[1201] | 1 | /* -*- mode: C++; indent-tabs-mode: nil; -*- |
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| 2 | * |
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| 3 | * This file is a part of LEMON, a generic C++ optimization library. |
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| 4 | * |
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| 5 | * Copyright (C) 2003-2010 |
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| 6 | * Egervary Jeno Kombinatorikus Optimalizalasi Kutatocsoport |
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| 7 | * (Egervary Research Group on Combinatorial Optimization, EGRES). |
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| 8 | * |
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| 9 | * Permission to use, modify and distribute this software is granted |
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| 10 | * provided that this copyright notice appears in all copies. For |
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| 11 | * precise terms see the accompanying LICENSE file. |
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| 12 | * |
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| 13 | * This software is provided "AS IS" with no warranty of any kind, |
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| 14 | * express or implied, and with no claim as to its suitability for any |
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| 15 | * purpose. |
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| 16 | * |
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| 17 | */ |
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| 18 | |
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[1199] | 19 | #ifndef LEMON_CHRISTOFIDES_TSP_H |
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| 20 | #define LEMON_CHRISTOFIDES_TSP_H |
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| 21 | |
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[1201] | 22 | /// \ingroup tsp |
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| 23 | /// \file |
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| 24 | /// \brief Christofides algorithm for symmetric TSP |
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| 25 | |
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[1199] | 26 | #include <lemon/full_graph.h> |
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| 27 | #include <lemon/smart_graph.h> |
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| 28 | #include <lemon/kruskal.h> |
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| 29 | #include <lemon/matching.h> |
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| 30 | #include <lemon/euler.h> |
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| 31 | |
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| 32 | namespace lemon { |
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| 33 | |
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[1202] | 34 | /// \ingroup tsp |
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| 35 | /// |
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[1201] | 36 | /// \brief Christofides algorithm for symmetric TSP. |
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| 37 | /// |
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| 38 | /// ChristofidesTsp implements Christofides' heuristic for solving |
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| 39 | /// symmetric \ref tsp "TSP". |
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| 40 | /// |
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| 41 | /// This a well-known approximation method for the TSP problem with |
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[1202] | 42 | /// metric cost function. |
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[1201] | 43 | /// It yields a tour whose total cost is at most 3/2 of the optimum, |
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| 44 | /// but it is usually much better. |
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| 45 | /// This implementation runs in O(n<sup>3</sup>log(n)) time. |
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| 46 | /// |
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| 47 | /// The algorithm starts with a \ref spantree "minimum cost spanning tree" and |
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| 48 | /// finds a \ref MaxWeightedPerfectMatching "minimum cost perfect matching" |
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| 49 | /// in the subgraph induced by the nodes that have odd degree in the |
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| 50 | /// spanning tree. |
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| 51 | /// Finally, it constructs the tour from the \ref EulerIt "Euler traversal" |
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| 52 | /// of the union of the spanning tree and the matching. |
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| 53 | /// During this last step, the algorithm simply skips the visited nodes |
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| 54 | /// (i.e. creates shortcuts) assuming that the triangle inequality holds |
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| 55 | /// for the cost function. |
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| 56 | /// |
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| 57 | /// \tparam CM Type of the cost map. |
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| 58 | /// |
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[1202] | 59 | /// \warning CM::Value must be a signed number type. |
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[1199] | 60 | template <typename CM> |
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[1201] | 61 | class ChristofidesTsp |
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| 62 | { |
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| 63 | public: |
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| 64 | |
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| 65 | /// Type of the cost map |
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| 66 | typedef CM CostMap; |
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| 67 | /// Type of the edge costs |
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| 68 | typedef typename CM::Value Cost; |
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| 69 | |
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[1199] | 70 | private: |
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[1201] | 71 | |
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| 72 | GRAPH_TYPEDEFS(FullGraph); |
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| 73 | |
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| 74 | const FullGraph &_gr; |
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| 75 | const CostMap &_cost; |
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| 76 | std::vector<Node> _path; |
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| 77 | Cost _sum; |
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[1199] | 78 | |
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| 79 | public: |
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| 80 | |
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[1201] | 81 | /// \brief Constructor |
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| 82 | /// |
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| 83 | /// Constructor. |
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| 84 | /// \param gr The \ref FullGraph "full graph" the algorithm runs on. |
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| 85 | /// \param cost The cost map. |
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| 86 | ChristofidesTsp(const FullGraph &gr, const CostMap &cost) |
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| 87 | : _gr(gr), _cost(cost) {} |
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| 88 | |
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| 89 | /// \name Execution Control |
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| 90 | /// @{ |
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| 91 | |
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| 92 | /// \brief Runs the algorithm. |
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| 93 | /// |
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| 94 | /// This function runs the algorithm. |
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| 95 | /// |
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| 96 | /// \return The total cost of the found tour. |
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[1199] | 97 | Cost run() { |
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| 98 | _path.clear(); |
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[1201] | 99 | |
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| 100 | if (_gr.nodeNum() == 0) return _sum = 0; |
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| 101 | else if (_gr.nodeNum() == 1) { |
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| 102 | _path.push_back(_gr(0)); |
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| 103 | return _sum = 0; |
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| 104 | } |
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| 105 | else if (_gr.nodeNum() == 2) { |
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| 106 | _path.push_back(_gr(0)); |
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| 107 | _path.push_back(_gr(1)); |
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| 108 | return _sum = 2 * _cost[_gr.edge(_gr(0), _gr(1))]; |
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| 109 | } |
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[1199] | 110 | |
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[1201] | 111 | // Compute min. cost spanning tree |
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| 112 | std::vector<Edge> tree; |
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| 113 | kruskal(_gr, _cost, std::back_inserter(tree)); |
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[1199] | 114 | |
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[1201] | 115 | FullGraph::NodeMap<int> deg(_gr, 0); |
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| 116 | for (int i = 0; i != int(tree.size()); ++i) { |
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| 117 | Edge e = tree[i]; |
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| 118 | ++deg[_gr.u(e)]; |
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| 119 | ++deg[_gr.v(e)]; |
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| 120 | } |
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| 121 | |
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| 122 | // Copy the induced subgraph of odd nodes |
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| 123 | std::vector<Node> odd_nodes; |
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| 124 | for (NodeIt u(_gr); u != INVALID; ++u) { |
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| 125 | if (deg[u] % 2 == 1) odd_nodes.push_back(u); |
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| 126 | } |
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| 127 | |
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| 128 | SmartGraph sgr; |
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| 129 | SmartGraph::EdgeMap<Cost> scost(sgr); |
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| 130 | for (int i = 0; i != int(odd_nodes.size()); ++i) { |
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| 131 | sgr.addNode(); |
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| 132 | } |
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| 133 | for (int i = 0; i != int(odd_nodes.size()); ++i) { |
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| 134 | for (int j = 0; j != int(odd_nodes.size()); ++j) { |
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| 135 | if (j == i) continue; |
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| 136 | SmartGraph::Edge e = |
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| 137 | sgr.addEdge(sgr.nodeFromId(i), sgr.nodeFromId(j)); |
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| 138 | scost[e] = -_cost[_gr.edge(odd_nodes[i], odd_nodes[j])]; |
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[1199] | 139 | } |
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| 140 | } |
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| 141 | |
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[1201] | 142 | // Compute min. cost perfect matching |
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| 143 | MaxWeightedPerfectMatching<SmartGraph, SmartGraph::EdgeMap<Cost> > |
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| 144 | mwpm(sgr, scost); |
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| 145 | mwpm.run(); |
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[1199] | 146 | |
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[1201] | 147 | for (SmartGraph::EdgeIt e(sgr); e != INVALID; ++e) { |
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| 148 | if (mwpm.matching(e)) { |
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| 149 | tree.push_back( _gr.edge(odd_nodes[sgr.id(sgr.u(e))], |
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| 150 | odd_nodes[sgr.id(sgr.v(e))]) ); |
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[1199] | 151 | } |
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| 152 | } |
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| 153 | |
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[1201] | 154 | // Join the spanning tree and the matching |
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| 155 | sgr.clear(); |
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| 156 | for (int i = 0; i != _gr.nodeNum(); ++i) { |
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| 157 | sgr.addNode(); |
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| 158 | } |
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| 159 | for (int i = 0; i != int(tree.size()); ++i) { |
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| 160 | int ui = _gr.id(_gr.u(tree[i])), |
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| 161 | vi = _gr.id(_gr.v(tree[i])); |
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| 162 | sgr.addEdge(sgr.nodeFromId(ui), sgr.nodeFromId(vi)); |
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| 163 | } |
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| 164 | |
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| 165 | // Compute the tour from the Euler traversal |
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| 166 | SmartGraph::NodeMap<bool> visited(sgr, false); |
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| 167 | for (EulerIt<SmartGraph> e(sgr); e != INVALID; ++e) { |
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| 168 | SmartGraph::Node n = sgr.target(e); |
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| 169 | if (!visited[n]) { |
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| 170 | _path.push_back(_gr(sgr.id(n))); |
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| 171 | visited[n] = true; |
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[1199] | 172 | } |
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| 173 | } |
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| 174 | |
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[1201] | 175 | _sum = _cost[_gr.edge(_path.back(), _path.front())]; |
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| 176 | for (int i = 0; i < int(_path.size())-1; ++i) { |
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| 177 | _sum += _cost[_gr.edge(_path[i], _path[i+1])]; |
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| 178 | } |
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[1199] | 179 | |
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| 180 | return _sum; |
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| 181 | } |
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| 182 | |
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[1201] | 183 | /// @} |
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[1199] | 184 | |
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[1201] | 185 | /// \name Query Functions |
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| 186 | /// @{ |
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[1199] | 187 | |
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[1201] | 188 | /// \brief The total cost of the found tour. |
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| 189 | /// |
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| 190 | /// This function returns the total cost of the found tour. |
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| 191 | /// |
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| 192 | /// \pre run() must be called before using this function. |
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| 193 | Cost tourCost() const { |
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[1199] | 194 | return _sum; |
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| 195 | } |
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| 196 | |
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[1201] | 197 | /// \brief Returns a const reference to the node sequence of the |
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| 198 | /// found tour. |
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| 199 | /// |
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[1202] | 200 | /// This function returns a const reference to a vector |
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[1201] | 201 | /// that stores the node sequence of the found tour. |
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| 202 | /// |
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| 203 | /// \pre run() must be called before using this function. |
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| 204 | const std::vector<Node>& tourNodes() const { |
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| 205 | return _path; |
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| 206 | } |
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[1199] | 207 | |
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[1201] | 208 | /// \brief Gives back the node sequence of the found tour. |
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| 209 | /// |
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| 210 | /// This function copies the node sequence of the found tour into |
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| 211 | /// the given standard container. |
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| 212 | /// |
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| 213 | /// \pre run() must be called before using this function. |
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| 214 | template <typename Container> |
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| 215 | void tourNodes(Container &container) const { |
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| 216 | container.assign(_path.begin(), _path.end()); |
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| 217 | } |
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| 218 | |
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| 219 | /// \brief Gives back the found tour as a path. |
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| 220 | /// |
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| 221 | /// This function copies the found tour as a list of arcs/edges into |
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| 222 | /// the given \ref concept::Path "path structure". |
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| 223 | /// |
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| 224 | /// \pre run() must be called before using this function. |
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| 225 | template <typename Path> |
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| 226 | void tour(Path &path) const { |
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| 227 | path.clear(); |
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| 228 | for (int i = 0; i < int(_path.size()) - 1; ++i) { |
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| 229 | path.addBack(_gr.arc(_path[i], _path[i+1])); |
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| 230 | } |
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| 231 | if (int(_path.size()) >= 2) { |
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| 232 | path.addBack(_gr.arc(_path.back(), _path.front())); |
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| 233 | } |
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| 234 | } |
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| 235 | |
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| 236 | /// @} |
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| 237 | |
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[1199] | 238 | }; |
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| 239 | |
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| 240 | }; // namespace lemon |
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| 241 | |
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| 242 | #endif |
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