1 | /* -*- mode: C++; indent-tabs-mode: nil; -*- |
---|
2 | * |
---|
3 | * This file is a part of LEMON, a generic C++ optimization library. |
---|
4 | * |
---|
5 | * Copyright (C) 2003-2013 |
---|
6 | * Egervary Jeno Kombinatorikus Optimalizalasi Kutatocsoport |
---|
7 | * (Egervary Research Group on Combinatorial Optimization, EGRES). |
---|
8 | * |
---|
9 | * Permission to use, modify and distribute this software is granted |
---|
10 | * provided that this copyright notice appears in all copies. For |
---|
11 | * precise terms see the accompanying LICENSE file. |
---|
12 | * |
---|
13 | * This software is provided "AS IS" with no warranty of any kind, |
---|
14 | * express or implied, and with no claim as to its suitability for any |
---|
15 | * purpose. |
---|
16 | * |
---|
17 | */ |
---|
18 | |
---|
19 | #ifndef LEMON_CHRISTOFIDES_TSP_H |
---|
20 | #define LEMON_CHRISTOFIDES_TSP_H |
---|
21 | |
---|
22 | /// \ingroup tsp |
---|
23 | /// \file |
---|
24 | /// \brief Christofides algorithm for symmetric TSP |
---|
25 | |
---|
26 | #include <lemon/full_graph.h> |
---|
27 | #include <lemon/smart_graph.h> |
---|
28 | #include <lemon/kruskal.h> |
---|
29 | #include <lemon/matching.h> |
---|
30 | #include <lemon/euler.h> |
---|
31 | |
---|
32 | namespace lemon { |
---|
33 | |
---|
34 | /// \ingroup tsp |
---|
35 | /// |
---|
36 | /// \brief Christofides algorithm for symmetric TSP. |
---|
37 | /// |
---|
38 | /// ChristofidesTsp implements Christofides' heuristic for solving |
---|
39 | /// symmetric \ref tsp "TSP". |
---|
40 | /// |
---|
41 | /// This a well-known approximation method for the TSP problem with |
---|
42 | /// metric cost function. |
---|
43 | /// It has a guaranteed approximation factor of 3/2 (i.e. it finds a tour |
---|
44 | /// whose total cost is at most 3/2 of the optimum), but it usually |
---|
45 | /// provides better solutions in practice. |
---|
46 | /// This implementation runs in O(n<sup>3</sup>log(n)) time. |
---|
47 | /// |
---|
48 | /// The algorithm starts with a \ref spantree "minimum cost spanning tree" and |
---|
49 | /// finds a \ref MaxWeightedPerfectMatching "minimum cost perfect matching" |
---|
50 | /// in the subgraph induced by the nodes that have odd degree in the |
---|
51 | /// spanning tree. |
---|
52 | /// Finally, it constructs the tour from the \ref EulerIt "Euler traversal" |
---|
53 | /// of the union of the spanning tree and the matching. |
---|
54 | /// During this last step, the algorithm simply skips the visited nodes |
---|
55 | /// (i.e. creates shortcuts) assuming that the triangle inequality holds |
---|
56 | /// for the cost function. |
---|
57 | /// |
---|
58 | /// \tparam CM Type of the cost map. |
---|
59 | /// |
---|
60 | /// \warning CM::Value must be a signed number type. |
---|
61 | template <typename CM> |
---|
62 | class ChristofidesTsp |
---|
63 | { |
---|
64 | public: |
---|
65 | |
---|
66 | /// Type of the cost map |
---|
67 | typedef CM CostMap; |
---|
68 | /// Type of the edge costs |
---|
69 | typedef typename CM::Value Cost; |
---|
70 | |
---|
71 | private: |
---|
72 | |
---|
73 | GRAPH_TYPEDEFS(FullGraph); |
---|
74 | |
---|
75 | const FullGraph &_gr; |
---|
76 | const CostMap &_cost; |
---|
77 | std::vector<Node> _path; |
---|
78 | Cost _sum; |
---|
79 | |
---|
80 | public: |
---|
81 | |
---|
82 | /// \brief Constructor |
---|
83 | /// |
---|
84 | /// Constructor. |
---|
85 | /// \param gr The \ref FullGraph "full graph" the algorithm runs on. |
---|
86 | /// \param cost The cost map. |
---|
87 | ChristofidesTsp(const FullGraph &gr, const CostMap &cost) |
---|
88 | : _gr(gr), _cost(cost) {} |
---|
89 | |
---|
90 | /// \name Execution Control |
---|
91 | /// @{ |
---|
92 | |
---|
93 | /// \brief Runs the algorithm. |
---|
94 | /// |
---|
95 | /// This function runs the algorithm. |
---|
96 | /// |
---|
97 | /// \return The total cost of the found tour. |
---|
98 | Cost run() { |
---|
99 | _path.clear(); |
---|
100 | |
---|
101 | if (_gr.nodeNum() == 0) return _sum = 0; |
---|
102 | else if (_gr.nodeNum() == 1) { |
---|
103 | _path.push_back(_gr(0)); |
---|
104 | return _sum = 0; |
---|
105 | } |
---|
106 | else if (_gr.nodeNum() == 2) { |
---|
107 | _path.push_back(_gr(0)); |
---|
108 | _path.push_back(_gr(1)); |
---|
109 | return _sum = 2 * _cost[_gr.edge(_gr(0), _gr(1))]; |
---|
110 | } |
---|
111 | |
---|
112 | // Compute min. cost spanning tree |
---|
113 | std::vector<Edge> tree; |
---|
114 | kruskal(_gr, _cost, std::back_inserter(tree)); |
---|
115 | |
---|
116 | FullGraph::NodeMap<int> deg(_gr, 0); |
---|
117 | for (int i = 0; i != int(tree.size()); ++i) { |
---|
118 | Edge e = tree[i]; |
---|
119 | ++deg[_gr.u(e)]; |
---|
120 | ++deg[_gr.v(e)]; |
---|
121 | } |
---|
122 | |
---|
123 | // Copy the induced subgraph of odd nodes |
---|
124 | std::vector<Node> odd_nodes; |
---|
125 | for (NodeIt u(_gr); u != INVALID; ++u) { |
---|
126 | if (deg[u] % 2 == 1) odd_nodes.push_back(u); |
---|
127 | } |
---|
128 | |
---|
129 | SmartGraph sgr; |
---|
130 | SmartGraph::EdgeMap<Cost> scost(sgr); |
---|
131 | for (int i = 0; i != int(odd_nodes.size()); ++i) { |
---|
132 | sgr.addNode(); |
---|
133 | } |
---|
134 | for (int i = 0; i != int(odd_nodes.size()); ++i) { |
---|
135 | for (int j = 0; j != int(odd_nodes.size()); ++j) { |
---|
136 | if (j == i) continue; |
---|
137 | SmartGraph::Edge e = |
---|
138 | sgr.addEdge(sgr.nodeFromId(i), sgr.nodeFromId(j)); |
---|
139 | scost[e] = -_cost[_gr.edge(odd_nodes[i], odd_nodes[j])]; |
---|
140 | } |
---|
141 | } |
---|
142 | |
---|
143 | // Compute min. cost perfect matching |
---|
144 | MaxWeightedPerfectMatching<SmartGraph, SmartGraph::EdgeMap<Cost> > |
---|
145 | mwpm(sgr, scost); |
---|
146 | mwpm.run(); |
---|
147 | |
---|
148 | for (SmartGraph::EdgeIt e(sgr); e != INVALID; ++e) { |
---|
149 | if (mwpm.matching(e)) { |
---|
150 | tree.push_back( _gr.edge(odd_nodes[sgr.id(sgr.u(e))], |
---|
151 | odd_nodes[sgr.id(sgr.v(e))]) ); |
---|
152 | } |
---|
153 | } |
---|
154 | |
---|
155 | // Join the spanning tree and the matching |
---|
156 | sgr.clear(); |
---|
157 | for (int i = 0; i != _gr.nodeNum(); ++i) { |
---|
158 | sgr.addNode(); |
---|
159 | } |
---|
160 | for (int i = 0; i != int(tree.size()); ++i) { |
---|
161 | int ui = _gr.id(_gr.u(tree[i])), |
---|
162 | vi = _gr.id(_gr.v(tree[i])); |
---|
163 | sgr.addEdge(sgr.nodeFromId(ui), sgr.nodeFromId(vi)); |
---|
164 | } |
---|
165 | |
---|
166 | // Compute the tour from the Euler traversal |
---|
167 | SmartGraph::NodeMap<bool> visited(sgr, false); |
---|
168 | for (EulerIt<SmartGraph> e(sgr); e != INVALID; ++e) { |
---|
169 | SmartGraph::Node n = sgr.target(e); |
---|
170 | if (!visited[n]) { |
---|
171 | _path.push_back(_gr(sgr.id(n))); |
---|
172 | visited[n] = true; |
---|
173 | } |
---|
174 | } |
---|
175 | |
---|
176 | _sum = _cost[_gr.edge(_path.back(), _path.front())]; |
---|
177 | for (int i = 0; i < int(_path.size())-1; ++i) { |
---|
178 | _sum += _cost[_gr.edge(_path[i], _path[i+1])]; |
---|
179 | } |
---|
180 | |
---|
181 | return _sum; |
---|
182 | } |
---|
183 | |
---|
184 | /// @} |
---|
185 | |
---|
186 | /// \name Query Functions |
---|
187 | /// @{ |
---|
188 | |
---|
189 | /// \brief The total cost of the found tour. |
---|
190 | /// |
---|
191 | /// This function returns the total cost of the found tour. |
---|
192 | /// |
---|
193 | /// \pre run() must be called before using this function. |
---|
194 | Cost tourCost() const { |
---|
195 | return _sum; |
---|
196 | } |
---|
197 | |
---|
198 | /// \brief Returns a const reference to the node sequence of the |
---|
199 | /// found tour. |
---|
200 | /// |
---|
201 | /// This function returns a const reference to a vector |
---|
202 | /// that stores the node sequence of the found tour. |
---|
203 | /// |
---|
204 | /// \pre run() must be called before using this function. |
---|
205 | const std::vector<Node>& tourNodes() const { |
---|
206 | return _path; |
---|
207 | } |
---|
208 | |
---|
209 | /// \brief Gives back the node sequence of the found tour. |
---|
210 | /// |
---|
211 | /// This function copies the node sequence of the found tour into |
---|
212 | /// an STL container through the given output iterator. The |
---|
213 | /// <tt>value_type</tt> of the container must be <tt>FullGraph::Node</tt>. |
---|
214 | /// For example, |
---|
215 | /// \code |
---|
216 | /// std::vector<FullGraph::Node> nodes(countNodes(graph)); |
---|
217 | /// tsp.tourNodes(nodes.begin()); |
---|
218 | /// \endcode |
---|
219 | /// or |
---|
220 | /// \code |
---|
221 | /// std::list<FullGraph::Node> nodes; |
---|
222 | /// tsp.tourNodes(std::back_inserter(nodes)); |
---|
223 | /// \endcode |
---|
224 | /// |
---|
225 | /// \pre run() must be called before using this function. |
---|
226 | template <typename Iterator> |
---|
227 | void tourNodes(Iterator out) const { |
---|
228 | std::copy(_path.begin(), _path.end(), out); |
---|
229 | } |
---|
230 | |
---|
231 | /// \brief Gives back the found tour as a path. |
---|
232 | /// |
---|
233 | /// This function copies the found tour as a list of arcs/edges into |
---|
234 | /// the given \ref lemon::concepts::Path "path structure". |
---|
235 | /// |
---|
236 | /// \pre run() must be called before using this function. |
---|
237 | template <typename Path> |
---|
238 | void tour(Path &path) const { |
---|
239 | path.clear(); |
---|
240 | for (int i = 0; i < int(_path.size()) - 1; ++i) { |
---|
241 | path.addBack(_gr.arc(_path[i], _path[i+1])); |
---|
242 | } |
---|
243 | if (int(_path.size()) >= 2) { |
---|
244 | path.addBack(_gr.arc(_path.back(), _path.front())); |
---|
245 | } |
---|
246 | } |
---|
247 | |
---|
248 | /// @} |
---|
249 | |
---|
250 | }; |
---|
251 | |
---|
252 | }; // namespace lemon |
---|
253 | |
---|
254 | #endif |
---|