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<a href="minimum__spanning__tree_8h.html">Go to the documentation of this file.</a><div class="fragment"><div class="line"><a name="l00001"></a><span class="lineno"> 1</span> <span class="comment">// Copyright 2010-2021 Google LLC</span></div><div class="line"><a name="l00002"></a><span class="lineno"> 2</span> <span class="comment">// Licensed under the Apache License, Version 2.0 (the "License");</span></div><div class="line"><a name="l00003"></a><span class="lineno"> 3</span> <span class="comment">// you may not use this file except in compliance with the License.</span></div><div class="line"><a name="l00004"></a><span class="lineno"> 4</span> <span class="comment">// You may obtain a copy of the License at</span></div><div class="line"><a name="l00005"></a><span class="lineno"> 5</span> <span class="comment">//</span></div><div class="line"><a name="l00006"></a><span class="lineno"> 6</span> <span class="comment">// http://www.apache.org/licenses/LICENSE-2.0</span></div><div class="line"><a name="l00007"></a><span class="lineno"> 7</span> <span class="comment">//</span></div><div class="line"><a name="l00008"></a><span class="lineno"> 8</span> <span class="comment">// Unless required by applicable law or agreed to in writing, software</span></div><div class="line"><a name="l00009"></a><span class="lineno"> 9</span> <span class="comment">// distributed under the License is distributed on an "AS IS" BASIS,</span></div><div class="line"><a name="l00010"></a><span class="lineno"> 10</span> <span class="comment">// WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.</span></div><div class="line"><a name="l00011"></a><span class="lineno"> 11</span> <span class="comment">// See the License for the specific language governing permissions and</span></div><div class="line"><a name="l00012"></a><span class="lineno"> 12</span> <span class="comment">// limitations under the License.</span></div><div class="line"><a name="l00013"></a><span class="lineno"> 13</span> </div><div class="line"><a name="l00014"></a><span class="lineno"> 14</span> <span class="preprocessor">#ifndef OR_TOOLS_GRAPH_MINIMUM_SPANNING_TREE_H_</span></div><div class="line"><a name="l00015"></a><span class="lineno"> 15</span> <span class="preprocessor">#define OR_TOOLS_GRAPH_MINIMUM_SPANNING_TREE_H_</span></div><div class="line"><a name="l00016"></a><span class="lineno"> 16</span> </div><div class="line"><a name="l00017"></a><span class="lineno"> 17</span> <span class="preprocessor">#include <queue></span></div><div class="line"><a name="l00018"></a><span class="lineno"> 18</span> <span class="preprocessor">#include <vector></span></div><div class="line"><a name="l00019"></a><span class="lineno"> 19</span> </div><div class="line"><a name="l00020"></a><span class="lineno"> 20</span> <span class="preprocessor">#include "ortools/base/adjustable_priority_queue-inl.h"</span></div><div class="line"><a name="l00021"></a><span class="lineno"> 21</span> <span class="preprocessor">#include "ortools/base/adjustable_priority_queue.h"</span></div><div class="line"><a name="l00022"></a><span class="lineno"> 22</span> <span class="preprocessor">#include "ortools/base/integral_types.h"</span></div><div class="line"><a name="l00023"></a><span class="lineno"> 23</span> <span class="preprocessor">#include "<a class="code" href="connected__components_8h.html">ortools/graph/connected_components.h</a>"</span></div><div class="line"><a name="l00024"></a><span class="lineno"> 24</span> <span class="preprocessor">#include "ortools/util/vector_or_function.h"</span></div><div class="line"><a name="l00025"></a><span class="lineno"> 25</span> </div><div class="line"><a name="l00026"></a><span class="lineno"> 26</span> <span class="keyword">namespace </span><a class="code" href="namespaceoperations__research.html">operations_research</a> {</div><div class="line"><a name="l00027"></a><span class="lineno"> 27</span> </div><div class="line"><a name="l00028"></a><span class="lineno"> 28</span> <span class="comment">// Implementation of Kruskal's mininumum spanning tree algorithm (c.f.</span></div><div class="line"><a name="l00029"></a><span class="lineno"> 29</span> <span class="comment">// https://en.wikipedia.org/wiki/Kruskal%27s_algorithm).</span></div><div class="line"><a name="l00030"></a><span class="lineno"> 30</span> <span class="comment">// Returns the index of the arcs appearing in the tree; will return a forest if</span></div><div class="line"><a name="l00031"></a><span class="lineno"> 31</span> <span class="comment">// the graph is disconnected. Nodes without any arcs will be ignored.</span></div><div class="line"><a name="l00032"></a><span class="lineno"> 32</span> <span class="comment">// Each arc of the graph is interpreted as an undirected arc.</span></div><div class="line"><a name="l00033"></a><span class="lineno"> 33</span> <span class="comment">// Complexity of the algorithm is O(E * log(E)) where E is the number of arcs</span></div><div class="line"><a name="l00034"></a><span class="lineno"> 34</span> <span class="comment">// in the graph. Memory usage is O(E * log(E)).</span></div><div class="line"><a name="l00035"></a><span class="lineno"> 35</span> </div><div class="line"><a name="l00036"></a><span class="lineno"> 36</span> <span class="comment">// TODO(user): Add a global Minimum Spanning Tree API automatically switching</span></div><div class="line"><a name="l00037"></a><span class="lineno"> 37</span> <span class="comment">// between Prim and Kruskal depending on problem size.</span></div><div class="line"><a name="l00038"></a><span class="lineno"> 38</span> </div><div class="line"><a name="l00039"></a><span class="lineno"> 39</span> <span class="comment">// Version taking sorted graph arcs. Allows somewhat incremental recomputation</span></div><div class="line"><a name="l00040"></a><span class="lineno"> 40</span> <span class="comment">// of minimum spanning trees as most of the processing time is spent sorting</span></div><div class="line"><a name="l00041"></a><span class="lineno"> 41</span> <span class="comment">// arcs.</span></div><div class="line"><a name="l00042"></a><span class="lineno"> 42</span> <span class="comment">// Usage:</span></div><div class="line"><a name="l00043"></a><span class="lineno"> 43</span> <span class="comment">// ListGraph<int, int> graph(...);</span></div><div class="line"><a name="l00044"></a><span class="lineno"> 44</span> <span class="comment">// std::vector<int> sorted_arcs = ...;</span></div><div class="line"><a name="l00045"></a><span class="lineno"> 45</span> <span class="comment">// std::vector<int> mst = BuildKruskalMinimumSpanningTreeFromSortedArcs(</span></div><div class="line"><a name="l00046"></a><span class="lineno"> 46</span> <span class="comment">// graph, sorted_arcs);</span></div><div class="line"><a name="l00047"></a><span class="lineno"> 47</span> <span class="comment">//</span></div><div class="line"><a name="l00048"></a><span class="lineno"> 48</span> <span class="keyword">template</span> <<span class="keyword">typename</span> Graph></div><div class="line"><a name="l00049"></a><span class="lineno"> 49</span> std::vector<typename Graph::ArcIndex></div><div class="line"><a name="l00050"></a><span class="lineno"><a class="line" href="namespaceoperations__research.html#a00ab79ee21ffd8dece0996e37f9faa7a"> 50</a></span> <a class="code" href="namespaceoperations__research.html#a00ab79ee21ffd8dece0996e37f9faa7a">BuildKruskalMinimumSpanningTreeFromSortedArcs</a>(</div><div class="line"><a name="l00051"></a><span class="lineno"> 51</span>  <span class="keyword">const</span> <a class="code" href="namespaceutil.html#a2f76166dbe0c4055a1f256235ad00478">Graph</a>& graph,</div><div class="line"><a name="l00052"></a><span class="lineno"> 52</span>  <span class="keyword">const</span> std::vector<typename Graph::ArcIndex>& sorted_arcs) {</div><div class="line"><a name="l00053"></a><span class="lineno"> 53</span>  <span class="keyword">using</span> <a class="code" href="namespaceoperations__research.html#aecf320bda6e95d1edaf3a546673e3e6b">ArcIndex</a> = <span class="keyword">typename</span> <a class="code" href="namespaceoperations__research.html#aecf320bda6e95d1edaf3a546673e3e6b">Graph::ArcIndex</a>;</div><div class="line"><a name="l00054"></a><span class="lineno"> 54</span>  <span class="keyword">using</span> <a class="code" href="namespaceoperations__research.html#a7ffcae041a5d746371436101400cdb6b">NodeIndex</a> = <span class="keyword">typename</span> <a class="code" href="namespaceoperations__research.html#a7ffcae041a5d746371436101400cdb6b">Graph::NodeIndex</a>;</div><div class="line"><a name="l00055"></a><span class="lineno"> 55</span>  <span class="keyword">const</span> <span class="keywordtype">int</span> num_arcs = graph.num_arcs();</div><div class="line"><a name="l00056"></a><span class="lineno"> 56</span>  <span class="keywordtype">int</span> arc_index = 0;</div><div class="line"><a name="l00057"></a><span class="lineno"> 57</span>  std::vector<ArcIndex> tree_arcs;</div><div class="line"><a name="l00058"></a><span class="lineno"> 58</span>  <span class="keywordflow">if</span> (graph.num_nodes() == 0) {</div><div class="line"><a name="l00059"></a><span class="lineno"> 59</span>  <span class="keywordflow">return</span> tree_arcs;</div><div class="line"><a name="l00060"></a><span class="lineno"> 60</span>  }</div><div class="line"><a name="l00061"></a><span class="lineno"> 61</span>  <span class="keyword">const</span> <span class="keywordtype">int</span> expected_tree_size = graph.num_nodes() - 1;</div><div class="line"><a name="l00062"></a><span class="lineno"> 62</span>  tree_arcs.reserve(expected_tree_size);</div><div class="line"><a name="l00063"></a><span class="lineno"> 63</span>  <a class="code" href="class_dense_connected_components_finder.html">DenseConnectedComponentsFinder</a> components;</div><div class="line"><a name="l00064"></a><span class="lineno"> 64</span>  components.<a class="code" href="class_dense_connected_components_finder.html#ad8e718920ab9683d39af650d714cffe1">SetNumberOfNodes</a>(graph.num_nodes());</div><div class="line"><a name="l00065"></a><span class="lineno"> 65</span>  <span class="keywordflow">while</span> (tree_arcs.size() != expected_tree_size && arc_index < num_arcs) {</div><div class="line"><a name="l00066"></a><span class="lineno"> 66</span>  <span class="keyword">const</span> <a class="code" href="namespaceoperations__research.html#aecf320bda6e95d1edaf3a546673e3e6b">ArcIndex</a> arc = sorted_arcs[arc_index];</div><div class="line"><a name="l00067"></a><span class="lineno"> 67</span>  <span class="keyword">const</span> <span class="keyword">auto</span> tail = graph.Tail(arc);</div><div class="line"><a name="l00068"></a><span class="lineno"> 68</span>  <span class="keyword">const</span> <span class="keyword">auto</span> head = graph.Head(arc);</div><div class="line"><a name="l00069"></a><span class="lineno"> 69</span>  <span class="keywordflow">if</span> (!components.<a class="code" href="class_dense_connected_components_finder.html#a962b54327591b21cc0a9273f78906a8b">Connected</a>(tail, head)) {</div><div class="line"><a name="l00070"></a><span class="lineno"> 70</span>  components.<a class="code" href="class_dense_connected_components_finder.html#a428ab6b7c944afe33bd86a6a1ae7e668">AddEdge</a>(tail, head);</div><div class="line"><a name="l00071"></a><span class="lineno"> 71</span>  tree_arcs.push_back(arc);</div><div class="line"><a name="l00072"></a><span class="lineno"> 72</span>  }</div><div class="line"><a name="l00073"></a><span class="lineno"> 73</span>  ++arc_index;</div><div class="line"><a name="l00074"></a><span class="lineno"> 74</span>  }</div><div class="line"><a name="l00075"></a><span class="lineno"> 75</span>  <span class="keywordflow">return</span> tree_arcs;</div><div class="line"><a name="l00076"></a><span class="lineno"> 76</span> }</div><div class="line"><a name="l00077"></a><span class="lineno"> 77</span> </div><div class="line"><a name="l00078"></a><span class="lineno"> 78</span> <span class="comment">// Version taking an arc comparator to sort graph arcs.</span></div><div class="line"><a name="l00079"></a><span class="lineno"> 79</span> <span class="comment">// Usage:</span></div><div class="line"><a name="l00080"></a><span class="lineno"> 80</span> <span class="comment">// ListGraph<int, int> graph(...);</span></div><div class="line"><a name="l00081"></a><span class="lineno"> 81</span> <span class="comment">// const auto arc_cost = [&graph](int arc) {</span></div><div class="line"><a name="l00082"></a><span class="lineno"> 82</span> <span class="comment">// return f(graph.Tail(arc), graph.Head(arc));</span></div><div class="line"><a name="l00083"></a><span class="lineno"> 83</span> <span class="comment">// };</span></div><div class="line"><a name="l00084"></a><span class="lineno"> 84</span> <span class="comment">// std::vector<int> mst = BuildKruskalMinimumSpanningTree(</span></div><div class="line"><a name="l00085"></a><span class="lineno"> 85</span> <span class="comment">// graph,</span></div><div class="line"><a name="l00086"></a><span class="lineno"> 86</span> <span class="comment">// [&arc_cost](int a, int b) { return arc_cost(a) < arc_cost(b); });</span></div><div class="line"><a name="l00087"></a><span class="lineno"> 87</span> <span class="comment">//</span></div><div class="line"><a name="l00088"></a><span class="lineno"> 88</span> <span class="keyword">template</span> <<span class="keyword">typename</span> Graph, <span class="keyword">typename</span> ArcComparator></div><div class="line"><a name="l00089"></a><span class="lineno"><a class="line" href="namespaceoperations__research.html#aa565a47a059ef32ef1aec39768e4ec98"> 89</a></span> std::vector<typename Graph::ArcIndex> <a class="code" href="namespaceoperations__research.html#aa565a47a059ef32ef1aec39768e4ec98">BuildKruskalMinimumSpanningTree</a>(</div><div class="line"><a name="l00090"></a><span class="lineno"> 90</span>  <span class="keyword">const</span> <a class="code" href="namespaceutil.html#a2f76166dbe0c4055a1f256235ad00478">Graph</a>& graph, <span class="keyword">const</span> ArcComparator& arc_comparator) {</div><div class="line"><a name="l00091"></a><span class="lineno"> 91</span>  <span class="keyword">using</span> <a class="code" href="namespaceoperations__research.html#aecf320bda6e95d1edaf3a546673e3e6b">ArcIndex</a> = <span class="keyword">typename</span> <a class="code" href="namespaceoperations__research.html#aecf320bda6e95d1edaf3a546673e3e6b">Graph::ArcIndex</a>;</div><div class="line"><a name="l00092"></a><span class="lineno"> 92</span>  std::vector<ArcIndex> sorted_arcs(graph.num_arcs());</div><div class="line"><a name="l00093"></a><span class="lineno"> 93</span>  <span class="keywordflow">for</span> (<span class="keyword">const</span> <a class="code" href="namespaceoperations__research.html#aecf320bda6e95d1edaf3a546673e3e6b">ArcIndex</a> arc : graph.AllForwardArcs()) {</div><div class="line"><a name="l00094"></a><span class="lineno"> 94</span>  sorted_arcs[arc] = arc;</div><div class="line"><a name="l00095"></a><span class="lineno"> 95</span>  }</div><div class="line"><a name="l00096"></a><span class="lineno"> 96</span>  std::sort(sorted_arcs.begin(), sorted_arcs.end(), arc_comparator);</div><div class="line"><a name="l00097"></a><span class="lineno"> 97</span>  <span class="keywordflow">return</span> <a class="code" href="namespaceoperations__research.html#a00ab79ee21ffd8dece0996e37f9faa7a">BuildKruskalMinimumSpanningTreeFromSortedArcs</a>(graph, sorted_arcs);</div><div class="line"><a name="l00098"></a><span class="lineno"> 98</span> }</div><div class="line"><a name="l00099"></a><span class="lineno"> 99</span> </div><div class="line"><a name="l00100"></a><span class="lineno"> 100</span> <span class="comment">// Implementation of Prim's mininumum spanning tree algorithm (c.f.</span></div><div class="line"><a name="l00101"></a><span class="lineno"> 101</span> <span class="comment">// https://en.wikipedia.org/wiki/Prim's_algorithm) on undirected connected</span></div><div class="line"><a name="l00102"></a><span class="lineno"> 102</span> <span class="comment">// graphs.</span></div><div class="line"><a name="l00103"></a><span class="lineno"> 103</span> <span class="comment">// Returns the index of the arcs appearing in the tree.</span></div><div class="line"><a name="l00104"></a><span class="lineno"> 104</span> <span class="comment">// Complexity of the algorithm is O(E * log(V)) where E is the number of arcs</span></div><div class="line"><a name="l00105"></a><span class="lineno"> 105</span> <span class="comment">// in the graph, V is the number of vertices. Memory usage is O(V) + memory</span></div><div class="line"><a name="l00106"></a><span class="lineno"> 106</span> <span class="comment">// taken by the graph.</span></div><div class="line"><a name="l00107"></a><span class="lineno"> 107</span> <span class="comment">// Usage:</span></div><div class="line"><a name="l00108"></a><span class="lineno"> 108</span> <span class="comment">// ListGraph<int, int> graph(...);</span></div><div class="line"><a name="l00109"></a><span class="lineno"> 109</span> <span class="comment">// const auto arc_cost = [&graph](int arc) -> int64_t {</span></div><div class="line"><a name="l00110"></a><span class="lineno"> 110</span> <span class="comment">// return f(graph.Tail(arc), graph.Head(arc));</span></div><div class="line"><a name="l00111"></a><span class="lineno"> 111</span> <span class="comment">// };</span></div><div class="line"><a name="l00112"></a><span class="lineno"> 112</span> <span class="comment">// std::vector<int> mst = BuildPrimMinimumSpanningTree(graph, arc_cost);</span></div><div class="line"><a name="l00113"></a><span class="lineno"> 113</span> <span class="comment">//</span></div><div class="line"><a name="l00114"></a><span class="lineno"> 114</span> <span class="keyword">template</span> <<span class="keyword">typename</span> Graph, <span class="keyword">typename</span> ArcValue></div><div class="line"><a name="l00115"></a><span class="lineno"><a class="line" href="namespaceoperations__research.html#a33a2f4c26fd60cd0fa98257b571c974f"> 115</a></span> std::vector<typename Graph::ArcIndex> <a class="code" href="namespaceoperations__research.html#a33a2f4c26fd60cd0fa98257b571c974f">BuildPrimMinimumSpanningTree</a>(</div><div class="line"><a name="l00116"></a><span class="lineno"> 116</span>  <span class="keyword">const</span> <a class="code" href="namespaceutil.html#a2f76166dbe0c4055a1f256235ad00478">Graph</a>& graph, <span class="keyword">const</span> ArcValue& arc_value) {</div><div class="line"><a name="l00117"></a><span class="lineno"> 117</span>  <span class="keyword">using</span> <a class="code" href="namespaceoperations__research.html#aecf320bda6e95d1edaf3a546673e3e6b">ArcIndex</a> = <span class="keyword">typename</span> <a class="code" href="namespaceoperations__research.html#aecf320bda6e95d1edaf3a546673e3e6b">Graph::ArcIndex</a>;</div><div class="line"><a name="l00118"></a><span class="lineno"> 118</span>  <span class="keyword">using</span> <a class="code" href="namespaceoperations__research.html#a7ffcae041a5d746371436101400cdb6b">NodeIndex</a> = <span class="keyword">typename</span> <a class="code" href="namespaceoperations__research.html#a7ffcae041a5d746371436101400cdb6b">Graph::NodeIndex</a>;</div><div class="line"><a name="l00119"></a><span class="lineno"> 119</span>  <span class="keyword">using</span> ArcValueType = decltype(arc_value(0));</div><div class="line"><a name="l00120"></a><span class="lineno"> 120</span>  std::vector<ArcIndex> tree_arcs;</div><div class="line"><a name="l00121"></a><span class="lineno"> 121</span>  <span class="keywordflow">if</span> (graph.num_nodes() == 0) {</div><div class="line"><a name="l00122"></a><span class="lineno"> 122</span>  <span class="keywordflow">return</span> tree_arcs;</div><div class="line"><a name="l00123"></a><span class="lineno"> 123</span>  }</div><div class="line"><a name="l00124"></a><span class="lineno"> 124</span>  <span class="keyword">const</span> <span class="keywordtype">int</span> expected_tree_size = graph.num_nodes() - 1;</div><div class="line"><a name="l00125"></a><span class="lineno"> 125</span>  tree_arcs.reserve(expected_tree_size);</div><div class="line"><a name="l00126"></a><span class="lineno"> 126</span>  std::vector<ArcIndex> node_neighbor(graph.num_nodes(), Graph::kNilArc);</div><div class="line"><a name="l00127"></a><span class="lineno"> 127</span>  std::vector<bool> node_active(graph.num_nodes(), <span class="keyword">true</span>);</div><div class="line"><a name="l00128"></a><span class="lineno"> 128</span> </div><div class="line"><a name="l00129"></a><span class="lineno"> 129</span>  <span class="comment">// This struct represents entries in the adjustable priority queue which</span></div><div class="line"><a name="l00130"></a><span class="lineno"> 130</span>  <span class="comment">// maintains active nodes (not added to the tree yet) in decreasing insertion</span></div><div class="line"><a name="l00131"></a><span class="lineno"> 131</span>  <span class="comment">// cost order. AdjustablePriorityQueue requires the existence of the</span></div><div class="line"><a name="l00132"></a><span class="lineno"> 132</span>  <span class="comment">// SetHeapIndex and GetHeapIndex methods.</span></div><div class="line"><a name="l00133"></a><span class="lineno"> 133</span>  <span class="keyword">struct </span>Entry {</div><div class="line"><a name="l00134"></a><span class="lineno"> 134</span>  <span class="keywordtype">void</span> SetHeapIndex(<span class="keywordtype">int</span> index) { heap_index = index; }</div><div class="line"><a name="l00135"></a><span class="lineno"> 135</span>  <span class="keywordtype">int</span> GetHeapIndex()<span class="keyword"> const </span>{ <span class="keywordflow">return</span> heap_index; }</div><div class="line"><a name="l00136"></a><span class="lineno"> 136</span>  <span class="keywordtype">bool</span> operator<(<span class="keyword">const</span> Entry& other)<span class="keyword"> const </span>{ <span class="keywordflow">return</span> value > other.value; }</div><div class="line"><a name="l00137"></a><span class="lineno"> 137</span> </div><div class="line"><a name="l00138"></a><span class="lineno"> 138</span>  <a class="code" href="namespaceoperations__research.html#a7ffcae041a5d746371436101400cdb6b">NodeIndex</a> node;</div><div class="line"><a name="l00139"></a><span class="lineno"> 139</span>  ArcValueType value;</div><div class="line"><a name="l00140"></a><span class="lineno"> 140</span>  <span class="keywordtype">int</span> heap_index;</div><div class="line"><a name="l00141"></a><span class="lineno"> 141</span>  };</div><div class="line"><a name="l00142"></a><span class="lineno"> 142</span> </div><div class="line"><a name="l00143"></a><span class="lineno"> 143</span>  AdjustablePriorityQueue<Entry> pq;</div><div class="line"><a name="l00144"></a><span class="lineno"> 144</span>  std::vector<Entry> entries;</div><div class="line"><a name="l00145"></a><span class="lineno"> 145</span>  std::vector<bool> touched_entry(graph.num_nodes(), <span class="keyword">false</span>);</div><div class="line"><a name="l00146"></a><span class="lineno"> 146</span>  <span class="keywordflow">for</span> (<a class="code" href="namespaceoperations__research.html#a7ffcae041a5d746371436101400cdb6b">NodeIndex</a> node : graph.AllNodes()) {</div><div class="line"><a name="l00147"></a><span class="lineno"> 147</span>  entries.push_back({node, std::numeric_limits<ArcValueType>::max(), -1});</div><div class="line"><a name="l00148"></a><span class="lineno"> 148</span>  }</div><div class="line"><a name="l00149"></a><span class="lineno"> 149</span>  entries[0].value = 0;</div><div class="line"><a name="l00150"></a><span class="lineno"> 150</span>  pq.Add(&entries[0]);</div><div class="line"><a name="l00151"></a><span class="lineno"> 151</span>  <span class="keywordflow">while</span> (!pq.IsEmpty() && tree_arcs.size() != expected_tree_size) {</div><div class="line"><a name="l00152"></a><span class="lineno"> 152</span>  <span class="keyword">const</span> Entry* best = pq.Top();</div><div class="line"><a name="l00153"></a><span class="lineno"> 153</span>  <span class="keyword">const</span> <a class="code" href="namespaceoperations__research.html#a7ffcae041a5d746371436101400cdb6b">NodeIndex</a> node = best->node;</div><div class="line"><a name="l00154"></a><span class="lineno"> 154</span>  pq.Pop();</div><div class="line"><a name="l00155"></a><span class="lineno"> 155</span>  node_active[node] = <span class="keyword">false</span>;</div><div class="line"><a name="l00156"></a><span class="lineno"> 156</span>  <span class="keywordflow">if</span> (node_neighbor[node] != Graph::kNilArc) {</div><div class="line"><a name="l00157"></a><span class="lineno"> 157</span>  tree_arcs.push_back(node_neighbor[node]);</div><div class="line"><a name="l00158"></a><span class="lineno"> 158</span>  }</div><div class="line"><a name="l00159"></a><span class="lineno"> 159</span>  <span class="keywordflow">for</span> (<span class="keyword">const</span> <a class="code" href="namespaceoperations__research.html#aecf320bda6e95d1edaf3a546673e3e6b">ArcIndex</a> arc : graph.OutgoingArcs(node)) {</div><div class="line"><a name="l00160"></a><span class="lineno"> 160</span>  <span class="keyword">const</span> <a class="code" href="namespaceoperations__research.html#a7ffcae041a5d746371436101400cdb6b">NodeIndex</a> neighbor = graph.Head(arc);</div><div class="line"><a name="l00161"></a><span class="lineno"> 161</span>  <span class="keywordflow">if</span> (node_active[neighbor]) {</div><div class="line"><a name="l00162"></a><span class="lineno"> 162</span>  <span class="keyword">const</span> ArcValueType value = arc_value(arc);</div><div class="line"><a name="l00163"></a><span class="lineno"> 163</span>  Entry& entry = entries[neighbor];</div><div class="line"><a name="l00164"></a><span class="lineno"> 164</span>  <span class="keywordflow">if</span> (value < entry.value || !touched_entry[neighbor]) {</div><div class="line"><a name="l00165"></a><span class="lineno"> 165</span>  node_neighbor[neighbor] = arc;</div><div class="line"><a name="l00166"></a><span class="lineno"> 166</span>  entry.value = value;</div><div class="line"><a name="l00167"></a><span class="lineno"> 167</span>  touched_entry[neighbor] = <span class="keyword">true</span>;</div><div class="line"><a name="l00168"></a><span class="lineno"> 168</span>  <span class="keywordflow">if</span> (pq.Contains(&entry)) {</div><div class="line"><a name="l00169"></a><span class="lineno"> 169</span>  pq.NoteChangedPriority(&entry);</div><div class="line"><a name="l00170"></a><span class="lineno"> 170</span>  } <span class="keywordflow">else</span> {</div><div class="line"><a name="l00171"></a><span class="lineno"> 171</span>  pq.Add(&entry);</div><div class="line"><a name="l00172"></a><span class="lineno"> 172</span>  }</div><div class="line"><a name="l00173"></a><span class="lineno"> 173</span>  }</div><div class="line"><a name="l00174"></a><span class="lineno"> 174</span>  }</div><div class="line"><a name="l00175"></a><span class="lineno"> 175</span>  }</div><div class="line"><a name="l00176"></a><span class="lineno"> 176</span>  }</div><div class="line"><a name="l00177"></a><span class="lineno"> 177</span>  <span class="keywordflow">return</span> tree_arcs;</div><div class="line"><a name="l00178"></a><span class="lineno"> 178</span> }</div><div class="line"><a name="l00179"></a><span class="lineno"> 179</span> </div><div class="line"><a name="l00180"></a><span class="lineno"> 180</span> } <span class="comment">// namespace operations_research</span></div><div class="line"><a name="l00181"></a><span class="lineno"> 181</span> <span class="preprocessor">#endif // OR_TOOLS_GRAPH_MINIMUM_SPANNING_TREE_H_</span></div><div class="ttc" id="class_dense_connected_components_finder_html"><div class="ttname"><a href="class_dense_connected_components_finder.html">DenseConnectedComponentsFinder</a></div><div class="ttdef"><b>Definition:</b> <a href="connected__components_8h_source.html#l00081">connected_components.h:81</a></div></div>
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<div class="ttc" id="namespaceoperations__research_html_a7ffcae041a5d746371436101400cdb6b"><div class="ttname"><a href="namespaceoperations__research.html#a7ffcae041a5d746371436101400cdb6b">operations_research::NodeIndex</a></div><div class="ttdeci">int32_t NodeIndex</div><div class="ttdef"><b>Definition:</b> <a href="ebert__graph_8h_source.html#l00193">ebert_graph.h:193</a></div></div>
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<div class="ttc" id="namespaceoperations__research_html_aa565a47a059ef32ef1aec39768e4ec98"><div class="ttname"><a href="namespaceoperations__research.html#aa565a47a059ef32ef1aec39768e4ec98">operations_research::BuildKruskalMinimumSpanningTree</a></div><div class="ttdeci">std::vector< typename Graph::ArcIndex > BuildKruskalMinimumSpanningTree(const Graph &graph, const ArcComparator &arc_comparator)</div><div class="ttdef"><b>Definition:</b> <a href="minimum__spanning__tree_8h_source.html#l00089">minimum_spanning_tree.h:89</a></div></div>
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<div class="ttc" id="namespaceoperations__research_html_a00ab79ee21ffd8dece0996e37f9faa7a"><div class="ttname"><a href="namespaceoperations__research.html#a00ab79ee21ffd8dece0996e37f9faa7a">operations_research::BuildKruskalMinimumSpanningTreeFromSortedArcs</a></div><div class="ttdeci">std::vector< typename Graph::ArcIndex > BuildKruskalMinimumSpanningTreeFromSortedArcs(const Graph &graph, const std::vector< typename Graph::ArcIndex > &sorted_arcs)</div><div class="ttdef"><b>Definition:</b> <a href="minimum__spanning__tree_8h_source.html#l00050">minimum_spanning_tree.h:50</a></div></div>
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<div class="ttc" id="class_dense_connected_components_finder_html_a962b54327591b21cc0a9273f78906a8b"><div class="ttname"><a href="class_dense_connected_components_finder.html#a962b54327591b21cc0a9273f78906a8b">DenseConnectedComponentsFinder::Connected</a></div><div class="ttdeci">bool Connected(int node1, int node2)</div></div>
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<div class="ttc" id="connected__components_8h_html"><div class="ttname"><a href="connected__components_8h.html">connected_components.h</a></div></div>
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<div class="ttc" id="namespaceoperations__research_html_aecf320bda6e95d1edaf3a546673e3e6b"><div class="ttname"><a href="namespaceoperations__research.html#aecf320bda6e95d1edaf3a546673e3e6b">operations_research::ArcIndex</a></div><div class="ttdeci">int32_t ArcIndex</div><div class="ttdef"><b>Definition:</b> <a href="ebert__graph_8h_source.html#l00202">ebert_graph.h:202</a></div></div>
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<div class="ttc" id="class_dense_connected_components_finder_html_ad8e718920ab9683d39af650d714cffe1"><div class="ttname"><a href="class_dense_connected_components_finder.html#ad8e718920ab9683d39af650d714cffe1">DenseConnectedComponentsFinder::SetNumberOfNodes</a></div><div class="ttdeci">void SetNumberOfNodes(int num_nodes)</div></div>
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<div class="ttc" id="namespaceutil_html_a2f76166dbe0c4055a1f256235ad00478"><div class="ttname"><a href="namespaceutil.html#a2f76166dbe0c4055a1f256235ad00478">util::Graph</a></div><div class="ttdeci">ListGraph Graph</div><div class="ttdef"><b>Definition:</b> <a href="graph_8h_source.html#l02361">graph.h:2361</a></div></div>
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<div class="ttc" id="namespaceoperations__research_html"><div class="ttname"><a href="namespaceoperations__research.html">operations_research</a></div><div class="ttdef"><b>Definition:</b> <a href="christofides_8h_source.html#l00043">christofides.h:43</a></div></div>
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<div class="ttc" id="class_dense_connected_components_finder_html_a428ab6b7c944afe33bd86a6a1ae7e668"><div class="ttname"><a href="class_dense_connected_components_finder.html#a428ab6b7c944afe33bd86a6a1ae7e668">DenseConnectedComponentsFinder::AddEdge</a></div><div class="ttdeci">bool AddEdge(int node1, int node2)</div></div>
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<div class="ttc" id="namespaceoperations__research_html_a33a2f4c26fd60cd0fa98257b571c974f"><div class="ttname"><a href="namespaceoperations__research.html#a33a2f4c26fd60cd0fa98257b571c974f">operations_research::BuildPrimMinimumSpanningTree</a></div><div class="ttdeci">std::vector< typename Graph::ArcIndex > BuildPrimMinimumSpanningTree(const Graph &graph, const ArcValue &arc_value)</div><div class="ttdef"><b>Definition:</b> <a href="minimum__spanning__tree_8h_source.html#l00115">minimum_spanning_tree.h:115</a></div></div>
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