RhodeCode

    /// \brief Problem type constants for the \c run() function.

    ///

    /// Enum type containing the problem type constants that can be

    /// returned by the \ref run() function of the algorithm.

    enum ProblemType {

      /// The problem has no feasible solution (flow).

      INFEASIBLE,

      /// The problem has optimal solution (i.e. it is feasible and

      /// bounded), and the algorithm has found optimal flow and node

      /// potentials (primal and dual solutions).

      OPTIMAL,

      /// The digraph contains an arc of negative cost and infinite

      /// upper bound. It means that the objective function is unbounded

      /// on that arc, however note that it could actually be bounded

      /// over the feasible flows, but this algroithm cannot handle

      /// these cases.

      UNBOUNDED

    };

  private:

    TEMPLATE_DIGRAPH_TYPEDEFS(GR);

    typedef std::vector<int> IntVector;

    typedef std::vector<Value> ValueVector;

    typedef std::vector<Cost> CostVector;

  private:

    // Data related to the underlying digraph

    const GR &_graph;

    int _node_num;

    int _arc_num;

    int _res_arc_num;

    int _root;

    // Parameters of the problem

    bool _have_lower;

    Value _sum_supply;

    // Data structures for storing the digraph

    IntNodeMap _node_id;

    IntArcMap _arc_idf;

    IntArcMap _arc_idb;

    IntVector _first_out;

    BoolVector _forward;

    IntVector _source;

    IntVector _target;

    Value _delta;

    int _factor;

    IntVector _pred;

  public:

    /// \brief Constant for infinite upper bounds (capacities).

    ///

    /// Constant for infinite upper bounds (capacities).

    /// It is \c std::numeric_limits<Value>::infinity() if available,

    /// \c std::numeric_limits<Value>::max() otherwise.

    const Value INF;

  private:

    // Special implementation of the Dijkstra algorithm for finding

    // shortest paths in the residual network of the digraph with

    // respect to the reduced arc costs and modifying the node

    // potentials according to the found distance labels.

    class ResidualDijkstra

    {

    private:

      int _node_num;

      const IntVector &_first_out;

      const IntVector &_target;

      const CostVector &_cost;

      const ValueVector &_res_cap;

      const ValueVector &_excess;

      CostVector &_pi;

      IntVector &_pred;

      IntVector _proc_nodes;

      CostVector _dist;

    public:

      ResidualDijkstra(CapacityScaling& cs) :

      {}

      int run(int s, Value delta = 1) {

        RangeMap<int> heap_cross_ref(_node_num, Heap::PRE_HEAP);

        Heap heap(heap_cross_ref);

        heap.push(s, 0);

        _pred[s] = -1;

        _proc_nodes.clear();

        // Process nodes

        while (!heap.empty() && _excess[heap.top()] > -delta) {

          int u = heap.top(), v;

          Cost d = heap.prio() + _pi[u], dn;

          _dist[u] = heap.prio();

          _proc_nodes.push_back(u);

          heap.pop();

          // Traverse outgoing residual arcs

            if (_res_cap[a] < delta) continue;

            v = _target[a];

            switch (heap.state(v)) {

              case Heap::PRE_HEAP:

                heap.push(v, d + _cost[a] - _pi[v]);

                _pred[v] = a;

                break;

              case Heap::IN_HEAP:

                dn = d + _cost[a] - _pi[v];

                if (dn < heap[v]) {

                  heap.decrease(v, dn);

                  _pred[v] = a;

                }

                break;

              case Heap::POST_HEAP:

                break;

            }

          }

        }

        if (heap.empty()) return -1;

        // Update potentials of processed nodes

        int t = heap.top();

        Cost dt = heap.prio();

    ///

    /// \pre \ref run() must be called before using this function.

    template <typename PotentialMap>

    void potentialMap(PotentialMap &map) const {

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

        map.set(n, _pi[_node_id[n]]);

      }

    }

    /// @}

  private:

    // Initialize the algorithm

    ProblemType init() {

      if (_node_num == 0) return INFEASIBLE;

      // Check the sum of supply values

      _sum_supply = 0;

      for (int i = 0; i != _root; ++i) {

        _sum_supply += _supply[i];

      }

      if (_sum_supply > 0) return INFEASIBLE;

      for (int i = 0; i != _root; ++i) {

        _pi[i] = 0;

        _excess[i] = _supply[i];

      }

      // Remove non-zero lower bounds

      if (_have_lower) {

        for (int i = 0; i != _root; ++i) {

            if (_forward[j]) {

              Value c = _lower[j];

              if (c >= 0) {

              } else {

              }

              _excess[i] -= c;

              _excess[_target[j]] += c;

            } else {

              _res_cap[j] = 0;

            }

          }

        }

      } else {

        for (int j = 0; j != _res_arc_num; ++j) {

          _res_cap[j] = _forward[j] ? _upper[j] : 0;

        }

      }

      // Handle negative costs

          }

        }

      }

      // Handle GEQ supply type

      if (_sum_supply < 0) {

        _pi[_root] = 0;

        _excess[_root] = -_sum_supply;

        for (int a = _first_out[_root]; a != _res_arc_num; ++a) {

          _cost[a] = 0;

        }

      } else {

        _pi[_root] = 0;

        _excess[_root] = 0;

        for (int a = _first_out[_root]; a != _res_arc_num; ++a) {

          _res_cap[a] = 1;

          _cost[a] = 0;

        }

      }

      // Initialize delta value

      if (_factor > 1) {

        // With scaling

        Value max_sup = 0, max_dem = 0;

        for (int i = 0; i != _node_num; ++i) {

        }

        Value max_cap = 0;

        for (int j = 0; j != _res_arc_num; ++j) {

          if (_res_cap[j] > max_cap) max_cap = _res_cap[j];

        }

        max_sup = std::min(std::min(max_sup, max_dem), max_cap);

        for (_delta = 1; 2 * _delta <= max_sup; _delta *= 2) ;

      } else {

        // Without scaling

        _delta = 1;

      }

      return OPTIMAL;

    }

    ProblemType start() {

      // Execute the algorithm

      ProblemType pt;

      if (_delta > 1)

        pt = startWithScaling();

      else

        pt = startWithoutScaling();

      // Handle non-zero lower bounds

      if (_have_lower) {

          if (!_forward[j]) _res_cap[j] += _lower[j];

        }

      }

      // Shift potentials if necessary

      Cost pr = _pi[_root];

      if (_sum_supply < 0 || pr > 0) {

        for (int i = 0; i != _node_num; ++i) {

          _pi[i] -= pr;

        }        

      }

      return pt;

    }

    // Execute the capacity scaling algorithm

    ProblemType startWithScaling() {

      // Perform capacity scaling phases

      int s, t;

      ResidualDijkstra _dijkstra(*this);

      while (true) {

        // Saturate all arcs not satisfying the optimality condition

        for (int u = 0; u != _node_num; ++u) {

            int v = _target[a];

            Cost c = _cost[a] + _pi[u] - _pi[v];

            Value rc = _res_cap[a];

            if (c < 0 && rc >= _delta) {

              _excess[u] -= rc;

              _excess[v] += rc;

              _res_cap[a] = 0;

              _res_cap[_reverse[a]] += rc;

            }

          }

        }

        // Find excess nodes and deficit nodes

        _excess_nodes.clear();

        _deficit_nodes.clear();

        for (int u = 0; u != _node_num; ++u) {

        }

        int next_node = 0, next_def_node = 0;

        // Find augmenting shortest paths

        while (next_node < int(_excess_nodes.size())) {

          // Check deficit nodes

          if (_delta > 1) {

            bool delta_deficit = false;

            for ( ; next_def_node < int(_deficit_nodes.size());

                    ++next_def_node ) {

              if (_excess[_deficit_nodes[next_def_node]] <= -_delta) {

                delta_deficit = true;

                break;

              }

            }

            if (!delta_deficit) break;

          }

          // Run Dijkstra in the residual network

          s = _excess_nodes[next_node];

          if ((t = _dijkstra.run(s, _delta)) == -1) {

            if (_delta > 1) {

              ++next_node;

              continue;

      /// The \e Block \e Search pivot rule.

      /// A specified number of arcs are examined in every iteration

      /// in a wraparound fashion and the best eligible arc is selected

      /// from this block.

      BLOCK_SEARCH,

      /// The \e Candidate \e List pivot rule.

      /// In a major iteration a candidate list is built from eligible arcs

      /// in a wraparound fashion and in the following minor iterations

      /// the best eligible arc is selected from this list.

      CANDIDATE_LIST,

      /// The \e Altering \e Candidate \e List pivot rule.

      /// It is a modified version of the Candidate List method.

      /// It keeps only the several best eligible arcs from the former

      /// candidate list and extends this list in every iteration.

      ALTERING_LIST

    };

  private:

    TEMPLATE_DIGRAPH_TYPEDEFS(GR);

    typedef std::vector<int> IntVector;

    typedef std::vector<Value> ValueVector;

    typedef std::vector<Cost> CostVector;

    // State constants for arcs

    enum ArcStateEnum {

      STATE_UPPER = -1,

      STATE_TREE  =  0,

      STATE_LOWER =  1

    };

  private:

    // Data related to the underlying digraph

    const GR &_graph;

    int _node_num;

    int _arc_num;

    int _all_arc_num;

    int _search_arc_num;

    // Parameters of the problem

    bool _have_lower;

    SupplyType _stype;

    Value _sum_supply;

    // Data structures for storing the digraph

    IntNodeMap _node_id;

    IntArcMap _arc_id;

    IntVector _source;

    IntVector _target;

    // Node and arc data

    ValueVector _lower;

    ValueVector _upper;

    ValueVector _cap;

    CostVector _cost;

    ValueVector _supply;

    ValueVector _flow;

    CostVector _pi;

    // Data for storing the spanning tree structure

    IntVector _parent;

    IntVector _pred;

    IntVector _thread;

    IntVector _rev_thread;

    IntVector _succ_num;

    IntVector _last_succ;

    IntVector _dirty_revs;

    int _root;

    // Temporary data used in the current pivot iteration

    int in_arc, join, u_in, v_in, u_out, v_out;

    int first, second, right, last;

    int stem, par_stem, new_stem;

    Value delta;

  public:

    /// \brief Constant for infinite upper bounds (capacities).

    ///

    /// Constant for infinite upper bounds (capacities).

    /// It is \c std::numeric_limits<Value>::infinity() if available,

    /// \c std::numeric_limits<Value>::max() otherwise.

    const Value INF;

  private:

    // Implementation of the First Eligible pivot rule

    class FirstEligiblePivotRule

    {

    private:

      // References to the NetworkSimplex class

      const IntVector  &_source;

      const IntVector  &_target;

      const CostVector &_cost;

      const CostVector &_pi;

      int &_in_arc;

      int _search_arc_num;

      // Pivot rule data

      int _next_arc;

    public:

      // Constructor

      FirstEligiblePivotRule(NetworkSimplex &ns) :

        _source(ns._source), _target(ns._target),

        _cost(ns._cost), _state(ns._state), _pi(ns._pi),

        _in_arc(ns.in_arc), _search_arc_num(ns._search_arc_num),

        _next_arc(0)

      {}

      // Find next entering arc

      bool findEnteringArc() {

        Cost c;

        for (int e = _next_arc; e < _search_arc_num; ++e) {

          c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);

          if (c < 0) {

            _in_arc = e;

        }

        for (int e = 0; e < _next_arc; ++e) {

          c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);

          if (c < 0) {

            _in_arc = e;

            _next_arc = e + 1;

            return true;

          }

        }

        return false;

      }

    }; //class FirstEligiblePivotRule

    // Implementation of the Best Eligible pivot rule

    class BestEligiblePivotRule

    {

    private:

      // References to the NetworkSimplex class

      const IntVector  &_source;

      const IntVector  &_target;

      const CostVector &_cost;

      const CostVector &_pi;

      int &_in_arc;

      int _search_arc_num;

    public:

      // Constructor

      BestEligiblePivotRule(NetworkSimplex &ns) :

        _source(ns._source), _target(ns._target),

        _cost(ns._cost), _state(ns._state), _pi(ns._pi),

        _in_arc(ns.in_arc), _search_arc_num(ns._search_arc_num)

      {}

      // Find next entering arc

      bool findEnteringArc() {

        Cost c, min = 0;

        for (int e = 0; e < _search_arc_num; ++e) {

          c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);

          if (c < min) {

            min = c;

            _in_arc = e;

          }

        }

        return min < 0;

      }

    }; //class BestEligiblePivotRule

    // Implementation of the Block Search pivot rule

    class BlockSearchPivotRule

    {

    private:

      // References to the NetworkSimplex class

      const IntVector  &_source;

      const IntVector  &_target;

      const CostVector &_cost;

      const CostVector &_pi;

      int &_in_arc;

      int _search_arc_num;

      // Pivot rule data

      int _block_size;

      int _next_arc;

    public:

      // Constructor

      BlockSearchPivotRule(NetworkSimplex &ns) :

        _source(ns._source), _target(ns._target),

        _cost(ns._cost), _state(ns._state), _pi(ns._pi),

        _in_arc(ns.in_arc), _search_arc_num(ns._search_arc_num),

        _next_arc(0)

      {

        // The main parameters of the pivot rule

        const double BLOCK_SIZE_FACTOR = 0.5;

        const int MIN_BLOCK_SIZE = 10;

        _block_size = std::max( int(BLOCK_SIZE_FACTOR *

                                    std::sqrt(double(_search_arc_num))),

                                MIN_BLOCK_SIZE );

          if (--cnt == 0) {

            if (min < 0) goto search_end;

            cnt = _block_size;

          }

        }

        if (min >= 0) return false;

      search_end:

        _next_arc = e;

        return true;

      }

    }; //class BlockSearchPivotRule

    // Implementation of the Candidate List pivot rule

    class CandidateListPivotRule

    {

    private:

      // References to the NetworkSimplex class

      const IntVector  &_source;

      const IntVector  &_target;

      const CostVector &_cost;

      const CostVector &_pi;

      int &_in_arc;

      int _search_arc_num;

      // Pivot rule data

      IntVector _candidates;

      int _list_length, _minor_limit;

      int _curr_length, _minor_count;

      int _next_arc;

    public:

      /// Constructor

      CandidateListPivotRule(NetworkSimplex &ns) :

        _source(ns._source), _target(ns._target),

        _cost(ns._cost), _state(ns._state), _pi(ns._pi),

        _in_arc(ns.in_arc), _search_arc_num(ns._search_arc_num),

        _next_arc(0)

      {

        // The main parameters of the pivot rule

        const double LIST_LENGTH_FACTOR = 0.25;

        const int MIN_LIST_LENGTH = 10;

        const double MINOR_LIMIT_FACTOR = 0.1;

        const int MIN_MINOR_LIMIT = 3;

            }

            if (_curr_length == _list_length) goto search_end;

          }

        }

        if (_curr_length == 0) return false;

      search_end:        

        _minor_count = 1;

        _next_arc = e;

        return true;

      }

    }; //class CandidateListPivotRule

    // Implementation of the Altering Candidate List pivot rule

    class AlteringListPivotRule

    {

    private:

      // References to the NetworkSimplex class

      const IntVector  &_source;

      const IntVector  &_target;

      const CostVector &_cost;

      const CostVector &_pi;

      int &_in_arc;

      int _search_arc_num;

      // Pivot rule data

      int _block_size, _head_length, _curr_length;

      int _next_arc;

      IntVector _candidates;

      CostVector _cand_cost;

      // Functor class to compare arcs during sort of the candidate list

      class SortFunc

      {

      private:

        const CostVector &_map;

      public:

        SortFunc(const CostVector &map) : _map(map) {}

        bool operator()(int left, int right) {

          return _map[left] > _map[right];

        }

      };

      SortFunc _sort_func;

        // Pop the first element of the heap

        _in_arc = _candidates[0];

        _next_arc = e;

        pop_heap( _candidates.begin(), _candidates.begin() + _curr_length,

                  _sort_func );

        _curr_length = std::min(_head_length, _curr_length - 1);

        return true;

      }

    }; //class AlteringListPivotRule

  public:

    /// \brief Constructor.

    ///

    /// The constructor of the class.

    ///

    /// \param graph The digraph the algorithm runs on.

    /// \param arc_mixing Indicate if the arcs have to be stored in a

    /// mixed order in the internal data structure. 

    /// In special cases, it could lead to better overall performance,

    /// but it is usually slower. Therefore it is disabled by default.

    NetworkSimplex(const GR& graph, bool arc_mixing = false) :

      _graph(graph), _node_id(graph), _arc_id(graph),

      INF(std::numeric_limits<Value>::has_infinity ?

    {

      // Check the value types

      LEMON_ASSERT(std::numeric_limits<Value>::is_signed,

        "The flow type of NetworkSimplex must be signed");

      LEMON_ASSERT(std::numeric_limits<Cost>::is_signed,

        "The cost type of NetworkSimplex must be signed");

      // Resize vectors

      _node_num = countNodes(_graph);

      _arc_num = countArcs(_graph);

      int all_node_num = _node_num + 1;

      int max_arc_num = _arc_num + 2 * _node_num;

      _source.resize(max_arc_num);

      _target.resize(max_arc_num);

      _lower.resize(_arc_num);

      _upper.resize(_arc_num);

      _cap.resize(max_arc_num);

      _cost.resize(max_arc_num);

      _supply.resize(all_node_num);

      _flow.resize(max_arc_num);

      _pi.resize(all_node_num);

      }

    }

    /// @}

  private:

    // Initialize internal data structures

    bool init() {

      if (_node_num == 0) return false;

      // Check the sum of supply values

      _sum_supply = 0;

      for (int i = 0; i != _node_num; ++i) {

        _sum_supply += _supply[i];

      }

      if ( !((_stype == GEQ && _sum_supply <= 0) ||

             (_stype == LEQ && _sum_supply >= 0)) ) return false;

      // Remove non-zero lower bounds

      if (_have_lower) {

        for (int i = 0; i != _arc_num; ++i) {

          Value c = _lower[i];

          if (c >= 0) {

          } else {

          }

          _supply[_source[i]] -= c;

          _supply[_target[i]] += c;

        }

      } else {

        for (int i = 0; i != _arc_num; ++i) {

          _cap[i] = _upper[i];

        }

      }

      // Initialize artifical cost

      Cost ART_COST;

      if (std::numeric_limits<Cost>::is_exact) {

        ART_COST = std::numeric_limits<Cost>::max() / 2 + 1;

      } else {

        ART_COST = std::numeric_limits<Cost>::min();

        for (int i = 0; i != _arc_num; ++i) {

          if (_cost[i] > ART_COST) ART_COST = _cost[i];

        }

        ART_COST = (ART_COST + 1) * _node_num;

      }

      // Initialize arc maps

      for (int i = 0; i != _arc_num; ++i) {

      join = u;

    }

    // Find the leaving arc of the cycle and returns true if the

    // leaving arc is not the same as the entering arc

    bool findLeavingArc() {

      // Initialize first and second nodes according to the direction

      // of the cycle

      if (_state[in_arc] == STATE_LOWER) {

        first  = _source[in_arc];

        second = _target[in_arc];

      } else {

        first  = _target[in_arc];

        second = _source[in_arc];

      }

      delta = _cap[in_arc];

      int result = 0;

      Value d;

      int e;

      // Search the cycle along the path form the first node to the root

      for (int u = first; u != join; u = _parent[u]) {

        e = _pred[u];

        d = _forward[u] ?

        if (d < delta) {

          delta = d;

          u_out = u;

          result = 1;

        }

      }

      // Search the cycle along the path form the second node to the root

      for (int u = second; u != join; u = _parent[u]) {

        e = _pred[u];

        d = _forward[u] ? 

        if (d <= delta) {

          delta = d;

          u_out = u;

          result = 2;

        }

      }

      if (result == 1) {

        u_in = first;

        v_in = second;

      } else {

        u_in = second;

        v_in = first;

      }

      return result != 0;

    }

    // Change _flow and _state vectors

    void changeFlow(bool change) {

      // Augment along the cycle

      if (delta > 0) {

        Value val = _state[in_arc] * delta;

        _flow[in_arc] += val;

        for (int u = _source[in_arc]; u != join; u = _parent[u]) {

      // Select the pivot rule implementation

      switch (pivot_rule) {

        case FIRST_ELIGIBLE:

          return start<FirstEligiblePivotRule>();

        case BEST_ELIGIBLE:

          return start<BestEligiblePivotRule>();

        case BLOCK_SEARCH:

          return start<BlockSearchPivotRule>();

        case CANDIDATE_LIST:

          return start<CandidateListPivotRule>();

        case ALTERING_LIST:

          return start<AlteringListPivotRule>();

      }

      return INFEASIBLE; // avoid warning

    }

    template <typename PivotRuleImpl>

    ProblemType start() {

      PivotRuleImpl pivot(*this);

      // Execute the Network Simplex algorithm

      while (pivot.findEnteringArc()) {

        findJoinNode();

        bool change = findLeavingArc();

        changeFlow(change);

        if (change) {

          updateTreeStructure();

          updatePotential();

        }

      }

      // Check feasibility

      for (int e = _search_arc_num; e != _all_arc_num; ++e) {

        if (_flow[e] != 0) return INFEASIBLE;

      }

      // Transform the solution and the supply map to the original form

      if (_have_lower) {

        for (int i = 0; i != _arc_num; ++i) {

          Value c = _lower[i];

          if (c != 0) {

            _flow[i] += c;

            _supply[_source[i]] += c;

            _supply[_target[i]] -= c;

          }

        }

      }