<ahref="cliques_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="l00015"name="l00015"></a><spanclass="lineno"> 15</span><spanclass="comment">// Maximal clique algorithms, based on the Bron-Kerbosch algorithm.</span></div>
<divclass="line"><aid="l00016"name="l00016"></a><spanclass="lineno"> 16</span><spanclass="comment">// See http://en.wikipedia.org/wiki/Bron-Kerbosch_algorithm</span></div>
<divclass="line"><aid="l00018"name="l00018"></a><spanclass="lineno"> 18</span><spanclass="comment">// C. Bron and J. Kerbosch, Joep, "Algorithm 457: finding all cliques of an</span></div>
<divclass="line"><aid="l00041"name="l00041"></a><spanclass="lineno"> 41</span><spanclass="comment">// Finds all maximal cliques, even of size 1, in the</span></div>
<divclass="line"><aid="l00042"name="l00042"></a><spanclass="lineno"> 42</span><spanclass="comment">// graph described by the graph callback. graph->Run(i, j) indicates</span></div>
<divclass="line"><aid="l00043"name="l00043"></a><spanclass="lineno"> 43</span><spanclass="comment">// if there is an arc between i and j.</span></div>
<divclass="line"><aid="l00044"name="l00044"></a><spanclass="lineno"> 44</span><spanclass="comment">// This function takes ownership of 'callback' and deletes it after it has run.</span></div>
<divclass="line"><aid="l00045"name="l00045"></a><spanclass="lineno"> 45</span><spanclass="comment">// If 'callback' returns true, then the search for cliques stops.</span></div>
<divclass="line"><aid="l00049"name="l00049"></a><spanclass="lineno"> 49</span><spanclass="comment">// Covers the maximum number of arcs of the graph with cliques. The graph</span></div>
<divclass="line"><aid="l00050"name="l00050"></a><spanclass="lineno"> 50</span><spanclass="comment">// is described by the graph callback. graph->Run(i, j) indicates if</span></div>
<divclass="line"><aid="l00051"name="l00051"></a><spanclass="lineno"> 51</span><spanclass="comment">// there is an arc between i and j.</span></div>
<divclass="line"><aid="l00052"name="l00052"></a><spanclass="lineno"> 52</span><spanclass="comment">// This function takes ownership of 'callback' and deletes it after it has run.</span></div>
<divclass="line"><aid="l00053"name="l00053"></a><spanclass="lineno"> 53</span><spanclass="comment">// It calls 'callback' upon each clique.</span></div>
<divclass="line"><aid="l00054"name="l00054"></a><spanclass="lineno"> 54</span><spanclass="comment">// It ignores cliques of size 1.</span></div>
<divclass="line"><aid="l00058"name="l00058"></a><spanclass="lineno"> 58</span><spanclass="comment">// Possible return values of the callback for reporting cliques. The returned</span></div>
<divclass="line"><aid="l00059"name="l00059"></a><spanclass="lineno"> 59</span><spanclass="comment">// value determines whether the algorithm will continue the search.</span></div>
<divclass="line"><aid="l00061"name="l00061"></a><spanclass="lineno"> 61</span><spanclass="comment">// The algorithm will continue searching for other maximal cliques.</span></div>
<divclass="line"><aid="l00063"name="l00063"></a><spanclass="lineno"> 63</span><spanclass="comment">// The algorithm will stop the search immediately. The search can be resumed</span></div>
<divclass="line"><aid="l00064"name="l00064"></a><spanclass="lineno"> 64</span><spanclass="comment">// by calling BronKerboschAlgorithm::Run (resp. RunIterations) again.</span></div>
<divclass="line"><aid="l00068"name="l00068"></a><spanclass="lineno"> 68</span><spanclass="comment">// The status value returned by BronKerboschAlgorithm::Run and</span></div>
<divclass="line"><aid="l00071"name="l00071"></a><spanclass="lineno"> 71</span><spanclass="comment">// The algorithm has enumerated all maximal cliques.</span></div>
<divclass="line"><aid="l00073"name="l00073"></a><spanclass="lineno"> 73</span><spanclass="comment">// The search algorithm was interrupted either because it reached the</span></div>
<divclass="line"><aid="l00074"name="l00074"></a><spanclass="lineno"> 74</span><spanclass="comment">// iteration limit or because the clique callback returned</span></div>
<divclass="line"><aid="l00079"name="l00079"></a><spanclass="lineno"> 79</span><spanclass="comment">// Implements the Bron-Kerbosch algorithm for finding maximal cliques.</span></div>
<divclass="line"><aid="l00080"name="l00080"></a><spanclass="lineno"> 80</span><spanclass="comment">// The graph is represented as a callback that gets two nodes as its arguments</span></div>
<divclass="line"><aid="l00081"name="l00081"></a><spanclass="lineno"> 81</span><spanclass="comment">// and it returns true if and only if there is an arc between the two nodes. The</span></div>
<divclass="line"><aid="l00082"name="l00082"></a><spanclass="lineno"> 82</span><spanclass="comment">// cliques are reported back to the user using a second callback.</span></div>
<divclass="line"><aid="l00098"name="l00098"></a><spanclass="lineno"> 98</span><spanclass="comment">// This is a non-recursive implementation of the Bron-Kerbosch algorithm with</span></div>
<divclass="line"><aid="l00099"name="l00099"></a><spanclass="lineno"> 99</span><spanclass="comment">// pivots as described in the paper by Bron and Kerbosch (1973) (the version 2</span></div>
<divclass="line"><aid="l00100"name="l00100"></a><spanclass="lineno"> 100</span><spanclass="comment">// algorithm in the paper).</span></div>
<divclass="line"><aid="l00101"name="l00101"></a><spanclass="lineno"> 101</span><spanclass="comment">// The basic idea of the algorithm is to incrementally build the cliques using</span></div>
<divclass="line"><aid="l00102"name="l00102"></a><spanclass="lineno"> 102</span><spanclass="comment">// depth-first search. During the search, the algorithm maintains two sets of</span></div>
<divclass="line"><aid="l00103"name="l00103"></a><spanclass="lineno"> 103</span><spanclass="comment">// candidates (nodes that are connected to all nodes in the current clique):</span></div>
<divclass="line"><aid="l00104"name="l00104"></a><spanclass="lineno"> 104</span><spanclass="comment">// - the "not" set - these are candidates that were already visited by the</span></div>
<divclass="line"><aid="l00105"name="l00105"></a><spanclass="lineno"> 105</span><spanclass="comment">// search and all the maximal cliques that contain them as a part of the</span></div>
<divclass="line"><aid="l00106"name="l00106"></a><spanclass="lineno"> 106</span><spanclass="comment">// current clique were already reported.</span></div>
<divclass="line"><aid="l00107"name="l00107"></a><spanclass="lineno"> 107</span><spanclass="comment">// - the actual candidates - these are candidates that were not visited yet, and</span></div>
<divclass="line"><aid="l00108"name="l00108"></a><spanclass="lineno"> 108</span><spanclass="comment">// they can be added to the clique.</span></div>
<divclass="line"><aid="l00109"name="l00109"></a><spanclass="lineno"> 109</span><spanclass="comment">// In each iteration, the algorithm does the first of the following actions that</span></div>
<divclass="line"><aid="l00111"name="l00111"></a><spanclass="lineno"> 111</span><spanclass="comment">// A. If there are no actual candidates and there are candidates in the "not"</span></div>
<divclass="line"><aid="l00112"name="l00112"></a><spanclass="lineno"> 112</span><spanclass="comment">// set, or if all actual candidates are connected to the same node in the</span></div>
<divclass="line"><aid="l00113"name="l00113"></a><spanclass="lineno"> 113</span><spanclass="comment">// "not" set, the current clique can't be extended to a maximal clique that</span></div>
<divclass="line"><aid="l00114"name="l00114"></a><spanclass="lineno"> 114</span><spanclass="comment">// was not already reported. Return from the recursive call and move the</span></div>
<divclass="line"><aid="l00115"name="l00115"></a><spanclass="lineno"> 115</span><spanclass="comment">// selected candidate to the set "not".</span></div>
<divclass="line"><aid="l00116"name="l00116"></a><spanclass="lineno"> 116</span><spanclass="comment">// B. If there are no candidates at all, it means that the current clique can't</span></div>
<divclass="line"><aid="l00117"name="l00117"></a><spanclass="lineno"> 117</span><spanclass="comment">// be extended and that it is in fact a maximal clique. Report it to the user</span></div>
<divclass="line"><aid="l00118"name="l00118"></a><spanclass="lineno"> 118</span><spanclass="comment">// and return from the recursive call. Move the selected candidate to the set</span></div>
<divclass="line"><aid="l00120"name="l00120"></a><spanclass="lineno"> 120</span><spanclass="comment">// C. Otherwise, there are actual candidates, extend the current clique with one</span></div>
<divclass="line"><aid="l00121"name="l00121"></a><spanclass="lineno"> 121</span><spanclass="comment">// of these candidates and process it recursively.</span></div>
<divclass="line"><aid="l00123"name="l00123"></a><spanclass="lineno"> 123</span><spanclass="comment">// To avoid unnecessary steps, the algorithm selects a pivot at each level of</span></div>
<divclass="line"><aid="l00124"name="l00124"></a><spanclass="lineno"> 124</span><spanclass="comment">// the recursion to guide the selection of candidates added to the current</span></div>
<divclass="line"><aid="l00125"name="l00125"></a><spanclass="lineno"> 125</span><spanclass="comment">// clique. The pivot can be either in the "not" set and among the actual</span></div>
<divclass="line"><aid="l00126"name="l00126"></a><spanclass="lineno"> 126</span><spanclass="comment">// candidates. The algorithm tries to move the pivot and all actual candidates</span></div>
<divclass="line"><aid="l00127"name="l00127"></a><spanclass="lineno"> 127</span><spanclass="comment">// connected to it to the set "not" as quickly as possible. This will fulfill</span></div>
<divclass="line"><aid="l00128"name="l00128"></a><spanclass="lineno"> 128</span><spanclass="comment">// the conditions of step A, and the search algorithm will be able to leave the</span></div>
<divclass="line"><aid="l00129"name="l00129"></a><spanclass="lineno"> 129</span><spanclass="comment">// current branch. Selecting a pivot that has the lowest number of disconnected</span></div>
<divclass="line"><aid="l00130"name="l00130"></a><spanclass="lineno"> 130</span><spanclass="comment">// nodes among the candidates can reduce the running time significantly.</span></div>
<divclass="line"><aid="l00132"name="l00132"></a><spanclass="lineno"> 132</span><spanclass="comment">// The worst-case maximal depth of the recursion is equal to the number of nodes</span></div>
<divclass="line"><aid="l00133"name="l00133"></a><spanclass="lineno"> 133</span><spanclass="comment">// in the graph, which makes the natural recursive implementation impractical</span></div>
<divclass="line"><aid="l00134"name="l00134"></a><spanclass="lineno"> 134</span><spanclass="comment">// for nodes with more than a few thousands of nodes. To avoid the limitation,</span></div>
<divclass="line"><aid="l00135"name="l00135"></a><spanclass="lineno"> 135</span><spanclass="comment">// this class simulates the recursion by maintaining a stack with the state at</span></div>
<divclass="line"><aid="l00136"name="l00136"></a><spanclass="lineno"> 136</span><spanclass="comment">// each level of the recursion. The algorithm then runs in a loop. In each</span></div>
<divclass="line"><aid="l00137"name="l00137"></a><spanclass="lineno"> 137</span><spanclass="comment">// iteration, the algorithm can do one or both of:</span></div>
<divclass="line"><aid="l00138"name="l00138"></a><spanclass="lineno"> 138</span><spanclass="comment">// 1. Return to the previous recursion level (step A or B of the algorithm) by</span></div>
<divclass="line"><aid="l00139"name="l00139"></a><spanclass="lineno"> 139</span><spanclass="comment">// removing the top state from the stack.</span></div>
<divclass="line"><aid="l00140"name="l00140"></a><spanclass="lineno"> 140</span><spanclass="comment">// 2. Select the next candidate and enter the next recursion level (step C of</span></div>
<divclass="line"><aid="l00141"name="l00141"></a><spanclass="lineno"> 141</span><spanclass="comment">// the algorithm) by adding a new state to the stack.</span></div>
<divclass="line"><aid="l00143"name="l00143"></a><spanclass="lineno"> 143</span><spanclass="comment">// The worst-case time complexity of the algorithm is O(3^(N/3)), and the memory</span></div>
<divclass="line"><aid="l00144"name="l00144"></a><spanclass="lineno"> 144</span><spanclass="comment">// complexity is O(N^2), where N is the number of nodes in the graph.</span></div>
<divclass="line"><aid="l00148"name="l00148"></a><spanclass="lineno"> 148</span><spanclass="comment">// A callback called by the algorithm to test if there is an arc between a</span></div>
<divclass="line"><aid="l00149"name="l00149"></a><spanclass="lineno"> 149</span><spanclass="comment">// pair of nodes. The callback must return true if and only if there is an</span></div>
<divclass="line"><aid="l00150"name="l00150"></a><spanclass="lineno"> 150</span><spanclass="comment">// arc. Note that to function properly, the function must be symmetrical</span></div>
<divclass="line"><aid="l00151"name="l00151"></a><spanclass="lineno"> 151</span><spanclass="comment">// (represent an undirected graph).</span></div>
<divclass="line"><aid="l00153"name="l00153"></a><spanclass="lineno"> 153</span><spanclass="comment">// A callback called by the algorithm to report a maximal clique to the user.</span></div>
<divclass="line"><aid="l00154"name="l00154"></a><spanclass="lineno"> 154</span><spanclass="comment">// The clique is returned as a list of nodes in the clique, in no particular</span></div>
<divclass="line"><aid="l00155"name="l00155"></a><spanclass="lineno"> 155</span><spanclass="comment">// order. The caller must make a copy of the vector if they want to keep the</span></div>
<divclass="line"><aid="l00158"name="l00158"></a><spanclass="lineno"> 158</span><spanclass="comment">// The return value of the callback controls how the algorithm continues after</span></div>
<divclass="line"><aid="l00159"name="l00159"></a><spanclass="lineno"> 159</span><spanclass="comment">// this clique. See the description of the values of 'CliqueResponse' for more</span></div>
<divclass="line"><aid="l00164"name="l00164"></a><spanclass="lineno"> 164</span><spanclass="comment">// Initializes the Bron-Kerbosch algorithm for the given graph and clique</span></div>
<divclass="line"><aid="l00172"name="l00172"></a><spanclass="lineno"> 172</span><spanclass="comment">// Runs the Bron-Kerbosch algorithm for kint64max iterations. In practice,</span></div>
<divclass="line"><aid="l00173"name="l00173"></a><spanclass="lineno"> 173</span><spanclass="comment">// this is equivalent to running until completion or until the clique callback</span></div>
<divclass="line"><aid="l00174"name="l00174"></a><spanclass="lineno"> 174</span><spanclass="comment">// returns BronKerboschAlgorithmStatus::STOP. If the method returned because</span></div>
<divclass="line"><aid="l00175"name="l00175"></a><spanclass="lineno"> 175</span><spanclass="comment">// the search is finished, it will return COMPLETED; otherwise, it will return</span></div>
<divclass="line"><aid="l00176"name="l00176"></a><spanclass="lineno"> 176</span><spanclass="comment">// INTERRUPTED and it can be resumed by calling this method again.</span></div>
<divclass="line"><aid="l00179"name="l00179"></a><spanclass="lineno"> 179</span><spanclass="comment">// Runs at most 'max_num_iterations' iterations of the Bron-Kerbosch</span></div>
<divclass="line"><aid="l00180"name="l00180"></a><spanclass="lineno"> 180</span><spanclass="comment">// algorithm. When this function returns INTERRUPTED, there is still work to</span></div>
<divclass="line"><aid="l00181"name="l00181"></a><spanclass="lineno"> 181</span><spanclass="comment">// be done to process all the cliques in the graph. In such case the method</span></div>
<divclass="line"><aid="l00182"name="l00182"></a><spanclass="lineno"> 182</span><spanclass="comment">// can be called again and it will resume the work where the previous call had</span></div>
<divclass="line"><aid="l00183"name="l00183"></a><spanclass="lineno"> 183</span><spanclass="comment">// stopped. When it returns COMPLETED any subsequent call to the method will</span></div>
<divclass="line"><aid="l00184"name="l00184"></a><spanclass="lineno"> 184</span><spanclass="comment">// resume the search from the beginning.</span></div>
<divclass="line"><aid="l00187"name="l00187"></a><spanclass="lineno"> 187</span><spanclass="comment">// Runs at most 'max_num_iterations' iterations of the Bron-Kerbosch</span></div>
<divclass="line"><aid="l00188"name="l00188"></a><spanclass="lineno"> 188</span><spanclass="comment">// algorithm, until the time limit is exceeded or until all cliques are</span></div>
<divclass="line"><aid="l00189"name="l00189"></a><spanclass="lineno"> 189</span><spanclass="comment">// enumerated. When this function returns INTERRUPTED, there is still work to</span></div>
<divclass="line"><aid="l00190"name="l00190"></a><spanclass="lineno"> 190</span><spanclass="comment">// be done to process all the cliques in the graph. In such case the method</span></div>
<divclass="line"><aid="l00191"name="l00191"></a><spanclass="lineno"> 191</span><spanclass="comment">// can be called again and it will resume the work where the previous call had</span></div>
<divclass="line"><aid="l00192"name="l00192"></a><spanclass="lineno"> 192</span><spanclass="comment">// stopped. When it returns COMPLETED any subsequent call to the method will</span></div>
<divclass="line"><aid="l00193"name="l00193"></a><spanclass="lineno"> 193</span><spanclass="comment">// resume the search from the beginning.</span></div>
<divclass="line"><aid="l00197"name="l00197"></a><spanclass="lineno"> 197</span><spanclass="comment">// Runs the Bron-Kerbosch algorithm for at most kint64max iterations, until</span></div>
<divclass="line"><aid="l00198"name="l00198"></a><spanclass="lineno"> 198</span><spanclass="comment">// the time limit is excceded or until all cliques are enumerated. In</span></div>
<divclass="line"><aid="l00199"name="l00199"></a><spanclass="lineno"> 199</span><spanclass="comment">// practice, running the algorithm for kint64max iterations is equivalent to</span></div>
<divclass="line"><aid="l00200"name="l00200"></a><spanclass="lineno"> 200</span><spanclass="comment">// running until completion or until the other stopping conditions apply. When</span></div>
<divclass="line"><aid="l00201"name="l00201"></a><spanclass="lineno"> 201</span><spanclass="comment">// this function returns INTERRUPTED, there is still work to be done to</span></div>
<divclass="line"><aid="l00202"name="l00202"></a><spanclass="lineno"> 202</span><spanclass="comment">// process all the cliques in the graph. In such case the method can be called</span></div>
<divclass="line"><aid="l00203"name="l00203"></a><spanclass="lineno"> 203</span><spanclass="comment">// again and it will resume the work where the previous call had stopped. When</span></div>
<divclass="line"><aid="l00204"name="l00204"></a><spanclass="lineno"> 204</span><spanclass="comment">// it returns COMPLETED any subsequent call to the method will resume the</span></div>
<divclass="line"><aid="l00205"name="l00205"></a><spanclass="lineno"> 205</span><spanclass="comment">// search from the beginning.</span></div>
<divclass="line"><aid="l00213"name="l00213"></a><spanclass="lineno"> 213</span><spanclass="comment">// A data structure that maintains the variables of one "iteration" of the</span></div>
<divclass="line"><aid="l00214"name="l00214"></a><spanclass="lineno"> 214</span><spanclass="comment">// search algorithm. These are the variables that would normally be allocated</span></div>
<divclass="line"><aid="l00215"name="l00215"></a><spanclass="lineno"> 215</span><spanclass="comment">// on the stack in the recursive implementation.</span></div>
<divclass="line"><aid="l00217"name="l00217"></a><spanclass="lineno"> 217</span><spanclass="comment">// Note that most of the variables in the structure are explicitly left</span></div>
<divclass="line"><aid="l00218"name="l00218"></a><spanclass="lineno"> 218</span><spanclass="comment">// uninitialized by the constructor to avoid wasting resources on values that</span></div>
<divclass="line"><aid="l00219"name="l00219"></a><spanclass="lineno"> 219</span><spanclass="comment">// will be overwritten anyway. Most of the initialization is done in</span></div>
<divclass="line"><aid="l00239"name="l00239"></a><spanclass="lineno"> 239</span><spanclass="comment">// Moves the first candidate in the state to the "not" set. Assumes that the</span></div>
<divclass="line"><aid="l00240"name="l00240"></a><spanclass="lineno"> 240</span><spanclass="comment">// first candidate is also the pivot or a candidate disconnected from the</span></div>
<divclass="line"><aid="l00241"name="l00241"></a><spanclass="lineno"> 241</span><spanclass="comment">// pivot (as done by RunIteration).</span></div>
<divclass="line"><aid="l00247"name="l00247"></a><spanclass="lineno"> 247</span><spanclass="comment">// Creates a human-readable representation of the current state.</span></div>
<divclass="line"><aid="l00263"name="l00263"></a><spanclass="lineno"> 263</span><spanclass="comment">// The pivot node selected for the given level of the recursion.</span></div>
<divclass="line"><aid="l00265"name="l00265"></a><spanclass="lineno"> 265</span><spanclass="comment">// The number of remaining candidates to be explored at the given level of</span></div>
<divclass="line"><aid="l00266"name="l00266"></a><spanclass="lineno"> 266</span><spanclass="comment">// the recursion; the number is computed as num_disconnected_nodes +</span></div>
<divclass="line"><aid="l00267"name="l00267"></a><spanclass="lineno"> 267</span><spanclass="comment">// pre_increment in the original algorithm.</span></div>
<divclass="line"><aid="l00269"name="l00269"></a><spanclass="lineno"> 269</span><spanclass="comment">// The list of nodes that are candidates for extending the current clique.</span></div>
<divclass="line"><aid="l00270"name="l00270"></a><spanclass="lineno"> 270</span><spanclass="comment">// This vector has the format proposed in the paper by Bron-Kerbosch; the</span></div>
<divclass="line"><aid="l00271"name="l00271"></a><spanclass="lineno"> 271</span><spanclass="comment">// first 'first_candidate_index' elements of the vector represent the</span></div>
<divclass="line"><aid="l00272"name="l00272"></a><spanclass="lineno"> 272</span><spanclass="comment">// "not" set of nodes that were already visited by the algorithm. The</span></div>
<divclass="line"><aid="l00273"name="l00273"></a><spanclass="lineno"> 273</span><spanclass="comment">// remaining elements are the actual candidates for extending the current</span></div>
<divclass="line"><aid="l00275"name="l00275"></a><spanclass="lineno"> 275</span><spanclass="comment">// NOTE(user): We could store the delta between the iterations; however,</span></div>
<divclass="line"><aid="l00276"name="l00276"></a><spanclass="lineno"> 276</span><spanclass="comment">// we need to evaluate the impact this would have on the performance.</span></div>
<divclass="line"><aid="l00278"name="l00278"></a><spanclass="lineno"> 278</span><spanclass="comment">// The index of the first actual candidate in 'candidates'. This number is</span></div>
<divclass="line"><aid="l00279"name="l00279"></a><spanclass="lineno"> 279</span><spanclass="comment">// also the number of elements of the "not" set stored at the beginning of</span></div>
<divclass="line"><aid="l00283"name="l00283"></a><spanclass="lineno"> 283</span><spanclass="comment">// The current position in candidates when looking for the pivot and/or the</span></div>
<divclass="line"><aid="l00284"name="l00284"></a><spanclass="lineno"> 284</span><spanclass="comment">// next candidate disconnected from the pivot.</span></div>
<divclass="line"><aid="l00288"name="l00288"></a><spanclass="lineno"> 288</span><spanclass="comment">// The deterministic time coefficients for the push and pop operations of the</span></div>
<divclass="line"><aid="l00289"name="l00289"></a><spanclass="lineno"> 289</span><spanclass="comment">// Bron-Kerbosch algorithm. The coefficients are set to match approximately</span></div>
<divclass="line"><aid="l00290"name="l00290"></a><spanclass="lineno"> 290</span><spanclass="comment">// the running time in seconds on a recent workstation on the random graph</span></div>
<divclass="line"><aid="l00292"name="l00292"></a><spanclass="lineno"> 292</span><spanclass="comment">// NOTE(user): PushState is not the only source of complexity in the</span></div>
<divclass="line"><aid="l00293"name="l00293"></a><spanclass="lineno"> 293</span><spanclass="comment">// algorithm, but non-negative linear least squares produced zero coefficients</span></div>
<divclass="line"><aid="l00294"name="l00294"></a><spanclass="lineno"> 294</span><spanclass="comment">// for all other deterministic counters tested during the benchmarking. When</span></div>
<divclass="line"><aid="l00295"name="l00295"></a><spanclass="lineno"> 295</span><spanclass="comment">// we optimize the algorithm, we might need to add deterministic time to the</span></div>
<divclass="line"><aid="l00296"name="l00296"></a><spanclass="lineno"> 296</span><spanclass="comment">// other places that may produce complexity, namely InitializeState, PopState</span></div>
<divclass="line"><aid="l00297"name="l00297"></a><spanclass="lineno"> 297</span><spanclass="comment">// and SelectCandidateIndexForRecursion.</span></div>
<divclass="line"><aid="l00300"name="l00300"></a><spanclass="lineno"> 300</span><spanclass="comment">// Initializes the root state of the algorithm.</span></div>
<divclass="line"><aid="l00303"name="l00303"></a><spanclass="lineno"> 303</span><spanclass="comment">// Removes the top state from the state stack. This is equivalent to returning</span></div>
<divclass="line"><aid="l00304"name="l00304"></a><spanclass="lineno"> 304</span><spanclass="comment">// in the recursive implementation of the algorithm.</span></div>
<divclass="line"><aid="l00307"name="l00307"></a><spanclass="lineno"> 307</span><spanclass="comment">// Adds a new state to the top of the stack, adding the node 'selected' to the</span></div>
<divclass="line"><aid="l00308"name="l00308"></a><spanclass="lineno"> 308</span><spanclass="comment">// current clique. This is equivalent to making a recurisve call in the</span></div>
<divclass="line"><aid="l00309"name="l00309"></a><spanclass="lineno"> 309</span><spanclass="comment">// recursive implementation of the algorithm.</span></div>
<divclass="line"><aid="l00312"name="l00312"></a><spanclass="lineno"> 312</span><spanclass="comment">// Initializes the given state. Runs the pivot selection algorithm in the</span></div>
<divclass="line"><aid="l00316"name="l00316"></a><spanclass="lineno"> 316</span><spanclass="comment">// Returns true if (node1, node2) is an arc in the graph or if node1 == node2.</span></div>
<divclass="line"><aid="l00321"name="l00321"></a><spanclass="lineno"> 321</span><spanclass="comment">// Selects the next node for recursion. The selected node is either the pivot</span></div>
<divclass="line"><aid="l00322"name="l00322"></a><spanclass="lineno"> 322</span><spanclass="comment">// (if it is not in the set "not") or a node that is disconnected from the</span></div>
<divclass="line"><aid="l00326"name="l00326"></a><spanclass="lineno"> 326</span><spanclass="comment">// Returns a human-readable string representation of the clique.</span></div>
<divclass="line"><aid="l00329"name="l00329"></a><spanclass="lineno"> 329</span><spanclass="comment">// The callback called when the algorithm needs to determine if (node1, node2)</span></div>
<divclass="line"><aid="l00330"name="l00330"></a><spanclass="lineno"> 330</span><spanclass="comment">// is an arc in the graph.</span></div>
<divclass="line"><aid="l00333"name="l00333"></a><spanclass="lineno"> 333</span><spanclass="comment">// The callback called when the algorithm discovers a maximal clique. The</span></div>
<divclass="line"><aid="l00334"name="l00334"></a><spanclass="lineno"> 334</span><spanclass="comment">// return value of the callback controls how the algorithm proceeds with the</span></div>
<divclass="line"><aid="l00341"name="l00341"></a><spanclass="lineno"> 341</span><spanclass="comment">// Contains the state of the aglorithm. The vector serves as an external stack</span></div>
<divclass="line"><aid="l00342"name="l00342"></a><spanclass="lineno"> 342</span><spanclass="comment">// for the recursive part of the algorithm - instead of using the C++ stack</span></div>
<divclass="line"><aid="l00343"name="l00343"></a><spanclass="lineno"> 343</span><spanclass="comment">// and natural recursion, it is implemented as a loop and new states are added</span></div>
<divclass="line"><aid="l00344"name="l00344"></a><spanclass="lineno"> 344</span><spanclass="comment">// to the top of the stack. The algorithm ends when the stack is empty.</span></div>
<divclass="line"><aid="l00347"name="l00347"></a><spanclass="lineno"> 347</span><spanclass="comment">// A vector that receives the current clique found by the algorithm.</span></div>
<divclass="line"><aid="l00350"name="l00350"></a><spanclass="lineno"> 350</span><spanclass="comment">// Set to true if the algorithm is active (it was not stopped by an the clique</span></div>
<divclass="line"><aid="l00354"name="l00354"></a><spanclass="lineno"> 354</span><spanclass="comment">// The current time limit used by the solver. The time limit is assigned by</span></div>
<divclass="line"><aid="l00355"name="l00355"></a><spanclass="lineno"> 355</span><spanclass="comment">// the Run methods and it can be different for each call to run.</span></div>
<divclass="line"><aid="l00461"name="l00461"></a><spanclass="lineno"> 461</span><spanclass="comment">// Add all candidates from previous_state->candidates that are connected to</span></div>
<divclass="line"><aid="l00462"name="l00462"></a><spanclass="lineno"> 462</span><spanclass="comment">// 'selected' in the graph to the vector 'new_candidates', skipping the node</span></div>
<divclass="line"><aid="l00463"name="l00463"></a><spanclass="lineno"> 463</span><spanclass="comment">// 'selected'; this node is always at the position</span></div>
<divclass="line"><aid="l00464"name="l00464"></a><spanclass="lineno"> 464</span><spanclass="comment">// 'previous_state->first_candidate_index', so we can skip it by skipping the</span></div>
<divclass="line"><aid="l00465"name="l00465"></a><spanclass="lineno"> 465</span><spanclass="comment">// element at this particular index.</span></div>
<divclass="line"><aid="l00484"name="l00484"></a><spanclass="lineno"> 484</span><spanclass="comment">// We've found a clique. Report it to the user, but do not push the state</span></div>
<divclass="line"><aid="l00485"name="l00485"></a><spanclass="lineno"> 485</span><spanclass="comment">// because it would be popped immediately anyway.</span></div>
<divclass="line"><aid="l00489"name="l00489"></a><spanclass="lineno"> 489</span><spanclass="comment">// The number of remaining iterations will be decremented at the end of</span></div>
<divclass="line"><aid="l00490"name="l00490"></a><spanclass="lineno"> 490</span><spanclass="comment">// the loop in RunIterations; setting it to 0 here would make it -1 at</span></div>
<divclass="line"><aid="l00491"name="l00491"></a><spanclass="lineno"> 491</span><spanclass="comment">// the end of the main loop.</span></div>
<divclass="line"><aid="l00499"name="l00499"></a><spanclass="lineno"> 499</span><spanclass="comment">// NOTE(user): The following line may invalidate previous_state (if the</span></div>
<divclass="line"><aid="l00500"name="l00500"></a><spanclass="lineno"> 500</span><spanclass="comment">// vector data was re-allocated in the process). We must avoid using</span></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_1_1math__opt_html_a5de89a1f6e3f80a49a0d76136d8044e2"><divclass="ttname"><ahref="namespaceoperations__research_1_1math__opt.html#a5de89a1f6e3f80a49a0d76136d8044e2">operations_research::math_opt::swap</a></div><divclass="ttdeci">void swap(IdMap< K, V >&a, IdMap< K, V >&b)</div><divclass="ttdef"><b>Definition:</b><ahref="id__map_8h_source.html#l00262">id_map.h:262</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>
<divclass="ttc"id="anamespaceoperations__research_html_a5986867bcb6d1470fd6c27438d289fcd"><divclass="ttname"><ahref="namespaceoperations__research.html#a5986867bcb6d1470fd6c27438d289fcd">operations_research::CoverArcsByCliques</a></div><divclass="ttdeci">void CoverArcsByCliques(std::function< bool(int, int)> graph, int node_count, std::function< bool(const std::vector< int >&)> callback)</div><divclass="ttdef"><b>Definition:</b><ahref="cliques_8cc_source.html#l00240">cliques.cc:240</a></div></div>
