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<div class="headertitle"><div class="title">minimum_spanning_tree.h</div></div>
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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 id="l00001" name="l00001"></a><span class="lineno"> 1</span><span class="comment">// Copyright 2010-2021 Google LLC</span></div>
<div class="line"><a id="l00002" name="l00002"></a><span class="lineno"> 2</span><span class="comment">// Licensed under the Apache License, Version 2.0 (the &quot;License&quot;);</span></div>
<div class="line"><a id="l00003" 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 id="l00004" 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 id="l00005" name="l00005"></a><span class="lineno"> 5</span><span class="comment">//</span></div>
<div class="line"><a id="l00006" 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 id="l00007" name="l00007"></a><span class="lineno"> 7</span><span class="comment">//</span></div>
<div class="line"><a id="l00008" 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 id="l00009" name="l00009"></a><span class="lineno"> 9</span><span class="comment">// distributed under the License is distributed on an &quot;AS IS&quot; BASIS,</span></div>
<div class="line"><a id="l00010" 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 id="l00011" 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 id="l00012" name="l00012"></a><span class="lineno"> 12</span><span class="comment">// limitations under the License.</span></div>
<div class="line"><a id="l00013" name="l00013"></a><span class="lineno"> 13</span> </div>
<div class="line"><a id="l00014" 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 id="l00015" 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 id="l00016" name="l00016"></a><span class="lineno"> 16</span> </div>
<div class="line"><a id="l00017" name="l00017"></a><span class="lineno"> 17</span><span class="preprocessor">#include &lt;queue&gt;</span></div>
<div class="line"><a id="l00018" name="l00018"></a><span class="lineno"> 18</span><span class="preprocessor">#include &lt;vector&gt;</span></div>
<div class="line"><a id="l00019" name="l00019"></a><span class="lineno"> 19</span> </div>
<div class="line"><a id="l00020" name="l00020"></a><span class="lineno"> 20</span><span class="preprocessor">#include &quot;<a class="code" href="adjustable__priority__queue-inl_8h.html">ortools/base/adjustable_priority_queue-inl.h</a>&quot;</span></div>
<div class="line"><a id="l00021" name="l00021"></a><span class="lineno"> 21</span><span class="preprocessor">#include &quot;<a class="code" href="adjustable__priority__queue_8h.html">ortools/base/adjustable_priority_queue.h</a>&quot;</span></div>
<div class="line"><a id="l00022" name="l00022"></a><span class="lineno"> 22</span><span class="preprocessor">#include &quot;<a class="code" href="integral__types_8h.html">ortools/base/integral_types.h</a>&quot;</span></div>
<div class="line"><a id="l00023" name="l00023"></a><span class="lineno"> 23</span><span class="preprocessor">#include &quot;<a class="code" href="connected__components_8h.html">ortools/graph/connected_components.h</a>&quot;</span></div>
<div class="line"><a id="l00024" name="l00024"></a><span class="lineno"> 24</span><span class="preprocessor">#include &quot;<a class="code" href="vector__or__function_8h.html">ortools/util/vector_or_function.h</a>&quot;</span></div>
<div class="line"><a id="l00025" name="l00025"></a><span class="lineno"> 25</span> </div>
<div class="line"><a id="l00026" name="l00026"></a><span class="lineno"> 26</span><span class="keyword">namespace </span><a class="code hl_namespace" href="namespaceoperations__research.html">operations_research</a> {</div>
<div class="line"><a id="l00027" name="l00027"></a><span class="lineno"> 27</span> </div>
<div class="line"><a id="l00028" name="l00028"></a><span class="lineno"> 28</span><span class="comment">// Implementation of Kruskal&#39;s mininumum spanning tree algorithm (c.f.</span></div>
<div class="line"><a id="l00029" 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 id="l00030" 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 id="l00031" 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 id="l00032" 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 id="l00033" 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 id="l00034" 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 id="l00035" name="l00035"></a><span class="lineno"> 35</span> </div>
<div class="line"><a id="l00036" 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 id="l00037" 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 id="l00038" name="l00038"></a><span class="lineno"> 38</span> </div>
<div class="line"><a id="l00039" 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 id="l00040" 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 id="l00041" name="l00041"></a><span class="lineno"> 41</span><span class="comment">// arcs.</span></div>
<div class="line"><a id="l00042" name="l00042"></a><span class="lineno"> 42</span><span class="comment">// Usage:</span></div>
<div class="line"><a id="l00043" name="l00043"></a><span class="lineno"> 43</span><span class="comment">// ListGraph&lt;int, int&gt; graph(...);</span></div>
<div class="line"><a id="l00044" name="l00044"></a><span class="lineno"> 44</span><span class="comment">// std::vector&lt;int&gt; sorted_arcs = ...;</span></div>
<div class="line"><a id="l00045" name="l00045"></a><span class="lineno"> 45</span><span class="comment">// std::vector&lt;int&gt; mst = BuildKruskalMinimumSpanningTreeFromSortedArcs(</span></div>
<div class="line"><a id="l00046" name="l00046"></a><span class="lineno"> 46</span><span class="comment">// graph, sorted_arcs);</span></div>
<div class="line"><a id="l00047" name="l00047"></a><span class="lineno"> 47</span><span class="comment">//</span></div>
<div class="line"><a id="l00048" name="l00048"></a><span class="lineno"> 48</span><span class="keyword">template</span> &lt;<span class="keyword">typename</span> Graph&gt;</div>
<div class="line"><a id="l00049" name="l00049"></a><span class="lineno"> 49</span>std::vector&lt;typename Graph::ArcIndex&gt;</div>
<div class="line"><a id="l00050" name="l00050"></a><span class="lineno"><a class="line" href="namespaceoperations__research.html#aefd088882d7ba8d27157eba391b02792"> 50</a></span><a class="code hl_function" href="namespaceoperations__research.html#aefd088882d7ba8d27157eba391b02792">BuildKruskalMinimumSpanningTreeFromSortedArcs</a>(</div>
<div class="line"><a id="l00051" name="l00051"></a><span class="lineno"> 51</span> <span class="keyword">const</span> <a class="code hl_typedef" href="namespaceutil.html#a2f76166dbe0c4055a1f256235ad00478">Graph</a>&amp; graph,</div>
<div class="line"><a id="l00052" name="l00052"></a><span class="lineno"> 52</span> <span class="keyword">const</span> std::vector&lt;typename Graph::ArcIndex&gt;&amp; sorted_arcs) {</div>
<div class="line"><a id="l00053" name="l00053"></a><span class="lineno"> 53</span> <span class="keyword">using</span> <a class="code hl_typedef" href="namespaceoperations__research.html#aecf320bda6e95d1edaf3a546673e3e6b">ArcIndex</a> = <span class="keyword">typename</span> <a class="code hl_typedef" href="namespaceoperations__research.html#aecf320bda6e95d1edaf3a546673e3e6b">Graph::ArcIndex</a>;</div>
<div class="line"><a id="l00054" name="l00054"></a><span class="lineno"> 54</span> <span class="keyword">using</span> <a class="code hl_typedef" href="namespaceoperations__research.html#a7ae31ba4c3b4899478e53ca13df35dfc">NodeIndex</a> = <span class="keyword">typename</span> <a class="code hl_typedef" href="namespaceoperations__research.html#a7ae31ba4c3b4899478e53ca13df35dfc">Graph::NodeIndex</a>;</div>
<div class="line"><a id="l00055" 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 id="l00056" name="l00056"></a><span class="lineno"> 56</span> <span class="keywordtype">int</span> arc_index = 0;</div>
<div class="line"><a id="l00057" name="l00057"></a><span class="lineno"> 57</span> std::vector&lt;ArcIndex&gt; tree_arcs;</div>
<div class="line"><a id="l00058" name="l00058"></a><span class="lineno"> 58</span> <span class="keywordflow">if</span> (graph.num_nodes() == 0) {</div>
<div class="line"><a id="l00059" name="l00059"></a><span class="lineno"> 59</span> <span class="keywordflow">return</span> tree_arcs;</div>
<div class="line"><a id="l00060" name="l00060"></a><span class="lineno"> 60</span> }</div>
<div class="line"><a id="l00061" 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 id="l00062" name="l00062"></a><span class="lineno"> 62</span> tree_arcs.reserve(expected_tree_size);</div>
<div class="line"><a id="l00063" name="l00063"></a><span class="lineno"> 63</span> <a class="code hl_class" href="class_dense_connected_components_finder.html">DenseConnectedComponentsFinder</a> components;</div>
<div class="line"><a id="l00064" name="l00064"></a><span class="lineno"> 64</span> components.<a class="code hl_function" href="class_dense_connected_components_finder.html#ad8e718920ab9683d39af650d714cffe1">SetNumberOfNodes</a>(graph.num_nodes());</div>
<div class="line"><a id="l00065" name="l00065"></a><span class="lineno"> 65</span> <span class="keywordflow">while</span> (tree_arcs.size() != expected_tree_size &amp;&amp; arc_index &lt; num_arcs) {</div>
<div class="line"><a id="l00066" name="l00066"></a><span class="lineno"> 66</span> <span class="keyword">const</span> <a class="code hl_typedef" href="namespaceoperations__research.html#aecf320bda6e95d1edaf3a546673e3e6b">ArcIndex</a> <a class="code hl_variable" href="local__search_8cc.html#a30b5726116ac85ea64777e5d6e333894">arc</a> = sorted_arcs[arc_index];</div>
<div class="line"><a id="l00067" name="l00067"></a><span class="lineno"> 67</span> <span class="keyword">const</span> <span class="keyword">auto</span> <a class="code hl_variable" href="routing__flow_8cc.html#a64e7efc5529154ba56903e75f5300990">tail</a> = graph.Tail(<a class="code hl_variable" href="local__search_8cc.html#a30b5726116ac85ea64777e5d6e333894">arc</a>);</div>
<div class="line"><a id="l00068" name="l00068"></a><span class="lineno"> 68</span> <span class="keyword">const</span> <span class="keyword">auto</span> <a class="code hl_variable" href="routing__flow_8cc.html#afca32f65388659a4b0956496169488b4">head</a> = graph.Head(<a class="code hl_variable" href="local__search_8cc.html#a30b5726116ac85ea64777e5d6e333894">arc</a>);</div>
<div class="line"><a id="l00069" name="l00069"></a><span class="lineno"> 69</span> <span class="keywordflow">if</span> (!components.<a class="code hl_function" href="class_dense_connected_components_finder.html#a962b54327591b21cc0a9273f78906a8b">Connected</a>(<a class="code hl_variable" href="routing__flow_8cc.html#a64e7efc5529154ba56903e75f5300990">tail</a>, <a class="code hl_variable" href="routing__flow_8cc.html#afca32f65388659a4b0956496169488b4">head</a>)) {</div>
<div class="line"><a id="l00070" name="l00070"></a><span class="lineno"> 70</span> components.<a class="code hl_function" href="class_dense_connected_components_finder.html#a428ab6b7c944afe33bd86a6a1ae7e668">AddEdge</a>(<a class="code hl_variable" href="routing__flow_8cc.html#a64e7efc5529154ba56903e75f5300990">tail</a>, <a class="code hl_variable" href="routing__flow_8cc.html#afca32f65388659a4b0956496169488b4">head</a>);</div>
<div class="line"><a id="l00071" name="l00071"></a><span class="lineno"> 71</span> tree_arcs.push_back(<a class="code hl_variable" href="local__search_8cc.html#a30b5726116ac85ea64777e5d6e333894">arc</a>);</div>
<div class="line"><a id="l00072" name="l00072"></a><span class="lineno"> 72</span> }</div>
<div class="line"><a id="l00073" name="l00073"></a><span class="lineno"> 73</span> ++arc_index;</div>
<div class="line"><a id="l00074" name="l00074"></a><span class="lineno"> 74</span> }</div>
<div class="line"><a id="l00075" name="l00075"></a><span class="lineno"> 75</span> <span class="keywordflow">return</span> tree_arcs;</div>
<div class="line"><a id="l00076" name="l00076"></a><span class="lineno"> 76</span>}</div>
<div class="line"><a id="l00077" name="l00077"></a><span class="lineno"> 77</span> </div>
<div class="line"><a id="l00078" 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 id="l00079" name="l00079"></a><span class="lineno"> 79</span><span class="comment">// Usage:</span></div>
<div class="line"><a id="l00080" name="l00080"></a><span class="lineno"> 80</span><span class="comment">// ListGraph&lt;int, int&gt; graph(...);</span></div>
<div class="line"><a id="l00081" name="l00081"></a><span class="lineno"> 81</span><span class="comment">// const auto arc_cost = [&amp;graph](int arc) {</span></div>
<div class="line"><a id="l00082" 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 id="l00083" name="l00083"></a><span class="lineno"> 83</span><span class="comment">// };</span></div>
<div class="line"><a id="l00084" name="l00084"></a><span class="lineno"> 84</span><span class="comment">// std::vector&lt;int&gt; mst = BuildKruskalMinimumSpanningTree(</span></div>
<div class="line"><a id="l00085" name="l00085"></a><span class="lineno"> 85</span><span class="comment">// graph,</span></div>
<div class="line"><a id="l00086" name="l00086"></a><span class="lineno"> 86</span><span class="comment">// [&amp;arc_cost](int a, int b) { return arc_cost(a) &lt; arc_cost(b); });</span></div>
<div class="line"><a id="l00087" name="l00087"></a><span class="lineno"> 87</span><span class="comment">//</span></div>
<div class="line"><a id="l00088" name="l00088"></a><span class="lineno"> 88</span><span class="keyword">template</span> &lt;<span class="keyword">typename</span> Graph, <span class="keyword">typename</span> ArcComparator&gt;</div>
<div class="line"><a id="l00089" name="l00089"></a><span class="lineno"><a class="line" href="namespaceoperations__research.html#aea7ea9ecb4c13ebf26b36a576c4fdc5f"> 89</a></span>std::vector&lt;typename Graph::ArcIndex&gt; <a class="code hl_function" href="namespaceoperations__research.html#aea7ea9ecb4c13ebf26b36a576c4fdc5f">BuildKruskalMinimumSpanningTree</a>(</div>
<div class="line"><a id="l00090" name="l00090"></a><span class="lineno"> 90</span> <span class="keyword">const</span> <a class="code hl_typedef" href="namespaceutil.html#a2f76166dbe0c4055a1f256235ad00478">Graph</a>&amp; graph, <span class="keyword">const</span> ArcComparator&amp; arc_comparator) {</div>
<div class="line"><a id="l00091" name="l00091"></a><span class="lineno"> 91</span> <span class="keyword">using</span> <a class="code hl_typedef" href="namespaceoperations__research.html#aecf320bda6e95d1edaf3a546673e3e6b">ArcIndex</a> = <span class="keyword">typename</span> <a class="code hl_typedef" href="namespaceoperations__research.html#aecf320bda6e95d1edaf3a546673e3e6b">Graph::ArcIndex</a>;</div>
<div class="line"><a id="l00092" name="l00092"></a><span class="lineno"> 92</span> std::vector&lt;ArcIndex&gt; sorted_arcs(graph.num_arcs());</div>
<div class="line"><a id="l00093" name="l00093"></a><span class="lineno"> 93</span> <span class="keywordflow">for</span> (<span class="keyword">const</span> <a class="code hl_typedef" href="namespaceoperations__research.html#aecf320bda6e95d1edaf3a546673e3e6b">ArcIndex</a> <a class="code hl_variable" href="local__search_8cc.html#a30b5726116ac85ea64777e5d6e333894">arc</a> : graph.AllForwardArcs()) {</div>
<div class="line"><a id="l00094" name="l00094"></a><span class="lineno"> 94</span> sorted_arcs[<a class="code hl_variable" href="local__search_8cc.html#a30b5726116ac85ea64777e5d6e333894">arc</a>] = <a class="code hl_variable" href="local__search_8cc.html#a30b5726116ac85ea64777e5d6e333894">arc</a>;</div>
<div class="line"><a id="l00095" name="l00095"></a><span class="lineno"> 95</span> }</div>
<div class="line"><a id="l00096" name="l00096"></a><span class="lineno"> 96</span> std::sort(sorted_arcs.begin(), sorted_arcs.end(), arc_comparator);</div>
<div class="line"><a id="l00097" name="l00097"></a><span class="lineno"> 97</span> <span class="keywordflow">return</span> <a class="code hl_function" href="namespaceoperations__research.html#aefd088882d7ba8d27157eba391b02792">BuildKruskalMinimumSpanningTreeFromSortedArcs</a>(graph, sorted_arcs);</div>
<div class="line"><a id="l00098" name="l00098"></a><span class="lineno"> 98</span>}</div>
<div class="line"><a id="l00099" name="l00099"></a><span class="lineno"> 99</span> </div>
<div class="line"><a id="l00100" name="l00100"></a><span class="lineno"> 100</span><span class="comment">// Implementation of Prim&#39;s mininumum spanning tree algorithm (c.f.</span></div>
<div class="line"><a id="l00101" name="l00101"></a><span class="lineno"> 101</span><span class="comment">// https://en.wikipedia.org/wiki/Prim&#39;s_algorithm) on undirected connected</span></div>
<div class="line"><a id="l00102" name="l00102"></a><span class="lineno"> 102</span><span class="comment">// graphs.</span></div>
<div class="line"><a id="l00103" 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 id="l00104" 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 id="l00105" 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 id="l00106" name="l00106"></a><span class="lineno"> 106</span><span class="comment">// taken by the graph.</span></div>
<div class="line"><a id="l00107" name="l00107"></a><span class="lineno"> 107</span><span class="comment">// Usage:</span></div>
<div class="line"><a id="l00108" name="l00108"></a><span class="lineno"> 108</span><span class="comment">// ListGraph&lt;int, int&gt; graph(...);</span></div>
<div class="line"><a id="l00109" name="l00109"></a><span class="lineno"> 109</span><span class="comment">// const auto arc_cost = [&amp;graph](int arc) -&gt; int64_t {</span></div>
<div class="line"><a id="l00110" 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 id="l00111" name="l00111"></a><span class="lineno"> 111</span><span class="comment">// };</span></div>
<div class="line"><a id="l00112" name="l00112"></a><span class="lineno"> 112</span><span class="comment">// std::vector&lt;int&gt; mst = BuildPrimMinimumSpanningTree(graph, arc_cost);</span></div>
<div class="line"><a id="l00113" name="l00113"></a><span class="lineno"> 113</span><span class="comment">//</span></div>
<div class="line"><a id="l00114" name="l00114"></a><span class="lineno"> 114</span><span class="keyword">template</span> &lt;<span class="keyword">typename</span> Graph, <span class="keyword">typename</span> ArcValue&gt;</div>
<div class="line"><a id="l00115" name="l00115"></a><span class="lineno"><a class="line" href="namespaceoperations__research.html#aa7f6b276e52d86253d0798bc37f4994e"> 115</a></span>std::vector&lt;typename Graph::ArcIndex&gt; <a class="code hl_function" href="namespaceoperations__research.html#aa7f6b276e52d86253d0798bc37f4994e">BuildPrimMinimumSpanningTree</a>(</div>
<div class="line"><a id="l00116" name="l00116"></a><span class="lineno"> 116</span> <span class="keyword">const</span> <a class="code hl_typedef" href="namespaceutil.html#a2f76166dbe0c4055a1f256235ad00478">Graph</a>&amp; graph, <span class="keyword">const</span> ArcValue&amp; arc_value) {</div>
<div class="line"><a id="l00117" name="l00117"></a><span class="lineno"> 117</span> <span class="keyword">using</span> <a class="code hl_typedef" href="namespaceoperations__research.html#aecf320bda6e95d1edaf3a546673e3e6b">ArcIndex</a> = <span class="keyword">typename</span> <a class="code hl_typedef" href="namespaceoperations__research.html#aecf320bda6e95d1edaf3a546673e3e6b">Graph::ArcIndex</a>;</div>
<div class="line"><a id="l00118" name="l00118"></a><span class="lineno"> 118</span> <span class="keyword">using</span> <a class="code hl_typedef" href="namespaceoperations__research.html#a7ae31ba4c3b4899478e53ca13df35dfc">NodeIndex</a> = <span class="keyword">typename</span> <a class="code hl_typedef" href="namespaceoperations__research.html#a7ae31ba4c3b4899478e53ca13df35dfc">Graph::NodeIndex</a>;</div>
<div class="line"><a id="l00119" name="l00119"></a><span class="lineno"> 119</span> <span class="keyword">using</span> ArcValueType = <span class="keyword">decltype</span>(arc_value(0));</div>
<div class="line"><a id="l00120" name="l00120"></a><span class="lineno"> 120</span> std::vector&lt;ArcIndex&gt; tree_arcs;</div>
<div class="line"><a id="l00121" name="l00121"></a><span class="lineno"> 121</span> <span class="keywordflow">if</span> (graph.num_nodes() == 0) {</div>
<div class="line"><a id="l00122" name="l00122"></a><span class="lineno"> 122</span> <span class="keywordflow">return</span> tree_arcs;</div>
<div class="line"><a id="l00123" name="l00123"></a><span class="lineno"> 123</span> }</div>
<div class="line"><a id="l00124" 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 id="l00125" name="l00125"></a><span class="lineno"> 125</span> tree_arcs.reserve(expected_tree_size);</div>
<div class="line"><a id="l00126" name="l00126"></a><span class="lineno"> 126</span> std::vector&lt;ArcIndex&gt; node_neighbor(graph.num_nodes(), Graph::kNilArc);</div>
<div class="line"><a id="l00127" name="l00127"></a><span class="lineno"> 127</span> std::vector&lt;bool&gt; node_active(graph.num_nodes(), <span class="keyword">true</span>);</div>
<div class="line"><a id="l00128" name="l00128"></a><span class="lineno"> 128</span> </div>
<div class="line"><a id="l00129" 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 id="l00130" 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 id="l00131" 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 id="l00132" name="l00132"></a><span class="lineno"> 132</span> <span class="comment">// SetHeapIndex and GetHeapIndex methods.</span></div>
<div class="line"><a id="l00133" name="l00133"></a><span class="lineno"> 133</span> <span class="keyword">struct </span>Entry {</div>
<div class="line"><a id="l00134" name="l00134"></a><span class="lineno"> 134</span> <span class="keywordtype">void</span> SetHeapIndex(<span class="keywordtype">int</span> <a class="code hl_variable" href="local__search_8cc.html#a750b5d744c39a06bfb13e6eb010e35d0">index</a>) { heap_index = <a class="code hl_variable" href="local__search_8cc.html#a750b5d744c39a06bfb13e6eb010e35d0">index</a>; }</div>
<div class="line"><a id="l00135" 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 id="l00136" name="l00136"></a><span class="lineno"> 136</span> <span class="keywordtype">bool</span> operator&lt;(<span class="keyword">const</span> Entry&amp; other)<span class="keyword"> const </span>{ <span class="keywordflow">return</span> <a class="code hl_variable" href="demon__profiler_8cc.html#ac072af30c4ffbc834bb4c681f6ecb514">value</a> &gt; other.value; }</div>
<div class="line"><a id="l00137" name="l00137"></a><span class="lineno"> 137</span> </div>
<div class="line"><a id="l00138" name="l00138"></a><span class="lineno"> 138</span> <a class="code hl_typedef" href="namespaceoperations__research.html#a7ae31ba4c3b4899478e53ca13df35dfc">NodeIndex</a> node;</div>
<div class="line"><a id="l00139" name="l00139"></a><span class="lineno"> 139</span> ArcValueType <a class="code hl_variable" href="demon__profiler_8cc.html#ac072af30c4ffbc834bb4c681f6ecb514">value</a>;</div>
<div class="line"><a id="l00140" name="l00140"></a><span class="lineno"> 140</span> <span class="keywordtype">int</span> heap_index;</div>
<div class="line"><a id="l00141" name="l00141"></a><span class="lineno"> 141</span> };</div>
<div class="line"><a id="l00142" name="l00142"></a><span class="lineno"> 142</span> </div>
<div class="line"><a id="l00143" name="l00143"></a><span class="lineno"> 143</span> <a class="code hl_class" href="class_adjustable_priority_queue.html">AdjustablePriorityQueue&lt;Entry&gt;</a> pq;</div>
<div class="line"><a id="l00144" name="l00144"></a><span class="lineno"> 144</span> std::vector&lt;Entry&gt; entries;</div>
<div class="line"><a id="l00145" name="l00145"></a><span class="lineno"> 145</span> std::vector&lt;bool&gt; touched_entry(graph.num_nodes(), <span class="keyword">false</span>);</div>
<div class="line"><a id="l00146" name="l00146"></a><span class="lineno"> 146</span> <span class="keywordflow">for</span> (<a class="code hl_typedef" href="namespaceoperations__research.html#a7ae31ba4c3b4899478e53ca13df35dfc">NodeIndex</a> node : graph.AllNodes()) {</div>
<div class="line"><a id="l00147" name="l00147"></a><span class="lineno"> 147</span> entries.push_back({node, <a class="code hl_variable" href="alldiff__cst_8cc.html#a26e6db9bcc64b584051ecc28171ed11f">std::numeric_limits&lt;ArcValueType&gt;::max</a>(), -1});</div>
<div class="line"><a id="l00148" name="l00148"></a><span class="lineno"> 148</span> }</div>
<div class="line"><a id="l00149" name="l00149"></a><span class="lineno"> 149</span> entries[0].value = 0;</div>
<div class="line"><a id="l00150" name="l00150"></a><span class="lineno"> 150</span> pq.<a class="code hl_function" href="class_adjustable_priority_queue.html#ac312fa979ea8afd5deabd6b581b3bb1f">Add</a>(&amp;entries[0]);</div>
<div class="line"><a id="l00151" name="l00151"></a><span class="lineno"> 151</span> <span class="keywordflow">while</span> (!pq.<a class="code hl_function" href="class_adjustable_priority_queue.html#a8e12342fc420701fbffd97025421575a">IsEmpty</a>() &amp;&amp; tree_arcs.size() != expected_tree_size) {</div>
<div class="line"><a id="l00152" name="l00152"></a><span class="lineno"> 152</span> <span class="keyword">const</span> Entry* best = pq.<a class="code hl_function" href="class_adjustable_priority_queue.html#a503e1bfef34fc7acf03340ddd74b5872">Top</a>();</div>
<div class="line"><a id="l00153" name="l00153"></a><span class="lineno"> 153</span> <span class="keyword">const</span> <a class="code hl_typedef" href="namespaceoperations__research.html#a7ae31ba4c3b4899478e53ca13df35dfc">NodeIndex</a> node = best-&gt;node;</div>
<div class="line"><a id="l00154" name="l00154"></a><span class="lineno"> 154</span> pq.<a class="code hl_function" href="class_adjustable_priority_queue.html#a701a584ce72cccbcce9cb0656b6c898b">Pop</a>();</div>
<div class="line"><a id="l00155" name="l00155"></a><span class="lineno"> 155</span> node_active[node] = <span class="keyword">false</span>;</div>
<div class="line"><a id="l00156" name="l00156"></a><span class="lineno"> 156</span> <span class="keywordflow">if</span> (node_neighbor[node] != Graph::kNilArc) {</div>
<div class="line"><a id="l00157" name="l00157"></a><span class="lineno"> 157</span> tree_arcs.push_back(node_neighbor[node]);</div>
<div class="line"><a id="l00158" name="l00158"></a><span class="lineno"> 158</span> }</div>
<div class="line"><a id="l00159" name="l00159"></a><span class="lineno"> 159</span> <span class="keywordflow">for</span> (<span class="keyword">const</span> <a class="code hl_typedef" href="namespaceoperations__research.html#aecf320bda6e95d1edaf3a546673e3e6b">ArcIndex</a> <a class="code hl_variable" href="local__search_8cc.html#a30b5726116ac85ea64777e5d6e333894">arc</a> : graph.OutgoingArcs(node)) {</div>
<div class="line"><a id="l00160" name="l00160"></a><span class="lineno"> 160</span> <span class="keyword">const</span> <a class="code hl_typedef" href="namespaceoperations__research.html#a7ae31ba4c3b4899478e53ca13df35dfc">NodeIndex</a> neighbor = graph.Head(<a class="code hl_variable" href="local__search_8cc.html#a30b5726116ac85ea64777e5d6e333894">arc</a>);</div>
<div class="line"><a id="l00161" name="l00161"></a><span class="lineno"> 161</span> <span class="keywordflow">if</span> (node_active[neighbor]) {</div>
<div class="line"><a id="l00162" name="l00162"></a><span class="lineno"> 162</span> <span class="keyword">const</span> ArcValueType <a class="code hl_variable" href="demon__profiler_8cc.html#ac072af30c4ffbc834bb4c681f6ecb514">value</a> = arc_value(<a class="code hl_variable" href="local__search_8cc.html#a30b5726116ac85ea64777e5d6e333894">arc</a>);</div>
<div class="line"><a id="l00163" name="l00163"></a><span class="lineno"> 163</span> Entry&amp; entry = entries[neighbor];</div>
<div class="line"><a id="l00164" name="l00164"></a><span class="lineno"> 164</span> <span class="keywordflow">if</span> (<a class="code hl_variable" href="demon__profiler_8cc.html#ac072af30c4ffbc834bb4c681f6ecb514">value</a> &lt; entry.value || !touched_entry[neighbor]) {</div>
<div class="line"><a id="l00165" name="l00165"></a><span class="lineno"> 165</span> node_neighbor[neighbor] = <a class="code hl_variable" href="local__search_8cc.html#a30b5726116ac85ea64777e5d6e333894">arc</a>;</div>
<div class="line"><a id="l00166" name="l00166"></a><span class="lineno"> 166</span> entry.value = <a class="code hl_variable" href="demon__profiler_8cc.html#ac072af30c4ffbc834bb4c681f6ecb514">value</a>;</div>
<div class="line"><a id="l00167" name="l00167"></a><span class="lineno"> 167</span> touched_entry[neighbor] = <span class="keyword">true</span>;</div>
<div class="line"><a id="l00168" name="l00168"></a><span class="lineno"> 168</span> <span class="keywordflow">if</span> (pq.<a class="code hl_function" href="class_adjustable_priority_queue.html#a7d1005d75f40525f91a2acb84348bc3c">Contains</a>(&amp;entry)) {</div>
<div class="line"><a id="l00169" name="l00169"></a><span class="lineno"> 169</span> pq.<a class="code hl_function" href="class_adjustable_priority_queue.html#a506169056c5f470fabdf77aa1934a5a3">NoteChangedPriority</a>(&amp;entry);</div>
<div class="line"><a id="l00170" name="l00170"></a><span class="lineno"> 170</span> } <span class="keywordflow">else</span> {</div>
<div class="line"><a id="l00171" name="l00171"></a><span class="lineno"> 171</span> pq.<a class="code hl_function" href="class_adjustable_priority_queue.html#ac312fa979ea8afd5deabd6b581b3bb1f">Add</a>(&amp;entry);</div>
<div class="line"><a id="l00172" name="l00172"></a><span class="lineno"> 172</span> }</div>
<div class="line"><a id="l00173" name="l00173"></a><span class="lineno"> 173</span> }</div>
<div class="line"><a id="l00174" name="l00174"></a><span class="lineno"> 174</span> }</div>
<div class="line"><a id="l00175" name="l00175"></a><span class="lineno"> 175</span> }</div>
<div class="line"><a id="l00176" name="l00176"></a><span class="lineno"> 176</span> }</div>
<div class="line"><a id="l00177" name="l00177"></a><span class="lineno"> 177</span> <span class="keywordflow">return</span> tree_arcs;</div>
<div class="line"><a id="l00178" name="l00178"></a><span class="lineno"> 178</span>}</div>
<div class="line"><a id="l00179" name="l00179"></a><span class="lineno"> 179</span> </div>
<div class="line"><a id="l00180" name="l00180"></a><span class="lineno"> 180</span>} <span class="comment">// namespace operations_research</span></div>
<div class="line"><a id="l00181" name="l00181"></a><span class="lineno"> 181</span><span class="preprocessor">#endif </span><span class="comment">// OR_TOOLS_GRAPH_MINIMUM_SPANNING_TREE_H_</span></div>
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<div class="ttc" id="aclass_adjustable_priority_queue_html"><div class="ttname"><a href="class_adjustable_priority_queue.html">AdjustablePriorityQueue</a></div><div class="ttdef"><b>Definition:</b> <a href="adjustable__priority__queue_8h_source.html#l00038">adjustable_priority_queue.h:38</a></div></div>
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<div class="ttc" id="aclass_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 class="ttdef"><b>Definition:</b> <a href="connected__components_8cc_source.html#l00036">connected_components.cc:36</a></div></div>
<div class="ttc" id="aconnected__components_8h_html"><div class="ttname"><a href="connected__components_8h.html">connected_components.h</a></div></div>
<div class="ttc" id="ademon__profiler_8cc_html_ac072af30c4ffbc834bb4c681f6ecb514"><div class="ttname"><a href="demon__profiler_8cc.html#ac072af30c4ffbc834bb4c681f6ecb514">value</a></div><div class="ttdeci">int64_t value</div><div class="ttdef"><b>Definition:</b> <a href="demon__profiler_8cc_source.html#l00044">demon_profiler.cc:44</a></div></div>
<div class="ttc" id="aintegral__types_8h_html"><div class="ttname"><a href="integral__types_8h.html">integral_types.h</a></div></div>
<div class="ttc" id="alocal__search_8cc_html_a30b5726116ac85ea64777e5d6e333894"><div class="ttname"><a href="local__search_8cc.html#a30b5726116ac85ea64777e5d6e333894">arc</a></div><div class="ttdeci">int arc</div><div class="ttdef"><b>Definition:</b> <a href="local__search_8cc_source.html#l02751">local_search.cc:2751</a></div></div>
<div class="ttc" id="alocal__search_8cc_html_a750b5d744c39a06bfb13e6eb010e35d0"><div class="ttname"><a href="local__search_8cc.html#a750b5d744c39a06bfb13e6eb010e35d0">index</a></div><div class="ttdeci">int index</div><div class="ttdef"><b>Definition:</b> <a href="local__search_8cc_source.html#l02750">local_search.cc:2750</a></div></div>
<div class="ttc" id="anamespaceoperations__research_html"><div class="ttname"><a href="namespaceoperations__research.html">operations_research</a></div><div class="ttdoc">Collection of objects used to extend the Constraint Solver library.</div><div class="ttdef"><b>Definition:</b> <a href="dense__doubly__linked__list_8h_source.html#l00021">dense_doubly_linked_list.h:21</a></div></div>
<div class="ttc" id="anamespaceoperations__research_html_a7ae31ba4c3b4899478e53ca13df35dfc"><div class="ttname"><a href="namespaceoperations__research.html#a7ae31ba4c3b4899478e53ca13df35dfc">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#l00201">ebert_graph.h:193</a></div></div>
<div class="ttc" id="anamespaceoperations__research_html_aa7f6b276e52d86253d0798bc37f4994e"><div class="ttname"><a href="namespaceoperations__research.html#aa7f6b276e52d86253d0798bc37f4994e">operations_research::BuildPrimMinimumSpanningTree</a></div><div class="ttdeci">std::vector&lt; typename Graph::ArcIndex &gt; BuildPrimMinimumSpanningTree(const Graph &amp;graph, const ArcValue &amp;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>
<div class="ttc" id="anamespaceoperations__research_html_aea7ea9ecb4c13ebf26b36a576c4fdc5f"><div class="ttname"><a href="namespaceoperations__research.html#aea7ea9ecb4c13ebf26b36a576c4fdc5f">operations_research::BuildKruskalMinimumSpanningTree</a></div><div class="ttdeci">std::vector&lt; typename Graph::ArcIndex &gt; BuildKruskalMinimumSpanningTree(const Graph &amp;graph, const ArcComparator &amp;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>
<div class="ttc" id="anamespaceoperations__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>
<div class="ttc" id="anamespaceoperations__research_html_aefd088882d7ba8d27157eba391b02792"><div class="ttname"><a href="namespaceoperations__research.html#aefd088882d7ba8d27157eba391b02792">operations_research::BuildKruskalMinimumSpanningTreeFromSortedArcs</a></div><div class="ttdeci">std::vector&lt; typename Graph::ArcIndex &gt; BuildKruskalMinimumSpanningTreeFromSortedArcs(const Graph &amp;graph, const std::vector&lt; typename Graph::ArcIndex &gt; &amp;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>
<div class="ttc" id="anamespaceutil_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#l02362">graph.h:2362</a></div></div>
<div class="ttc" id="arouting__flow_8cc_html_a64e7efc5529154ba56903e75f5300990"><div class="ttname"><a href="routing__flow_8cc.html#a64e7efc5529154ba56903e75f5300990">tail</a></div><div class="ttdeci">int64_t tail</div><div class="ttdef"><b>Definition:</b> <a href="routing__flow_8cc_source.html#l00149">routing_flow.cc:149</a></div></div>
<div class="ttc" id="arouting__flow_8cc_html_afca32f65388659a4b0956496169488b4"><div class="ttname"><a href="routing__flow_8cc.html#afca32f65388659a4b0956496169488b4">head</a></div><div class="ttdeci">int64_t head</div><div class="ttdef"><b>Definition:</b> <a href="routing__flow_8cc_source.html#l00150">routing_flow.cc:150</a></div></div>
<div class="ttc" id="avector__or__function_8h_html"><div class="ttname"><a href="vector__or__function_8h.html">vector_or_function.h</a></div></div>
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