Mercurial > repos > shellac > sam_consensus_v3
annotate env/lib/python3.9/site-packages/networkx/algorithms/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 """Functions for computing and verifying matchings in a graph.""" |
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2 from collections import Counter |
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3 from itertools import combinations |
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4 from itertools import repeat |
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5 |
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6 __all__ = [ |
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7 "is_matching", |
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8 "is_maximal_matching", |
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9 "is_perfect_matching", |
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10 "max_weight_matching", |
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11 "maximal_matching", |
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12 ] |
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13 |
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14 |
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15 def maximal_matching(G): |
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16 r"""Find a maximal matching in the graph. |
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17 |
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18 A matching is a subset of edges in which no node occurs more than once. |
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19 A maximal matching cannot add more edges and still be a matching. |
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20 |
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21 Parameters |
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22 ---------- |
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23 G : NetworkX graph |
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24 Undirected graph |
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25 |
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26 Returns |
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27 ------- |
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28 matching : set |
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29 A maximal matching of the graph. |
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30 |
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31 Notes |
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32 ----- |
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33 The algorithm greedily selects a maximal matching M of the graph G |
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34 (i.e. no superset of M exists). It runs in $O(|E|)$ time. |
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35 """ |
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36 matching = set() |
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37 nodes = set() |
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38 for u, v in G.edges(): |
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39 # If the edge isn't covered, add it to the matching |
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40 # then remove neighborhood of u and v from consideration. |
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41 if u not in nodes and v not in nodes and u != v: |
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42 matching.add((u, v)) |
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43 nodes.add(u) |
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44 nodes.add(v) |
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45 return matching |
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46 |
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47 |
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48 def matching_dict_to_set(matching): |
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49 """Converts a dictionary representing a matching (as returned by |
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50 :func:`max_weight_matching`) to a set representing a matching (as |
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51 returned by :func:`maximal_matching`). |
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52 |
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53 In the definition of maximal matching adopted by NetworkX, |
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54 self-loops are not allowed, so the provided dictionary is expected |
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55 to never have any mapping from a key to itself. However, the |
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56 dictionary is expected to have mirrored key/value pairs, for |
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57 example, key ``u`` with value ``v`` and key ``v`` with value ``u``. |
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58 |
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59 """ |
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60 # Need to compensate for the fact that both pairs (u, v) and (v, u) |
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61 # appear in `matching.items()`, so we use a set of sets. This way, |
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62 # only the (frozen)set `{u, v}` appears as an element in the |
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63 # returned set. |
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64 |
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65 return {(u, v) for (u, v) in set(map(frozenset, matching.items()))} |
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66 |
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67 |
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68 def is_matching(G, matching): |
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69 """Decides whether the given set or dictionary represents a valid |
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70 matching in ``G``. |
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71 |
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72 A *matching* in a graph is a set of edges in which no two distinct |
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73 edges share a common endpoint. |
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74 |
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75 Parameters |
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76 ---------- |
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77 G : NetworkX graph |
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78 |
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79 matching : dict or set |
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80 A dictionary or set representing a matching. If a dictionary, it |
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81 must have ``matching[u] == v`` and ``matching[v] == u`` for each |
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82 edge ``(u, v)`` in the matching. If a set, it must have elements |
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83 of the form ``(u, v)``, where ``(u, v)`` is an edge in the |
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84 matching. |
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85 |
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86 Returns |
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87 ------- |
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88 bool |
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89 Whether the given set or dictionary represents a valid matching |
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90 in the graph. |
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91 |
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92 """ |
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93 if isinstance(matching, dict): |
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94 matching = matching_dict_to_set(matching) |
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95 # TODO This is parallelizable. |
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96 return all(len(set(e1) & set(e2)) == 0 for e1, e2 in combinations(matching, 2)) |
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97 |
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98 |
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99 def is_maximal_matching(G, matching): |
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100 """Decides whether the given set or dictionary represents a valid |
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101 maximal matching in ``G``. |
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102 |
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103 A *maximal matching* in a graph is a matching in which adding any |
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104 edge would cause the set to no longer be a valid matching. |
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105 |
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106 Parameters |
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107 ---------- |
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108 G : NetworkX graph |
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109 |
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110 matching : dict or set |
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111 A dictionary or set representing a matching. If a dictionary, it |
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112 must have ``matching[u] == v`` and ``matching[v] == u`` for each |
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113 edge ``(u, v)`` in the matching. If a set, it must have elements |
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114 of the form ``(u, v)``, where ``(u, v)`` is an edge in the |
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115 matching. |
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116 |
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117 Returns |
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118 ------- |
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119 bool |
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120 Whether the given set or dictionary represents a valid maximal |
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121 matching in the graph. |
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122 |
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123 """ |
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124 if isinstance(matching, dict): |
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125 matching = matching_dict_to_set(matching) |
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126 # If the given set is not a matching, then it is not a maximal matching. |
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127 if not is_matching(G, matching): |
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128 return False |
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129 # A matching is maximal if adding any unmatched edge to it causes |
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130 # the resulting set to *not* be a valid matching. |
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131 # |
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132 # HACK Since the `matching_dict_to_set` function returns a set of |
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133 # sets, we need to convert the list of edges to a set of sets in |
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134 # order for the set difference function to work. Ideally, we would |
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135 # just be able to do `set(G.edges()) - matching`. |
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136 all_edges = set(map(frozenset, G.edges())) |
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137 matched_edges = set(map(frozenset, matching)) |
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138 unmatched_edges = all_edges - matched_edges |
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139 # TODO This is parallelizable. |
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140 return all(not is_matching(G, matching | {e}) for e in unmatched_edges) |
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141 |
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142 |
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143 def is_perfect_matching(G, matching): |
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144 """Decides whether the given set represents a valid perfect matching in |
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145 ``G``. |
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146 |
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147 A *perfect matching* in a graph is a matching in which exactly one edge |
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148 is incident upon each vertex. |
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149 |
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150 Parameters |
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151 ---------- |
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152 G : NetworkX graph |
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153 |
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154 matching : dict or set |
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155 A dictionary or set representing a matching. If a dictionary, it |
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156 must have ``matching[u] == v`` and ``matching[v] == u`` for each |
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157 edge ``(u, v)`` in the matching. If a set, it must have elements |
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158 of the form ``(u, v)``, where ``(u, v)`` is an edge in the |
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159 matching. |
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160 |
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161 Returns |
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162 ------- |
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163 bool |
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164 Whether the given set or dictionary represents a valid perfect |
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165 matching in the graph. |
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166 |
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167 """ |
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168 if isinstance(matching, dict): |
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169 matching = matching_dict_to_set(matching) |
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170 |
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171 if not is_matching(G, matching): |
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172 return False |
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173 |
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174 counts = Counter(sum(matching, ())) |
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175 |
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176 return all(counts[v] == 1 for v in G) |
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177 |
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178 |
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179 def max_weight_matching(G, maxcardinality=False, weight="weight"): |
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180 """Compute a maximum-weighted matching of G. |
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181 |
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182 A matching is a subset of edges in which no node occurs more than once. |
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183 The weight of a matching is the sum of the weights of its edges. |
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184 A maximal matching cannot add more edges and still be a matching. |
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185 The cardinality of a matching is the number of matched edges. |
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186 |
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187 Parameters |
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188 ---------- |
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189 G : NetworkX graph |
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190 Undirected graph |
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191 |
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192 maxcardinality: bool, optional (default=False) |
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193 If maxcardinality is True, compute the maximum-cardinality matching |
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194 with maximum weight among all maximum-cardinality matchings. |
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195 |
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196 weight: string, optional (default='weight') |
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197 Edge data key corresponding to the edge weight. |
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198 If key not found, uses 1 as weight. |
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199 |
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200 |
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201 Returns |
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202 ------- |
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203 matching : set |
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204 A maximal matching of the graph. |
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205 |
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206 Notes |
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207 ----- |
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208 If G has edges with weight attributes the edge data are used as |
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209 weight values else the weights are assumed to be 1. |
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210 |
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211 This function takes time O(number_of_nodes ** 3). |
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212 |
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213 If all edge weights are integers, the algorithm uses only integer |
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214 computations. If floating point weights are used, the algorithm |
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215 could return a slightly suboptimal matching due to numeric |
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216 precision errors. |
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217 |
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218 This method is based on the "blossom" method for finding augmenting |
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219 paths and the "primal-dual" method for finding a matching of maximum |
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220 weight, both methods invented by Jack Edmonds [1]_. |
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221 |
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222 Bipartite graphs can also be matched using the functions present in |
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223 :mod:`networkx.algorithms.bipartite.matching`. |
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224 |
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225 References |
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226 ---------- |
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227 .. [1] "Efficient Algorithms for Finding Maximum Matching in Graphs", |
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228 Zvi Galil, ACM Computing Surveys, 1986. |
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229 """ |
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230 # |
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231 # The algorithm is taken from "Efficient Algorithms for Finding Maximum |
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232 # Matching in Graphs" by Zvi Galil, ACM Computing Surveys, 1986. |
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233 # It is based on the "blossom" method for finding augmenting paths and |
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234 # the "primal-dual" method for finding a matching of maximum weight, both |
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235 # methods invented by Jack Edmonds. |
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236 # |
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237 # A C program for maximum weight matching by Ed Rothberg was used |
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238 # extensively to validate this new code. |
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239 # |
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240 # Many terms used in the code comments are explained in the paper |
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241 # by Galil. You will probably need the paper to make sense of this code. |
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242 # |
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243 |
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244 class NoNode: |
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245 """Dummy value which is different from any node.""" |
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246 |
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247 pass |
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248 |
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249 class Blossom: |
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250 """Representation of a non-trivial blossom or sub-blossom.""" |
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251 |
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252 __slots__ = ["childs", "edges", "mybestedges"] |
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253 |
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254 # b.childs is an ordered list of b's sub-blossoms, starting with |
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255 # the base and going round the blossom. |
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256 |
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257 # b.edges is the list of b's connecting edges, such that |
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258 # b.edges[i] = (v, w) where v is a vertex in b.childs[i] |
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259 # and w is a vertex in b.childs[wrap(i+1)]. |
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260 |
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261 # If b is a top-level S-blossom, |
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262 # b.mybestedges is a list of least-slack edges to neighbouring |
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263 # S-blossoms, or None if no such list has been computed yet. |
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264 # This is used for efficient computation of delta3. |
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265 |
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266 # Generate the blossom's leaf vertices. |
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267 def leaves(self): |
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268 for t in self.childs: |
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269 if isinstance(t, Blossom): |
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270 yield from t.leaves() |
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271 else: |
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272 yield t |
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273 |
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274 # Get a list of vertices. |
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275 gnodes = list(G) |
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276 if not gnodes: |
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277 return set() # don't bother with empty graphs |
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278 |
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279 # Find the maximum edge weight. |
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280 maxweight = 0 |
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281 allinteger = True |
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282 for i, j, d in G.edges(data=True): |
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283 wt = d.get(weight, 1) |
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284 if i != j and wt > maxweight: |
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285 maxweight = wt |
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286 allinteger = allinteger and (str(type(wt)).split("'")[1] in ("int", "long")) |
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287 |
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288 # If v is a matched vertex, mate[v] is its partner vertex. |
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289 # If v is a single vertex, v does not occur as a key in mate. |
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290 # Initially all vertices are single; updated during augmentation. |
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291 mate = {} |
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292 |
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293 # If b is a top-level blossom, |
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294 # label.get(b) is None if b is unlabeled (free), |
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295 # 1 if b is an S-blossom, |
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296 # 2 if b is a T-blossom. |
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297 # The label of a vertex is found by looking at the label of its top-level |
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298 # containing blossom. |
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299 # If v is a vertex inside a T-blossom, label[v] is 2 iff v is reachable |
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300 # from an S-vertex outside the blossom. |
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301 # Labels are assigned during a stage and reset after each augmentation. |
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302 label = {} |
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303 |
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304 # If b is a labeled top-level blossom, |
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305 # labeledge[b] = (v, w) is the edge through which b obtained its label |
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306 # such that w is a vertex in b, or None if b's base vertex is single. |
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307 # If w is a vertex inside a T-blossom and label[w] == 2, |
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308 # labeledge[w] = (v, w) is an edge through which w is reachable from |
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309 # outside the blossom. |
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310 labeledge = {} |
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311 |
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312 # If v is a vertex, inblossom[v] is the top-level blossom to which v |
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313 # belongs. |
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314 # If v is a top-level vertex, inblossom[v] == v since v is itself |
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315 # a (trivial) top-level blossom. |
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316 # Initially all vertices are top-level trivial blossoms. |
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317 inblossom = dict(zip(gnodes, gnodes)) |
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318 |
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319 # If b is a sub-blossom, |
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320 # blossomparent[b] is its immediate parent (sub-)blossom. |
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321 # If b is a top-level blossom, blossomparent[b] is None. |
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322 blossomparent = dict(zip(gnodes, repeat(None))) |
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323 |
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324 # If b is a (sub-)blossom, |
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325 # blossombase[b] is its base VERTEX (i.e. recursive sub-blossom). |
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326 blossombase = dict(zip(gnodes, gnodes)) |
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327 |
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328 # If w is a free vertex (or an unreached vertex inside a T-blossom), |
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329 # bestedge[w] = (v, w) is the least-slack edge from an S-vertex, |
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330 # or None if there is no such edge. |
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331 # If b is a (possibly trivial) top-level S-blossom, |
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332 # bestedge[b] = (v, w) is the least-slack edge to a different S-blossom |
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333 # (v inside b), or None if there is no such edge. |
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334 # This is used for efficient computation of delta2 and delta3. |
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335 bestedge = {} |
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336 |
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337 # If v is a vertex, |
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338 # dualvar[v] = 2 * u(v) where u(v) is the v's variable in the dual |
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339 # optimization problem (if all edge weights are integers, multiplication |
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340 # by two ensures that all values remain integers throughout the algorithm). |
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341 # Initially, u(v) = maxweight / 2. |
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342 dualvar = dict(zip(gnodes, repeat(maxweight))) |
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343 |
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344 # If b is a non-trivial blossom, |
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345 # blossomdual[b] = z(b) where z(b) is b's variable in the dual |
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346 # optimization problem. |
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347 blossomdual = {} |
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348 |
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349 # If (v, w) in allowedge or (w, v) in allowedg, then the edge |
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350 # (v, w) is known to have zero slack in the optimization problem; |
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351 # otherwise the edge may or may not have zero slack. |
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352 allowedge = {} |
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353 |
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354 # Queue of newly discovered S-vertices. |
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355 queue = [] |
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356 |
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357 # Return 2 * slack of edge (v, w) (does not work inside blossoms). |
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358 def slack(v, w): |
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359 return dualvar[v] + dualvar[w] - 2 * G[v][w].get(weight, 1) |
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360 |
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361 # Assign label t to the top-level blossom containing vertex w, |
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362 # coming through an edge from vertex v. |
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363 def assignLabel(w, t, v): |
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364 b = inblossom[w] |
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365 assert label.get(w) is None and label.get(b) is None |
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366 label[w] = label[b] = t |
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367 if v is not None: |
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368 labeledge[w] = labeledge[b] = (v, w) |
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369 else: |
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370 labeledge[w] = labeledge[b] = None |
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371 bestedge[w] = bestedge[b] = None |
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372 if t == 1: |
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373 # b became an S-vertex/blossom; add it(s vertices) to the queue. |
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374 if isinstance(b, Blossom): |
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375 queue.extend(b.leaves()) |
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376 else: |
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377 queue.append(b) |
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378 elif t == 2: |
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379 # b became a T-vertex/blossom; assign label S to its mate. |
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380 # (If b is a non-trivial blossom, its base is the only vertex |
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381 # with an external mate.) |
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382 base = blossombase[b] |
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383 assignLabel(mate[base], 1, base) |
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384 |
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385 # Trace back from vertices v and w to discover either a new blossom |
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386 # or an augmenting path. Return the base vertex of the new blossom, |
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387 # or NoNode if an augmenting path was found. |
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388 def scanBlossom(v, w): |
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389 # Trace back from v and w, placing breadcrumbs as we go. |
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390 path = [] |
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391 base = NoNode |
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392 while v is not NoNode: |
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393 # Look for a breadcrumb in v's blossom or put a new breadcrumb. |
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394 b = inblossom[v] |
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395 if label[b] & 4: |
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396 base = blossombase[b] |
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397 break |
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398 assert label[b] == 1 |
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399 path.append(b) |
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400 label[b] = 5 |
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401 # Trace one step back. |
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402 if labeledge[b] is None: |
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403 # The base of blossom b is single; stop tracing this path. |
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404 assert blossombase[b] not in mate |
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405 v = NoNode |
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406 else: |
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407 assert labeledge[b][0] == mate[blossombase[b]] |
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408 v = labeledge[b][0] |
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409 b = inblossom[v] |
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410 assert label[b] == 2 |
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411 # b is a T-blossom; trace one more step back. |
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412 v = labeledge[b][0] |
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413 # Swap v and w so that we alternate between both paths. |
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414 if w is not NoNode: |
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415 v, w = w, v |
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416 # Remove breadcrumbs. |
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417 for b in path: |
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418 label[b] = 1 |
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419 # Return base vertex, if we found one. |
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420 return base |
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421 |
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422 # Construct a new blossom with given base, through S-vertices v and w. |
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423 # Label the new blossom as S; set its dual variable to zero; |
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424 # relabel its T-vertices to S and add them to the queue. |
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425 def addBlossom(base, v, w): |
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426 bb = inblossom[base] |
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427 bv = inblossom[v] |
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428 bw = inblossom[w] |
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429 # Create blossom. |
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430 b = Blossom() |
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431 blossombase[b] = base |
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432 blossomparent[b] = None |
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433 blossomparent[bb] = b |
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434 # Make list of sub-blossoms and their interconnecting edge endpoints. |
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435 b.childs = path = [] |
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436 b.edges = edgs = [(v, w)] |
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437 # Trace back from v to base. |
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438 while bv != bb: |
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439 # Add bv to the new blossom. |
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440 blossomparent[bv] = b |
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441 path.append(bv) |
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442 edgs.append(labeledge[bv]) |
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443 assert label[bv] == 2 or ( |
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444 label[bv] == 1 and labeledge[bv][0] == mate[blossombase[bv]] |
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445 ) |
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446 # Trace one step back. |
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447 v = labeledge[bv][0] |
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448 bv = inblossom[v] |
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449 # Add base sub-blossom; reverse lists. |
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450 path.append(bb) |
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451 path.reverse() |
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452 edgs.reverse() |
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453 # Trace back from w to base. |
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454 while bw != bb: |
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455 # Add bw to the new blossom. |
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456 blossomparent[bw] = b |
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457 path.append(bw) |
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458 edgs.append((labeledge[bw][1], labeledge[bw][0])) |
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459 assert label[bw] == 2 or ( |
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460 label[bw] == 1 and labeledge[bw][0] == mate[blossombase[bw]] |
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461 ) |
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462 # Trace one step back. |
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463 w = labeledge[bw][0] |
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464 bw = inblossom[w] |
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465 # Set label to S. |
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466 assert label[bb] == 1 |
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467 label[b] = 1 |
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468 labeledge[b] = labeledge[bb] |
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469 # Set dual variable to zero. |
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470 blossomdual[b] = 0 |
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471 # Relabel vertices. |
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472 for v in b.leaves(): |
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473 if label[inblossom[v]] == 2: |
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474 # This T-vertex now turns into an S-vertex because it becomes |
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475 # part of an S-blossom; add it to the queue. |
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476 queue.append(v) |
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477 inblossom[v] = b |
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478 # Compute b.mybestedges. |
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479 bestedgeto = {} |
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480 for bv in path: |
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481 if isinstance(bv, Blossom): |
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482 if bv.mybestedges is not None: |
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483 # Walk this subblossom's least-slack edges. |
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484 nblist = bv.mybestedges |
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485 # The sub-blossom won't need this data again. |
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486 bv.mybestedges = None |
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487 else: |
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488 # This subblossom does not have a list of least-slack |
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489 # edges; get the information from the vertices. |
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490 nblist = [ |
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491 (v, w) for v in bv.leaves() for w in G.neighbors(v) if v != w |
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492 ] |
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493 else: |
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494 nblist = [(bv, w) for w in G.neighbors(bv) if bv != w] |
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495 for k in nblist: |
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496 (i, j) = k |
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497 if inblossom[j] == b: |
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498 i, j = j, i |
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499 bj = inblossom[j] |
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500 if ( |
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501 bj != b |
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502 and label.get(bj) == 1 |
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503 and ((bj not in bestedgeto) or slack(i, j) < slack(*bestedgeto[bj])) |
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504 ): |
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505 bestedgeto[bj] = k |
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506 # Forget about least-slack edge of the subblossom. |
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507 bestedge[bv] = None |
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508 b.mybestedges = list(bestedgeto.values()) |
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509 # Select bestedge[b]. |
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510 mybestedge = None |
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511 bestedge[b] = None |
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512 for k in b.mybestedges: |
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513 kslack = slack(*k) |
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514 if mybestedge is None or kslack < mybestslack: |
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515 mybestedge = k |
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516 mybestslack = kslack |
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517 bestedge[b] = mybestedge |
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518 |
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519 # Expand the given top-level blossom. |
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520 def expandBlossom(b, endstage): |
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521 # Convert sub-blossoms into top-level blossoms. |
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522 for s in b.childs: |
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523 blossomparent[s] = None |
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524 if isinstance(s, Blossom): |
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525 if endstage and blossomdual[s] == 0: |
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526 # Recursively expand this sub-blossom. |
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527 expandBlossom(s, endstage) |
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528 else: |
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529 for v in s.leaves(): |
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530 inblossom[v] = s |
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531 else: |
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532 inblossom[s] = s |
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533 # If we expand a T-blossom during a stage, its sub-blossoms must be |
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534 # relabeled. |
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535 if (not endstage) and label.get(b) == 2: |
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536 # Start at the sub-blossom through which the expanding |
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537 # blossom obtained its label, and relabel sub-blossoms untili |
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538 # we reach the base. |
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539 # Figure out through which sub-blossom the expanding blossom |
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540 # obtained its label initially. |
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541 entrychild = inblossom[labeledge[b][1]] |
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542 # Decide in which direction we will go round the blossom. |
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543 j = b.childs.index(entrychild) |
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544 if j & 1: |
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545 # Start index is odd; go forward and wrap. |
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546 j -= len(b.childs) |
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547 jstep = 1 |
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548 else: |
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549 # Start index is even; go backward. |
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550 jstep = -1 |
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551 # Move along the blossom until we get to the base. |
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552 v, w = labeledge[b] |
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553 while j != 0: |
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554 # Relabel the T-sub-blossom. |
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555 if jstep == 1: |
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556 p, q = b.edges[j] |
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557 else: |
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558 q, p = b.edges[j - 1] |
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559 label[w] = None |
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560 label[q] = None |
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561 assignLabel(w, 2, v) |
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562 # Step to the next S-sub-blossom and note its forward edge. |
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563 allowedge[(p, q)] = allowedge[(q, p)] = True |
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564 j += jstep |
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565 if jstep == 1: |
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566 v, w = b.edges[j] |
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567 else: |
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568 w, v = b.edges[j - 1] |
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569 # Step to the next T-sub-blossom. |
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570 allowedge[(v, w)] = allowedge[(w, v)] = True |
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571 j += jstep |
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572 # Relabel the base T-sub-blossom WITHOUT stepping through to |
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573 # its mate (so don't call assignLabel). |
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574 bw = b.childs[j] |
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575 label[w] = label[bw] = 2 |
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576 labeledge[w] = labeledge[bw] = (v, w) |
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577 bestedge[bw] = None |
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578 # Continue along the blossom until we get back to entrychild. |
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579 j += jstep |
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580 while b.childs[j] != entrychild: |
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581 # Examine the vertices of the sub-blossom to see whether |
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582 # it is reachable from a neighbouring S-vertex outside the |
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583 # expanding blossom. |
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584 bv = b.childs[j] |
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585 if label.get(bv) == 1: |
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586 # This sub-blossom just got label S through one of its |
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587 # neighbours; leave it be. |
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588 j += jstep |
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589 continue |
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590 if isinstance(bv, Blossom): |
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591 for v in bv.leaves(): |
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592 if label.get(v): |
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593 break |
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594 else: |
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595 v = bv |
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596 # If the sub-blossom contains a reachable vertex, assign |
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597 # label T to the sub-blossom. |
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598 if label.get(v): |
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599 assert label[v] == 2 |
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600 assert inblossom[v] == bv |
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601 label[v] = None |
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602 label[mate[blossombase[bv]]] = None |
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603 assignLabel(v, 2, labeledge[v][0]) |
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604 j += jstep |
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605 # Remove the expanded blossom entirely. |
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606 label.pop(b, None) |
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607 labeledge.pop(b, None) |
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608 bestedge.pop(b, None) |
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609 del blossomparent[b] |
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610 del blossombase[b] |
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611 del blossomdual[b] |
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612 |
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613 # Swap matched/unmatched edges over an alternating path through blossom b |
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614 # between vertex v and the base vertex. Keep blossom bookkeeping |
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615 # consistent. |
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616 def augmentBlossom(b, v): |
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617 # Bubble up through the blossom tree from vertex v to an immediate |
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618 # sub-blossom of b. |
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619 t = v |
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620 while blossomparent[t] != b: |
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621 t = blossomparent[t] |
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622 # Recursively deal with the first sub-blossom. |
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623 if isinstance(t, Blossom): |
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624 augmentBlossom(t, v) |
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625 # Decide in which direction we will go round the blossom. |
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626 i = j = b.childs.index(t) |
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627 if i & 1: |
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628 # Start index is odd; go forward and wrap. |
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629 j -= len(b.childs) |
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630 jstep = 1 |
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631 else: |
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632 # Start index is even; go backward. |
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633 jstep = -1 |
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634 # Move along the blossom until we get to the base. |
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635 while j != 0: |
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636 # Step to the next sub-blossom and augment it recursively. |
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637 j += jstep |
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638 t = b.childs[j] |
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639 if jstep == 1: |
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640 w, x = b.edges[j] |
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641 else: |
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642 x, w = b.edges[j - 1] |
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643 if isinstance(t, Blossom): |
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644 augmentBlossom(t, w) |
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645 # Step to the next sub-blossom and augment it recursively. |
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646 j += jstep |
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647 t = b.childs[j] |
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648 if isinstance(t, Blossom): |
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649 augmentBlossom(t, x) |
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650 # Match the edge connecting those sub-blossoms. |
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651 mate[w] = x |
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652 mate[x] = w |
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653 # Rotate the list of sub-blossoms to put the new base at the front. |
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654 b.childs = b.childs[i:] + b.childs[:i] |
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655 b.edges = b.edges[i:] + b.edges[:i] |
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656 blossombase[b] = blossombase[b.childs[0]] |
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657 assert blossombase[b] == v |
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658 |
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659 # Swap matched/unmatched edges over an alternating path between two |
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660 # single vertices. The augmenting path runs through S-vertices v and w. |
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661 def augmentMatching(v, w): |
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662 for (s, j) in ((v, w), (w, v)): |
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663 # Match vertex s to vertex j. Then trace back from s |
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664 # until we find a single vertex, swapping matched and unmatched |
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665 # edges as we go. |
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666 while 1: |
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667 bs = inblossom[s] |
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668 assert label[bs] == 1 |
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669 assert (labeledge[bs] is None and blossombase[bs] not in mate) or ( |
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670 labeledge[bs][0] == mate[blossombase[bs]] |
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671 ) |
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672 # Augment through the S-blossom from s to base. |
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673 if isinstance(bs, Blossom): |
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674 augmentBlossom(bs, s) |
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675 # Update mate[s] |
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676 mate[s] = j |
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677 # Trace one step back. |
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678 if labeledge[bs] is None: |
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679 # Reached single vertex; stop. |
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680 break |
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681 t = labeledge[bs][0] |
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682 bt = inblossom[t] |
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683 assert label[bt] == 2 |
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684 # Trace one more step back. |
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685 s, j = labeledge[bt] |
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686 # Augment through the T-blossom from j to base. |
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687 assert blossombase[bt] == t |
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688 if isinstance(bt, Blossom): |
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689 augmentBlossom(bt, j) |
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690 # Update mate[j] |
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691 mate[j] = s |
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692 |
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693 # Verify that the optimum solution has been reached. |
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694 def verifyOptimum(): |
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695 if maxcardinality: |
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696 # Vertices may have negative dual; |
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697 # find a constant non-negative number to add to all vertex duals. |
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698 vdualoffset = max(0, -min(dualvar.values())) |
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699 else: |
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700 vdualoffset = 0 |
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701 # 0. all dual variables are non-negative |
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702 assert min(dualvar.values()) + vdualoffset >= 0 |
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703 assert len(blossomdual) == 0 or min(blossomdual.values()) >= 0 |
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704 # 0. all edges have non-negative slack and |
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705 # 1. all matched edges have zero slack; |
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706 for i, j, d in G.edges(data=True): |
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707 wt = d.get(weight, 1) |
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708 if i == j: |
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709 continue # ignore self-loops |
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710 s = dualvar[i] + dualvar[j] - 2 * wt |
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711 iblossoms = [i] |
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712 jblossoms = [j] |
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713 while blossomparent[iblossoms[-1]] is not None: |
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714 iblossoms.append(blossomparent[iblossoms[-1]]) |
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715 while blossomparent[jblossoms[-1]] is not None: |
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716 jblossoms.append(blossomparent[jblossoms[-1]]) |
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717 iblossoms.reverse() |
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718 jblossoms.reverse() |
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719 for (bi, bj) in zip(iblossoms, jblossoms): |
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720 if bi != bj: |
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721 break |
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722 s += 2 * blossomdual[bi] |
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723 assert s >= 0 |
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724 if mate.get(i) == j or mate.get(j) == i: |
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725 assert mate[i] == j and mate[j] == i |
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726 assert s == 0 |
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727 # 2. all single vertices have zero dual value; |
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728 for v in gnodes: |
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729 assert (v in mate) or dualvar[v] + vdualoffset == 0 |
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730 # 3. all blossoms with positive dual value are full. |
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731 for b in blossomdual: |
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732 if blossomdual[b] > 0: |
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733 assert len(b.edges) % 2 == 1 |
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734 for (i, j) in b.edges[1::2]: |
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735 assert mate[i] == j and mate[j] == i |
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736 # Ok. |
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737 |
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738 # Main loop: continue until no further improvement is possible. |
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739 while 1: |
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740 |
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741 # Each iteration of this loop is a "stage". |
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742 # A stage finds an augmenting path and uses that to improve |
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743 # the matching. |
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744 |
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745 # Remove labels from top-level blossoms/vertices. |
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746 label.clear() |
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747 labeledge.clear() |
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748 |
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749 # Forget all about least-slack edges. |
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750 bestedge.clear() |
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751 for b in blossomdual: |
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752 b.mybestedges = None |
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753 |
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754 # Loss of labeling means that we can not be sure that currently |
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755 # allowable edges remain allowable throughout this stage. |
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756 allowedge.clear() |
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757 |
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758 # Make queue empty. |
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759 queue[:] = [] |
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760 |
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761 # Label single blossoms/vertices with S and put them in the queue. |
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762 for v in gnodes: |
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763 if (v not in mate) and label.get(inblossom[v]) is None: |
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764 assignLabel(v, 1, None) |
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765 |
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766 # Loop until we succeed in augmenting the matching. |
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767 augmented = 0 |
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768 while 1: |
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769 |
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770 # Each iteration of this loop is a "substage". |
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771 # A substage tries to find an augmenting path; |
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772 # if found, the path is used to improve the matching and |
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773 # the stage ends. If there is no augmenting path, the |
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774 # primal-dual method is used to pump some slack out of |
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775 # the dual variables. |
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776 |
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777 # Continue labeling until all vertices which are reachable |
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778 # through an alternating path have got a label. |
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779 while queue and not augmented: |
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780 |
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781 # Take an S vertex from the queue. |
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782 v = queue.pop() |
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783 assert label[inblossom[v]] == 1 |
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784 |
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785 # Scan its neighbours: |
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786 for w in G.neighbors(v): |
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787 if w == v: |
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788 continue # ignore self-loops |
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789 # w is a neighbour to v |
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790 bv = inblossom[v] |
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791 bw = inblossom[w] |
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792 if bv == bw: |
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793 # this edge is internal to a blossom; ignore it |
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794 continue |
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795 if (v, w) not in allowedge: |
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796 kslack = slack(v, w) |
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797 if kslack <= 0: |
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798 # edge k has zero slack => it is allowable |
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799 allowedge[(v, w)] = allowedge[(w, v)] = True |
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800 if (v, w) in allowedge: |
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801 if label.get(bw) is None: |
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802 # (C1) w is a free vertex; |
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803 # label w with T and label its mate with S (R12). |
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804 assignLabel(w, 2, v) |
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805 elif label.get(bw) == 1: |
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806 # (C2) w is an S-vertex (not in the same blossom); |
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807 # follow back-links to discover either an |
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808 # augmenting path or a new blossom. |
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809 base = scanBlossom(v, w) |
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810 if base is not NoNode: |
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811 # Found a new blossom; add it to the blossom |
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812 # bookkeeping and turn it into an S-blossom. |
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813 addBlossom(base, v, w) |
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814 else: |
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815 # Found an augmenting path; augment the |
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816 # matching and end this stage. |
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817 augmentMatching(v, w) |
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818 augmented = 1 |
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819 break |
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820 elif label.get(w) is None: |
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821 # w is inside a T-blossom, but w itself has not |
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822 # yet been reached from outside the blossom; |
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823 # mark it as reached (we need this to relabel |
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824 # during T-blossom expansion). |
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825 assert label[bw] == 2 |
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826 label[w] = 2 |
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827 labeledge[w] = (v, w) |
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828 elif label.get(bw) == 1: |
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829 # keep track of the least-slack non-allowable edge to |
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830 # a different S-blossom. |
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831 if bestedge.get(bv) is None or kslack < slack(*bestedge[bv]): |
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832 bestedge[bv] = (v, w) |
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833 elif label.get(w) is None: |
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834 # w is a free vertex (or an unreached vertex inside |
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835 # a T-blossom) but we can not reach it yet; |
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836 # keep track of the least-slack edge that reaches w. |
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837 if bestedge.get(w) is None or kslack < slack(*bestedge[w]): |
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838 bestedge[w] = (v, w) |
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839 |
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840 if augmented: |
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841 break |
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842 |
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843 # There is no augmenting path under these constraints; |
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844 # compute delta and reduce slack in the optimization problem. |
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845 # (Note that our vertex dual variables, edge slacks and delta's |
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846 # are pre-multiplied by two.) |
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847 deltatype = -1 |
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848 delta = deltaedge = deltablossom = None |
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849 |
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850 # Compute delta1: the minimum value of any vertex dual. |
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851 if not maxcardinality: |
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852 deltatype = 1 |
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853 delta = min(dualvar.values()) |
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854 |
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855 # Compute delta2: the minimum slack on any edge between |
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856 # an S-vertex and a free vertex. |
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857 for v in G.nodes(): |
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858 if label.get(inblossom[v]) is None and bestedge.get(v) is not None: |
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859 d = slack(*bestedge[v]) |
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860 if deltatype == -1 or d < delta: |
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861 delta = d |
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862 deltatype = 2 |
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863 deltaedge = bestedge[v] |
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864 |
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865 # Compute delta3: half the minimum slack on any edge between |
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866 # a pair of S-blossoms. |
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867 for b in blossomparent: |
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868 if ( |
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869 blossomparent[b] is None |
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870 and label.get(b) == 1 |
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871 and bestedge.get(b) is not None |
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872 ): |
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873 kslack = slack(*bestedge[b]) |
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874 if allinteger: |
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875 assert (kslack % 2) == 0 |
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876 d = kslack // 2 |
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877 else: |
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878 d = kslack / 2.0 |
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879 if deltatype == -1 or d < delta: |
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880 delta = d |
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881 deltatype = 3 |
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882 deltaedge = bestedge[b] |
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883 |
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884 # Compute delta4: minimum z variable of any T-blossom. |
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885 for b in blossomdual: |
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886 if ( |
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887 blossomparent[b] is None |
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888 and label.get(b) == 2 |
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889 and (deltatype == -1 or blossomdual[b] < delta) |
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890 ): |
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891 delta = blossomdual[b] |
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892 deltatype = 4 |
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893 deltablossom = b |
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894 |
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895 if deltatype == -1: |
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896 # No further improvement possible; max-cardinality optimum |
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897 # reached. Do a final delta update to make the optimum |
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898 # verifyable. |
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899 assert maxcardinality |
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900 deltatype = 1 |
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901 delta = max(0, min(dualvar.values())) |
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902 |
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903 # Update dual variables according to delta. |
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904 for v in gnodes: |
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905 if label.get(inblossom[v]) == 1: |
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906 # S-vertex: 2*u = 2*u - 2*delta |
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907 dualvar[v] -= delta |
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908 elif label.get(inblossom[v]) == 2: |
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909 # T-vertex: 2*u = 2*u + 2*delta |
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910 dualvar[v] += delta |
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911 for b in blossomdual: |
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912 if blossomparent[b] is None: |
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913 if label.get(b) == 1: |
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914 # top-level S-blossom: z = z + 2*delta |
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915 blossomdual[b] += delta |
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916 elif label.get(b) == 2: |
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917 # top-level T-blossom: z = z - 2*delta |
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918 blossomdual[b] -= delta |
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919 |
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920 # Take action at the point where minimum delta occurred. |
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921 if deltatype == 1: |
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922 # No further improvement possible; optimum reached. |
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923 break |
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924 elif deltatype == 2: |
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925 # Use the least-slack edge to continue the search. |
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926 (v, w) = deltaedge |
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927 assert label[inblossom[v]] == 1 |
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928 allowedge[(v, w)] = allowedge[(w, v)] = True |
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929 queue.append(v) |
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930 elif deltatype == 3: |
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931 # Use the least-slack edge to continue the search. |
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932 (v, w) = deltaedge |
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933 allowedge[(v, w)] = allowedge[(w, v)] = True |
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934 assert label[inblossom[v]] == 1 |
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935 queue.append(v) |
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936 elif deltatype == 4: |
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937 # Expand the least-z blossom. |
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938 expandBlossom(deltablossom, False) |
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939 |
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940 # End of a this substage. |
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941 |
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942 # Paranoia check that the matching is symmetric. |
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943 for v in mate: |
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944 assert mate[mate[v]] == v |
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945 |
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946 # Stop when no more augmenting path can be found. |
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947 if not augmented: |
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948 break |
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949 |
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950 # End of a stage; expand all S-blossoms which have zero dual. |
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951 for b in list(blossomdual.keys()): |
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952 if b not in blossomdual: |
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953 continue # already expanded |
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954 if blossomparent[b] is None and label.get(b) == 1 and blossomdual[b] == 0: |
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955 expandBlossom(b, True) |
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956 |
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957 # Verify that we reached the optimum solution (only for integer weights). |
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958 if allinteger: |
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959 verifyOptimum() |
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960 |
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961 return matching_dict_to_set(mate) |