[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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[1270] | 5 | * Copyright (C) 2003-2013 |
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[1201] | 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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[1270] | 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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[1204] | 43 | /// It has a guaranteed approximation factor of 3/2 (i.e. it finds a tour |
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| 44 | /// whose total cost is at most 3/2 of the optimum), but it usually |
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| 45 | /// provides better solutions in practice. |
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[1201] | 46 | /// This implementation runs in O(n<sup>3</sup>log(n)) time. |
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| 47 | /// |
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| 48 | /// The algorithm starts with a \ref spantree "minimum cost spanning tree" and |
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| 49 | /// finds a \ref MaxWeightedPerfectMatching "minimum cost perfect matching" |
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| 50 | /// in the subgraph induced by the nodes that have odd degree in the |
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| 51 | /// spanning tree. |
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| 52 | /// Finally, it constructs the tour from the \ref EulerIt "Euler traversal" |
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| 53 | /// of the union of the spanning tree and the matching. |
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| 54 | /// During this last step, the algorithm simply skips the visited nodes |
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| 55 | /// (i.e. creates shortcuts) assuming that the triangle inequality holds |
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| 56 | /// for the cost function. |
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| 57 | /// |
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| 58 | /// \tparam CM Type of the cost map. |
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| 59 | /// |
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[1202] | 60 | /// \warning CM::Value must be a signed number type. |
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[1199] | 61 | template <typename CM> |
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[1201] | 62 | class ChristofidesTsp |
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| 63 | { |
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| 64 | public: |
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| 65 | |
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| 66 | /// Type of the cost map |
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| 67 | typedef CM CostMap; |
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| 68 | /// Type of the edge costs |
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| 69 | typedef typename CM::Value Cost; |
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| 70 | |
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[1199] | 71 | private: |
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[1201] | 72 | |
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| 73 | GRAPH_TYPEDEFS(FullGraph); |
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| 74 | |
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| 75 | const FullGraph &_gr; |
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| 76 | const CostMap &_cost; |
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| 77 | std::vector<Node> _path; |
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| 78 | Cost _sum; |
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[1199] | 79 | |
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| 80 | public: |
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| 81 | |
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[1201] | 82 | /// \brief Constructor |
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| 83 | /// |
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| 84 | /// Constructor. |
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| 85 | /// \param gr The \ref FullGraph "full graph" the algorithm runs on. |
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| 86 | /// \param cost The cost map. |
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| 87 | ChristofidesTsp(const FullGraph &gr, const CostMap &cost) |
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| 88 | : _gr(gr), _cost(cost) {} |
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| 89 | |
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| 90 | /// \name Execution Control |
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| 91 | /// @{ |
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| 92 | |
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| 93 | /// \brief Runs the algorithm. |
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| 94 | /// |
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| 95 | /// This function runs the algorithm. |
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| 96 | /// |
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| 97 | /// \return The total cost of the found tour. |
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[1199] | 98 | Cost run() { |
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| 99 | _path.clear(); |
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[1201] | 100 | |
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| 101 | if (_gr.nodeNum() == 0) return _sum = 0; |
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| 102 | else if (_gr.nodeNum() == 1) { |
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| 103 | _path.push_back(_gr(0)); |
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| 104 | return _sum = 0; |
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| 105 | } |
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| 106 | else if (_gr.nodeNum() == 2) { |
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| 107 | _path.push_back(_gr(0)); |
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| 108 | _path.push_back(_gr(1)); |
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| 109 | return _sum = 2 * _cost[_gr.edge(_gr(0), _gr(1))]; |
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| 110 | } |
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[1270] | 111 | |
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[1201] | 112 | // Compute min. cost spanning tree |
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| 113 | std::vector<Edge> tree; |
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| 114 | kruskal(_gr, _cost, std::back_inserter(tree)); |
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[1270] | 115 | |
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[1201] | 116 | FullGraph::NodeMap<int> deg(_gr, 0); |
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| 117 | for (int i = 0; i != int(tree.size()); ++i) { |
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| 118 | Edge e = tree[i]; |
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| 119 | ++deg[_gr.u(e)]; |
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| 120 | ++deg[_gr.v(e)]; |
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| 121 | } |
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| 122 | |
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| 123 | // Copy the induced subgraph of odd nodes |
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| 124 | std::vector<Node> odd_nodes; |
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| 125 | for (NodeIt u(_gr); u != INVALID; ++u) { |
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| 126 | if (deg[u] % 2 == 1) odd_nodes.push_back(u); |
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| 127 | } |
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[1270] | 128 | |
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[1201] | 129 | SmartGraph sgr; |
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| 130 | SmartGraph::EdgeMap<Cost> scost(sgr); |
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| 131 | for (int i = 0; i != int(odd_nodes.size()); ++i) { |
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| 132 | sgr.addNode(); |
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| 133 | } |
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| 134 | for (int i = 0; i != int(odd_nodes.size()); ++i) { |
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| 135 | for (int j = 0; j != int(odd_nodes.size()); ++j) { |
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| 136 | if (j == i) continue; |
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| 137 | SmartGraph::Edge e = |
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| 138 | sgr.addEdge(sgr.nodeFromId(i), sgr.nodeFromId(j)); |
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| 139 | scost[e] = -_cost[_gr.edge(odd_nodes[i], odd_nodes[j])]; |
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[1199] | 140 | } |
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| 141 | } |
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[1270] | 142 | |
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[1201] | 143 | // Compute min. cost perfect matching |
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| 144 | MaxWeightedPerfectMatching<SmartGraph, SmartGraph::EdgeMap<Cost> > |
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| 145 | mwpm(sgr, scost); |
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| 146 | mwpm.run(); |
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[1270] | 147 | |
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[1201] | 148 | for (SmartGraph::EdgeIt e(sgr); e != INVALID; ++e) { |
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| 149 | if (mwpm.matching(e)) { |
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| 150 | tree.push_back( _gr.edge(odd_nodes[sgr.id(sgr.u(e))], |
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| 151 | odd_nodes[sgr.id(sgr.v(e))]) ); |
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[1199] | 152 | } |
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| 153 | } |
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[1270] | 154 | |
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| 155 | // Join the spanning tree and the matching |
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[1201] | 156 | sgr.clear(); |
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| 157 | for (int i = 0; i != _gr.nodeNum(); ++i) { |
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| 158 | sgr.addNode(); |
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| 159 | } |
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| 160 | for (int i = 0; i != int(tree.size()); ++i) { |
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| 161 | int ui = _gr.id(_gr.u(tree[i])), |
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| 162 | vi = _gr.id(_gr.v(tree[i])); |
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| 163 | sgr.addEdge(sgr.nodeFromId(ui), sgr.nodeFromId(vi)); |
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| 164 | } |
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| 165 | |
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| 166 | // Compute the tour from the Euler traversal |
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| 167 | SmartGraph::NodeMap<bool> visited(sgr, false); |
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| 168 | for (EulerIt<SmartGraph> e(sgr); e != INVALID; ++e) { |
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| 169 | SmartGraph::Node n = sgr.target(e); |
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| 170 | if (!visited[n]) { |
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| 171 | _path.push_back(_gr(sgr.id(n))); |
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| 172 | visited[n] = true; |
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[1199] | 173 | } |
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| 174 | } |
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| 175 | |
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[1201] | 176 | _sum = _cost[_gr.edge(_path.back(), _path.front())]; |
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| 177 | for (int i = 0; i < int(_path.size())-1; ++i) { |
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| 178 | _sum += _cost[_gr.edge(_path[i], _path[i+1])]; |
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| 179 | } |
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[1199] | 180 | |
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| 181 | return _sum; |
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| 182 | } |
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| 183 | |
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[1201] | 184 | /// @} |
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[1270] | 185 | |
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[1201] | 186 | /// \name Query Functions |
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| 187 | /// @{ |
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[1270] | 188 | |
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[1201] | 189 | /// \brief The total cost of the found tour. |
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| 190 | /// |
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| 191 | /// This function returns the total cost of the found tour. |
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| 192 | /// |
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| 193 | /// \pre run() must be called before using this function. |
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| 194 | Cost tourCost() const { |
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[1199] | 195 | return _sum; |
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| 196 | } |
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[1270] | 197 | |
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[1201] | 198 | /// \brief Returns a const reference to the node sequence of the |
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| 199 | /// found tour. |
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| 200 | /// |
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[1202] | 201 | /// This function returns a const reference to a vector |
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[1201] | 202 | /// that stores the node sequence of the found tour. |
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| 203 | /// |
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| 204 | /// \pre run() must be called before using this function. |
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| 205 | const std::vector<Node>& tourNodes() const { |
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| 206 | return _path; |
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| 207 | } |
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[1199] | 208 | |
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[1201] | 209 | /// \brief Gives back the node sequence of the found tour. |
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| 210 | /// |
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| 211 | /// This function copies the node sequence of the found tour into |
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[1205] | 212 | /// an STL container through the given output iterator. The |
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| 213 | /// <tt>value_type</tt> of the container must be <tt>FullGraph::Node</tt>. |
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| 214 | /// For example, |
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| 215 | /// \code |
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| 216 | /// std::vector<FullGraph::Node> nodes(countNodes(graph)); |
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| 217 | /// tsp.tourNodes(nodes.begin()); |
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| 218 | /// \endcode |
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| 219 | /// or |
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| 220 | /// \code |
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| 221 | /// std::list<FullGraph::Node> nodes; |
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| 222 | /// tsp.tourNodes(std::back_inserter(nodes)); |
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| 223 | /// \endcode |
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[1201] | 224 | /// |
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| 225 | /// \pre run() must be called before using this function. |
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[1205] | 226 | template <typename Iterator> |
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| 227 | void tourNodes(Iterator out) const { |
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| 228 | std::copy(_path.begin(), _path.end(), out); |
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[1201] | 229 | } |
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[1270] | 230 | |
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[1201] | 231 | /// \brief Gives back the found tour as a path. |
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| 232 | /// |
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| 233 | /// This function copies the found tour as a list of arcs/edges into |
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[1250] | 234 | /// the given \ref lemon::concepts::Path "path structure". |
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[1201] | 235 | /// |
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| 236 | /// \pre run() must be called before using this function. |
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| 237 | template <typename Path> |
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| 238 | void tour(Path &path) const { |
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| 239 | path.clear(); |
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| 240 | for (int i = 0; i < int(_path.size()) - 1; ++i) { |
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| 241 | path.addBack(_gr.arc(_path[i], _path[i+1])); |
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| 242 | } |
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| 243 | if (int(_path.size()) >= 2) { |
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| 244 | path.addBack(_gr.arc(_path.back(), _path.front())); |
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| 245 | } |
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| 246 | } |
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[1270] | 247 | |
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[1201] | 248 | /// @} |
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[1270] | 249 | |
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[1199] | 250 | }; |
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| 251 | |
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| 252 | }; // namespace lemon |
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| 253 | |
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| 254 | #endif |
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