kpeter@1033: /* -*- mode: C++; indent-tabs-mode: nil; -*-
kpeter@1033: *
kpeter@1033: * This file is a part of LEMON, a generic C++ optimization library.
kpeter@1033: *
kpeter@1033: * Copyright (C) 2003-2010
kpeter@1033: * Egervary Jeno Kombinatorikus Optimalizalasi Kutatocsoport
kpeter@1033: * (Egervary Research Group on Combinatorial Optimization, EGRES).
kpeter@1033: *
kpeter@1033: * Permission to use, modify and distribute this software is granted
kpeter@1033: * provided that this copyright notice appears in all copies. For
kpeter@1033: * precise terms see the accompanying LICENSE file.
kpeter@1033: *
kpeter@1033: * This software is provided "AS IS" with no warranty of any kind,
kpeter@1033: * express or implied, and with no claim as to its suitability for any
kpeter@1033: * purpose.
kpeter@1033: *
kpeter@1033: */
kpeter@1033:
f4c3@1031: #ifndef LEMON_CHRISTOFIDES_TSP_H
f4c3@1031: #define LEMON_CHRISTOFIDES_TSP_H
f4c3@1031:
kpeter@1033: /// \ingroup tsp
kpeter@1033: /// \file
kpeter@1033: /// \brief Christofides algorithm for symmetric TSP
kpeter@1033:
f4c3@1031: #include <lemon/full_graph.h>
f4c3@1031: #include <lemon/smart_graph.h>
f4c3@1031: #include <lemon/kruskal.h>
f4c3@1031: #include <lemon/matching.h>
f4c3@1031: #include <lemon/euler.h>
f4c3@1031:
f4c3@1031: namespace lemon {
f4c3@1031:
kpeter@1034: /// \ingroup tsp
kpeter@1034: ///
kpeter@1033: /// \brief Christofides algorithm for symmetric TSP.
kpeter@1033: ///
kpeter@1033: /// ChristofidesTsp implements Christofides' heuristic for solving
kpeter@1033: /// symmetric \ref tsp "TSP".
kpeter@1033: ///
kpeter@1033: /// This a well-known approximation method for the TSP problem with
kpeter@1034: /// metric cost function.
kpeter@1036: /// It has a guaranteed approximation factor of 3/2 (i.e. it finds a tour
kpeter@1036: /// whose total cost is at most 3/2 of the optimum), but it usually
kpeter@1036: /// provides better solutions in practice.
kpeter@1033: /// This implementation runs in O(n<sup>3</sup>log(n)) time.
kpeter@1033: ///
kpeter@1033: /// The algorithm starts with a \ref spantree "minimum cost spanning tree" and
kpeter@1033: /// finds a \ref MaxWeightedPerfectMatching "minimum cost perfect matching"
kpeter@1033: /// in the subgraph induced by the nodes that have odd degree in the
kpeter@1033: /// spanning tree.
kpeter@1033: /// Finally, it constructs the tour from the \ref EulerIt "Euler traversal"
kpeter@1033: /// of the union of the spanning tree and the matching.
kpeter@1033: /// During this last step, the algorithm simply skips the visited nodes
kpeter@1033: /// (i.e. creates shortcuts) assuming that the triangle inequality holds
kpeter@1033: /// for the cost function.
kpeter@1033: ///
kpeter@1033: /// \tparam CM Type of the cost map.
kpeter@1033: ///
kpeter@1034: /// \warning CM::Value must be a signed number type.
f4c3@1031: template <typename CM>
kpeter@1033: class ChristofidesTsp
kpeter@1033: {
kpeter@1033: public:
kpeter@1033:
kpeter@1033: /// Type of the cost map
kpeter@1033: typedef CM CostMap;
kpeter@1033: /// Type of the edge costs
kpeter@1033: typedef typename CM::Value Cost;
kpeter@1033:
f4c3@1031: private:
kpeter@1033:
kpeter@1033: GRAPH_TYPEDEFS(FullGraph);
kpeter@1033:
kpeter@1033: const FullGraph &_gr;
kpeter@1033: const CostMap &_cost;
kpeter@1033: std::vector<Node> _path;
kpeter@1033: Cost _sum;
f4c3@1031:
f4c3@1031: public:
f4c3@1031:
kpeter@1033: /// \brief Constructor
kpeter@1033: ///
kpeter@1033: /// Constructor.
kpeter@1033: /// \param gr The \ref FullGraph "full graph" the algorithm runs on.
kpeter@1033: /// \param cost The cost map.
kpeter@1033: ChristofidesTsp(const FullGraph &gr, const CostMap &cost)
kpeter@1033: : _gr(gr), _cost(cost) {}
kpeter@1033:
kpeter@1033: /// \name Execution Control
kpeter@1033: /// @{
kpeter@1033:
kpeter@1033: /// \brief Runs the algorithm.
kpeter@1033: ///
kpeter@1033: /// This function runs the algorithm.
kpeter@1033: ///
kpeter@1033: /// \return The total cost of the found tour.
f4c3@1031: Cost run() {
f4c3@1031: _path.clear();
kpeter@1033:
kpeter@1033: if (_gr.nodeNum() == 0) return _sum = 0;
kpeter@1033: else if (_gr.nodeNum() == 1) {
kpeter@1033: _path.push_back(_gr(0));
kpeter@1033: return _sum = 0;
kpeter@1033: }
kpeter@1033: else if (_gr.nodeNum() == 2) {
kpeter@1033: _path.push_back(_gr(0));
kpeter@1033: _path.push_back(_gr(1));
kpeter@1033: return _sum = 2 * _cost[_gr.edge(_gr(0), _gr(1))];
kpeter@1033: }
f4c3@1031:
kpeter@1033: // Compute min. cost spanning tree
kpeter@1033: std::vector<Edge> tree;
kpeter@1033: kruskal(_gr, _cost, std::back_inserter(tree));
f4c3@1031:
kpeter@1033: FullGraph::NodeMap<int> deg(_gr, 0);
kpeter@1033: for (int i = 0; i != int(tree.size()); ++i) {
kpeter@1033: Edge e = tree[i];
kpeter@1033: ++deg[_gr.u(e)];
kpeter@1033: ++deg[_gr.v(e)];
kpeter@1033: }
kpeter@1033:
kpeter@1033: // Copy the induced subgraph of odd nodes
kpeter@1033: std::vector<Node> odd_nodes;
kpeter@1033: for (NodeIt u(_gr); u != INVALID; ++u) {
kpeter@1033: if (deg[u] % 2 == 1) odd_nodes.push_back(u);
kpeter@1033: }
kpeter@1033:
kpeter@1033: SmartGraph sgr;
kpeter@1033: SmartGraph::EdgeMap<Cost> scost(sgr);
kpeter@1033: for (int i = 0; i != int(odd_nodes.size()); ++i) {
kpeter@1033: sgr.addNode();
kpeter@1033: }
kpeter@1033: for (int i = 0; i != int(odd_nodes.size()); ++i) {
kpeter@1033: for (int j = 0; j != int(odd_nodes.size()); ++j) {
kpeter@1033: if (j == i) continue;
kpeter@1033: SmartGraph::Edge e =
kpeter@1033: sgr.addEdge(sgr.nodeFromId(i), sgr.nodeFromId(j));
kpeter@1033: scost[e] = -_cost[_gr.edge(odd_nodes[i], odd_nodes[j])];
f4c3@1031: }
f4c3@1031: }
f4c3@1031:
kpeter@1033: // Compute min. cost perfect matching
kpeter@1033: MaxWeightedPerfectMatching<SmartGraph, SmartGraph::EdgeMap<Cost> >
kpeter@1033: mwpm(sgr, scost);
kpeter@1033: mwpm.run();
f4c3@1031:
kpeter@1033: for (SmartGraph::EdgeIt e(sgr); e != INVALID; ++e) {
kpeter@1033: if (mwpm.matching(e)) {
kpeter@1033: tree.push_back( _gr.edge(odd_nodes[sgr.id(sgr.u(e))],
kpeter@1033: odd_nodes[sgr.id(sgr.v(e))]) );
f4c3@1031: }
f4c3@1031: }
f4c3@1031:
kpeter@1033: // Join the spanning tree and the matching
kpeter@1033: sgr.clear();
kpeter@1033: for (int i = 0; i != _gr.nodeNum(); ++i) {
kpeter@1033: sgr.addNode();
kpeter@1033: }
kpeter@1033: for (int i = 0; i != int(tree.size()); ++i) {
kpeter@1033: int ui = _gr.id(_gr.u(tree[i])),
kpeter@1033: vi = _gr.id(_gr.v(tree[i]));
kpeter@1033: sgr.addEdge(sgr.nodeFromId(ui), sgr.nodeFromId(vi));
kpeter@1033: }
kpeter@1033:
kpeter@1033: // Compute the tour from the Euler traversal
kpeter@1033: SmartGraph::NodeMap<bool> visited(sgr, false);
kpeter@1033: for (EulerIt<SmartGraph> e(sgr); e != INVALID; ++e) {
kpeter@1033: SmartGraph::Node n = sgr.target(e);
kpeter@1033: if (!visited[n]) {
kpeter@1033: _path.push_back(_gr(sgr.id(n)));
kpeter@1033: visited[n] = true;
f4c3@1031: }
f4c3@1031: }
f4c3@1031:
kpeter@1033: _sum = _cost[_gr.edge(_path.back(), _path.front())];
kpeter@1033: for (int i = 0; i < int(_path.size())-1; ++i) {
kpeter@1033: _sum += _cost[_gr.edge(_path[i], _path[i+1])];
kpeter@1033: }
f4c3@1031:
f4c3@1031: return _sum;
f4c3@1031: }
f4c3@1031:
kpeter@1033: /// @}
f4c3@1031:
kpeter@1033: /// \name Query Functions
kpeter@1033: /// @{
f4c3@1031:
kpeter@1033: /// \brief The total cost of the found tour.
kpeter@1033: ///
kpeter@1033: /// This function returns the total cost of the found tour.
kpeter@1033: ///
kpeter@1033: /// \pre run() must be called before using this function.
kpeter@1033: Cost tourCost() const {
f4c3@1031: return _sum;
f4c3@1031: }
f4c3@1031:
kpeter@1033: /// \brief Returns a const reference to the node sequence of the
kpeter@1033: /// found tour.
kpeter@1033: ///
kpeter@1034: /// This function returns a const reference to a vector
kpeter@1033: /// that stores the node sequence of the found tour.
kpeter@1033: ///
kpeter@1033: /// \pre run() must be called before using this function.
kpeter@1033: const std::vector<Node>& tourNodes() const {
kpeter@1033: return _path;
kpeter@1033: }
f4c3@1031:
kpeter@1033: /// \brief Gives back the node sequence of the found tour.
kpeter@1033: ///
kpeter@1033: /// This function copies the node sequence of the found tour into
kpeter@1037: /// an STL container through the given output iterator. The
kpeter@1037: /// <tt>value_type</tt> of the container must be <tt>FullGraph::Node</tt>.
kpeter@1037: /// For example,
kpeter@1037: /// \code
kpeter@1037: /// std::vector<FullGraph::Node> nodes(countNodes(graph));
kpeter@1037: /// tsp.tourNodes(nodes.begin());
kpeter@1037: /// \endcode
kpeter@1037: /// or
kpeter@1037: /// \code
kpeter@1037: /// std::list<FullGraph::Node> nodes;
kpeter@1037: /// tsp.tourNodes(std::back_inserter(nodes));
kpeter@1037: /// \endcode
kpeter@1033: ///
kpeter@1033: /// \pre run() must be called before using this function.
kpeter@1037: template <typename Iterator>
kpeter@1037: void tourNodes(Iterator out) const {
kpeter@1037: std::copy(_path.begin(), _path.end(), out);
kpeter@1033: }
kpeter@1033:
kpeter@1033: /// \brief Gives back the found tour as a path.
kpeter@1033: ///
kpeter@1033: /// This function copies the found tour as a list of arcs/edges into
alpar@1076: /// the given \ref lemon::concepts::Path "path structure".
kpeter@1033: ///
kpeter@1033: /// \pre run() must be called before using this function.
kpeter@1033: template <typename Path>
kpeter@1033: void tour(Path &path) const {
kpeter@1033: path.clear();
kpeter@1033: for (int i = 0; i < int(_path.size()) - 1; ++i) {
kpeter@1033: path.addBack(_gr.arc(_path[i], _path[i+1]));
kpeter@1033: }
kpeter@1033: if (int(_path.size()) >= 2) {
kpeter@1033: path.addBack(_gr.arc(_path.back(), _path.front()));
kpeter@1033: }
kpeter@1033: }
kpeter@1033:
kpeter@1033: /// @}
kpeter@1033:
f4c3@1031: };
f4c3@1031:
f4c3@1031: }; // namespace lemon
f4c3@1031:
f4c3@1031: #endif