Mercurial > repos > shellac > sam_consensus_v3
annotate env/lib/python3.9/site-packages/networkx/algorithms/bipartite/matching.py @ 0:4f3585e2f14b draft default tip
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author | shellac |
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date | Mon, 22 Mar 2021 18:12:50 +0000 |
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1 # 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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"planemo upload commit 60cee0fc7c0cda8592644e1aad72851dec82c959"
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568 V = list(right) |
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"planemo upload commit 60cee0fc7c0cda8592644e1aad72851dec82c959"
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569 # We explicitly create the biadjancency matrix having infinities |
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"planemo upload commit 60cee0fc7c0cda8592644e1aad72851dec82c959"
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570 # where edges are missing (as opposed to zeros, which is what one would |
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"planemo upload commit 60cee0fc7c0cda8592644e1aad72851dec82c959"
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571 # get by using toarray on the sparse matrix). |
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"planemo upload commit 60cee0fc7c0cda8592644e1aad72851dec82c959"
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572 weights_sparse = biadjacency_matrix( |
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"planemo upload commit 60cee0fc7c0cda8592644e1aad72851dec82c959"
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573 G, row_order=U, column_order=V, weight=weight, format="coo" |
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"planemo upload commit 60cee0fc7c0cda8592644e1aad72851dec82c959"
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574 ) |
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"planemo upload commit 60cee0fc7c0cda8592644e1aad72851dec82c959"
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575 weights = np.full(weights_sparse.shape, np.inf) |
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"planemo upload commit 60cee0fc7c0cda8592644e1aad72851dec82c959"
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576 weights[weights_sparse.row, weights_sparse.col] = weights_sparse.data |
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"planemo upload commit 60cee0fc7c0cda8592644e1aad72851dec82c959"
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577 left_matches = scipy.optimize.linear_sum_assignment(weights) |
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"planemo upload commit 60cee0fc7c0cda8592644e1aad72851dec82c959"
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578 d = {U[u]: V[v] for u, v in zip(*left_matches)} |
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"planemo upload commit 60cee0fc7c0cda8592644e1aad72851dec82c959"
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579 # d will contain the matching from edges in left to right; we need to |
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"planemo upload commit 60cee0fc7c0cda8592644e1aad72851dec82c959"
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580 # add the ones from right to left as well. |
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"planemo upload commit 60cee0fc7c0cda8592644e1aad72851dec82c959"
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581 d.update({v: u for u, v in d.items()}) |
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"planemo upload commit 60cee0fc7c0cda8592644e1aad72851dec82c959"
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582 return d |