Mercurial > repos > shellac > sam_consensus_v3
annotate env/lib/python3.9/site-packages/networkx/algorithms/isomorphism/ismags.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 """ |
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2 **************** |
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3 ISMAGS Algorithm |
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4 **************** |
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5 |
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6 Provides a Python implementation of the ISMAGS algorithm. [1]_ |
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7 |
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8 It is capable of finding (subgraph) isomorphisms between two graphs, taking the |
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9 symmetry of the subgraph into account. In most cases the VF2 algorithm is |
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10 faster (at least on small graphs) than this implementation, but in some cases |
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11 there is an exponential number of isomorphisms that are symmetrically |
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12 equivalent. In that case, the ISMAGS algorithm will provide only one solution |
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13 per symmetry group. |
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14 |
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15 >>> petersen = nx.petersen_graph() |
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16 >>> ismags = nx.isomorphism.ISMAGS(petersen, petersen) |
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17 >>> isomorphisms = list(ismags.isomorphisms_iter(symmetry=False)) |
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18 >>> len(isomorphisms) |
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19 120 |
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20 >>> isomorphisms = list(ismags.isomorphisms_iter(symmetry=True)) |
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21 >>> answer = [{0: 0, 1: 1, 2: 2, 3: 3, 4: 4, 5: 5, 6: 6, 7: 7, 8: 8, 9: 9}] |
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22 >>> answer == isomorphisms |
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23 True |
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24 |
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25 In addition, this implementation also provides an interface to find the |
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26 largest common induced subgraph [2]_ between any two graphs, again taking |
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27 symmetry into account. Given `graph` and `subgraph` the algorithm will remove |
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28 nodes from the `subgraph` until `subgraph` is isomorphic to a subgraph of |
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29 `graph`. Since only the symmetry of `subgraph` is taken into account it is |
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30 worth thinking about how you provide your graphs: |
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31 |
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32 >>> graph1 = nx.path_graph(4) |
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33 >>> graph2 = nx.star_graph(3) |
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34 >>> ismags = nx.isomorphism.ISMAGS(graph1, graph2) |
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35 >>> ismags.is_isomorphic() |
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36 False |
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37 >>> largest_common_subgraph = list(ismags.largest_common_subgraph()) |
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38 >>> answer = [{1: 0, 0: 1, 2: 2}, {2: 0, 1: 1, 3: 2}] |
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39 >>> answer == largest_common_subgraph |
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40 True |
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41 >>> ismags2 = nx.isomorphism.ISMAGS(graph2, graph1) |
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42 >>> largest_common_subgraph = list(ismags2.largest_common_subgraph()) |
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43 >>> answer = [ |
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44 ... {1: 0, 0: 1, 2: 2}, |
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45 ... {1: 0, 0: 1, 3: 2}, |
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46 ... {2: 0, 0: 1, 1: 2}, |
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47 ... {2: 0, 0: 1, 3: 2}, |
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48 ... {3: 0, 0: 1, 1: 2}, |
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49 ... {3: 0, 0: 1, 2: 2}, |
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50 ... ] |
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51 >>> answer == largest_common_subgraph |
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52 True |
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53 |
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54 However, when not taking symmetry into account, it doesn't matter: |
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55 |
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56 >>> largest_common_subgraph = list(ismags.largest_common_subgraph(symmetry=False)) |
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57 >>> answer = [ |
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58 ... {1: 0, 0: 1, 2: 2}, |
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59 ... {1: 0, 2: 1, 0: 2}, |
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60 ... {2: 0, 1: 1, 3: 2}, |
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61 ... {2: 0, 3: 1, 1: 2}, |
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62 ... {1: 0, 0: 1, 2: 3}, |
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63 ... {1: 0, 2: 1, 0: 3}, |
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64 ... {2: 0, 1: 1, 3: 3}, |
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65 ... {2: 0, 3: 1, 1: 3}, |
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66 ... {1: 0, 0: 2, 2: 3}, |
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67 ... {1: 0, 2: 2, 0: 3}, |
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68 ... {2: 0, 1: 2, 3: 3}, |
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69 ... {2: 0, 3: 2, 1: 3}, |
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70 ... ] |
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71 >>> answer == largest_common_subgraph |
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72 True |
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73 >>> largest_common_subgraph = list(ismags2.largest_common_subgraph(symmetry=False)) |
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74 >>> answer = [ |
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75 ... {1: 0, 0: 1, 2: 2}, |
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76 ... {1: 0, 0: 1, 3: 2}, |
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77 ... {2: 0, 0: 1, 1: 2}, |
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78 ... {2: 0, 0: 1, 3: 2}, |
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79 ... {3: 0, 0: 1, 1: 2}, |
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80 ... {3: 0, 0: 1, 2: 2}, |
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81 ... {1: 1, 0: 2, 2: 3}, |
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82 ... {1: 1, 0: 2, 3: 3}, |
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83 ... {2: 1, 0: 2, 1: 3}, |
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84 ... {2: 1, 0: 2, 3: 3}, |
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85 ... {3: 1, 0: 2, 1: 3}, |
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86 ... {3: 1, 0: 2, 2: 3}, |
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87 ... ] |
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88 >>> answer == largest_common_subgraph |
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89 True |
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90 |
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91 Notes |
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92 ----- |
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93 - The current implementation works for undirected graphs only. The algorithm |
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94 in general should work for directed graphs as well though. |
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95 - Node keys for both provided graphs need to be fully orderable as well as |
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96 hashable. |
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97 - Node and edge equality is assumed to be transitive: if A is equal to B, and |
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98 B is equal to C, then A is equal to C. |
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99 |
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100 References |
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101 ---------- |
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102 .. [1] M. Houbraken, S. Demeyer, T. Michoel, P. Audenaert, D. Colle, |
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103 M. Pickavet, "The Index-Based Subgraph Matching Algorithm with General |
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104 Symmetries (ISMAGS): Exploiting Symmetry for Faster Subgraph |
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105 Enumeration", PLoS One 9(5): e97896, 2014. |
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106 https://doi.org/10.1371/journal.pone.0097896 |
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107 .. [2] https://en.wikipedia.org/wiki/Maximum_common_induced_subgraph |
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108 """ |
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109 |
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110 __all__ = ["ISMAGS"] |
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111 |
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112 from collections import defaultdict, Counter |
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113 from functools import reduce, wraps |
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114 import itertools |
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115 |
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116 |
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117 def are_all_equal(iterable): |
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118 """ |
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119 Returns ``True`` if and only if all elements in `iterable` are equal; and |
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120 ``False`` otherwise. |
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121 |
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122 Parameters |
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123 ---------- |
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124 iterable: collections.abc.Iterable |
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125 The container whose elements will be checked. |
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126 |
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127 Returns |
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128 ------- |
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129 bool |
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130 ``True`` iff all elements in `iterable` compare equal, ``False`` |
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131 otherwise. |
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132 """ |
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133 try: |
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134 shape = iterable.shape |
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135 except AttributeError: |
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136 pass |
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137 else: |
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138 if len(shape) > 1: |
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139 message = "The function does not works on multidimension arrays." |
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140 raise NotImplementedError(message) from None |
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141 |
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142 iterator = iter(iterable) |
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143 first = next(iterator, None) |
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144 return all(item == first for item in iterator) |
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145 |
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146 |
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147 def make_partitions(items, test): |
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148 """ |
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149 Partitions items into sets based on the outcome of ``test(item1, item2)``. |
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150 Pairs of items for which `test` returns `True` end up in the same set. |
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151 |
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152 Parameters |
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153 ---------- |
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154 items : collections.abc.Iterable[collections.abc.Hashable] |
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155 Items to partition |
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156 test : collections.abc.Callable[collections.abc.Hashable, collections.abc.Hashable] |
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157 A function that will be called with 2 arguments, taken from items. |
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158 Should return `True` if those 2 items need to end up in the same |
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159 partition, and `False` otherwise. |
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160 |
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161 Returns |
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162 ------- |
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163 list[set] |
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164 A list of sets, with each set containing part of the items in `items`, |
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165 such that ``all(test(*pair) for pair in itertools.combinations(set, 2)) |
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166 == True`` |
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167 |
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168 Notes |
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169 ----- |
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170 The function `test` is assumed to be transitive: if ``test(a, b)`` and |
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171 ``test(b, c)`` return ``True``, then ``test(a, c)`` must also be ``True``. |
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172 """ |
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173 partitions = [] |
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174 for item in items: |
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175 for partition in partitions: |
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176 p_item = next(iter(partition)) |
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177 if test(item, p_item): |
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178 partition.add(item) |
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179 break |
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180 else: # No break |
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181 partitions.append({item}) |
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182 return partitions |
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183 |
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184 |
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185 def partition_to_color(partitions): |
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186 """ |
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187 Creates a dictionary with for every item in partition for every partition |
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188 in partitions the index of partition in partitions. |
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189 |
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190 Parameters |
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191 ---------- |
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192 partitions: collections.abc.Sequence[collections.abc.Iterable] |
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193 As returned by :func:`make_partitions`. |
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194 |
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195 Returns |
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196 ------- |
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197 dict |
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198 """ |
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199 colors = dict() |
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200 for color, keys in enumerate(partitions): |
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201 for key in keys: |
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202 colors[key] = color |
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203 return colors |
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204 |
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205 |
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206 def intersect(collection_of_sets): |
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207 """ |
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208 Given an collection of sets, returns the intersection of those sets. |
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209 |
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210 Parameters |
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211 ---------- |
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212 collection_of_sets: collections.abc.Collection[set] |
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213 A collection of sets. |
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214 |
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215 Returns |
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216 ------- |
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217 set |
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218 An intersection of all sets in `collection_of_sets`. Will have the same |
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219 type as the item initially taken from `collection_of_sets`. |
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220 """ |
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221 collection_of_sets = list(collection_of_sets) |
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222 first = collection_of_sets.pop() |
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223 out = reduce(set.intersection, collection_of_sets, set(first)) |
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224 return type(first)(out) |
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225 |
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226 |
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227 class ISMAGS: |
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228 """ |
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229 Implements the ISMAGS subgraph matching algorith. [1]_ ISMAGS stands for |
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230 "Index-based Subgraph Matching Algorithm with General Symmetries". As the |
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231 name implies, it is symmetry aware and will only generate non-symmetric |
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232 isomorphisms. |
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233 |
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234 Notes |
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235 ----- |
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236 The implementation imposes additional conditions compared to the VF2 |
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237 algorithm on the graphs provided and the comparison functions |
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238 (:attr:`node_equality` and :attr:`edge_equality`): |
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239 |
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240 - Node keys in both graphs must be orderable as well as hashable. |
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241 - Equality must be transitive: if A is equal to B, and B is equal to C, |
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242 then A must be equal to C. |
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243 |
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244 Attributes |
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245 ---------- |
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246 graph: networkx.Graph |
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247 subgraph: networkx.Graph |
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248 node_equality: collections.abc.Callable |
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249 The function called to see if two nodes should be considered equal. |
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250 It's signature looks like this: |
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251 ``f(graph1: networkx.Graph, node1, graph2: networkx.Graph, node2) -> bool``. |
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252 `node1` is a node in `graph1`, and `node2` a node in `graph2`. |
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253 Constructed from the argument `node_match`. |
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254 edge_equality: collections.abc.Callable |
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255 The function called to see if two edges should be considered equal. |
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256 It's signature looks like this: |
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257 ``f(graph1: networkx.Graph, edge1, graph2: networkx.Graph, edge2) -> bool``. |
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258 `edge1` is an edge in `graph1`, and `edge2` an edge in `graph2`. |
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259 Constructed from the argument `edge_match`. |
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260 |
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261 References |
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262 ---------- |
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263 .. [1] M. Houbraken, S. Demeyer, T. Michoel, P. Audenaert, D. Colle, |
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264 M. Pickavet, "The Index-Based Subgraph Matching Algorithm with General |
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265 Symmetries (ISMAGS): Exploiting Symmetry for Faster Subgraph |
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266 Enumeration", PLoS One 9(5): e97896, 2014. |
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267 https://doi.org/10.1371/journal.pone.0097896 |
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268 """ |
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269 |
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270 def __init__(self, graph, subgraph, node_match=None, edge_match=None, cache=None): |
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271 """ |
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272 Parameters |
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273 ---------- |
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274 graph: networkx.Graph |
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275 subgraph: networkx.Graph |
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276 node_match: collections.abc.Callable or None |
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277 Function used to determine whether two nodes are equivalent. Its |
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278 signature should look like ``f(n1: dict, n2: dict) -> bool``, with |
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279 `n1` and `n2` node property dicts. See also |
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280 :func:`~networkx.algorithms.isomorphism.categorical_node_match` and |
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281 friends. |
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282 If `None`, all nodes are considered equal. |
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283 edge_match: collections.abc.Callable or None |
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284 Function used to determine whether two edges are equivalent. Its |
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285 signature should look like ``f(e1: dict, e2: dict) -> bool``, with |
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286 `e1` and `e2` edge property dicts. See also |
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287 :func:`~networkx.algorithms.isomorphism.categorical_edge_match` and |
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288 friends. |
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289 If `None`, all edges are considered equal. |
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290 cache: collections.abc.Mapping |
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291 A cache used for caching graph symmetries. |
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292 """ |
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293 # TODO: graph and subgraph setter methods that invalidate the caches. |
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294 # TODO: allow for precomputed partitions and colors |
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295 self.graph = graph |
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296 self.subgraph = subgraph |
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297 self._symmetry_cache = cache |
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298 # Naming conventions are taken from the original paper. For your |
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299 # sanity: |
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300 # sg: subgraph |
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301 # g: graph |
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302 # e: edge(s) |
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303 # n: node(s) |
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304 # So: sgn means "subgraph nodes". |
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305 self._sgn_partitions_ = None |
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306 self._sge_partitions_ = None |
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307 |
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308 self._sgn_colors_ = None |
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309 self._sge_colors_ = None |
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310 |
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311 self._gn_partitions_ = None |
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312 self._ge_partitions_ = None |
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313 |
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314 self._gn_colors_ = None |
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315 self._ge_colors_ = None |
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316 |
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317 self._node_compat_ = None |
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318 self._edge_compat_ = None |
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319 |
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320 if node_match is None: |
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321 self.node_equality = self._node_match_maker(lambda n1, n2: True) |
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322 self._sgn_partitions_ = [set(self.subgraph.nodes)] |
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323 self._gn_partitions_ = [set(self.graph.nodes)] |
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324 self._node_compat_ = {0: 0} |
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325 else: |
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326 self.node_equality = self._node_match_maker(node_match) |
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327 if edge_match is None: |
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328 self.edge_equality = self._edge_match_maker(lambda e1, e2: True) |
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329 self._sge_partitions_ = [set(self.subgraph.edges)] |
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330 self._ge_partitions_ = [set(self.graph.edges)] |
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331 self._edge_compat_ = {0: 0} |
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332 else: |
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333 self.edge_equality = self._edge_match_maker(edge_match) |
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334 |
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335 @property |
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336 def _sgn_partitions(self): |
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337 if self._sgn_partitions_ is None: |
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338 |
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339 def nodematch(node1, node2): |
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340 return self.node_equality(self.subgraph, node1, self.subgraph, node2) |
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341 |
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342 self._sgn_partitions_ = make_partitions(self.subgraph.nodes, nodematch) |
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343 return self._sgn_partitions_ |
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344 |
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345 @property |
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346 def _sge_partitions(self): |
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347 if self._sge_partitions_ is None: |
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348 |
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349 def edgematch(edge1, edge2): |
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350 return self.edge_equality(self.subgraph, edge1, self.subgraph, edge2) |
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351 |
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352 self._sge_partitions_ = make_partitions(self.subgraph.edges, edgematch) |
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353 return self._sge_partitions_ |
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354 |
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355 @property |
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356 def _gn_partitions(self): |
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357 if self._gn_partitions_ is None: |
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358 |
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359 def nodematch(node1, node2): |
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360 return self.node_equality(self.graph, node1, self.graph, node2) |
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361 |
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362 self._gn_partitions_ = make_partitions(self.graph.nodes, nodematch) |
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363 return self._gn_partitions_ |
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364 |
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365 @property |
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366 def _ge_partitions(self): |
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367 if self._ge_partitions_ is None: |
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368 |
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369 def edgematch(edge1, edge2): |
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370 return self.edge_equality(self.graph, edge1, self.graph, edge2) |
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371 |
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372 self._ge_partitions_ = make_partitions(self.graph.edges, edgematch) |
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373 return self._ge_partitions_ |
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374 |
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375 @property |
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376 def _sgn_colors(self): |
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377 if self._sgn_colors_ is None: |
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378 self._sgn_colors_ = partition_to_color(self._sgn_partitions) |
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379 return self._sgn_colors_ |
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380 |
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381 @property |
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382 def _sge_colors(self): |
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383 if self._sge_colors_ is None: |
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384 self._sge_colors_ = partition_to_color(self._sge_partitions) |
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385 return self._sge_colors_ |
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386 |
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387 @property |
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388 def _gn_colors(self): |
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389 if self._gn_colors_ is None: |
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390 self._gn_colors_ = partition_to_color(self._gn_partitions) |
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391 return self._gn_colors_ |
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392 |
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393 @property |
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394 def _ge_colors(self): |
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395 if self._ge_colors_ is None: |
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396 self._ge_colors_ = partition_to_color(self._ge_partitions) |
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397 return self._ge_colors_ |
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398 |
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399 @property |
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400 def _node_compatibility(self): |
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401 if self._node_compat_ is not None: |
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402 return self._node_compat_ |
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403 self._node_compat_ = {} |
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404 for sgn_part_color, gn_part_color in itertools.product( |
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405 range(len(self._sgn_partitions)), range(len(self._gn_partitions)) |
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406 ): |
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407 sgn = next(iter(self._sgn_partitions[sgn_part_color])) |
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408 gn = next(iter(self._gn_partitions[gn_part_color])) |
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409 if self.node_equality(self.subgraph, sgn, self.graph, gn): |
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410 self._node_compat_[sgn_part_color] = gn_part_color |
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411 return self._node_compat_ |
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412 |
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413 @property |
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414 def _edge_compatibility(self): |
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415 if self._edge_compat_ is not None: |
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416 return self._edge_compat_ |
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417 self._edge_compat_ = {} |
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418 for sge_part_color, ge_part_color in itertools.product( |
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419 range(len(self._sge_partitions)), range(len(self._ge_partitions)) |
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420 ): |
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421 sge = next(iter(self._sge_partitions[sge_part_color])) |
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422 ge = next(iter(self._ge_partitions[ge_part_color])) |
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423 if self.edge_equality(self.subgraph, sge, self.graph, ge): |
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424 self._edge_compat_[sge_part_color] = ge_part_color |
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425 return self._edge_compat_ |
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426 |
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427 @staticmethod |
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428 def _node_match_maker(cmp): |
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429 @wraps(cmp) |
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430 def comparer(graph1, node1, graph2, node2): |
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431 return cmp(graph1.nodes[node1], graph2.nodes[node2]) |
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432 |
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433 return comparer |
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434 |
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435 @staticmethod |
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436 def _edge_match_maker(cmp): |
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437 @wraps(cmp) |
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438 def comparer(graph1, edge1, graph2, edge2): |
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439 return cmp(graph1.edges[edge1], graph2.edges[edge2]) |
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440 |
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441 return comparer |
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442 |
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443 def find_isomorphisms(self, symmetry=True): |
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444 """Find all subgraph isomorphisms between subgraph and graph |
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445 |
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446 Finds isomorphisms where :attr:`subgraph` <= :attr:`graph`. |
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447 |
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448 Parameters |
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449 ---------- |
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450 symmetry: bool |
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451 Whether symmetry should be taken into account. If False, found |
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452 isomorphisms may be symmetrically equivalent. |
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453 |
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454 Yields |
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455 ------ |
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456 dict |
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457 The found isomorphism mappings of {graph_node: subgraph_node}. |
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458 """ |
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459 # The networkx VF2 algorithm is slightly funny in when it yields an |
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460 # empty dict and when not. |
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461 if not self.subgraph: |
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462 yield {} |
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463 return |
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464 elif not self.graph: |
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465 return |
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466 elif len(self.graph) < len(self.subgraph): |
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467 return |
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468 |
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469 if symmetry: |
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470 _, cosets = self.analyze_symmetry( |
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471 self.subgraph, self._sgn_partitions, self._sge_colors |
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472 ) |
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473 constraints = self._make_constraints(cosets) |
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474 else: |
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475 constraints = [] |
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476 |
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477 candidates = self._find_nodecolor_candidates() |
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478 la_candidates = self._get_lookahead_candidates() |
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479 for sgn in self.subgraph: |
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480 extra_candidates = la_candidates[sgn] |
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481 if extra_candidates: |
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482 candidates[sgn] = candidates[sgn] | {frozenset(extra_candidates)} |
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483 |
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484 if any(candidates.values()): |
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485 start_sgn = min(candidates, key=lambda n: min(candidates[n], key=len)) |
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486 candidates[start_sgn] = (intersect(candidates[start_sgn]),) |
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487 yield from self._map_nodes(start_sgn, candidates, constraints) |
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488 else: |
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489 return |
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490 |
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491 @staticmethod |
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492 def _find_neighbor_color_count(graph, node, node_color, edge_color): |
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493 """ |
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494 For `node` in `graph`, count the number of edges of a specific color |
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495 it has to nodes of a specific color. |
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496 """ |
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497 counts = Counter() |
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498 neighbors = graph[node] |
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499 for neighbor in neighbors: |
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500 n_color = node_color[neighbor] |
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501 if (node, neighbor) in edge_color: |
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502 e_color = edge_color[node, neighbor] |
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503 else: |
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504 e_color = edge_color[neighbor, node] |
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505 counts[e_color, n_color] += 1 |
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506 return counts |
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507 |
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508 def _get_lookahead_candidates(self): |
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509 """ |
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510 Returns a mapping of {subgraph node: collection of graph nodes} for |
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511 which the graph nodes are feasible candidates for the subgraph node, as |
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512 determined by looking ahead one edge. |
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513 """ |
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514 g_counts = {} |
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515 for gn in self.graph: |
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516 g_counts[gn] = self._find_neighbor_color_count( |
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517 self.graph, gn, self._gn_colors, self._ge_colors |
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518 ) |
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519 candidates = defaultdict(set) |
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520 for sgn in self.subgraph: |
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521 sg_count = self._find_neighbor_color_count( |
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522 self.subgraph, sgn, self._sgn_colors, self._sge_colors |
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523 ) |
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524 new_sg_count = Counter() |
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525 for (sge_color, sgn_color), count in sg_count.items(): |
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526 try: |
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527 ge_color = self._edge_compatibility[sge_color] |
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528 gn_color = self._node_compatibility[sgn_color] |
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529 except KeyError: |
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530 pass |
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531 else: |
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532 new_sg_count[ge_color, gn_color] = count |
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533 |
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534 for gn, g_count in g_counts.items(): |
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535 if all(new_sg_count[x] <= g_count[x] for x in new_sg_count): |
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536 # Valid candidate |
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537 candidates[sgn].add(gn) |
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538 return candidates |
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539 |
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540 def largest_common_subgraph(self, symmetry=True): |
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541 """ |
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542 Find the largest common induced subgraphs between :attr:`subgraph` and |
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543 :attr:`graph`. |
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544 |
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545 Parameters |
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546 ---------- |
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547 symmetry: bool |
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548 Whether symmetry should be taken into account. If False, found |
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549 largest common subgraphs may be symmetrically equivalent. |
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550 |
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551 Yields |
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552 ------ |
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553 dict |
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554 The found isomorphism mappings of {graph_node: subgraph_node}. |
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555 """ |
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556 # The networkx VF2 algorithm is slightly funny in when it yields an |
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557 # empty dict and when not. |
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558 if not self.subgraph: |
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559 yield {} |
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560 return |
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561 elif not self.graph: |
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562 return |
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563 |
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564 if symmetry: |
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565 _, cosets = self.analyze_symmetry( |
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566 self.subgraph, self._sgn_partitions, self._sge_colors |
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567 ) |
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568 constraints = self._make_constraints(cosets) |
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569 else: |
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570 constraints = [] |
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571 |
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572 candidates = self._find_nodecolor_candidates() |
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573 |
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574 if any(candidates.values()): |
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575 yield from self._largest_common_subgraph(candidates, constraints) |
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576 else: |
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577 return |
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578 |
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579 def analyze_symmetry(self, graph, node_partitions, edge_colors): |
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580 """ |
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581 Find a minimal set of permutations and corresponding co-sets that |
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582 describe the symmetry of :attr:`subgraph`. |
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583 |
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584 Returns |
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585 ------- |
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586 set[frozenset] |
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587 The found permutations. This is a set of frozenset of pairs of node |
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588 keys which can be exchanged without changing :attr:`subgraph`. |
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589 dict[collections.abc.Hashable, set[collections.abc.Hashable]] |
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590 The found co-sets. The co-sets is a dictionary of {node key: |
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591 set of node keys}. Every key-value pair describes which `values` |
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592 can be interchanged without changing nodes less than `key`. |
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593 """ |
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594 if self._symmetry_cache is not None: |
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595 key = hash( |
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596 ( |
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597 tuple(graph.nodes), |
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598 tuple(graph.edges), |
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599 tuple(map(tuple, node_partitions)), |
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600 tuple(edge_colors.items()), |
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601 ) |
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602 ) |
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603 if key in self._symmetry_cache: |
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604 return self._symmetry_cache[key] |
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605 node_partitions = list( |
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606 self._refine_node_partitions(graph, node_partitions, edge_colors) |
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607 ) |
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608 assert len(node_partitions) == 1 |
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609 node_partitions = node_partitions[0] |
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610 permutations, cosets = self._process_ordered_pair_partitions( |
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611 graph, node_partitions, node_partitions, edge_colors |
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612 ) |
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613 if self._symmetry_cache is not None: |
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614 self._symmetry_cache[key] = permutations, cosets |
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615 return permutations, cosets |
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616 |
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617 def is_isomorphic(self, symmetry=False): |
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618 """ |
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619 Returns True if :attr:`graph` is isomorphic to :attr:`subgraph` and |
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620 False otherwise. |
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621 |
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622 Returns |
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623 ------- |
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624 bool |
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625 """ |
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626 return len(self.subgraph) == len(self.graph) and self.subgraph_is_isomorphic( |
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627 symmetry |
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628 ) |
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629 |
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630 def subgraph_is_isomorphic(self, symmetry=False): |
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631 """ |
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632 Returns True if a subgraph of :attr:`graph` is isomorphic to |
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633 :attr:`subgraph` and False otherwise. |
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634 |
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635 Returns |
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636 ------- |
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637 bool |
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638 """ |
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639 # symmetry=False, since we only need to know whether there is any |
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640 # example; figuring out all symmetry elements probably costs more time |
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641 # than it gains. |
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642 isom = next(self.subgraph_isomorphisms_iter(symmetry=symmetry), None) |
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643 return isom is not None |
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644 |
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645 def isomorphisms_iter(self, symmetry=True): |
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646 """ |
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647 Does the same as :meth:`find_isomorphisms` if :attr:`graph` and |
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648 :attr:`subgraph` have the same number of nodes. |
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649 """ |
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650 if len(self.graph) == len(self.subgraph): |
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651 yield from self.subgraph_isomorphisms_iter(symmetry=symmetry) |
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652 |
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653 def subgraph_isomorphisms_iter(self, symmetry=True): |
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654 """Alternative name for :meth:`find_isomorphisms`.""" |
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655 return self.find_isomorphisms(symmetry) |
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656 |
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657 def _find_nodecolor_candidates(self): |
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658 """ |
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659 Per node in subgraph find all nodes in graph that have the same color. |
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660 """ |
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661 candidates = defaultdict(set) |
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662 for sgn in self.subgraph.nodes: |
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663 sgn_color = self._sgn_colors[sgn] |
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664 if sgn_color in self._node_compatibility: |
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665 gn_color = self._node_compatibility[sgn_color] |
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666 candidates[sgn].add(frozenset(self._gn_partitions[gn_color])) |
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667 else: |
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668 candidates[sgn].add(frozenset()) |
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669 candidates = dict(candidates) |
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670 for sgn, options in candidates.items(): |
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671 candidates[sgn] = frozenset(options) |
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672 return candidates |
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673 |
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674 @staticmethod |
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675 def _make_constraints(cosets): |
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676 """ |
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677 Turn cosets into constraints. |
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678 """ |
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679 constraints = [] |
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680 for node_i, node_ts in cosets.items(): |
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681 for node_t in node_ts: |
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682 if node_i != node_t: |
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683 # Node i must be smaller than node t. |
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684 constraints.append((node_i, node_t)) |
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685 return constraints |
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686 |
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687 @staticmethod |
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688 def _find_node_edge_color(graph, node_colors, edge_colors): |
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689 """ |
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690 For every node in graph, come up with a color that combines 1) the |
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691 color of the node, and 2) the number of edges of a color to each type |
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692 of node. |
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693 """ |
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694 counts = defaultdict(lambda: defaultdict(int)) |
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695 for node1, node2 in graph.edges: |
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696 if (node1, node2) in edge_colors: |
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697 # FIXME directed graphs |
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698 ecolor = edge_colors[node1, node2] |
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699 else: |
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700 ecolor = edge_colors[node2, node1] |
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701 # Count per node how many edges it has of what color to nodes of |
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702 # what color |
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703 counts[node1][ecolor, node_colors[node2]] += 1 |
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704 counts[node2][ecolor, node_colors[node1]] += 1 |
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705 |
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706 node_edge_colors = dict() |
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707 for node in graph.nodes: |
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708 node_edge_colors[node] = node_colors[node], set(counts[node].items()) |
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709 |
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710 return node_edge_colors |
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711 |
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712 @staticmethod |
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713 def _get_permutations_by_length(items): |
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714 """ |
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715 Get all permutations of items, but only permute items with the same |
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716 length. |
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717 |
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718 >>> found = list(ISMAGS._get_permutations_by_length([[1], [2], [3, 4], [4, 5]])) |
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719 >>> answer = [ |
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720 ... (([1], [2]), ([3, 4], [4, 5])), |
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721 ... (([1], [2]), ([4, 5], [3, 4])), |
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722 ... (([2], [1]), ([3, 4], [4, 5])), |
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723 ... (([2], [1]), ([4, 5], [3, 4])), |
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724 ... ] |
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725 >>> found == answer |
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726 True |
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727 """ |
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728 by_len = defaultdict(list) |
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729 for item in items: |
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730 by_len[len(item)].append(item) |
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731 |
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732 yield from itertools.product( |
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733 *(itertools.permutations(by_len[l]) for l in sorted(by_len)) |
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734 ) |
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735 |
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736 @classmethod |
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737 def _refine_node_partitions(cls, graph, node_partitions, edge_colors, branch=False): |
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738 """ |
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739 Given a partition of nodes in graph, make the partitions smaller such |
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740 that all nodes in a partition have 1) the same color, and 2) the same |
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741 number of edges to specific other partitions. |
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742 """ |
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743 |
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744 def equal_color(node1, node2): |
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745 return node_edge_colors[node1] == node_edge_colors[node2] |
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746 |
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747 node_partitions = list(node_partitions) |
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748 node_colors = partition_to_color(node_partitions) |
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749 node_edge_colors = cls._find_node_edge_color(graph, node_colors, edge_colors) |
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750 if all( |
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751 are_all_equal(node_edge_colors[node] for node in partition) |
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752 for partition in node_partitions |
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753 ): |
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754 yield node_partitions |
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755 return |
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756 |
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757 new_partitions = [] |
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758 output = [new_partitions] |
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759 for partition in node_partitions: |
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760 if not are_all_equal(node_edge_colors[node] for node in partition): |
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761 refined = make_partitions(partition, equal_color) |
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762 if ( |
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763 branch |
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764 and len(refined) != 1 |
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765 and len({len(r) for r in refined}) != len([len(r) for r in refined]) |
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766 ): |
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767 # This is where it breaks. There are multiple new cells |
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768 # in refined with the same length, and their order |
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769 # matters. |
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770 # So option 1) Hit it with a big hammer and simply make all |
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771 # orderings. |
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772 permutations = cls._get_permutations_by_length(refined) |
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773 new_output = [] |
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774 for n_p in output: |
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775 for permutation in permutations: |
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776 new_output.append(n_p + list(permutation[0])) |
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777 output = new_output |
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778 else: |
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779 for n_p in output: |
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780 n_p.extend(sorted(refined, key=len)) |
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781 else: |
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782 for n_p in output: |
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783 n_p.append(partition) |
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784 for n_p in output: |
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785 yield from cls._refine_node_partitions(graph, n_p, edge_colors, branch) |
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786 |
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787 def _edges_of_same_color(self, sgn1, sgn2): |
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788 """ |
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789 Returns all edges in :attr:`graph` that have the same colour as the |
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790 edge between sgn1 and sgn2 in :attr:`subgraph`. |
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791 """ |
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792 if (sgn1, sgn2) in self._sge_colors: |
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793 # FIXME directed graphs |
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794 sge_color = self._sge_colors[sgn1, sgn2] |
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795 else: |
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796 sge_color = self._sge_colors[sgn2, sgn1] |
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797 if sge_color in self._edge_compatibility: |
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798 ge_color = self._edge_compatibility[sge_color] |
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799 g_edges = self._ge_partitions[ge_color] |
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800 else: |
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801 g_edges = [] |
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802 return g_edges |
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803 |
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804 def _map_nodes(self, sgn, candidates, constraints, mapping=None, to_be_mapped=None): |
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805 """ |
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806 Find all subgraph isomorphisms honoring constraints. |
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807 """ |
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808 if mapping is None: |
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809 mapping = {} |
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810 else: |
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811 mapping = mapping.copy() |
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812 if to_be_mapped is None: |
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813 to_be_mapped = set(self.subgraph.nodes) |
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814 |
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815 # Note, we modify candidates here. Doesn't seem to affect results, but |
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816 # remember this. |
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817 # candidates = candidates.copy() |
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818 sgn_candidates = intersect(candidates[sgn]) |
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819 candidates[sgn] = frozenset([sgn_candidates]) |
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820 for gn in sgn_candidates: |
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821 # We're going to try to map sgn to gn. |
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822 if gn in mapping.values() or sgn not in to_be_mapped: |
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823 # gn is already mapped to something |
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824 continue # pragma: no cover |
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825 |
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826 # REDUCTION and COMBINATION |
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827 mapping[sgn] = gn |
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828 # BASECASE |
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829 if to_be_mapped == set(mapping.keys()): |
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830 yield {v: k for k, v in mapping.items()} |
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831 continue |
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832 left_to_map = to_be_mapped - set(mapping.keys()) |
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833 |
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834 new_candidates = candidates.copy() |
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835 sgn_neighbours = set(self.subgraph[sgn]) |
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836 not_gn_neighbours = set(self.graph.nodes) - set(self.graph[gn]) |
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837 for sgn2 in left_to_map: |
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838 if sgn2 not in sgn_neighbours: |
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839 gn2_options = not_gn_neighbours |
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840 else: |
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841 # Get all edges to gn of the right color: |
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842 g_edges = self._edges_of_same_color(sgn, sgn2) |
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843 # FIXME directed graphs |
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844 # And all nodes involved in those which are connected to gn |
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845 gn2_options = {n for e in g_edges for n in e if gn in e} |
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846 # Node color compatibility should be taken care of by the |
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847 # initial candidate lists made by find_subgraphs |
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848 |
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849 # Add gn2_options to the right collection. Since new_candidates |
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850 # is a dict of frozensets of frozensets of node indices it's |
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851 # a bit clunky. We can't do .add, and + also doesn't work. We |
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852 # could do |, but I deem union to be clearer. |
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853 new_candidates[sgn2] = new_candidates[sgn2].union( |
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854 [frozenset(gn2_options)] |
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855 ) |
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856 |
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857 if (sgn, sgn2) in constraints: |
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858 gn2_options = {gn2 for gn2 in self.graph if gn2 > gn} |
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859 elif (sgn2, sgn) in constraints: |
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860 gn2_options = {gn2 for gn2 in self.graph if gn2 < gn} |
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861 else: |
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862 continue # pragma: no cover |
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863 new_candidates[sgn2] = new_candidates[sgn2].union( |
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864 [frozenset(gn2_options)] |
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865 ) |
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866 |
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867 # The next node is the one that is unmapped and has fewest |
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868 # candidates |
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869 # Pylint disables because it's a one-shot function. |
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870 next_sgn = min( |
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871 left_to_map, key=lambda n: min(new_candidates[n], key=len) |
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872 ) # pylint: disable=cell-var-from-loop |
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873 yield from self._map_nodes( |
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874 next_sgn, |
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875 new_candidates, |
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876 constraints, |
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877 mapping=mapping, |
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878 to_be_mapped=to_be_mapped, |
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879 ) |
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880 # Unmap sgn-gn. Strictly not necessary since it'd get overwritten |
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881 # when making a new mapping for sgn. |
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882 # del mapping[sgn] |
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883 |
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884 def _largest_common_subgraph(self, candidates, constraints, to_be_mapped=None): |
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885 """ |
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886 Find all largest common subgraphs honoring constraints. |
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887 """ |
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888 if to_be_mapped is None: |
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889 to_be_mapped = {frozenset(self.subgraph.nodes)} |
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890 |
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891 # The LCS problem is basically a repeated subgraph isomorphism problem |
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892 # with smaller and smaller subgraphs. We store the nodes that are |
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893 # "part of" the subgraph in to_be_mapped, and we make it a little |
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894 # smaller every iteration. |
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895 |
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896 # pylint disable becuase it's guarded against by default value |
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897 current_size = len( |
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898 next(iter(to_be_mapped), []) |
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899 ) # pylint: disable=stop-iteration-return |
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900 |
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|
901 found_iso = False |
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|
902 if current_size <= len(self.graph): |
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|
903 # There's no point in trying to find isomorphisms of |
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904 # graph >= subgraph if subgraph has more nodes than graph. |
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905 |
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|
906 # Try the isomorphism first with the nodes with lowest ID. So sort |
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|
907 # them. Those are more likely to be part of the final |
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|
908 # correspondence. This makes finding the first answer(s) faster. In |
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|
909 # theory. |
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|
910 for nodes in sorted(to_be_mapped, key=sorted): |
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|
911 # Find the isomorphism between subgraph[to_be_mapped] <= graph |
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|
912 next_sgn = min(nodes, key=lambda n: min(candidates[n], key=len)) |
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913 isomorphs = self._map_nodes( |
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|
914 next_sgn, candidates, constraints, to_be_mapped=nodes |
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915 ) |
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916 |
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|
917 # This is effectively `yield from isomorphs`, except that we look |
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918 # whether an item was yielded. |
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|
919 try: |
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920 item = next(isomorphs) |
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921 except StopIteration: |
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922 pass |
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923 else: |
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924 yield item |
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925 yield from isomorphs |
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926 found_iso = True |
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927 |
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928 # BASECASE |
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929 if found_iso or current_size == 1: |
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930 # Shrinking has no point because either 1) we end up with a smaller |
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931 # common subgraph (and we want the largest), or 2) there'll be no |
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932 # more subgraph. |
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|
933 return |
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934 |
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935 left_to_be_mapped = set() |
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936 for nodes in to_be_mapped: |
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937 for sgn in nodes: |
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938 # We're going to remove sgn from to_be_mapped, but subject to |
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939 # symmetry constraints. We know that for every constraint we |
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940 # have those subgraph nodes are equal. So whenever we would |
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941 # remove the lower part of a constraint, remove the higher |
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942 # instead. This is all dealth with by _remove_node. And because |
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943 # left_to_be_mapped is a set, we don't do double work. |
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944 |
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945 # And finally, make the subgraph one node smaller. |
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946 # REDUCTION |
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947 new_nodes = self._remove_node(sgn, nodes, constraints) |
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948 left_to_be_mapped.add(new_nodes) |
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949 # COMBINATION |
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950 yield from self._largest_common_subgraph( |
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951 candidates, constraints, to_be_mapped=left_to_be_mapped |
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952 ) |
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953 |
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954 @staticmethod |
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955 def _remove_node(node, nodes, constraints): |
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956 """ |
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957 Returns a new set where node has been removed from nodes, subject to |
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958 symmetry constraints. We know, that for every constraint we have |
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959 those subgraph nodes are equal. So whenever we would remove the |
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960 lower part of a constraint, remove the higher instead. |
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961 """ |
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962 while True: |
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963 for low, high in constraints: |
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964 if low == node and high in nodes: |
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965 node = high |
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966 break |
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967 else: # no break, couldn't find node in constraints |
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968 break |
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969 return frozenset(nodes - {node}) |
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970 |
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971 @staticmethod |
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972 def _find_permutations(top_partitions, bottom_partitions): |
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973 """ |
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974 Return the pairs of top/bottom partitions where the partitions are |
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975 different. Ensures that all partitions in both top and bottom |
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976 partitions have size 1. |
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977 """ |
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978 # Find permutations |
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979 permutations = set() |
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980 for top, bot in zip(top_partitions, bottom_partitions): |
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981 # top and bot have only one element |
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982 if len(top) != 1 or len(bot) != 1: |
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983 raise IndexError( |
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984 "Not all nodes are coupled. This is" |
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985 f" impossible: {top_partitions}, {bottom_partitions}" |
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986 ) |
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987 if top != bot: |
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988 permutations.add(frozenset((next(iter(top)), next(iter(bot))))) |
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989 return permutations |
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990 |
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991 @staticmethod |
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992 def _update_orbits(orbits, permutations): |
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993 """ |
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994 Update orbits based on permutations. Orbits is modified in place. |
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995 For every pair of items in permutations their respective orbits are |
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996 merged. |
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997 """ |
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998 for permutation in permutations: |
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999 node, node2 = permutation |
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1000 # Find the orbits that contain node and node2, and replace the |
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1001 # orbit containing node with the union |
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1002 first = second = None |
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1003 for idx, orbit in enumerate(orbits): |
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1004 if first is not None and second is not None: |
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1005 break |
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1006 if node in orbit: |
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1007 first = idx |
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1008 if node2 in orbit: |
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1009 second = idx |
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1010 if first != second: |
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1011 orbits[first].update(orbits[second]) |
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1012 del orbits[second] |
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1013 |
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1014 def _couple_nodes( |
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1015 self, |
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1016 top_partitions, |
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1017 bottom_partitions, |
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1018 pair_idx, |
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1019 t_node, |
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1020 b_node, |
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1021 graph, |
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1022 edge_colors, |
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1023 ): |
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1024 """ |
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1025 Generate new partitions from top and bottom_partitions where t_node is |
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1026 coupled to b_node. pair_idx is the index of the partitions where t_ and |
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1027 b_node can be found. |
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1028 """ |
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1029 t_partition = top_partitions[pair_idx] |
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1030 b_partition = bottom_partitions[pair_idx] |
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1031 assert t_node in t_partition and b_node in b_partition |
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1032 # Couple node to node2. This means they get their own partition |
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1033 new_top_partitions = [top.copy() for top in top_partitions] |
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1034 new_bottom_partitions = [bot.copy() for bot in bottom_partitions] |
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1035 new_t_groups = {t_node}, t_partition - {t_node} |
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1036 new_b_groups = {b_node}, b_partition - {b_node} |
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1037 # Replace the old partitions with the coupled ones |
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1038 del new_top_partitions[pair_idx] |
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1039 del new_bottom_partitions[pair_idx] |
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1040 new_top_partitions[pair_idx:pair_idx] = new_t_groups |
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1041 new_bottom_partitions[pair_idx:pair_idx] = new_b_groups |
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1042 |
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1043 new_top_partitions = self._refine_node_partitions( |
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1044 graph, new_top_partitions, edge_colors |
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1045 ) |
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1046 new_bottom_partitions = self._refine_node_partitions( |
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1047 graph, new_bottom_partitions, edge_colors, branch=True |
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1048 ) |
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1049 new_top_partitions = list(new_top_partitions) |
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1050 assert len(new_top_partitions) == 1 |
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1051 new_top_partitions = new_top_partitions[0] |
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1052 for bot in new_bottom_partitions: |
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1053 yield list(new_top_partitions), bot |
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1054 |
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1055 def _process_ordered_pair_partitions( |
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1056 self, |
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1057 graph, |
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1058 top_partitions, |
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1059 bottom_partitions, |
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1060 edge_colors, |
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1061 orbits=None, |
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1062 cosets=None, |
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1063 ): |
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1064 """ |
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1065 Processes ordered pair partitions as per the reference paper. Finds and |
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1066 returns all permutations and cosets that leave the graph unchanged. |
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1067 """ |
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1068 if orbits is None: |
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1069 orbits = [{node} for node in graph.nodes] |
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1070 else: |
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1071 # Note that we don't copy orbits when we are given one. This means |
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1072 # we leak information between the recursive branches. This is |
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1073 # intentional! |
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1074 orbits = orbits |
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1075 if cosets is None: |
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1076 cosets = {} |
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1077 else: |
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1078 cosets = cosets.copy() |
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1079 |
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1080 assert all( |
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1081 len(t_p) == len(b_p) for t_p, b_p in zip(top_partitions, bottom_partitions) |
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1082 ) |
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1083 |
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1084 # BASECASE |
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1085 if all(len(top) == 1 for top in top_partitions): |
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1086 # All nodes are mapped |
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1087 permutations = self._find_permutations(top_partitions, bottom_partitions) |
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1088 self._update_orbits(orbits, permutations) |
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1089 if permutations: |
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1090 return [permutations], cosets |
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1091 else: |
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1092 return [], cosets |
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1093 |
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1094 permutations = [] |
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1095 unmapped_nodes = { |
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1096 (node, idx) |
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1097 for idx, t_partition in enumerate(top_partitions) |
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1098 for node in t_partition |
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1099 if len(t_partition) > 1 |
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1100 } |
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1101 node, pair_idx = min(unmapped_nodes) |
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1102 b_partition = bottom_partitions[pair_idx] |
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1103 |
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1104 for node2 in sorted(b_partition): |
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1105 if len(b_partition) == 1: |
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1106 # Can never result in symmetry |
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1107 continue |
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1108 if node != node2 and any( |
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1109 node in orbit and node2 in orbit for orbit in orbits |
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1110 ): |
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1111 # Orbit prune branch |
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1112 continue |
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1113 # REDUCTION |
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1114 # Couple node to node2 |
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1115 partitions = self._couple_nodes( |
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1116 top_partitions, |
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1117 bottom_partitions, |
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1118 pair_idx, |
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1119 node, |
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1120 node2, |
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1121 graph, |
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1122 edge_colors, |
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1123 ) |
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1124 for opp in partitions: |
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1125 new_top_partitions, new_bottom_partitions = opp |
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1126 |
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1127 new_perms, new_cosets = self._process_ordered_pair_partitions( |
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1128 graph, |
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1129 new_top_partitions, |
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1130 new_bottom_partitions, |
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1131 edge_colors, |
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1132 orbits, |
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1133 cosets, |
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1134 ) |
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1135 # COMBINATION |
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1136 permutations += new_perms |
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1137 cosets.update(new_cosets) |
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1138 |
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1139 mapped = { |
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1140 k |
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1141 for top, bottom in zip(top_partitions, bottom_partitions) |
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1142 for k in top |
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1143 if len(top) == 1 and top == bottom |
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1144 } |
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1145 ks = {k for k in graph.nodes if k < node} |
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1146 # Have all nodes with ID < node been mapped? |
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1147 find_coset = ks <= mapped and node not in cosets |
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1148 if find_coset: |
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1149 # Find the orbit that contains node |
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1150 for orbit in orbits: |
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1151 if node in orbit: |
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1152 cosets[node] = orbit.copy() |
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1153 return permutations, cosets |