/* -*- mode: C++; indent-tabs-mode: nil; -*-

 *

 * This file is a part of LEMON, a generic C++ optimization library.

 *

 * Copyright (C) 2003-2013

 * Egervary Jeno Kombinatorikus Optimalizalasi Kutatocsoport

 * (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.

 *

 */

#ifndef LEMON_HAO_ORLIN_H

#define LEMON_HAO_ORLIN_H

#include <vector>

#include <list>

#include <limits>

#include <lemon/maps.h>

#include <lemon/core.h>

#include <lemon/tolerance.h>

/// \file

/// \ingroup min_cut

/// \brief Implementation of the Hao-Orlin algorithm.

///

/// Implementation of the Hao-Orlin algorithm for finding a minimum cut

/// in a digraph.

namespace lemon {

  /// \ingroup min_cut

  ///

  /// \brief Hao-Orlin algorithm for finding a minimum cut in a digraph.

  ///

  /// This class implements the Hao-Orlin algorithm for finding a minimum

  /// value cut in a directed graph \f$D=(V,A)\f$.

  /// It takes a fixed node \f$ source \in V \f$ and

  /// consists of two phases: in the first phase it determines a

  /// minimum cut with \f$ source \f$ on the source-side (i.e. a set

  /// \f$ X\subsetneq V \f$ with \f$ source \in X \f$ and minimal outgoing

  /// capacity) and in the second phase it determines a minimum cut

  /// with \f$ source \f$ on the sink-side (i.e. a set

  /// \f$ X\subsetneq V \f$ with \f$ source \notin X \f$ and minimal outgoing

  /// capacity). Obviously, the smaller of these two cuts will be a

  /// minimum cut of \f$ D \f$. The algorithm is a modified

  /// preflow push-relabel algorithm. Our implementation calculates

  /// the minimum cut in \f$ O(n^2\sqrt{m}) \f$ time (we use the

  /// highest-label rule), or in \f$O(nm)\f$ for unit capacities. A notable

  /// use of this algorithm is testing network reliability.

  ///

  /// For an undirected graph you can run just the first phase of the

  /// algorithm or you can use the algorithm of Nagamochi and Ibaraki,

  /// which solves the undirected problem in \f$ O(nm + n^2 \log n) \f$

  /// time. It is implemented in the NagamochiIbaraki algorithm class.

  ///

  /// \tparam GR The type of the digraph the algorithm runs on.

  /// \tparam CAP The type of the arc map containing the capacities,

  /// which can be any numreric type. The default map type is

  /// \ref concepts::Digraph::ArcMap "GR::ArcMap<int>".

  /// \tparam TOL Tolerance class for handling inexact computations. The

  /// default tolerance type is \ref Tolerance "Tolerance<CAP::Value>".

#ifdef DOXYGEN

  template <typename GR, typename CAP, typename TOL>

#else

  template <typename GR,

            typename CAP = typename GR::template ArcMap<int>,

            typename TOL = Tolerance<typename CAP::Value> >

#endif

  class HaoOrlin {

  public:

    /// The digraph type of the algorithm

    typedef GR Digraph;

    /// The capacity map type of the algorithm

    typedef CAP CapacityMap;

    /// The tolerance type of the algorithm

    typedef TOL Tolerance;

  private:

    typedef typename CapacityMap::Value Value;

    TEMPLATE_DIGRAPH_TYPEDEFS(Digraph);

    const Digraph& _graph;

    const CapacityMap* _capacity;

    typedef typename Digraph::template ArcMap<Value> FlowMap;

    FlowMap* _flow;

    Node _source;

    int _node_num;

    // Bucketing structure

    std::vector<Node> _first, _last;

    typename Digraph::template NodeMap<Node>* _next;

    typename Digraph::template NodeMap<Node>* _prev;

    typename Digraph::template NodeMap<bool>* _active;

    typename Digraph::template NodeMap<int>* _bucket;

    std::vector<bool> _dormant;

    std::list<std::list<int> > _sets;

    std::list<int>::iterator _highest;

    typedef typename Digraph::template NodeMap<Value> ExcessMap;

    ExcessMap* _excess;

    typedef typename Digraph::template NodeMap<bool> SourceSetMap;

    SourceSetMap* _source_set;

    Value _min_cut;

    typedef typename Digraph::template NodeMap<bool> MinCutMap;

    MinCutMap* _min_cut_map;

    Tolerance _tolerance;

  public:

    /// \brief Constructor

    ///

    /// Constructor of the algorithm class.

    HaoOrlin(const Digraph& graph, const CapacityMap& capacity,

             const Tolerance& tolerance = Tolerance()) :

      _graph(graph), _capacity(&capacity), _flow(0), _source(),

      _node_num(), _first(), _last(), _next(0), _prev(0),

      _active(0), _bucket(0), _dormant(), _sets(), _highest(),

      _excess(0), _source_set(0), _min_cut(), _min_cut_map(0),

      _tolerance(tolerance) {}

    ~HaoOrlin() {

      if (_min_cut_map) {

        delete _min_cut_map;

      }

      if (_source_set) {

        delete _source_set;

      }

      if (_excess) {

        delete _excess;

      }

      if (_next) {

        delete _next;

      }

      if (_prev) {

        delete _prev;

      }

      if (_active) {

        delete _active;

      }

      if (_bucket) {

        delete _bucket;

      }

      if (_flow) {

        delete _flow;

      }

    }

    /// \brief Set the tolerance used by the algorithm.

    ///

    /// This function sets the tolerance object used by the algorithm.

    /// \return <tt>(*this)</tt>

    HaoOrlin& tolerance(const Tolerance& tolerance) {

      _tolerance = tolerance;

      return *this;

    }

    /// \brief Returns a const reference to the tolerance.

    ///

    /// This function returns a const reference to the tolerance object

    /// used by the algorithm.

    const Tolerance& tolerance() const {

      return _tolerance;

    }

  private:

    void activate(const Node& i) {

      (*_active)[i] = true;

      int bucket = (*_bucket)[i];

      if ((*_prev)[i] == INVALID || (*_active)[(*_prev)[i]]) return;

      //unlace

      (*_next)[(*_prev)[i]] = (*_next)[i];

      if ((*_next)[i] != INVALID) {

        (*_prev)[(*_next)[i]] = (*_prev)[i];

      } else {

        _last[bucket] = (*_prev)[i];

      }

      //lace

      (*_next)[i] = _first[bucket];

      (*_prev)[_first[bucket]] = i;

      (*_prev)[i] = INVALID;

      _first[bucket] = i;

    }

    void deactivate(const Node& i) {

      (*_active)[i] = false;

      int bucket = (*_bucket)[i];

      if ((*_next)[i] == INVALID || !(*_active)[(*_next)[i]]) return;

      //unlace

      (*_prev)[(*_next)[i]] = (*_prev)[i];

      if ((*_prev)[i] != INVALID) {

        (*_next)[(*_prev)[i]] = (*_next)[i];

      } else {

        _first[bucket] = (*_next)[i];

      }

      //lace

      (*_prev)[i] = _last[bucket];

      (*_next)[_last[bucket]] = i;

      (*_next)[i] = INVALID;

      _last[bucket] = i;

    }

    void addItem(const Node& i, int bucket) {

      (*_bucket)[i] = bucket;

      if (_last[bucket] != INVALID) {

        (*_prev)[i] = _last[bucket];

        (*_next)[_last[bucket]] = i;

        (*_next)[i] = INVALID;

        _last[bucket] = i;

      } else {

        (*_prev)[i] = INVALID;

        _first[bucket] = i;

        (*_next)[i] = INVALID;

        _last[bucket] = i;

      }

    }

    void findMinCutOut() {

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

        (*_excess)[n] = 0;

        (*_source_set)[n] = false;

      }

      for (ArcIt a(_graph); a != INVALID; ++a) {

        (*_flow)[a] = 0;

      }

      int bucket_num = 0;

      std::vector<Node> queue(_node_num);

      int qfirst = 0, qlast = 0, qsep = 0;

      {

        typename Digraph::template NodeMap<bool> reached(_graph, false);

        reached[_source] = true;

        bool first_set = true;

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

          if (reached[t]) continue;

          _sets.push_front(std::list<int>());

          queue[qlast++] = t;

          reached[t] = true;

          while (qfirst != qlast) {

            if (qsep == qfirst) {

              ++bucket_num;

              _sets.front().push_front(bucket_num);

              _dormant[bucket_num] = !first_set;

              _first[bucket_num] = _last[bucket_num] = INVALID;

              qsep = qlast;

            }

            Node n = queue[qfirst++];

            addItem(n, bucket_num);

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

              Node u = _graph.source(a);

              if (!reached[u] && _tolerance.positive((*_capacity)[a])) {

                reached[u] = true;

                queue[qlast++] = u;

              }

            }

          }

          first_set = false;

        }

        ++bucket_num;

        (*_bucket)[_source] = 0;

        _dormant[0] = true;

      }

      (*_source_set)[_source] = true;

      Node target = _last[_sets.back().back()];

      {

        for (OutArcIt a(_graph, _source); a != INVALID; ++a) {

          if (_tolerance.positive((*_capacity)[a])) {

            Node u = _graph.target(a);

            (*_flow)[a] = (*_capacity)[a];

            (*_excess)[u] += (*_capacity)[a];

            if (!(*_active)[u] && u != _source) {

              activate(u);

            }

          }

        }

        if ((*_active)[target]) {

          deactivate(target);

        }

        _highest = _sets.back().begin();

        while (_highest != _sets.back().end() &&

               !(*_active)[_first[*_highest]]) {

          ++_highest;

        }

      }

      while (true) {

        while (_highest != _sets.back().end()) {

          Node n = _first[*_highest];

          Value excess = (*_excess)[n];

          int next_bucket = _node_num;

          int under_bucket;

          if (++std::list<int>::iterator(_highest) == _sets.back().end()) {

            under_bucket = -1;

          } else {

            under_bucket = *(++std::list<int>::iterator(_highest));

          }

          for (OutArcIt a(_graph, n); a != INVALID; ++a) {

            Node v = _graph.target(a);

            if (_dormant[(*_bucket)[v]]) continue;

            Value rem = (*_capacity)[a] - (*_flow)[a];

            if (!_tolerance.positive(rem)) continue;

            if ((*_bucket)[v] == under_bucket) {

              if (!(*_active)[v] && v != target) {

                activate(v);

              }

              if (!_tolerance.less(rem, excess)) {

                (*_flow)[a] += excess;

                (*_excess)[v] += excess;

                excess = 0;

                goto no_more_push;

              } else {

                excess -= rem;

                (*_excess)[v] += rem;

                (*_flow)[a] = (*_capacity)[a];

              }

            } else if (next_bucket > (*_bucket)[v]) {

              next_bucket = (*_bucket)[v];

            }

          }

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

            Node v = _graph.source(a);

            if (_dormant[(*_bucket)[v]]) continue;

            Value rem = (*_flow)[a];

            if (!_tolerance.positive(rem)) continue;

            if ((*_bucket)[v] == under_bucket) {

              if (!(*_active)[v] && v != target) {

                activate(v);

              }

              if (!_tolerance.less(rem, excess)) {

                (*_flow)[a] -= excess;

                (*_excess)[v] += excess;

                excess = 0;

                goto no_more_push;

              } else {

                excess -= rem;

                (*_excess)[v] += rem;

                (*_flow)[a] = 0;

              }

            } else if (next_bucket > (*_bucket)[v]) {

              next_bucket = (*_bucket)[v];

            }

          }

        no_more_push:

          (*_excess)[n] = excess;

          if (excess != 0) {

            if ((*_next)[n] == INVALID) {

              typename std::list<std::list<int> >::iterator new_set =

                _sets.insert(--_sets.end(), std::list<int>());

              new_set->splice(new_set->end(), _sets.back(),

                              _sets.back().begin(), ++_highest);

              for (std::list<int>::iterator it = new_set->begin();

                   it != new_set->end(); ++it) {

                _dormant[*it] = true;

              }

              while (_highest != _sets.back().end() &&

                     !(*_active)[_first[*_highest]]) {

                ++_highest;

              }

            } else if (next_bucket == _node_num) {

              _first[(*_bucket)[n]] = (*_next)[n];

              (*_prev)[(*_next)[n]] = INVALID;

              std::list<std::list<int> >::iterator new_set =

                _sets.insert(--_sets.end(), std::list<int>());

              new_set->push_front(bucket_num);

              (*_bucket)[n] = bucket_num;

              _first[bucket_num] = _last[bucket_num] = n;

              (*_next)[n] = INVALID;

              (*_prev)[n] = INVALID;

              _dormant[bucket_num] = true;

              ++bucket_num;

              while (_highest != _sets.back().end() &&

                     !(*_active)[_first[*_highest]]) {

                ++_highest;

              }

            } else {

              _first[*_highest] = (*_next)[n];

              (*_prev)[(*_next)[n]] = INVALID;

              while (next_bucket != *_highest) {

                --_highest;

              }

              if (_highest == _sets.back().begin()) {

                _sets.back().push_front(bucket_num);

                _dormant[bucket_num] = false;

                _first[bucket_num] = _last[bucket_num] = INVALID;

                ++bucket_num;

              }

              --_highest;

              (*_bucket)[n] = *_highest;

              (*_next)[n] = _first[*_highest];

              if (_first[*_highest] != INVALID) {

                (*_prev)[_first[*_highest]] = n;

              } else {

                _last[*_highest] = n;

              }

              _first[*_highest] = n;

            }

          } else {

            deactivate(n);

            if (!(*_active)[_first[*_highest]]) {

              ++_highest;

              if (_highest != _sets.back().end() &&

                  !(*_active)[_first[*_highest]]) {

                _highest = _sets.back().end();

              }

            }

          }

        }

        if ((*_excess)[target] < _min_cut) {

          _min_cut = (*_excess)[target];

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

            (*_min_cut_map)[i] = true;

          }

          for (std::list<int>::iterator it = _sets.back().begin();

               it != _sets.back().end(); ++it) {

            Node n = _first[*it];

            while (n != INVALID) {

              (*_min_cut_map)[n] = false;

              n = (*_next)[n];

            }

          }

        }

        {

          Node new_target;

          if ((*_prev)[target] != INVALID || (*_next)[target] != INVALID) {

            if ((*_next)[target] == INVALID) {

              _last[(*_bucket)[target]] = (*_prev)[target];

              new_target = (*_prev)[target];

            } else {

              (*_prev)[(*_next)[target]] = (*_prev)[target];

              new_target = (*_next)[target];

            }

            if ((*_prev)[target] == INVALID) {

              _first[(*_bucket)[target]] = (*_next)[target];

            } else {

              (*_next)[(*_prev)[target]] = (*_next)[target];

            }

          } else {

            _sets.back().pop_back();

            if (_sets.back().empty()) {

              _sets.pop_back();

              if (_sets.empty())

                break;

              for (std::list<int>::iterator it = _sets.back().begin();

                   it != _sets.back().end(); ++it) {

                _dormant[*it] = false;

              }

            }

            new_target = _last[_sets.back().back()];

          }

          (*_bucket)[target] = 0;

          (*_source_set)[target] = true;

          for (OutArcIt a(_graph, target); a != INVALID; ++a) {

            Value rem = (*_capacity)[a] - (*_flow)[a];

            if (!_tolerance.positive(rem)) continue;

            Node v = _graph.target(a);

            if (!(*_active)[v] && !(*_source_set)[v]) {

              activate(v);

            }

            (*_excess)[v] += rem;

            (*_flow)[a] = (*_capacity)[a];

          }

          for (InArcIt a(_graph, target); a != INVALID; ++a) {

            Value rem = (*_flow)[a];

            if (!_tolerance.positive(rem)) continue;

            Node v = _graph.source(a);

            if (!(*_active)[v] && !(*_source_set)[v]) {

              activate(v);

            }

            (*_excess)[v] += rem;

            (*_flow)[a] = 0;

          }

          target = new_target;

          if ((*_active)[target]) {

            deactivate(target);

          }

          _highest = _sets.back().begin();

          while (_highest != _sets.back().end() &&

                 !(*_active)[_first[*_highest]]) {

            ++_highest;

          }

        }

      }

    }

    void findMinCutIn() {

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

        (*_excess)[n] = 0;

        (*_source_set)[n] = false;

      }

      for (ArcIt a(_graph); a != INVALID; ++a) {

        (*_flow)[a] = 0;

      }

      int bucket_num = 0;

      std::vector<Node> queue(_node_num);

      int qfirst = 0, qlast = 0, qsep = 0;

      {

        typename Digraph::template NodeMap<bool> reached(_graph, false);

        reached[_source] = true;

        bool first_set = true;

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

          if (reached[t]) continue;

          _sets.push_front(std::list<int>());

          queue[qlast++] = t;

          reached[t] = true;

          while (qfirst != qlast) {

            if (qsep == qfirst) {

              ++bucket_num;

              _sets.front().push_front(bucket_num);

              _dormant[bucket_num] = !first_set;

              _first[bucket_num] = _last[bucket_num] = INVALID;

              qsep = qlast;

            }

            Node n = queue[qfirst++];

            addItem(n, bucket_num);

            for (OutArcIt a(_graph, n); a != INVALID; ++a) {

              Node u = _graph.target(a);

              if (!reached[u] && _tolerance.positive((*_capacity)[a])) {

                reached[u] = true;

                queue[qlast++] = u;

              }

            }

          }

          first_set = false;

        }

        ++bucket_num;

        (*_bucket)[_source] = 0;

        _dormant[0] = true;

      }

      (*_source_set)[_source] = true;

      Node target = _last[_sets.back().back()];

      {

        for (InArcIt a(_graph, _source); a != INVALID; ++a) {

          if (_tolerance.positive((*_capacity)[a])) {

            Node u = _graph.source(a);

            (*_flow)[a] = (*_capacity)[a];

            (*_excess)[u] += (*_capacity)[a];

            if (!(*_active)[u] && u != _source) {

              activate(u);

            }

          }

        }

        if ((*_active)[target]) {

          deactivate(target);

        }

        _highest = _sets.back().begin();

        while (_highest != _sets.back().end() &&

               !(*_active)[_first[*_highest]]) {

          ++_highest;

        }

      }

      while (true) {

        while (_highest != _sets.back().end()) {

          Node n = _first[*_highest];

          Value excess = (*_excess)[n];

          int next_bucket = _node_num;

          int under_bucket;

          if (++std::list<int>::iterator(_highest) == _sets.back().end()) {

            under_bucket = -1;

          } else {

            under_bucket = *(++std::list<int>::iterator(_highest));

          }

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

            Node v = _graph.source(a);

            if (_dormant[(*_bucket)[v]]) continue;

            Value rem = (*_capacity)[a] - (*_flow)[a];

            if (!_tolerance.positive(rem)) continue;

            if ((*_bucket)[v] == under_bucket) {

              if (!(*_active)[v] && v != target) {

                activate(v);

              }

              if (!_tolerance.less(rem, excess)) {

                (*_flow)[a] += excess;

                (*_excess)[v] += excess;

                excess = 0;

                goto no_more_push;

              } else {

                excess -= rem;

                (*_excess)[v] += rem;

                (*_flow)[a] = (*_capacity)[a];

              }

            } else if (next_bucket > (*_bucket)[v]) {

              next_bucket = (*_bucket)[v];

            }

          }

          for (OutArcIt a(_graph, n); a != INVALID; ++a) {

            Node v = _graph.target(a);

            if (_dormant[(*_bucket)[v]]) continue;

            Value rem = (*_flow)[a];

            if (!_tolerance.positive(rem)) continue;

            if ((*_bucket)[v] == under_bucket) {

              if (!(*_active)[v] && v != target) {

                activate(v);

              }

              if (!_tolerance.less(rem, excess)) {

                (*_flow)[a] -= excess;

                (*_excess)[v] += excess;

                excess = 0;

                goto no_more_push;

              } else {

                excess -= rem;

                (*_excess)[v] += rem;

                (*_flow)[a] = 0;

              }

            } else if (next_bucket > (*_bucket)[v]) {

              next_bucket = (*_bucket)[v];

            }

          }

        no_more_push:

          (*_excess)[n] = excess;

          if (excess != 0) {

            if ((*_next)[n] == INVALID) {

              typename std::list<std::list<int> >::iterator new_set =

                _sets.insert(--_sets.end(), std::list<int>());

              new_set->splice(new_set->end(), _sets.back(),

                              _sets.back().begin(), ++_highest);

              for (std::list<int>::iterator it = new_set->begin();

                   it != new_set->end(); ++it) {

                _dormant[*it] = true;

              }

              while (_highest != _sets.back().end() &&

                     !(*_active)[_first[*_highest]]) {

                ++_highest;

              }

            } else if (next_bucket == _node_num) {

              _first[(*_bucket)[n]] = (*_next)[n];

              (*_prev)[(*_next)[n]] = INVALID;

              std::list<std::list<int> >::iterator new_set =

                _sets.insert(--_sets.end(), std::list<int>());

              new_set->push_front(bucket_num);

              (*_bucket)[n] = bucket_num;

              _first[bucket_num] = _last[bucket_num] = n;

              (*_next)[n] = INVALID;

              (*_prev)[n] = INVALID;

              _dormant[bucket_num] = true;

              ++bucket_num;

              while (_highest != _sets.back().end() &&

                     !(*_active)[_first[*_highest]]) {

                ++_highest;

              }

            } else {

              _first[*_highest] = (*_next)[n];

              (*_prev)[(*_next)[n]] = INVALID;

              while (next_bucket != *_highest) {

                --_highest;

              }

              if (_highest == _sets.back().begin()) {

                _sets.back().push_front(bucket_num);

                _dormant[bucket_num] = false;

                _first[bucket_num] = _last[bucket_num] = INVALID;

                ++bucket_num;

              }

              --_highest;

              (*_bucket)[n] = *_highest;

              (*_next)[n] = _first[*_highest];

              if (_first[*_highest] != INVALID) {

                (*_prev)[_first[*_highest]] = n;

              } else {

                _last[*_highest] = n;

              }

              _first[*_highest] = n;

            }

          } else {

            deactivate(n);

            if (!(*_active)[_first[*_highest]]) {

              ++_highest;

              if (_highest != _sets.back().end() &&

                  !(*_active)[_first[*_highest]]) {

                _highest = _sets.back().end();

              }

            }

          }

        }

        if ((*_excess)[target] < _min_cut) {

          _min_cut = (*_excess)[target];

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

            (*_min_cut_map)[i] = false;

          }

          for (std::list<int>::iterator it = _sets.back().begin();

               it != _sets.back().end(); ++it) {

            Node n = _first[*it];

            while (n != INVALID) {

              (*_min_cut_map)[n] = true;

              n = (*_next)[n];

            }

          }

        }

        {

          Node new_target;

          if ((*_prev)[target] != INVALID || (*_next)[target] != INVALID) {

            if ((*_next)[target] == INVALID) {

              _last[(*_bucket)[target]] = (*_prev)[target];

              new_target = (*_prev)[target];

            } else {

              (*_prev)[(*_next)[target]] = (*_prev)[target];

              new_target = (*_next)[target];

            }

            if ((*_prev)[target] == INVALID) {

              _first[(*_bucket)[target]] = (*_next)[target];

            } else {

              (*_next)[(*_prev)[target]] = (*_next)[target];

            }

          } else {

            _sets.back().pop_back();

            if (_sets.back().empty()) {

              _sets.pop_back();

              if (_sets.empty())

                break;

              for (std::list<int>::iterator it = _sets.back().begin();

                   it != _sets.back().end(); ++it) {

                _dormant[*it] = false;

              }

            }

            new_target = _last[_sets.back().back()];

          }

          (*_bucket)[target] = 0;

          (*_source_set)[target] = true;

          for (InArcIt a(_graph, target); a != INVALID; ++a) {

            Value rem = (*_capacity)[a] - (*_flow)[a];

            if (!_tolerance.positive(rem)) continue;

            Node v = _graph.source(a);

            if (!(*_active)[v] && !(*_source_set)[v]) {

              activate(v);

            }

            (*_excess)[v] += rem;

            (*_flow)[a] = (*_capacity)[a];

          }

          for (OutArcIt a(_graph, target); a != INVALID; ++a) {

            Value rem = (*_flow)[a];

            if (!_tolerance.positive(rem)) continue;

            Node v = _graph.target(a);

            if (!(*_active)[v] && !(*_source_set)[v]) {

              activate(v);

            }

            (*_excess)[v] += rem;

            (*_flow)[a] = 0;

          }

          target = new_target;

          if ((*_active)[target]) {

            deactivate(target);

          }

          _highest = _sets.back().begin();

          while (_highest != _sets.back().end() &&

                 !(*_active)[_first[*_highest]]) {

            ++_highest;

          }

        }

      }

    }

  public:

    /// \name Execution Control

    /// The simplest way to execute the algorithm is to use

    /// one of the member functions called \ref run().

    /// \n

    /// If you need better control on the execution,

    /// you have to call one of the \ref init() functions first, then

    /// \ref calculateOut() and/or \ref calculateIn().

    /// @{

    /// \brief Initialize the internal data structures.

    ///

    /// This function initializes the internal data structures. It creates

    /// the maps and some bucket structures for the algorithm.

    /// The first node is used as the source node for the push-relabel

    /// algorithm.

    void init() {

      init(NodeIt(_graph));

    }

    /// \brief Initialize the internal data structures.

    ///

    /// This function initializes the internal data structures. It creates

    /// the maps and some bucket structures for the algorithm.

    /// The given node is used as the source node for the push-relabel

    /// algorithm.

    void init(const Node& source) {

      _source = source;

      _node_num = countNodes(_graph);

      _first.resize(_node_num);

      _last.resize(_node_num);

      _dormant.resize(_node_num);

      if (!_flow) {

        _flow = new FlowMap(_graph);

      }

      if (!_next) {

        _next = new typename Digraph::template NodeMap<Node>(_graph);

      }

      if (!_prev) {

        _prev = new typename Digraph::template NodeMap<Node>(_graph);

      }

      if (!_active) {

        _active = new typename Digraph::template NodeMap<bool>(_graph);

      }

      if (!_bucket) {

        _bucket = new typename Digraph::template NodeMap<int>(_graph);

      }

      if (!_excess) {

        _excess = new ExcessMap(_graph);

      }

      if (!_source_set) {

        _source_set = new SourceSetMap(_graph);

      }

      if (!_min_cut_map) {

        _min_cut_map = new MinCutMap(_graph);

      }

      _min_cut = std::numeric_limits<Value>::max();

    }

    /// \brief Calculate a minimum cut with \f$ source \f$ on the

    /// source-side.

    ///

    /// This function calculates a minimum cut with \f$ source \f$ on the

    /// source-side (i.e. a set \f$ X\subsetneq V \f$ with

    /// \f$ source \in X \f$ and minimal outgoing capacity).

    /// It updates the stored cut if (and only if) the newly found one

    /// is better.

    ///

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

    void calculateOut() {

      findMinCutOut();

    }

    /// \brief Calculate a minimum cut with \f$ source \f$ on the

    /// sink-side.

    ///

    /// This function calculates a minimum cut with \f$ source \f$ on the

    /// sink-side (i.e. a set \f$ X\subsetneq V \f$ with

    /// \f$ source \notin X \f$ and minimal outgoing capacity).

    /// It updates the stored cut if (and only if) the newly found one

    /// is better.

    ///

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

    void calculateIn() {

      findMinCutIn();

    }

    /// \brief Run the algorithm.

    ///

    /// This function runs the algorithm. It chooses source node,

    /// then calls \ref init(), \ref calculateOut()

    /// and \ref calculateIn().

    void run() {

      init();

      calculateOut();

      calculateIn();

    }

    /// \brief Run the algorithm.

    ///

    /// This function runs the algorithm. It calls \ref init(),

    /// \ref calculateOut() and \ref calculateIn() with the given

    /// source node.

    void run(const Node& s) {

      init(s);

      calculateOut();

      calculateIn();

    }

    /// @}

    /// \name Query Functions

    /// The result of the %HaoOrlin algorithm

    /// can be obtained using these functions.\n

    /// \ref run(), \ref calculateOut() or \ref calculateIn()

    /// should be called before using them.

    /// @{

    /// \brief Return the value of the minimum cut.

    ///

    /// This function returns the value of the best cut found by the

    /// previously called \ref run(), \ref calculateOut() or \ref

    /// calculateIn().

    ///

    /// \pre \ref run(), \ref calculateOut() or \ref calculateIn()

    /// must be called before using this function.

    Value minCutValue() const {

      return _min_cut;

    }

    /// \brief Return a minimum cut.

    ///

    /// This function gives the best cut found by the

    /// previously called \ref run(), \ref calculateOut() or \ref

    /// calculateIn().

    ///

    /// It sets \c cutMap to the characteristic vector of the found

    /// minimum value cut - a non-empty set \f$ X\subsetneq V \f$

    /// of minimum outgoing capacity (i.e. \c cutMap will be \c true exactly

    /// for the nodes of \f$ X \f$).

    ///