RhodeCode

 * (Egervary Research Group on Combinatorial Optimization, EGRES).

 *

 * Permission to use, modify and distribute this software is granted

 * provided that this copyright notice appears in all copies. For

 * precise terms see the accompanying LICENSE file.

 *

 * This software is provided "AS IS" with no warranty of any kind,

 * express or implied, and with no claim as to its suitability for any

 * purpose.

 *

 */

#include <vector>

#include <queue>

#include <set>

#include <limits>

#include <lemon/core.h>

#include <lemon/unionfind.h>

#include <lemon/bin_heap.h>

#include <lemon/maps.h>

///\ingroup matching

///\file

///\brief Maximum matching algorithms in general graphs.

namespace lemon {

  /// \ingroup matching

  ///

  /// \brief Maximum cardinality matching in general graphs

  ///

  /// This class implements Edmonds' alternating forest matching algorithm

  /// for finding a maximum cardinality matching in a general undirected graph.

  /// (the default is the empty one).

  ///

  /// The dual solution of the problem is a map of the nodes to

  /// \ref MaxMatching::Status "Status", having values \c EVEN (or \c D),

  /// \c ODD (or \c A) and \c MATCHED (or \c C) defining the Gallai-Edmonds

  /// decomposition of the graph. The nodes in \c EVEN/D induce a subgraph

  /// with factor-critical components, the nodes in \c ODD/A form the

  /// canonical barrier, and the nodes in \c MATCHED/C induce a graph having

  /// a perfect matching. The number of the factor-critical components

  /// minus the number of barrier nodes is a lower bound on the

  /// unmatched nodes, and the matching is optimal if and only if this bound is

  /// tight. This decomposition can be obtained using \ref status() or

  template <typename GR>

  class MaxMatching {

  public:

    /// The graph type of the algorithm

    typedef GR Graph;

    /// The type of the matching map

    typedef typename Graph::template NodeMap<typename Graph::Arc>

    MatchingMap;

    ///\brief Status constants for Gallai-Edmonds decomposition.

    ///

    ///decomposition of a graph. The nodes with status \c EVEN (or \c D)

    ///induce a subgraph with factor-critical components, the nodes with

    ///status \c ODD (or \c A) form the canonical barrier, and the nodes

    ///perfect matching.

    enum Status {

      EVEN = 1,       ///< = 1. (\c D is an alias for \c EVEN.)

      D = 1,

      MATCHED = 0,    ///< = 0. (\c C is an alias for \c MATCHED.)

      C = 0,

      ODD = -1,       ///< = -1. (\c A is an alias for \c ODD.)

      A = -1,

      UNMATCHED = -2  ///< = -2.

    };

    /// The type of the status map

    /// called before using this function.

    void startSparse() {

      for(NodeIt n(_graph); n != INVALID; ++n) {

        if ((*_status)[n] == UNMATCHED) {

          (*_blossom_rep)[_blossom_set->insert(n)] = n;

          _tree_set->insert(n);

          (*_status)[n] = EVEN;

          processSparse(n);

        }

      }

    }

    /// for dense graphs

    ///

    /// This function runs Edmonds' algorithm with a heuristic of postponing

    /// shrinks, therefore resulting in a faster algorithm for dense graphs.

    ///

    /// \pre \ref init(), \ref greedyInit() or \ref matchingInit() must be

    /// called before using this function.

    void startDense() {

      for(NodeIt n(_graph); n != INVALID; ++n) {

        if ((*_status)[n] == UNMATCHED) {

          (*_blossom_rep)[_blossom_set->insert(n)] = n;

          _tree_set->insert(n);

          (*_status)[n] = EVEN;

          processDense(n);

        }

      }

    }

    /// \brief Run Edmonds' algorithm

    ///

    /// (for which <tt>m>=2*n</tt> holds).

    void run() {

      if (countEdges(_graph) < 2 * countNodes(_graph)) {

        greedyInit();

        startSparse();

      } else {

        init();

        startDense();

      }

    }

    /// @}

    /// \name Primal Solution

    /// Functions to get the primal solution, i.e. the maximum matching.

    /// @{

    /// \brief Return the size (cardinality) of the matching.

    ///

    /// After run() it returns the size of the maximum matching in the graph.

    int matchingSize() const {

      int size = 0;

      for (NodeIt n(_graph); n != INVALID; ++n) {

        if ((*_matching)[n] != INVALID) {

          ++size;

        }

      }

      return size / 2;

    }

    /// \brief Return \c true if the given edge is in the matching.

    ///

    /// matching.

    bool matching(const Edge& edge) const {

      return edge == (*_matching)[_graph.u(edge)];

    }

    /// \brief Return the matching arc (or edge) incident to the given node.

    ///

    /// This function returns the matching arc (or edge) incident to the

    /// not covered by the matching.

    Arc matching(const Node& n) const {

      return (*_matching)[n];

    }

    /// \brief Return a const reference to the matching map.

    ///

    /// This function returns a const reference to a node map that stores

    /// the matching arc (or edge) incident to each node.

    const MatchingMap& matchingMap() const {

      return *_matching;

    }

    /// \brief Return the mate of the given node.

    ///

    /// matching or \c INVALID if the node is not covered by the matching.

    Node mate(const Node& n) const {

      return (*_matching)[n] != INVALID ?

        _graph.target((*_matching)[n]) : INVALID;

    }

    /// @}

    /// \name Dual Solution

    /// decomposition.

    /// @{

    /// \brief Return the status of the given node in the Edmonds-Gallai

    /// decomposition.

    ///

    /// This function returns the \ref Status "status" of the given node

    /// in the Edmonds-Gallai decomposition.

    Status status(const Node& n) const {

      return (*_status)[n];

    }

  };

  /// \ingroup matching

  ///

  /// \brief Weighted matching in general graphs

  ///

  /// This class provides an efficient implementation of Edmond's

  /// maximum weighted matching algorithm. The implementation is based

  /// on extensive use of priority queues and provides

  /// \f$O(nm\log n)\f$ time complexity.

  ///

  /// each node has at most one incident edge.

  /// It can be formulated with the following linear program.

  /// \f[ \sum_{e \in \delta(u)}x_e \le 1 \quad \forall u\in V\f]

  /** \f[ \sum_{e \in \gamma(B)}x_e \le \frac{\vert B \vert - 1}{2}

      \quad \forall B\in\mathcal{O}\f] */

  /// \f[x_e \ge 0\quad \forall e\in E\f]

  /// \f[\max \sum_{e\in E}x_ew_e\f]

  /// where \f$\delta(X)\f$ is the set of edges incident to a node in

  /// \f$X\f$, \f$\gamma(X)\f$ is the set of edges with both ends in

  /// \f$X\f$ and \f$\mathcal{O}\f$ is the set of odd cardinality

  /// subsets of the nodes.

  ///

  /// The algorithm calculates an optimal matching and a proof of the

  /// optimality. The solution of the dual problem can be used to check

  /// the result of the algorithm. The dual linear problem is the

  /// following.

  /** \f[ y_u + y_v + \sum_{B \in \mathcal{O}, uv \in \gamma(B)}

      z_B \ge w_{uv} \quad \forall uv\in E\f] */

  /// \f[y_u \ge 0 \quad \forall u \in V\f]

  /// \f[z_B \ge 0 \quad \forall B \in \mathcal{O}\f]

  /** \f[\min \sum_{u \in V}y_u + \sum_{B \in \mathcal{O}}

      \frac{\vert B \vert - 1}{2}z_B\f] */

  ///

  /// After it the matching (the primal solution) and the dual solution

  /// If the value type is integer, then the dual solution is multiplied

  /// by \ref MaxWeightedMatching::dualScale "4".

  ///

  /// \tparam GR The undirected graph type the algorithm runs on.

  /// \ref concepts::Graph::EdgeMap "GR::EdgeMap<int>".

#ifdef DOXYGEN

  template <typename GR, typename WM>

#else

  template <typename GR,

            typename WM = typename GR::template EdgeMap<int> >

#endif

  class MaxWeightedMatching {

  public:

    /// The graph type of the algorithm

    typedef GR Graph;

    NodePotential* _node_potential;

    BlossomPotential _blossom_potential;

    BlossomNodeList _blossom_node_list;

    int _node_num;

    int _blossom_num;

    typedef RangeMap<int> IntIntMap;

    enum Status {

    };

    typedef HeapUnionFind<Value, IntNodeMap> BlossomSet;

    struct BlossomData {

      int tree;

      Status status;

      Arc pred, next;

      Value pot, offset;

      Node base;

    };

    IntNodeMap *_blossom_index;

      }

      if (!_delta3) {

        _delta3_index = new IntEdgeMap(_graph);

        _delta3 = new BinHeap<Value, IntEdgeMap>(*_delta3_index);

      }

      if (!_delta4) {

        _delta4_index = new IntIntMap(_blossom_num);

        _delta4 = new BinHeap<Value, IntIntMap>(*_delta4_index);

      }

    }

    void destroyStructures() {

      if (_matching) {

        delete _matching;

      }

      if (_node_potential) {

        delete _node_potential;

      }

      if (_blossom_set) {

        delete _blossom_index;

        delete _blossom_set;

        delete _blossom_data;

      }

        for (InArcIt e(_graph, n); e != INVALID; ++e) {

          Node v = _graph.source(e);

          int vb = _blossom_set->find(v);

          int vi = (*_node_index)[v];

          Value rw = (*_node_data)[ni].pot + (*_node_data)[vi].pot -

            dualScale * _weight[e];

          if ((*_blossom_data)[vb].status == EVEN) {

            if (_delta3->state(e) != _delta3->IN_HEAP && blossom != vb) {

              _delta3->push(e, rw / 2);

            }

          } else {

            typename std::map<int, Arc>::iterator it =

              (*_node_data)[vi].heap_index.find(tree);

            if (it != (*_node_data)[vi].heap_index.end()) {

              if ((*_node_data)[vi].heap[it->second] > rw) {

                (*_node_data)[vi].heap.replace(it->second, e);

                (*_node_data)[vi].heap.decrease(e, rw);

                it->second = e;

              }

            } else {

              (*_node_data)[vi].heap.push(e, rw);

              (*_node_data)[vi].heap_index.insert(std::make_pair(tree, e));

            }

            if ((*_blossom_set)[v] > (*_node_data)[vi].heap.prio()) {

              _blossom_set->decrease(v, (*_node_data)[vi].heap.prio());

              if ((*_blossom_data)[vb].status == MATCHED) {

                if (_delta2->state(vb) != _delta2->IN_HEAP) {

                  _delta2->push(vb, _blossom_set->classPrio(vb) -

                               (*_blossom_data)[vb].offset);

                } else if ((*_delta2)[vb] > _blossom_set->classPrio(vb) -

                           (*_blossom_data)[vb].offset) {

                  _delta2->decrease(vb, _blossom_set->classPrio(vb) -

                                   (*_blossom_data)[vb].offset);

                }

              }

            }

          }

        }

      }

      (*_blossom_data)[blossom].offset = 0;

    }

      if (_delta2->state(blossom) == _delta2->IN_HEAP) {

        _delta2->erase(blossom);

      }

      for (typename BlossomSet::ItemIt n(*_blossom_set, blossom);

           n != INVALID; ++n) {

        int ni = (*_node_index)[n];

          int vb = _blossom_set->find(v);

          int vi = (*_node_index)[v];

          Value rw = (*_node_data)[ni].pot + (*_node_data)[vi].pot -

            dualScale * _weight[e];

            }

          }

        }

      }

    }

      for (typename BlossomSet::ItemIt n(*_blossom_set, blossom);

           n != INVALID; ++n) {

        int ni = (*_node_index)[n];

        for (InArcIt e(_graph, n); e != INVALID; ++e) {

          Node v = _graph.source(e);

          int vb = _blossom_set->find(v);

          int vi = (*_node_index)[v];

          Value rw = (*_node_data)[ni].pot + (*_node_data)[vi].pot -

            dualScale * _weight[e];

            }

            typename std::map<int, Arc>::iterator it =

              }

            } else {

            }

              }

            }

          }

        }

      }

    }

    void alternatePath(int even, int tree) {

      int odd;

      evenToMatched(even, tree);

      (*_blossom_data)[even].status = MATCHED;

      while ((*_blossom_data)[even].pred != INVALID) {

        odd = _blossom_set->find(_graph.target((*_blossom_data)[even].pred));

        (*_blossom_data)[odd].status = MATCHED;

        oddToMatched(odd);

      }

      _tree_set->eraseClass(tree);

    }

    void unmatchNode(const Node& node) {

      int blossom = _blossom_set->find(node);

      int tree = _tree_set->find(blossom);

      alternatePath(blossom, tree);

      destroyTree(tree);

      (*_blossom_data)[blossom].base = node;

    }

    void augmentOnEdge(const Edge& edge) {

      int left = _blossom_set->find(_graph.u(edge));

      int right = _blossom_set->find(_graph.v(edge));

      (*_blossom_data)[left].next = _graph.direct(edge, true);

      (*_blossom_data)[right].next = _graph.direct(edge, false);

    }

    void extendOnArc(const Arc& arc) {

      int base = _blossom_set->find(_graph.target(arc));

      int tree = _tree_set->find(base);

      int odd = _blossom_set->find(_graph.source(arc));

      _tree_set->insert(odd, tree);

      (*_blossom_data)[odd].status = ODD;

      matchedToOdd(odd);

      (*_blossom_data)[odd].pred = arc;

      int even = _blossom_set->find(_graph.target((*_blossom_data)[odd].next));

      (*_blossom_data)[even].pred = (*_blossom_data)[even].next;

        }

        for (int i = ib; i != id; i = (i + 2) % subblossoms.size()) {

          int sb = subblossoms[i];

          int tb = subblossoms[(i + 1) % subblossoms.size()];

          int ub = subblossoms[(i + 2) % subblossoms.size()];

          (*_blossom_data)[sb].status = ODD;

          matchedToOdd(sb);

          _tree_set->insert(sb, tree);

          (*_blossom_data)[sb].pred = pred;

          (*_blossom_data)[sb].next =

          pred = (*_blossom_data)[ub].next;

          (*_blossom_data)[tb].status = EVEN;

          matchedToEven(tb, tree);

          _tree_set->insert(tb, tree);

          (*_blossom_data)[tb].pred = (*_blossom_data)[tb].next;

        }

        (*_blossom_data)[subblossoms[id]].status = ODD;

        matchedToOdd(subblossoms[id]);

        _tree_set->insert(subblossoms[id], tree);

        _blossom_potential.push_back(BlossomVariable(bn, en, pot));

      }

    }

    void extractMatching() {

      std::vector<int> blossoms;

      for (typename BlossomSet::ClassIt c(*_blossom_set); c != INVALID; ++c) {

        blossoms.push_back(c);

      }

      for (int i = 0; i < int(blossoms.size()); ++i) {

          Value offset = (*_blossom_data)[blossoms[i]].offset;

          (*_blossom_data)[blossoms[i]].pot += 2 * offset;

          for (typename BlossomSet::ItemIt n(*_blossom_set, blossoms[i]);

               n != INVALID; ++n) {

            (*_node_data)[(*_node_index)[n]].pot -= offset;

          }

          Arc matching = (*_blossom_data)[blossoms[i]].next;

          Node base = _graph.source(matching);

          extractBlossom(blossoms[i], base, matching);

        } else {

        Value d1 = !_delta1->empty() ?

          _delta1->prio() : std::numeric_limits<Value>::max();

        Value d2 = !_delta2->empty() ?

          _delta2->prio() : std::numeric_limits<Value>::max();

        Value d3 = !_delta3->empty() ?

          _delta3->prio() : std::numeric_limits<Value>::max();

        Value d4 = !_delta4->empty() ?

          _delta4->prio() : std::numeric_limits<Value>::max();

        if (d2 < _delta_sum) { _delta_sum = d2; ot = D2; }

        if (d4 < _delta_sum) { _delta_sum = d4; ot = D4; }

        switch (ot) {

        case D1:

          {

            Node n = _delta1->top();

            unmatchNode(n);

            --unmatched;

          }

          break;

        case D2:

          {

            int blossom = _delta2->top();

            Node n = _blossom_set->classTop(blossom);

          }

          break;

        case D3:

          {

            Edge e = _delta3->top();

            int left_blossom = _blossom_set->find(_graph.u(e));

            int right_blossom = _blossom_set->find(_graph.v(e));

            if (left_blossom == right_blossom) {

              _delta3->pop();

            } else {

              if (left_tree == right_tree) {

                shrinkOnEdge(e, left_tree);

              } else {

                augmentOnEdge(e);

                unmatched -= 2;

              }

            }

          } break;

        case D4:

          splitBlossom(_delta4->top());

          break;

    /// \code

    ///   mwm.init();

    ///   mwm.start();

    /// \endcode

    void run() {

      init();

      start();

    }

    /// @}

    /// \name Primal Solution

    /// matching.\n

    /// Either \ref run() or \ref start() function should be called before

    /// using them.

    /// @{

    /// \brief Return the weight of the matching.

    ///

    /// This function returns the weight of the found matching.

    ///

    /// \pre Either run() or start() must be called before using this function.

    Value matchingWeight() const {

      Value sum = 0;

      for (NodeIt n(_graph); n != INVALID; ++n) {

        if ((*_matching)[n] != INVALID) {

          sum += _weight[(*_matching)[n]];

        }

      }

    }

    /// \brief Return the size (cardinality) of the matching.

    ///

    /// This function returns the size (cardinality) of the found matching.

    ///

    /// \pre Either run() or start() must be called before using this function.

    int matchingSize() const {

      int num = 0;

      for (NodeIt n(_graph); n != INVALID; ++n) {

        if ((*_matching)[n] != INVALID) {

          ++num;

        }

      }

      return num /= 2;

    }

    /// \brief Return \c true if the given edge is in the matching.

    ///

    /// matching.

    ///

    /// \pre Either run() or start() must be called before using this function.

    bool matching(const Edge& edge) const {

      return edge == (*_matching)[_graph.u(edge)];