node `w`. Unmatched 
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535 nodes do not occur as a key in `matches`. 
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536 
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537 Raises 
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538  
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539 ValueError 
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540 Raised if no full matching exists. 
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541 
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542 ImportError 
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543 Raised if SciPy is not available. 
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544 
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545 Notes 
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546  
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547 The problem of determining a minimum weight full matching is also known as 
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548 the rectangular linear assignment problem. This implementation defers the 
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549 calculation of the assignment to SciPy. 
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550 
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551 References 
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552  
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553 .. [1] Richard Manning Karp: 
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554 An algorithm to Solve the m x n Assignment Problem in Expected Time 
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555 O(mn log n). 
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556 Networks, 10(2):143–152, 1980. 
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557 
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558 """ 
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559 try: 
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560 import numpy as np 
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561 import scipy.optimize 
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562 except ImportError as e: 
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563 raise ImportError( 
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564 "minimum_weight_full_matching requires SciPy: " + "https://scipy.org/" 
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565 ) from e 
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566 left, right = nx.bipartite.sets(G, top_nodes) 
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567 U = list(left) 
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568 V = list(right) 
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569 # We explicitly create the biadjancency matrix having infinities 
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570 # where edges are missing (as opposed to zeros, which is what one would 
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571 # get by using toarray on the sparse matrix). 
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572 weights_sparse = biadjacency_matrix( 
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573 G, row_order=U, column_order=V, weight=weight, format="coo" 
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574 ) 
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575 weights = np.full(weights_sparse.shape, np.inf) 
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576 weights[weights_sparse.row, weights_sparse.col] = weights_sparse.data 
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577 left_matches = scipy.optimize.linear_sum_assignment(weights) 
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578 d = {U[u]: V[v] for u, v in zip(*left_matches)} 
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579 # d will contain the matching from edges in left to right; we need to 
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580 # add the ones from right to left as well. 
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581 d.update({v: u for u, v in d.items()}) 
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582 return d 