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<a href="linear__assignment_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="comment">//</span></div><div class="line"><a name="l00015"></a><span class="lineno"> 15</span> <span class="comment">// An implementation of a cost-scaling push-relabel algorithm for the</span></div><div class="line"><a name="l00016"></a><span class="lineno"> 16</span> <span class="comment">// assignment problem (minimum-cost perfect bipartite matching), from</span></div><div class="line"><a name="l00017"></a><span class="lineno"> 17</span> <span class="comment">// the paper of Goldberg and Kennedy (1995).</span></div><div class="line"><a name="l00018"></a><span class="lineno"> 18</span> <span class="comment">//</span></div><div class="line"><a name="l00019"></a><span class="lineno"> 19</span> <span class="comment">//</span></div><div class="line"><a name="l00020"></a><span class="lineno"> 20</span> <span class="comment">// This implementation finds the minimum-cost perfect assignment in</span></div><div class="line"><a name="l00021"></a><span class="lineno"> 21</span> <span class="comment">// the given graph with integral edge weights set through the</span></div><div class="line"><a name="l00022"></a><span class="lineno"> 22</span> <span class="comment">// SetArcCost method.</span></div><div class="line"><a name="l00023"></a><span class="lineno"> 23</span> <span class="comment">//</span></div><div class="line"><a name="l00024"></a><span class="lineno"> 24</span> <span class="comment">// The running time is O(n*m*log(nC)) where n is the number of nodes,</span></div><div class="line"><a name="l00025"></a><span class="lineno"> 25</span> <span class="comment">// m is the number of edges, and C is the largest magnitude of an edge cost.</span></div><div class="line"><a name="l00026"></a><span class="lineno"> 26</span> <span class="comment">// In principle it can be worse than the Hungarian algorithm but we don't know</span></div><div class="line"><a name="l00027"></a><span class="lineno"> 27</span> <span class="comment">// of any class of problems where that actually happens. An additional sqrt(n)</span></div><div class="line"><a name="l00028"></a><span class="lineno"> 28</span> <span class="comment">// factor could be shaved off the running time bound using the technique</span></div><div class="line"><a name="l00029"></a><span class="lineno"> 29</span> <span class="comment">// described in http://dx.doi.org/10.1137/S0895480194281185</span></div><div class="line"><a name="l00030"></a><span class="lineno"> 30</span> <span class="comment">// (see also http://theory.stanford.edu/~robert/papers/glob_upd.ps).</span></div><div class="line"><a name="l00031"></a><span class="lineno"> 31</span> <span class="comment">//</span></div><div class="line"><a name="l00032"></a><span class="lineno"> 32</span> <span class="comment">//</span></div><div class="line"><a name="l00033"></a><span class="lineno"> 33</span> <span class="comment">// Example usage:</span></div><div class="line"><a name="l00034"></a><span class="lineno"> 34</span> <span class="comment">//</span></div><div class="line"><a name="l00035"></a><span class="lineno"> 35</span> <span class="comment">// #include "ortools/graph/graph.h"</span></div><div class="line"><a name="l00036"></a><span class="lineno"> 36</span> <span class="comment">// #include "ortools/graph/linear_assignment.h"</span></div><div class="line"><a name="l00037"></a><span class="lineno"> 37</span> <span class="comment">//</span></div><div class="line"><a name="l00038"></a><span class="lineno"> 38</span> <span class="comment">// // Choose a graph implementation (we recommend StaticGraph<>).</span></div><div class="line"><a name="l00039"></a><span class="lineno"> 39</span> <span class="comment">// typedef util::StaticGraph<> Graph;</span></div><div class="line"><a name="l00040"></a><span class="lineno"> 40</span> <span class="comment">//</span></div><div class="line"><a name="l00041"></a><span class="lineno"> 41</span> <span class="comment">// // Define a num_nodes / 2 by num_nodes / 2 assignment problem:</span></div><div class="line"><a name="l00042"></a><span class="lineno"> 42</span> <span class="comment">// const int num_nodes = ...</span></div><div class="line"><a name="l00043"></a><span class="lineno"> 43</span> <span class="comment">// const int num_arcs = ...</span></div><div class="line"><a name="l00044"></a><span class="lineno"> 44</span> <span class="comment">// const int num_left_nodes = num_nodes / 2;</span></div><div class="line"><a name="l00045"></a><span class="lineno"> 45</span> <span class="comment">// Graph graph(num_nodes, num_arcs);</span></div><div class="line"><a name="l00046"></a><span class="lineno"> 46</span> <span class="comment">// std::vector<operations_research::CostValue> arc_costs(num_arcs);</span></div><div class="line"><a name="l00047"></a><span class="lineno"> 47</span> <span class="comment">// for (int arc = 0; arc < num_arcs; ++arc) {</span></div><div class="line"><a name="l00048"></a><span class="lineno"> 48</span> <span class="comment">// const int arc_tail = ... // must be in [0, num_left_nodes)</span></div><div class="line"><a name="l00049"></a><span class="lineno"> 49</span> <span class="comment">// const int arc_head = ... // must be in [num_left_nodes, num_nodes)</span></div><div class="line"><a name="l00050"></a><span class="lineno"> 50</span> <span class="comment">// graph.AddArc(arc_tail, arc_head);</span></div><div class="line"><a name="l00051"></a><span class="lineno"> 51</span> <span class="comment">// arc_costs[arc] = ...</span></div><div class="line"><a name="l00052"></a><span class="lineno"> 52</span> <span class="comment">// }</span></div><div class="line"><a name="l00053"></a><span class="lineno"> 53</span> <span class="comment">//</span></div><div class="line"><a name="l00054"></a><span class="lineno"> 54</span> <span class="comment">// // Build the StaticGraph. You can skip this step by using a ListGraph<></span></div><div class="line"><a name="l00055"></a><span class="lineno"> 55</span> <span class="comment">// // instead, but then the ComputeAssignment() below will be slower. It is</span></div><div class="line"><a name="l00056"></a><span class="lineno"> 56</span> <span class="comment">// // okay if your graph is small and performance is not critical though.</span></div><div class="line"><a name="l00057"></a><span class="lineno"> 57</span> <span class="comment">// {</span></div><div class="line"><a name="l00058"></a><span class="lineno"> 58</span> <span class="comment">// std::vector<Graph::ArcIndex> arc_permutation;</span></div><div class="line"><a name="l00059"></a><span class="lineno"> 59</span> <span class="comment">// graph.Build(&arc_permutation);</span></div><div class="line"><a name="l00060"></a><span class="lineno"> 60</span> <span class="comment">// util::Permute(arc_permutation, &arc_costs);</span></div><div class="line"><a name="l00061"></a><span class="lineno"> 61</span> <span class="comment">// }</span></div><div class="line"><a name="l00062"></a><span class="lineno"> 62</span> <span class="comment">//</span></div><div class="line"><a name="l00063"></a><span class="lineno"> 63</span> <span class="comment">// // Construct the LinearSumAssignment.</span></div><div class="line"><a name="l00064"></a><span class="lineno"> 64</span> <span class="comment">// ::operations_research::LinearSumAssignment<Graph> a(graph, num_left_nodes);</span></div><div class="line"><a name="l00065"></a><span class="lineno"> 65</span> <span class="comment">// for (int arc = 0; arc < num_arcs; ++arc) {</span></div><div class="line"><a name="l00066"></a><span class="lineno"> 66</span> <span class="comment">// // You can also replace 'arc_costs[arc]' by something like</span></div><div class="line"><a name="l00067"></a><span class="lineno"> 67</span> <span class="comment">// // ComputeArcCost(permutation.empty() ? arc : permutation[arc])</span></div><div class="line"><a name="l00068"></a><span class="lineno"> 68</span> <span class="comment">// // if you don't want to store the costs in arc_costs to save memory.</span></div><div class="line"><a name="l00069"></a><span class="lineno"> 69</span> <span class="comment">// a.SetArcCost(arc, arc_costs[arc]);</span></div><div class="line"><a name="l00070"></a><span class="lineno"> 70</span> <span class="comment">// }</span></div><div class="line"><a name="l00071"></a><span class="lineno"> 71</span> <span class="comment">//</span></div><div class="line"><a name="l00072"></a><span class="lineno"> 72</span> <span class="comment">// // Compute the optimum assignment.</span></div><div class="line"><a name="l00073"></a><span class="lineno"> 73</span> <span class="comment">// bool success = a.ComputeAssignment();</span></div><div class="line"><a name="l00074"></a><span class="lineno"> 74</span> <span class="comment">// // Retrieve the cost of the optimum assignment.</span></div><div class="line"><a name="l00075"></a><span class="lineno"> 75</span> <span class="comment">// operations_research::CostValue optimum_cost = a.GetCost();</span></div><div class="line"><a name="l00076"></a><span class="lineno"> 76</span> <span class="comment">// // Retrieve the node-node correspondence of the optimum assignment and the</span></div><div class="line"><a name="l00077"></a><span class="lineno"> 77</span> <span class="comment">// // cost of each node pairing.</span></div><div class="line"><a name="l00078"></a><span class="lineno"> 78</span> <span class="comment">// for (int left_node = 0; left_node < num_left_nodes; ++left_node) {</span></div><div class="line"><a name="l00079"></a><span class="lineno"> 79</span> <span class="comment">// const int right_node = a.GetMate(left_node);</span></div><div class="line"><a name="l00080"></a><span class="lineno"> 80</span> <span class="comment">// operations_research::CostValue node_pair_cost =</span></div><div class="line"><a name="l00081"></a><span class="lineno"> 81</span> <span class="comment">// a.GetAssignmentCost(left_node);</span></div><div class="line"><a name="l00082"></a><span class="lineno"> 82</span> <span class="comment">// ...</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">//</span></div><div class="line"><a name="l00085"></a><span class="lineno"> 85</span> <span class="comment">// In the following, we consider a bipartite graph</span></div><div class="line"><a name="l00086"></a><span class="lineno"> 86</span> <span class="comment">// G = (V = X union Y, E subset XxY),</span></div><div class="line"><a name="l00087"></a><span class="lineno"> 87</span> <span class="comment">// where V denotes the set of nodes (vertices) in the graph, E denotes</span></div><div class="line"><a name="l00088"></a><span class="lineno"> 88</span> <span class="comment">// the set of arcs (edges), n = |V| denotes the number of nodes in the</span></div><div class="line"><a name="l00089"></a><span class="lineno"> 89</span> <span class="comment">// graph, and m = |E| denotes the number of arcs in the graph.</span></div><div class="line"><a name="l00090"></a><span class="lineno"> 90</span> <span class="comment">//</span></div><div class="line"><a name="l00091"></a><span class="lineno"> 91</span> <span class="comment">// The set of nodes is divided into two parts, X and Y, and every arc</span></div><div class="line"><a name="l00092"></a><span class="lineno"> 92</span> <span class="comment">// must go between a node of X and a node of Y. With each arc is</span></div><div class="line"><a name="l00093"></a><span class="lineno"> 93</span> <span class="comment">// associated a cost c(v, w). A matching M is a subset of E with the</span></div><div class="line"><a name="l00094"></a><span class="lineno"> 94</span> <span class="comment">// property that no two arcs in M have a head or tail node in common,</span></div><div class="line"><a name="l00095"></a><span class="lineno"> 95</span> <span class="comment">// and a perfect matching is a matching that touches every node in the</span></div><div class="line"><a name="l00096"></a><span class="lineno"> 96</span> <span class="comment">// graph. The cost of a matching M is the sum of the costs of all the</span></div><div class="line"><a name="l00097"></a><span class="lineno"> 97</span> <span class="comment">// arcs in M.</span></div><div class="line"><a name="l00098"></a><span class="lineno"> 98</span> <span class="comment">//</span></div><div class="line"><a name="l00099"></a><span class="lineno"> 99</span> <span class="comment">// The assignment problem is to find a perfect matching of minimum</span></div><div class="line"><a name="l00100"></a><span class="lineno"> 100</span> <span class="comment">// cost in the given bipartite graph. The present algorithm reduces</span></div><div class="line"><a name="l00101"></a><span class="lineno"> 101</span> <span class="comment">// the assignment problem to an instance of the minimum-cost flow</span></div><div class="line"><a name="l00102"></a><span class="lineno"> 102</span> <span class="comment">// problem and takes advantage of special properties of the resulting</span></div><div class="line"><a name="l00103"></a><span class="lineno"> 103</span> <span class="comment">// minimum-cost flow problem to solve it efficiently using a</span></div><div class="line"><a name="l00104"></a><span class="lineno"> 104</span> <span class="comment">// push-relabel method. For more information about minimum-cost flow</span></div><div class="line"><a name="l00105"></a><span class="lineno"> 105</span> <span class="comment">// see ortools/graph/min_cost_flow.h</span></div><div class="line"><a name="l00106"></a><span class="lineno"> 106</span> <span class="comment">//</span></div><div class="line"><a name="l00107"></a><span class="lineno"> 107</span> <span class="comment">// The method used here is the cost-scaling approach for the</span></div><div class="line"><a name="l00108"></a><span class="lineno"> 108</span> <span class="comment">// minimum-cost circulation problem as described in [Goldberg and</span></div><div class="line"><a name="l00109"></a><span class="lineno"> 109</span> <span class="comment">// Tarjan] with some technical modifications:</span></div><div class="line"><a name="l00110"></a><span class="lineno"> 110</span> <span class="comment">// 1. For efficiency, we solve a transportation problem instead of</span></div><div class="line"><a name="l00111"></a><span class="lineno"> 111</span> <span class="comment">// minimum-cost circulation. We might revisit this decision if it</span></div><div class="line"><a name="l00112"></a><span class="lineno"> 112</span> <span class="comment">// is important to handle problems in which no perfect matching</span></div><div class="line"><a name="l00113"></a><span class="lineno"> 113</span> <span class="comment">// exists.</span></div><div class="line"><a name="l00114"></a><span class="lineno"> 114</span> <span class="comment">// 2. We use a modified "asymmetric" notion of epsilon-optimality in</span></div><div class="line"><a name="l00115"></a><span class="lineno"> 115</span> <span class="comment">// which left-to-right residual arcs are required to have reduced</span></div><div class="line"><a name="l00116"></a><span class="lineno"> 116</span> <span class="comment">// cost bounded below by zero and right-to-left residual arcs are</span></div><div class="line"><a name="l00117"></a><span class="lineno"> 117</span> <span class="comment">// required to have reduced cost bounded below by -epsilon. For</span></div><div class="line"><a name="l00118"></a><span class="lineno"> 118</span> <span class="comment">// each residual arc direction, the reduced-cost threshold for</span></div><div class="line"><a name="l00119"></a><span class="lineno"> 119</span> <span class="comment">// admissibility is epsilon/2 above the threshold for epsilon</span></div><div class="line"><a name="l00120"></a><span class="lineno"> 120</span> <span class="comment">// optimality.</span></div><div class="line"><a name="l00121"></a><span class="lineno"> 121</span> <span class="comment">// 3. We do not limit the applicability of the relabeling operation to</span></div><div class="line"><a name="l00122"></a><span class="lineno"> 122</span> <span class="comment">// nodes with excess. Instead we use the double-push operation</span></div><div class="line"><a name="l00123"></a><span class="lineno"> 123</span> <span class="comment">// (discussed in the Goldberg and Kennedy CSA paper and Kennedy's</span></div><div class="line"><a name="l00124"></a><span class="lineno"> 124</span> <span class="comment">// thesis) which relabels right-side nodes just *after* they have</span></div><div class="line"><a name="l00125"></a><span class="lineno"> 125</span> <span class="comment">// been discharged.</span></div><div class="line"><a name="l00126"></a><span class="lineno"> 126</span> <span class="comment">// The above differences are explained in detail in [Kennedy's thesis]</span></div><div class="line"><a name="l00127"></a><span class="lineno"> 127</span> <span class="comment">// and explained not quite as cleanly in [Goldberg and Kennedy's CSA</span></div><div class="line"><a name="l00128"></a><span class="lineno"> 128</span> <span class="comment">// paper]. But note that the thesis explanation uses a value of</span></div><div class="line"><a name="l00129"></a><span class="lineno"> 129</span> <span class="comment">// epsilon that's double what we use here.</span></div><div class="line"><a name="l00130"></a><span class="lineno"> 130</span> <span class="comment">//</span></div><div class="line"><a name="l00131"></a><span class="lineno"> 131</span> <span class="comment">// Some definitions:</span></div><div class="line"><a name="l00132"></a><span class="lineno"> 132</span> <span class="comment">// Active: A node is called active when it has excess. It is</span></div><div class="line"><a name="l00133"></a><span class="lineno"> 133</span> <span class="comment">// eligible to be pushed from. In this implementation, every active</span></div><div class="line"><a name="l00134"></a><span class="lineno"> 134</span> <span class="comment">// node is on the left side of the graph where prices are determined</span></div><div class="line"><a name="l00135"></a><span class="lineno"> 135</span> <span class="comment">// implicitly, so no left-side relabeling is necessary before</span></div><div class="line"><a name="l00136"></a><span class="lineno"> 136</span> <span class="comment">// pushing from an active node. We do, however, need to compute</span></div><div class="line"><a name="l00137"></a><span class="lineno"> 137</span> <span class="comment">// the implications for price changes on the affected right-side</span></div><div class="line"><a name="l00138"></a><span class="lineno"> 138</span> <span class="comment">// nodes.</span></div><div class="line"><a name="l00139"></a><span class="lineno"> 139</span> <span class="comment">// Admissible: A residual arc (one that can carry more flow) is</span></div><div class="line"><a name="l00140"></a><span class="lineno"> 140</span> <span class="comment">// called admissible when its reduced cost is small enough. We can</span></div><div class="line"><a name="l00141"></a><span class="lineno"> 141</span> <span class="comment">// push additional flow along such an arc without violating</span></div><div class="line"><a name="l00142"></a><span class="lineno"> 142</span> <span class="comment">// epsilon-optimality. In the case of a left-to-right residual</span></div><div class="line"><a name="l00143"></a><span class="lineno"> 143</span> <span class="comment">// arc, the reduced cost must be at most epsilon/2. In the case of</span></div><div class="line"><a name="l00144"></a><span class="lineno"> 144</span> <span class="comment">// a right-to-left residual arc, the reduced cost must be at</span></div><div class="line"><a name="l00145"></a><span class="lineno"> 145</span> <span class="comment">// most -epsilon/2. The careful reader will note that these thresholds</span></div><div class="line"><a name="l00146"></a><span class="lineno"> 146</span> <span class="comment">// are not used explicitly anywhere in this implementation, and</span></div><div class="line"><a name="l00147"></a><span class="lineno"> 147</span> <span class="comment">// the reason is the implicit pricing of left-side nodes.</span></div><div class="line"><a name="l00148"></a><span class="lineno"> 148</span> <span class="comment">// Reduced cost: Essentially an arc's reduced cost is its</span></div><div class="line"><a name="l00149"></a><span class="lineno"> 149</span> <span class="comment">// complementary slackness. In push-relabel algorithms this is</span></div><div class="line"><a name="l00150"></a><span class="lineno"> 150</span> <span class="comment">// c_p(v, w) = p(v) + c(v, w) - p(w),</span></div><div class="line"><a name="l00151"></a><span class="lineno"> 151</span> <span class="comment">// where p() is the node price function and c(v, w) is the cost of</span></div><div class="line"><a name="l00152"></a><span class="lineno"> 152</span> <span class="comment">// the arc from v to w. See min_cost_flow.h for more details.</span></div><div class="line"><a name="l00153"></a><span class="lineno"> 153</span> <span class="comment">// Partial reduced cost: We maintain prices implicitly for left-side</span></div><div class="line"><a name="l00154"></a><span class="lineno"> 154</span> <span class="comment">// nodes in this implementation, so instead of reduced costs we</span></div><div class="line"><a name="l00155"></a><span class="lineno"> 155</span> <span class="comment">// work with partial reduced costs, defined as</span></div><div class="line"><a name="l00156"></a><span class="lineno"> 156</span> <span class="comment">// c'_p(v, w) = c(v, w) - p(w).</span></div><div class="line"><a name="l00157"></a><span class="lineno"> 157</span> <span class="comment">//</span></div><div class="line"><a name="l00158"></a><span class="lineno"> 158</span> <span class="comment">// We check at initialization time for the possibility of arithmetic</span></div><div class="line"><a name="l00159"></a><span class="lineno"> 159</span> <span class="comment">// overflow and warn if the given costs are too large. In many cases</span></div><div class="line"><a name="l00160"></a><span class="lineno"> 160</span> <span class="comment">// the bound we use to trigger the warning is pessimistic so the given</span></div><div class="line"><a name="l00161"></a><span class="lineno"> 161</span> <span class="comment">// problem can often be solved even if we warn that overflow is</span></div><div class="line"><a name="l00162"></a><span class="lineno"> 162</span> <span class="comment">// possible.</span></div><div class="line"><a name="l00163"></a><span class="lineno"> 163</span> <span class="comment">//</span></div><div class="line"><a name="l00164"></a><span class="lineno"> 164</span> <span class="comment">// We don't use the interface from</span></div><div class="line"><a name="l00165"></a><span class="lineno"> 165</span> <span class="comment">// operations_research/algorithms/hungarian.h because we want to be</span></div><div class="line"><a name="l00166"></a><span class="lineno"> 166</span> <span class="comment">// able to express sparse problems efficiently.</span></div><div class="line"><a name="l00167"></a><span class="lineno"> 167</span> <span class="comment">//</span></div><div class="line"><a name="l00168"></a><span class="lineno"> 168</span> <span class="comment">// When asked to solve the given assignment problem we return a</span></div><div class="line"><a name="l00169"></a><span class="lineno"> 169</span> <span class="comment">// boolean to indicate whether the given problem was feasible.</span></div><div class="line"><a name="l00170"></a><span class="lineno"> 170</span> <span class="comment">//</span></div><div class="line"><a name="l00171"></a><span class="lineno"> 171</span> <span class="comment">// References:</span></div><div class="line"><a name="l00172"></a><span class="lineno"> 172</span> <span class="comment">// [ Goldberg and Kennedy's CSA paper ] A. V. Goldberg and R. Kennedy,</span></div><div class="line"><a name="l00173"></a><span class="lineno"> 173</span> <span class="comment">// "An Efficient Cost Scaling Algorithm for the Assignment Problem."</span></div><div class="line"><a name="l00174"></a><span class="lineno"> 174</span> <span class="comment">// Mathematical Programming, Vol. 71, pages 153-178, December 1995.</span></div><div class="line"><a name="l00175"></a><span class="lineno"> 175</span> <span class="comment">//</span></div><div class="line"><a name="l00176"></a><span class="lineno"> 176</span> <span class="comment">// [ Goldberg and Tarjan ] A. V. Goldberg and R. E. Tarjan, "Finding</span></div><div class="line"><a name="l00177"></a><span class="lineno"> 177</span> <span class="comment">// Minimum-Cost Circulations by Successive Approximation." Mathematics</span></div><div class="line"><a name="l00178"></a><span class="lineno"> 178</span> <span class="comment">// of Operations Research, Vol. 15, No. 3, pages 430-466, August 1990.</span></div><div class="line"><a name="l00179"></a><span class="lineno"> 179</span> <span class="comment">//</span></div><div class="line"><a name="l00180"></a><span class="lineno"> 180</span> <span class="comment">// [ Kennedy's thesis ] J. R. Kennedy, Jr., "Solving Unweighted and</span></div><div class="line"><a name="l00181"></a><span class="lineno"> 181</span> <span class="comment">// Weighted Bipartite Matching Problems in Theory and Practice."</span></div><div class="line"><a name="l00182"></a><span class="lineno"> 182</span> <span class="comment">// Stanford University Doctoral Dissertation, Department of Computer</span></div><div class="line"><a name="l00183"></a><span class="lineno"> 183</span> <span class="comment">// Science, 1995.</span></div><div class="line"><a name="l00184"></a><span class="lineno"> 184</span> <span class="comment">//</span></div><div class="line"><a name="l00185"></a><span class="lineno"> 185</span> <span class="comment">// [ Burkard et al. ] R. Burkard, M. Dell'Amico, S. Martello, "Assignment</span></div><div class="line"><a name="l00186"></a><span class="lineno"> 186</span> <span class="comment">// Problems", SIAM, 2009, ISBN: 978-0898716634,</span></div><div class="line"><a name="l00187"></a><span class="lineno"> 187</span> <span class="comment">// http://www.amazon.com/dp/0898716632/</span></div><div class="line"><a name="l00188"></a><span class="lineno"> 188</span> <span class="comment">//</span></div><div class="line"><a name="l00189"></a><span class="lineno"> 189</span> <span class="comment">// [ Ahuja et al. ] R. K. Ahuja, T. L. Magnanti, J. B. Orlin, "Network Flows:</span></div><div class="line"><a name="l00190"></a><span class="lineno"> 190</span> <span class="comment">// Theory, Algorithms, and Applications," Prentice Hall, 1993,</span></div><div class="line"><a name="l00191"></a><span class="lineno"> 191</span> <span class="comment">// ISBN: 978-0136175490, http://www.amazon.com/dp/013617549X.</span></div><div class="line"><a name="l00192"></a><span class="lineno"> 192</span> <span class="comment">//</span></div><div class="line"><a name="l00193"></a><span class="lineno"> 193</span> <span class="comment">// Keywords: linear sum assignment problem, Hungarian method, Goldberg, Kennedy.</span></div><div class="line"><a name="l00194"></a><span class="lineno"> 194</span> </div><div class="line"><a name="l00195"></a><span class="lineno"> 195</span> <span class="preprocessor">#ifndef OR_TOOLS_GRAPH_LINEAR_ASSIGNMENT_H_</span></div><div class="line"><a name="l00196"></a><span class="lineno"> 196</span> <span class="preprocessor">#define OR_TOOLS_GRAPH_LINEAR_ASSIGNMENT_H_</span></div><div class="line"><a name="l00197"></a><span class="lineno"> 197</span> </div><div class="line"><a name="l00198"></a><span class="lineno"> 198</span> <span class="preprocessor">#include <algorithm></span></div><div class="line"><a name="l00199"></a><span class="lineno"> 199</span> <span class="preprocessor">#include <cstdint></span></div><div class="line"><a name="l00200"></a><span class="lineno"> 200</span> <span class="preprocessor">#include <cstdlib></span></div><div class="line"><a name="l00201"></a><span class="lineno"> 201</span> <span class="preprocessor">#include <deque></span></div><div class="line"><a name="l00202"></a><span class="lineno"> 202</span> <span class="preprocessor">#include <limits></span></div><div class="line"><a name="l00203"></a><span class="lineno"> 203</span> <span class="preprocessor">#include <memory></span></div><div class="line"><a name="l00204"></a><span class="lineno"> 204</span> <span class="preprocessor">#include <string></span></div><div class="line"><a name="l00205"></a><span class="lineno"> 205</span> <span class="preprocessor">#include <utility></span></div><div class="line"><a name="l00206"></a><span class="lineno"> 206</span> <span class="preprocessor">#include <vector></span></div><div class="line"><a name="l00207"></a><span class="lineno"> 207</span> </div><div class="line"><a name="l00208"></a><span class="lineno"> 208</span> <span class="preprocessor">#include "absl/strings/str_format.h"</span></div><div class="line"><a name="l00209"></a><span class="lineno"> 209</span> <span class="preprocessor">#include "<a class="code" href="commandlineflags_8h.html">ortools/base/commandlineflags.h</a>"</span></div><div class="line"><a name="l00210"></a><span class="lineno"> 210</span> <span class="preprocessor">#include "<a class="code" href="integral__types_8h.html">ortools/base/integral_types.h</a>"</span></div><div class="line"><a name="l00211"></a><span class="lineno"> 211</span> <span class="preprocessor">#include "<a class="code" href="base_2logging_8h.html">ortools/base/logging.h</a>"</span></div><div class="line"><a name="l00212"></a><span class="lineno"> 212</span> <span class="preprocessor">#include "<a class="code" href="macros_8h.html">ortools/base/macros.h</a>"</span></div><div class="line"><a name="l00213"></a><span class="lineno"> 213</span> <span class="preprocessor">#include "<a class="code" href="ebert__graph_8h.html">ortools/graph/ebert_graph.h</a>"</span></div><div class="line"><a name="l00214"></a><span class="lineno"> 214</span> <span class="preprocessor">#include "<a class="code" href="util_2permutation_8h.html">ortools/util/permutation.h</a>"</span></div><div class="line"><a name="l00215"></a><span class="lineno"> 215</span> <span class="preprocessor">#include "<a class="code" href="zvector_8h.html">ortools/util/zvector.h</a>"</span></div><div class="line"><a name="l00216"></a><span class="lineno"> 216</span> </div><div class="line"><a name="l00217"></a><span class="lineno"> 217</span> <span class="preprocessor">#ifndef SWIG</span></div><div class="line"><a name="l00218"></a><span class="lineno"> 218</span> <a class="code" href="linear__assignment_8h.html#aaa4eff2168771ea36e49d66a895fb504">ABSL_DECLARE_FLAG</a>(int64_t, assignment_alpha);</div><div class="line"><a name="l00219"></a><span class="lineno"> 219</span> <a class="code" href="linear__assignment_8h.html#aaa4eff2168771ea36e49d66a895fb504">ABSL_DECLARE_FLAG</a>(<span class="keywordtype">int</span>, assignment_progress_logging_period);</div><div class="line"><a name="l00220"></a><span class="lineno"> 220</span> <a class="code" href="linear__assignment_8h.html#aaa4eff2168771ea36e49d66a895fb504">ABSL_DECLARE_FLAG</a>(<span class="keywordtype">bool</span>, assignment_stack_order);</div><div class="line"><a name="l00221"></a><span class="lineno"> 221</span> <span class="preprocessor">#endif</span></div><div class="line"><a name="l00222"></a><span class="lineno"> 222</span> </div><div class="line"><a name="l00223"></a><span class="lineno"> 223</span> <span class="keyword">namespace </span><a class="code" href="namespaceoperations__research.html">operations_research</a> {</div><div class="line"><a name="l00224"></a><span class="lineno"> 224</span> </div><div class="line"><a name="l00225"></a><span class="lineno"> 225</span> <span class="comment">// This class does not take ownership of its underlying graph.</span></div><div class="line"><a name="l00226"></a><span class="lineno"> 226</span> <span class="keyword">template</span> <<span class="keyword">typename</span> GraphType></div><div class="line"><a name="l00227"></a><span class="lineno"><a class="line" href="classoperations__research_1_1_linear_sum_assignment.html"> 227</a></span> <span class="keyword">class </span><a class="code" href="classoperations__research_1_1_linear_sum_assignment.html">LinearSumAssignment</a> {</div><div class="line"><a name="l00228"></a><span class="lineno"> 228</span>  <span class="keyword">public</span>:</div><div class="line"><a name="l00229"></a><span class="lineno"><a class="line" href="classoperations__research_1_1_linear_sum_assignment.html#aca73212d30b08a7c287c311b74311a6e"> 229</a></span>  <span class="keyword">typedef</span> <span class="keyword">typename</span> <a class="code" href="namespaceoperations__research.html#a7ae31ba4c3b4899478e53ca13df35dfc">GraphType::NodeIndex</a> <a class="code" href="classoperations__research_1_1_linear_sum_assignment.html#aca73212d30b08a7c287c311b74311a6e">NodeIndex</a>;</div><div class="line"><a name="l00230"></a><span class="lineno"><a class="line" href="classoperations__research_1_1_linear_sum_assignment.html#a28178bb3e97b7c81dc157fe283b8cd64"> 230</a></span>  <span class="keyword">typedef</span> <span class="keyword">typename</span> <a class="code" href="namespaceoperations__research.html#aecf320bda6e95d1edaf3a546673e3e6b">GraphType::ArcIndex</a> <a class="code" href="classoperations__research_1_1_linear_sum_assignment.html#a28178bb3e97b7c81dc157fe283b8cd64">ArcIndex</a>;</div><div class="line"><a name="l00231"></a><span class="lineno"> 231</span> </div><div class="line"><a name="l00232"></a><span class="lineno"> 232</span>  <span class="comment">// Constructor for the case in which we will build the graph</span></div><div class="line"><a name="l00233"></a><span class="lineno"> 233</span>  <span class="comment">// incrementally as we discover arc costs, as might be done with any</span></div><div class="line"><a name="l00234"></a><span class="lineno"> 234</span>  <span class="comment">// of the dynamic graph representations such as StarGraph or ForwardStarGraph.</span></div><div class="line"><a name="l00235"></a><span class="lineno"> 235</span>  <a class="code" href="classoperations__research_1_1_linear_sum_assignment.html#af45fdf861fabd6a46ce230a9f6101eda">LinearSumAssignment</a>(<span class="keyword">const</span> GraphType& graph, <a class="code" href="classoperations__research_1_1_linear_sum_assignment.html#aca73212d30b08a7c287c311b74311a6e">NodeIndex</a> num_left_nodes);</div><div class="line"><a name="l00236"></a><span class="lineno"> 236</span> </div><div class="line"><a name="l00237"></a><span class="lineno"> 237</span>  <span class="comment">// Constructor for the case in which the underlying graph cannot be</span></div><div class="line"><a name="l00238"></a><span class="lineno"> 238</span>  <span class="comment">// built until after all the arc costs are known, as is the case</span></div><div class="line"><a name="l00239"></a><span class="lineno"> 239</span>  <span class="comment">// with ForwardStarStaticGraph. In this case, the graph is passed to</span></div><div class="line"><a name="l00240"></a><span class="lineno"> 240</span>  <span class="comment">// us later via the SetGraph() method, below.</span></div><div class="line"><a name="l00241"></a><span class="lineno"> 241</span>  <a class="code" href="classoperations__research_1_1_linear_sum_assignment.html#af45fdf861fabd6a46ce230a9f6101eda">LinearSumAssignment</a>(<a class="code" href="classoperations__research_1_1_linear_sum_assignment.html#aca73212d30b08a7c287c311b74311a6e">NodeIndex</a> num_left_nodes, <a class="code" href="classoperations__research_1_1_linear_sum_assignment.html#a28178bb3e97b7c81dc157fe283b8cd64">ArcIndex</a> num_arcs);</div><div class="line"><a name="l00242"></a><span class="lineno"> 242</span> </div><div class="line"><a name="l00243"></a><span class="lineno"><a class="line" href="classoperations__research_1_1_linear_sum_assignment.html#a4f658e59dd3bd0b8cd1269e19b730cee"> 243</a></span>  <a class="code" href="classoperations__research_1_1_linear_sum_assignment.html#a4f658e59dd3bd0b8cd1269e19b730cee">~LinearSumAssignment</a>() {}</div><div class="line"><a name="l00244"></a><span class="lineno"> 244</span> </div><div class="line"><a name="l00245"></a><span class="lineno"> 245</span>  <span class="comment">// Sets the graph used by the LinearSumAssignment instance, for use</span></div><div class="line"><a name="l00246"></a><span class="lineno"> 246</span>  <span class="comment">// when the graph layout can be determined only after arc costs are</span></div><div class="line"><a name="l00247"></a><span class="lineno"> 247</span>  <span class="comment">// set. This happens, for example, when we use a ForwardStarStaticGraph.</span></div><div class="line"><a name="l00248"></a><span class="lineno"><a class="line" href="classoperations__research_1_1_linear_sum_assignment.html#aececfe5b0affea1dd1b8a38d8c1fb769"> 248</a></span>  <span class="keywordtype">void</span> <a class="code" href="classoperations__research_1_1_linear_sum_assignment.html#aececfe5b0affea1dd1b8a38d8c1fb769">SetGraph</a>(<span class="keyword">const</span> GraphType* graph) {</div><div class="line"><a name="l00249"></a><span class="lineno"> 249</span>  <a class="code" href="base_2logging_8h.html#ae17f8119c108cf3070bad3449c7e0006">DCHECK</a>(graph_ == <span class="keyword">nullptr</span>);</div><div class="line"><a name="l00250"></a><span class="lineno"> 250</span>  graph_ = graph;</div><div class="line"><a name="l00251"></a><span class="lineno"> 251</span>  }</div><div class="line"><a name="l00252"></a><span class="lineno"> 252</span> </div><div class="line"><a name="l00253"></a><span class="lineno"> 253</span>  <span class="comment">// Sets the cost-scaling divisor, i.e., the amount by which we</span></div><div class="line"><a name="l00254"></a><span class="lineno"> 254</span>  <span class="comment">// divide the scaling parameter on each iteration.</span></div><div class="line"><a name="l00255"></a><span class="lineno"><a class="line" href="classoperations__research_1_1_linear_sum_assignment.html#accac1fc7c4ac9bff1591ec627a59a4f7"> 255</a></span>  <span class="keywordtype">void</span> <a class="code" href="classoperations__research_1_1_linear_sum_assignment.html#accac1fc7c4ac9bff1591ec627a59a4f7">SetCostScalingDivisor</a>(<a class="code" href="namespaceoperations__research.html#aee97ac67f280d35acdef2c5d461a85c3">CostValue</a> factor) { alpha_ = factor; }</div><div class="line"><a name="l00256"></a><span class="lineno"> 256</span> </div><div class="line"><a name="l00257"></a><span class="lineno"> 257</span>  <span class="comment">// Returns a permutation cycle handler that can be passed to the</span></div><div class="line"><a name="l00258"></a><span class="lineno"> 258</span>  <span class="comment">// TransformToForwardStaticGraph method so that arc costs get</span></div><div class="line"><a name="l00259"></a><span class="lineno"> 259</span>  <span class="comment">// permuted along with arcs themselves.</span></div><div class="line"><a name="l00260"></a><span class="lineno"> 260</span>  <span class="comment">//</span></div><div class="line"><a name="l00261"></a><span class="lineno"> 261</span>  <span class="comment">// Passes ownership of the cycle handler to the caller.</span></div><div class="line"><a name="l00262"></a><span class="lineno"> 262</span>  <span class="comment">//</span></div><div class="line"><a name="l00263"></a><span class="lineno"> 263</span>  <a class="code" href="classoperations__research_1_1_permutation_cycle_handler.html">operations_research::PermutationCycleHandler<typename GraphType::ArcIndex></a>*</div><div class="line"><a name="l00264"></a><span class="lineno"> 264</span>  <a class="code" href="classoperations__research_1_1_linear_sum_assignment.html#ab70deec1644ce0de2f383c97830391e7">ArcAnnotationCycleHandler</a>();</div><div class="line"><a name="l00265"></a><span class="lineno"> 265</span> </div><div class="line"><a name="l00266"></a><span class="lineno"> 266</span>  <span class="comment">// Optimizes the layout of the graph for the access pattern our</span></div><div class="line"><a name="l00267"></a><span class="lineno"> 267</span>  <span class="comment">// implementation will use.</span></div><div class="line"><a name="l00268"></a><span class="lineno"> 268</span>  <span class="comment">//</span></div><div class="line"><a name="l00269"></a><span class="lineno"> 269</span>  <span class="comment">// REQUIRES for LinearSumAssignment template instantiation if a call</span></div><div class="line"><a name="l00270"></a><span class="lineno"> 270</span>  <span class="comment">// to the OptimizeGraphLayout() method is compiled: GraphType is a</span></div><div class="line"><a name="l00271"></a><span class="lineno"> 271</span>  <span class="comment">// dynamic graph, i.e., one that implements the</span></div><div class="line"><a name="l00272"></a><span class="lineno"> 272</span>  <span class="comment">// GroupForwardArcsByFunctor() member template method.</span></div><div class="line"><a name="l00273"></a><span class="lineno"> 273</span>  <span class="comment">//</span></div><div class="line"><a name="l00274"></a><span class="lineno"> 274</span>  <span class="comment">// If analogous optimization is needed for LinearSumAssignment</span></div><div class="line"><a name="l00275"></a><span class="lineno"> 275</span>  <span class="comment">// instances based on static graphs, the graph layout should be</span></div><div class="line"><a name="l00276"></a><span class="lineno"> 276</span>  <span class="comment">// constructed such that each node's outgoing arcs are sorted by</span></div><div class="line"><a name="l00277"></a><span class="lineno"> 277</span>  <span class="comment">// head node index before the</span></div><div class="line"><a name="l00278"></a><span class="lineno"> 278</span>  <span class="comment">// LinearSumAssignment<GraphType>::SetGraph() method is called.</span></div><div class="line"><a name="l00279"></a><span class="lineno"> 279</span>  <span class="keywordtype">void</span> <a class="code" href="classoperations__research_1_1_linear_sum_assignment.html#aa4b556b7bee4c7f3fb60781e7ebda1b2">OptimizeGraphLayout</a>(GraphType* graph);</div><div class="line"><a name="l00280"></a><span class="lineno"> 280</span> </div><div class="line"><a name="l00281"></a><span class="lineno"> 281</span>  <span class="comment">// Allows tests, iterators, etc., to inspect our underlying graph.</span></div><div class="line"><a name="l00282"></a><span class="lineno"><a class="line" href="classoperations__research_1_1_linear_sum_assignment.html#adb325b7bb71f0f00df13383ad0b2adae"> 282</a></span>  <span class="keyword">inline</span> <span class="keyword">const</span> GraphType& <a class="code" href="classoperations__research_1_1_linear_sum_assignment.html#adb325b7bb71f0f00df13383ad0b2adae">Graph</a>()<span class="keyword"> const </span>{ <span class="keywordflow">return</span> *graph_; }</div><div class="line"><a name="l00283"></a><span class="lineno"> 283</span> </div><div class="line"><a name="l00284"></a><span class="lineno"> 284</span>  <span class="comment">// These handy member functions make the code more compact, and we</span></div><div class="line"><a name="l00285"></a><span class="lineno"> 285</span>  <span class="comment">// expose them to clients so that client code that doesn't have</span></div><div class="line"><a name="l00286"></a><span class="lineno"> 286</span>  <span class="comment">// direct access to the graph can learn about the optimum assignment</span></div><div class="line"><a name="l00287"></a><span class="lineno"> 287</span>  <span class="comment">// once it is computed.</span></div><div class="line"><a name="l00288"></a><span class="lineno"><a class="line" href="classoperations__research_1_1_linear_sum_assignment.html#aa5f729a6274027e5e5478e4bd76603ca"> 288</a></span>  <span class="keyword">inline</span> <a class="code" href="classoperations__research_1_1_linear_sum_assignment.html#aca73212d30b08a7c287c311b74311a6e">NodeIndex</a> <a class="code" href="classoperations__research_1_1_linear_sum_assignment.html#aa5f729a6274027e5e5478e4bd76603ca">Head</a>(<a class="code" href="classoperations__research_1_1_linear_sum_assignment.html#a28178bb3e97b7c81dc157fe283b8cd64">ArcIndex</a> arc)<span class="keyword"> const </span>{ <span class="keywordflow">return</span> graph_->Head(arc); }</div><div class="line"><a name="l00289"></a><span class="lineno"> 289</span> </div><div class="line"><a name="l00290"></a><span class="lineno"> 290</span>  <span class="comment">// Returns the original arc cost for use by a client that's</span></div><div class="line"><a name="l00291"></a><span class="lineno"> 291</span>  <span class="comment">// iterating over the optimum assignment.</span></div><div class="line"><a name="l00292"></a><span class="lineno"><a class="line" href="classoperations__research_1_1_linear_sum_assignment.html#aa64fbe18913fc55f5d41182666751728"> 292</a></span>  <a class="code" href="namespaceoperations__research.html#aee97ac67f280d35acdef2c5d461a85c3">CostValue</a> <a class="code" href="classoperations__research_1_1_linear_sum_assignment.html#aa64fbe18913fc55f5d41182666751728">ArcCost</a>(<a class="code" href="classoperations__research_1_1_linear_sum_assignment.html#a28178bb3e97b7c81dc157fe283b8cd64">ArcIndex</a> arc)<span class="keyword"> const </span>{</div><div class="line"><a name="l00293"></a><span class="lineno"> 293</span>  <a class="code" href="base_2logging_8h.html#ae89df3243bbb8341130c7b3f44145ea0">DCHECK_EQ</a>(0, scaled_arc_cost_[arc] % cost_scaling_factor_);</div><div class="line"><a name="l00294"></a><span class="lineno"> 294</span>  <span class="keywordflow">return</span> scaled_arc_cost_[arc] / cost_scaling_factor_;</div><div class="line"><a name="l00295"></a><span class="lineno"> 295</span>  }</div><div class="line"><a name="l00296"></a><span class="lineno"> 296</span> </div><div class="line"><a name="l00297"></a><span class="lineno"> 297</span>  <span class="comment">// Sets the cost of an arc already present in the given graph.</span></div><div class="line"><a name="l00298"></a><span class="lineno"> 298</span>  <span class="keywordtype">void</span> <a class="code" href="classoperations__research_1_1_linear_sum_assignment.html#a77a0519df5fb71834593bb661b72921c">SetArcCost</a>(<a class="code" href="namespaceoperations__research.html#aecf320bda6e95d1edaf3a546673e3e6b">ArcIndex</a> arc, <a class="code" href="namespaceoperations__research.html#aee97ac67f280d35acdef2c5d461a85c3">CostValue</a> <a class="code" href="routing__flow_8cc.html#a75d7b5e4cab1e156cc7a2c5eba1e16f1">cost</a>);</div><div class="line"><a name="l00299"></a><span class="lineno"> 299</span> </div><div class="line"><a name="l00300"></a><span class="lineno"> 300</span>  <span class="comment">// Completes initialization after the problem is fully specified.</span></div><div class="line"><a name="l00301"></a><span class="lineno"> 301</span>  <span class="comment">// Returns true if we successfully prove that arithmetic</span></div><div class="line"><a name="l00302"></a><span class="lineno"> 302</span>  <span class="comment">// calculations are guaranteed not to overflow. ComputeAssignment()</span></div><div class="line"><a name="l00303"></a><span class="lineno"> 303</span>  <span class="comment">// calls this method itself, so only clients that care about</span></div><div class="line"><a name="l00304"></a><span class="lineno"> 304</span>  <span class="comment">// obtaining a warning about the possibility of arithmetic precision</span></div><div class="line"><a name="l00305"></a><span class="lineno"> 305</span>  <span class="comment">// problems need to call this method explicitly.</span></div><div class="line"><a name="l00306"></a><span class="lineno"> 306</span>  <span class="comment">//</span></div><div class="line"><a name="l00307"></a><span class="lineno"> 307</span>  <span class="comment">// Separate from ComputeAssignment() for white-box testing and for</span></div><div class="line"><a name="l00308"></a><span class="lineno"> 308</span>  <span class="comment">// clients that need to react to the possibility that arithmetic</span></div><div class="line"><a name="l00309"></a><span class="lineno"> 309</span>  <span class="comment">// overflow is not ruled out.</span></div><div class="line"><a name="l00310"></a><span class="lineno"> 310</span>  <span class="comment">//</span></div><div class="line"><a name="l00311"></a><span class="lineno"> 311</span>  <span class="comment">// FinalizeSetup() is idempotent.</span></div><div class="line"><a name="l00312"></a><span class="lineno"> 312</span>  <span class="keywordtype">bool</span> <a class="code" href="classoperations__research_1_1_linear_sum_assignment.html#afb41339280a9b3dd128f9a376ec38007">FinalizeSetup</a>();</div><div class="line"><a name="l00313"></a><span class="lineno"> 313</span> </div><div class="line"><a name="l00314"></a><span class="lineno"> 314</span>  <span class="comment">// Computes the optimum assignment. Returns true on success. Return</span></div><div class="line"><a name="l00315"></a><span class="lineno"> 315</span>  <span class="comment">// value of false implies the given problem is infeasible.</span></div><div class="line"><a name="l00316"></a><span class="lineno"> 316</span>  <span class="keywordtype">bool</span> <a class="code" href="classoperations__research_1_1_linear_sum_assignment.html#a63b3d12e721188086870cc42cc46a258">ComputeAssignment</a>();</div><div class="line"><a name="l00317"></a><span class="lineno"> 317</span> </div><div class="line"><a name="l00318"></a><span class="lineno"> 318</span>  <span class="comment">// Returns the cost of the minimum-cost perfect matching.</span></div><div class="line"><a name="l00319"></a><span class="lineno"> 319</span>  <span class="comment">// Precondition: success_ == true, signifying that we computed the</span></div><div class="line"><a name="l00320"></a><span class="lineno"> 320</span>  <span class="comment">// optimum assignment for a feasible problem.</span></div><div class="line"><a name="l00321"></a><span class="lineno"> 321</span>  <a class="code" href="namespaceoperations__research.html#aee97ac67f280d35acdef2c5d461a85c3">CostValue</a> <a class="code" href="classoperations__research_1_1_linear_sum_assignment.html#a5afb9dde1c31f5f053cc83bf6e594db0">GetCost</a>() <span class="keyword">const</span>;</div><div class="line"><a name="l00322"></a><span class="lineno"> 322</span> </div><div class="line"><a name="l00323"></a><span class="lineno"> 323</span>  <span class="comment">// Returns the total number of nodes in the given problem.</span></div><div class="line"><a name="l00324"></a><span class="lineno"><a class="line" href="classoperations__research_1_1_linear_sum_assignment.html#a3be0f6fcc44bc6a4a1e976c1e60b31d0"> 324</a></span>  <a class="code" href="classoperations__research_1_1_linear_sum_assignment.html#aca73212d30b08a7c287c311b74311a6e">NodeIndex</a> <a class="code" href="classoperations__research_1_1_linear_sum_assignment.html#a3be0f6fcc44bc6a4a1e976c1e60b31d0">NumNodes</a>()<span class="keyword"> const </span>{</div><div class="line"><a name="l00325"></a><span class="lineno"> 325</span>  <span class="keywordflow">if</span> (graph_ == <span class="keyword">nullptr</span>) {</div><div class="line"><a name="l00326"></a><span class="lineno"> 326</span>  <span class="comment">// Return a guess that must be true if ultimately we are given a</span></div><div class="line"><a name="l00327"></a><span class="lineno"> 327</span>  <span class="comment">// feasible problem to solve.</span></div><div class="line"><a name="l00328"></a><span class="lineno"> 328</span>  <span class="keywordflow">return</span> 2 * <a class="code" href="classoperations__research_1_1_linear_sum_assignment.html#a11815dc60d6275c8272be0771883d573">NumLeftNodes</a>();</div><div class="line"><a name="l00329"></a><span class="lineno"> 329</span>  } <span class="keywordflow">else</span> {</div><div class="line"><a name="l00330"></a><span class="lineno"> 330</span>  <span class="keywordflow">return</span> graph_->num_nodes();</div><div class="line"><a name="l00331"></a><span class="lineno"> 331</span>  }</div><div class="line"><a name="l00332"></a><span class="lineno"> 332</span>  }</div><div class="line"><a name="l00333"></a><span class="lineno"> 333</span> </div><div class="line"><a name="l00334"></a><span class="lineno"> 334</span>  <span class="comment">// Returns the number of nodes on the left side of the given</span></div><div class="line"><a name="l00335"></a><span class="lineno"> 335</span>  <span class="comment">// problem.</span></div><div class="line"><a name="l00336"></a><span class="lineno"><a class="line" href="classoperations__research_1_1_linear_sum_assignment.html#a11815dc60d6275c8272be0771883d573"> 336</a></span>  <a class="code" href="classoperations__research_1_1_linear_sum_assignment.html#aca73212d30b08a7c287c311b74311a6e">NodeIndex</a> <a class="code" href="classoperations__research_1_1_linear_sum_assignment.html#a11815dc60d6275c8272be0771883d573">NumLeftNodes</a>()<span class="keyword"> const </span>{ <span class="keywordflow">return</span> num_left_nodes_; }</div><div class="line"><a name="l00337"></a><span class="lineno"> 337</span> </div><div class="line"><a name="l00338"></a><span class="lineno"> 338</span>  <span class="comment">// Returns the arc through which the given node is matched.</span></div><div class="line"><a name="l00339"></a><span class="lineno"><a class="line" href="classoperations__research_1_1_linear_sum_assignment.html#a13537686f25d401f7112b9fa4f63d1ab"> 339</a></span>  <span class="keyword">inline</span> <a class="code" href="classoperations__research_1_1_linear_sum_assignment.html#a28178bb3e97b7c81dc157fe283b8cd64">ArcIndex</a> <a class="code" href="classoperations__research_1_1_linear_sum_assignment.html#a13537686f25d401f7112b9fa4f63d1ab">GetAssignmentArc</a>(<a class="code" href="classoperations__research_1_1_linear_sum_assignment.html#aca73212d30b08a7c287c311b74311a6e">NodeIndex</a> left_node)<span class="keyword"> const </span>{</div><div class="line"><a name="l00340"></a><span class="lineno"> 340</span>  <a class="code" href="base_2logging_8h.html#ab62f5ed8f2d48e29802be0cbbcd1359a">DCHECK_LT</a>(left_node, num_left_nodes_);</div><div class="line"><a name="l00341"></a><span class="lineno"> 341</span>  <span class="keywordflow">return</span> matched_arc_[left_node];</div><div class="line"><a name="l00342"></a><span class="lineno"> 342</span>  }</div><div class="line"><a name="l00343"></a><span class="lineno"> 343</span> </div><div class="line"><a name="l00344"></a><span class="lineno"> 344</span>  <span class="comment">// Returns the cost of the assignment arc incident to the given</span></div><div class="line"><a name="l00345"></a><span class="lineno"> 345</span>  <span class="comment">// node.</span></div><div class="line"><a name="l00346"></a><span class="lineno"><a class="line" href="classoperations__research_1_1_linear_sum_assignment.html#aa9b122f6d2e28bf78a0339f26cfbd432"> 346</a></span>  <span class="keyword">inline</span> <a class="code" href="namespaceoperations__research.html#aee97ac67f280d35acdef2c5d461a85c3">CostValue</a> <a class="code" href="classoperations__research_1_1_linear_sum_assignment.html#aa9b122f6d2e28bf78a0339f26cfbd432">GetAssignmentCost</a>(<a class="code" href="classoperations__research_1_1_linear_sum_assignment.html#aca73212d30b08a7c287c311b74311a6e">NodeIndex</a> node)<span class="keyword"> const </span>{</div><div class="line"><a name="l00347"></a><span class="lineno"> 347</span>  <span class="keywordflow">return</span> <a class="code" href="classoperations__research_1_1_linear_sum_assignment.html#aa64fbe18913fc55f5d41182666751728">ArcCost</a>(<a class="code" href="classoperations__research_1_1_linear_sum_assignment.html#a13537686f25d401f7112b9fa4f63d1ab">GetAssignmentArc</a>(node));</div><div class="line"><a name="l00348"></a><span class="lineno"> 348</span>  }</div><div class="line"><a name="l00349"></a><span class="lineno"> 349</span> </div><div class="line"><a name="l00350"></a><span class="lineno"> 350</span>  <span class="comment">// Returns the node to which the given node is matched.</span></div><div class="line"><a name="l00351"></a><span class="lineno"><a class="line" href="classoperations__research_1_1_linear_sum_assignment.html#ad253fe3f1dd016a976d0713d8a54a0ab"> 351</a></span>  <span class="keyword">inline</span> <a class="code" href="classoperations__research_1_1_linear_sum_assignment.html#aca73212d30b08a7c287c311b74311a6e">NodeIndex</a> <a class="code" href="classoperations__research_1_1_linear_sum_assignment.html#ad253fe3f1dd016a976d0713d8a54a0ab">GetMate</a>(<a class="code" href="classoperations__research_1_1_linear_sum_assignment.html#aca73212d30b08a7c287c311b74311a6e">NodeIndex</a> left_node)<span class="keyword"> const </span>{</div><div class="line"><a name="l00352"></a><span class="lineno"> 352</span>  <a class="code" href="base_2logging_8h.html#ab62f5ed8f2d48e29802be0cbbcd1359a">DCHECK_LT</a>(left_node, num_left_nodes_);</div><div class="line"><a name="l00353"></a><span class="lineno"> 353</span>  <a class="code" href="classoperations__research_1_1_linear_sum_assignment.html#a28178bb3e97b7c81dc157fe283b8cd64">ArcIndex</a> matching_arc = <a class="code" href="classoperations__research_1_1_linear_sum_assignment.html#a13537686f25d401f7112b9fa4f63d1ab">GetAssignmentArc</a>(left_node);</div><div class="line"><a name="l00354"></a><span class="lineno"> 354</span>  <a class="code" href="base_2logging_8h.html#a46e69120fbd3b36e6960e096d23b66f0">DCHECK_NE</a>(GraphType::kNilArc, matching_arc);</div><div class="line"><a name="l00355"></a><span class="lineno"> 355</span>  <span class="keywordflow">return</span> <a class="code" href="classoperations__research_1_1_linear_sum_assignment.html#aa5f729a6274027e5e5478e4bd76603ca">Head</a>(matching_arc);</div><div class="line"><a name="l00356"></a><span class="lineno"> 356</span>  }</div><div class="line"><a name="l00357"></a><span class="lineno"> 357</span> </div><div class="line"><a name="l00358"></a><span class="lineno"><a class="line" href="classoperations__research_1_1_linear_sum_assignment.html#a1286f5a02e4b2a9e89431626e12fd498"> 358</a></span>  std::string <a class="code" href="classoperations__research_1_1_linear_sum_assignment.html#a1286f5a02e4b2a9e89431626e12fd498">StatsString</a>()<span class="keyword"> const </span>{ <span class="keywordflow">return</span> total_stats_.StatsString(); }</div><div class="line"><a name="l00359"></a><span class="lineno"> 359</span> </div><div class="line"><a name="l00360"></a><span class="lineno"><a class="line" href="classoperations__research_1_1_linear_sum_assignment_1_1_bipartite_left_node_iterator.html"> 360</a></span>  <span class="keyword">class </span><a class="code" href="classoperations__research_1_1_linear_sum_assignment_1_1_bipartite_left_node_iterator.html">BipartiteLeftNodeIterator</a> {</div><div class="line"><a name="l00361"></a><span class="lineno"> 361</span>  <span class="keyword">public</span>:</div><div class="line"><a name="l00362"></a><span class="lineno"><a class="line" href="classoperations__research_1_1_linear_sum_assignment_1_1_bipartite_left_node_iterator.html#a64734b3e0ff85820aba52f5109432f51"> 362</a></span>  <a class="code" href="classoperations__research_1_1_linear_sum_assignment_1_1_bipartite_left_node_iterator.html#a64734b3e0ff85820aba52f5109432f51">BipartiteLeftNodeIterator</a>(<span class="keyword">const</span> GraphType& graph, <a class="code" href="classoperations__research_1_1_linear_sum_assignment.html#aca73212d30b08a7c287c311b74311a6e">NodeIndex</a> num_left_nodes)</div><div class="line"><a name="l00363"></a><span class="lineno"> 363</span>  : num_left_nodes_(num_left_nodes), node_iterator_(0) {}</div><div class="line"><a name="l00364"></a><span class="lineno"> 364</span> </div><div class="line"><a name="l00365"></a><span class="lineno"><a class="line" href="classoperations__research_1_1_linear_sum_assignment_1_1_bipartite_left_node_iterator.html#ab6dfabf93ff38b84d807d6190e88011f"> 365</a></span>  <span class="keyword">explicit</span> <a class="code" href="classoperations__research_1_1_linear_sum_assignment_1_1_bipartite_left_node_iterator.html#ab6dfabf93ff38b84d807d6190e88011f">BipartiteLeftNodeIterator</a>(<span class="keyword">const</span> <a class="code" href="classoperations__research_1_1_linear_sum_assignment.html">LinearSumAssignment</a>& assignment)</div><div class="line"><a name="l00366"></a><span class="lineno"> 366</span>  : num_left_nodes_(assignment.<a class="code" href="classoperations__research_1_1_linear_sum_assignment.html#a11815dc60d6275c8272be0771883d573">NumLeftNodes</a>()), node_iterator_(0) {}</div><div class="line"><a name="l00367"></a><span class="lineno"> 367</span> </div><div class="line"><a name="l00368"></a><span class="lineno"><a class="line" href="classoperations__research_1_1_linear_sum_assignment_1_1_bipartite_left_node_iterator.html#afadba1101a25565c8c207f7e81a67950"> 368</a></span>  <a class="code" href="classoperations__research_1_1_linear_sum_assignment.html#aca73212d30b08a7c287c311b74311a6e">NodeIndex</a> <a class="code" href="classoperations__research_1_1_linear_sum_assignment_1_1_bipartite_left_node_iterator.html#afadba1101a25565c8c207f7e81a67950">Index</a>()<span class="keyword"> const </span>{ <span class="keywordflow">return</span> node_iterator_; }</div><div class="line"><a name="l00369"></a><span class="lineno"> 369</span> </div><div class="line"><a name="l00370"></a><span class="lineno"><a class="line" href="classoperations__research_1_1_linear_sum_assignment_1_1_bipartite_left_node_iterator.html#aa7e07ffe21ad6b4c71a0d22c65f30347"> 370</a></span>  <span class="keywordtype">bool</span> <a class="code" href="classoperations__research_1_1_linear_sum_assignment_1_1_bipartite_left_node_iterator.html#aa7e07ffe21ad6b4c71a0d22c65f30347">Ok</a>()<span class="keyword"> const </span>{ <span class="keywordflow">return</span> node_iterator_ < num_left_nodes_; }</div><div class="line"><a name="l00371"></a><span class="lineno"> 371</span> </div><div class="line"><a name="l00372"></a><span class="lineno"><a class="line" href="classoperations__research_1_1_linear_sum_assignment_1_1_bipartite_left_node_iterator.html#a659a293dd51073a1b9560bb80f687705"> 372</a></span>  <span class="keywordtype">void</span> <a class="code" href="classoperations__research_1_1_linear_sum_assignment_1_1_bipartite_left_node_iterator.html#a659a293dd51073a1b9560bb80f687705">Next</a>() { ++node_iterator_; }</div><div class="line"><a name="l00373"></a><span class="lineno"> 373</span> </div><div class="line"><a name="l00374"></a><span class="lineno"> 374</span>  <span class="keyword">private</span>:</div><div class="line"><a name="l00375"></a><span class="lineno"> 375</span>  <span class="keyword">const</span> <a class="code" href="classoperations__research_1_1_linear_sum_assignment.html#aca73212d30b08a7c287c311b74311a6e">NodeIndex</a> num_left_nodes_;</div><div class="line"><a name="l00376"></a><span class="lineno"> 376</span>  <span class="keyword">typename</span> <a class="code" href="namespaceoperations__research.html#a7ae31ba4c3b4899478e53ca13df35dfc">GraphType::NodeIndex</a> node_iterator_;</div><div class="line"><a name="l00377"></a><span class="lineno"> 377</span>  };</div><div class="line"><a name="l00378"></a><span class="lineno"> 378</span> </div><div class="line"><a name="l00379"></a><span class="lineno"> 379</span>  <span class="keyword">private</span>:</div><div class="line"><a name="l00380"></a><span class="lineno"> 380</span>  <span class="keyword">struct </span>Stats {</div><div class="line"><a name="l00381"></a><span class="lineno"> 381</span>  Stats() : pushes_(0), double_pushes_(0), relabelings_(0), refinements_(0) {}</div><div class="line"><a name="l00382"></a><span class="lineno"> 382</span>  <span class="keywordtype">void</span> Clear() {</div><div class="line"><a name="l00383"></a><span class="lineno"> 383</span>  pushes_ = 0;</div><div class="line"><a name="l00384"></a><span class="lineno"> 384</span>  double_pushes_ = 0;</div><div class="line"><a name="l00385"></a><span class="lineno"> 385</span>  relabelings_ = 0;</div><div class="line"><a name="l00386"></a><span class="lineno"> 386</span>  refinements_ = 0;</div><div class="line"><a name="l00387"></a><span class="lineno"> 387</span>  }</div><div class="line"><a name="l00388"></a><span class="lineno"> 388</span>  <span class="keywordtype">void</span> Add(<span class="keyword">const</span> Stats& that) {</div><div class="line"><a name="l00389"></a><span class="lineno"> 389</span>  pushes_ += that.pushes_;</div><div class="line"><a name="l00390"></a><span class="lineno"> 390</span>  double_pushes_ += that.double_pushes_;</div><div class="line"><a name="l00391"></a><span class="lineno"> 391</span>  relabelings_ += that.relabelings_;</div><div class="line"><a name="l00392"></a><span class="lineno"> 392</span>  refinements_ += that.refinements_;</div><div class="line"><a name="l00393"></a><span class="lineno"> 393</span>  }</div><div class="line"><a name="l00394"></a><span class="lineno"> 394</span>  std::string <a class="code" href="classoperations__research_1_1_linear_sum_assignment.html#a1286f5a02e4b2a9e89431626e12fd498">StatsString</a>()<span class="keyword"> const </span>{</div><div class="line"><a name="l00395"></a><span class="lineno"> 395</span>  <span class="keywordflow">return</span> absl::StrFormat(</div><div class="line"><a name="l00396"></a><span class="lineno"> 396</span>  <span class="stringliteral">"%d refinements; %d relabelings; "</span></div><div class="line"><a name="l00397"></a><span class="lineno"> 397</span>  <span class="stringliteral">"%d double pushes; %d pushes"</span>,</div><div class="line"><a name="l00398"></a><span class="lineno"> 398</span>  refinements_, relabelings_, double_pushes_, pushes_);</div><div class="line"><a name="l00399"></a><span class="lineno"> 399</span>  }</div><div class="line"><a name="l00400"></a><span class="lineno"> 400</span>  int64_t pushes_;</div><div class="line"><a name="l00401"></a><span class="lineno"> 401</span>  int64_t double_pushes_;</div><div class="line"><a name="l00402"></a><span class="lineno"> 402</span>  int64_t relabelings_;</div><div class="line"><a name="l00403"></a><span class="lineno"> 403</span>  int64_t refinements_;</div><div class="line"><a name="l00404"></a><span class="lineno"> 404</span>  };</div><div class="line"><a name="l00405"></a><span class="lineno"> 405</span> </div><div class="line"><a name="l00406"></a><span class="lineno"> 406</span> <span class="preprocessor">#ifndef SWIG</span></div><div class="line"><a name="l00407"></a><span class="lineno"> 407</span>  <span class="keyword">class </span>ActiveNodeContainerInterface {</div><div class="line"><a name="l00408"></a><span class="lineno"> 408</span>  <span class="keyword">public</span>:</div><div class="line"><a name="l00409"></a><span class="lineno"> 409</span>  <span class="keyword">virtual</span> ~ActiveNodeContainerInterface() {}</div><div class="line"><a name="l00410"></a><span class="lineno"> 410</span>  <span class="keyword">virtual</span> <span class="keywordtype">bool</span> Empty() <span class="keyword">const</span> = 0;</div><div class="line"><a name="l00411"></a><span class="lineno"> 411</span>  <span class="keyword">virtual</span> <span class="keywordtype">void</span> Add(<a class="code" href="classoperations__research_1_1_linear_sum_assignment.html#aca73212d30b08a7c287c311b74311a6e">NodeIndex</a> node) = 0;</div><div class="line"><a name="l00412"></a><span class="lineno"> 412</span>  <span class="keyword">virtual</span> <a class="code" href="classoperations__research_1_1_linear_sum_assignment.html#aca73212d30b08a7c287c311b74311a6e">NodeIndex</a> Get() = 0;</div><div class="line"><a name="l00413"></a><span class="lineno"> 413</span>  };</div><div class="line"><a name="l00414"></a><span class="lineno"> 414</span> </div><div class="line"><a name="l00415"></a><span class="lineno"> 415</span>  <span class="keyword">class </span>ActiveNodeStack : <span class="keyword">public</span> ActiveNodeContainerInterface {</div><div class="line"><a name="l00416"></a><span class="lineno"> 416</span>  <span class="keyword">public</span>:</div><div class="line"><a name="l00417"></a><span class="lineno"> 417</span>  ~ActiveNodeStack()<span class="keyword"> override </span>{}</div><div class="line"><a name="l00418"></a><span class="lineno"> 418</span> </div><div class="line"><a name="l00419"></a><span class="lineno"> 419</span>  <span class="keywordtype">bool</span> Empty()<span class="keyword"> const override </span>{ <span class="keywordflow">return</span> v_.empty(); }</div><div class="line"><a name="l00420"></a><span class="lineno"> 420</span> </div><div class="line"><a name="l00421"></a><span class="lineno"> 421</span>  <span class="keywordtype">void</span> Add(<a class="code" href="classoperations__research_1_1_linear_sum_assignment.html#aca73212d30b08a7c287c311b74311a6e">NodeIndex</a> node)<span class="keyword"> override </span>{ v_.push_back(node); }</div><div class="line"><a name="l00422"></a><span class="lineno"> 422</span> </div><div class="line"><a name="l00423"></a><span class="lineno"> 423</span>  <a class="code" href="classoperations__research_1_1_linear_sum_assignment.html#aca73212d30b08a7c287c311b74311a6e">NodeIndex</a> Get()<span class="keyword"> override </span>{</div><div class="line"><a name="l00424"></a><span class="lineno"> 424</span>  <a class="code" href="base_2logging_8h.html#ae17f8119c108cf3070bad3449c7e0006">DCHECK</a>(!Empty());</div><div class="line"><a name="l00425"></a><span class="lineno"> 425</span>  <a class="code" href="classoperations__research_1_1_linear_sum_assignment.html#aca73212d30b08a7c287c311b74311a6e">NodeIndex</a> result = v_.back();</div><div class="line"><a name="l00426"></a><span class="lineno"> 426</span>  v_.pop_back();</div><div class="line"><a name="l00427"></a><span class="lineno"> 427</span>  <span class="keywordflow">return</span> result;</div><div class="line"><a name="l00428"></a><span class="lineno"> 428</span>  }</div><div class="line"><a name="l00429"></a><span class="lineno"> 429</span> </div><div class="line"><a name="l00430"></a><span class="lineno"> 430</span>  <span class="keyword">private</span>:</div><div class="line"><a name="l00431"></a><span class="lineno"> 431</span>  std::vector<NodeIndex> v_;</div><div class="line"><a name="l00432"></a><span class="lineno"> 432</span>  };</div><div class="line"><a name="l00433"></a><span class="lineno"> 433</span> </div><div class="line"><a name="l00434"></a><span class="lineno"> 434</span>  <span class="keyword">class </span>ActiveNodeQueue : <span class="keyword">public</span> ActiveNodeContainerInterface {</div><div class="line"><a name="l00435"></a><span class="lineno"> 435</span>  <span class="keyword">public</span>:</div><div class="line"><a name="l00436"></a><span class="lineno"> 436</span>  ~ActiveNodeQueue()<span class="keyword"> override </span>{}</div><div class="line"><a name="l00437"></a><span class="lineno"> 437</span> </div><div class="line"><a name="l00438"></a><span class="lineno"> 438</span>  <span class="keywordtype">bool</span> Empty()<span class="keyword"> const override </span>{ <span class="keywordflow">return</span> q_.empty(); }</div><div class="line"><a name="l00439"></a><span class="lineno"> 439</span> </div><div class="line"><a name="l00440"></a><span class="lineno"> 440</span>  <span class="keywordtype">void</span> Add(<a class="code" href="classoperations__research_1_1_linear_sum_assignment.html#aca73212d30b08a7c287c311b74311a6e">NodeIndex</a> node)<span class="keyword"> override </span>{ q_.push_front(node); }</div><div class="line"><a name="l00441"></a><span class="lineno"> 441</span> </div><div class="line"><a name="l00442"></a><span class="lineno"> 442</span>  <a class="code" href="classoperations__research_1_1_linear_sum_assignment.html#aca73212d30b08a7c287c311b74311a6e">NodeIndex</a> Get()<span class="keyword"> override </span>{</div><div class="line"><a name="l00443"></a><span class="lineno"> 443</span>  <a class="code" href="base_2logging_8h.html#ae17f8119c108cf3070bad3449c7e0006">DCHECK</a>(!Empty());</div><div class="line"><a name="l00444"></a><span class="lineno"> 444</span>  <a class="code" href="classoperations__research_1_1_linear_sum_assignment.html#aca73212d30b08a7c287c311b74311a6e">NodeIndex</a> result = q_.back();</div><div class="line"><a name="l00445"></a><span class="lineno"> 445</span>  q_.pop_back();</div><div class="line"><a name="l00446"></a><span class="lineno"> 446</span>  <span class="keywordflow">return</span> result;</div><div class="line"><a name="l00447"></a><span class="lineno"> 447</span>  }</div><div class="line"><a name="l00448"></a><span class="lineno"> 448</span> </div><div class="line"><a name="l00449"></a><span class="lineno"> 449</span>  <span class="keyword">private</span>:</div><div class="line"><a name="l00450"></a><span class="lineno"> 450</span>  std::deque<NodeIndex> q_;</div><div class="line"><a name="l00451"></a><span class="lineno"> 451</span>  };</div><div class="line"><a name="l00452"></a><span class="lineno"> 452</span> <span class="preprocessor">#endif</span></div><div class="line"><a name="l00453"></a><span class="lineno"> 453</span> </div><div class="line"><a name="l00454"></a><span class="lineno"> 454</span>  <span class="comment">// Type definition for a pair</span></div><div class="line"><a name="l00455"></a><span class="lineno"> 455</span>  <span class="comment">// (arc index, reduced cost gap)</span></div><div class="line"><a name="l00456"></a><span class="lineno"> 456</span>  <span class="comment">// giving the arc along which we will push from a given left-side</span></div><div class="line"><a name="l00457"></a><span class="lineno"> 457</span>  <span class="comment">// node and the gap between that arc's partial reduced cost and the</span></div><div class="line"><a name="l00458"></a><span class="lineno"> 458</span>  <span class="comment">// reduced cost of the next-best (necessarily residual) arc out of</span></div><div class="line"><a name="l00459"></a><span class="lineno"> 459</span>  <span class="comment">// the node. This information helps us efficiently relabel</span></div><div class="line"><a name="l00460"></a><span class="lineno"> 460</span>  <span class="comment">// right-side nodes during DoublePush operations.</span></div><div class="line"><a name="l00461"></a><span class="lineno"> 461</span>  <span class="keyword">typedef</span> std::pair<ArcIndex, CostValue> ImplicitPriceSummary;</div><div class="line"><a name="l00462"></a><span class="lineno"> 462</span> </div><div class="line"><a name="l00463"></a><span class="lineno"> 463</span>  <span class="comment">// Returns true if and only if the current pseudoflow is</span></div><div class="line"><a name="l00464"></a><span class="lineno"> 464</span>  <span class="comment">// epsilon-optimal. To be used in a DCHECK.</span></div><div class="line"><a name="l00465"></a><span class="lineno"> 465</span>  <span class="keywordtype">bool</span> EpsilonOptimal() <span class="keyword">const</span>;</div><div class="line"><a name="l00466"></a><span class="lineno"> 466</span> </div><div class="line"><a name="l00467"></a><span class="lineno"> 467</span>  <span class="comment">// Checks that all nodes are matched.</span></div><div class="line"><a name="l00468"></a><span class="lineno"> 468</span>  <span class="comment">// To be used in a DCHECK.</span></div><div class="line"><a name="l00469"></a><span class="lineno"> 469</span>  <span class="keywordtype">bool</span> AllMatched() <span class="keyword">const</span>;</div><div class="line"><a name="l00470"></a><span class="lineno"> 470</span> </div><div class="line"><a name="l00471"></a><span class="lineno"> 471</span>  <span class="comment">// Calculates the implicit price of the given node.</span></div><div class="line"><a name="l00472"></a><span class="lineno"> 472</span>  <span class="comment">// Only for debugging, for use in EpsilonOptimal().</span></div><div class="line"><a name="l00473"></a><span class="lineno"> 473</span>  <span class="keyword">inline</span> <a class="code" href="namespaceoperations__research.html#aee97ac67f280d35acdef2c5d461a85c3">CostValue</a> ImplicitPrice(<a class="code" href="classoperations__research_1_1_linear_sum_assignment.html#aca73212d30b08a7c287c311b74311a6e">NodeIndex</a> left_node) <span class="keyword">const</span>;</div><div class="line"><a name="l00474"></a><span class="lineno"> 474</span> </div><div class="line"><a name="l00475"></a><span class="lineno"> 475</span>  <span class="comment">// For use by DoublePush()</span></div><div class="line"><a name="l00476"></a><span class="lineno"> 476</span>  <span class="keyword">inline</span> ImplicitPriceSummary BestArcAndGap(<a class="code" href="classoperations__research_1_1_linear_sum_assignment.html#aca73212d30b08a7c287c311b74311a6e">NodeIndex</a> left_node) <span class="keyword">const</span>;</div><div class="line"><a name="l00477"></a><span class="lineno"> 477</span> </div><div class="line"><a name="l00478"></a><span class="lineno"> 478</span>  <span class="comment">// Accumulates stats between iterations and reports them if the</span></div><div class="line"><a name="l00479"></a><span class="lineno"> 479</span>  <span class="comment">// verbosity level is high enough.</span></div><div class="line"><a name="l00480"></a><span class="lineno"> 480</span>  <span class="keywordtype">void</span> ReportAndAccumulateStats();</div><div class="line"><a name="l00481"></a><span class="lineno"> 481</span> </div><div class="line"><a name="l00482"></a><span class="lineno"> 482</span>  <span class="comment">// Utility function to compute the next error parameter value. This</span></div><div class="line"><a name="l00483"></a><span class="lineno"> 483</span>  <span class="comment">// is used to ensure that the same sequence of error parameter</span></div><div class="line"><a name="l00484"></a><span class="lineno"> 484</span>  <span class="comment">// values is used for computation of price bounds as is used for</span></div><div class="line"><a name="l00485"></a><span class="lineno"> 485</span>  <span class="comment">// computing the optimum assignment.</span></div><div class="line"><a name="l00486"></a><span class="lineno"> 486</span>  <a class="code" href="namespaceoperations__research.html#aee97ac67f280d35acdef2c5d461a85c3">CostValue</a> NewEpsilon(<a class="code" href="namespaceoperations__research.html#aee97ac67f280d35acdef2c5d461a85c3">CostValue</a> current_epsilon) <span class="keyword">const</span>;</div><div class="line"><a name="l00487"></a><span class="lineno"> 487</span> </div><div class="line"><a name="l00488"></a><span class="lineno"> 488</span>  <span class="comment">// Advances internal state to prepare for the next scaling</span></div><div class="line"><a name="l00489"></a><span class="lineno"> 489</span>  <span class="comment">// iteration. Returns false if infeasibility is detected, true</span></div><div class="line"><a name="l00490"></a><span class="lineno"> 490</span>  <span class="comment">// otherwise.</span></div><div class="line"><a name="l00491"></a><span class="lineno"> 491</span>  <span class="keywordtype">bool</span> UpdateEpsilon();</div><div class="line"><a name="l00492"></a><span class="lineno"> 492</span> </div><div class="line"><a name="l00493"></a><span class="lineno"> 493</span>  <span class="comment">// Indicates whether the given left_node has positive excess. Called</span></div><div class="line"><a name="l00494"></a><span class="lineno"> 494</span>  <span class="comment">// only for nodes on the left side.</span></div><div class="line"><a name="l00495"></a><span class="lineno"> 495</span>  <span class="keyword">inline</span> <span class="keywordtype">bool</span> IsActive(<a class="code" href="classoperations__research_1_1_linear_sum_assignment.html#aca73212d30b08a7c287c311b74311a6e">NodeIndex</a> left_node) <span class="keyword">const</span>;</div><div class="line"><a name="l00496"></a><span class="lineno"> 496</span> </div><div class="line"><a name="l00497"></a><span class="lineno"> 497</span>  <span class="comment">// Indicates whether the given node has nonzero excess. The idea</span></div><div class="line"><a name="l00498"></a><span class="lineno"> 498</span>  <span class="comment">// here is the same as the IsActive method above, but that method</span></div><div class="line"><a name="l00499"></a><span class="lineno"> 499</span>  <span class="comment">// contains a safety DCHECK() that its argument is a left-side node,</span></div><div class="line"><a name="l00500"></a><span class="lineno"> 500</span>  <span class="comment">// while this method is usable for any node.</span></div><div class="line"><a name="l00501"></a><span class="lineno"> 501</span>  <span class="comment">// To be used in a DCHECK.</span></div><div class="line"><a name="l00502"></a><span class="lineno"> 502</span>  <span class="keyword">inline</span> <span class="keywordtype">bool</span> IsActiveForDebugging(<a class="code" href="classoperations__research_1_1_linear_sum_assignment.html#aca73212d30b08a7c287c311b74311a6e">NodeIndex</a> node) <span class="keyword">const</span>;</div><div class="line"><a name="l00503"></a><span class="lineno"> 503</span> </div><div class="line"><a name="l00504"></a><span class="lineno"> 504</span>  <span class="comment">// Performs the push/relabel work for one scaling iteration.</span></div><div class="line"><a name="l00505"></a><span class="lineno"> 505</span>  <span class="keywordtype">bool</span> Refine();</div><div class="line"><a name="l00506"></a><span class="lineno"> 506</span> </div><div class="line"><a name="l00507"></a><span class="lineno"> 507</span>  <span class="comment">// Puts all left-side nodes in the active set in preparation for the</span></div><div class="line"><a name="l00508"></a><span class="lineno"> 508</span>  <span class="comment">// first scaling iteration.</span></div><div class="line"><a name="l00509"></a><span class="lineno"> 509</span>  <span class="keywordtype">void</span> InitializeActiveNodeContainer();</div><div class="line"><a name="l00510"></a><span class="lineno"> 510</span> </div><div class="line"><a name="l00511"></a><span class="lineno"> 511</span>  <span class="comment">// Saturates all negative-reduced-cost arcs at the beginning of each</span></div><div class="line"><a name="l00512"></a><span class="lineno"> 512</span>  <span class="comment">// scaling iteration. Note that according to the asymmetric</span></div><div class="line"><a name="l00513"></a><span class="lineno"> 513</span>  <span class="comment">// definition of admissibility, this action is different from</span></div><div class="line"><a name="l00514"></a><span class="lineno"> 514</span>  <span class="comment">// saturating all admissible arcs (which we never do). All negative</span></div><div class="line"><a name="l00515"></a><span class="lineno"> 515</span>  <span class="comment">// arcs are admissible, but not all admissible arcs are negative. It</span></div><div class="line"><a name="l00516"></a><span class="lineno"> 516</span>  <span class="comment">// is alwsys enough to saturate only the negative ones.</span></div><div class="line"><a name="l00517"></a><span class="lineno"> 517</span>  <span class="keywordtype">void</span> SaturateNegativeArcs();</div><div class="line"><a name="l00518"></a><span class="lineno"> 518</span> </div><div class="line"><a name="l00519"></a><span class="lineno"> 519</span>  <span class="comment">// Performs an optimized sequence of pushing a unit of excess out of</span></div><div class="line"><a name="l00520"></a><span class="lineno"> 520</span>  <span class="comment">// the left-side node v and back to another left-side node if no</span></div><div class="line"><a name="l00521"></a><span class="lineno"> 521</span>  <span class="comment">// deficit is cancelled with the first push.</span></div><div class="line"><a name="l00522"></a><span class="lineno"> 522</span>  <span class="keywordtype">bool</span> DoublePush(<a class="code" href="classoperations__research_1_1_linear_sum_assignment.html#aca73212d30b08a7c287c311b74311a6e">NodeIndex</a> source);</div><div class="line"><a name="l00523"></a><span class="lineno"> 523</span> </div><div class="line"><a name="l00524"></a><span class="lineno"> 524</span>  <span class="comment">// Returns the partial reduced cost of the given arc.</span></div><div class="line"><a name="l00525"></a><span class="lineno"> 525</span>  <span class="keyword">inline</span> <a class="code" href="namespaceoperations__research.html#aee97ac67f280d35acdef2c5d461a85c3">CostValue</a> PartialReducedCost(<a class="code" href="classoperations__research_1_1_linear_sum_assignment.html#a28178bb3e97b7c81dc157fe283b8cd64">ArcIndex</a> arc)<span class="keyword"> const </span>{</div><div class="line"><a name="l00526"></a><span class="lineno"> 526</span>  <span class="keywordflow">return</span> scaled_arc_cost_[arc] - price_[<a class="code" href="classoperations__research_1_1_linear_sum_assignment.html#aa5f729a6274027e5e5478e4bd76603ca">Head</a>(arc)];</div><div class="line"><a name="l00527"></a><span class="lineno"> 527</span>  }</div><div class="line"><a name="l00528"></a><span class="lineno"> 528</span> </div><div class="line"><a name="l00529"></a><span class="lineno"> 529</span>  <span class="comment">// The graph underlying the problem definition we are given. Not</span></div><div class="line"><a name="l00530"></a><span class="lineno"> 530</span>  <span class="comment">// owned by *this.</span></div><div class="line"><a name="l00531"></a><span class="lineno"> 531</span>  <span class="keyword">const</span> GraphType* graph_;</div><div class="line"><a name="l00532"></a><span class="lineno"> 532</span> </div><div class="line"><a name="l00533"></a><span class="lineno"> 533</span>  <span class="comment">// The number of nodes on the left side of the graph we are given.</span></div><div class="line"><a name="l00534"></a><span class="lineno"> 534</span>  <a class="code" href="classoperations__research_1_1_linear_sum_assignment.html#aca73212d30b08a7c287c311b74311a6e">NodeIndex</a> num_left_nodes_;</div><div class="line"><a name="l00535"></a><span class="lineno"> 535</span> </div><div class="line"><a name="l00536"></a><span class="lineno"> 536</span>  <span class="comment">// A flag indicating, after FinalizeSetup() has run, whether the</span></div><div class="line"><a name="l00537"></a><span class="lineno"> 537</span>  <span class="comment">// arc-incidence precondition required by BestArcAndGap() is</span></div><div class="line"><a name="l00538"></a><span class="lineno"> 538</span>  <span class="comment">// satisfied by every left-side node. If not, the problem is</span></div><div class="line"><a name="l00539"></a><span class="lineno"> 539</span>  <span class="comment">// infeasible.</span></div><div class="line"><a name="l00540"></a><span class="lineno"> 540</span>  <span class="keywordtype">bool</span> incidence_precondition_satisfied_;</div><div class="line"><a name="l00541"></a><span class="lineno"> 541</span> </div><div class="line"><a name="l00542"></a><span class="lineno"> 542</span>  <span class="comment">// A flag indicating that an optimal perfect matching has been computed.</span></div><div class="line"><a name="l00543"></a><span class="lineno"> 543</span>  <span class="keywordtype">bool</span> success_;</div><div class="line"><a name="l00544"></a><span class="lineno"> 544</span> </div><div class="line"><a name="l00545"></a><span class="lineno"> 545</span>  <span class="comment">// The value by which we multiply all the arc costs we are given in</span></div><div class="line"><a name="l00546"></a><span class="lineno"> 546</span>  <span class="comment">// order to be able to use integer arithmetic in all our</span></div><div class="line"><a name="l00547"></a><span class="lineno"> 547</span>  <span class="comment">// computations. In order to establish optimality of the final</span></div><div class="line"><a name="l00548"></a><span class="lineno"> 548</span>  <span class="comment">// matching we compute, we need that</span></div><div class="line"><a name="l00549"></a><span class="lineno"> 549</span>  <span class="comment">// (cost_scaling_factor_ / kMinEpsilon) > graph_->num_nodes().</span></div><div class="line"><a name="l00550"></a><span class="lineno"> 550</span>  <span class="keyword">const</span> <a class="code" href="namespaceoperations__research.html#aee97ac67f280d35acdef2c5d461a85c3">CostValue</a> cost_scaling_factor_;</div><div class="line"><a name="l00551"></a><span class="lineno"> 551</span> </div><div class="line"><a name="l00552"></a><span class="lineno"> 552</span>  <span class="comment">// Scaling divisor.</span></div><div class="line"><a name="l00553"></a><span class="lineno"> 553</span>  <a class="code" href="namespaceoperations__research.html#aee97ac67f280d35acdef2c5d461a85c3">CostValue</a> alpha_;</div><div class="line"><a name="l00554"></a><span class="lineno"> 554</span> </div><div class="line"><a name="l00555"></a><span class="lineno"> 555</span>  <span class="comment">// Minimum value of epsilon. When a flow is epsilon-optimal for</span></div><div class="line"><a name="l00556"></a><span class="lineno"> 556</span>  <span class="comment">// epsilon == kMinEpsilon, the flow is optimal.</span></div><div class="line"><a name="l00557"></a><span class="lineno"> 557</span>  <span class="keyword">static</span> <span class="keyword">const</span> <a class="code" href="namespaceoperations__research.html#aee97ac67f280d35acdef2c5d461a85c3">CostValue</a> kMinEpsilon;</div><div class="line"><a name="l00558"></a><span class="lineno"> 558</span> </div><div class="line"><a name="l00559"></a><span class="lineno"> 559</span>  <span class="comment">// Current value of epsilon, the cost scaling parameter.</span></div><div class="line"><a name="l00560"></a><span class="lineno"> 560</span>  <a class="code" href="namespaceoperations__research.html#aee97ac67f280d35acdef2c5d461a85c3">CostValue</a> epsilon_;</div><div class="line"><a name="l00561"></a><span class="lineno"> 561</span> </div><div class="line"><a name="l00562"></a><span class="lineno"> 562</span>  <span class="comment">// The following two data members, price_lower_bound_ and</span></div><div class="line"><a name="l00563"></a><span class="lineno"> 563</span>  <span class="comment">// slack_relabeling_price_, have to do with bounds on the amount by</span></div><div class="line"><a name="l00564"></a><span class="lineno"> 564</span>  <span class="comment">// which node prices can change during execution of the algorithm.</span></div><div class="line"><a name="l00565"></a><span class="lineno"> 565</span>  <span class="comment">// We need some detailed discussion of this topic because we violate</span></div><div class="line"><a name="l00566"></a><span class="lineno"> 566</span>  <span class="comment">// several simplifying assumptions typically made in the theoretical</span></div><div class="line"><a name="l00567"></a><span class="lineno"> 567</span>  <span class="comment">// literature. In particular, we use integer arithmetic, we use a</span></div><div class="line"><a name="l00568"></a><span class="lineno"> 568</span>  <span class="comment">// reduction to the transportation problem rather than min-cost</span></div><div class="line"><a name="l00569"></a><span class="lineno"> 569</span>  <span class="comment">// circulation, we provide detection of infeasible problems rather</span></div><div class="line"><a name="l00570"></a><span class="lineno"> 570</span>  <span class="comment">// than assume feasibility, we detect when our computations might</span></div><div class="line"><a name="l00571"></a><span class="lineno"> 571</span>  <span class="comment">// exceed the range of representable cost values, and we use the</span></div><div class="line"><a name="l00572"></a><span class="lineno"> 572</span>  <span class="comment">// double-push heuristic which relabels nodes that do not have</span></div><div class="line"><a name="l00573"></a><span class="lineno"> 573</span>  <span class="comment">// excess.</span></div><div class="line"><a name="l00574"></a><span class="lineno"> 574</span>  <span class="comment">//</span></div><div class="line"><a name="l00575"></a><span class="lineno"> 575</span>  <span class="comment">// In the following discussion, we prove the following propositions:</span></div><div class="line"><a name="l00576"></a><span class="lineno"> 576</span>  <span class="comment">// Proposition 1. [Fidelity of arithmetic precision guarantee] If</span></div><div class="line"><a name="l00577"></a><span class="lineno"> 577</span>  <span class="comment">// FinalizeSetup() returns true, no arithmetic</span></div><div class="line"><a name="l00578"></a><span class="lineno"> 578</span>  <span class="comment">// overflow occurs during ComputeAssignment().</span></div><div class="line"><a name="l00579"></a><span class="lineno"> 579</span>  <span class="comment">// Proposition 2. [Fidelity of feasibility detection] If no</span></div><div class="line"><a name="l00580"></a><span class="lineno"> 580</span>  <span class="comment">// arithmetic overflow occurs during</span></div><div class="line"><a name="l00581"></a><span class="lineno"> 581</span>  <span class="comment">// ComputeAssignment(), the return value of</span></div><div class="line"><a name="l00582"></a><span class="lineno"> 582</span>  <span class="comment">// ComputeAssignment() faithfully indicates whether</span></div><div class="line"><a name="l00583"></a><span class="lineno"> 583</span>  <span class="comment">// the given problem is feasible.</span></div><div class="line"><a name="l00584"></a><span class="lineno"> 584</span>  <span class="comment">//</span></div><div class="line"><a name="l00585"></a><span class="lineno"> 585</span>  <span class="comment">// We begin with some general discussion.</span></div><div class="line"><a name="l00586"></a><span class="lineno"> 586</span>  <span class="comment">//</span></div><div class="line"><a name="l00587"></a><span class="lineno"> 587</span>  <span class="comment">// The ideas used to prove our two propositions are essentially</span></div><div class="line"><a name="l00588"></a><span class="lineno"> 588</span>  <span class="comment">// those that appear in [Goldberg and Tarjan], but several details</span></div><div class="line"><a name="l00589"></a><span class="lineno"> 589</span>  <span class="comment">// are different: [Goldberg and Tarjan] assumes a feasible problem,</span></div><div class="line"><a name="l00590"></a><span class="lineno"> 590</span>  <span class="comment">// uses a symmetric notion of epsilon-optimality, considers only</span></div><div class="line"><a name="l00591"></a><span class="lineno"> 591</span>  <span class="comment">// nodes with excess eligible for relabeling, and does not treat the</span></div><div class="line"><a name="l00592"></a><span class="lineno"> 592</span>  <span class="comment">// question of arithmetic overflow. This implementation, on the</span></div><div class="line"><a name="l00593"></a><span class="lineno"> 593</span>  <span class="comment">// other hand, detects and reports infeasible problems, uses</span></div><div class="line"><a name="l00594"></a><span class="lineno"> 594</span>  <span class="comment">// asymmetric epsilon-optimality, relabels nodes with no excess in</span></div><div class="line"><a name="l00595"></a><span class="lineno"> 595</span>  <span class="comment">// the course of the double-push operation, and gives a reasonably</span></div><div class="line"><a name="l00596"></a><span class="lineno"> 596</span>  <span class="comment">// tight guarantee of arithmetic precision. No fundamentally new</span></div><div class="line"><a name="l00597"></a><span class="lineno"> 597</span>  <span class="comment">// ideas are involved, but the details are a bit tricky so they are</span></div><div class="line"><a name="l00598"></a><span class="lineno"> 598</span>  <span class="comment">// explained here.</span></div><div class="line"><a name="l00599"></a><span class="lineno"> 599</span>  <span class="comment">//</span></div><div class="line"><a name="l00600"></a><span class="lineno"> 600</span>  <span class="comment">// We have two intertwined needs that lead us to compute bounds on</span></div><div class="line"><a name="l00601"></a><span class="lineno"> 601</span>  <span class="comment">// the prices nodes can have during the assignment computation, on</span></div><div class="line"><a name="l00602"></a><span class="lineno"> 602</span>  <span class="comment">// the assumption that the given problem is feasible:</span></div><div class="line"><a name="l00603"></a><span class="lineno"> 603</span>  <span class="comment">// 1. Infeasibility detection: Infeasibility is detected by</span></div><div class="line"><a name="l00604"></a><span class="lineno"> 604</span>  <span class="comment">// observing that some node's price has been reduced too much by</span></div><div class="line"><a name="l00605"></a><span class="lineno"> 605</span>  <span class="comment">// relabeling operations (see [Goldberg and Tarjan] for the</span></div><div class="line"><a name="l00606"></a><span class="lineno"> 606</span>  <span class="comment">// argument -- duplicated in modified form below -- bounding the</span></div><div class="line"><a name="l00607"></a><span class="lineno"> 607</span>  <span class="comment">// running time of the push/relabel min-cost flow algorithm for</span></div><div class="line"><a name="l00608"></a><span class="lineno"> 608</span>  <span class="comment">// feasible problems); and</span></div><div class="line"><a name="l00609"></a><span class="lineno"> 609</span>  <span class="comment">// 2. Aggressively relabeling nodes and arcs whose matching is</span></div><div class="line"><a name="l00610"></a><span class="lineno"> 610</span>  <span class="comment">// forced: When a left-side node is incident to only one arc a,</span></div><div class="line"><a name="l00611"></a><span class="lineno"> 611</span>  <span class="comment">// any feasible solution must include a, and reducing the price</span></div><div class="line"><a name="l00612"></a><span class="lineno"> 612</span>  <span class="comment">// of Head(a) by any nonnegative amount preserves epsilon-</span></div><div class="line"><a name="l00613"></a><span class="lineno"> 613</span>  <span class="comment">// optimality. Because of this freedom, we'll call this sort of</span></div><div class="line"><a name="l00614"></a><span class="lineno"> 614</span>  <span class="comment">// relabeling (i.e., a relabeling of a right-side node that is</span></div><div class="line"><a name="l00615"></a><span class="lineno"> 615</span>  <span class="comment">// the only neighbor of the left-side node to which it has been</span></div><div class="line"><a name="l00616"></a><span class="lineno"> 616</span>  <span class="comment">// matched in the present double-push operation) a "slack"</span></div><div class="line"><a name="l00617"></a><span class="lineno"> 617</span>  <span class="comment">// relabeling. Relabelings that are not slack relabelings are</span></div><div class="line"><a name="l00618"></a><span class="lineno"> 618</span>  <span class="comment">// called "confined" relabelings. By relabeling Head(a) to have</span></div><div class="line"><a name="l00619"></a><span class="lineno"> 619</span>  <span class="comment">// p(Head(a))=-infinity, we could guarantee that a never becomes</span></div><div class="line"><a name="l00620"></a><span class="lineno"> 620</span>  <span class="comment">// unmatched during the current iteration, and this would prevent</span></div><div class="line"><a name="l00621"></a><span class="lineno"> 621</span>  <span class="comment">// our wasting time repeatedly unmatching and rematching a. But</span></div><div class="line"><a name="l00622"></a><span class="lineno"> 622</span>  <span class="comment">// there are some details we need to handle:</span></div><div class="line"><a name="l00623"></a><span class="lineno"> 623</span>  <span class="comment">// a. The CostValue type cannot represent -infinity;</span></div><div class="line"><a name="l00624"></a><span class="lineno"> 624</span>  <span class="comment">// b. Low node prices are precisely the signal we use to detect</span></div><div class="line"><a name="l00625"></a><span class="lineno"> 625</span>  <span class="comment">// infeasibility (see (1)), so we must be careful not to</span></div><div class="line"><a name="l00626"></a><span class="lineno"> 626</span>  <span class="comment">// falsely conclude that the problem is infeasible as a result</span></div><div class="line"><a name="l00627"></a><span class="lineno"> 627</span>  <span class="comment">// of the low price we gave Head(a); and</span></div><div class="line"><a name="l00628"></a><span class="lineno"> 628</span>  <span class="comment">// c. We need to indicate accurately to the client when our best</span></div><div class="line"><a name="l00629"></a><span class="lineno"> 629</span>  <span class="comment">// understanding indicates that we can't rule out arithmetic</span></div><div class="line"><a name="l00630"></a><span class="lineno"> 630</span>  <span class="comment">// overflow in our calculations. Most importantly, if we don't</span></div><div class="line"><a name="l00631"></a><span class="lineno"> 631</span>  <span class="comment">// warn the client, we must be certain to avoid overflow. This</span></div><div class="line"><a name="l00632"></a><span class="lineno"> 632</span>  <span class="comment">// means our slack relabelings must not be so aggressive as to</span></div><div class="line"><a name="l00633"></a><span class="lineno"> 633</span>  <span class="comment">// create the possibility of unforeseen overflow. Although we</span></div><div class="line"><a name="l00634"></a><span class="lineno"> 634</span>  <span class="comment">// will not achieve this in practice, slack relabelings would</span></div><div class="line"><a name="l00635"></a><span class="lineno"> 635</span>  <span class="comment">// ideally not introduce overflow unless overflow was</span></div><div class="line"><a name="l00636"></a><span class="lineno"> 636</span>  <span class="comment">// inevitable were even the smallest reasonable price change</span></div><div class="line"><a name="l00637"></a><span class="lineno"> 637</span>  <span class="comment">// (== epsilon) used for slack relabelings.</span></div><div class="line"><a name="l00638"></a><span class="lineno"> 638</span>  <span class="comment">// Using the analysis below, we choose a finite amount of price</span></div><div class="line"><a name="l00639"></a><span class="lineno"> 639</span>  <span class="comment">// change for slack relabelings aggressive enough that we don't</span></div><div class="line"><a name="l00640"></a><span class="lineno"> 640</span>  <span class="comment">// waste time doing repeated slack relabelings in a single</span></div><div class="line"><a name="l00641"></a><span class="lineno"> 641</span>  <span class="comment">// iteration, yet modest enough that we keep a good handle on</span></div><div class="line"><a name="l00642"></a><span class="lineno"> 642</span>  <span class="comment">// arithmetic precision and our ability to detect infeasible</span></div><div class="line"><a name="l00643"></a><span class="lineno"> 643</span>  <span class="comment">// problems.</span></div><div class="line"><a name="l00644"></a><span class="lineno"> 644</span>  <span class="comment">//</span></div><div class="line"><a name="l00645"></a><span class="lineno"> 645</span>  <span class="comment">// To provide faithful detection of infeasibility, a dependable</span></div><div class="line"><a name="l00646"></a><span class="lineno"> 646</span>  <span class="comment">// guarantee of arithmetic precision whenever possible, and good</span></div><div class="line"><a name="l00647"></a><span class="lineno"> 647</span>  <span class="comment">// performance by aggressively relabeling nodes whose matching is</span></div><div class="line"><a name="l00648"></a><span class="lineno"> 648</span>  <span class="comment">// forced, we exploit these facts:</span></div><div class="line"><a name="l00649"></a><span class="lineno"> 649</span>  <span class="comment">// 1. Beyond the first iteration, infeasibility detection isn't needed</span></div><div class="line"><a name="l00650"></a><span class="lineno"> 650</span>  <span class="comment">// because a problem is feasible in some iteration if and only if</span></div><div class="line"><a name="l00651"></a><span class="lineno"> 651</span>  <span class="comment">// it's feasible in all others. Therefore we are free to use an</span></div><div class="line"><a name="l00652"></a><span class="lineno"> 652</span>  <span class="comment">// infeasibility detection mechanism that might work in just one</span></div><div class="line"><a name="l00653"></a><span class="lineno"> 653</span>  <span class="comment">// iteration and switch it off in all other iterations.</span></div><div class="line"><a name="l00654"></a><span class="lineno"> 654</span>  <span class="comment">// 2. When we do a slack relabeling, we must choose the amount of</span></div><div class="line"><a name="l00655"></a><span class="lineno"> 655</span>  <span class="comment">// price reduction to use. We choose an amount large enough to</span></div><div class="line"><a name="l00656"></a><span class="lineno"> 656</span>  <span class="comment">// guarantee putting the node's matching to rest, yet (although</span></div><div class="line"><a name="l00657"></a><span class="lineno"> 657</span>  <span class="comment">// we don't bother to prove this explicitly) small enough that</span></div><div class="line"><a name="l00658"></a><span class="lineno"> 658</span>  <span class="comment">// the node's price obeys the overall lower bound that holds if</span></div><div class="line"><a name="l00659"></a><span class="lineno"> 659</span>  <span class="comment">// the slack relabeling amount is small.</span></div><div class="line"><a name="l00660"></a><span class="lineno"> 660</span>  <span class="comment">//</span></div><div class="line"><a name="l00661"></a><span class="lineno"> 661</span>  <span class="comment">// We will establish Propositions (1) and (2) above according to the</span></div><div class="line"><a name="l00662"></a><span class="lineno"> 662</span>  <span class="comment">// following steps:</span></div><div class="line"><a name="l00663"></a><span class="lineno"> 663</span>  <span class="comment">// First, we prove Lemma 1, which is a modified form of lemma 5.8 of</span></div><div class="line"><a name="l00664"></a><span class="lineno"> 664</span>  <span class="comment">// [Goldberg and Tarjan] giving a bound on the difference in price</span></div><div class="line"><a name="l00665"></a><span class="lineno"> 665</span>  <span class="comment">// between the end nodes of certain paths in the residual graph.</span></div><div class="line"><a name="l00666"></a><span class="lineno"> 666</span>  <span class="comment">// Second, we prove Lemma 2, which is technical lemma to establish</span></div><div class="line"><a name="l00667"></a><span class="lineno"> 667</span>  <span class="comment">// reachability of certain "anchor" nodes in the residual graph from</span></div><div class="line"><a name="l00668"></a><span class="lineno"> 668</span>  <span class="comment">// any node where a relabeling takes place.</span></div><div class="line"><a name="l00669"></a><span class="lineno"> 669</span>  <span class="comment">// Third, we apply the first two lemmas to prove Lemma 3 and Lemma</span></div><div class="line"><a name="l00670"></a><span class="lineno"> 670</span>  <span class="comment">// 4, which give two similar bounds that hold whenever the given</span></div><div class="line"><a name="l00671"></a><span class="lineno"> 671</span>  <span class="comment">// problem is feasible: (for feasibility detection) a bound on the</span></div><div class="line"><a name="l00672"></a><span class="lineno"> 672</span>  <span class="comment">// price of any node we relabel during any iteration (and the first</span></div><div class="line"><a name="l00673"></a><span class="lineno"> 673</span>  <span class="comment">// iteration in particular), and (for arithmetic precision) a bound</span></div><div class="line"><a name="l00674"></a><span class="lineno"> 674</span>  <span class="comment">// on the price of any node we relabel during the entire algorithm.</span></div><div class="line"><a name="l00675"></a><span class="lineno"> 675</span>  <span class="comment">//</span></div><div class="line"><a name="l00676"></a><span class="lineno"> 676</span>  <span class="comment">// Finally, we note that if the whole-algorithm price bound can be</span></div><div class="line"><a name="l00677"></a><span class="lineno"> 677</span>  <span class="comment">// represented precisely by the CostValue type, arithmetic overflow</span></div><div class="line"><a name="l00678"></a><span class="lineno"> 678</span>  <span class="comment">// cannot occur (establishing Proposition 1), and assuming no</span></div><div class="line"><a name="l00679"></a><span class="lineno"> 679</span>  <span class="comment">// overflow occurs during the first iteration, any violation of the</span></div><div class="line"><a name="l00680"></a><span class="lineno"> 680</span>  <span class="comment">// first-iteration price bound establishes infeasibility</span></div><div class="line"><a name="l00681"></a><span class="lineno"> 681</span>  <span class="comment">// (Proposition 2).</span></div><div class="line"><a name="l00682"></a><span class="lineno"> 682</span>  <span class="comment">//</span></div><div class="line"><a name="l00683"></a><span class="lineno"> 683</span>  <span class="comment">// The statement of Lemma 1 is perhaps easier to understand when the</span></div><div class="line"><a name="l00684"></a><span class="lineno"> 684</span>  <span class="comment">// reader knows how it will be used. To wit: In this lemma, f' and</span></div><div class="line"><a name="l00685"></a><span class="lineno"> 685</span>  <span class="comment">// e_0 are the flow and error parameter (epsilon) at the beginning</span></div><div class="line"><a name="l00686"></a><span class="lineno"> 686</span>  <span class="comment">// of the current iteration, while f and e_1 are the current</span></div><div class="line"><a name="l00687"></a><span class="lineno"> 687</span>  <span class="comment">// pseudoflow and error parameter when a relabeling of interest</span></div><div class="line"><a name="l00688"></a><span class="lineno"> 688</span>  <span class="comment">// occurs. Without loss of generality, c is the reduced cost</span></div><div class="line"><a name="l00689"></a><span class="lineno"> 689</span>  <span class="comment">// function at the beginning of the current iteration and p is the</span></div><div class="line"><a name="l00690"></a><span class="lineno"> 690</span>  <span class="comment">// change in prices that has taken place in the current iteration.</span></div><div class="line"><a name="l00691"></a><span class="lineno"> 691</span>  <span class="comment">//</span></div><div class="line"><a name="l00692"></a><span class="lineno"> 692</span>  <span class="comment">// Lemma 1 (a variant of lemma 5.8 from [Goldberg and Tarjan]): Let</span></div><div class="line"><a name="l00693"></a><span class="lineno"> 693</span>  <span class="comment">// f be a pseudoflow and let f' be a flow. Suppose P is a simple</span></div><div class="line"><a name="l00694"></a><span class="lineno"> 694</span>  <span class="comment">// path from right-side node v to right-side node w such that P is</span></div><div class="line"><a name="l00695"></a><span class="lineno"> 695</span>  <span class="comment">// residual with respect to f and reverse(P) is residual with</span></div><div class="line"><a name="l00696"></a><span class="lineno"> 696</span>  <span class="comment">// respect to f'. Further, suppose c is an arc cost function with</span></div><div class="line"><a name="l00697"></a><span class="lineno"> 697</span>  <span class="comment">// respect to which f' is e_0-optimal with the zero price function</span></div><div class="line"><a name="l00698"></a><span class="lineno"> 698</span>  <span class="comment">// and p is a price function with respect to which f is e_1-optimal</span></div><div class="line"><a name="l00699"></a><span class="lineno"> 699</span>  <span class="comment">// with respect to p. Then</span></div><div class="line"><a name="l00700"></a><span class="lineno"> 700</span>  <span class="comment">// p(v) - p(w) >= -(e_0 + e_1) * (n-2)/2. (***)</span></div><div class="line"><a name="l00701"></a><span class="lineno"> 701</span>  <span class="comment">//</span></div><div class="line"><a name="l00702"></a><span class="lineno"> 702</span>  <span class="comment">// Proof: We have c_p(P) = p(v) + c(P) - p(w) and hence</span></div><div class="line"><a name="l00703"></a><span class="lineno"> 703</span>  <span class="comment">// p(v) - p(w) = c_p(P) - c(P).</span></div><div class="line"><a name="l00704"></a><span class="lineno"> 704</span>  <span class="comment">// So we seek a bound on c_p(P) - c(P).</span></div><div class="line"><a name="l00705"></a><span class="lineno"> 705</span>  <span class="comment">// p(v) = c_p(P) - c(P).</span></div><div class="line"><a name="l00706"></a><span class="lineno"> 706</span>  <span class="comment">// Let arc a lie on P, which implies that a is residual with respect</span></div><div class="line"><a name="l00707"></a><span class="lineno"> 707</span>  <span class="comment">// to f and reverse(a) is residual with respect to f'.</span></div><div class="line"><a name="l00708"></a><span class="lineno"> 708</span>  <span class="comment">// Case 1: a is a forward arc. Then by e_1-optimality of f with</span></div><div class="line"><a name="l00709"></a><span class="lineno"> 709</span>  <span class="comment">// respect to p, c_p(a) >= 0 and reverse(a) is residual with</span></div><div class="line"><a name="l00710"></a><span class="lineno"> 710</span>  <span class="comment">// respect to f'. By e_0-optimality of f', c(a) <= e_0. So</span></div><div class="line"><a name="l00711"></a><span class="lineno"> 711</span>  <span class="comment">// c_p(a) - c(a) >= -e_0.</span></div><div class="line"><a name="l00712"></a><span class="lineno"> 712</span>  <span class="comment">// Case 2: a is a reverse arc. Then by e_1-optimality of f with</span></div><div class="line"><a name="l00713"></a><span class="lineno"> 713</span>  <span class="comment">// respect to p, c_p(a) >= -e_1 and reverse(a) is residual</span></div><div class="line"><a name="l00714"></a><span class="lineno"> 714</span>  <span class="comment">// with respect to f'. By e_0-optimality of f', c(a) <= 0.</span></div><div class="line"><a name="l00715"></a><span class="lineno"> 715</span>  <span class="comment">// So</span></div><div class="line"><a name="l00716"></a><span class="lineno"> 716</span>  <span class="comment">// c_p(a) - c(a) >= -e_1.</span></div><div class="line"><a name="l00717"></a><span class="lineno"> 717</span>  <span class="comment">// We assumed v and w are both right-side nodes, so there are at</span></div><div class="line"><a name="l00718"></a><span class="lineno"> 718</span>  <span class="comment">// most n - 2 arcs on the path P, of which at most (n-2)/2 are</span></div><div class="line"><a name="l00719"></a><span class="lineno"> 719</span>  <span class="comment">// forward arcs and at most (n-2)/2 are reverse arcs, so</span></div><div class="line"><a name="l00720"></a><span class="lineno"> 720</span>  <span class="comment">// p(v) - p(w) = c_p(P) - c(P)</span></div><div class="line"><a name="l00721"></a><span class="lineno"> 721</span>  <span class="comment">// >= -(e_0 + e_1) * (n-2)/2. (***)</span></div><div class="line"><a name="l00722"></a><span class="lineno"> 722</span>  <span class="comment">//</span></div><div class="line"><a name="l00723"></a><span class="lineno"> 723</span>  <span class="comment">// Some of the rest of our argument is given as a sketch, omitting</span></div><div class="line"><a name="l00724"></a><span class="lineno"> 724</span>  <span class="comment">// several details. Also elided here are some minor technical issues</span></div><div class="line"><a name="l00725"></a><span class="lineno"> 725</span>  <span class="comment">// related to the first iteration, inasmuch as our arguments assume</span></div><div class="line"><a name="l00726"></a><span class="lineno"> 726</span>  <span class="comment">// on the surface a "previous iteration" that doesn't exist in that</span></div><div class="line"><a name="l00727"></a><span class="lineno"> 727</span>  <span class="comment">// case. The issues are not substantial, just a bit messy.</span></div><div class="line"><a name="l00728"></a><span class="lineno"> 728</span>  <span class="comment">//</span></div><div class="line"><a name="l00729"></a><span class="lineno"> 729</span>  <span class="comment">// Lemma 2 is analogous to lemma 5.7 of [Goldberg and Tarjan], where</span></div><div class="line"><a name="l00730"></a><span class="lineno"> 730</span>  <span class="comment">// they have only relabelings that take place at nodes with excess</span></div><div class="line"><a name="l00731"></a><span class="lineno"> 731</span>  <span class="comment">// while we have only relabelings that take place as part of the</span></div><div class="line"><a name="l00732"></a><span class="lineno"> 732</span>  <span class="comment">// double-push operation at nodes without excess.</span></div><div class="line"><a name="l00733"></a><span class="lineno"> 733</span>  <span class="comment">//</span></div><div class="line"><a name="l00734"></a><span class="lineno"> 734</span>  <span class="comment">// Lemma 2: If the problem is feasible, for any node v with excess,</span></div><div class="line"><a name="l00735"></a><span class="lineno"> 735</span>  <span class="comment">// there exists a path P from v to a node w with deficit such that P</span></div><div class="line"><a name="l00736"></a><span class="lineno"> 736</span>  <span class="comment">// is residual with respect to the current pseudoflow, and</span></div><div class="line"><a name="l00737"></a><span class="lineno"> 737</span>  <span class="comment">// reverse(P) is residual with respect to the flow at the beginning</span></div><div class="line"><a name="l00738"></a><span class="lineno"> 738</span>  <span class="comment">// of the current iteration. (Note that such a path exactly</span></div><div class="line"><a name="l00739"></a><span class="lineno"> 739</span>  <span class="comment">// satisfies the conditions of Lemma 1.)</span></div><div class="line"><a name="l00740"></a><span class="lineno"> 740</span>  <span class="comment">//</span></div><div class="line"><a name="l00741"></a><span class="lineno"> 741</span>  <span class="comment">// Let the bound from Lemma 1 with p(w) = 0 be called B(e_0, e_1),</span></div><div class="line"><a name="l00742"></a><span class="lineno"> 742</span>  <span class="comment">// and let us say that when a slack relabeling of a node v occurs,</span></div><div class="line"><a name="l00743"></a><span class="lineno"> 743</span>  <span class="comment">// we will change the price of v by B(e_0, e_1) such that v tightly</span></div><div class="line"><a name="l00744"></a><span class="lineno"> 744</span>  <span class="comment">// satisfies the bound of Lemma 1. Explicitly, we define</span></div><div class="line"><a name="l00745"></a><span class="lineno"> 745</span>  <span class="comment">// B(e_0, e_1) = -(e_0 + e_1) * (n-2)/2.</span></div><div class="line"><a name="l00746"></a><span class="lineno"> 746</span>  <span class="comment">//</span></div><div class="line"><a name="l00747"></a><span class="lineno"> 747</span>  <span class="comment">// Lemma 1 and Lemma 2 combine to bound the price change during an</span></div><div class="line"><a name="l00748"></a><span class="lineno"> 748</span>  <span class="comment">// iteration for any node with excess. Viewed a different way, Lemma</span></div><div class="line"><a name="l00749"></a><span class="lineno"> 749</span>  <span class="comment">// 1 and Lemma 2 tell us that if epsilon-optimality can be preserved</span></div><div class="line"><a name="l00750"></a><span class="lineno"> 750</span>  <span class="comment">// by changing the price of a node by B(e_0, e_1), that node will</span></div><div class="line"><a name="l00751"></a><span class="lineno"> 751</span>  <span class="comment">// never have excess again during the current iteration unless the</span></div><div class="line"><a name="l00752"></a><span class="lineno"> 752</span>  <span class="comment">// problem is infeasible. This insight gives us an approach to</span></div><div class="line"><a name="l00753"></a><span class="lineno"> 753</span>  <span class="comment">// detect infeasibility (by observing prices on nodes with excess</span></div><div class="line"><a name="l00754"></a><span class="lineno"> 754</span>  <span class="comment">// that violate this bound) and to relabel nodes aggressively enough</span></div><div class="line"><a name="l00755"></a><span class="lineno"> 755</span>  <span class="comment">// to avoid unnecessary future work while we also avoid falsely</span></div><div class="line"><a name="l00756"></a><span class="lineno"> 756</span>  <span class="comment">// concluding the problem is infeasible.</span></div><div class="line"><a name="l00757"></a><span class="lineno"> 757</span>  <span class="comment">//</span></div><div class="line"><a name="l00758"></a><span class="lineno"> 758</span>  <span class="comment">// From Lemma 1 and Lemma 2, and taking into account our knowledge</span></div><div class="line"><a name="l00759"></a><span class="lineno"> 759</span>  <span class="comment">// of the slack relabeling amount, we have Lemma 3.</span></div><div class="line"><a name="l00760"></a><span class="lineno"> 760</span>  <span class="comment">//</span></div><div class="line"><a name="l00761"></a><span class="lineno"> 761</span>  <span class="comment">// Lemma 3: During any iteration, if the given problem is feasible</span></div><div class="line"><a name="l00762"></a><span class="lineno"> 762</span>  <span class="comment">// the price of any node is reduced by less than</span></div><div class="line"><a name="l00763"></a><span class="lineno"> 763</span>  <span class="comment">// -2 * B(e_0, e_1) = (e_0 + e_1) * (n-2).</span></div><div class="line"><a name="l00764"></a><span class="lineno"> 764</span>  <span class="comment">//</span></div><div class="line"><a name="l00765"></a><span class="lineno"> 765</span>  <span class="comment">// Proof: Straightforward, omitted for expedience.</span></div><div class="line"><a name="l00766"></a><span class="lineno"> 766</span>  <span class="comment">//</span></div><div class="line"><a name="l00767"></a><span class="lineno"> 767</span>  <span class="comment">// In the case where e_0 = e_1 * alpha, we can express the bound</span></div><div class="line"><a name="l00768"></a><span class="lineno"> 768</span>  <span class="comment">// just in terms of e_1, the current iteration's value of epsilon_:</span></div><div class="line"><a name="l00769"></a><span class="lineno"> 769</span>  <span class="comment">// B(e_1) = B(e_1 * alpha, e_1) = -(1 + alpha) * e_1 * (n-2)/2,</span></div><div class="line"><a name="l00770"></a><span class="lineno"> 770</span>  <span class="comment">// so we have that p(v) is reduced by less than 2 * B(e_1).</span></div><div class="line"><a name="l00771"></a><span class="lineno"> 771</span>  <span class="comment">//</span></div><div class="line"><a name="l00772"></a><span class="lineno"> 772</span>  <span class="comment">// Because we use truncating division to compute each iteration's error</span></div><div class="line"><a name="l00773"></a><span class="lineno"> 773</span>  <span class="comment">// parameter from that of the previous iteration, it isn't exactly</span></div><div class="line"><a name="l00774"></a><span class="lineno"> 774</span>  <span class="comment">// the case that e_0 = e_1 * alpha as we just assumed. To patch this</span></div><div class="line"><a name="l00775"></a><span class="lineno"> 775</span>  <span class="comment">// up, we can use the observation that</span></div><div class="line"><a name="l00776"></a><span class="lineno"> 776</span>  <span class="comment">// e_1 = floor(e_0 / alpha),</span></div><div class="line"><a name="l00777"></a><span class="lineno"> 777</span>  <span class="comment">// which implies</span></div><div class="line"><a name="l00778"></a><span class="lineno"> 778</span>  <span class="comment">// -e_0 > -(e_1 + 1) * alpha</span></div><div class="line"><a name="l00779"></a><span class="lineno"> 779</span>  <span class="comment">// to rewrite from (***):</span></div><div class="line"><a name="l00780"></a><span class="lineno"> 780</span>  <span class="comment">// p(v) > 2 * B(e_0, e_1) > 2 * B((e_1 + 1) * alpha, e_1)</span></div><div class="line"><a name="l00781"></a><span class="lineno"> 781</span>  <span class="comment">// = 2 * -((e_1 + 1) * alpha + e_1) * (n-2)/2</span></div><div class="line"><a name="l00782"></a><span class="lineno"> 782</span>  <span class="comment">// = 2 * -(1 + alpha) * e_1 * (n-2)/2 - alpha * (n-2)</span></div><div class="line"><a name="l00783"></a><span class="lineno"> 783</span>  <span class="comment">// = 2 * B(e_1) - alpha * (n-2)</span></div><div class="line"><a name="l00784"></a><span class="lineno"> 784</span>  <span class="comment">// = -((1 + alpha) * e_1 + alpha) * (n-2).</span></div><div class="line"><a name="l00785"></a><span class="lineno"> 785</span>  <span class="comment">//</span></div><div class="line"><a name="l00786"></a><span class="lineno"> 786</span>  <span class="comment">// We sum up the bounds for all the iterations to get Lemma 4:</span></div><div class="line"><a name="l00787"></a><span class="lineno"> 787</span>  <span class="comment">//</span></div><div class="line"><a name="l00788"></a><span class="lineno"> 788</span>  <span class="comment">// Lemma 4: If the given problem is feasible, after k iterations the</span></div><div class="line"><a name="l00789"></a><span class="lineno"> 789</span>  <span class="comment">// price of any node is always greater than</span></div><div class="line"><a name="l00790"></a><span class="lineno"> 790</span>  <span class="comment">// -((1 + alpha) * C + (k * alpha)) * (n-2)</span></div><div class="line"><a name="l00791"></a><span class="lineno"> 791</span>  <span class="comment">//</span></div><div class="line"><a name="l00792"></a><span class="lineno"> 792</span>  <span class="comment">// Proof: Suppose the price decrease of every node in the iteration</span></div><div class="line"><a name="l00793"></a><span class="lineno"> 793</span>  <span class="comment">// with epsilon_ == x is bounded by B(x) which is proportional to x</span></div><div class="line"><a name="l00794"></a><span class="lineno"> 794</span>  <span class="comment">// (not surprisingly, this will be the same function B() as</span></div><div class="line"><a name="l00795"></a><span class="lineno"> 795</span>  <span class="comment">// above). Assume for simplicity that C, the largest cost magnitude,</span></div><div class="line"><a name="l00796"></a><span class="lineno"> 796</span>  <span class="comment">// is a power of alpha. Then the price of each node, tallied across</span></div><div class="line"><a name="l00797"></a><span class="lineno"> 797</span>  <span class="comment">// all iterations is bounded</span></div><div class="line"><a name="l00798"></a><span class="lineno"> 798</span>  <span class="comment">// p(v) > 2 * B(C/alpha) + 2 * B(C/alpha^2) + ... + 2 * B(kMinEpsilon)</span></div><div class="line"><a name="l00799"></a><span class="lineno"> 799</span>  <span class="comment">// == 2 * B(C/alpha) * alpha / (alpha - 1)</span></div><div class="line"><a name="l00800"></a><span class="lineno"> 800</span>  <span class="comment">// == 2 * B(C) / (alpha - 1).</span></div><div class="line"><a name="l00801"></a><span class="lineno"> 801</span>  <span class="comment">// As above, this needs some patching up to handle the fact that we</span></div><div class="line"><a name="l00802"></a><span class="lineno"> 802</span>  <span class="comment">// use truncating arithmetic. We saw that each iteration effectively</span></div><div class="line"><a name="l00803"></a><span class="lineno"> 803</span>  <span class="comment">// reduces the price bound by alpha * (n-2), hence if there are k</span></div><div class="line"><a name="l00804"></a><span class="lineno"> 804</span>  <span class="comment">// iterations, the bound is</span></div><div class="line"><a name="l00805"></a><span class="lineno"> 805</span>  <span class="comment">// p(v) > 2 * B(C) / (alpha - 1) - k * alpha * (n-2)</span></div><div class="line"><a name="l00806"></a><span class="lineno"> 806</span>  <span class="comment">// = -(1 + alpha) * C * (n-2) / (alpha - 1) - k * alpha * (n-2)</span></div><div class="line"><a name="l00807"></a><span class="lineno"> 807</span>  <span class="comment">// = (n-2) * (C * (1 + alpha) / (1 - alpha) - k * alpha).</span></div><div class="line"><a name="l00808"></a><span class="lineno"> 808</span>  <span class="comment">//</span></div><div class="line"><a name="l00809"></a><span class="lineno"> 809</span>  <span class="comment">// The bound of lemma 4 can be used to warn for possible overflow of</span></div><div class="line"><a name="l00810"></a><span class="lineno"> 810</span>  <span class="comment">// arithmetic precision. But because it involves the number of</span></div><div class="line"><a name="l00811"></a><span class="lineno"> 811</span>  <span class="comment">// iterations, k, we might as well count through the iterations</span></div><div class="line"><a name="l00812"></a><span class="lineno"> 812</span>  <span class="comment">// simply adding up the bounds given by Lemma 3 to get a tighter</span></div><div class="line"><a name="l00813"></a><span class="lineno"> 813</span>  <span class="comment">// result. This is what the implementation does.</span></div><div class="line"><a name="l00814"></a><span class="lineno"> 814</span> </div><div class="line"><a name="l00815"></a><span class="lineno"> 815</span>  <span class="comment">// A lower bound on the price of any node at any time throughout the</span></div><div class="line"><a name="l00816"></a><span class="lineno"> 816</span>  <span class="comment">// computation. A price below this level proves infeasibility; this</span></div><div class="line"><a name="l00817"></a><span class="lineno"> 817</span>  <span class="comment">// value is used for feasibility detection. We use this value also</span></div><div class="line"><a name="l00818"></a><span class="lineno"> 818</span>  <span class="comment">// to rule out the possibility of arithmetic overflow or warn the</span></div><div class="line"><a name="l00819"></a><span class="lineno"> 819</span>  <span class="comment">// client that we have not been able to rule out that possibility.</span></div><div class="line"><a name="l00820"></a><span class="lineno"> 820</span>  <span class="comment">//</span></div><div class="line"><a name="l00821"></a><span class="lineno"> 821</span>  <span class="comment">// We can use the value implied by Lemma 4 here, but note that that</span></div><div class="line"><a name="l00822"></a><span class="lineno"> 822</span>  <span class="comment">// value includes k, the number of iterations. It's plenty fast if</span></div><div class="line"><a name="l00823"></a><span class="lineno"> 823</span>  <span class="comment">// we count through the iterations to compute that value, but if</span></div><div class="line"><a name="l00824"></a><span class="lineno"> 824</span>  <span class="comment">// we're going to count through the iterations, we might as well use</span></div><div class="line"><a name="l00825"></a><span class="lineno"> 825</span>  <span class="comment">// the two-parameter bound from Lemma 3, summing up as we go. This</span></div><div class="line"><a name="l00826"></a><span class="lineno"> 826</span>  <span class="comment">// gives us a tighter bound and more comprehensible code.</span></div><div class="line"><a name="l00827"></a><span class="lineno"> 827</span>  <span class="comment">//</span></div><div class="line"><a name="l00828"></a><span class="lineno"> 828</span>  <span class="comment">// While computing this bound, if we find the value justified by the</span></div><div class="line"><a name="l00829"></a><span class="lineno"> 829</span>  <span class="comment">// theory lies outside the representable range of CostValue, we</span></div><div class="line"><a name="l00830"></a><span class="lineno"> 830</span>  <span class="comment">// conclude that the given arc costs have magnitudes so large that</span></div><div class="line"><a name="l00831"></a><span class="lineno"> 831</span>  <span class="comment">// we cannot guarantee our calculations don't overflow. If the value</span></div><div class="line"><a name="l00832"></a><span class="lineno"> 832</span>  <span class="comment">// justified by the theory lies inside the representable range of</span></div><div class="line"><a name="l00833"></a><span class="lineno"> 833</span>  <span class="comment">// CostValue, we commit that our calculation will not overflow. This</span></div><div class="line"><a name="l00834"></a><span class="lineno"> 834</span>  <span class="comment">// commitment means we need to be careful with the amount by which</span></div><div class="line"><a name="l00835"></a><span class="lineno"> 835</span>  <span class="comment">// we relabel right-side nodes that are incident to any node with</span></div><div class="line"><a name="l00836"></a><span class="lineno"> 836</span>  <span class="comment">// only one neighbor.</span></div><div class="line"><a name="l00837"></a><span class="lineno"> 837</span>  <a class="code" href="namespaceoperations__research.html#aee97ac67f280d35acdef2c5d461a85c3">CostValue</a> price_lower_bound_;</div><div class="line"><a name="l00838"></a><span class="lineno"> 838</span> </div><div class="line"><a name="l00839"></a><span class="lineno"> 839</span>  <span class="comment">// A bound on the amount by which a node's price can be reduced</span></div><div class="line"><a name="l00840"></a><span class="lineno"> 840</span>  <span class="comment">// during the current iteration, used only for slack</span></div><div class="line"><a name="l00841"></a><span class="lineno"> 841</span>  <span class="comment">// relabelings. Where epsilon is the first iteration's error</span></div><div class="line"><a name="l00842"></a><span class="lineno"> 842</span>  <span class="comment">// parameter and C is the largest magnitude of an arc cost, we set</span></div><div class="line"><a name="l00843"></a><span class="lineno"> 843</span>  <span class="comment">// slack_relabeling_price_ = -B(C, epsilon)</span></div><div class="line"><a name="l00844"></a><span class="lineno"> 844</span>  <span class="comment">// = (C + epsilon) * (n-2)/2.</span></div><div class="line"><a name="l00845"></a><span class="lineno"> 845</span>  <span class="comment">//</span></div><div class="line"><a name="l00846"></a><span class="lineno"> 846</span>  <span class="comment">// We could use slack_relabeling_price_ for feasibility detection</span></div><div class="line"><a name="l00847"></a><span class="lineno"> 847</span>  <span class="comment">// but the feasibility threshold is double the slack relabeling</span></div><div class="line"><a name="l00848"></a><span class="lineno"> 848</span>  <span class="comment">// amount and we judge it not to be worth having to multiply by two</span></div><div class="line"><a name="l00849"></a><span class="lineno"> 849</span>  <span class="comment">// gratuitously to check feasibility in each double push</span></div><div class="line"><a name="l00850"></a><span class="lineno"> 850</span>  <span class="comment">// operation. Instead we settle for feasibility detection using</span></div><div class="line"><a name="l00851"></a><span class="lineno"> 851</span>  <span class="comment">// price_lower_bound_ instead, which is somewhat slower in the</span></div><div class="line"><a name="l00852"></a><span class="lineno"> 852</span>  <span class="comment">// infeasible case because more relabelings will be required for</span></div><div class="line"><a name="l00853"></a><span class="lineno"> 853</span>  <span class="comment">// some node price to attain the looser bound.</span></div><div class="line"><a name="l00854"></a><span class="lineno"> 854</span>  <a class="code" href="namespaceoperations__research.html#aee97ac67f280d35acdef2c5d461a85c3">CostValue</a> slack_relabeling_price_;</div><div class="line"><a name="l00855"></a><span class="lineno"> 855</span> </div><div class="line"><a name="l00856"></a><span class="lineno"> 856</span>  <span class="comment">// Computes the value of the bound on price reduction for an</span></div><div class="line"><a name="l00857"></a><span class="lineno"> 857</span>  <span class="comment">// iteration, given the old and new values of epsilon_. Because the</span></div><div class="line"><a name="l00858"></a><span class="lineno"> 858</span>  <span class="comment">// expression computed here is used in at least one place where we</span></div><div class="line"><a name="l00859"></a><span class="lineno"> 859</span>  <span class="comment">// want an additional factor in the denominator, we take that factor</span></div><div class="line"><a name="l00860"></a><span class="lineno"> 860</span>  <span class="comment">// as an argument. If extra_divisor == 1, this function computes of</span></div><div class="line"><a name="l00861"></a><span class="lineno"> 861</span>  <span class="comment">// the function B() discussed above.</span></div><div class="line"><a name="l00862"></a><span class="lineno"> 862</span>  <span class="comment">//</span></div><div class="line"><a name="l00863"></a><span class="lineno"> 863</span>  <span class="comment">// Avoids overflow in computing the bound, and sets *in_range =</span></div><div class="line"><a name="l00864"></a><span class="lineno"> 864</span>  <span class="comment">// false if the value of the bound doesn't fit in CostValue.</span></div><div class="line"><a name="l00865"></a><span class="lineno"> 865</span>  <span class="keyword">inline</span> <a class="code" href="namespaceoperations__research.html#aee97ac67f280d35acdef2c5d461a85c3">CostValue</a> PriceChangeBound(<a class="code" href="namespaceoperations__research.html#aee97ac67f280d35acdef2c5d461a85c3">CostValue</a> old_epsilon,</div><div class="line"><a name="l00866"></a><span class="lineno"> 866</span>  <a class="code" href="namespaceoperations__research.html#aee97ac67f280d35acdef2c5d461a85c3">CostValue</a> new_epsilon,</div><div class="line"><a name="l00867"></a><span class="lineno"> 867</span>  <span class="keywordtype">bool</span>* in_range)<span class="keyword"> const </span>{</div><div class="line"><a name="l00868"></a><span class="lineno"> 868</span>  <span class="keyword">const</span> <a class="code" href="namespaceoperations__research.html#aee97ac67f280d35acdef2c5d461a85c3">CostValue</a> n = graph_->num_nodes();</div><div class="line"><a name="l00869"></a><span class="lineno"> 869</span>  <span class="comment">// We work in double-precision floating point to determine whether</span></div><div class="line"><a name="l00870"></a><span class="lineno"> 870</span>  <span class="comment">// we'll overflow the integral CostValue type's range of</span></div><div class="line"><a name="l00871"></a><span class="lineno"> 871</span>  <span class="comment">// representation. Switching between integer and double is a</span></div><div class="line"><a name="l00872"></a><span class="lineno"> 872</span>  <span class="comment">// rather expensive operation, but we do this only twice per</span></div><div class="line"><a name="l00873"></a><span class="lineno"> 873</span>  <span class="comment">// scaling iteration, so we can afford it rather than resort to</span></div><div class="line"><a name="l00874"></a><span class="lineno"> 874</span>  <span class="comment">// complex and subtle tricks within the bounds of integer</span></div><div class="line"><a name="l00875"></a><span class="lineno"> 875</span>  <span class="comment">// arithmetic.</span></div><div class="line"><a name="l00876"></a><span class="lineno"> 876</span>  <span class="comment">//</span></div><div class="line"><a name="l00877"></a><span class="lineno"> 877</span>  <span class="comment">// You will want to read the comments above about</span></div><div class="line"><a name="l00878"></a><span class="lineno"> 878</span>  <span class="comment">// price_lower_bound_ and slack_relabeling_price_, and have a</span></div><div class="line"><a name="l00879"></a><span class="lineno"> 879</span>  <span class="comment">// pencil handy. :-)</span></div><div class="line"><a name="l00880"></a><span class="lineno"> 880</span>  <span class="keyword">const</span> <span class="keywordtype">double</span> result =</div><div class="line"><a name="l00881"></a><span class="lineno"> 881</span>  static_cast<double>(std::max<CostValue>(1, n / 2 - 1)) *</div><div class="line"><a name="l00882"></a><span class="lineno"> 882</span>  (static_cast<double>(old_epsilon) + static_cast<double>(new_epsilon));</div><div class="line"><a name="l00883"></a><span class="lineno"> 883</span>  <span class="keyword">const</span> <span class="keywordtype">double</span> limit =</div><div class="line"><a name="l00884"></a><span class="lineno"> 884</span>  static_cast<double>(<a class="code" href="alldiff__cst_8cc.html#a26e6db9bcc64b584051ecc28171ed11f">std::numeric_limits<CostValue>::max</a>());</div><div class="line"><a name="l00885"></a><span class="lineno"> 885</span>  <span class="keywordflow">if</span> (result > limit) {</div><div class="line"><a name="l00886"></a><span class="lineno"> 886</span>  <span class="comment">// Our integer computations could overflow.</span></div><div class="line"><a name="l00887"></a><span class="lineno"> 887</span>  <span class="keywordflow">if</span> (in_range != <span class="keyword">nullptr</span>) *in_range = <span class="keyword">false</span>;</div><div class="line"><a name="l00888"></a><span class="lineno"> 888</span>  <span class="keywordflow">return</span> <a class="code" href="alldiff__cst_8cc.html#a26e6db9bcc64b584051ecc28171ed11f">std::numeric_limits<CostValue>::max</a>();</div><div class="line"><a name="l00889"></a><span class="lineno"> 889</span>  } <span class="keywordflow">else</span> {</div><div class="line"><a name="l00890"></a><span class="lineno"> 890</span>  <span class="comment">// Don't touch *in_range; other computations could already have</span></div><div class="line"><a name="l00891"></a><span class="lineno"> 891</span>  <span class="comment">// set it to false and we don't want to overwrite that result.</span></div><div class="line"><a name="l00892"></a><span class="lineno"> 892</span>  <span class="keywordflow">return</span> static_cast<CostValue>(result);</div><div class="line"><a name="l00893"></a><span class="lineno"> 893</span>  }</div><div class="line"><a name="l00894"></a><span class="lineno"> 894</span>  }</div><div class="line"><a name="l00895"></a><span class="lineno"> 895</span> </div><div class="line"><a name="l00896"></a><span class="lineno"> 896</span>  <span class="comment">// A scaled record of the largest arc-cost magnitude we've been</span></div><div class="line"><a name="l00897"></a><span class="lineno"> 897</span>  <span class="comment">// given during problem setup. This is used to set the initial value</span></div><div class="line"><a name="l00898"></a><span class="lineno"> 898</span>  <span class="comment">// of epsilon_, which in turn is used not only as the error</span></div><div class="line"><a name="l00899"></a><span class="lineno"> 899</span>  <span class="comment">// parameter but also to determine whether we risk arithmetic</span></div><div class="line"><a name="l00900"></a><span class="lineno"> 900</span>  <span class="comment">// overflow during the algorithm.</span></div><div class="line"><a name="l00901"></a><span class="lineno"> 901</span>  <span class="comment">//</span></div><div class="line"><a name="l00902"></a><span class="lineno"> 902</span>  <span class="comment">// Note: Our treatment of arithmetic overflow assumes the following</span></div><div class="line"><a name="l00903"></a><span class="lineno"> 903</span>  <span class="comment">// property of CostValue:</span></div><div class="line"><a name="l00904"></a><span class="lineno"> 904</span>  <span class="comment">// -std::numeric_limits<CostValue>::max() is a representable</span></div><div class="line"><a name="l00905"></a><span class="lineno"> 905</span>  <span class="comment">// CostValue.</span></div><div class="line"><a name="l00906"></a><span class="lineno"> 906</span>  <span class="comment">// That property is satisfied if CostValue uses a two's-complement</span></div><div class="line"><a name="l00907"></a><span class="lineno"> 907</span>  <span class="comment">// representation.</span></div><div class="line"><a name="l00908"></a><span class="lineno"> 908</span>  <a class="code" href="namespaceoperations__research.html#aee97ac67f280d35acdef2c5d461a85c3">CostValue</a> largest_scaled_cost_magnitude_;</div><div class="line"><a name="l00909"></a><span class="lineno"> 909</span> </div><div class="line"><a name="l00910"></a><span class="lineno"> 910</span>  <span class="comment">// The total excess in the graph. Given our asymmetric definition of</span></div><div class="line"><a name="l00911"></a><span class="lineno"> 911</span>  <span class="comment">// epsilon-optimality and our use of the double-push operation, this</span></div><div class="line"><a name="l00912"></a><span class="lineno"> 912</span>  <span class="comment">// equals the number of unmatched left-side nodes.</span></div><div class="line"><a name="l00913"></a><span class="lineno"> 913</span>  <a class="code" href="classoperations__research_1_1_linear_sum_assignment.html#aca73212d30b08a7c287c311b74311a6e">NodeIndex</a> total_excess_;</div><div class="line"><a name="l00914"></a><span class="lineno"> 914</span> </div><div class="line"><a name="l00915"></a><span class="lineno"> 915</span>  <span class="comment">// Indexed by node index, the price_ values are maintained only for</span></div><div class="line"><a name="l00916"></a><span class="lineno"> 916</span>  <span class="comment">// right-side nodes.</span></div><div class="line"><a name="l00917"></a><span class="lineno"> 917</span>  <span class="comment">//</span></div><div class="line"><a name="l00918"></a><span class="lineno"> 918</span>  <span class="comment">// Note: We use a ZVector to only allocate a vector of size num_left_nodes_</span></div><div class="line"><a name="l00919"></a><span class="lineno"> 919</span>  <span class="comment">// instead of 2*num_left_nodes_ since the right-side node indices start at</span></div><div class="line"><a name="l00920"></a><span class="lineno"> 920</span>  <span class="comment">// num_left_nodes_.</span></div><div class="line"><a name="l00921"></a><span class="lineno"> 921</span>  ZVector<CostValue> price_;</div><div class="line"><a name="l00922"></a><span class="lineno"> 922</span> </div><div class="line"><a name="l00923"></a><span class="lineno"> 923</span>  <span class="comment">// Indexed by left-side node index, the matched_arc_ array gives the</span></div><div class="line"><a name="l00924"></a><span class="lineno"> 924</span>  <span class="comment">// arc index of the arc matching any given left-side node, or</span></div><div class="line"><a name="l00925"></a><span class="lineno"> 925</span>  <span class="comment">// GraphType::kNilArc if the node is unmatched.</span></div><div class="line"><a name="l00926"></a><span class="lineno"> 926</span>  std::vector<ArcIndex> matched_arc_;</div><div class="line"><a name="l00927"></a><span class="lineno"> 927</span> </div><div class="line"><a name="l00928"></a><span class="lineno"> 928</span>  <span class="comment">// Indexed by right-side node index, the matched_node_ array gives</span></div><div class="line"><a name="l00929"></a><span class="lineno"> 929</span>  <span class="comment">// the node index of the left-side node matching any given</span></div><div class="line"><a name="l00930"></a><span class="lineno"> 930</span>  <span class="comment">// right-side node, or GraphType::kNilNode if the right-side node is</span></div><div class="line"><a name="l00931"></a><span class="lineno"> 931</span>  <span class="comment">// unmatched.</span></div><div class="line"><a name="l00932"></a><span class="lineno"> 932</span>  <span class="comment">//</span></div><div class="line"><a name="l00933"></a><span class="lineno"> 933</span>  <span class="comment">// Note: We use a ZVector for the same reason as for price_.</span></div><div class="line"><a name="l00934"></a><span class="lineno"> 934</span>  ZVector<NodeIndex> matched_node_;</div><div class="line"><a name="l00935"></a><span class="lineno"> 935</span> </div><div class="line"><a name="l00936"></a><span class="lineno"> 936</span>  <span class="comment">// The array of arc costs as given in the problem definition, except</span></div><div class="line"><a name="l00937"></a><span class="lineno"> 937</span>  <span class="comment">// that they are scaled up by the number of nodes in the graph so we</span></div><div class="line"><a name="l00938"></a><span class="lineno"> 938</span>  <span class="comment">// can use integer arithmetic throughout.</span></div><div class="line"><a name="l00939"></a><span class="lineno"> 939</span>  std::vector<CostValue> scaled_arc_cost_;</div><div class="line"><a name="l00940"></a><span class="lineno"> 940</span> </div><div class="line"><a name="l00941"></a><span class="lineno"> 941</span>  <span class="comment">// The container of active nodes (i.e., unmatched nodes). This can</span></div><div class="line"><a name="l00942"></a><span class="lineno"> 942</span>  <span class="comment">// be switched easily between ActiveNodeStack and ActiveNodeQueue</span></div><div class="line"><a name="l00943"></a><span class="lineno"> 943</span>  <span class="comment">// for experimentation.</span></div><div class="line"><a name="l00944"></a><span class="lineno"> 944</span>  std::unique_ptr<ActiveNodeContainerInterface> active_nodes_;</div><div class="line"><a name="l00945"></a><span class="lineno"> 945</span> </div><div class="line"><a name="l00946"></a><span class="lineno"> 946</span>  <span class="comment">// Statistics giving the overall numbers of various operations the</span></div><div class="line"><a name="l00947"></a><span class="lineno"> 947</span>  <span class="comment">// algorithm performs.</span></div><div class="line"><a name="l00948"></a><span class="lineno"> 948</span>  Stats total_stats_;</div><div class="line"><a name="l00949"></a><span class="lineno"> 949</span> </div><div class="line"><a name="l00950"></a><span class="lineno"> 950</span>  <span class="comment">// Statistics giving the numbers of various operations the algorithm</span></div><div class="line"><a name="l00951"></a><span class="lineno"> 951</span>  <span class="comment">// has performed in the current iteration.</span></div><div class="line"><a name="l00952"></a><span class="lineno"> 952</span>  Stats iteration_stats_;</div><div class="line"><a name="l00953"></a><span class="lineno"> 953</span> </div><div class="line"><a name="l00954"></a><span class="lineno"> 954</span>  DISALLOW_COPY_AND_ASSIGN(<a class="code" href="classoperations__research_1_1_linear_sum_assignment.html#af45fdf861fabd6a46ce230a9f6101eda">LinearSumAssignment</a>);</div><div class="line"><a name="l00955"></a><span class="lineno"> 955</span> };</div><div class="line"><a name="l00956"></a><span class="lineno"> 956</span> </div><div class="line"><a name="l00957"></a><span class="lineno"> 957</span> <span class="comment">// Implementation of out-of-line LinearSumAssignment template member</span></div><div class="line"><a name="l00958"></a><span class="lineno"> 958</span> <span class="comment">// functions.</span></div><div class="line"><a name="l00959"></a><span class="lineno"> 959</span> </div><div class="line"><a name="l00960"></a><span class="lineno"> 960</span> <span class="keyword">template</span> <<span class="keyword">typename</span> GraphType></div><div class="line"><a name="l00961"></a><span class="lineno"> 961</span> <span class="keyword">const</span> <a class="code" href="namespaceoperations__research.html#aee97ac67f280d35acdef2c5d461a85c3">CostValue</a> LinearSumAssignment<GraphType>::kMinEpsilon = 1;</div><div class="line"><a name="l00962"></a><span class="lineno"> 962</span> </div><div class="line"><a name="l00963"></a><span class="lineno"> 963</span> <span class="keyword">template</span> <<span class="keyword">typename</span> GraphType></div><div class="line"><a name="l00964"></a><span class="lineno"><a class="line" href="classoperations__research_1_1_linear_sum_assignment.html#af45fdf861fabd6a46ce230a9f6101eda"> 964</a></span> <a class="code" href="classoperations__research_1_1_linear_sum_assignment.html#af45fdf861fabd6a46ce230a9f6101eda">LinearSumAssignment<GraphType>::LinearSumAssignment</a>(</div><div class="line"><a name="l00965"></a><span class="lineno"> 965</span>  <span class="keyword">const</span> GraphType& graph, <span class="keyword">const</span> <a class="code" href="classoperations__research_1_1_linear_sum_assignment.html#aca73212d30b08a7c287c311b74311a6e">NodeIndex</a> num_left_nodes)</div><div class="line"><a name="l00966"></a><span class="lineno"> 966</span>  : graph_(&graph),</div><div class="line"><a name="l00967"></a><span class="lineno"> 967</span>  num_left_nodes_(num_left_nodes),</div><div class="line"><a name="l00968"></a><span class="lineno"> 968</span>  success_(false),</div><div class="line"><a name="l00969"></a><span class="lineno"> 969</span>  cost_scaling_factor_(1 + num_left_nodes),</div><div class="line"><a name="l00970"></a><span class="lineno"> 970</span>  alpha_(<a class="code" href="namespaceabsl.html">absl</a>::GetFlag(FLAGS_assignment_alpha)),</div><div class="line"><a name="l00971"></a><span class="lineno"> 971</span>  epsilon_(0),</div><div class="line"><a name="l00972"></a><span class="lineno"> 972</span>  price_lower_bound_(0),</div><div class="line"><a name="l00973"></a><span class="lineno"> 973</span>  slack_relabeling_price_(0),</div><div class="line"><a name="l00974"></a><span class="lineno"> 974</span>  largest_scaled_cost_magnitude_(0),</div><div class="line"><a name="l00975"></a><span class="lineno"> 975</span>  total_excess_(0),</div><div class="line"><a name="l00976"></a><span class="lineno"> 976</span>  price_(num_left_nodes, 2 * num_left_nodes - 1),</div><div class="line"><a name="l00977"></a><span class="lineno"> 977</span>  matched_arc_(num_left_nodes, 0),</div><div class="line"><a name="l00978"></a><span class="lineno"> 978</span>  matched_node_(num_left_nodes, 2 * num_left_nodes - 1),</div><div class="line"><a name="l00979"></a><span class="lineno"> 979</span>  scaled_arc_cost_(graph.max_end_arc_index(), 0),</div><div class="line"><a name="l00980"></a><span class="lineno"> 980</span>  active_nodes_(<a class="code" href="namespaceabsl.html">absl</a>::GetFlag(FLAGS_assignment_stack_order)</div><div class="line"><a name="l00981"></a><span class="lineno"> 981</span>  ? static_cast<ActiveNodeContainerInterface*>(</div><div class="line"><a name="l00982"></a><span class="lineno"> 982</span>  new ActiveNodeStack())</div><div class="line"><a name="l00983"></a><span class="lineno"> 983</span>  : static_cast<ActiveNodeContainerInterface*>(</div><div class="line"><a name="l00984"></a><span class="lineno"> 984</span>  new ActiveNodeQueue())) {}</div><div class="line"><a name="l00985"></a><span class="lineno"> 985</span> </div><div class="line"><a name="l00986"></a><span class="lineno"> 986</span> <span class="keyword">template</span> <<span class="keyword">typename</span> GraphType></div><div class="line"><a name="l00987"></a><span class="lineno"><a class="line" href="classoperations__research_1_1_linear_sum_assignment.html#adf78bd1dd4c9cf7158ea23d06dcc89a4"> 987</a></span> <a class="code" href="classoperations__research_1_1_linear_sum_assignment.html#af45fdf861fabd6a46ce230a9f6101eda">LinearSumAssignment<GraphType>::LinearSumAssignment</a>(</div><div class="line"><a name="l00988"></a><span class="lineno"> 988</span>  <span class="keyword">const</span> <a class="code" href="classoperations__research_1_1_linear_sum_assignment.html#aca73212d30b08a7c287c311b74311a6e">NodeIndex</a> num_left_nodes, <span class="keyword">const</span> <a class="code" href="classoperations__research_1_1_linear_sum_assignment.html#a28178bb3e97b7c81dc157fe283b8cd64">ArcIndex</a> num_arcs)</div><div class="line"><a name="l00989"></a><span class="lineno"> 989</span>  : graph_(nullptr),</div><div class="line"><a name="l00990"></a><span class="lineno"> 990</span>  num_left_nodes_(num_left_nodes),</div><div class="line"><a name="l00991"></a><span class="lineno"> 991</span>  success_(false),</div><div class="line"><a name="l00992"></a><span class="lineno"> 992</span>  cost_scaling_factor_(1 + num_left_nodes),</div><div class="line"><a name="l00993"></a><span class="lineno"> 993</span>  alpha_(<a class="code" href="namespaceabsl.html">absl</a>::GetFlag(FLAGS_assignment_alpha)),</div><div class="line"><a name="l00994"></a><span class="lineno"> 994</span>  epsilon_(0),</div><div class="line"><a name="l00995"></a><span class="lineno"> 995</span>  price_lower_bound_(0),</div><div class="line"><a name="l00996"></a><span class="lineno"> 996</span>  slack_relabeling_price_(0),</div><div class="line"><a name="l00997"></a><span class="lineno"> 997</span>  largest_scaled_cost_magnitude_(0),</div><div class="line"><a name="l00998"></a><span class="lineno"> 998</span>  total_excess_(0),</div><div class="line"><a name="l00999"></a><span class="lineno"> 999</span>  price_(num_left_nodes, 2 * num_left_nodes - 1),</div><div class="line"><a name="l01000"></a><span class="lineno"> 1000</span>  matched_arc_(num_left_nodes, 0),</div><div class="line"><a name="l01001"></a><span class="lineno"> 1001</span>  matched_node_(num_left_nodes, 2 * num_left_nodes - 1),</div><div class="line"><a name="l01002"></a><span class="lineno"> 1002</span>  scaled_arc_cost_(num_arcs, 0),</div><div class="line"><a name="l01003"></a><span class="lineno"> 1003</span>  active_nodes_(<a class="code" href="namespaceabsl.html">absl</a>::GetFlag(FLAGS_assignment_stack_order)</div><div class="line"><a name="l01004"></a><span class="lineno"> 1004</span>  ? static_cast<ActiveNodeContainerInterface*>(</div><div class="line"><a name="l01005"></a><span class="lineno"> 1005</span>  new ActiveNodeStack())</div><div class="line"><a name="l01006"></a><span class="lineno"> 1006</span>  : static_cast<ActiveNodeContainerInterface*>(</div><div class="line"><a name="l01007"></a><span class="lineno"> 1007</span>  new ActiveNodeQueue())) {}</div><div class="line"><a name="l01008"></a><span class="lineno"> 1008</span> </div><div class="line"><a name="l01009"></a><span class="lineno"> 1009</span> <span class="keyword">template</span> <<span class="keyword">typename</span> GraphType></div><div class="line"><a name="l01010"></a><span class="lineno"><a class="line" href="classoperations__research_1_1_linear_sum_assignment.html#a77a0519df5fb71834593bb661b72921c"> 1010</a></span> <span class="keywordtype">void</span> <a class="code" href="classoperations__research_1_1_linear_sum_assignment.html#a77a0519df5fb71834593bb661b72921c">LinearSumAssignment<GraphType>::SetArcCost</a>(<a class="code" href="classoperations__research_1_1_linear_sum_assignment.html#a28178bb3e97b7c81dc157fe283b8cd64">ArcIndex</a> arc, <a class="code" href="namespaceoperations__research.html#aee97ac67f280d35acdef2c5d461a85c3">CostValue</a> <a class="code" href="routing__flow_8cc.html#a75d7b5e4cab1e156cc7a2c5eba1e16f1">cost</a>) {</div><div class="line"><a name="l01011"></a><span class="lineno"> 1011</span>  <span class="keywordflow">if</span> (graph_ != <span class="keyword">nullptr</span>) {</div><div class="line"><a name="l01012"></a><span class="lineno"> 1012</span>  <a class="code" href="base_2logging_8h.html#aae2dc65d9ea248d54bf39daa986dd295">DCHECK_GE</a>(arc, 0);</div><div class="line"><a name="l01013"></a><span class="lineno"> 1013</span>  <a class="code" href="base_2logging_8h.html#ab62f5ed8f2d48e29802be0cbbcd1359a">DCHECK_LT</a>(arc, graph_->num_arcs());</div><div class="line"><a name="l01014"></a><span class="lineno"> 1014</span>  <a class="code" href="classoperations__research_1_1_linear_sum_assignment.html#aca73212d30b08a7c287c311b74311a6e">NodeIndex</a> <a class="code" href="routing__flow_8cc.html#afca32f65388659a4b0956496169488b4">head</a> = Head(arc);</div><div class="line"><a name="l01015"></a><span class="lineno"> 1015</span>  <a class="code" href="base_2logging_8h.html#a4395e95bab44e222cb2e77251017a0e2">DCHECK_LE</a>(num_left_nodes_, <a class="code" href="routing__flow_8cc.html#afca32f65388659a4b0956496169488b4">head</a>);</div><div class="line"><a name="l01016"></a><span class="lineno"> 1016</span>  }</div><div class="line"><a name="l01017"></a><span class="lineno"> 1017</span>  <a class="code" href="routing__flow_8cc.html#a75d7b5e4cab1e156cc7a2c5eba1e16f1">cost</a> *= cost_scaling_factor_;</div><div class="line"><a name="l01018"></a><span class="lineno"> 1018</span>  <span class="keyword">const</span> <a class="code" href="namespaceoperations__research.html#aee97ac67f280d35acdef2c5d461a85c3">CostValue</a> cost_magnitude = std::abs(<a class="code" href="routing__flow_8cc.html#a75d7b5e4cab1e156cc7a2c5eba1e16f1">cost</a>);</div><div class="line"><a name="l01019"></a><span class="lineno"> 1019</span>  largest_scaled_cost_magnitude_ =</div><div class="line"><a name="l01020"></a><span class="lineno"> 1020</span>  <a class="code" href="alldiff__cst_8cc.html#a26e6db9bcc64b584051ecc28171ed11f">std::max</a>(largest_scaled_cost_magnitude_, cost_magnitude);</div><div class="line"><a name="l01021"></a><span class="lineno"> 1021</span>  scaled_arc_cost_[arc] = <a class="code" href="routing__flow_8cc.html#a75d7b5e4cab1e156cc7a2c5eba1e16f1">cost</a>;</div><div class="line"><a name="l01022"></a><span class="lineno"> 1022</span> }</div><div class="line"><a name="l01023"></a><span class="lineno"> 1023</span> </div><div class="line"><a name="l01024"></a><span class="lineno"> 1024</span> <span class="keyword">template</span> <<span class="keyword">typename</span> ArcIndexType></div><div class="line"><a name="l01025"></a><span class="lineno"><a class="line" href="classoperations__research_1_1_cost_value_cycle_handler.html"> 1025</a></span> <span class="keyword">class </span><a class="code" href="classoperations__research_1_1_cost_value_cycle_handler.html">CostValueCycleHandler</a> : <span class="keyword">public</span> <a class="code" href="classoperations__research_1_1_permutation_cycle_handler.html">PermutationCycleHandler</a><ArcIndexType> {</div><div class="line"><a name="l01026"></a><span class="lineno"> 1026</span>  <span class="keyword">public</span>:</div><div class="line"><a name="l01027"></a><span class="lineno"><a class="line" href="classoperations__research_1_1_cost_value_cycle_handler.html#a8bd36eabd11be9f5c4e3094418412544"> 1027</a></span>  <span class="keyword">explicit</span> <a class="code" href="classoperations__research_1_1_cost_value_cycle_handler.html#a8bd36eabd11be9f5c4e3094418412544">CostValueCycleHandler</a>(std::vector<CostValue>* <a class="code" href="routing__flow_8cc.html#a75d7b5e4cab1e156cc7a2c5eba1e16f1">cost</a>)</div><div class="line"><a name="l01028"></a><span class="lineno"> 1028</span>  : temp_(0), cost_(<a class="code" href="routing__flow_8cc.html#a75d7b5e4cab1e156cc7a2c5eba1e16f1">cost</a>) {}</div><div class="line"><a name="l01029"></a><span class="lineno"> 1029</span> </div><div class="line"><a name="l01030"></a><span class="lineno"><a class="line" href="classoperations__research_1_1_cost_value_cycle_handler.html#a999f9e59b55e3f184efa317522f2eeb0"> 1030</a></span>  <span class="keywordtype">void</span> <a class="code" href="classoperations__research_1_1_cost_value_cycle_handler.html#a999f9e59b55e3f184efa317522f2eeb0">SetTempFromIndex</a>(ArcIndexType source)<span class="keyword"> override </span>{</div><div class="line"><a name="l01031"></a><span class="lineno"> 1031</span>  temp_ = (*cost_)[source];</div><div class="line"><a name="l01032"></a><span class="lineno"> 1032</span>  }</div><div class="line"><a name="l01033"></a><span class="lineno"> 1033</span> </div><div class="line"><a name="l01034"></a><span class="lineno"><a class="line" href="classoperations__research_1_1_cost_value_cycle_handler.html#a2fd5ecc6414d07e3456e96c0d665ae9b"> 1034</a></span>  <span class="keywordtype">void</span> <a class="code" href="classoperations__research_1_1_cost_value_cycle_handler.html#a2fd5ecc6414d07e3456e96c0d665ae9b">SetIndexFromIndex</a>(ArcIndexType source,</div><div class="line"><a name="l01035"></a><span class="lineno"> 1035</span>  ArcIndexType destination)<span class="keyword"> const override </span>{</div><div class="line"><a name="l01036"></a><span class="lineno"> 1036</span>  (*cost_)[destination] = (*cost_)[source];</div><div class="line"><a name="l01037"></a><span class="lineno"> 1037</span>  }</div><div class="line"><a name="l01038"></a><span class="lineno"> 1038</span> </div><div class="line"><a name="l01039"></a><span class="lineno"><a class="line" href="classoperations__research_1_1_cost_value_cycle_handler.html#ab8c76fdd7493de9946f7551ed3ca16bc"> 1039</a></span>  <span class="keywordtype">void</span> <a class="code" href="classoperations__research_1_1_cost_value_cycle_handler.html#ab8c76fdd7493de9946f7551ed3ca16bc">SetIndexFromTemp</a>(ArcIndexType destination)<span class="keyword"> const override </span>{</div><div class="line"><a name="l01040"></a><span class="lineno"> 1040</span>  (*cost_)[destination] = temp_;</div><div class="line"><a name="l01041"></a><span class="lineno"> 1041</span>  }</div><div class="line"><a name="l01042"></a><span class="lineno"> 1042</span> </div><div class="line"><a name="l01043"></a><span class="lineno"><a class="line" href="classoperations__research_1_1_cost_value_cycle_handler.html#a11007fbe576099d4e9ba5a433036a8ed"> 1043</a></span>  <a class="code" href="classoperations__research_1_1_cost_value_cycle_handler.html#a11007fbe576099d4e9ba5a433036a8ed">~CostValueCycleHandler</a>()<span class="keyword"> override </span>{}</div><div class="line"><a name="l01044"></a><span class="lineno"> 1044</span> </div><div class="line"><a name="l01045"></a><span class="lineno"> 1045</span>  <span class="keyword">private</span>:</div><div class="line"><a name="l01046"></a><span class="lineno"> 1046</span>  <a class="code" href="namespaceoperations__research.html#aee97ac67f280d35acdef2c5d461a85c3">CostValue</a> temp_;</div><div class="line"><a name="l01047"></a><span class="lineno"> 1047</span>  std::vector<CostValue>* <span class="keyword">const</span> cost_;</div><div class="line"><a name="l01048"></a><span class="lineno"> 1048</span> </div><div class="line"><a name="l01049"></a><span class="lineno"> 1049</span>  DISALLOW_COPY_AND_ASSIGN(<a class="code" href="classoperations__research_1_1_cost_value_cycle_handler.html">CostValueCycleHandler</a>);</div><div class="line"><a name="l01050"></a><span class="lineno"> 1050</span> };</div><div class="line"><a name="l01051"></a><span class="lineno"> 1051</span> </div><div class="line"><a name="l01052"></a><span class="lineno"> 1052</span> <span class="comment">// Logically this class should be defined inside OptimizeGraphLayout,</span></div><div class="line"><a name="l01053"></a><span class="lineno"> 1053</span> <span class="comment">// but compilation fails if we do that because C++98 doesn't allow</span></div><div class="line"><a name="l01054"></a><span class="lineno"> 1054</span> <span class="comment">// instantiation of member templates with function-scoped types as</span></div><div class="line"><a name="l01055"></a><span class="lineno"> 1055</span> <span class="comment">// template parameters, which in turn is because those function-scoped</span></div><div class="line"><a name="l01056"></a><span class="lineno"> 1056</span> <span class="comment">// types lack linkage.</span></div><div class="line"><a name="l01057"></a><span class="lineno"> 1057</span> <span class="keyword">template</span> <<span class="keyword">typename</span> GraphType></div><div class="line"><a name="l01058"></a><span class="lineno"><a class="line" href="classoperations__research_1_1_arc_index_ordering_by_tail_node.html"> 1058</a></span> <span class="keyword">class </span><a class="code" href="classoperations__research_1_1_arc_index_ordering_by_tail_node.html">ArcIndexOrderingByTailNode</a> {</div><div class="line"><a name="l01059"></a><span class="lineno"> 1059</span>  <span class="keyword">public</span>:</div><div class="line"><a name="l01060"></a><span class="lineno"><a class="line" href="classoperations__research_1_1_arc_index_ordering_by_tail_node.html#a6d178bcb4ffb6f497b77a9faedd8601c"> 1060</a></span>  <span class="keyword">explicit</span> <a class="code" href="classoperations__research_1_1_arc_index_ordering_by_tail_node.html#a6d178bcb4ffb6f497b77a9faedd8601c">ArcIndexOrderingByTailNode</a>(<span class="keyword">const</span> GraphType& graph) : graph_(graph) {}</div><div class="line"><a name="l01061"></a><span class="lineno"> 1061</span> </div><div class="line"><a name="l01062"></a><span class="lineno"> 1062</span>  <span class="comment">// Says ArcIndex a is less than ArcIndex b if arc a's tail is less</span></div><div class="line"><a name="l01063"></a><span class="lineno"> 1063</span>  <span class="comment">// than arc b's tail. If their tails are equal, orders according to</span></div><div class="line"><a name="l01064"></a><span class="lineno"> 1064</span>  <span class="comment">// heads.</span></div><div class="line"><a name="l01065"></a><span class="lineno"><a class="line" href="classoperations__research_1_1_arc_index_ordering_by_tail_node.html#a7fc8cebeaaae309b2282772d6cac1888"> 1065</a></span>  <span class="keywordtype">bool</span> <a class="code" href="classoperations__research_1_1_arc_index_ordering_by_tail_node.html#a7fc8cebeaaae309b2282772d6cac1888">operator()</a>(<span class="keyword">typename</span> <a class="code" href="namespaceoperations__research.html#aecf320bda6e95d1edaf3a546673e3e6b">GraphType::ArcIndex</a> <a class="code" href="constraint__solver_2table_8cc.html#acb18315d548212835cd8ed4287e6c0b6">a</a>,</div><div class="line"><a name="l01066"></a><span class="lineno"> 1066</span>  <span class="keyword">typename</span> <a class="code" href="namespaceoperations__research.html#aecf320bda6e95d1edaf3a546673e3e6b">GraphType::ArcIndex</a> <a class="code" href="constraint__solver_2table_8cc.html#a9293e4d29cac928301645070dd307683">b</a>)<span class="keyword"> const </span>{</div><div class="line"><a name="l01067"></a><span class="lineno"> 1067</span>  <span class="keywordflow">return</span> ((graph_.Tail(<a class="code" href="constraint__solver_2table_8cc.html#acb18315d548212835cd8ed4287e6c0b6">a</a>) < graph_.Tail(<a class="code" href="constraint__solver_2table_8cc.html#a9293e4d29cac928301645070dd307683">b</a>)) ||</div><div class="line"><a name="l01068"></a><span class="lineno"> 1068</span>  ((graph_.Tail(<a class="code" href="constraint__solver_2table_8cc.html#acb18315d548212835cd8ed4287e6c0b6">a</a>) == graph_.Tail(<a class="code" href="constraint__solver_2table_8cc.html#a9293e4d29cac928301645070dd307683">b</a>)) &&</div><div class="line"><a name="l01069"></a><span class="lineno"> 1069</span>  (graph_.Head(<a class="code" href="constraint__solver_2table_8cc.html#acb18315d548212835cd8ed4287e6c0b6">a</a>) < graph_.Head(<a class="code" href="constraint__solver_2table_8cc.html#a9293e4d29cac928301645070dd307683">b</a>))));</div><div class="line"><a name="l01070"></a><span class="lineno"> 1070</span>  }</div><div class="line"><a name="l01071"></a><span class="lineno"> 1071</span> </div><div class="line"><a name="l01072"></a><span class="lineno"> 1072</span>  <span class="keyword">private</span>:</div><div class="line"><a name="l01073"></a><span class="lineno"> 1073</span>  <span class="keyword">const</span> GraphType& graph_;</div><div class="line"><a name="l01074"></a><span class="lineno"> 1074</span> </div><div class="line"><a name="l01075"></a><span class="lineno"> 1075</span>  <span class="comment">// Copy and assign are allowed; they have to be for STL to work</span></div><div class="line"><a name="l01076"></a><span class="lineno"> 1076</span>  <span class="comment">// with this functor, although it seems like a bug for STL to be</span></div><div class="line"><a name="l01077"></a><span class="lineno"> 1077</span>  <span class="comment">// written that way.</span></div><div class="line"><a name="l01078"></a><span class="lineno"> 1078</span> };</div><div class="line"><a name="l01079"></a><span class="lineno"> 1079</span> </div><div class="line"><a name="l01080"></a><span class="lineno"> 1080</span> <span class="comment">// Passes ownership of the cycle handler to the caller.</span></div><div class="line"><a name="l01081"></a><span class="lineno"> 1081</span> <span class="keyword">template</span> <<span class="keyword">typename</span> GraphType></div><div class="line"><a name="l01082"></a><span class="lineno"> 1082</span> PermutationCycleHandler<typename GraphType::ArcIndex>*</div><div class="line"><a name="l01083"></a><span class="lineno"><a class="line" href="classoperations__research_1_1_linear_sum_assignment.html#ab70deec1644ce0de2f383c97830391e7"> 1083</a></span> <a class="code" href="classoperations__research_1_1_linear_sum_assignment.html#ab70deec1644ce0de2f383c97830391e7">LinearSumAssignment<GraphType>::ArcAnnotationCycleHandler</a>() {</div><div class="line"><a name="l01084"></a><span class="lineno"> 1084</span>  <span class="keywordflow">return</span> <span class="keyword">new</span> <a class="code" href="classoperations__research_1_1_cost_value_cycle_handler.html">CostValueCycleHandler<typename GraphType::ArcIndex></a>(</div><div class="line"><a name="l01085"></a><span class="lineno"> 1085</span>  &scaled_arc_cost_);</div><div class="line"><a name="l01086"></a><span class="lineno"> 1086</span> }</div><div class="line"><a name="l01087"></a><span class="lineno"> 1087</span> </div><div class="line"><a name="l01088"></a><span class="lineno"> 1088</span> <span class="keyword">template</span> <<span class="keyword">typename</span> GraphType></div><div class="line"><a name="l01089"></a><span class="lineno"><a class="line" href="classoperations__research_1_1_linear_sum_assignment.html#aa4b556b7bee4c7f3fb60781e7ebda1b2"> 1089</a></span> <span class="keywordtype">void</span> <a class="code" href="classoperations__research_1_1_linear_sum_assignment.html#aa4b556b7bee4c7f3fb60781e7ebda1b2">LinearSumAssignment<GraphType>::OptimizeGraphLayout</a>(GraphType* graph) {</div><div class="line"><a name="l01090"></a><span class="lineno"> 1090</span>  <span class="comment">// The graph argument is only to give us a non-const-qualified</span></div><div class="line"><a name="l01091"></a><span class="lineno"> 1091</span>  <span class="comment">// handle on the graph we already have. Any different graph is</span></div><div class="line"><a name="l01092"></a><span class="lineno"> 1092</span>  <span class="comment">// nonsense.</span></div><div class="line"><a name="l01093"></a><span class="lineno"> 1093</span>  <a class="code" href="base_2logging_8h.html#ae89df3243bbb8341130c7b3f44145ea0">DCHECK_EQ</a>(graph_, graph);</div><div class="line"><a name="l01094"></a><span class="lineno"> 1094</span>  <span class="keyword">const</span> <a class="code" href="classoperations__research_1_1_arc_index_ordering_by_tail_node.html">ArcIndexOrderingByTailNode<GraphType></a> compare(*graph_);</div><div class="line"><a name="l01095"></a><span class="lineno"> 1095</span>  <a class="code" href="classoperations__research_1_1_cost_value_cycle_handler.html">CostValueCycleHandler<typename GraphType::ArcIndex></a> cycle_handler(</div><div class="line"><a name="l01096"></a><span class="lineno"> 1096</span>  &scaled_arc_cost_);</div><div class="line"><a name="l01097"></a><span class="lineno"> 1097</span>  <a class="code" href="classoperations__research_1_1_tail_array_manager.html">TailArrayManager<GraphType></a> tail_array_manager(graph);</div><div class="line"><a name="l01098"></a><span class="lineno"> 1098</span>  tail_array_manager.<a class="code" href="classoperations__research_1_1_tail_array_manager.html#a5b9e3e11b5999e1e2265f9f14a824214">BuildTailArrayFromAdjacencyListsIfForwardGraph</a>();</div><div class="line"><a name="l01099"></a><span class="lineno"> 1099</span>  graph->GroupForwardArcsByFunctor(compare, &cycle_handler);</div><div class="line"><a name="l01100"></a><span class="lineno"> 1100</span>  tail_array_manager.<a class="code" href="classoperations__research_1_1_tail_array_manager.html#a930399f60b3c83a155ca86db090655c4">ReleaseTailArrayIfForwardGraph</a>();</div><div class="line"><a name="l01101"></a><span class="lineno"> 1101</span> }</div><div class="line"><a name="l01102"></a><span class="lineno"> 1102</span> </div><div class="line"><a name="l01103"></a><span class="lineno"> 1103</span> <span class="keyword">template</span> <<span class="keyword">typename</span> GraphType></div><div class="line"><a name="l01104"></a><span class="lineno"> 1104</span> <a class="code" href="namespaceoperations__research.html#aee97ac67f280d35acdef2c5d461a85c3">CostValue</a> <a class="code" href="classoperations__research_1_1_linear_sum_assignment.html">LinearSumAssignment<GraphType>::NewEpsilon</a>(</div><div class="line"><a name="l01105"></a><span class="lineno"> 1105</span>  <span class="keyword">const</span> <a class="code" href="namespaceoperations__research.html#aee97ac67f280d35acdef2c5d461a85c3">CostValue</a> current_epsilon)<span class="keyword"> const </span>{</div><div class="line"><a name="l01106"></a><span class="lineno"> 1106</span>  <span class="keywordflow">return</span> <a class="code" href="alldiff__cst_8cc.html#a26e6db9bcc64b584051ecc28171ed11f">std::max</a>(current_epsilon / alpha_, kMinEpsilon);</div><div class="line"><a name="l01107"></a><span class="lineno"> 1107</span> }</div><div class="line"><a name="l01108"></a><span class="lineno"> 1108</span> </div><div class="line"><a name="l01109"></a><span class="lineno"> 1109</span> <span class="keyword">template</span> <<span class="keyword">typename</span> GraphType></div><div class="line"><a name="l01110"></a><span class="lineno"> 1110</span> <span class="keywordtype">bool</span> LinearSumAssignment<GraphType>::UpdateEpsilon() {</div><div class="line"><a name="l01111"></a><span class="lineno"> 1111</span>  <a class="code" href="namespaceoperations__research.html#aee97ac67f280d35acdef2c5d461a85c3">CostValue</a> new_epsilon = NewEpsilon(epsilon_);</div><div class="line"><a name="l01112"></a><span class="lineno"> 1112</span>  slack_relabeling_price_ = PriceChangeBound(epsilon_, new_epsilon, <span class="keyword">nullptr</span>);</div><div class="line"><a name="l01113"></a><span class="lineno"> 1113</span>  epsilon_ = new_epsilon;</div><div class="line"><a name="l01114"></a><span class="lineno"> 1114</span>  <a class="code" href="base_2logging_8h.html#afcaa7cadd41741bb855c2ada1d2ef927">VLOG</a>(3) << <span class="stringliteral">"Updated: epsilon_ == "</span> << epsilon_;</div><div class="line"><a name="l01115"></a><span class="lineno"> 1115</span>  <a class="code" href="base_2logging_8h.html#afcaa7cadd41741bb855c2ada1d2ef927">VLOG</a>(4) << <span class="stringliteral">"slack_relabeling_price_ == "</span> << slack_relabeling_price_;</div><div class="line"><a name="l01116"></a><span class="lineno"> 1116</span>  <a class="code" href="base_2logging_8h.html#ab4f56aa24c4c9cddc47a6abd2d747f9a">DCHECK_GT</a>(slack_relabeling_price_, 0);</div><div class="line"><a name="l01117"></a><span class="lineno"> 1117</span>  <span class="comment">// For today we always return true; in the future updating epsilon</span></div><div class="line"><a name="l01118"></a><span class="lineno"> 1118</span>  <span class="comment">// in sophisticated ways could conceivably detect infeasibility</span></div><div class="line"><a name="l01119"></a><span class="lineno"> 1119</span>  <span class="comment">// before the first iteration of Refine().</span></div><div class="line"><a name="l01120"></a><span class="lineno"> 1120</span>  <span class="keywordflow">return</span> <span class="keyword">true</span>;</div><div class="line"><a name="l01121"></a><span class="lineno"> 1121</span> }</div><div class="line"><a name="l01122"></a><span class="lineno"> 1122</span> </div><div class="line"><a name="l01123"></a><span class="lineno"> 1123</span> <span class="comment">// For production code that checks whether a left-side node is active.</span></div><div class="line"><a name="l01124"></a><span class="lineno"> 1124</span> <span class="keyword">template</span> <<span class="keyword">typename</span> GraphType></div><div class="line"><a name="l01125"></a><span class="lineno"> 1125</span> <span class="keyword">inline</span> <span class="keywordtype">bool</span> LinearSumAssignment<GraphType>::IsActive(</div><div class="line"><a name="l01126"></a><span class="lineno"> 1126</span>  <a class="code" href="namespaceoperations__research.html#a7ae31ba4c3b4899478e53ca13df35dfc">NodeIndex</a> left_node)<span class="keyword"> const </span>{</div><div class="line"><a name="l01127"></a><span class="lineno"> 1127</span>  <a class="code" href="base_2logging_8h.html#ab62f5ed8f2d48e29802be0cbbcd1359a">DCHECK_LT</a>(left_node, num_left_nodes_);</div><div class="line"><a name="l01128"></a><span class="lineno"> 1128</span>  <span class="keywordflow">return</span> matched_arc_[left_node] == GraphType::kNilArc;</div><div class="line"><a name="l01129"></a><span class="lineno"> 1129</span> }</div><div class="line"><a name="l01130"></a><span class="lineno"> 1130</span> </div><div class="line"><a name="l01131"></a><span class="lineno"> 1131</span> <span class="comment">// Only for debugging. Separate from the production IsActive() method</span></div><div class="line"><a name="l01132"></a><span class="lineno"> 1132</span> <span class="comment">// so that method can assert that its argument is a left-side node,</span></div><div class="line"><a name="l01133"></a><span class="lineno"> 1133</span> <span class="comment">// while for debugging we need to be able to test any node.</span></div><div class="line"><a name="l01134"></a><span class="lineno"> 1134</span> <span class="keyword">template</span> <<span class="keyword">typename</span> GraphType></div><div class="line"><a name="l01135"></a><span class="lineno"> 1135</span> <span class="keyword">inline</span> <span class="keywordtype">bool</span> LinearSumAssignment<GraphType>::IsActiveForDebugging(</div><div class="line"><a name="l01136"></a><span class="lineno"> 1136</span>  <a class="code" href="namespaceoperations__research.html#a7ae31ba4c3b4899478e53ca13df35dfc">NodeIndex</a> node)<span class="keyword"> const </span>{</div><div class="line"><a name="l01137"></a><span class="lineno"> 1137</span>  <span class="keywordflow">if</span> (node < num_left_nodes_) {</div><div class="line"><a name="l01138"></a><span class="lineno"> 1138</span>  <span class="keywordflow">return</span> IsActive(node);</div><div class="line"><a name="l01139"></a><span class="lineno"> 1139</span>  } <span class="keywordflow">else</span> {</div><div class="line"><a name="l01140"></a><span class="lineno"> 1140</span>  <span class="keywordflow">return</span> matched_node_[node] == GraphType::kNilNode;</div><div class="line"><a name="l01141"></a><span class="lineno"> 1141</span>  }</div><div class="line"><a name="l01142"></a><span class="lineno"> 1142</span> }</div><div class="line"><a name="l01143"></a><span class="lineno"> 1143</span> </div><div class="line"><a name="l01144"></a><span class="lineno"> 1144</span> <span class="keyword">template</span> <<span class="keyword">typename</span> GraphType></div><div class="line"><a name="l01145"></a><span class="lineno"> 1145</span> <span class="keywordtype">void</span> LinearSumAssignment<GraphType>::InitializeActiveNodeContainer() {</div><div class="line"><a name="l01146"></a><span class="lineno"> 1146</span>  <a class="code" href="base_2logging_8h.html#ae17f8119c108cf3070bad3449c7e0006">DCHECK</a>(active_nodes_->Empty());</div><div class="line"><a name="l01147"></a><span class="lineno"> 1147</span>  <span class="keywordflow">for</span> (BipartiteLeftNodeIterator node_it(*graph_, num_left_nodes_);</div><div class="line"><a name="l01148"></a><span class="lineno"> 1148</span>  node_it.Ok(); node_it.Next()) {</div><div class="line"><a name="l01149"></a><span class="lineno"> 1149</span>  <span class="keyword">const</span> <a class="code" href="namespaceoperations__research.html#a7ae31ba4c3b4899478e53ca13df35dfc">NodeIndex</a> node = node_it.Index();</div><div class="line"><a name="l01150"></a><span class="lineno"> 1150</span>  <span class="keywordflow">if</span> (IsActive(node)) {</div><div class="line"><a name="l01151"></a><span class="lineno"> 1151</span>  active_nodes_->Add(node);</div><div class="line"><a name="l01152"></a><span class="lineno"> 1152</span>  }</div><div class="line"><a name="l01153"></a><span class="lineno"> 1153</span>  }</div><div class="line"><a name="l01154"></a><span class="lineno"> 1154</span> }</div><div class="line"><a name="l01155"></a><span class="lineno"> 1155</span> </div><div class="line"><a name="l01156"></a><span class="lineno"> 1156</span> <span class="comment">// There exists a price function such that the admissible arcs at the</span></div><div class="line"><a name="l01157"></a><span class="lineno"> 1157</span> <span class="comment">// beginning of an iteration are exactly the reverse arcs of all</span></div><div class="line"><a name="l01158"></a><span class="lineno"> 1158</span> <span class="comment">// matching arcs. Saturating all admissible arcs with respect to that</span></div><div class="line"><a name="l01159"></a><span class="lineno"> 1159</span> <span class="comment">// price function therefore means simply unmatching every matched</span></div><div class="line"><a name="l01160"></a><span class="lineno"> 1160</span> <span class="comment">// node.</span></div><div class="line"><a name="l01161"></a><span class="lineno"> 1161</span> <span class="comment">//</span></div><div class="line"><a name="l01162"></a><span class="lineno"> 1162</span> <span class="comment">// In the future we will price out arcs, which will reduce the set of</span></div><div class="line"><a name="l01163"></a><span class="lineno"> 1163</span> <span class="comment">// nodes we unmatch here. If a matching arc is priced out, we will not</span></div><div class="line"><a name="l01164"></a><span class="lineno"> 1164</span> <span class="comment">// unmatch its endpoints since that element of the matching is</span></div><div class="line"><a name="l01165"></a><span class="lineno"> 1165</span> <span class="comment">// guaranteed not to change.</span></div><div class="line"><a name="l01166"></a><span class="lineno"> 1166</span> <span class="keyword">template</span> <<span class="keyword">typename</span> GraphType></div><div class="line"><a name="l01167"></a><span class="lineno"> 1167</span> <span class="keywordtype">void</span> LinearSumAssignment<GraphType>::SaturateNegativeArcs() {</div><div class="line"><a name="l01168"></a><span class="lineno"> 1168</span>  total_excess_ = 0;</div><div class="line"><a name="l01169"></a><span class="lineno"> 1169</span>  <span class="keywordflow">for</span> (BipartiteLeftNodeIterator node_it(*graph_, num_left_nodes_);</div><div class="line"><a name="l01170"></a><span class="lineno"> 1170</span>  node_it.Ok(); node_it.Next()) {</div><div class="line"><a name="l01171"></a><span class="lineno"> 1171</span>  <span class="keyword">const</span> <a class="code" href="namespaceoperations__research.html#a7ae31ba4c3b4899478e53ca13df35dfc">NodeIndex</a> node = node_it.Index();</div><div class="line"><a name="l01172"></a><span class="lineno"> 1172</span>  <span class="keywordflow">if</span> (IsActive(node)) {</div><div class="line"><a name="l01173"></a><span class="lineno"> 1173</span>  <span class="comment">// This can happen in the first iteration when nothing is</span></div><div class="line"><a name="l01174"></a><span class="lineno"> 1174</span>  <span class="comment">// matched yet.</span></div><div class="line"><a name="l01175"></a><span class="lineno"> 1175</span>  total_excess_ += 1;</div><div class="line"><a name="l01176"></a><span class="lineno"> 1176</span>  } <span class="keywordflow">else</span> {</div><div class="line"><a name="l01177"></a><span class="lineno"> 1177</span>  <span class="comment">// We're about to create a unit of excess by unmatching these nodes.</span></div><div class="line"><a name="l01178"></a><span class="lineno"> 1178</span>  total_excess_ += 1;</div><div class="line"><a name="l01179"></a><span class="lineno"> 1179</span>  <span class="keyword">const</span> <a class="code" href="namespaceoperations__research.html#a7ae31ba4c3b4899478e53ca13df35dfc">NodeIndex</a> mate = GetMate(node);</div><div class="line"><a name="l01180"></a><span class="lineno"> 1180</span>  matched_arc_[node] = GraphType::kNilArc;</div><div class="line"><a name="l01181"></a><span class="lineno"> 1181</span>  matched_node_[mate] = GraphType::kNilNode;</div><div class="line"><a name="l01182"></a><span class="lineno"> 1182</span>  }</div><div class="line"><a name="l01183"></a><span class="lineno"> 1183</span>  }</div><div class="line"><a name="l01184"></a><span class="lineno"> 1184</span> }</div><div class="line"><a name="l01185"></a><span class="lineno"> 1185</span> </div><div class="line"><a name="l01186"></a><span class="lineno"> 1186</span> <span class="comment">// Returns true for success, false for infeasible.</span></div><div class="line"><a name="l01187"></a><span class="lineno"> 1187</span> <span class="keyword">template</span> <<span class="keyword">typename</span> GraphType></div><div class="line"><a name="l01188"></a><span class="lineno"> 1188</span> <span class="keywordtype">bool</span> LinearSumAssignment<GraphType>::DoublePush(<a class="code" href="namespaceoperations__research.html#a7ae31ba4c3b4899478e53ca13df35dfc">NodeIndex</a> source) {</div><div class="line"><a name="l01189"></a><span class="lineno"> 1189</span>  <a class="code" href="base_2logging_8h.html#ab4f56aa24c4c9cddc47a6abd2d747f9a">DCHECK_GT</a>(num_left_nodes_, source);</div><div class="line"><a name="l01190"></a><span class="lineno"> 1190</span>  <a class="code" href="base_2logging_8h.html#ae17f8119c108cf3070bad3449c7e0006">DCHECK</a>(IsActive(source)) << <span class="stringliteral">"Node "</span> << source</div><div class="line"><a name="l01191"></a><span class="lineno"> 1191</span>  << <span class="stringliteral">"must be active (unmatched)!"</span>;</div><div class="line"><a name="l01192"></a><span class="lineno"> 1192</span>  ImplicitPriceSummary summary = BestArcAndGap(source);</div><div class="line"><a name="l01193"></a><span class="lineno"> 1193</span>  <span class="keyword">const</span> <a class="code" href="namespaceoperations__research.html#aecf320bda6e95d1edaf3a546673e3e6b">ArcIndex</a> best_arc = summary.first;</div><div class="line"><a name="l01194"></a><span class="lineno"> 1194</span>  <span class="keyword">const</span> <a class="code" href="namespaceoperations__research.html#aee97ac67f280d35acdef2c5d461a85c3">CostValue</a> gap = summary.second;</div><div class="line"><a name="l01195"></a><span class="lineno"> 1195</span>  <span class="comment">// Now we have the best arc incident to source, i.e., the one with</span></div><div class="line"><a name="l01196"></a><span class="lineno"> 1196</span>  <span class="comment">// minimum reduced cost. Match that arc, unmatching its head if</span></div><div class="line"><a name="l01197"></a><span class="lineno"> 1197</span>  <span class="comment">// necessary.</span></div><div class="line"><a name="l01198"></a><span class="lineno"> 1198</span>  <span class="keywordflow">if</span> (best_arc == GraphType::kNilArc) {</div><div class="line"><a name="l01199"></a><span class="lineno"> 1199</span>  <span class="keywordflow">return</span> <span class="keyword">false</span>;</div><div class="line"><a name="l01200"></a><span class="lineno"> 1200</span>  }</div><div class="line"><a name="l01201"></a><span class="lineno"> 1201</span>  <span class="keyword">const</span> <a class="code" href="namespaceoperations__research.html#a7ae31ba4c3b4899478e53ca13df35dfc">NodeIndex</a> new_mate = Head(best_arc);</div><div class="line"><a name="l01202"></a><span class="lineno"> 1202</span>  <span class="keyword">const</span> <a class="code" href="namespaceoperations__research.html#a7ae31ba4c3b4899478e53ca13df35dfc">NodeIndex</a> to_unmatch = matched_node_[new_mate];</div><div class="line"><a name="l01203"></a><span class="lineno"> 1203</span>  <span class="keywordflow">if</span> (to_unmatch != GraphType::kNilNode) {</div><div class="line"><a name="l01204"></a><span class="lineno"> 1204</span>  <span class="comment">// Unmatch new_mate from its current mate, pushing the unit of</span></div><div class="line"><a name="l01205"></a><span class="lineno"> 1205</span>  <span class="comment">// flow back to a node on the left side as a unit of excess.</span></div><div class="line"><a name="l01206"></a><span class="lineno"> 1206</span>  matched_arc_[to_unmatch] = GraphType::kNilArc;</div><div class="line"><a name="l01207"></a><span class="lineno"> 1207</span>  active_nodes_->Add(to_unmatch);</div><div class="line"><a name="l01208"></a><span class="lineno"> 1208</span>  <span class="comment">// This counts as a double push.</span></div><div class="line"><a name="l01209"></a><span class="lineno"> 1209</span>  iteration_stats_.double_pushes_ += 1;</div><div class="line"><a name="l01210"></a><span class="lineno"> 1210</span>  } <span class="keywordflow">else</span> {</div><div class="line"><a name="l01211"></a><span class="lineno"> 1211</span>  <span class="comment">// We are about to increase the cardinality of the matching.</span></div><div class="line"><a name="l01212"></a><span class="lineno"> 1212</span>  total_excess_ -= 1;</div><div class="line"><a name="l01213"></a><span class="lineno"> 1213</span>  <span class="comment">// This counts as a single push.</span></div><div class="line"><a name="l01214"></a><span class="lineno"> 1214</span>  iteration_stats_.pushes_ += 1;</div><div class="line"><a name="l01215"></a><span class="lineno"> 1215</span>  }</div><div class="line"><a name="l01216"></a><span class="lineno"> 1216</span>  matched_arc_[source] = best_arc;</div><div class="line"><a name="l01217"></a><span class="lineno"> 1217</span>  matched_node_[new_mate] = source;</div><div class="line"><a name="l01218"></a><span class="lineno"> 1218</span>  <span class="comment">// Finally, relabel new_mate.</span></div><div class="line"><a name="l01219"></a><span class="lineno"> 1219</span>  iteration_stats_.relabelings_ += 1;</div><div class="line"><a name="l01220"></a><span class="lineno"> 1220</span>  <span class="keyword">const</span> <a class="code" href="namespaceoperations__research.html#aee97ac67f280d35acdef2c5d461a85c3">CostValue</a> new_price = price_[new_mate] - gap - epsilon_;</div><div class="line"><a name="l01221"></a><span class="lineno"> 1221</span>  price_[new_mate] = new_price;</div><div class="line"><a name="l01222"></a><span class="lineno"> 1222</span>  <span class="keywordflow">return</span> new_price >= price_lower_bound_;</div><div class="line"><a name="l01223"></a><span class="lineno"> 1223</span> }</div><div class="line"><a name="l01224"></a><span class="lineno"> 1224</span> </div><div class="line"><a name="l01225"></a><span class="lineno"> 1225</span> <span class="keyword">template</span> <<span class="keyword">typename</span> GraphType></div><div class="line"><a name="l01226"></a><span class="lineno"> 1226</span> <span class="keywordtype">bool</span> LinearSumAssignment<GraphType>::Refine() {</div><div class="line"><a name="l01227"></a><span class="lineno"> 1227</span>  SaturateNegativeArcs();</div><div class="line"><a name="l01228"></a><span class="lineno"> 1228</span>  InitializeActiveNodeContainer();</div><div class="line"><a name="l01229"></a><span class="lineno"> 1229</span>  <span class="keywordflow">while</span> (total_excess_ > 0) {</div><div class="line"><a name="l01230"></a><span class="lineno"> 1230</span>  <span class="comment">// Get an active node (i.e., one with excess == 1) and discharge</span></div><div class="line"><a name="l01231"></a><span class="lineno"> 1231</span>  <span class="comment">// it using DoublePush.</span></div><div class="line"><a name="l01232"></a><span class="lineno"> 1232</span>  <span class="keyword">const</span> <a class="code" href="namespaceoperations__research.html#a7ae31ba4c3b4899478e53ca13df35dfc">NodeIndex</a> node = active_nodes_->Get();</div><div class="line"><a name="l01233"></a><span class="lineno"> 1233</span>  <span class="keywordflow">if</span> (!DoublePush(node)) {</div><div class="line"><a name="l01234"></a><span class="lineno"> 1234</span>  <span class="comment">// Infeasibility detected.</span></div><div class="line"><a name="l01235"></a><span class="lineno"> 1235</span>  <span class="comment">//</span></div><div class="line"><a name="l01236"></a><span class="lineno"> 1236</span>  <span class="comment">// If infeasibility is detected after the first iteration, we</span></div><div class="line"><a name="l01237"></a><span class="lineno"> 1237</span>  <span class="comment">// have a bug. We don't crash production code in this case but</span></div><div class="line"><a name="l01238"></a><span class="lineno"> 1238</span>  <span class="comment">// we know we're returning a wrong answer so we we leave a</span></div><div class="line"><a name="l01239"></a><span class="lineno"> 1239</span>  <span class="comment">// message in the logs to increase our hope of chasing down the</span></div><div class="line"><a name="l01240"></a><span class="lineno"> 1240</span>  <span class="comment">// problem.</span></div><div class="line"><a name="l01241"></a><span class="lineno"> 1241</span>  <a class="code" href="base_2logging_8h.html#a09f7d88282cf92c9f231270ac113e5c6">LOG_IF</a>(DFATAL, total_stats_.refinements_ > 0)</div><div class="line"><a name="l01242"></a><span class="lineno"> 1242</span>  << <span class="stringliteral">"Infeasibility detection triggered after first iteration found "</span></div><div class="line"><a name="l01243"></a><span class="lineno"> 1243</span>  << <span class="stringliteral">"a feasible assignment!"</span>;</div><div class="line"><a name="l01244"></a><span class="lineno"> 1244</span>  <span class="keywordflow">return</span> <span class="keyword">false</span>;</div><div class="line"><a name="l01245"></a><span class="lineno"> 1245</span>  }</div><div class="line"><a name="l01246"></a><span class="lineno"> 1246</span>  }</div><div class="line"><a name="l01247"></a><span class="lineno"> 1247</span>  <a class="code" href="base_2logging_8h.html#ae17f8119c108cf3070bad3449c7e0006">DCHECK</a>(active_nodes_->Empty());</div><div class="line"><a name="l01248"></a><span class="lineno"> 1248</span>  iteration_stats_.refinements_ += 1;</div><div class="line"><a name="l01249"></a><span class="lineno"> 1249</span>  <span class="keywordflow">return</span> <span class="keyword">true</span>;</div><div class="line"><a name="l01250"></a><span class="lineno"> 1250</span> }</div><div class="line"><a name="l01251"></a><span class="lineno"> 1251</span> </div><div class="line"><a name="l01252"></a><span class="lineno"> 1252</span> <span class="comment">// Computes best_arc, the minimum reduced-cost arc incident to</span></div><div class="line"><a name="l01253"></a><span class="lineno"> 1253</span> <span class="comment">// left_node and admissibility_gap, the amount by which the reduced</span></div><div class="line"><a name="l01254"></a><span class="lineno"> 1254</span> <span class="comment">// cost of best_arc must be increased to make it equal in reduced cost</span></div><div class="line"><a name="l01255"></a><span class="lineno"> 1255</span> <span class="comment">// to another residual arc incident to left_node.</span></div><div class="line"><a name="l01256"></a><span class="lineno"> 1256</span> <span class="comment">//</span></div><div class="line"><a name="l01257"></a><span class="lineno"> 1257</span> <span class="comment">// Precondition: left_node is unmatched and has at least one incident</span></div><div class="line"><a name="l01258"></a><span class="lineno"> 1258</span> <span class="comment">// arc. This allows us to simplify the code. The debug-only</span></div><div class="line"><a name="l01259"></a><span class="lineno"> 1259</span> <span class="comment">// counterpart to this routine is LinearSumAssignment::ImplicitPrice()</span></div><div class="line"><a name="l01260"></a><span class="lineno"> 1260</span> <span class="comment">// and it assumes there is an incident arc but does not assume the</span></div><div class="line"><a name="l01261"></a><span class="lineno"> 1261</span> <span class="comment">// node is unmatched. The condition that each left node has at least</span></div><div class="line"><a name="l01262"></a><span class="lineno"> 1262</span> <span class="comment">// one incident arc is explicitly computed during FinalizeSetup().</span></div><div class="line"><a name="l01263"></a><span class="lineno"> 1263</span> <span class="comment">//</span></div><div class="line"><a name="l01264"></a><span class="lineno"> 1264</span> <span class="comment">// This function is large enough that our suggestion that the compiler</span></div><div class="line"><a name="l01265"></a><span class="lineno"> 1265</span> <span class="comment">// inline it might be pointless.</span></div><div class="line"><a name="l01266"></a><span class="lineno"> 1266</span> <span class="keyword">template</span> <<span class="keyword">typename</span> GraphType></div><div class="line"><a name="l01267"></a><span class="lineno"> 1267</span> <span class="keyword">inline</span> <span class="keyword">typename</span> LinearSumAssignment<GraphType>::ImplicitPriceSummary</div><div class="line"><a name="l01268"></a><span class="lineno"> 1268</span> LinearSumAssignment<GraphType>::BestArcAndGap(<a class="code" href="namespaceoperations__research.html#a7ae31ba4c3b4899478e53ca13df35dfc">NodeIndex</a> left_node)<span class="keyword"> const </span>{</div><div class="line"><a name="l01269"></a><span class="lineno"> 1269</span>  <a class="code" href="base_2logging_8h.html#ae17f8119c108cf3070bad3449c7e0006">DCHECK</a>(IsActive(left_node))</div><div class="line"><a name="l01270"></a><span class="lineno"> 1270</span>  << <span class="stringliteral">"Node "</span> << left_node << <span class="stringliteral">" must be active (unmatched)!"</span>;</div><div class="line"><a name="l01271"></a><span class="lineno"> 1271</span>  <a class="code" href="base_2logging_8h.html#ab4f56aa24c4c9cddc47a6abd2d747f9a">DCHECK_GT</a>(epsilon_, 0);</div><div class="line"><a name="l01272"></a><span class="lineno"> 1272</span>  <span class="keyword">typename</span> GraphType::OutgoingArcIterator arc_it(*graph_, left_node);</div><div class="line"><a name="l01273"></a><span class="lineno"> 1273</span>  <a class="code" href="namespaceoperations__research.html#aecf320bda6e95d1edaf3a546673e3e6b">ArcIndex</a> best_arc = arc_it.Index();</div><div class="line"><a name="l01274"></a><span class="lineno"> 1274</span>  <a class="code" href="namespaceoperations__research.html#aee97ac67f280d35acdef2c5d461a85c3">CostValue</a> min_partial_reduced_cost = PartialReducedCost(best_arc);</div><div class="line"><a name="l01275"></a><span class="lineno"> 1275</span>  <span class="comment">// We choose second_min_partial_reduced_cost so that in the case of</span></div><div class="line"><a name="l01276"></a><span class="lineno"> 1276</span>  <span class="comment">// the largest possible gap (which results from a left-side node</span></div><div class="line"><a name="l01277"></a><span class="lineno"> 1277</span>  <span class="comment">// with only a single incident residual arc), the corresponding</span></div><div class="line"><a name="l01278"></a><span class="lineno"> 1278</span>  <span class="comment">// right-side node will be relabeled by an amount that exactly</span></div><div class="line"><a name="l01279"></a><span class="lineno"> 1279</span>  <span class="comment">// matches slack_relabeling_price_.</span></div><div class="line"><a name="l01280"></a><span class="lineno"> 1280</span>  <span class="keyword">const</span> <a class="code" href="namespaceoperations__research.html#aee97ac67f280d35acdef2c5d461a85c3">CostValue</a> max_gap = slack_relabeling_price_ - epsilon_;</div><div class="line"><a name="l01281"></a><span class="lineno"> 1281</span>  <a class="code" href="namespaceoperations__research.html#aee97ac67f280d35acdef2c5d461a85c3">CostValue</a> second_min_partial_reduced_cost =</div><div class="line"><a name="l01282"></a><span class="lineno"> 1282</span>  min_partial_reduced_cost + max_gap;</div><div class="line"><a name="l01283"></a><span class="lineno"> 1283</span>  <span class="keywordflow">for</span> (arc_it.Next(); arc_it.Ok(); arc_it.Next()) {</div><div class="line"><a name="l01284"></a><span class="lineno"> 1284</span>  <span class="keyword">const</span> <a class="code" href="namespaceoperations__research.html#aecf320bda6e95d1edaf3a546673e3e6b">ArcIndex</a> arc = arc_it.Index();</div><div class="line"><a name="l01285"></a><span class="lineno"> 1285</span>  <span class="keyword">const</span> <a class="code" href="namespaceoperations__research.html#aee97ac67f280d35acdef2c5d461a85c3">CostValue</a> partial_reduced_cost = PartialReducedCost(arc);</div><div class="line"><a name="l01286"></a><span class="lineno"> 1286</span>  <span class="keywordflow">if</span> (partial_reduced_cost < second_min_partial_reduced_cost) {</div><div class="line"><a name="l01287"></a><span class="lineno"> 1287</span>  <span class="keywordflow">if</span> (partial_reduced_cost < min_partial_reduced_cost) {</div><div class="line"><a name="l01288"></a><span class="lineno"> 1288</span>  best_arc = arc;</div><div class="line"><a name="l01289"></a><span class="lineno"> 1289</span>  second_min_partial_reduced_cost = min_partial_reduced_cost;</div><div class="line"><a name="l01290"></a><span class="lineno"> 1290</span>  min_partial_reduced_cost = partial_reduced_cost;</div><div class="line"><a name="l01291"></a><span class="lineno"> 1291</span>  } <span class="keywordflow">else</span> {</div><div class="line"><a name="l01292"></a><span class="lineno"> 1292</span>  second_min_partial_reduced_cost = partial_reduced_cost;</div><div class="line"><a name="l01293"></a><span class="lineno"> 1293</span>  }</div><div class="line"><a name="l01294"></a><span class="lineno"> 1294</span>  }</div><div class="line"><a name="l01295"></a><span class="lineno"> 1295</span>  }</div><div class="line"><a name="l01296"></a><span class="lineno"> 1296</span>  <span class="keyword">const</span> <a class="code" href="namespaceoperations__research.html#aee97ac67f280d35acdef2c5d461a85c3">CostValue</a> gap = std::min<CostValue>(</div><div class="line"><a name="l01297"></a><span class="lineno"> 1297</span>  second_min_partial_reduced_cost - min_partial_reduced_cost, max_gap);</div><div class="line"><a name="l01298"></a><span class="lineno"> 1298</span>  <a class="code" href="base_2logging_8h.html#aae2dc65d9ea248d54bf39daa986dd295">DCHECK_GE</a>(gap, 0);</div><div class="line"><a name="l01299"></a><span class="lineno"> 1299</span>  <span class="keywordflow">return</span> std::make_pair(best_arc, gap);</div><div class="line"><a name="l01300"></a><span class="lineno"> 1300</span> }</div><div class="line"><a name="l01301"></a><span class="lineno"> 1301</span> </div><div class="line"><a name="l01302"></a><span class="lineno"> 1302</span> <span class="comment">// Only for debugging.</span></div><div class="line"><a name="l01303"></a><span class="lineno"> 1303</span> <span class="comment">//</span></div><div class="line"><a name="l01304"></a><span class="lineno"> 1304</span> <span class="comment">// Requires the precondition, explicitly computed in FinalizeSetup(),</span></div><div class="line"><a name="l01305"></a><span class="lineno"> 1305</span> <span class="comment">// that every left-side node has at least one incident arc.</span></div><div class="line"><a name="l01306"></a><span class="lineno"> 1306</span> <span class="keyword">template</span> <<span class="keyword">typename</span> GraphType></div><div class="line"><a name="l01307"></a><span class="lineno"> 1307</span> <span class="keyword">inline</span> <a class="code" href="namespaceoperations__research.html#aee97ac67f280d35acdef2c5d461a85c3">CostValue</a> LinearSumAssignment<GraphType>::ImplicitPrice(</div><div class="line"><a name="l01308"></a><span class="lineno"> 1308</span>  <a class="code" href="namespaceoperations__research.html#a7ae31ba4c3b4899478e53ca13df35dfc">NodeIndex</a> left_node)<span class="keyword"> const </span>{</div><div class="line"><a name="l01309"></a><span class="lineno"> 1309</span>  <a class="code" href="base_2logging_8h.html#ab4f56aa24c4c9cddc47a6abd2d747f9a">DCHECK_GT</a>(num_left_nodes_, left_node);</div><div class="line"><a name="l01310"></a><span class="lineno"> 1310</span>  <a class="code" href="base_2logging_8h.html#ab4f56aa24c4c9cddc47a6abd2d747f9a">DCHECK_GT</a>(epsilon_, 0);</div><div class="line"><a name="l01311"></a><span class="lineno"> 1311</span>  <span class="keyword">typename</span> GraphType::OutgoingArcIterator arc_it(*graph_, left_node);</div><div class="line"><a name="l01312"></a><span class="lineno"> 1312</span>  <span class="comment">// We must not execute this method if left_node has no incident arc.</span></div><div class="line"><a name="l01313"></a><span class="lineno"> 1313</span>  <a class="code" href="base_2logging_8h.html#ae17f8119c108cf3070bad3449c7e0006">DCHECK</a>(arc_it.Ok());</div><div class="line"><a name="l01314"></a><span class="lineno"> 1314</span>  <a class="code" href="namespaceoperations__research.html#aecf320bda6e95d1edaf3a546673e3e6b">ArcIndex</a> best_arc = arc_it.Index();</div><div class="line"><a name="l01315"></a><span class="lineno"> 1315</span>  <span class="keywordflow">if</span> (best_arc == matched_arc_[left_node]) {</div><div class="line"><a name="l01316"></a><span class="lineno"> 1316</span>  arc_it.Next();</div><div class="line"><a name="l01317"></a><span class="lineno"> 1317</span>  <span class="keywordflow">if</span> (arc_it.Ok()) {</div><div class="line"><a name="l01318"></a><span class="lineno"> 1318</span>  best_arc = arc_it.Index();</div><div class="line"><a name="l01319"></a><span class="lineno"> 1319</span>  }</div><div class="line"><a name="l01320"></a><span class="lineno"> 1320</span>  }</div><div class="line"><a name="l01321"></a><span class="lineno"> 1321</span>  <a class="code" href="namespaceoperations__research.html#aee97ac67f280d35acdef2c5d461a85c3">CostValue</a> min_partial_reduced_cost = PartialReducedCost(best_arc);</div><div class="line"><a name="l01322"></a><span class="lineno"> 1322</span>  <span class="keywordflow">if</span> (!arc_it.Ok()) {</div><div class="line"><a name="l01323"></a><span class="lineno"> 1323</span>  <span class="comment">// Only one arc is incident to left_node, and the node is</span></div><div class="line"><a name="l01324"></a><span class="lineno"> 1324</span>  <span class="comment">// currently matched along that arc, which must be the case in any</span></div><div class="line"><a name="l01325"></a><span class="lineno"> 1325</span>  <span class="comment">// feasible solution. Therefore we implicitly price this node so</span></div><div class="line"><a name="l01326"></a><span class="lineno"> 1326</span>  <span class="comment">// low that we will never consider unmatching it.</span></div><div class="line"><a name="l01327"></a><span class="lineno"> 1327</span>  <span class="keywordflow">return</span> -(min_partial_reduced_cost + slack_relabeling_price_);</div><div class="line"><a name="l01328"></a><span class="lineno"> 1328</span>  }</div><div class="line"><a name="l01329"></a><span class="lineno"> 1329</span>  <span class="keywordflow">for</span> (arc_it.Next(); arc_it.Ok(); arc_it.Next()) {</div><div class="line"><a name="l01330"></a><span class="lineno"> 1330</span>  <span class="keyword">const</span> <a class="code" href="namespaceoperations__research.html#aecf320bda6e95d1edaf3a546673e3e6b">ArcIndex</a> arc = arc_it.Index();</div><div class="line"><a name="l01331"></a><span class="lineno"> 1331</span>  <span class="keywordflow">if</span> (arc != matched_arc_[left_node]) {</div><div class="line"><a name="l01332"></a><span class="lineno"> 1332</span>  <span class="keyword">const</span> <a class="code" href="namespaceoperations__research.html#aee97ac67f280d35acdef2c5d461a85c3">CostValue</a> partial_reduced_cost = PartialReducedCost(arc);</div><div class="line"><a name="l01333"></a><span class="lineno"> 1333</span>  <span class="keywordflow">if</span> (partial_reduced_cost < min_partial_reduced_cost) {</div><div class="line"><a name="l01334"></a><span class="lineno"> 1334</span>  min_partial_reduced_cost = partial_reduced_cost;</div><div class="line"><a name="l01335"></a><span class="lineno"> 1335</span>  }</div><div class="line"><a name="l01336"></a><span class="lineno"> 1336</span>  }</div><div class="line"><a name="l01337"></a><span class="lineno"> 1337</span>  }</div><div class="line"><a name="l01338"></a><span class="lineno"> 1338</span>  <span class="keywordflow">return</span> -min_partial_reduced_cost;</div><div class="line"><a name="l01339"></a><span class="lineno"> 1339</span> }</div><div class="line"><a name="l01340"></a><span class="lineno"> 1340</span> </div><div class="line"><a name="l01341"></a><span class="lineno"> 1341</span> <span class="comment">// Only for debugging.</span></div><div class="line"><a name="l01342"></a><span class="lineno"> 1342</span> <span class="keyword">template</span> <<span class="keyword">typename</span> GraphType></div><div class="line"><a name="l01343"></a><span class="lineno"> 1343</span> <span class="keywordtype">bool</span> LinearSumAssignment<GraphType>::AllMatched()<span class="keyword"> const </span>{</div><div class="line"><a name="l01344"></a><span class="lineno"> 1344</span>  <span class="keywordflow">for</span> (<a class="code" href="namespaceoperations__research.html#a7ae31ba4c3b4899478e53ca13df35dfc">NodeIndex</a> node = 0; node < graph_->num_nodes(); ++node) {</div><div class="line"><a name="l01345"></a><span class="lineno"> 1345</span>  <span class="keywordflow">if</span> (IsActiveForDebugging(node)) {</div><div class="line"><a name="l01346"></a><span class="lineno"> 1346</span>  <span class="keywordflow">return</span> <span class="keyword">false</span>;</div><div class="line"><a name="l01347"></a><span class="lineno"> 1347</span>  }</div><div class="line"><a name="l01348"></a><span class="lineno"> 1348</span>  }</div><div class="line"><a name="l01349"></a><span class="lineno"> 1349</span>  <span class="keywordflow">return</span> <span class="keyword">true</span>;</div><div class="line"><a name="l01350"></a><span class="lineno"> 1350</span> }</div><div class="line"><a name="l01351"></a><span class="lineno"> 1351</span> </div><div class="line"><a name="l01352"></a><span class="lineno"> 1352</span> <span class="comment">// Only for debugging.</span></div><div class="line"><a name="l01353"></a><span class="lineno"> 1353</span> <span class="keyword">template</span> <<span class="keyword">typename</span> GraphType></div><div class="line"><a name="l01354"></a><span class="lineno"> 1354</span> <span class="keywordtype">bool</span> LinearSumAssignment<GraphType>::EpsilonOptimal()<span class="keyword"> const </span>{</div><div class="line"><a name="l01355"></a><span class="lineno"> 1355</span>  <span class="keywordflow">for</span> (BipartiteLeftNodeIterator node_it(*graph_, num_left_nodes_);</div><div class="line"><a name="l01356"></a><span class="lineno"> 1356</span>  node_it.Ok(); node_it.Next()) {</div><div class="line"><a name="l01357"></a><span class="lineno"> 1357</span>  <span class="keyword">const</span> <a class="code" href="namespaceoperations__research.html#a7ae31ba4c3b4899478e53ca13df35dfc">NodeIndex</a> left_node = node_it.Index();</div><div class="line"><a name="l01358"></a><span class="lineno"> 1358</span>  <span class="comment">// Get the implicit price of left_node and make sure the reduced</span></div><div class="line"><a name="l01359"></a><span class="lineno"> 1359</span>  <span class="comment">// costs of left_node's incident arcs are in bounds.</span></div><div class="line"><a name="l01360"></a><span class="lineno"> 1360</span>  <a class="code" href="namespaceoperations__research.html#aee97ac67f280d35acdef2c5d461a85c3">CostValue</a> left_node_price = ImplicitPrice(left_node);</div><div class="line"><a name="l01361"></a><span class="lineno"> 1361</span>  <span class="keywordflow">for</span> (<span class="keyword">typename</span> GraphType::OutgoingArcIterator arc_it(*graph_, left_node);</div><div class="line"><a name="l01362"></a><span class="lineno"> 1362</span>  arc_it.Ok(); arc_it.Next()) {</div><div class="line"><a name="l01363"></a><span class="lineno"> 1363</span>  <span class="keyword">const</span> <a class="code" href="namespaceoperations__research.html#aecf320bda6e95d1edaf3a546673e3e6b">ArcIndex</a> arc = arc_it.Index();</div><div class="line"><a name="l01364"></a><span class="lineno"> 1364</span>  <span class="keyword">const</span> <a class="code" href="namespaceoperations__research.html#aee97ac67f280d35acdef2c5d461a85c3">CostValue</a> reduced_cost = left_node_price + PartialReducedCost(arc);</div><div class="line"><a name="l01365"></a><span class="lineno"> 1365</span>  <span class="comment">// Note the asymmetric definition of epsilon-optimality that we</span></div><div class="line"><a name="l01366"></a><span class="lineno"> 1366</span>  <span class="comment">// use because it means we can saturate all admissible arcs in</span></div><div class="line"><a name="l01367"></a><span class="lineno"> 1367</span>  <span class="comment">// the beginning of Refine() just by unmatching all matched</span></div><div class="line"><a name="l01368"></a><span class="lineno"> 1368</span>  <span class="comment">// nodes.</span></div><div class="line"><a name="l01369"></a><span class="lineno"> 1369</span>  <span class="keywordflow">if</span> (matched_arc_[left_node] == arc) {</div><div class="line"><a name="l01370"></a><span class="lineno"> 1370</span>  <span class="comment">// The reverse arc is residual. Epsilon-optimality requires</span></div><div class="line"><a name="l01371"></a><span class="lineno"> 1371</span>  <span class="comment">// that the reduced cost of the forward arc be at most</span></div><div class="line"><a name="l01372"></a><span class="lineno"> 1372</span>  <span class="comment">// epsilon_.</span></div><div class="line"><a name="l01373"></a><span class="lineno"> 1373</span>  <span class="keywordflow">if</span> (reduced_cost > epsilon_) {</div><div class="line"><a name="l01374"></a><span class="lineno"> 1374</span>  <span class="keywordflow">return</span> <span class="keyword">false</span>;</div><div class="line"><a name="l01375"></a><span class="lineno"> 1375</span>  }</div><div class="line"><a name="l01376"></a><span class="lineno"> 1376</span>  } <span class="keywordflow">else</span> {</div><div class="line"><a name="l01377"></a><span class="lineno"> 1377</span>  <span class="comment">// The forward arc is residual. Epsilon-optimality requires</span></div><div class="line"><a name="l01378"></a><span class="lineno"> 1378</span>  <span class="comment">// that the reduced cost of the forward arc be at least zero.</span></div><div class="line"><a name="l01379"></a><span class="lineno"> 1379</span>  <span class="keywordflow">if</span> (reduced_cost < 0) {</div><div class="line"><a name="l01380"></a><span class="lineno"> 1380</span>  <span class="keywordflow">return</span> <span class="keyword">false</span>;</div><div class="line"><a name="l01381"></a><span class="lineno"> 1381</span>  }</div><div class="line"><a name="l01382"></a><span class="lineno"> 1382</span>  }</div><div class="line"><a name="l01383"></a><span class="lineno"> 1383</span>  }</div><div class="line"><a name="l01384"></a><span class="lineno"> 1384</span>  }</div><div class="line"><a name="l01385"></a><span class="lineno"> 1385</span>  <span class="keywordflow">return</span> <span class="keyword">true</span>;</div><div class="line"><a name="l01386"></a><span class="lineno"> 1386</span> }</div><div class="line"><a name="l01387"></a><span class="lineno"> 1387</span> </div><div class="line"><a name="l01388"></a><span class="lineno"> 1388</span> <span class="keyword">template</span> <<span class="keyword">typename</span> GraphType></div><div class="line"><a name="l01389"></a><span class="lineno"><a class="line" href="classoperations__research_1_1_linear_sum_assignment.html#afb41339280a9b3dd128f9a376ec38007"> 1389</a></span> <span class="keywordtype">bool</span> <a class="code" href="classoperations__research_1_1_linear_sum_assignment.html#afb41339280a9b3dd128f9a376ec38007">LinearSumAssignment<GraphType>::FinalizeSetup</a>() {</div><div class="line"><a name="l01390"></a><span class="lineno"> 1390</span>  incidence_precondition_satisfied_ = <span class="keyword">true</span>;</div><div class="line"><a name="l01391"></a><span class="lineno"> 1391</span>  <span class="comment">// epsilon_ must be greater than kMinEpsilon so that in the case</span></div><div class="line"><a name="l01392"></a><span class="lineno"> 1392</span>  <span class="comment">// where the largest arc cost is zero, we still do a Refine()</span></div><div class="line"><a name="l01393"></a><span class="lineno"> 1393</span>  <span class="comment">// iteration.</span></div><div class="line"><a name="l01394"></a><span class="lineno"> 1394</span>  epsilon_ = <a class="code" href="alldiff__cst_8cc.html#a26e6db9bcc64b584051ecc28171ed11f">std::max</a>(largest_scaled_cost_magnitude_, kMinEpsilon + 1);</div><div class="line"><a name="l01395"></a><span class="lineno"> 1395</span>  <a class="code" href="base_2logging_8h.html#afcaa7cadd41741bb855c2ada1d2ef927">VLOG</a>(2) << <span class="stringliteral">"Largest given cost magnitude: "</span></div><div class="line"><a name="l01396"></a><span class="lineno"> 1396</span>  << largest_scaled_cost_magnitude_ / cost_scaling_factor_;</div><div class="line"><a name="l01397"></a><span class="lineno"> 1397</span>  <span class="comment">// Initialize left-side node-indexed arrays and check incidence</span></div><div class="line"><a name="l01398"></a><span class="lineno"> 1398</span>  <span class="comment">// precondition.</span></div><div class="line"><a name="l01399"></a><span class="lineno"> 1399</span>  <span class="keywordflow">for</span> (<a class="code" href="classoperations__research_1_1_linear_sum_assignment.html#aca73212d30b08a7c287c311b74311a6e">NodeIndex</a> node = 0; node < num_left_nodes_; ++node) {</div><div class="line"><a name="l01400"></a><span class="lineno"> 1400</span>  matched_arc_[node] = GraphType::kNilArc;</div><div class="line"><a name="l01401"></a><span class="lineno"> 1401</span>  <span class="keyword">typename</span> GraphType::OutgoingArcIterator arc_it(*graph_, node);</div><div class="line"><a name="l01402"></a><span class="lineno"> 1402</span>  <span class="keywordflow">if</span> (!arc_it.Ok()) {</div><div class="line"><a name="l01403"></a><span class="lineno"> 1403</span>  incidence_precondition_satisfied_ = <span class="keyword">false</span>;</div><div class="line"><a name="l01404"></a><span class="lineno"> 1404</span>  }</div><div class="line"><a name="l01405"></a><span class="lineno"> 1405</span>  }</div><div class="line"><a name="l01406"></a><span class="lineno"> 1406</span>  <span class="comment">// Initialize right-side node-indexed arrays. Example: prices are</span></div><div class="line"><a name="l01407"></a><span class="lineno"> 1407</span>  <span class="comment">// stored only for right-side nodes.</span></div><div class="line"><a name="l01408"></a><span class="lineno"> 1408</span>  <span class="keywordflow">for</span> (<a class="code" href="classoperations__research_1_1_linear_sum_assignment.html#aca73212d30b08a7c287c311b74311a6e">NodeIndex</a> node = num_left_nodes_; node < graph_->num_nodes(); ++node) {</div><div class="line"><a name="l01409"></a><span class="lineno"> 1409</span>  price_[node] = 0;</div><div class="line"><a name="l01410"></a><span class="lineno"> 1410</span>  matched_node_[node] = GraphType::kNilNode;</div><div class="line"><a name="l01411"></a><span class="lineno"> 1411</span>  }</div><div class="line"><a name="l01412"></a><span class="lineno"> 1412</span>  <span class="keywordtype">bool</span> in_range = <span class="keyword">true</span>;</div><div class="line"><a name="l01413"></a><span class="lineno"> 1413</span>  <span class="keywordtype">double</span> double_price_lower_bound = 0.0;</div><div class="line"><a name="l01414"></a><span class="lineno"> 1414</span>  <a class="code" href="namespaceoperations__research.html#aee97ac67f280d35acdef2c5d461a85c3">CostValue</a> new_error_parameter;</div><div class="line"><a name="l01415"></a><span class="lineno"> 1415</span>  <a class="code" href="namespaceoperations__research.html#aee97ac67f280d35acdef2c5d461a85c3">CostValue</a> old_error_parameter = epsilon_;</div><div class="line"><a name="l01416"></a><span class="lineno"> 1416</span>  <span class="keywordflow">do</span> {</div><div class="line"><a name="l01417"></a><span class="lineno"> 1417</span>  new_error_parameter = NewEpsilon(old_error_parameter);</div><div class="line"><a name="l01418"></a><span class="lineno"> 1418</span>  double_price_lower_bound -=</div><div class="line"><a name="l01419"></a><span class="lineno"> 1419</span>  2.0 * static_cast<double>(PriceChangeBound(</div><div class="line"><a name="l01420"></a><span class="lineno"> 1420</span>  old_error_parameter, new_error_parameter, &in_range));</div><div class="line"><a name="l01421"></a><span class="lineno"> 1421</span>  old_error_parameter = new_error_parameter;</div><div class="line"><a name="l01422"></a><span class="lineno"> 1422</span>  } <span class="keywordflow">while</span> (new_error_parameter != kMinEpsilon);</div><div class="line"><a name="l01423"></a><span class="lineno"> 1423</span>  <span class="keyword">const</span> <span class="keywordtype">double</span> limit =</div><div class="line"><a name="l01424"></a><span class="lineno"> 1424</span>  -static_cast<double>(<a class="code" href="alldiff__cst_8cc.html#a26e6db9bcc64b584051ecc28171ed11f">std::numeric_limits<CostValue>::max</a>());</div><div class="line"><a name="l01425"></a><span class="lineno"> 1425</span>  <span class="keywordflow">if</span> (double_price_lower_bound < limit) {</div><div class="line"><a name="l01426"></a><span class="lineno"> 1426</span>  in_range = <span class="keyword">false</span>;</div><div class="line"><a name="l01427"></a><span class="lineno"> 1427</span>  price_lower_bound_ = -<a class="code" href="alldiff__cst_8cc.html#a26e6db9bcc64b584051ecc28171ed11f">std::numeric_limits<CostValue>::max</a>();</div><div class="line"><a name="l01428"></a><span class="lineno"> 1428</span>  } <span class="keywordflow">else</span> {</div><div class="line"><a name="l01429"></a><span class="lineno"> 1429</span>  price_lower_bound_ = static_cast<CostValue>(double_price_lower_bound);</div><div class="line"><a name="l01430"></a><span class="lineno"> 1430</span>  }</div><div class="line"><a name="l01431"></a><span class="lineno"> 1431</span>  <a class="code" href="base_2logging_8h.html#afcaa7cadd41741bb855c2ada1d2ef927">VLOG</a>(4) << <span class="stringliteral">"price_lower_bound_ == "</span> << price_lower_bound_;</div><div class="line"><a name="l01432"></a><span class="lineno"> 1432</span>  <a class="code" href="base_2logging_8h.html#a4395e95bab44e222cb2e77251017a0e2">DCHECK_LE</a>(price_lower_bound_, 0);</div><div class="line"><a name="l01433"></a><span class="lineno"> 1433</span>  <span class="keywordflow">if</span> (!in_range) {</div><div class="line"><a name="l01434"></a><span class="lineno"> 1434</span>  <a class="code" href="base_2logging_8h.html#accad43a85d781d53381cd53a9894b6ae">LOG</a>(<a class="code" href="log__severity_8h.html#a50e5762f38854b37ee3e2851bc1bb0e7">WARNING</a>) << <span class="stringliteral">"Price change bound exceeds range of representable "</span></div><div class="line"><a name="l01435"></a><span class="lineno"> 1435</span>  << <span class="stringliteral">"costs; arithmetic overflow is not ruled out and "</span></div><div class="line"><a name="l01436"></a><span class="lineno"> 1436</span>  << <span class="stringliteral">"infeasibility might go undetected."</span>;</div><div class="line"><a name="l01437"></a><span class="lineno"> 1437</span>  }</div><div class="line"><a name="l01438"></a><span class="lineno"> 1438</span>  <span class="keywordflow">return</span> in_range;</div><div class="line"><a name="l01439"></a><span class="lineno"> 1439</span> }</div><div class="line"><a name="l01440"></a><span class="lineno"> 1440</span> </div><div class="line"><a name="l01441"></a><span class="lineno"> 1441</span> <span class="keyword">template</span> <<span class="keyword">typename</span> GraphType></div><div class="line"><a name="l01442"></a><span class="lineno"> 1442</span> <span class="keywordtype">void</span> <a class="code" href="classoperations__research_1_1_linear_sum_assignment.html">LinearSumAssignment<GraphType>::ReportAndAccumulateStats</a>() {</div><div class="line"><a name="l01443"></a><span class="lineno"> 1443</span>  total_stats_.Add(iteration_stats_);</div><div class="line"><a name="l01444"></a><span class="lineno"> 1444</span>  <a class="code" href="base_2logging_8h.html#afcaa7cadd41741bb855c2ada1d2ef927">VLOG</a>(3) << <span class="stringliteral">"Iteration stats: "</span> << iteration_stats_.StatsString();</div><div class="line"><a name="l01445"></a><span class="lineno"> 1445</span>  iteration_stats_.Clear();</div><div class="line"><a name="l01446"></a><span class="lineno"> 1446</span> }</div><div class="line"><a name="l01447"></a><span class="lineno"> 1447</span> </div><div class="line"><a name="l01448"></a><span class="lineno"> 1448</span> <span class="keyword">template</span> <<span class="keyword">typename</span> GraphType></div><div class="line"><a name="l01449"></a><span class="lineno"><a class="line" href="classoperations__research_1_1_linear_sum_assignment.html#a63b3d12e721188086870cc42cc46a258"> 1449</a></span> <span class="keywordtype">bool</span> <a class="code" href="classoperations__research_1_1_linear_sum_assignment.html#a63b3d12e721188086870cc42cc46a258">LinearSumAssignment<GraphType>::ComputeAssignment</a>() {</div><div class="line"><a name="l01450"></a><span class="lineno"> 1450</span>  <a class="code" href="base_2logging_8h.html#a3e1cfef60e774a81f30eaddf26a3a274">CHECK</a>(graph_ != <span class="keyword">nullptr</span>);</div><div class="line"><a name="l01451"></a><span class="lineno"> 1451</span>  <span class="keywordtype">bool</span> ok = graph_->num_nodes() == 2 * num_left_nodes_;</div><div class="line"><a name="l01452"></a><span class="lineno"> 1452</span>  <span class="keywordflow">if</span> (!ok) <span class="keywordflow">return</span> <span class="keyword">false</span>;</div><div class="line"><a name="l01453"></a><span class="lineno"> 1453</span>  <span class="comment">// Note: FinalizeSetup() might have been called already by white-box</span></div><div class="line"><a name="l01454"></a><span class="lineno"> 1454</span>  <span class="comment">// test code or by a client that wants to react to the possibility</span></div><div class="line"><a name="l01455"></a><span class="lineno"> 1455</span>  <span class="comment">// of overflow before solving the given problem, but FinalizeSetup()</span></div><div class="line"><a name="l01456"></a><span class="lineno"> 1456</span>  <span class="comment">// is idempotent and reasonably fast, so we call it unconditionally</span></div><div class="line"><a name="l01457"></a><span class="lineno"> 1457</span>  <span class="comment">// here.</span></div><div class="line"><a name="l01458"></a><span class="lineno"> 1458</span>  FinalizeSetup();</div><div class="line"><a name="l01459"></a><span class="lineno"> 1459</span>  ok = ok && incidence_precondition_satisfied_;</div><div class="line"><a name="l01460"></a><span class="lineno"> 1460</span>  <a class="code" href="base_2logging_8h.html#ae17f8119c108cf3070bad3449c7e0006">DCHECK</a>(!ok || EpsilonOptimal());</div><div class="line"><a name="l01461"></a><span class="lineno"> 1461</span>  <span class="keywordflow">while</span> (ok && epsilon_ > kMinEpsilon) {</div><div class="line"><a name="l01462"></a><span class="lineno"> 1462</span>  ok = ok && UpdateEpsilon();</div><div class="line"><a name="l01463"></a><span class="lineno"> 1463</span>  ok = ok && Refine();</div><div class="line"><a name="l01464"></a><span class="lineno"> 1464</span>  ReportAndAccumulateStats();</div><div class="line"><a name="l01465"></a><span class="lineno"> 1465</span>  <a class="code" href="base_2logging_8h.html#ae17f8119c108cf3070bad3449c7e0006">DCHECK</a>(!ok || EpsilonOptimal());</div><div class="line"><a name="l01466"></a><span class="lineno"> 1466</span>  <a class="code" href="base_2logging_8h.html#ae17f8119c108cf3070bad3449c7e0006">DCHECK</a>(!ok || AllMatched());</div><div class="line"><a name="l01467"></a><span class="lineno"> 1467</span>  }</div><div class="line"><a name="l01468"></a><span class="lineno"> 1468</span>  success_ = ok;</div><div class="line"><a name="l01469"></a><span class="lineno"> 1469</span>  <a class="code" href="base_2logging_8h.html#afcaa7cadd41741bb855c2ada1d2ef927">VLOG</a>(1) << <span class="stringliteral">"Overall stats: "</span> << total_stats_.StatsString();</div><div class="line"><a name="l01470"></a><span class="lineno"> 1470</span>  <span class="keywordflow">return</span> ok;</div><div class="line"><a name="l01471"></a><span class="lineno"> 1471</span> }</div><div class="line"><a name="l01472"></a><span class="lineno"> 1472</span> </div><div class="line"><a name="l01473"></a><span class="lineno"> 1473</span> <span class="keyword">template</span> <<span class="keyword">typename</span> GraphType></div><div class="line"><a name="l01474"></a><span class="lineno"><a class="line" href="classoperations__research_1_1_linear_sum_assignment.html#a5afb9dde1c31f5f053cc83bf6e594db0"> 1474</a></span> <a class="code" href="namespaceoperations__research.html#aee97ac67f280d35acdef2c5d461a85c3">CostValue</a> <a class="code" href="classoperations__research_1_1_linear_sum_assignment.html#a5afb9dde1c31f5f053cc83bf6e594db0">LinearSumAssignment<GraphType>::GetCost</a>()<span class="keyword"> const </span>{</div><div class="line"><a name="l01475"></a><span class="lineno"> 1475</span>  <span class="comment">// It is illegal to call this method unless we successfully computed</span></div><div class="line"><a name="l01476"></a><span class="lineno"> 1476</span>  <span class="comment">// an optimum assignment.</span></div><div class="line"><a name="l01477"></a><span class="lineno"> 1477</span>  <a class="code" href="base_2logging_8h.html#ae17f8119c108cf3070bad3449c7e0006">DCHECK</a>(success_);</div><div class="line"><a name="l01478"></a><span class="lineno"> 1478</span>  <a class="code" href="namespaceoperations__research.html#aee97ac67f280d35acdef2c5d461a85c3">CostValue</a> <a class="code" href="routing__flow_8cc.html#a75d7b5e4cab1e156cc7a2c5eba1e16f1">cost</a> = 0;</div><div class="line"><a name="l01479"></a><span class="lineno"> 1479</span>  <span class="keywordflow">for</span> (<a class="code" href="classoperations__research_1_1_linear_sum_assignment_1_1_bipartite_left_node_iterator.html">BipartiteLeftNodeIterator</a> node_it(*<span class="keyword">this</span>); node_it.<a class="code" href="classoperations__research_1_1_linear_sum_assignment_1_1_bipartite_left_node_iterator.html#aa7e07ffe21ad6b4c71a0d22c65f30347">Ok</a>(); node_it.Next()) {</div><div class="line"><a name="l01480"></a><span class="lineno"> 1480</span>  <a class="code" href="routing__flow_8cc.html#a75d7b5e4cab1e156cc7a2c5eba1e16f1">cost</a> += GetAssignmentCost(node_it.Index());</div><div class="line"><a name="l01481"></a><span class="lineno"> 1481</span>  }</div><div class="line"><a name="l01482"></a><span class="lineno"> 1482</span>  <span class="keywordflow">return</span> <a class="code" href="routing__flow_8cc.html#a75d7b5e4cab1e156cc7a2c5eba1e16f1">cost</a>;</div><div class="line"><a name="l01483"></a><span class="lineno"> 1483</span> }</div><div class="line"><a name="l01484"></a><span class="lineno"> 1484</span> </div><div class="line"><a name="l01485"></a><span class="lineno"> 1485</span> } <span class="comment">// namespace operations_research</span></div><div class="line"><a name="l01486"></a><span class="lineno"> 1486</span> </div><div class="line"><a name="l01487"></a><span class="lineno"> 1487</span> <span class="preprocessor">#endif // OR_TOOLS_GRAPH_LINEAR_ASSIGNMENT_H_</span></div><div class="ttc" id="routing__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>
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<div class="ttc" id="namespaceoperations__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#l00193">ebert_graph.h:193</a></div></div>
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<div class="ttc" id="base_2logging_8h_html_a3e1cfef60e774a81f30eaddf26a3a274"><div class="ttname"><a href="base_2logging_8h.html#a3e1cfef60e774a81f30eaddf26a3a274">CHECK</a></div><div class="ttdeci">#define CHECK(condition)</div><div class="ttdef"><b>Definition:</b> <a href="base_2logging_8h_source.html#l00491">base/logging.h:491</a></div></div>
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<div class="ttc" id="classoperations__research_1_1_linear_sum_assignment_html_aa9b122f6d2e28bf78a0339f26cfbd432"><div class="ttname"><a href="classoperations__research_1_1_linear_sum_assignment.html#aa9b122f6d2e28bf78a0339f26cfbd432">operations_research::LinearSumAssignment::GetAssignmentCost</a></div><div class="ttdeci">CostValue GetAssignmentCost(NodeIndex node) const</div><div class="ttdef"><b>Definition:</b> <a href="linear__assignment_8h_source.html#l00346">linear_assignment.h:346</a></div></div>
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<div class="ttc" id="util_2permutation_8h_html"><div class="ttname"><a href="util_2permutation_8h.html">permutation.h</a></div></div>
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<div class="ttc" id="classoperations__research_1_1_linear_sum_assignment_html_afb41339280a9b3dd128f9a376ec38007"><div class="ttname"><a href="classoperations__research_1_1_linear_sum_assignment.html#afb41339280a9b3dd128f9a376ec38007">operations_research::LinearSumAssignment::FinalizeSetup</a></div><div class="ttdeci">bool FinalizeSetup()</div><div class="ttdef"><b>Definition:</b> <a href="linear__assignment_8h_source.html#l01389">linear_assignment.h:1389</a></div></div>
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<div class="ttc" id="base_2logging_8h_html_afcaa7cadd41741bb855c2ada1d2ef927"><div class="ttname"><a href="base_2logging_8h.html#afcaa7cadd41741bb855c2ada1d2ef927">VLOG</a></div><div class="ttdeci">#define VLOG(verboselevel)</div><div class="ttdef"><b>Definition:</b> <a href="base_2logging_8h_source.html#l00979">base/logging.h:979</a></div></div>
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<div class="ttc" id="classoperations__research_1_1_linear_sum_assignment_1_1_bipartite_left_node_iterator_html_afadba1101a25565c8c207f7e81a67950"><div class="ttname"><a href="classoperations__research_1_1_linear_sum_assignment_1_1_bipartite_left_node_iterator.html#afadba1101a25565c8c207f7e81a67950">operations_research::LinearSumAssignment::BipartiteLeftNodeIterator::Index</a></div><div class="ttdeci">NodeIndex Index() const</div><div class="ttdef"><b>Definition:</b> <a href="linear__assignment_8h_source.html#l00368">linear_assignment.h:368</a></div></div>
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<div class="ttc" id="classoperations__research_1_1_linear_sum_assignment_html_aca73212d30b08a7c287c311b74311a6e"><div class="ttname"><a href="classoperations__research_1_1_linear_sum_assignment.html#aca73212d30b08a7c287c311b74311a6e">operations_research::LinearSumAssignment::NodeIndex</a></div><div class="ttdeci">GraphType::NodeIndex NodeIndex</div><div class="ttdef"><b>Definition:</b> <a href="linear__assignment_8h_source.html#l00229">linear_assignment.h:229</a></div></div>
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<div class="ttc" id="classoperations__research_1_1_linear_sum_assignment_html_a4f658e59dd3bd0b8cd1269e19b730cee"><div class="ttname"><a href="classoperations__research_1_1_linear_sum_assignment.html#a4f658e59dd3bd0b8cd1269e19b730cee">operations_research::LinearSumAssignment::~LinearSumAssignment</a></div><div class="ttdeci">~LinearSumAssignment()</div><div class="ttdef"><b>Definition:</b> <a href="linear__assignment_8h_source.html#l00243">linear_assignment.h:243</a></div></div>
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<div class="ttc" id="classoperations__research_1_1_cost_value_cycle_handler_html_a999f9e59b55e3f184efa317522f2eeb0"><div class="ttname"><a href="classoperations__research_1_1_cost_value_cycle_handler.html#a999f9e59b55e3f184efa317522f2eeb0">operations_research::CostValueCycleHandler::SetTempFromIndex</a></div><div class="ttdeci">void SetTempFromIndex(ArcIndexType source) override</div><div class="ttdef"><b>Definition:</b> <a href="linear__assignment_8h_source.html#l01030">linear_assignment.h:1030</a></div></div>
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<div class="ttc" id="namespaceoperations__research_html_aee97ac67f280d35acdef2c5d461a85c3"><div class="ttname"><a href="namespaceoperations__research.html#aee97ac67f280d35acdef2c5d461a85c3">operations_research::CostValue</a></div><div class="ttdeci">int64_t CostValue</div><div class="ttdef"><b>Definition:</b> <a href="ebert__graph_8h_source.html#l00204">ebert_graph.h:204</a></div></div>
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<div class="ttc" id="base_2logging_8h_html_accad43a85d781d53381cd53a9894b6ae"><div class="ttname"><a href="base_2logging_8h.html#accad43a85d781d53381cd53a9894b6ae">LOG</a></div><div class="ttdeci">#define LOG(severity)</div><div class="ttdef"><b>Definition:</b> <a href="base_2logging_8h_source.html#l00416">base/logging.h:416</a></div></div>
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<div class="ttc" id="classoperations__research_1_1_linear_sum_assignment_html_aa4b556b7bee4c7f3fb60781e7ebda1b2"><div class="ttname"><a href="classoperations__research_1_1_linear_sum_assignment.html#aa4b556b7bee4c7f3fb60781e7ebda1b2">operations_research::LinearSumAssignment::OptimizeGraphLayout</a></div><div class="ttdeci">void OptimizeGraphLayout(GraphType *graph)</div><div class="ttdef"><b>Definition:</b> <a href="linear__assignment_8h_source.html#l01089">linear_assignment.h:1089</a></div></div>
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<div class="ttc" id="classoperations__research_1_1_linear_sum_assignment_html_aa64fbe18913fc55f5d41182666751728"><div class="ttname"><a href="classoperations__research_1_1_linear_sum_assignment.html#aa64fbe18913fc55f5d41182666751728">operations_research::LinearSumAssignment::ArcCost</a></div><div class="ttdeci">CostValue ArcCost(ArcIndex arc) const</div><div class="ttdef"><b>Definition:</b> <a href="linear__assignment_8h_source.html#l00292">linear_assignment.h:292</a></div></div>
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<div class="ttc" id="classoperations__research_1_1_linear_sum_assignment_html_a1286f5a02e4b2a9e89431626e12fd498"><div class="ttname"><a href="classoperations__research_1_1_linear_sum_assignment.html#a1286f5a02e4b2a9e89431626e12fd498">operations_research::LinearSumAssignment::StatsString</a></div><div class="ttdeci">std::string StatsString() const</div><div class="ttdef"><b>Definition:</b> <a href="linear__assignment_8h_source.html#l00358">linear_assignment.h:358</a></div></div>
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<div class="ttc" id="base_2logging_8h_html_ab4f56aa24c4c9cddc47a6abd2d747f9a"><div class="ttname"><a href="base_2logging_8h.html#ab4f56aa24c4c9cddc47a6abd2d747f9a">DCHECK_GT</a></div><div class="ttdeci">#define DCHECK_GT(val1, val2)</div><div class="ttdef"><b>Definition:</b> <a href="base_2logging_8h_source.html#l00891">base/logging.h:891</a></div></div>
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<div class="ttc" id="classoperations__research_1_1_linear_sum_assignment_html_af45fdf861fabd6a46ce230a9f6101eda"><div class="ttname"><a href="classoperations__research_1_1_linear_sum_assignment.html#af45fdf861fabd6a46ce230a9f6101eda">operations_research::LinearSumAssignment::LinearSumAssignment</a></div><div class="ttdeci">LinearSumAssignment(const GraphType &graph, NodeIndex num_left_nodes)</div><div class="ttdef"><b>Definition:</b> <a href="linear__assignment_8h_source.html#l00964">linear_assignment.h:964</a></div></div>
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<div class="ttc" id="classoperations__research_1_1_linear_sum_assignment_html_a77a0519df5fb71834593bb661b72921c"><div class="ttname"><a href="classoperations__research_1_1_linear_sum_assignment.html#a77a0519df5fb71834593bb661b72921c">operations_research::LinearSumAssignment::SetArcCost</a></div><div class="ttdeci">void SetArcCost(ArcIndex arc, CostValue cost)</div><div class="ttdef"><b>Definition:</b> <a href="linear__assignment_8h_source.html#l01010">linear_assignment.h:1010</a></div></div>
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<div class="ttc" id="constraint__solver_2table_8cc_html_a9293e4d29cac928301645070dd307683"><div class="ttname"><a href="constraint__solver_2table_8cc.html#a9293e4d29cac928301645070dd307683">b</a></div><div class="ttdeci">int64_t b</div><div class="ttdef"><b>Definition:</b> <a href="constraint__solver_2table_8cc_source.html#l00047">constraint_solver/table.cc:47</a></div></div>
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<div class="ttc" id="integral__types_8h_html"><div class="ttname"><a href="integral__types_8h.html">integral_types.h</a></div></div>
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<div class="ttc" id="macros_8h_html"><div class="ttname"><a href="macros_8h.html">macros.h</a></div></div>
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<div class="ttc" id="classoperations__research_1_1_linear_sum_assignment_1_1_bipartite_left_node_iterator_html_aa7e07ffe21ad6b4c71a0d22c65f30347"><div class="ttname"><a href="classoperations__research_1_1_linear_sum_assignment_1_1_bipartite_left_node_iterator.html#aa7e07ffe21ad6b4c71a0d22c65f30347">operations_research::LinearSumAssignment::BipartiteLeftNodeIterator::Ok</a></div><div class="ttdeci">bool Ok() const</div><div class="ttdef"><b>Definition:</b> <a href="linear__assignment_8h_source.html#l00370">linear_assignment.h:370</a></div></div>
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<div class="ttc" id="classoperations__research_1_1_linear_sum_assignment_1_1_bipartite_left_node_iterator_html_a64734b3e0ff85820aba52f5109432f51"><div class="ttname"><a href="classoperations__research_1_1_linear_sum_assignment_1_1_bipartite_left_node_iterator.html#a64734b3e0ff85820aba52f5109432f51">operations_research::LinearSumAssignment::BipartiteLeftNodeIterator::BipartiteLeftNodeIterator</a></div><div class="ttdeci">BipartiteLeftNodeIterator(const GraphType &graph, NodeIndex num_left_nodes)</div><div class="ttdef"><b>Definition:</b> <a href="linear__assignment_8h_source.html#l00362">linear_assignment.h:362</a></div></div>
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<div class="ttc" id="alldiff__cst_8cc_html_a26e6db9bcc64b584051ecc28171ed11f"><div class="ttname"><a href="alldiff__cst_8cc.html#a26e6db9bcc64b584051ecc28171ed11f">max</a></div><div class="ttdeci">int64_t max</div><div class="ttdef"><b>Definition:</b> <a href="alldiff__cst_8cc_source.html#l00140">alldiff_cst.cc:140</a></div></div>
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<div class="ttc" id="classoperations__research_1_1_linear_sum_assignment_1_1_bipartite_left_node_iterator_html_a659a293dd51073a1b9560bb80f687705"><div class="ttname"><a href="classoperations__research_1_1_linear_sum_assignment_1_1_bipartite_left_node_iterator.html#a659a293dd51073a1b9560bb80f687705">operations_research::LinearSumAssignment::BipartiteLeftNodeIterator::Next</a></div><div class="ttdeci">void Next()</div><div class="ttdef"><b>Definition:</b> <a href="linear__assignment_8h_source.html#l00372">linear_assignment.h:372</a></div></div>
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<div class="ttc" id="namespaceabsl_html"><div class="ttname"><a href="namespaceabsl.html">absl</a></div><div class="ttdef"><b>Definition:</b> <a href="cleanup_8h_source.html#l00022">cleanup.h:22</a></div></div>
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<div class="ttc" id="classoperations__research_1_1_cost_value_cycle_handler_html_ab8c76fdd7493de9946f7551ed3ca16bc"><div class="ttname"><a href="classoperations__research_1_1_cost_value_cycle_handler.html#ab8c76fdd7493de9946f7551ed3ca16bc">operations_research::CostValueCycleHandler::SetIndexFromTemp</a></div><div class="ttdeci">void SetIndexFromTemp(ArcIndexType destination) const override</div><div class="ttdef"><b>Definition:</b> <a href="linear__assignment_8h_source.html#l01039">linear_assignment.h:1039</a></div></div>
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<div class="ttc" id="log__severity_8h_html_a50e5762f38854b37ee3e2851bc1bb0e7"><div class="ttname"><a href="log__severity_8h.html#a50e5762f38854b37ee3e2851bc1bb0e7">WARNING</a></div><div class="ttdeci">const int WARNING</div><div class="ttdef"><b>Definition:</b> <a href="log__severity_8h_source.html#l00031">log_severity.h:31</a></div></div>
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<div class="ttc" id="classoperations__research_1_1_tail_array_manager_html_a930399f60b3c83a155ca86db090655c4"><div class="ttname"><a href="classoperations__research_1_1_tail_array_manager.html#a930399f60b3c83a155ca86db090655c4">operations_research::TailArrayManager::ReleaseTailArrayIfForwardGraph</a></div><div class="ttdeci">void ReleaseTailArrayIfForwardGraph() const</div><div class="ttdef"><b>Definition:</b> <a href="ebert__graph_8h_source.html#l01928">ebert_graph.h:1928</a></div></div>
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<div class="ttc" id="classoperations__research_1_1_arc_index_ordering_by_tail_node_html_a7fc8cebeaaae309b2282772d6cac1888"><div class="ttname"><a href="classoperations__research_1_1_arc_index_ordering_by_tail_node.html#a7fc8cebeaaae309b2282772d6cac1888">operations_research::ArcIndexOrderingByTailNode::operator()</a></div><div class="ttdeci">bool operator()(typename GraphType::ArcIndex a, typename GraphType::ArcIndex b) const</div><div class="ttdef"><b>Definition:</b> <a href="linear__assignment_8h_source.html#l01065">linear_assignment.h:1065</a></div></div>
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<div class="ttc" id="classoperations__research_1_1_tail_array_manager_html_a5b9e3e11b5999e1e2265f9f14a824214"><div class="ttname"><a href="classoperations__research_1_1_tail_array_manager.html#a5b9e3e11b5999e1e2265f9f14a824214">operations_research::TailArrayManager::BuildTailArrayFromAdjacencyListsIfForwardGraph</a></div><div class="ttdeci">bool BuildTailArrayFromAdjacencyListsIfForwardGraph() const</div><div class="ttdef"><b>Definition:</b> <a href="ebert__graph_8h_source.html#l01921">ebert_graph.h:1921</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="classoperations__research_1_1_linear_sum_assignment_html"><div class="ttname"><a href="classoperations__research_1_1_linear_sum_assignment.html">operations_research::LinearSumAssignment</a></div><div class="ttdef"><b>Definition:</b> <a href="linear__assignment_8h_source.html#l00227">linear_assignment.h:227</a></div></div>
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<div class="ttc" id="base_2logging_8h_html_a46e69120fbd3b36e6960e096d23b66f0"><div class="ttname"><a href="base_2logging_8h.html#a46e69120fbd3b36e6960e096d23b66f0">DCHECK_NE</a></div><div class="ttdeci">#define DCHECK_NE(val1, val2)</div><div class="ttdef"><b>Definition:</b> <a href="base_2logging_8h_source.html#l00887">base/logging.h:887</a></div></div>
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<div class="ttc" id="zvector_8h_html"><div class="ttname"><a href="zvector_8h.html">zvector.h</a></div></div>
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<div class="ttc" id="classoperations__research_1_1_linear_sum_assignment_html_aececfe5b0affea1dd1b8a38d8c1fb769"><div class="ttname"><a href="classoperations__research_1_1_linear_sum_assignment.html#aececfe5b0affea1dd1b8a38d8c1fb769">operations_research::LinearSumAssignment::SetGraph</a></div><div class="ttdeci">void SetGraph(const GraphType *graph)</div><div class="ttdef"><b>Definition:</b> <a href="linear__assignment_8h_source.html#l00248">linear_assignment.h:248</a></div></div>
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<div class="ttc" id="classoperations__research_1_1_linear_sum_assignment_html_adb325b7bb71f0f00df13383ad0b2adae"><div class="ttname"><a href="classoperations__research_1_1_linear_sum_assignment.html#adb325b7bb71f0f00df13383ad0b2adae">operations_research::LinearSumAssignment::Graph</a></div><div class="ttdeci">const GraphType & Graph() const</div><div class="ttdef"><b>Definition:</b> <a href="linear__assignment_8h_source.html#l00282">linear_assignment.h:282</a></div></div>
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<div class="ttc" id="classoperations__research_1_1_arc_index_ordering_by_tail_node_html_a6d178bcb4ffb6f497b77a9faedd8601c"><div class="ttname"><a href="classoperations__research_1_1_arc_index_ordering_by_tail_node.html#a6d178bcb4ffb6f497b77a9faedd8601c">operations_research::ArcIndexOrderingByTailNode::ArcIndexOrderingByTailNode</a></div><div class="ttdeci">ArcIndexOrderingByTailNode(const GraphType &graph)</div><div class="ttdef"><b>Definition:</b> <a href="linear__assignment_8h_source.html#l01060">linear_assignment.h:1060</a></div></div>
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<div class="ttc" id="classoperations__research_1_1_linear_sum_assignment_html_a13537686f25d401f7112b9fa4f63d1ab"><div class="ttname"><a href="classoperations__research_1_1_linear_sum_assignment.html#a13537686f25d401f7112b9fa4f63d1ab">operations_research::LinearSumAssignment::GetAssignmentArc</a></div><div class="ttdeci">ArcIndex GetAssignmentArc(NodeIndex left_node) const</div><div class="ttdef"><b>Definition:</b> <a href="linear__assignment_8h_source.html#l00339">linear_assignment.h:339</a></div></div>
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<div class="ttc" id="classoperations__research_1_1_linear_sum_assignment_html_a5afb9dde1c31f5f053cc83bf6e594db0"><div class="ttname"><a href="classoperations__research_1_1_linear_sum_assignment.html#a5afb9dde1c31f5f053cc83bf6e594db0">operations_research::LinearSumAssignment::GetCost</a></div><div class="ttdeci">CostValue GetCost() const</div><div class="ttdef"><b>Definition:</b> <a href="linear__assignment_8h_source.html#l01474">linear_assignment.h:1474</a></div></div>
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<div class="ttc" id="classoperations__research_1_1_linear_sum_assignment_html_aa5f729a6274027e5e5478e4bd76603ca"><div class="ttname"><a href="classoperations__research_1_1_linear_sum_assignment.html#aa5f729a6274027e5e5478e4bd76603ca">operations_research::LinearSumAssignment::Head</a></div><div class="ttdeci">NodeIndex Head(ArcIndex arc) const</div><div class="ttdef"><b>Definition:</b> <a href="linear__assignment_8h_source.html#l00288">linear_assignment.h:288</a></div></div>
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<div class="ttc" id="base_2logging_8h_html_aae2dc65d9ea248d54bf39daa986dd295"><div class="ttname"><a href="base_2logging_8h.html#aae2dc65d9ea248d54bf39daa986dd295">DCHECK_GE</a></div><div class="ttdeci">#define DCHECK_GE(val1, val2)</div><div class="ttdef"><b>Definition:</b> <a href="base_2logging_8h_source.html#l00890">base/logging.h:890</a></div></div>
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<div class="ttc" id="routing__flow_8cc_html_a75d7b5e4cab1e156cc7a2c5eba1e16f1"><div class="ttname"><a href="routing__flow_8cc.html#a75d7b5e4cab1e156cc7a2c5eba1e16f1">cost</a></div><div class="ttdeci">int64_t cost</div><div class="ttdef"><b>Definition:</b> <a href="routing__flow_8cc_source.html#l00152">routing_flow.cc:152</a></div></div>
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<div class="ttc" id="base_2logging_8h_html"><div class="ttname"><a href="base_2logging_8h.html">logging.h</a></div></div>
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<div class="ttc" id="classoperations__research_1_1_linear_sum_assignment_html_accac1fc7c4ac9bff1591ec627a59a4f7"><div class="ttname"><a href="classoperations__research_1_1_linear_sum_assignment.html#accac1fc7c4ac9bff1591ec627a59a4f7">operations_research::LinearSumAssignment::SetCostScalingDivisor</a></div><div class="ttdeci">void SetCostScalingDivisor(CostValue factor)</div><div class="ttdef"><b>Definition:</b> <a href="linear__assignment_8h_source.html#l00255">linear_assignment.h:255</a></div></div>
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<div class="ttc" id="classoperations__research_1_1_cost_value_cycle_handler_html_a2fd5ecc6414d07e3456e96c0d665ae9b"><div class="ttname"><a href="classoperations__research_1_1_cost_value_cycle_handler.html#a2fd5ecc6414d07e3456e96c0d665ae9b">operations_research::CostValueCycleHandler::SetIndexFromIndex</a></div><div class="ttdeci">void SetIndexFromIndex(ArcIndexType source, ArcIndexType destination) const override</div><div class="ttdef"><b>Definition:</b> <a href="linear__assignment_8h_source.html#l01034">linear_assignment.h:1034</a></div></div>
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<div class="ttc" id="base_2logging_8h_html_a09f7d88282cf92c9f231270ac113e5c6"><div class="ttname"><a href="base_2logging_8h.html#a09f7d88282cf92c9f231270ac113e5c6">LOG_IF</a></div><div class="ttdeci">#define LOG_IF(severity, condition)</div><div class="ttdef"><b>Definition:</b> <a href="base_2logging_8h_source.html#l00475">base/logging.h:475</a></div></div>
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<div class="ttc" id="base_2logging_8h_html_ae17f8119c108cf3070bad3449c7e0006"><div class="ttname"><a href="base_2logging_8h.html#ae17f8119c108cf3070bad3449c7e0006">DCHECK</a></div><div class="ttdeci">#define DCHECK(condition)</div><div class="ttdef"><b>Definition:</b> <a href="base_2logging_8h_source.html#l00885">base/logging.h:885</a></div></div>
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<div class="ttc" id="classoperations__research_1_1_cost_value_cycle_handler_html"><div class="ttname"><a href="classoperations__research_1_1_cost_value_cycle_handler.html">operations_research::CostValueCycleHandler</a></div><div class="ttdef"><b>Definition:</b> <a href="linear__assignment_8h_source.html#l01025">linear_assignment.h:1025</a></div></div>
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<div class="ttc" id="classoperations__research_1_1_linear_sum_assignment_1_1_bipartite_left_node_iterator_html"><div class="ttname"><a href="classoperations__research_1_1_linear_sum_assignment_1_1_bipartite_left_node_iterator.html">operations_research::LinearSumAssignment::BipartiteLeftNodeIterator</a></div><div class="ttdef"><b>Definition:</b> <a href="linear__assignment_8h_source.html#l00360">linear_assignment.h:360</a></div></div>
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<div class="ttc" id="classoperations__research_1_1_linear_sum_assignment_html_a28178bb3e97b7c81dc157fe283b8cd64"><div class="ttname"><a href="classoperations__research_1_1_linear_sum_assignment.html#a28178bb3e97b7c81dc157fe283b8cd64">operations_research::LinearSumAssignment::ArcIndex</a></div><div class="ttdeci">GraphType::ArcIndex ArcIndex</div><div class="ttdef"><b>Definition:</b> <a href="linear__assignment_8h_source.html#l00230">linear_assignment.h:230</a></div></div>
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<div class="ttc" id="base_2logging_8h_html_ae89df3243bbb8341130c7b3f44145ea0"><div class="ttname"><a href="base_2logging_8h.html#ae89df3243bbb8341130c7b3f44145ea0">DCHECK_EQ</a></div><div class="ttdeci">#define DCHECK_EQ(val1, val2)</div><div class="ttdef"><b>Definition:</b> <a href="base_2logging_8h_source.html#l00886">base/logging.h:886</a></div></div>
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<div class="ttc" id="classoperations__research_1_1_linear_sum_assignment_html_ab70deec1644ce0de2f383c97830391e7"><div class="ttname"><a href="classoperations__research_1_1_linear_sum_assignment.html#ab70deec1644ce0de2f383c97830391e7">operations_research::LinearSumAssignment::ArcAnnotationCycleHandler</a></div><div class="ttdeci">operations_research::PermutationCycleHandler< typename GraphType::ArcIndex > * ArcAnnotationCycleHandler()</div><div class="ttdef"><b>Definition:</b> <a href="linear__assignment_8h_source.html#l01083">linear_assignment.h:1083</a></div></div>
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<div class="ttc" id="base_2logging_8h_html_a4395e95bab44e222cb2e77251017a0e2"><div class="ttname"><a href="base_2logging_8h.html#a4395e95bab44e222cb2e77251017a0e2">DCHECK_LE</a></div><div class="ttdeci">#define DCHECK_LE(val1, val2)</div><div class="ttdef"><b>Definition:</b> <a href="base_2logging_8h_source.html#l00888">base/logging.h:888</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="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>
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<div class="ttc" id="commandlineflags_8h_html"><div class="ttname"><a href="commandlineflags_8h.html">commandlineflags.h</a></div></div>
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<div class="ttc" id="classoperations__research_1_1_permutation_cycle_handler_html"><div class="ttname"><a href="classoperations__research_1_1_permutation_cycle_handler.html">operations_research::PermutationCycleHandler</a></div><div class="ttdef"><b>Definition:</b> <a href="util_2permutation_8h_source.html#l00094">util/permutation.h:94</a></div></div>
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<div class="ttc" id="ebert__graph_8h_html"><div class="ttname"><a href="ebert__graph_8h.html">ebert_graph.h</a></div></div>
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<div class="ttc" id="classoperations__research_1_1_cost_value_cycle_handler_html_a11007fbe576099d4e9ba5a433036a8ed"><div class="ttname"><a href="classoperations__research_1_1_cost_value_cycle_handler.html#a11007fbe576099d4e9ba5a433036a8ed">operations_research::CostValueCycleHandler::~CostValueCycleHandler</a></div><div class="ttdeci">~CostValueCycleHandler() override</div><div class="ttdef"><b>Definition:</b> <a href="linear__assignment_8h_source.html#l01043">linear_assignment.h:1043</a></div></div>
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<div class="ttc" id="classoperations__research_1_1_tail_array_manager_html"><div class="ttname"><a href="classoperations__research_1_1_tail_array_manager.html">operations_research::TailArrayManager</a></div><div class="ttdef"><b>Definition:</b> <a href="ebert__graph_8h_source.html#l01917">ebert_graph.h:1917</a></div></div>
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<div class="ttc" id="classoperations__research_1_1_arc_index_ordering_by_tail_node_html"><div class="ttname"><a href="classoperations__research_1_1_arc_index_ordering_by_tail_node.html">operations_research::ArcIndexOrderingByTailNode</a></div><div class="ttdef"><b>Definition:</b> <a href="linear__assignment_8h_source.html#l01058">linear_assignment.h:1058</a></div></div>
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<div class="ttc" id="classoperations__research_1_1_linear_sum_assignment_html_ad253fe3f1dd016a976d0713d8a54a0ab"><div class="ttname"><a href="classoperations__research_1_1_linear_sum_assignment.html#ad253fe3f1dd016a976d0713d8a54a0ab">operations_research::LinearSumAssignment::GetMate</a></div><div class="ttdeci">NodeIndex GetMate(NodeIndex left_node) const</div><div class="ttdef"><b>Definition:</b> <a href="linear__assignment_8h_source.html#l00351">linear_assignment.h:351</a></div></div>
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<div class="ttc" id="linear__assignment_8h_html_aaa4eff2168771ea36e49d66a895fb504"><div class="ttname"><a href="linear__assignment_8h.html#aaa4eff2168771ea36e49d66a895fb504">ABSL_DECLARE_FLAG</a></div><div class="ttdeci">ABSL_DECLARE_FLAG(int64_t, assignment_alpha)</div></div>
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<div class="ttc" id="classoperations__research_1_1_linear_sum_assignment_html_a63b3d12e721188086870cc42cc46a258"><div class="ttname"><a href="classoperations__research_1_1_linear_sum_assignment.html#a63b3d12e721188086870cc42cc46a258">operations_research::LinearSumAssignment::ComputeAssignment</a></div><div class="ttdeci">bool ComputeAssignment()</div><div class="ttdef"><b>Definition:</b> <a href="linear__assignment_8h_source.html#l01449">linear_assignment.h:1449</a></div></div>
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<div class="ttc" id="classoperations__research_1_1_linear_sum_assignment_html_a3be0f6fcc44bc6a4a1e976c1e60b31d0"><div class="ttname"><a href="classoperations__research_1_1_linear_sum_assignment.html#a3be0f6fcc44bc6a4a1e976c1e60b31d0">operations_research::LinearSumAssignment::NumNodes</a></div><div class="ttdeci">NodeIndex NumNodes() const</div><div class="ttdef"><b>Definition:</b> <a href="linear__assignment_8h_source.html#l00324">linear_assignment.h:324</a></div></div>
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<div class="ttc" id="classoperations__research_1_1_linear_sum_assignment_html_a11815dc60d6275c8272be0771883d573"><div class="ttname"><a href="classoperations__research_1_1_linear_sum_assignment.html#a11815dc60d6275c8272be0771883d573">operations_research::LinearSumAssignment::NumLeftNodes</a></div><div class="ttdeci">NodeIndex NumLeftNodes() const</div><div class="ttdef"><b>Definition:</b> <a href="linear__assignment_8h_source.html#l00336">linear_assignment.h:336</a></div></div>
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<div class="ttc" id="classoperations__research_1_1_linear_sum_assignment_1_1_bipartite_left_node_iterator_html_ab6dfabf93ff38b84d807d6190e88011f"><div class="ttname"><a href="classoperations__research_1_1_linear_sum_assignment_1_1_bipartite_left_node_iterator.html#ab6dfabf93ff38b84d807d6190e88011f">operations_research::LinearSumAssignment::BipartiteLeftNodeIterator::BipartiteLeftNodeIterator</a></div><div class="ttdeci">BipartiteLeftNodeIterator(const LinearSumAssignment &assignment)</div><div class="ttdef"><b>Definition:</b> <a href="linear__assignment_8h_source.html#l00365">linear_assignment.h:365</a></div></div>
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<div class="ttc" id="classoperations__research_1_1_cost_value_cycle_handler_html_a8bd36eabd11be9f5c4e3094418412544"><div class="ttname"><a href="classoperations__research_1_1_cost_value_cycle_handler.html#a8bd36eabd11be9f5c4e3094418412544">operations_research::CostValueCycleHandler::CostValueCycleHandler</a></div><div class="ttdeci">CostValueCycleHandler(std::vector< CostValue > *cost)</div><div class="ttdef"><b>Definition:</b> <a href="linear__assignment_8h_source.html#l01027">linear_assignment.h:1027</a></div></div>
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<div class="ttc" id="base_2logging_8h_html_ab62f5ed8f2d48e29802be0cbbcd1359a"><div class="ttname"><a href="base_2logging_8h.html#ab62f5ed8f2d48e29802be0cbbcd1359a">DCHECK_LT</a></div><div class="ttdeci">#define DCHECK_LT(val1, val2)</div><div class="ttdef"><b>Definition:</b> <a href="base_2logging_8h_source.html#l00889">base/logging.h:889</a></div></div>
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<div class="ttc" id="constraint__solver_2table_8cc_html_acb18315d548212835cd8ed4287e6c0b6"><div class="ttname"><a href="constraint__solver_2table_8cc.html#acb18315d548212835cd8ed4287e6c0b6">a</a></div><div class="ttdeci">int64_t a</div><div class="ttdef"><b>Definition:</b> <a href="constraint__solver_2table_8cc_source.html#l00046">constraint_solver/table.cc:46</a></div></div>
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