RhodeCode

/* -*- C++ -*-

 *

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

 *

 * Copyright (C) 2003-2008

 * 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 auxdat

///\brief Binomial Heap implementation.

#include <vector>

#include <functional>

#include <lemon/math.h>

#include <lemon/counter.h>

namespace lemon {

  /// \ingroup auxdat

  ///

  ///\brief Binomial Heap.

  ///

  ///This class implements the \e Binomial \e heap data structure. A \e heap

  ///is a data structure for storing items with specified values called \e

  ///priorities in such a way that finding the item with minimum priority is

  ///efficient. \c Compare specifies the ordering of the priorities. In a heap

  ///one can change the priority of an item, add or erase an item, etc.

  ///

  ///The methods \ref increase and \ref erase are not efficient in a Binomial

  ///heap. In case of many calls to these operations, it is better to use a

  ///\ref BinHeap "binary heap".

  ///

  ///\param _Prio Type of the priority of the items.

  ///\param _ItemIntMap A read and writable Item int map, used internally

  ///to handle the cross references.

  ///\param _Compare A class for the ordering of the priorities. The

  ///default is \c std::less<_Prio>.

  ///

  ///\sa BinHeap

  ///\sa Dijkstra

  ///\author Dorian Batha

#ifdef DOXYGEN

  template <typename _Prio,

            typename _ItemIntMap,

            typename _Compare>

#else

  template <typename _Prio,

            typename _ItemIntMap,

            typename _Compare = std::less<_Prio> >

#endif

  class BinomHeap {

  public:

    typedef _ItemIntMap ItemIntMap;

    typedef _Prio Prio;

    typedef typename ItemIntMap::Key Item;

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

    typedef _Compare Compare;

  private:

    class store;

    std::vector<store> container;

    int minimum, head;

    ItemIntMap &iimap;

    Compare comp;

    int num_items;

  public:

    ///Status of the nodes

    enum State {

      ///The node is in the heap

      IN_HEAP = 0,

      ///The node has never been in the heap

      PRE_HEAP = -1,

      ///The node was in the heap but it got out of it

      POST_HEAP = -2

    };

    /// \brief The constructor

    ///

    /// \c _iimap should be given to the constructor, since it is

    ///   used internally to handle the cross references.

    explicit BinomHeap(ItemIntMap &_iimap)

      : minimum(0), head(-1), iimap(_iimap), num_items() {}

    /// \brief The constructor

    ///

    /// \c _iimap should be given to the constructor, since it is used

    /// internally to handle the cross references. \c _comp is an

    /// object for ordering of the priorities.

    BinomHeap(ItemIntMap &_iimap, const Compare &_comp)

      : minimum(0), head(-1), iimap(_iimap), comp(_comp), num_items() {}

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

    ///

    /// Returns the number of items stored in the heap.

    int size() const { return num_items; }

    /// \brief Checks if the heap stores no items.

    ///

    ///   Returns \c true if and only if the heap stores no items.

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

    /// \brief Make empty this heap.

    ///

    /// Make empty this heap. It does not change the cross reference

    /// map.  If you want to reuse a heap what is not surely empty you

    /// should first clear the heap and after that you should set the

    /// cross reference map for each item to \c PRE_HEAP.

    void clear() {

      container.clear(); minimum=0; num_items=0; head=-1;

    }

    /// \brief \c item gets to the heap with priority \c value independently

    /// if \c item was already there.

    ///

    /// This method calls \ref push(\c item, \c value) if \c item is not

    /// stored in the heap and it calls \ref decrease(\c item, \c value) or

    /// \ref increase(\c item, \c value) otherwise.

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

      int i=iimap[item];

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

        if ( comp(value, container[i].prio) ) decrease(item, value);

        if ( comp(container[i].prio, value) ) increase(item, value);

      } else push(item, value);

    }

    /// \brief Adds \c item to the heap with priority \c value.

    ///

    /// Adds \c item to the heap with priority \c value.

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

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

      int i=iimap[item];

      if ( i<0 ) {

        int s=container.size();

        iimap.set( item,s );

        store st;

        st.name=item;

        container.push_back(st);

        i=s;

      }

      else {

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

        container[i].degree=0;

        container[i].in=true;

      }

      container[i].prio=value;

      if( 0==num_items ) { head=i; minimum=i; }

      else { merge(i); }

      minimum = find_min();

      ++num_items;

    }

    /// \brief Returns the item with minimum priority relative to \c Compare.

    ///

    /// This method returns the item with minimum priority relative to \c

    /// Compare.

    /// \pre The heap must be nonempty.

    Item top() const { return container[minimum].name; }

    /// \brief Returns the minimum priority relative to \c Compare.

    ///

    /// It returns the minimum priority relative to \c Compare.

    /// \pre The heap must be nonempty.

    const Prio& prio() const { return container[minimum].prio; }

    /// \brief Returns the priority of \c item.

    ///

    /// It returns the priority of \c item.

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

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

      return container[iimap[item]].prio;

    }

    /// \brief Deletes the item with minimum priority relative to \c Compare.

    ///

    /// This method deletes the item with minimum priority relative to \c

    /// Compare from the heap.

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

    void pop() {

      container[minimum].in=false;

      int head_child=-1;

      if ( container[minimum].child!=-1 ) {

        int child=container[minimum].child;

        int neighb;

        int prev=-1;

        while( child!=-1 ) {

          neighb=container[child].right_neighbor;

          container[child].parent=-1;

          container[child].right_neighbor=prev;

          head_child=child;

          prev=child;

          child=neighb;

        }

      }

      // The first case is that there are only one root.

      if ( -1==container[head].right_neighbor ) {

        head=head_child;

      }

      // The case where there are more roots.

      else {

        if( head!=minimum )  { unlace(minimum); }

        else { head=container[head].right_neighbor; }

        merge(head_child);

      }

      minimum=find_min();

      --num_items;

    }

    /// \brief Deletes \c item from the heap.

    ///

    /// This method deletes \c item from the heap, if \c item was already

    /// stored in the heap. It is quite inefficient in Binomial heaps.

    void erase (const Item& item) {

      int i=iimap[item];

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

        decrease( item, container[minimum].prio-1 );

        pop();

      }

    }

    /// \brief Decreases the priority of \c item to \c value.

    ///

    /// This method decreases the priority of \c item to \c value.

    /// \pre \c item must be stored in the heap with priority at least \c

    ///   value relative to \c Compare.

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

      int i=iimap[item];

      if( comp( value,container[i].prio ) ) {

        container[i].prio=value;

        int p_loc=container[i].parent, loc=i;

        int parent, child, neighb;

        while( -1!=p_loc && comp(container[loc].prio,container[p_loc].prio) ) {

          // parent set for other loc_child

          child=container[loc].child;

          while( -1!=child ) {

            container[child].parent=p_loc;

            child=container[child].right_neighbor;

          }

          // parent set for other p_loc_child

          child=container[p_loc].child;

          while( -1!=child ) {

            container[child].parent=loc;

            child=container[child].right_neighbor;

          }

          child=container[p_loc].child;

          container[p_loc].child=container[loc].child;

          if( child==loc )

            child=p_loc;

          container[loc].child=child;

          // left_neighb set for p_loc

          if( container[loc].child!=p_loc ) {

            while( container[child].right_neighbor!=loc )

              child=container[child].right_neighbor;

            container[child].right_neighbor=p_loc;

          }

          // left_neighb set for loc

          parent=container[p_loc].parent;

          if( -1!=parent ) child=container[parent].child;

          else child=head;

          if( child!=p_loc ) {

            while( container[child].right_neighbor!=p_loc )

              child=container[child].right_neighbor;

            container[child].right_neighbor=loc;

          }

          neighb=container[p_loc].right_neighbor;

          container[p_loc].right_neighbor=container[loc].right_neighbor;

          container[loc].right_neighbor=neighb;

          container[p_loc].parent=loc;

          container[loc].parent=parent;

          if( -1!=parent && container[parent].child==p_loc )

            container[parent].child=loc;

          /*if new parent will be the first root*/

          if( head==p_loc )

            head=loc;

          p_loc=container[loc].parent;

        }

      }

      if( comp(value,container[minimum].prio) ) {

        minimum=i;

      }

    }

    /// \brief Increases the priority of \c item to \c value.

    ///

    /// This method sets the priority of \c item to \c value. Though

    /// there is no precondition on the priority of \c item, this

    /// method should be used only if it is indeed necessary to increase

    /// (relative to \c Compare) the priority of \c item, because this

    /// method is inefficient.

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

      erase(item);

      push(item, value);

    }

    /// \brief Returns if \c item is in, has already been in, or has never

    /// been in the heap.

    ///

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

    /// heap, IN_HEAP if it is in the heap at the moment, and POST_HEAP

    /// otherwise. In the latter case it is possible that \c item will

    /// get back to the heap again.

    State state(const Item &item) const {

      int i=iimap[item];

      if( i>=0 ) {

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

        else i=-2;

      }

      return State(i);

    }

    /// \brief Sets the state of the \c item in the heap.

    ///

    /// Sets the state of the \c item in the heap. It can be used to

    /// manually clear the heap when it is important to achive the

    /// 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);

        }

        iimap[i] = st;

        break;

      case IN_HEAP:

        break;

      }

    }

  private:

    int find_min() {

      int min_loc=-1, min_val;

      int x=head;

      if( x!=-1 ) {

        min_val=container[x].prio;

        min_loc=x;

        x=container[x].right_neighbor;

        while( x!=-1 ) {

          if( comp( container[x].prio,min_val ) ) {

            min_val=container[x].prio;

            min_loc=x;

          }

          x=container[x].right_neighbor;

        }

      }

      return min_loc;

    }

    void merge(int a) {

      interleave(a);

      int x=head;

      if( -1!=x ) {

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

        while( -1!=x_next ) {

          if( container[x].degree!=container[x_next].degree || ( -1!=container[x_next].right_neighbor && container[container[x_next].right_neighbor].degree==container[x].degree ) ) {

            x_prev=x;

            x=x_next;

          }

          else {

            if( comp(container[x].prio,container[x_next].prio) ) {

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

              fuse(x_next,x);

            }

            else {

              if( -1==x_prev ) { head=x_next; }

              else {

                container[x_prev].right_neighbor=x_next;

              }

              fuse(x,x_next);

              x=x_next;

            }

          }

          x_next=container[x].right_neighbor;

        }

      }

    }

    void interleave(int a) {

      int other=-1, head_other=-1;

      while( -1!=a || -1!=head ) {

        if( -1==a ) {

          if( -1==head_other ) {

            head_other=head;

          }

          else {

            container[other].right_neighbor=head;

          }

          head=-1;

        }

        else if( -1==head ) {

          if( -1==head_other ) {

            head_other=a;

          }

          else {

            container[other].right_neighbor=a;

          }

          a=-1;

        }

        else {

          if( container[a].degree<container[head].degree ) {

            if( -1==head_other ) {

              head_other=a;

            }

            else {

              container[other].right_neighbor=a;

            }

            other=a;

            a=container[a].right_neighbor;

          }

          else {

            if( -1==head_other ) {

              head_other=head;

            }

            else {

              container[other].right_neighbor=head;

            }

            other=head;

            head=container[head].right_neighbor;

          }

        }

      }

      head=head_other;

    }

    // Lacing a under b

    void fuse(int a, int b) {

      container[a].parent=b;

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

      container[b].child=a;

      ++container[b].degree;

    }

    // It is invoked only if a has siblings.

    void unlace(int a) {

      int neighb=container[a].right_neighbor;

      int other=head;

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

        other=container[other].right_neighbor;

      container[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

/* -*- C++ -*-

 *

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

 *

 * Copyright (C) 2003-2008

 * 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 auxdat

///\file

///\brief 4ary Heap implementation.

#include <iostream>

#include <vector>

#include <utility>

#include <functional>

namespace lemon {

  ///\ingroup auxdat

  ///

  ///\brief A 4ary Heap implementation.

  ///

  ///This class implements the \e 4ary \e heap data structure. A \e heap

  ///is a data structure for storing items with specified values called \e

  ///priorities in such a way that finding the item with minimum priority is

  ///efficient. \c Compare specifies the ordering of the priorities. In a heap

  ///one can change the priority of an item, add or erase an item, etc.

  ///

  ///\param _Prio Type of the priority of the items.

  ///\param _ItemIntMap A read and writable Item int map, used internally

  ///to handle the cross references.

  ///\param _Compare A class for the ordering of the priorities. The

  ///default is \c std::less<_Prio>.

  ///

  ///\sa FibHeap

  ///\sa Dijkstra

  ///\author Dorian Batha

  template <typename _Prio, typename _ItemIntMap,

            typename _Compare = std::less<_Prio> >

  class FouraryHeap {

  public:

    ///\e

    typedef _ItemIntMap ItemIntMap;

    ///\e

    typedef _Prio Prio;

    ///\e

    typedef typename ItemIntMap::Key Item;

    ///\e

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

    ///\e

    typedef _Compare Compare;

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

    ///

    /// Each Item element have a state associated to it. It may 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 ItemIntMap \e should be initialized in such way that it maps

    /// PRE_HEAP (-1) to any element to be put in the heap...

    enum State {

      IN_HEAP = 0,

      PRE_HEAP = -1,

      POST_HEAP = -2

    };

  private:

    std::vector<Pair> data;

    Compare comp;

    ItemIntMap &iim;

  public:

    /// \brief The constructor.

    ///

    /// The constructor.

    /// \param _iim should be given to the constructor, since it is used

    /// internally to handle the cross references. The value of the map

    /// should be PRE_HEAP (-1) for each element.

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

    /// \brief The constructor.

    ///

    /// The constructor.

    /// \param _iim should be given to the constructor, since it is used

    /// internally to handle the cross references. The value of the map

    /// should be PRE_HEAP (-1) for each element.

    ///

    /// \param _comp The comparator function object.

    FouraryHeap(ItemIntMap &_iim, const Compare &_comp)

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

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

    ///

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

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

    /// \brief Checks if the heap stores no items.

    ///

    /// Returns \c true if and only if the heap stores no items.

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

    /// \brief Make empty this heap.

    ///

    /// Make empty this heap. It does not change the cross reference map.

    /// If you want to reuse what is not surely empty you should first clear

    /// the heap and after that you should set the cross reference map for

    /// each item to \c PRE_HEAP.

    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);

    }

    int find_min(const int child, const int length) {

      int min=child;

      if( child+3<length ) {

        if( less(data[child+3], data[min]) )

          min=child+3;

        if( less(data[child+2], data[min]) )

          min=child+2;

        if( less(data[child+1], data[min]) )

          min=child+1;

      }

      else if( child+2<length ) {

        if( less(data[child+2], data[min]) )

          min=child+2;

        if( less(data[child+1], data[min]) )

          min=child+1;

      }

      else if( child+1<length ) {

        if( less(data[child+1], data[min]) )

          min=child+1;

      }

      return min;

    }

    void bubble_up(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 bubble_down(int hole, Pair p, int length) {

      int child = firstChild(hole);

      while( child<length && length>1 ) {

        child = find_min(child,length);

        if( !less(data[child], p) )

          goto ok;

        move(data[child], hole);

        hole = child;

        child = firstChild(hole);

      }

    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.

    ///

    /// Adds \c p.first to the heap with priority \c p.second.

    /// \param p The pair to insert.

    void push(const Pair &p) {

      int n = data.size();

      data.resize(n+1);

      bubble_up(n, p);

    }

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

    ///

    /// Adds \c i to the heap with priority \c p.

    /// \param i The item to insert.

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

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

    /// \brief Returns the item with minimum priority relative to \c Compare.

    ///

    /// This method returns the item with minimum priority relative to \c

    /// Compare.

    /// \pre The heap must be nonempty.

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

    /// \brief Returns the minimum priority relative to \c Compare.

    ///

    /// It returns the minimum priority relative to \c Compare.

    /// \pre The heap must be nonempty.

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

    /// \brief Deletes the item with minimum priority relative to \c Compare.

    ///

    /// This method deletes the item with minimum priority relative to \c

    /// Compare from the heap.

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

    void pop() {

      int n = data.size()-1;

      iim.set(data[0].first, POST_HEAP);

      if (n>0) bubble_down(0, data[n], n);

      data.pop_back();

    }

    /// \brief Deletes \c i from the heap.

    ///

    /// This method deletes item \c i from the heap.

    /// \param i The item to erase.

    /// \pre The item should 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]) )

          bubble_down(h, data[n], n);

        else

          bubble_up(h, data[n]);

      }

      data.pop_back();

    }

    /// \brief Returns the priority of \c i.

    ///

    /// This function returns the priority of item \c i.

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

    /// \param i The item.

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

      int idx = iim[i];

      return data[idx].second;

    }

    /// \brief \c i gets to the heap with priority \c p independently

    /// if \c i was already there.

    ///

    /// This method calls \ref push(\c i, \c p) if \c i is not stored

    /// in the heap and sets the priority of \c i to \c p otherwise.

    /// \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) )

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

      else

        bubble_down(idx, Pair(i,p), data.size());

    }

    /// \brief Decreases the priority of \c i to \c p.

    ///

    /// This method decreases the priority of item \c i to \c p.

    /// \pre \c i must be stored in the heap with priority at least \c

    /// p relative to \c Compare.

    /// \param i The item.

    /// \param p The priority.

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

      int idx = iim[i];

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

    }

    /// \brief Increases the priority of \c i to \c p.

    ///

    /// This method sets the priority of item \c i to \c p.

    /// \pre \c i must be stored in the heap with priority at most \c

    /// p relative to \c Compare.

    /// \param i The item.

    /// \param p The priority.

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

      int idx = iim[i];

      bubble_down(idx, Pair(i,p), data.size());

    }

    /// \brief Returns if \c item is in, has already been in, or has

    /// never been in the heap.

    ///

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

    /// heap, IN_HEAP if it is in the heap at the moment, and POST_HEAP

    /// otherwise. In the latter case it is possible that \c 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 Sets the state of the \c item in the heap.

    ///

    /// Sets the state of the \c item in the heap. It can be used to

    /// manually clear the heap when it is important to achive the

    /// 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 Replaces an item in the heap.

    ///

    /// The \c i item is replaced with \c j item. The \c i item should

    /// be in the heap, while the \c j should be out of the heap. The

    /// \c i item will out of the heap and \c j will be in the heap

    /// with the same prioriority as prevoiusly the \c i item.

    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

/* -*- C++ -*-

 *

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

 *

 * Copyright (C) 2003-2008

 * 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 auxdat

///\file

///\brief Kary Heap implementation.

#include <iostream>

#include <vector>

#include <utility>

#include <functional>

namespace lemon {

  ///\ingroup auxdat

  ///

  ///\brief A Kary Heap implementation.

  ///

  ///This class implements the \e Kary \e heap data structure. A \e heap

  ///is a data structure for storing items with specified values called \e

  ///priorities in such a way that finding the item with minimum priority is

  ///efficient. \c Compare specifies the ordering of the priorities. In a heap

  ///one can change the priority of an item, add or erase an item, etc.

  ///

  ///\param _Prio Type of the priority of the items.

  ///\param _ItemIntMap A read and writable Item int map, used internally

  ///to handle the cross references.

  ///\param _Compare A class for the ordering of the priorities. The

  ///default is \c std::less<_Prio>.

  ///

  ///\sa FibHeap

  ///\sa Dijkstra

  ///\author Dorian Batha

  template <typename _Prio, typename _ItemIntMap,

            typename _Compare = std::less<_Prio> >

  class KaryHeap {

  public:

    ///\e

    typedef _ItemIntMap ItemIntMap;

    ///\e

    typedef _Prio Prio;

    ///\e

    typedef typename ItemIntMap::Key Item;

    ///\e

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

    ///\e

    typedef _Compare Compare;

    ///\e

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

    ///

    /// Each Item element have a state associated to it. It may 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 ItemIntMap \e should be initialized in such way that it maps

    /// PRE_HEAP (-1) to any element to be put in the heap...

    enum State {

      IN_HEAP = 0,

      PRE_HEAP = -1,

      POST_HEAP = -2

    };

  private:

    std::vector<Pair> data;

    Compare comp;

    ItemIntMap &iim;

    int K;

  public:

    /// \brief The constructor.

    ///

    /// The constructor.

    /// \param _iim should be given to the constructor, since it is used

    /// internally to handle the cross references. The value of the map

    /// should be PRE_HEAP (-1) for each element.

    explicit KaryHeap(ItemIntMap &_iim, const int &_K=32) : iim(_iim), K(_K) {}

    /// \brief The constructor.

    ///

    /// The constructor.

    /// \param _iim should be given to the constructor, since it is used

    /// internally to handle the cross references. The value of the map

    /// should be PRE_HEAP (-1) for each element.

    ///

    /// \param _comp The comparator function object.

    KaryHeap(ItemIntMap &_iim, const Compare &_comp, const int &_K=32)

      : iim(_iim), comp(_comp), K(_K) {}

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

    ///

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

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

    /// \brief Checks if the heap stores no items.

    ///

    /// Returns \c true if and only if the heap stores no items.

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

    /// \brief Make empty this heap.

    ///

    /// Make empty this heap. It does not change the cross reference map.

    /// If you want to reuse what is not surely empty you should first clear

    /// the heap and after that you should set the cross reference map for

    /// each item to \c PRE_HEAP.

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

  private:

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

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

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

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

    }

    int find_min(const int child, const int length) {

      int min=child, i=1;

      while( i<K && child+i<length ) {

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

          min=child+i;

        ++i;

      }

      return min;

    }

    void bubble_up(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 bubble_down(int hole, Pair p, int length) {

      if( length>1 ) {

        int child = first_child(hole);

        while( child<length ) {

          child = find_min(child, length);

          if( !less(data[child], p) )

            goto ok;

          move(data[child], hole);

          hole = child;

          child = first_child(hole);

        }

      }

    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.

    ///

    /// Adds \c p.first to the heap with priority \c p.second.

    /// \param p The pair to insert.

    void push(const Pair &p) {

      int n = data.size();

      data.resize(n+1);

      bubble_up(n, p);

    }

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

    ///

    /// Adds \c i to the heap with priority \c p.

    /// \param i The item to insert.

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

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

    /// \brief Returns the item with minimum priority relative to \c Compare.

    ///

    /// This method returns the item with minimum priority relative to \c

    /// Compare.

    /// \pre The heap must be nonempty.

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

    /// \brief Returns the minimum priority relative to \c Compare.

    ///

    /// It returns the minimum priority relative to \c Compare.

    /// \pre The heap must be nonempty.

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

    /// \brief Deletes the item with minimum priority relative to \c Compare.

    ///

    /// This method deletes the item with minimum priority relative to \c

    /// Compare from the heap.

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

    void pop() {

      int n = data.size()-1;

      iim.set(data[0].first, POST_HEAP);

      if (n>0) bubble_down(0, data[n], n);

      data.pop_back();

    }

    /// \brief Deletes \c i from the heap.

    ///

    /// This method deletes item \c i from the heap.

    /// \param i The item to erase.

    /// \pre The item should 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]) )

          bubble_down(h, data[n], n);

        else

          bubble_up(h, data[n]);

      }

      data.pop_back();

    }

    /// \brief Returns the priority of \c i.

    ///

    /// This function returns the priority of item \c i.

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

    /// \param i The item.

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

      int idx = iim[i];

      return data[idx].second;

    }

    /// \brief \c i gets to the heap with priority \c p independently