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<li><a href="../cpp_algorithms/annotated.html">Algorithms</a></li>
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<h1 style="color: #145A32;">C++ Reference: Graph</h1>
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<div class="title">eulerian_path.h</div> </div>
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<a href="eulerian__path_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 2010-2018 Google LLC</span></div>
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<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>
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<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>
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<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>
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<div class="line"><a name="l00005"></a><span class="lineno"> 5</span> <span class="comment">//</span></div>
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<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>
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<div class="line"><a name="l00007"></a><span class="lineno"> 7</span> <span class="comment">//</span></div>
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<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>
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<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>
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<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>
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<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>
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<div class="line"><a name="l00012"></a><span class="lineno"> 12</span> <span class="comment">// limitations under the License.</span></div>
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<div class="line"><a name="l00013"></a><span class="lineno"> 13</span>  </div>
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<div class="line"><a name="l00014"></a><span class="lineno"> 14</span> <span class="comment">// Utility to build Eulerian paths and tours on a graph. For more information,</span></div>
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<div class="line"><a name="l00015"></a><span class="lineno"> 15</span> <span class="comment">// see https://en.wikipedia.org/wiki/Eulerian_path.</span></div>
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<div class="line"><a name="l00016"></a><span class="lineno"> 16</span> <span class="comment">// As of 10/2015, only undirected graphs are supported.</span></div>
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<div class="line"><a name="l00017"></a><span class="lineno"> 17</span> <span class="comment">//</span></div>
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<div class="line"><a name="l00018"></a><span class="lineno"> 18</span> <span class="comment">// Usage:</span></div>
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<div class="line"><a name="l00019"></a><span class="lineno"> 19</span> <span class="comment">// - Building an Eulerian tour on a ReverseArcListGraph:</span></div>
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<div class="line"><a name="l00020"></a><span class="lineno"> 20</span> <span class="comment">// ReverseArcListGraph<int, int> graph;</span></div>
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<div class="line"><a name="l00021"></a><span class="lineno"> 21</span> <span class="comment">// // Fill graph</span></div>
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<div class="line"><a name="l00022"></a><span class="lineno"> 22</span> <span class="comment">// std::vector<int> tour = BuildEulerianTour(graph);</span></div>
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<div class="line"><a name="l00023"></a><span class="lineno"> 23</span> <span class="comment">//</span></div>
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<div class="line"><a name="l00024"></a><span class="lineno"> 24</span> <span class="comment">// - Building an Eulerian path on a ReverseArcListGraph:</span></div>
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<div class="line"><a name="l00025"></a><span class="lineno"> 25</span> <span class="comment">// ReverseArcListGraph<int, int> graph;</span></div>
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<div class="line"><a name="l00026"></a><span class="lineno"> 26</span> <span class="comment">// // Fill graph</span></div>
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<div class="line"><a name="l00027"></a><span class="lineno"> 27</span> <span class="comment">// std::vector<int> tour = BuildEulerianPath(graph);</span></div>
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<div class="line"><a name="l00028"></a><span class="lineno"> 28</span> <span class="comment">//</span></div>
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<div class="line"><a name="l00029"></a><span class="lineno"> 29</span> <span class="preprocessor">#ifndef OR_TOOLS_GRAPH_EULERIAN_PATH_H_</span></div>
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<div class="line"><a name="l00030"></a><span class="lineno"> 30</span> <span class="preprocessor">#define OR_TOOLS_GRAPH_EULERIAN_PATH_H_</span></div>
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<div class="line"><a name="l00031"></a><span class="lineno"> 31</span>  </div>
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<div class="line"><a name="l00032"></a><span class="lineno"> 32</span> <span class="preprocessor">#include <vector></span></div>
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<div class="line"><a name="l00033"></a><span class="lineno"> 33</span>  </div>
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<div class="line"><a name="l00034"></a><span class="lineno"> 34</span> <span class="preprocessor">#include "ortools/base/logging.h"</span></div>
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<div class="line"><a name="l00035"></a><span class="lineno"> 35</span>  </div>
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<div class="line"><a name="l00036"></a><span class="lineno"> 36</span> <span class="keyword">namespace </span><a class="code" href="namespaceoperations__research.html">operations_research</a> {</div>
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<div class="line"><a name="l00037"></a><span class="lineno"> 37</span>  </div>
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<div class="line"><a name="l00038"></a><span class="lineno"> 38</span> <span class="comment">// Returns true if a graph is Eulerian, aka all its nodes are of even degree.</span></div>
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<div class="line"><a name="l00039"></a><span class="lineno"> 39</span> <span class="keyword">template</span> <<span class="keyword">typename</span> Graph></div>
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<div class="line"><a name="l00040"></a><span class="lineno"><a class="line" href="namespaceoperations__research.html#ab1cf773de0cae72d0c44efe5b8f4bb89"> 40</a></span> <span class="keywordtype">bool</span> <a class="code" href="namespaceoperations__research.html#ab1cf773de0cae72d0c44efe5b8f4bb89">IsEulerianGraph</a>(<span class="keyword">const</span> <a class="code" href="namespaceutil.html#a2f76166dbe0c4055a1f256235ad00478">Graph</a>& graph) {</div>
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<div class="line"><a name="l00041"></a><span class="lineno"> 41</span>  <span class="keyword">typedef</span> <span class="keyword">typename</span> <a class="code" href="namespaceoperations__research.html#a272bcc4d6ee0e87a7fc430a5cb13c582">Graph::NodeIndex</a> <a class="code" href="namespaceoperations__research.html#a272bcc4d6ee0e87a7fc430a5cb13c582">NodeIndex</a>;</div>
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<div class="line"><a name="l00042"></a><span class="lineno"> 42</span>  <span class="keywordflow">for</span> (<span class="keyword">const</span> <a class="code" href="namespaceoperations__research.html#a272bcc4d6ee0e87a7fc430a5cb13c582">NodeIndex</a> node : graph.AllNodes()) {</div>
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<div class="line"><a name="l00043"></a><span class="lineno"> 43</span>  <span class="keywordflow">if</span> ((graph.OutDegree(node) + graph.InDegree(node)) % 2 != 0) {</div>
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<div class="line"><a name="l00044"></a><span class="lineno"> 44</span>  <span class="keywordflow">return</span> <span class="keyword">false</span>;</div>
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<div class="line"><a name="l00045"></a><span class="lineno"> 45</span>  }</div>
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<div class="line"><a name="l00046"></a><span class="lineno"> 46</span>  }</div>
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<div class="line"><a name="l00047"></a><span class="lineno"> 47</span>  <span class="comment">// TODO(user): Check graph connectivity.</span></div>
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<div class="line"><a name="l00048"></a><span class="lineno"> 48</span>  <span class="keywordflow">return</span> <span class="keyword">true</span>;</div>
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<div class="line"><a name="l00049"></a><span class="lineno"> 49</span> }</div>
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<div class="line"><a name="l00050"></a><span class="lineno"> 50</span>  </div>
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<div class="line"><a name="l00051"></a><span class="lineno"> 51</span> <span class="comment">// Returns true if a graph is Semi-Eulerian, aka at most two of its nodes are of</span></div>
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<div class="line"><a name="l00052"></a><span class="lineno"> 52</span> <span class="comment">// odd degree.</span></div>
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<div class="line"><a name="l00053"></a><span class="lineno"> 53</span> <span class="comment">// odd_nodes is filled with odd nodes of the graph.</span></div>
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<div class="line"><a name="l00054"></a><span class="lineno"> 54</span> <span class="keyword">template</span> <<span class="keyword">typename</span> NodeIndex, <span class="keyword">typename</span> Graph></div>
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<div class="line"><a name="l00055"></a><span class="lineno"><a class="line" href="namespaceoperations__research.html#a6b312dd19c90b2af099e6f159869f7ee"> 55</a></span> <span class="keywordtype">bool</span> <a class="code" href="namespaceoperations__research.html#a6b312dd19c90b2af099e6f159869f7ee">IsSemiEulerianGraph</a>(<span class="keyword">const</span> <a class="code" href="namespaceutil.html#a2f76166dbe0c4055a1f256235ad00478">Graph</a>& graph,</div>
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<div class="line"><a name="l00056"></a><span class="lineno"> 56</span>  std::vector<NodeIndex>* odd_nodes) {</div>
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<div class="line"><a name="l00057"></a><span class="lineno"> 57</span>  CHECK(odd_nodes != <span class="keyword">nullptr</span>);</div>
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<div class="line"><a name="l00058"></a><span class="lineno"> 58</span>  <span class="keywordflow">for</span> (<span class="keyword">const</span> <a class="code" href="namespaceoperations__research.html#a272bcc4d6ee0e87a7fc430a5cb13c582">NodeIndex</a> node : graph.AllNodes()) {</div>
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<div class="line"><a name="l00059"></a><span class="lineno"> 59</span>  <span class="keyword">const</span> <span class="keywordtype">int</span> degree = graph.OutDegree(node) + graph.InDegree(node);</div>
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<div class="line"><a name="l00060"></a><span class="lineno"> 60</span>  <span class="keywordflow">if</span> (degree % 2 != 0) {</div>
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<div class="line"><a name="l00061"></a><span class="lineno"> 61</span>  odd_nodes->push_back(node);</div>
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<div class="line"><a name="l00062"></a><span class="lineno"> 62</span>  }</div>
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<div class="line"><a name="l00063"></a><span class="lineno"> 63</span>  }</div>
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<div class="line"><a name="l00064"></a><span class="lineno"> 64</span>  <span class="comment">// TODO(user): Check graph connectivity.</span></div>
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<div class="line"><a name="l00065"></a><span class="lineno"> 65</span>  <span class="keywordflow">return</span> odd_nodes->size() <= 2;</div>
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<div class="line"><a name="l00066"></a><span class="lineno"> 66</span> }</div>
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<div class="line"><a name="l00067"></a><span class="lineno"> 67</span>  </div>
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<div class="line"><a name="l00068"></a><span class="lineno"> 68</span> <span class="comment">// Builds an Eulerian path/trail on an undirected graph starting from node root.</span></div>
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<div class="line"><a name="l00069"></a><span class="lineno"> 69</span> <span class="comment">// Supposes the graph is connected and is eulerian or semi-eulerian.</span></div>
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<div class="line"><a name="l00070"></a><span class="lineno"> 70</span> <span class="comment">// This is an implementation of Hierholzer's algorithm.</span></div>
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<div class="line"><a name="l00071"></a><span class="lineno"> 71</span> <span class="comment">// If m is the number of edges in the graph and n the number of nodes, time</span></div>
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<div class="line"><a name="l00072"></a><span class="lineno"> 72</span> <span class="comment">// and memory complexity is O(n + m).</span></div>
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<div class="line"><a name="l00073"></a><span class="lineno"> 73</span> <span class="keyword">template</span> <<span class="keyword">typename</span> NodeIndex, <span class="keyword">typename</span> Graph></div>
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<div class="line"><a name="l00074"></a><span class="lineno"><a class="line" href="namespaceoperations__research.html#a743d8c9d6f64531bdeb7bbf18023e9d4"> 74</a></span> std::vector<NodeIndex> <a class="code" href="namespaceoperations__research.html#a743d8c9d6f64531bdeb7bbf18023e9d4">BuildEulerianPathFromNode</a>(<span class="keyword">const</span> <a class="code" href="namespaceutil.html#a2f76166dbe0c4055a1f256235ad00478">Graph</a>& graph,</div>
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<div class="line"><a name="l00075"></a><span class="lineno"> 75</span>  <a class="code" href="namespaceoperations__research.html#a272bcc4d6ee0e87a7fc430a5cb13c582">NodeIndex</a> root) {</div>
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<div class="line"><a name="l00076"></a><span class="lineno"> 76</span>  <span class="keyword">typedef</span> <span class="keyword">typename</span> <a class="code" href="namespaceoperations__research.html#ad90a9fc7d16bb2c98a088a4162bbbc9d">Graph::ArcIndex</a> <a class="code" href="namespaceoperations__research.html#ad90a9fc7d16bb2c98a088a4162bbbc9d">ArcIndex</a>;</div>
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<div class="line"><a name="l00077"></a><span class="lineno"> 77</span>  std::vector<bool> unvisited_edges(graph.num_arcs(), <span class="keyword">true</span>);</div>
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<div class="line"><a name="l00078"></a><span class="lineno"> 78</span>  std::vector<NodeIndex> tour;</div>
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<div class="line"><a name="l00079"></a><span class="lineno"> 79</span>  <span class="keywordflow">if</span> (graph.IsNodeValid(root)) {</div>
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<div class="line"><a name="l00080"></a><span class="lineno"> 80</span>  std::vector<NodeIndex> tour_stack = {root};</div>
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<div class="line"><a name="l00081"></a><span class="lineno"> 81</span>  std::vector<ArcIndex> active_arcs(graph.num_nodes());</div>
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<div class="line"><a name="l00082"></a><span class="lineno"> 82</span>  <span class="keywordflow">for</span> (<span class="keyword">const</span> <a class="code" href="namespaceoperations__research.html#a272bcc4d6ee0e87a7fc430a5cb13c582">NodeIndex</a> node : graph.AllNodes()) {</div>
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<div class="line"><a name="l00083"></a><span class="lineno"> 83</span>  active_arcs[node] = *(graph.OutgoingOrOppositeIncomingArcs(node)).begin();</div>
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<div class="line"><a name="l00084"></a><span class="lineno"> 84</span>  }</div>
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<div class="line"><a name="l00085"></a><span class="lineno"> 85</span>  <span class="keywordflow">while</span> (!tour_stack.empty()) {</div>
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<div class="line"><a name="l00086"></a><span class="lineno"> 86</span>  <span class="keyword">const</span> <a class="code" href="namespaceoperations__research.html#a272bcc4d6ee0e87a7fc430a5cb13c582">NodeIndex</a> node = tour_stack.back();</div>
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<div class="line"><a name="l00087"></a><span class="lineno"> 87</span>  <span class="keywordtype">bool</span> has_unvisited_edges = <span class="keyword">false</span>;</div>
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<div class="line"><a name="l00088"></a><span class="lineno"> 88</span>  <span class="keywordflow">for</span> (<span class="keyword">const</span> <a class="code" href="namespaceoperations__research.html#ad90a9fc7d16bb2c98a088a4162bbbc9d">ArcIndex</a> arc :</div>
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<div class="line"><a name="l00089"></a><span class="lineno"> 89</span>  graph.OutgoingOrOppositeIncomingArcsStartingFrom(</div>
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<div class="line"><a name="l00090"></a><span class="lineno"> 90</span>  node, active_arcs[node])) {</div>
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<div class="line"><a name="l00091"></a><span class="lineno"> 91</span>  <span class="keyword">const</span> <a class="code" href="namespaceoperations__research.html#ad90a9fc7d16bb2c98a088a4162bbbc9d">ArcIndex</a> edge = arc < 0 ? graph.OppositeArc(arc) : arc;</div>
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<div class="line"><a name="l00092"></a><span class="lineno"> 92</span>  <span class="keywordflow">if</span> (unvisited_edges[edge]) {</div>
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<div class="line"><a name="l00093"></a><span class="lineno"> 93</span>  has_unvisited_edges = <span class="keyword">true</span>;</div>
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<div class="line"><a name="l00094"></a><span class="lineno"> 94</span>  active_arcs[node] = arc;</div>
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<div class="line"><a name="l00095"></a><span class="lineno"> 95</span>  tour_stack.push_back(graph.Head(arc));</div>
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<div class="line"><a name="l00096"></a><span class="lineno"> 96</span>  unvisited_edges[edge] = <span class="keyword">false</span>;</div>
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<div class="line"><a name="l00097"></a><span class="lineno"> 97</span>  <span class="keywordflow">break</span>;</div>
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<div class="line"><a name="l00098"></a><span class="lineno"> 98</span>  }</div>
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<div class="line"><a name="l00099"></a><span class="lineno"> 99</span>  }</div>
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<div class="line"><a name="l00100"></a><span class="lineno"> 100</span>  <span class="keywordflow">if</span> (!has_unvisited_edges) {</div>
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<div class="line"><a name="l00101"></a><span class="lineno"> 101</span>  tour.push_back(node);</div>
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<div class="line"><a name="l00102"></a><span class="lineno"> 102</span>  tour_stack.pop_back();</div>
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<div class="line"><a name="l00103"></a><span class="lineno"> 103</span>  }</div>
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<div class="line"><a name="l00104"></a><span class="lineno"> 104</span>  }</div>
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<div class="line"><a name="l00105"></a><span class="lineno"> 105</span>  }</div>
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<div class="line"><a name="l00106"></a><span class="lineno"> 106</span>  <span class="keywordflow">return</span> tour;</div>
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<div class="line"><a name="l00107"></a><span class="lineno"> 107</span> }</div>
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<div class="line"><a name="l00108"></a><span class="lineno"> 108</span>  </div>
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<div class="line"><a name="l00109"></a><span class="lineno"> 109</span> <span class="comment">// Builds an Eulerian tour/circuit/cycle starting and ending at node root on an</span></div>
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<div class="line"><a name="l00110"></a><span class="lineno"> 110</span> <span class="comment">// undirected graph.</span></div>
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<div class="line"><a name="l00111"></a><span class="lineno"> 111</span> <span class="comment">// This function works only on Reverse graphs</span></div>
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<div class="line"><a name="l00112"></a><span class="lineno"> 112</span> <span class="comment">// (cf. ortools/graph/graph.h).</span></div>
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<div class="line"><a name="l00113"></a><span class="lineno"> 113</span> <span class="comment">// Returns an empty tour if either root is invalid or if a tour cannot be built.</span></div>
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<div class="line"><a name="l00114"></a><span class="lineno"> 114</span> <span class="comment">// As of 10/2015, assumes the graph is connected.</span></div>
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<div class="line"><a name="l00115"></a><span class="lineno"> 115</span> <span class="keyword">template</span> <<span class="keyword">typename</span> NodeIndex, <span class="keyword">typename</span> Graph></div>
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<div class="line"><a name="l00116"></a><span class="lineno"><a class="line" href="namespaceoperations__research.html#aa63055860fc53f8eed56d23d2571c180"> 116</a></span> std::vector<NodeIndex> <a class="code" href="namespaceoperations__research.html#aa63055860fc53f8eed56d23d2571c180">BuildEulerianTourFromNode</a>(<span class="keyword">const</span> <a class="code" href="namespaceutil.html#a2f76166dbe0c4055a1f256235ad00478">Graph</a>& graph,</div>
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<div class="line"><a name="l00117"></a><span class="lineno"> 117</span>  <a class="code" href="namespaceoperations__research.html#a272bcc4d6ee0e87a7fc430a5cb13c582">NodeIndex</a> root) {</div>
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<div class="line"><a name="l00118"></a><span class="lineno"> 118</span>  std::vector<NodeIndex> tour;</div>
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<div class="line"><a name="l00119"></a><span class="lineno"> 119</span>  <span class="keywordflow">if</span> (<a class="code" href="namespaceoperations__research.html#ab1cf773de0cae72d0c44efe5b8f4bb89">IsEulerianGraph</a>(graph)) {</div>
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<div class="line"><a name="l00120"></a><span class="lineno"> 120</span>  tour = <a class="code" href="namespaceoperations__research.html#a743d8c9d6f64531bdeb7bbf18023e9d4">BuildEulerianPathFromNode</a>(graph, root);</div>
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<div class="line"><a name="l00121"></a><span class="lineno"> 121</span>  }</div>
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<div class="line"><a name="l00122"></a><span class="lineno"> 122</span>  <span class="keywordflow">return</span> tour;</div>
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<div class="line"><a name="l00123"></a><span class="lineno"> 123</span> }</div>
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<div class="line"><a name="l00124"></a><span class="lineno"> 124</span>  </div>
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<div class="line"><a name="l00125"></a><span class="lineno"> 125</span> <span class="comment">// Same as above but without specifying a start/end root node (node 0 is taken</span></div>
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<div class="line"><a name="l00126"></a><span class="lineno"> 126</span> <span class="comment">// as default root).</span></div>
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<div class="line"><a name="l00127"></a><span class="lineno"> 127</span> <span class="keyword">template</span> <<span class="keyword">typename</span> Graph></div>
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<div class="line"><a name="l00128"></a><span class="lineno"><a class="line" href="namespaceoperations__research.html#a034666fe63ca105b735272974006362a"> 128</a></span> std::vector<typename Graph::NodeIndex> <a class="code" href="namespaceoperations__research.html#a034666fe63ca105b735272974006362a">BuildEulerianTour</a>(<span class="keyword">const</span> <a class="code" href="namespaceutil.html#a2f76166dbe0c4055a1f256235ad00478">Graph</a>& graph) {</div>
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<div class="line"><a name="l00129"></a><span class="lineno"> 129</span>  <span class="keywordflow">return</span> <a class="code" href="namespaceoperations__research.html#aa63055860fc53f8eed56d23d2571c180">BuildEulerianTourFromNode</a>(graph, 0);</div>
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<div class="line"><a name="l00130"></a><span class="lineno"> 130</span> }</div>
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<div class="line"><a name="l00131"></a><span class="lineno"> 131</span>  </div>
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<div class="line"><a name="l00132"></a><span class="lineno"> 132</span> <span class="comment">// Builds an Eulerian path/trail on an undirected graph.</span></div>
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<div class="line"><a name="l00133"></a><span class="lineno"> 133</span> <span class="comment">// This function works only on Reverse graphs</span></div>
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<div class="line"><a name="l00134"></a><span class="lineno"> 134</span> <span class="comment">// (cf. ortools/graph/graph.h).</span></div>
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<div class="line"><a name="l00135"></a><span class="lineno"> 135</span> <span class="comment">// Returns an empty tour if a tour cannot be built.</span></div>
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<div class="line"><a name="l00136"></a><span class="lineno"> 136</span> <span class="comment">// As of 10/2015, assumes the graph is connected.</span></div>
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<div class="line"><a name="l00137"></a><span class="lineno"> 137</span> <span class="keyword">template</span> <<span class="keyword">typename</span> Graph></div>
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<div class="line"><a name="l00138"></a><span class="lineno"><a class="line" href="namespaceoperations__research.html#a49b170b2d03863c465331e67b21f0c33"> 138</a></span> std::vector<typename Graph::NodeIndex> <a class="code" href="namespaceoperations__research.html#a49b170b2d03863c465331e67b21f0c33">BuildEulerianPath</a>(<span class="keyword">const</span> <a class="code" href="namespaceutil.html#a2f76166dbe0c4055a1f256235ad00478">Graph</a>& graph) {</div>
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<div class="line"><a name="l00139"></a><span class="lineno"> 139</span>  <span class="keyword">typedef</span> <span class="keyword">typename</span> <a class="code" href="namespaceoperations__research.html#a272bcc4d6ee0e87a7fc430a5cb13c582">Graph::NodeIndex</a> <a class="code" href="namespaceoperations__research.html#a272bcc4d6ee0e87a7fc430a5cb13c582">NodeIndex</a>;</div>
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<div class="line"><a name="l00140"></a><span class="lineno"> 140</span>  std::vector<NodeIndex> path;</div>
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<div class="line"><a name="l00141"></a><span class="lineno"> 141</span>  std::vector<NodeIndex> roots;</div>
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<div class="line"><a name="l00142"></a><span class="lineno"> 142</span>  <span class="keywordflow">if</span> (<a class="code" href="namespaceoperations__research.html#a6b312dd19c90b2af099e6f159869f7ee">IsSemiEulerianGraph</a>(graph, &roots)) {</div>
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<div class="line"><a name="l00143"></a><span class="lineno"> 143</span>  <span class="keyword">const</span> <a class="code" href="namespaceoperations__research.html#a272bcc4d6ee0e87a7fc430a5cb13c582">NodeIndex</a> root = roots.empty() ? 0 : roots.back();</div>
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<div class="line"><a name="l00144"></a><span class="lineno"> 144</span>  path = <a class="code" href="namespaceoperations__research.html#a743d8c9d6f64531bdeb7bbf18023e9d4">BuildEulerianPathFromNode</a>(graph, root);</div>
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<div class="line"><a name="l00145"></a><span class="lineno"> 145</span>  }</div>
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<div class="line"><a name="l00146"></a><span class="lineno"> 146</span>  <span class="keywordflow">return</span> path;</div>
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<div class="line"><a name="l00147"></a><span class="lineno"> 147</span> }</div>
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<div class="line"><a name="l00148"></a><span class="lineno"> 148</span> } <span class="comment">// namespace operations_research</span></div>
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<div class="line"><a name="l00149"></a><span class="lineno"> 149</span>  </div>
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<div class="line"><a name="l00150"></a><span class="lineno"> 150</span> <span class="preprocessor">#endif // OR_TOOLS_GRAPH_EULERIAN_PATH_H_</span></div>
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</div><!-- fragment --></div><!-- contents -->
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<div class="ttc" id="anamespaceoperations__research_html_a034666fe63ca105b735272974006362a"><div class="ttname"><a href="namespaceoperations__research.html#a034666fe63ca105b735272974006362a">operations_research::BuildEulerianTour</a></div><div class="ttdeci">std::vector< typename Graph::NodeIndex > BuildEulerianTour(const Graph &graph)</div><div class="ttdef"><b>Definition:</b> <a href="eulerian__path_8h_source.html#l00128">eulerian_path.h:128</a></div></div>
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<div class="ttc" id="anamespaceoperations__research_html_ab1cf773de0cae72d0c44efe5b8f4bb89"><div class="ttname"><a href="namespaceoperations__research.html#ab1cf773de0cae72d0c44efe5b8f4bb89">operations_research::IsEulerianGraph</a></div><div class="ttdeci">bool IsEulerianGraph(const Graph &graph)</div><div class="ttdef"><b>Definition:</b> <a href="eulerian__path_8h_source.html#l00040">eulerian_path.h:40</a></div></div>
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<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#l02356">graph.h:2356</a></div></div>
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<div class="ttc" id="anamespaceoperations__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#l00033">christofides.h:33</a></div></div>
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<div class="ttc" id="anamespaceoperations__research_html_a6b312dd19c90b2af099e6f159869f7ee"><div class="ttname"><a href="namespaceoperations__research.html#a6b312dd19c90b2af099e6f159869f7ee">operations_research::IsSemiEulerianGraph</a></div><div class="ttdeci">bool IsSemiEulerianGraph(const Graph &graph, std::vector< NodeIndex > *odd_nodes)</div><div class="ttdef"><b>Definition:</b> <a href="eulerian__path_8h_source.html#l00055">eulerian_path.h:55</a></div></div>
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<div class="ttc" id="anamespaceoperations__research_html_ad90a9fc7d16bb2c98a088a4162bbbc9d"><div class="ttname"><a href="namespaceoperations__research.html#ad90a9fc7d16bb2c98a088a4162bbbc9d">operations_research::ArcIndex</a></div><div class="ttdeci">int32 ArcIndex</div><div class="ttdef"><b>Definition:</b> <a href="ebert__graph_8h_source.html#l00201">ebert_graph.h:201</a></div></div>
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<div class="ttc" id="anamespaceoperations__research_html_aa63055860fc53f8eed56d23d2571c180"><div class="ttname"><a href="namespaceoperations__research.html#aa63055860fc53f8eed56d23d2571c180">operations_research::BuildEulerianTourFromNode</a></div><div class="ttdeci">std::vector< NodeIndex > BuildEulerianTourFromNode(const Graph &graph, NodeIndex root)</div><div class="ttdef"><b>Definition:</b> <a href="eulerian__path_8h_source.html#l00116">eulerian_path.h:116</a></div></div>
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<div class="ttc" id="anamespaceoperations__research_html_a272bcc4d6ee0e87a7fc430a5cb13c582"><div class="ttname"><a href="namespaceoperations__research.html#a272bcc4d6ee0e87a7fc430a5cb13c582">operations_research::NodeIndex</a></div><div class="ttdeci">int32 NodeIndex</div><div class="ttdef"><b>Definition:</b> <a href="ebert__graph_8h_source.html#l00192">ebert_graph.h:192</a></div></div>
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<div class="ttc" id="anamespaceoperations__research_html_a743d8c9d6f64531bdeb7bbf18023e9d4"><div class="ttname"><a href="namespaceoperations__research.html#a743d8c9d6f64531bdeb7bbf18023e9d4">operations_research::BuildEulerianPathFromNode</a></div><div class="ttdeci">std::vector< NodeIndex > BuildEulerianPathFromNode(const Graph &graph, NodeIndex root)</div><div class="ttdef"><b>Definition:</b> <a href="eulerian__path_8h_source.html#l00074">eulerian_path.h:74</a></div></div>
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<div class="ttc" id="anamespaceoperations__research_html_a49b170b2d03863c465331e67b21f0c33"><div class="ttname"><a href="namespaceoperations__research.html#a49b170b2d03863c465331e67b21f0c33">operations_research::BuildEulerianPath</a></div><div class="ttdeci">std::vector< typename Graph::NodeIndex > BuildEulerianPath(const Graph &graph)</div><div class="ttdef"><b>Definition:</b> <a href="eulerian__path_8h_source.html#l00138">eulerian_path.h:138</a></div></div>
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