<ahref="find__graph__symmetries_8h.html">Go to the documentation of this file.</a><divclass="fragment"><divclass="line"><aid="l00001"name="l00001"></a><spanclass="lineno"> 1</span><spanclass="comment">// Copyright 2010-2021 Google LLC</span></div>
<divclass="line"><aid="l00002"name="l00002"></a><spanclass="lineno"> 2</span><spanclass="comment">// Licensed under the Apache License, Version 2.0 (the "License");</span></div>
<divclass="line"><aid="l00003"name="l00003"></a><spanclass="lineno"> 3</span><spanclass="comment">// you may not use this file except in compliance with the License.</span></div>
<divclass="line"><aid="l00004"name="l00004"></a><spanclass="lineno"> 4</span><spanclass="comment">// You may obtain a copy of the License at</span></div>
<divclass="line"><aid="l00008"name="l00008"></a><spanclass="lineno"> 8</span><spanclass="comment">// Unless required by applicable law or agreed to in writing, software</span></div>
<divclass="line"><aid="l00009"name="l00009"></a><spanclass="lineno"> 9</span><spanclass="comment">// distributed under the License is distributed on an "AS IS" BASIS,</span></div>
<divclass="line"><aid="l00010"name="l00010"></a><spanclass="lineno"> 10</span><spanclass="comment">// WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.</span></div>
<divclass="line"><aid="l00011"name="l00011"></a><spanclass="lineno"> 11</span><spanclass="comment">// See the License for the specific language governing permissions and</span></div>
<divclass="line"><aid="l00012"name="l00012"></a><spanclass="lineno"> 12</span><spanclass="comment">// limitations under the License.</span></div>
<divclass="line"><aid="l00014"name="l00014"></a><spanclass="lineno"> 14</span><spanclass="comment">// This class solves the graph automorphism problem</span></div>
<divclass="line"><aid="l00015"name="l00015"></a><spanclass="lineno"> 15</span><spanclass="comment">// (https://en.wikipedia.org/wiki/Graph_automorphism), a variant of the famous</span></div>
<divclass="line"><aid="l00016"name="l00016"></a><spanclass="lineno"> 16</span><spanclass="comment">// graph isomorphism problem (https://en.wikipedia.org/wiki/Graph_isomorphism).</span></div>
<divclass="line"><aid="l00018"name="l00018"></a><spanclass="lineno"> 18</span><spanclass="comment">// The algorithm is largely based on the following article, published in 2008:</span></div>
<divclass="line"><aid="l00019"name="l00019"></a><spanclass="lineno"> 19</span><spanclass="comment">// "Faster Symmetry Discovery using Sparsity of Symmetries" by Darga, Sakallah</span></div>
<divclass="line"><aid="l00020"name="l00020"></a><spanclass="lineno"> 20</span><spanclass="comment">// and Markov. http://web.eecs.umich.edu/~imarkov/pubs/conf/dac08-sym.pdf.</span></div>
<divclass="line"><aid="l00022"name="l00022"></a><spanclass="lineno"> 22</span><spanclass="comment">// See the comments on the class below for more details.</span></div>
<divclass="line"><aid="l00048"name="l00048"></a><spanclass="lineno"> 48</span><spanclass="comment">// If the Graph passed to the GraphSymmetryFinder is undirected, i.e.</span></div>
<divclass="line"><aid="l00049"name="l00049"></a><spanclass="lineno"> 49</span><spanclass="comment">// for every arc a->b, b->a is also present, then you should set</span></div>
<divclass="line"><aid="l00050"name="l00050"></a><spanclass="lineno"> 50</span><spanclass="comment">// "is_undirected" to true.</span></div>
<divclass="line"><aid="l00051"name="l00051"></a><spanclass="lineno"> 51</span><spanclass="comment">// This will, in effect, DCHECK() that the graph is indeed undirected,</span></div>
<divclass="line"><aid="l00052"name="l00052"></a><spanclass="lineno"> 52</span><spanclass="comment">// and bypass the need for reverse adjacency lists.</span></div>
<divclass="line"><aid="l00054"name="l00054"></a><spanclass="lineno"> 54</span><spanclass="comment">// If you don't know this in advance, you may use GraphIsSymmetric() from</span></div>
<divclass="line"><aid="l00057"name="l00057"></a><spanclass="lineno"> 57</span><spanclass="comment">// "graph" must not have multi-arcs.</span></div>
<divclass="line"><aid="l00058"name="l00058"></a><spanclass="lineno"> 58</span><spanclass="comment">// TODO(user): support multi-arcs.</span></div>
<divclass="line"><aid="l00061"name="l00061"></a><spanclass="lineno"> 61</span><spanclass="comment">// Whether the given permutation is an automorphism of the graph given at</span></div>
<divclass="line"><aid="l00062"name="l00062"></a><spanclass="lineno"> 62</span><spanclass="comment">// construction. This costs O(sum(degree(x))) (the sum is over all nodes x</span></div>
<divclass="line"><aid="l00063"name="l00063"></a><spanclass="lineno"> 63</span><spanclass="comment">// that are displaced by the permutation).</span></div>
<divclass="line"><aid="l00066"name="l00066"></a><spanclass="lineno"> 66</span><spanclass="comment">// Find a set of generators of the automorphism subgroup of the graph that</span></div>
<divclass="line"><aid="l00067"name="l00067"></a><spanclass="lineno"> 67</span><spanclass="comment">// respects the given node equivalence classes. The generators are themselves</span></div>
<divclass="line"><aid="l00068"name="l00068"></a><spanclass="lineno"> 68</span><spanclass="comment">// permutations of the nodes: see http://en.wikipedia.org/wiki/Automorphism.</span></div>
<divclass="line"><aid="l00069"name="l00069"></a><spanclass="lineno"> 69</span><spanclass="comment">// These permutations may only map a node onto a node of its equivalence</span></div>
<divclass="line"><aid="l00070"name="l00070"></a><spanclass="lineno"> 70</span><spanclass="comment">// class: two nodes i and j are in the same equivalence class iff</span></div>
<divclass="line"><aid="l00073"name="l00073"></a><spanclass="lineno"> 73</span><spanclass="comment">// This set of generators is not necessarily the smallest possible (neither in</span></div>
<divclass="line"><aid="l00074"name="l00074"></a><spanclass="lineno"> 74</span><spanclass="comment">// the number of generators, nor in the size of these generators), but it is</span></div>
<divclass="line"><aid="l00075"name="l00075"></a><spanclass="lineno"> 75</span><spanclass="comment">// minimal in that no generator can be removed while keeping the generated</span></div>
<divclass="line"><aid="l00076"name="l00076"></a><spanclass="lineno"> 76</span><spanclass="comment">// group intact.</span></div>
<divclass="line"><aid="l00077"name="l00077"></a><spanclass="lineno"> 77</span><spanclass="comment">// TODO(user): verify the minimality in unit tests.</span></div>
<divclass="line"><aid="l00079"name="l00079"></a><spanclass="lineno"> 79</span><spanclass="comment">// Note that if "generators" is empty, then the graph has no symmetry: the</span></div>
<divclass="line"><aid="l00080"name="l00080"></a><spanclass="lineno"> 80</span><spanclass="comment">// only automorphism is the identity.</span></div>
<divclass="line"><aid="l00082"name="l00082"></a><spanclass="lineno"> 82</span><spanclass="comment">// The equivalence classes are actually an input/output: they are refined</span></div>
<divclass="line"><aid="l00083"name="l00083"></a><spanclass="lineno"> 83</span><spanclass="comment">// according to all asymmetries found. In the end, n1 and n2 will be</span></div>
<divclass="line"><aid="l00084"name="l00084"></a><spanclass="lineno"> 84</span><spanclass="comment">// considered equivalent (i.e. node_equivalence_classes_io[n1] ==</span></div>
<divclass="line"><aid="l00085"name="l00085"></a><spanclass="lineno"> 85</span><spanclass="comment">// node_equivalence_classes_io[n2]) if and only if there exists a</span></div>
<divclass="line"><aid="l00086"name="l00086"></a><spanclass="lineno"> 86</span><spanclass="comment">// permutation of nodes that:</span></div>
<divclass="line"><aid="l00087"name="l00087"></a><spanclass="lineno"> 87</span><spanclass="comment">// - keeps the graph invariant</span></div>
<divclass="line"><aid="l00089"name="l00089"></a><spanclass="lineno"> 89</span><spanclass="comment">// - maps each node to a node of its original equivalence class.</span></div>
<divclass="line"><aid="l00091"name="l00091"></a><spanclass="lineno"> 91</span><spanclass="comment">// This method also outputs the size of the automorphism group, expressed as</span></div>
<divclass="line"><aid="l00092"name="l00092"></a><spanclass="lineno"> 92</span><spanclass="comment">// a factorized product of integers (note that the size itself may be as</span></div>
<divclass="line"><aid="l00093"name="l00093"></a><spanclass="lineno"> 93</span><spanclass="comment">// large as N!).</span></div>
<divclass="line"><aid="l00095"name="l00095"></a><spanclass="lineno"> 95</span><spanclass="comment">// DEADLINE AND PARTIAL COMPLETION:</span></div>
<divclass="line"><aid="l00096"name="l00096"></a><spanclass="lineno"> 96</span><spanclass="comment">// If the deadline passed as argument (via TimeLimit) is reached, this method</span></div>
<divclass="line"><aid="l00097"name="l00097"></a><spanclass="lineno"> 97</span><spanclass="comment">// will return quickly (within a few milliseconds of the limit). The outputs</span></div>
<divclass="line"><aid="l00098"name="l00098"></a><spanclass="lineno"> 98</span><spanclass="comment">// may be partially filled:</span></div>
<divclass="line"><aid="l00099"name="l00099"></a><spanclass="lineno"> 99</span><spanclass="comment">// - Each element of "generators", if non-empty, will be a valid permutation.</span></div>
<divclass="line"><aid="l00100"name="l00100"></a><spanclass="lineno"> 100</span><spanclass="comment">// - "node_equivalence_classes_io" will contain the equivalence classes</span></div>
<divclass="line"><aid="l00101"name="l00101"></a><spanclass="lineno"> 101</span><spanclass="comment">// corresponding to the orbits under all the generators in "generators".</span></div>
<divclass="line"><aid="l00102"name="l00102"></a><spanclass="lineno"> 102</span><spanclass="comment">// - "factorized_automorphism_group_size" will also be incomplete, and</span></div>
<divclass="line"><aid="l00103"name="l00103"></a><spanclass="lineno"> 103</span><spanclass="comment">// partially valid: its last element may be undervalued. But all prior</span></div>
<divclass="line"><aid="l00104"name="l00104"></a><spanclass="lineno"> 104</span><spanclass="comment">// elements are valid factors of the automorphism group size.</span></div>
<divclass="line"><aid="l00111"name="l00111"></a><spanclass="lineno"> 111</span><spanclass="comment">// Fully refine the partition of nodes, using the graph as symmetry breaker.</span></div>
<divclass="line"><aid="l00112"name="l00112"></a><spanclass="lineno"> 112</span><spanclass="comment">// This means applying the following steps on each part P of the partition:</span></div>
<divclass="line"><aid="l00113"name="l00113"></a><spanclass="lineno"> 113</span><spanclass="comment">// - Compute the aggregated in-degree of all nodes of the graph, only looking</span></div>
<divclass="line"><aid="l00114"name="l00114"></a><spanclass="lineno"> 114</span><spanclass="comment">// at arcs originating from nodes in P.</span></div>
<divclass="line"><aid="l00115"name="l00115"></a><spanclass="lineno"> 115</span><spanclass="comment">// - For each in-degree d=1...max_in_degree, refine the partition by the set</span></div>
<divclass="line"><aid="l00116"name="l00116"></a><spanclass="lineno"> 116</span><spanclass="comment">// of nodes with in-degree d.</span></div>
<divclass="line"><aid="l00117"name="l00117"></a><spanclass="lineno"> 117</span><spanclass="comment">// And recursively applying it on all new or modified parts.</span></div>
<divclass="line"><aid="l00119"name="l00119"></a><spanclass="lineno"> 119</span><spanclass="comment">// In our use cases, we may call this in a scenario where the partition was</span></div>
<divclass="line"><aid="l00120"name="l00120"></a><spanclass="lineno"> 120</span><spanclass="comment">// already partially refined on all parts #0...#K, then you should set</span></div>
<divclass="line"><aid="l00121"name="l00121"></a><spanclass="lineno"> 121</span><spanclass="comment">// "first_unrefined_part_index" to K+1.</span></div>
<divclass="line"><aid="l00125"name="l00125"></a><spanclass="lineno"> 125</span><spanclass="comment">// **** Methods below are public FOR TESTING ONLY. ****</span></div>
<divclass="line"><aid="l00127"name="l00127"></a><spanclass="lineno"> 127</span><spanclass="comment">// Special wrapper of the above method: assuming that partition is already</span></div>
<divclass="line"><aid="l00128"name="l00128"></a><spanclass="lineno"> 128</span><spanclass="comment">// fully refined, further refine it by {node}, and propagate by adjacency.</span></div>
<divclass="line"><aid="l00129"name="l00129"></a><spanclass="lineno"> 129</span><spanclass="comment">// Also, optionally collect all the new singletons of the partition in</span></div>
<divclass="line"><aid="l00130"name="l00130"></a><spanclass="lineno"> 130</span><spanclass="comment">// "new_singletons", sorted by their part number in the partition.</span></div>
<divclass="line"><aid="l00139"name="l00139"></a><spanclass="lineno"> 139</span><spanclass="comment">// If the graph isn't symmetric, then we store the reverse adjacency lists</span></div>
<divclass="line"><aid="l00140"name="l00140"></a><spanclass="lineno"> 140</span><spanclass="comment">// here: for each i in 0..NumNodes()-1, the list of nodes that have an</span></div>
<divclass="line"><aid="l00141"name="l00141"></a><spanclass="lineno"> 141</span><spanclass="comment">// outgoing arc to i is stored (sorted by node) in:</span></div>
<divclass="line"><aid="l00144"name="l00144"></a><spanclass="lineno"> 144</span><spanclass="comment">// and can be iterated on easily with:</span></div>
<divclass="line"><aid="l00145"name="l00145"></a><spanclass="lineno"> 145</span><spanclass="comment">// for (const int tail : TailsOfIncomingArcsTo(node)) ...</span></div>
<divclass="line"><aid="l00147"name="l00147"></a><spanclass="lineno"> 147</span><spanclass="comment">// If the graph was specified as symmetric upon construction, both these</span></div>
<divclass="line"><aid="l00148"name="l00148"></a><spanclass="lineno"> 148</span><spanclass="comment">// vectors are empty, and TailsOfIncomingArcsTo() crashes.</span></div>
<divclass="line"><aid="l00154"name="l00154"></a><spanclass="lineno"> 154</span><spanclass="comment">// Deadline management. Populated upon FindSymmetries(). If the passed</span></div>
<divclass="line"><aid="l00155"name="l00155"></a><spanclass="lineno"> 155</span><spanclass="comment">// time limit is nullptr, time_limit_ will point to dummy_time_limit_ which</span></div>
<divclass="line"><aid="l00156"name="l00156"></a><spanclass="lineno"> 156</span><spanclass="comment">// is an object with infinite limits by default.</span></div>
<divclass="line"><aid="l00160"name="l00160"></a><spanclass="lineno"> 160</span><spanclass="comment">// Internal search code used in FindSymmetries(), split out for readability:</span></div>
<divclass="line"><aid="l00161"name="l00161"></a><spanclass="lineno"> 161</span><spanclass="comment">// find one permutation (if it exists) that maps root_node to root_image_node</span></div>
<divclass="line"><aid="l00162"name="l00162"></a><spanclass="lineno"> 162</span><spanclass="comment">// and such that the image of "base_partition" by that permutation is equal to</span></div>
<divclass="line"><aid="l00163"name="l00163"></a><spanclass="lineno"> 163</span><spanclass="comment">// the "image_partition". If no such permutation exists, returns nullptr.</span></div>
<divclass="line"><aid="l00165"name="l00165"></a><spanclass="lineno"> 165</span><spanclass="comment">// "generators_found_so_far" and "permutations_displacing_node" are used for</span></div>
<divclass="line"><aid="l00166"name="l00166"></a><spanclass="lineno"> 166</span><spanclass="comment">// pruning in the search. The former is just the "generators" vector of</span></div>
<divclass="line"><aid="l00167"name="l00167"></a><spanclass="lineno"> 167</span><spanclass="comment">// FindGraphSymmetries(), with the permutations found so far; and the latter</span></div>
<divclass="line"><aid="l00168"name="l00168"></a><spanclass="lineno"> 168</span><spanclass="comment">// is an inverted index from each node to all permutations (that we found)</span></div>
<divclass="line"><aid="l00169"name="l00169"></a><spanclass="lineno"> 169</span><spanclass="comment">// that displace it.</span></div>
<divclass="line"><aid="l00177"name="l00177"></a><spanclass="lineno"> 177</span><spanclass="comment">// Data structure used by FindOneSuitablePermutation(). See the .cc</span></div>
<divclass="line"><aid="l00181"name="l00181"></a><spanclass="lineno"> 181</span><spanclass="comment">// We're tentatively mapping "base_node" to some image node. At first, we</span></div>
<divclass="line"><aid="l00182"name="l00182"></a><spanclass="lineno"> 182</span><spanclass="comment">// just pick a single candidate: we fill "first_image_node". If this</span></div>
<divclass="line"><aid="l00183"name="l00183"></a><spanclass="lineno"> 183</span><spanclass="comment">// candidate doesn't work out, we'll select all other candidates in the same</span></div>
<divclass="line"><aid="l00184"name="l00184"></a><spanclass="lineno"> 184</span><spanclass="comment">// image part, prune them by the symmetries we found already, and put them</span></div>
<divclass="line"><aid="l00185"name="l00185"></a><spanclass="lineno"> 185</span><spanclass="comment">// in "remaining_pruned_image_nodes" (and set "first_image_node" to -1).</span></div>
<divclass="line"><aid="l00191"name="l00191"></a><spanclass="lineno"> 191</span><spanclass="comment">// Only parts that are at or beyond this index, or their parent parts, may</span></div>
<divclass="line"><aid="l00192"name="l00192"></a><spanclass="lineno"> 192</span><spanclass="comment">// be mismatching between the base and the image partitions.</span></div>
<divclass="line"><aid="l00205"name="l00205"></a><spanclass="lineno"> 205</span><spanclass="comment">// Subroutine of FindOneSuitablePermutation(), split out for modularity:</span></div>
<divclass="line"><aid="l00206"name="l00206"></a><spanclass="lineno"> 206</span><spanclass="comment">// With the partial candidate mapping given by "base_partition",</span></div>
<divclass="line"><aid="l00207"name="l00207"></a><spanclass="lineno"> 207</span><spanclass="comment">// "image_partition" and "current_permutation_candidate", determine whether</span></div>
<divclass="line"><aid="l00208"name="l00208"></a><spanclass="lineno"> 208</span><spanclass="comment">// we have a full match (eg. the permutation is a valid candidate).</span></div>
<divclass="line"><aid="l00209"name="l00209"></a><spanclass="lineno"> 209</span><spanclass="comment">// If so, simply return true. If not, return false but also fill</span></div>
<divclass="line"><aid="l00210"name="l00210"></a><spanclass="lineno"> 210</span><spanclass="comment">// "next_base_node" and "next_image_node" with what should be the next mapping</span></div>
<divclass="line"><aid="l00213"name="l00213"></a><spanclass="lineno"> 213</span><spanclass="comment">// This also uses and updates "min_potential_mismatching_part_index_io"</span></div>
<divclass="line"><aid="l00214"name="l00214"></a><spanclass="lineno"> 214</span><spanclass="comment">// to incrementally search for mismatching parts along the partitions.</span></div>
<divclass="line"><aid="l00216"name="l00216"></a><spanclass="lineno"> 216</span><spanclass="comment">// Note(user): there may be false positives, i.e. this method may return true</span></div>
<divclass="line"><aid="l00217"name="l00217"></a><spanclass="lineno"> 217</span><spanclass="comment">// even if the partitions aren't actually a full match, because it uses</span></div>
<divclass="line"><aid="l00218"name="l00218"></a><spanclass="lineno"> 218</span><spanclass="comment">// fingerprints to compare part. This should almost never happen.</span></div>
<divclass="line"><aid="l00226"name="l00226"></a><spanclass="lineno"> 226</span><spanclass="comment">// Subroutine of FindOneSuitablePermutation(), split out for modularity:</span></div>
<divclass="line"><aid="l00227"name="l00227"></a><spanclass="lineno"> 227</span><spanclass="comment">// Keep only one node of "nodes" per orbit, where the orbits are described</span></div>
<divclass="line"><aid="l00228"name="l00228"></a><spanclass="lineno"> 228</span><spanclass="comment">// by a subset of "all_permutations": the ones with indices in</span></div>
<divclass="line"><aid="l00229"name="l00229"></a><spanclass="lineno"> 229</span><spanclass="comment">// "permutation_indices" and that are compatible with "partition".</span></div>
<divclass="line"><aid="l00230"name="l00230"></a><spanclass="lineno"> 230</span><spanclass="comment">// For each orbit, keep the first node that appears in "nodes".</span></div>
<divclass="line"><aid="l00236"name="l00236"></a><spanclass="lineno"> 236</span><spanclass="comment">// Temporary objects used by some of the class methods, and owned by the</span></div>
<divclass="line"><aid="l00237"name="l00237"></a><spanclass="lineno"> 237</span><spanclass="comment">// class to avoid (costly) re-allocation. Their resting states are described</span></div>
<divclass="line"><aid="l00238"name="l00238"></a><spanclass="lineno"> 238</span><spanclass="comment">// in the side comments; with N = NumNodes().</span></div>
<divclass="line"><aid="l00247"name="l00247"></a><spanclass="lineno"> 247</span><spanclass="comment">// Internal statistics, used for performance tuning and debugging.</span></div>
<divclass="line"><aid="l00268"name="l00268"></a><spanclass="lineno"> 268</span><spanclass="stringliteral">"p ┣╸Mapping election / full match detection"</span>, this),</div>
<divclass="ttc"id="aclassoperations__research_1_1_time_limit_html"><divclass="ttname"><ahref="classoperations__research_1_1_time_limit.html">operations_research::TimeLimit</a></div><divclass="ttdoc">A simple class to enforce both an elapsed time limit and a deterministic time limit in the same threa...</div><divclass="ttdef"><b>Definition:</b><ahref="time__limit_8h_source.html#l00106">time_limit.h:106</a></div></div>
<divclass="ttc"id="anamespaceoperations__research_html"><divclass="ttname"><ahref="namespaceoperations__research.html">operations_research</a></div><divclass="ttdoc">Collection of objects used to extend the Constraint Solver library.</div><divclass="ttdef"><b>Definition:</b><ahref="dense__doubly__linked__list_8h_source.html#l00021">dense_doubly_linked_list.h:21</a></div></div>
