annotate env/lib/python3.9/site-packages/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 Hopcroft--Karp 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 # Attribution-Share-Alike License 3.0
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5 # <http://creativecommons.org/licenses/by-sa/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/ABOUT-PADS.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 `Hopcroft--Karp 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. 225--231. <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 Hopcroft--Karp 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/ABOUT-PADS.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 three-tuple:
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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 # three-tuple 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 depth-first 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