RhodeCode

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

 *

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

 *

 * Copyright (C) 2003-2009

 * 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_BINOM_HEAP_H

#define LEMON_BINOM_HEAP_H

///\file

///\ingroup heaps

///\brief Binomial Heap implementation.

#include <vector>

#include <utility>

#include <functional>

#include <lemon/math.h>

#include <lemon/counter.h>

namespace lemon {

  /// \ingroup heaps

  ///

  ///\brief Binomial heap data structure.

  ///

  /// This class implements the \e binomial \e heap data structure.

  /// It fully conforms to the \ref concepts::Heap "heap concept".

  ///

  /// The methods \ref increase() and \ref erase() are not efficient

  /// in a binomial heap. In case of many calls of these operations,

  /// it is better to use other heap structure, e.g. \ref BinHeap

  /// "binary heap".

  ///

  /// \tparam PR Type of the priorities of the items.

  /// \tparam IM A read-writable item map with \c int values, used

  /// internally to handle the cross references.

  /// \tparam CMP A functor class for comparing the priorities.

  /// The default is \c std::less<PR>.

#ifdef DOXYGEN

  template <typename PR, typename IM, typename CMP>

#else

  template <typename PR, typename IM, typename CMP = std::less<PR> >

#endif

  class BinomHeap {

  public:

    /// Type of the item-int map.

    typedef IM ItemIntMap;

    /// Type of the priorities.

    typedef PR Prio;

    /// Type of the items stored in the heap.

    typedef typename ItemIntMap::Key Item;

    /// Functor type for comparing the priorities.

    typedef CMP Compare;

    /// \brief Type to represent the states of the items.

    ///

    /// Each item has a state associated to it. It can be "in heap",

    /// "pre-heap" or "post-heap". The latter two are indifferent from the

    /// heap's point of view, but may be useful to the user.

    ///

    /// The item-int map must be initialized in such way that it assigns

    /// \c PRE_HEAP (<tt>-1</tt>) to any element to be put in the heap.

    enum State {

      IN_HEAP = 0,    ///< = 0.

      PRE_HEAP = -1,  ///< = -1.

      POST_HEAP = -2  ///< = -2.

    };

  private:

    class Store;

    std::vector<Store> _data;

    int _min, _head;

    ItemIntMap &_iim;

    Compare _comp;

    int _num_items;

  public:

    /// \brief Constructor.

    ///

    /// Constructor.

    /// \param map A map that assigns \c int values to the items.

    /// It is used internally to handle the cross references.

    /// The assigned value must be \c PRE_HEAP (<tt>-1</tt>) for each item.

    explicit BinomHeap(ItemIntMap &map)

      : _min(0), _head(-1), _iim(map), _num_items(0) {}

    /// \brief Constructor.

    ///

    /// Constructor.

    /// \param map A map that assigns \c int values to the items.

    /// It is used internally to handle the cross references.

    /// The assigned value must be \c PRE_HEAP (<tt>-1</tt>) for each item.

    /// \param comp The function object used for comparing the priorities.

    BinomHeap(ItemIntMap &map, const Compare &comp)

      : _min(0), _head(-1), _iim(map), _comp(comp), _num_items(0) {}

    /// \brief The number of items stored in the heap.

    ///

    /// This function returns the number of items stored in the heap.

    int size() const { return _num_items; }

    /// \brief Check if the heap is empty.

    ///

    /// This function returns \c true if the heap is empty.

    bool empty() const { return _num_items==0; }

    /// \brief Make the heap empty.

    ///

    /// This functon makes the heap empty.

    /// It does not change the cross reference map. If you want to reuse

    /// a heap that is not surely empty, you should first clear it and

    /// then you should set the cross reference map to \c PRE_HEAP

    /// for each item.

    void clear() {

      _data.clear(); _min=0; _num_items=0; _head=-1;

    }

    /// \brief Set the priority of an item or insert it, if it is

    /// not stored in the heap.

    ///

    /// This method sets the priority of the given item if it is

    /// already stored in the heap. Otherwise it inserts the given

    /// item into the heap with the given priority.

    /// \param item The item.

    /// \param value The priority.

    void set (const Item& item, const Prio& value) {

      int i=_iim[item];

      if ( i >= 0 && _data[i].in ) {

        if ( _comp(value, _data[i].prio) ) decrease(item, value);

        if ( _comp(_data[i].prio, value) ) increase(item, value);

      } else push(item, value);

    }

    /// \brief Insert an item into the heap with the given priority.

    ///

    /// This function inserts the given item into the heap with the

    /// given priority.

    /// \param item The item to insert.

    /// \param value The priority of the item.

    /// \pre \e item must not be stored in the heap.

    void push (const Item& item, const Prio& value) {

      int i=_iim[item];

      if ( i<0 ) {

        int s=_data.size();

        _iim.set( item,s );

        Store st;

        st.name=item;

        st.prio=value;

        _data.push_back(st);

        i=s;

      }

      else {

        _data[i].parent=_data[i].right_neighbor=_data[i].child=-1;

        _data[i].degree=0;

        _data[i].in=true;

        _data[i].prio=value;

      }

      if( 0==_num_items ) {

        _head=i;

        _min=i;

      } else {

        merge(i);

        if( _comp(_data[i].prio, _data[_min].prio) ) _min=i;

      }

      ++_num_items;

    }

    /// \brief Return the item having minimum priority.

    ///

    /// This function returns the item having minimum priority.

    /// \pre The heap must be non-empty.

    Item top() const { return _data[_min].name; }

    /// \brief The minimum priority.

    ///

    /// This function returns the minimum priority.

    /// \pre The heap must be non-empty.

    Prio prio() const { return _data[_min].prio; }

    /// \brief The priority of the given item.

    ///

    /// This function returns the priority of the given item.

    /// \param item The item.

    /// \pre \e item must be in the heap.

    const Prio& operator[](const Item& item) const {

      return _data[_iim[item]].prio;

    }

    /// \brief Remove the item having minimum priority.

    ///

    /// This function removes the item having minimum priority.

    /// \pre The heap must be non-empty.

    void pop() {

      _data[_min].in=false;

      int head_child=-1;

      if ( _data[_min].child!=-1 ) {

        int child=_data[_min].child;

        int neighb;

        while( child!=-1 ) {

          neighb=_data[child].right_neighbor;

          _data[child].parent=-1;

          _data[child].right_neighbor=head_child;

          head_child=child;

          child=neighb;

        }

      }

      if ( _data[_head].right_neighbor==-1 ) {

        // there was only one root

        _head=head_child;

      }

      else {

        // there were more roots

        if( _head!=_min )  { unlace(_min); }

        else { _head=_data[_head].right_neighbor; }

        merge(head_child);

      }

      _min=findMin();

      --_num_items;

    }

    /// \brief Remove the given item from the heap.

    ///

    /// This function removes the given item from the heap if it is

    /// already stored.

    /// \param item The item to delete.

    /// \pre \e item must be in the heap.

    void erase (const Item& item) {

      int i=_iim[item];

      if ( i >= 0 && _data[i].in ) {

        decrease( item, _data[_min].prio-1 );

        pop();

      }

    }

    /// \brief Decrease the priority of an item to the given value.

    ///

    /// This function decreases the priority of an item to the given value.

    /// \param item The item.

    /// \param value The priority.

    /// \pre \e item must be stored in the heap with priority at least \e value.

    void decrease (Item item, const Prio& value) {

      int i=_iim[item];

      int p=_data[i].parent;

      _data[i].prio=value;

      while( p!=-1 && _comp(value, _data[p].prio) ) {

        _data[i].name=_data[p].name;

        _data[i].prio=_data[p].prio;

        _data[p].name=item;

        _data[p].prio=value;

        _iim[_data[i].name]=i;

        i=p;

        p=_data[p].parent;

      }

      _iim[item]=i;

      if ( _comp(value, _data[_min].prio) ) _min=i;

    }

    /// \brief Increase the priority of an item to the given value.

    ///

    /// This function increases the priority of an item to the given value.

    /// \param item The item.

    /// \param value The priority.

    /// \pre \e item must be stored in the heap with priority at most \e value.

    void increase (Item item, const Prio& value) {

      erase(item);

      push(item, value);

    }

    /// \brief Return the state of an item.

    ///

    /// This method returns \c PRE_HEAP if the given item has never

    /// been in the heap, \c IN_HEAP if it is in the heap at the moment,

    /// and \c POST_HEAP otherwise.

    /// In the latter case it is possible that the item will get back

    /// to the heap again.

    /// \param item The item.

    State state(const Item &item) const {

      int i=_iim[item];

      if( i>=0 ) {

        if ( _data[i].in ) i=0;

        else i=-2;

      }

      return State(i);

    }

    /// \brief Set the state of an item in the heap.

    ///

    /// This function sets the state of the given item in the heap.

    /// It can be used to manually clear the heap when it is important

    /// to achive better time complexity.

    /// \param i The item.

    /// \param st The state. It should not be \c IN_HEAP.

    void state(const Item& i, State st) {

      switch (st) {

      case POST_HEAP:

      case PRE_HEAP:

        if (state(i) == IN_HEAP) {

          erase(i);

        }

        _iim[i] = st;

        break;

      case IN_HEAP:

        break;

      }

    }

  private:

    // Find the minimum of the roots

    int findMin() {

      if( _head!=-1 ) {

        int min_loc=_head, min_val=_data[_head].prio;

        for( int x=_data[_head].right_neighbor; x!=-1;

             x=_data[x].right_neighbor ) {

          if( _comp( _data[x].prio,min_val ) ) {

            min_val=_data[x].prio;

            min_loc=x;

          }

        }

        return min_loc;

      }

      else return -1;

    }

    // Merge the heap with another heap starting at the given position

    void merge(int a) {

      if( _head==-1 || a==-1 ) return;

      if( _data[a].right_neighbor==-1 &&

          _data[a].degree<=_data[_head].degree ) {

        _data[a].right_neighbor=_head;

        _head=a;

      } else {

        interleave(a);

      }

      if( _data[_head].right_neighbor==-1 ) return;

      int x=_head;

      int x_prev=-1, x_next=_data[x].right_neighbor;

      while( x_next!=-1 ) {

        if( _data[x].degree!=_data[x_next].degree ||

            ( _data[x_next].right_neighbor!=-1 &&

              _data[_data[x_next].right_neighbor].degree==_data[x].degree ) ) {

          x_prev=x;

          x=x_next;

        }

        else {

          if( _comp(_data[x_next].prio,_data[x].prio) ) {

            if( x_prev==-1 ) {

              _head=x_next;

            } else {

              _data[x_prev].right_neighbor=x_next;

            }

            fuse(x,x_next);

            x=x_next;

          }

          else {

            _data[x].right_neighbor=_data[x_next].right_neighbor;

            fuse(x_next,x);

          }

        }

        x_next=_data[x].right_neighbor;

      }

    }

    // Interleave the elements of the given list into the list of the roots

    void interleave(int a) {

      int p=_head, q=a;

      int curr=_data.size();

      _data.push_back(Store());

      while( p!=-1 || q!=-1 ) {

        if( q==-1 || ( p!=-1 && _data[p].degree<_data[q].degree ) ) {

          _data[curr].right_neighbor=p;

          curr=p;

          p=_data[p].right_neighbor;

        }

        else {

          _data[curr].right_neighbor=q;

          curr=q;

          q=_data[q].right_neighbor;

        }

      }

      _head=_data.back().right_neighbor;

      _data.pop_back();

    }

    // Lace node a under node b

    void fuse(int a, int b) {

      _data[a].parent=b;

      _data[a].right_neighbor=_data[b].child;

      _data[b].child=a;

      ++_data[b].degree;

    }

    // Unlace node a (if it has siblings)

    void unlace(int a) {

      int neighb=_data[a].right_neighbor;

      int other=_head;

      while( _data[other].right_neighbor!=a )

        other=_data[other].right_neighbor;

      _data[other].right_neighbor=neighb;

    }

  private:

    class Store {

      friend class BinomHeap;

      Item name;

      int parent;

      int right_neighbor;

      int child;

      int degree;

      bool in;

      Prio prio;

      Store() : parent(-1), right_neighbor(-1), child(-1), degree(0),

        in(true) {}

    };

  };

} //namespace lemon

#endif //LEMON_BINOM_HEAP_H

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

 *

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

 *

 * Copyright (C) 2003-2009

 * 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_FOURARY_HEAP_H

#define LEMON_FOURARY_HEAP_H

///\ingroup heaps

///\file

///\brief Fourary heap implementation.

#include <vector>

#include <utility>

#include <functional>

namespace lemon {

  /// \ingroup heaps

  ///

  ///\brief Fourary heap data structure.

  ///

  /// This class implements the \e fourary \e heap data structure.

  /// It fully conforms to the \ref concepts::Heap "heap concept".

  ///

  /// The fourary heap is a specialization of the \ref KaryHeap "K-ary heap"

  /// for <tt>K=4</tt>. It is similar to the \ref BinHeap "binary heap",

  /// but its nodes have at most four children, instead of two.

  ///

  /// \tparam PR Type of the priorities of the items.

  /// \tparam IM A read-writable item map with \c int values, used

  /// internally to handle the cross references.

  /// \tparam CMP A functor class for comparing the priorities.

  /// The default is \c std::less<PR>.

  ///

  ///\sa BinHeap

  ///\sa KaryHeap

#ifdef DOXYGEN

  template <typename PR, typename IM, typename CMP>

#else

  template <typename PR, typename IM, typename CMP = std::less<PR> >

#endif

  class FouraryHeap {

  public:

    /// Type of the item-int map.

    typedef IM ItemIntMap;

    /// Type of the priorities.

    typedef PR Prio;

    /// Type of the items stored in the heap.

    typedef typename ItemIntMap::Key Item;

    /// Type of the item-priority pairs.

    typedef std::pair<Item,Prio> Pair;

    /// Functor type for comparing the priorities.

    typedef CMP Compare;

    /// \brief Type to represent the states of the items.

    ///

    /// Each item has a state associated to it. It can be "in heap",

    /// "pre-heap" or "post-heap". The latter two are indifferent from the

    /// heap's point of view, but may be useful to the user.

    ///

    /// The item-int map must be initialized in such way that it assigns

    /// \c PRE_HEAP (<tt>-1</tt>) to any element to be put in the heap.

    enum State {

      IN_HEAP = 0,    ///< = 0.

      PRE_HEAP = -1,  ///< = -1.

      POST_HEAP = -2  ///< = -2.

    };

  private:

    std::vector<Pair> _data;

    Compare _comp;

    ItemIntMap &_iim;

  public:

    /// \brief Constructor.

    ///

    /// Constructor.

    /// \param map A map that assigns \c int values to the items.

    /// It is used internally to handle the cross references.

    /// The assigned value must be \c PRE_HEAP (<tt>-1</tt>) for each item.

    explicit FouraryHeap(ItemIntMap &map) : _iim(map) {}

    /// \brief Constructor.

    ///

    /// Constructor.

    /// \param map A map that assigns \c int values to the items.

    /// It is used internally to handle the cross references.

    /// The assigned value must be \c PRE_HEAP (<tt>-1</tt>) for each item.

    /// \param comp The function object used for comparing the priorities.

    FouraryHeap(ItemIntMap &map, const Compare &comp)

      : _iim(map), _comp(comp) {}

    /// \brief The number of items stored in the heap.

    ///

    /// This function returns the number of items stored in the heap.

    int size() const { return _data.size(); }

    /// \brief Check if the heap is empty.

    ///

    /// This function returns \c true if the heap is empty.

    bool empty() const { return _data.empty(); }

    /// \brief Make the heap empty.

    ///

    /// This functon makes the heap empty.

    /// It does not change the cross reference map. If you want to reuse

    /// a heap that is not surely empty, you should first clear it and

    /// then you should set the cross reference map to \c PRE_HEAP

    /// for each item.

    void clear() { _data.clear(); }

  private:

    static int parent(int i) { return (i-1)/4; }

    static int firstChild(int i) { return 4*i+1; }

    bool less(const Pair &p1, const Pair &p2) const {

      return _comp(p1.second, p2.second);

    }

    void bubbleUp(int hole, Pair p) {

      int par = parent(hole);

      while( hole>0 && less(p,_data[par]) ) {

        move(_data[par],hole);

        hole = par;

        par = parent(hole);

      }

      move(p, hole);

    }

    void bubbleDown(int hole, Pair p, int length) {

      if( length>1 ) {

        int child = firstChild(hole);

        while( child+3<length ) {

          int min=child;

          if( less(_data[++child], _data[min]) ) min=child;

          if( less(_data[++child], _data[min]) ) min=child;

          if( less(_data[++child], _data[min]) ) min=child;

          if( !less(_data[min], p) )

            goto ok;

          move(_data[min], hole);

          hole = min;

          child = firstChild(hole);

        }

        if ( child<length ) {

          int min = child;

          if( ++child<length && less(_data[child], _data[min]) ) min=child;

          if( ++child<length && less(_data[child], _data[min]) ) min=child;

          if( less(_data[min], p) ) {

            move(_data[min], hole);

            hole = min;

          }

        }

      }

    ok:

      move(p, hole);

    }

    void move(const Pair &p, int i) {

      _data[i] = p;

      _iim.set(p.first, i);

    }

  public:

    /// \brief Insert a pair of item and priority into the heap.

    ///

    /// This function inserts \c p.first to the heap with priority

    /// \c p.second.

    /// \param p The pair to insert.

    /// \pre \c p.first must not be stored in the heap.

    void push(const Pair &p) {

      int n = _data.size();

      _data.resize(n+1);

      bubbleUp(n, p);

    }

    /// \brief Insert an item into the heap with the given priority.

    ///

    /// This function inserts the given item into the heap with the

    /// given priority.

    /// \param i The item to insert.

    /// \param p The priority of the item.

    /// \pre \e i must not be stored in the heap.

    void push(const Item &i, const Prio &p) { push(Pair(i,p)); }

    /// \brief Return the item having minimum priority.

    ///

    /// This function returns the item having minimum priority.

    /// \pre The heap must be non-empty.

    Item top() const { return _data[0].first; }

    /// \brief The minimum priority.

    ///

    /// This function returns the minimum priority.

    /// \pre The heap must be non-empty.

    Prio prio() const { return _data[0].second; }

    /// \brief Remove the item having minimum priority.

    ///

    /// This function removes the item having minimum priority.

    /// \pre The heap must be non-empty.

    void pop() {

      int n = _data.size()-1;

      _iim.set(_data[0].first, POST_HEAP);

      if (n>0) bubbleDown(0, _data[n], n);

      _data.pop_back();

    }

    /// \brief Remove the given item from the heap.

    ///

    /// This function removes the given item from the heap if it is

    /// already stored.

    /// \param i The item to delete.

    /// \pre \e i must be in the heap.

    void erase(const Item &i) {

      int h = _iim[i];

      int n = _data.size()-1;

      _iim.set(_data[h].first, POST_HEAP);

      if( h<n ) {

        if( less(_data[parent(h)], _data[n]) )

          bubbleDown(h, _data[n], n);

        else

          bubbleUp(h, _data[n]);

      }

      _data.pop_back();

    }

    /// \brief The priority of the given item.

    ///

    /// This function returns the priority of the given item.

    /// \param i The item.

    /// \pre \e i must be in the heap.

    Prio operator[](const Item &i) const {

      int idx = _iim[i];

      return _data[idx].second;

    }

    /// \brief Set the priority of an item or insert it, if it is

    /// not stored in the heap.

    ///

    /// This method sets the priority of the given item if it is

    /// already stored in the heap. Otherwise it inserts the given

    /// item into the heap with the given priority.

    /// \param i The item.

    /// \param p The priority.

    void set(const Item &i, const Prio &p) {

      int idx = _iim[i];

      if( idx < 0 )

        push(i,p);

      else if( _comp(p, _data[idx].second) )

        bubbleUp(idx, Pair(i,p));

      else

        bubbleDown(idx, Pair(i,p), _data.size());

    }

    /// \brief Decrease the priority of an item to the given value.

    ///

    /// This function decreases the priority of an item to the given value.

    /// \param i The item.

    /// \param p The priority.

    /// \pre \e i must be stored in the heap with priority at least \e p.

    void decrease(const Item &i, const Prio &p) {

      int idx = _iim[i];

      bubbleUp(idx, Pair(i,p));

    }

    /// \brief Increase the priority of an item to the given value.

    ///

    /// This function increases the priority of an item to the given value.

    /// \param i The item.

    /// \param p The priority.

    /// \pre \e i must be stored in the heap with priority at most \e p.

    void increase(const Item &i, const Prio &p) {

      int idx = _iim[i];

      bubbleDown(idx, Pair(i,p), _data.size());

    }

    /// \brief Return the state of an item.

    ///

    /// This method returns \c PRE_HEAP if the given item has never

    /// been in the heap, \c IN_HEAP if it is in the heap at the moment,

    /// and \c POST_HEAP otherwise.

    /// In the latter case it is possible that the item will get back

    /// to the heap again.

    /// \param i The item.

    State state(const Item &i) const {

      int s = _iim[i];

      if (s>=0) s=0;

      return State(s);

    }

    /// \brief Set the state of an item in the heap.

    ///

    /// This function sets the state of the given item in the heap.

    /// It can be used to manually clear the heap when it is important

    /// to achive better time complexity.

    /// \param i The item.

    /// \param st The state. It should not be \c IN_HEAP.

    void state(const Item& i, State st) {

      switch (st) {

        case POST_HEAP:

        case PRE_HEAP:

          if (state(i) == IN_HEAP) erase(i);

          _iim[i] = st;

          break;

        case IN_HEAP:

          break;

      }

    }

    /// \brief Replace an item in the heap.

    ///

    /// This function replaces item \c i with item \c j.

    /// Item \c i must be in the heap, while \c j must be out of the heap.

    /// After calling this method, item \c i will be out of the

    /// heap and \c j will be in the heap with the same prioriority

    /// as item \c i had before.

    void replace(const Item& i, const Item& j) {

      int idx = _iim[i];

      _iim.set(i, _iim[j]);

      _iim.set(j, idx);

      _data[idx].first = j;

    }

  }; // class FouraryHeap

} // namespace lemon

#endif // LEMON_FOURARY_HEAP_H

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

 *

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

 *

 * Copyright (C) 2003-2009

 * 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_KARY_HEAP_H

#define LEMON_KARY_HEAP_H

///\ingroup heaps

///\file

///\brief Fourary heap implementation.

#include <vector>

#include <utility>

#include <functional>

namespace lemon {

  /// \ingroup heaps

  ///

  ///\brief K-ary heap data structure.

  ///

  /// This class implements the \e K-ary \e heap data structure.

  /// It fully conforms to the \ref concepts::Heap "heap concept".

  ///

  /// The \ref KaryHeap "K-ary heap" is a generalization of the

  /// \ref BinHeap "binary heap" structure, its nodes have at most

  /// \c K children, instead of two.

  /// \ref BinHeap and \ref FouraryHeap are specialized implementations

  /// of this structure for <tt>K=2</tt> and <tt>K=4</tt>, respectively.

  ///

  /// \tparam PR Type of the priorities of the items.

  /// \tparam IM A read-writable item map with \c int values, used

  /// internally to handle the cross references.

  /// \tparam K The degree of the heap, each node have at most \e K

  /// children. The default is 16. Powers of two are suggested to use

  /// so that the multiplications and divisions needed to traverse the

  /// nodes of the heap could be performed faster.

  /// \tparam CMP A functor class for comparing the priorities.

  /// The default is \c std::less<PR>.

  ///

  ///\sa BinHeap

  ///\sa FouraryHeap

#ifdef DOXYGEN

  template <typename PR, typename IM, int K, typename CMP>

#else

  template <typename PR, typename IM, int K = 16,

            typename CMP = std::less<PR> >

#endif

  class KaryHeap {

  public:

    /// Type of the item-int map.

    typedef IM ItemIntMap;

    /// Type of the priorities.

    typedef PR Prio;

    /// Type of the items stored in the heap.

    typedef typename ItemIntMap::Key Item;

    /// Type of the item-priority pairs.

    typedef std::pair<Item,Prio> Pair;

    /// Functor type for comparing the priorities.

    typedef CMP Compare;

    /// \brief Type to represent the states of the items.

    ///

    /// Each item has a state associated to it. It can be "in heap",

    /// "pre-heap" or "post-heap". The latter two are indifferent from the

    /// heap's point of view, but may be useful to the user.

    ///

    /// The item-int map must be initialized in such way that it assigns

    /// \c PRE_HEAP (<tt>-1</tt>) to any element to be put in the heap.

    enum State {

      IN_HEAP = 0,    ///< = 0.

      PRE_HEAP = -1,  ///< = -1.

      POST_HEAP = -2  ///< = -2.

    };

  private:

    std::vector<Pair> _data;

    Compare _comp;

    ItemIntMap &_iim;

  public:

    /// \brief Constructor.

    ///

    /// Constructor.

    /// \param map A map that assigns \c int values to the items.

    /// It is used internally to handle the cross references.

    /// The assigned value must be \c PRE_HEAP (<tt>-1</tt>) for each item.

    explicit KaryHeap(ItemIntMap &map) : _iim(map) {}

    /// \brief Constructor.

    ///

    /// Constructor.

    /// \param map A map that assigns \c int values to the items.

    /// It is used internally to handle the cross references.

    /// The assigned value must be \c PRE_HEAP (<tt>-1</tt>) for each item.

    /// \param comp The function object used for comparing the priorities.

    KaryHeap(ItemIntMap &map, const Compare &comp)

      : _iim(map), _comp(comp) {}

    /// \brief The number of items stored in the heap.

    ///

    /// This function returns the number of items stored in the heap.

    int size() const { return _data.size(); }

    /// \brief Check if the heap is empty.

    ///

    /// This function returns \c true if the heap is empty.

    bool empty() const { return _data.empty(); }

    /// \brief Make the heap empty.

    ///

    /// This functon makes the heap empty.

    /// It does not change the cross reference map. If you want to reuse

    /// a heap that is not surely empty, you should first clear it and

    /// then you should set the cross reference map to \c PRE_HEAP

    /// for each item.

    void clear() { _data.clear(); }

  private:

    int parent(int i) { return (i-1)/K; }

    int firstChild(int i) { return K*i+1; }

    bool less(const Pair &p1, const Pair &p2) const {

      return _comp(p1.second, p2.second);

    }

    void bubbleUp(int hole, Pair p) {

      int par = parent(hole);

      while( hole>0 && less(p,_data[par]) ) {

        move(_data[par],hole);

        hole = par;

        par = parent(hole);

      }

      move(p, hole);

    }

    void bubbleDown(int hole, Pair p, int length) {

      if( length>1 ) {

        int child = firstChild(hole);

        while( child+K<=length ) {

          int min=child;

          for (int i=1; i<K; ++i) {

            if( less(_data[child+i], _data[min]) )

              min=child+i;

          }

          if( !less(_data[min], p) )

            goto ok;

          move(_data[min], hole);

          hole = min;

          child = firstChild(hole);

        }

        if ( child<length ) {

          int min = child;

          while (++child < length) {

            if( less(_data[child], _data[min]) )

              min=child;

          }

          if( less(_data[min], p) ) {

            move(_data[min], hole);

            hole = min;

          }

        }

      }

    ok:

      move(p, hole);

    }

    void move(const Pair &p, int i) {

      _data[i] = p;

      _iim.set(p.first, i);

    }

  public:

    /// \brief Insert a pair of item and priority into the heap.

    ///

    /// This function inserts \c p.first to the heap with priority

    /// \c p.second.

    /// \param p The pair to insert.

    /// \pre \c p.first must not be stored in the heap.

    void push(const Pair &p) {

      int n = _data.size();

      _data.resize(n+1);

      bubbleUp(n, p);

    }

    /// \brief Insert an item into the heap with the given priority.

    ///

    /// This function inserts the given item into the heap with the

    /// given priority.

    /// \param i The item to insert.

    /// \param p The priority of the item.

    /// \pre \e i must not be stored in the heap.

    void push(const Item &i, const Prio &p) { push(Pair(i,p)); }

    /// \brief Return the item having minimum priority.

    ///

    /// This function returns the item having minimum priority.

    /// \pre The heap must be non-empty.

    Item top() const { return _data[0].first; }

    /// \brief The minimum priority.

    ///

    /// This function returns the minimum priority.

    /// \pre The heap must be non-empty.

    Prio prio() const { return _data[0].second; }

    /// \brief Remove the item having minimum priority.

    ///

    /// This function removes the item having minimum priority.

    /// \pre The heap must be non-empty.

    void pop() {

      int n = _data.size()-1;

      _iim.set(_data[0].first, POST_HEAP);

      if (n>0) bubbleDown(0, _data[n], n);

      _data.pop_back();

    }

    /// \brief Remove the given item from the heap.

    ///

    /// This function removes the given item from the heap if it is

    /// already stored.

    /// \param i The item to delete.

    /// \pre \e i must be in the heap.

    void erase(const Item &i) {

      int h = _iim[i];

      int n = _data.size()-1;

      _iim.set(_data[h].first, POST_HEAP);

      if( h<n ) {

        if( less(_data[parent(h)], _data[n]) )

          bubbleDown(h, _data[n], n);

        else

          bubbleUp(h, _data[n]);

      }

      _data.pop_back();

    }

    /// \brief The priority of the given item.

    ///

    /// This function returns the priority of the given item.

    /// \param i The item.

    /// \pre \e i must be in the heap.

    Prio operator[](const Item &i) const {

      int idx = _iim[i];

      return _data[idx].second;

    }

    /// \brief Set the priority of an item or insert it, if it is

    /// not stored in the heap.

    ///

    /// This method sets the priority of the given item if it is

    /// already stored in the heap. Otherwise it inserts the given

    /// item into the heap with the given priority.

    /// \param i The item.

    /// \param p The priority.

    void set(const Item &i, const Prio &p) {

      int idx = _iim[i];

      if( idx<0 )

        push(i,p);

      else if( _comp(p, _data[idx].second) )

        bubbleUp(idx, Pair(i,p));

      else

        bubbleDown(idx, Pair(i,p), _data.size());

    }

    /// \brief Decrease the priority of an item to the given value.

    ///

    /// This function decreases the priority of an item to the given value.

    /// \param i The item.

    /// \param p The priority.

    /// \pre \e i must be stored in the heap with priority at least \e p.

    void decrease(const Item &i, const Prio &p) {

      int idx = _iim[i];

      bubbleUp(idx, Pair(i,p));

    }

    /// \brief Increase the priority of an item to the given value.

    ///

    /// This function increases the priority of an item to the given value.

    /// \param i The item.

    /// \param p The priority.

    /// \pre \e i must be stored in the heap with priority at most \e p.

    void increase(const Item &i, const Prio &p) {

      int idx = _iim[i];

      bubbleDown(idx, Pair(i,p), _data.size());

    }

    /// \brief Return the state of an item.

    ///

    /// This method returns \c PRE_HEAP if the given item has never

    /// been in the heap, \c IN_HEAP if it is in the heap at the moment,

    /// and \c POST_HEAP otherwise.

    /// In the latter case it is possible that the item will get back

    /// to the heap again.

    /// \param i The item.

    State state(const Item &i) const {

      int s = _iim[i];

      if (s>=0) s=0;

      return State(s);

    }

    /// \brief Set the state of an item in the heap.

    ///

    /// This function sets the state of the given item in the heap.

    /// It can be used to manually clear the heap when it is important

    /// to achive better time complexity.

    /// \param i The item.