lemon/binom_heap.h
changeset 705 39a5b48bcace
parent 701 d1a9224f1e30
child 707 3887d6f994d7
equal deleted inserted replaced
0:7b6512325b3e 1:4a920e9e25a5
     1 /* -*- C++ -*-
     1 /* -*- mode: C++; indent-tabs-mode: nil; -*-
     2  *
     2  *
     3  * This file is a part of LEMON, a generic C++ optimization library
     3  * This file is a part of LEMON, a generic C++ optimization library.
     4  *
     4  *
     5  * Copyright (C) 2003-2008
     5  * Copyright (C) 2003-2009
     6  * Egervary Jeno Kombinatorikus Optimalizalasi Kutatocsoport
     6  * Egervary Jeno Kombinatorikus Optimalizalasi Kutatocsoport
     7  * (Egervary Research Group on Combinatorial Optimization, EGRES).
     7  * (Egervary Research Group on Combinatorial Optimization, EGRES).
     8  *
     8  *
     9  * Permission to use, modify and distribute this software is granted
     9  * Permission to use, modify and distribute this software is granted
    10  * provided that this copyright notice appears in all copies. For
    10  * provided that this copyright notice appears in all copies. For
    18 
    18 
    19 #ifndef LEMON_BINOM_HEAP_H
    19 #ifndef LEMON_BINOM_HEAP_H
    20 #define LEMON_BINOM_HEAP_H
    20 #define LEMON_BINOM_HEAP_H
    21 
    21 
    22 ///\file
    22 ///\file
    23 ///\ingroup auxdat
    23 ///\ingroup heaps
    24 ///\brief Binomial Heap implementation.
    24 ///\brief Binomial Heap implementation.
    25 
    25 
    26 #include <vector>
    26 #include <vector>
       
    27 #include <utility>
    27 #include <functional>
    28 #include <functional>
    28 #include <lemon/math.h>
    29 #include <lemon/math.h>
    29 #include <lemon/counter.h>
    30 #include <lemon/counter.h>
    30 
    31 
    31 namespace lemon {
    32 namespace lemon {
    32 
    33 
    33   /// \ingroup auxdat
    34   /// \ingroup heaps
    34   ///
    35   ///
    35   ///\brief Binomial Heap.
    36   ///\brief Binomial heap data structure.
    36   ///
    37   ///
    37   ///This class implements the \e Binomial \e heap data structure. A \e heap
    38   /// This class implements the \e binomial \e heap data structure.
    38   ///is a data structure for storing items with specified values called \e
    39   /// It fully conforms to the \ref concepts::Heap "heap concept".
    39   ///priorities in such a way that finding the item with minimum priority is
       
    40   ///efficient. \c Compare specifies the ordering of the priorities. In a heap
       
    41   ///one can change the priority of an item, add or erase an item, etc.
       
    42   ///
    40   ///
    43   ///The methods \ref increase and \ref erase are not efficient in a Binomial
    41   /// The methods \ref increase() and \ref erase() are not efficient
    44   ///heap. In case of many calls to these operations, it is better to use a
    42   /// in a binomial heap. In case of many calls of these operations,
    45   ///\ref BinHeap "binary heap".
    43   /// it is better to use other heap structure, e.g. \ref BinHeap
       
    44   /// "binary heap".
    46   ///
    45   ///
    47   ///\param _Prio Type of the priority of the items.
    46   /// \tparam PR Type of the priorities of the items.
    48   ///\param _ItemIntMap A read and writable Item int map, used internally
    47   /// \tparam IM A read-writable item map with \c int values, used
    49   ///to handle the cross references.
    48   /// internally to handle the cross references.
    50   ///\param _Compare A class for the ordering of the priorities. The
    49   /// \tparam CMP A functor class for comparing the priorities.
    51   ///default is \c std::less<_Prio>.
    50   /// The default is \c std::less<PR>.
    52   ///
       
    53   ///\sa BinHeap
       
    54   ///\sa Dijkstra
       
    55   ///\author Dorian Batha
       
    56 
       
    57 #ifdef DOXYGEN
    51 #ifdef DOXYGEN
    58   template <typename _Prio,
    52   template <typename PR, typename IM, typename CMP>
    59             typename _ItemIntMap,
       
    60             typename _Compare>
       
    61 #else
    53 #else
    62   template <typename _Prio,
    54   template <typename PR, typename IM, typename CMP = std::less<PR> >
    63             typename _ItemIntMap,
       
    64             typename _Compare = std::less<_Prio> >
       
    65 #endif
    55 #endif
    66   class BinomHeap {
    56   class BinomHeap {
    67   public:
    57   public:
    68     typedef _ItemIntMap ItemIntMap;
    58     /// Type of the item-int map.
    69     typedef _Prio Prio;
    59     typedef IM ItemIntMap;
       
    60     /// Type of the priorities.
       
    61     typedef PR Prio;
       
    62     /// Type of the items stored in the heap.
    70     typedef typename ItemIntMap::Key Item;
    63     typedef typename ItemIntMap::Key Item;
    71     typedef std::pair<Item,Prio> Pair;
    64     /// Functor type for comparing the priorities.
    72     typedef _Compare Compare;
    65     typedef CMP Compare;
       
    66 
       
    67     /// \brief Type to represent the states of the items.
       
    68     ///
       
    69     /// Each item has a state associated to it. It can be "in heap",
       
    70     /// "pre-heap" or "post-heap". The latter two are indifferent from the
       
    71     /// heap's point of view, but may be useful to the user.
       
    72     ///
       
    73     /// The item-int map must be initialized in such way that it assigns
       
    74     /// \c PRE_HEAP (<tt>-1</tt>) to any element to be put in the heap.
       
    75     enum State {
       
    76       IN_HEAP = 0,    ///< = 0.
       
    77       PRE_HEAP = -1,  ///< = -1.
       
    78       POST_HEAP = -2  ///< = -2.
       
    79     };
    73 
    80 
    74   private:
    81   private:
    75     class store;
    82     class store;
    76 
    83 
    77     std::vector<store> container;
    84     std::vector<store> _data;
    78     int minimum, head;
    85     int _min, _head;
    79     ItemIntMap &iimap;
    86     ItemIntMap &_iim;
    80     Compare comp;
    87     Compare _comp;
    81     int num_items;
    88     int _num_items;
    82 
    89 
    83   public:
    90   public:
    84     ///Status of the nodes
    91     /// \brief Constructor.
    85     enum State {
    92     ///
    86       ///The node is in the heap
    93     /// Constructor.
    87       IN_HEAP = 0,
    94     /// \param map A map that assigns \c int values to the items.
    88       ///The node has never been in the heap
    95     /// It is used internally to handle the cross references.
    89       PRE_HEAP = -1,
    96     /// The assigned value must be \c PRE_HEAP (<tt>-1</tt>) for each item.
    90       ///The node was in the heap but it got out of it
    97     explicit BinomHeap(ItemIntMap &map)
    91       POST_HEAP = -2
    98       : _min(0), _head(-1), _iim(map), _num_items(0) {}
    92     };
    99 
    93 
   100     /// \brief Constructor.
    94     /// \brief The constructor
   101     ///
    95     ///
   102     /// Constructor.
    96     /// \c _iimap should be given to the constructor, since it is
   103     /// \param map A map that assigns \c int values to the items.
    97     ///   used internally to handle the cross references.
   104     /// It is used internally to handle the cross references.
    98     explicit BinomHeap(ItemIntMap &_iimap)
   105     /// The assigned value must be \c PRE_HEAP (<tt>-1</tt>) for each item.
    99       : minimum(0), head(-1), iimap(_iimap), num_items() {}
   106     /// \param comp The function object used for comparing the priorities.
   100 
   107     BinomHeap(ItemIntMap &map, const Compare &comp)
   101     /// \brief The constructor
   108       : _min(0), _head(-1), _iim(map), _comp(comp), _num_items(0) {}
   102     ///
       
   103     /// \c _iimap should be given to the constructor, since it is used
       
   104     /// internally to handle the cross references. \c _comp is an
       
   105     /// object for ordering of the priorities.
       
   106     BinomHeap(ItemIntMap &_iimap, const Compare &_comp)
       
   107       : minimum(0), head(-1), iimap(_iimap), comp(_comp), num_items() {}
       
   108 
   109 
   109     /// \brief The number of items stored in the heap.
   110     /// \brief The number of items stored in the heap.
   110     ///
   111     ///
   111     /// Returns the number of items stored in the heap.
   112     /// This function returns the number of items stored in the heap.
   112     int size() const { return num_items; }
   113     int size() const { return _num_items; }
   113 
   114 
   114     /// \brief Checks if the heap stores no items.
   115     /// \brief Check if the heap is empty.
   115     ///
   116     ///
   116     ///   Returns \c true if and only if the heap stores no items.
   117     /// This function returns \c true if the heap is empty.
   117     bool empty() const { return num_items==0; }
   118     bool empty() const { return _num_items==0; }
   118 
   119 
   119     /// \brief Make empty this heap.
   120     /// \brief Make the heap empty.
   120     ///
   121     ///
   121     /// Make empty this heap. It does not change the cross reference
   122     /// This functon makes the heap empty.
   122     /// map.  If you want to reuse a heap what is not surely empty you
   123     /// It does not change the cross reference map. If you want to reuse
   123     /// should first clear the heap and after that you should set the
   124     /// a heap that is not surely empty, you should first clear it and
   124     /// cross reference map for each item to \c PRE_HEAP.
   125     /// then you should set the cross reference map to \c PRE_HEAP
       
   126     /// for each item.
   125     void clear() {
   127     void clear() {
   126       container.clear(); minimum=0; num_items=0; head=-1;
   128       _data.clear(); _min=0; _num_items=0; _head=-1;
   127     }
   129     }
   128 
   130 
   129     /// \brief \c item gets to the heap with priority \c value independently
   131     /// \brief Set the priority of an item or insert it, if it is
   130     /// if \c item was already there.
   132     /// not stored in the heap.
   131     ///
   133     ///
   132     /// This method calls \ref push(\c item, \c value) if \c item is not
   134     /// This method sets the priority of the given item if it is
   133     /// stored in the heap and it calls \ref decrease(\c item, \c value) or
   135     /// already stored in the heap. Otherwise it inserts the given
   134     /// \ref increase(\c item, \c value) otherwise.
   136     /// item into the heap with the given priority.
       
   137     /// \param item The item.
       
   138     /// \param value The priority.
   135     void set (const Item& item, const Prio& value) {
   139     void set (const Item& item, const Prio& value) {
   136       int i=iimap[item];
   140       int i=_iim[item];
   137       if ( i >= 0 && container[i].in ) {
   141       if ( i >= 0 && _data[i].in ) {
   138         if ( comp(value, container[i].prio) ) decrease(item, value);
   142         if ( _comp(value, _data[i].prio) ) decrease(item, value);
   139         if ( comp(container[i].prio, value) ) increase(item, value);
   143         if ( _comp(_data[i].prio, value) ) increase(item, value);
   140       } else push(item, value);
   144       } else push(item, value);
   141     }
   145     }
   142 
   146 
   143     /// \brief Adds \c item to the heap with priority \c value.
   147     /// \brief Insert an item into the heap with the given priority.
   144     ///
   148     ///
   145     /// Adds \c item to the heap with priority \c value.
   149     /// This function inserts the given item into the heap with the
   146     /// \pre \c item must not be stored in the heap.
   150     /// given priority.
       
   151     /// \param item The item to insert.
       
   152     /// \param value The priority of the item.
       
   153     /// \pre \e item must not be stored in the heap.
   147     void push (const Item& item, const Prio& value) {
   154     void push (const Item& item, const Prio& value) {
   148       int i=iimap[item];
   155       int i=_iim[item];
   149       if ( i<0 ) {
   156       if ( i<0 ) {
   150         int s=container.size();
   157         int s=_data.size();
   151         iimap.set( item,s );
   158         _iim.set( item,s );
   152         store st;
   159         store st;
   153         st.name=item;
   160         st.name=item;
   154         container.push_back(st);
   161         _data.push_back(st);
   155         i=s;
   162         i=s;
   156       }
   163       }
   157       else {
   164       else {
   158         container[i].parent=container[i].right_neighbor=container[i].child=-1;
   165         _data[i].parent=_data[i].right_neighbor=_data[i].child=-1;
   159         container[i].degree=0;
   166         _data[i].degree=0;
   160         container[i].in=true;
   167         _data[i].in=true;
   161       }
   168       }
   162       container[i].prio=value;
   169       _data[i].prio=value;
   163 
   170 
   164       if( 0==num_items ) { head=i; minimum=i; }
   171       if( 0==_num_items ) { _head=i; _min=i; }
   165       else { merge(i); }
   172       else { merge(i); }
   166 
   173 
   167       minimum = find_min();
   174       _min = findMin();
   168 
   175 
   169       ++num_items;
   176       ++_num_items;
   170     }
   177     }
   171 
   178 
   172     /// \brief Returns the item with minimum priority relative to \c Compare.
   179     /// \brief Return the item having minimum priority.
   173     ///
   180     ///
   174     /// This method returns the item with minimum priority relative to \c
   181     /// This function returns the item having minimum priority.
   175     /// Compare.
   182     /// \pre The heap must be non-empty.
   176     /// \pre The heap must be nonempty.
   183     Item top() const { return _data[_min].name; }
   177     Item top() const { return container[minimum].name; }
   184 
   178 
   185     /// \brief The minimum priority.
   179     /// \brief Returns the minimum priority relative to \c Compare.
   186     ///
   180     ///
   187     /// This function returns the minimum priority.
   181     /// It returns the minimum priority relative to \c Compare.
   188     /// \pre The heap must be non-empty.
   182     /// \pre The heap must be nonempty.
   189     Prio prio() const { return _data[_min].prio; }
   183     const Prio& prio() const { return container[minimum].prio; }
   190 
   184 
   191     /// \brief The priority of the given item.
   185     /// \brief Returns the priority of \c item.
   192     ///
   186     ///
   193     /// This function returns the priority of the given item.
   187     /// It returns the priority of \c item.
   194     /// \param item The item.
   188     /// \pre \c item must be in the heap.
   195     /// \pre \e item must be in the heap.
   189     const Prio& operator[](const Item& item) const {
   196     const Prio& operator[](const Item& item) const {
   190       return container[iimap[item]].prio;
   197       return _data[_iim[item]].prio;
   191     }
   198     }
   192 
   199 
   193     /// \brief Deletes the item with minimum priority relative to \c Compare.
   200     /// \brief Remove the item having minimum priority.
   194     ///
   201     ///
   195     /// This method deletes the item with minimum priority relative to \c
   202     /// This function removes the item having minimum priority.
   196     /// Compare from the heap.
       
   197     /// \pre The heap must be non-empty.
   203     /// \pre The heap must be non-empty.
   198     void pop() {
   204     void pop() {
   199       container[minimum].in=false;
   205       _data[_min].in=false;
   200 
   206 
   201       int head_child=-1;
   207       int head_child=-1;
   202       if ( container[minimum].child!=-1 ) {
   208       if ( _data[_min].child!=-1 ) {
   203         int child=container[minimum].child;
   209         int child=_data[_min].child;
   204         int neighb;
   210         int neighb;
   205         int prev=-1;
   211         int prev=-1;
   206         while( child!=-1 ) {
   212         while( child!=-1 ) {
   207           neighb=container[child].right_neighbor;
   213           neighb=_data[child].right_neighbor;
   208           container[child].parent=-1;
   214           _data[child].parent=-1;
   209           container[child].right_neighbor=prev;
   215           _data[child].right_neighbor=prev;
   210           head_child=child;
   216           head_child=child;
   211           prev=child;
   217           prev=child;
   212           child=neighb;
   218           child=neighb;
   213         }
   219         }
   214       }
   220       }
   215 
   221 
   216       // The first case is that there are only one root.
   222       // The first case is that there are only one root.
   217       if ( -1==container[head].right_neighbor ) {
   223       if ( -1==_data[_head].right_neighbor ) {
   218         head=head_child;
   224         _head=head_child;
   219       }
   225       }
   220       // The case where there are more roots.
   226       // The case where there are more roots.
   221       else {
   227       else {
   222         if( head!=minimum )  { unlace(minimum); }
   228         if( _head!=_min )  { unlace(_min); }
   223         else { head=container[head].right_neighbor; }
   229         else { _head=_data[_head].right_neighbor; }
   224 
   230 
   225         merge(head_child);
   231         merge(head_child);
   226       }
   232       }
   227       minimum=find_min();
   233       _min=findMin();
   228       --num_items;
   234       --_num_items;
   229     }
   235     }
   230 
   236 
   231     /// \brief Deletes \c item from the heap.
   237     /// \brief Remove the given item from the heap.
   232     ///
   238     ///
   233     /// This method deletes \c item from the heap, if \c item was already
   239     /// This function removes the given item from the heap if it is
   234     /// stored in the heap. It is quite inefficient in Binomial heaps.
   240     /// already stored.
       
   241     /// \param item The item to delete.
       
   242     /// \pre \e item must be in the heap.
   235     void erase (const Item& item) {
   243     void erase (const Item& item) {
   236       int i=iimap[item];
   244       int i=_iim[item];
   237       if ( i >= 0 && container[i].in ) {
   245       if ( i >= 0 && _data[i].in ) {
   238         decrease( item, container[minimum].prio-1 );
   246         decrease( item, _data[_min].prio-1 );
   239         pop();
   247         pop();
   240       }
   248       }
   241     }
   249     }
   242 
   250 
   243     /// \brief Decreases the priority of \c item to \c value.
   251     /// \brief Decrease the priority of an item to the given value.
   244     ///
   252     ///
   245     /// This method decreases the priority of \c item to \c value.
   253     /// This function decreases the priority of an item to the given value.
   246     /// \pre \c item must be stored in the heap with priority at least \c
   254     /// \param item The item.
   247     ///   value relative to \c Compare.
   255     /// \param value The priority.
       
   256     /// \pre \e item must be stored in the heap with priority at least \e value.
   248     void decrease (Item item, const Prio& value) {
   257     void decrease (Item item, const Prio& value) {
   249       int i=iimap[item];
   258       int i=_iim[item];
   250 
   259 
   251       if( comp( value,container[i].prio ) ) {
   260       if( _comp( value,_data[i].prio ) ) {
   252         container[i].prio=value;
   261         _data[i].prio=value;
   253 
   262 
   254         int p_loc=container[i].parent, loc=i;
   263         int p_loc=_data[i].parent, loc=i;
   255         int parent, child, neighb;
   264         int parent, child, neighb;
   256 
   265 
   257         while( -1!=p_loc && comp(container[loc].prio,container[p_loc].prio) ) {
   266         while( -1!=p_loc && _comp(_data[loc].prio,_data[p_loc].prio) ) {
   258 
   267 
   259           // parent set for other loc_child
   268           // parent set for other loc_child
   260           child=container[loc].child;
   269           child=_data[loc].child;
   261           while( -1!=child ) {
   270           while( -1!=child ) {
   262             container[child].parent=p_loc;
   271             _data[child].parent=p_loc;
   263             child=container[child].right_neighbor;
   272             child=_data[child].right_neighbor;
   264           }
   273           }
   265 
   274 
   266           // parent set for other p_loc_child
   275           // parent set for other p_loc_child
   267           child=container[p_loc].child;
   276           child=_data[p_loc].child;
   268           while( -1!=child ) {
   277           while( -1!=child ) {
   269             container[child].parent=loc;
   278             _data[child].parent=loc;
   270             child=container[child].right_neighbor;
   279             child=_data[child].right_neighbor;
   271           }
   280           }
   272 
   281 
   273           child=container[p_loc].child;
   282           child=_data[p_loc].child;
   274           container[p_loc].child=container[loc].child;
   283           _data[p_loc].child=_data[loc].child;
   275           if( child==loc )
   284           if( child==loc )
   276             child=p_loc;
   285             child=p_loc;
   277           container[loc].child=child;
   286           _data[loc].child=child;
   278 
   287 
   279           // left_neighb set for p_loc
   288           // left_neighb set for p_loc
   280           if( container[loc].child!=p_loc ) {
   289           if( _data[loc].child!=p_loc ) {
   281             while( container[child].right_neighbor!=loc )
   290             while( _data[child].right_neighbor!=loc )
   282               child=container[child].right_neighbor;
   291               child=_data[child].right_neighbor;
   283             container[child].right_neighbor=p_loc;
   292             _data[child].right_neighbor=p_loc;
   284           }
   293           }
   285 
   294 
   286           // left_neighb set for loc
   295           // left_neighb set for loc
   287           parent=container[p_loc].parent;
   296           parent=_data[p_loc].parent;
   288           if( -1!=parent ) child=container[parent].child;
   297           if( -1!=parent ) child=_data[parent].child;
   289           else child=head;
   298           else child=_head;
   290 
   299 
   291           if( child!=p_loc ) {
   300           if( child!=p_loc ) {
   292             while( container[child].right_neighbor!=p_loc )
   301             while( _data[child].right_neighbor!=p_loc )
   293               child=container[child].right_neighbor;
   302               child=_data[child].right_neighbor;
   294             container[child].right_neighbor=loc;
   303             _data[child].right_neighbor=loc;
   295           }
   304           }
   296 
   305 
   297           neighb=container[p_loc].right_neighbor;
   306           neighb=_data[p_loc].right_neighbor;
   298           container[p_loc].right_neighbor=container[loc].right_neighbor;
   307           _data[p_loc].right_neighbor=_data[loc].right_neighbor;
   299           container[loc].right_neighbor=neighb;
   308           _data[loc].right_neighbor=neighb;
   300 
   309 
   301           container[p_loc].parent=loc;
   310           _data[p_loc].parent=loc;
   302           container[loc].parent=parent;
   311           _data[loc].parent=parent;
   303 
   312 
   304           if( -1!=parent && container[parent].child==p_loc )
   313           if( -1!=parent && _data[parent].child==p_loc )
   305             container[parent].child=loc;
   314             _data[parent].child=loc;
   306 
   315 
   307           /*if new parent will be the first root*/
   316           /*if new parent will be the first root*/
   308           if( head==p_loc )
   317           if( _head==p_loc )
   309             head=loc;
   318             _head=loc;
   310 
   319 
   311           p_loc=container[loc].parent;
   320           p_loc=_data[loc].parent;
   312         }
   321         }
   313       }
   322       }
   314       if( comp(value,container[minimum].prio) ) {
   323       if( _comp(value,_data[_min].prio) ) {
   315         minimum=i;
   324         _min=i;
   316       }
   325       }
   317     }
   326     }
   318 
   327 
   319     /// \brief Increases the priority of \c item to \c value.
   328     /// \brief Increase the priority of an item to the given value.
   320     ///
   329     ///
   321     /// This method sets the priority of \c item to \c value. Though
   330     /// This function increases the priority of an item to the given value.
   322     /// there is no precondition on the priority of \c item, this
   331     /// \param item The item.
   323     /// method should be used only if it is indeed necessary to increase
   332     /// \param value The priority.
   324     /// (relative to \c Compare) the priority of \c item, because this
   333     /// \pre \e item must be stored in the heap with priority at most \e value.
   325     /// method is inefficient.
       
   326     void increase (Item item, const Prio& value) {
   334     void increase (Item item, const Prio& value) {
   327       erase(item);
   335       erase(item);
   328       push(item, value);
   336       push(item, value);
   329     }
   337     }
   330 
   338 
   331 
   339     /// \brief Return the state of an item.
   332     /// \brief Returns if \c item is in, has already been in, or has never
   340     ///
   333     /// been in the heap.
   341     /// This method returns \c PRE_HEAP if the given item has never
   334     ///
   342     /// been in the heap, \c IN_HEAP if it is in the heap at the moment,
   335     /// This method returns PRE_HEAP if \c item has never been in the
   343     /// and \c POST_HEAP otherwise.
   336     /// heap, IN_HEAP if it is in the heap at the moment, and POST_HEAP
   344     /// In the latter case it is possible that the item will get back
   337     /// otherwise. In the latter case it is possible that \c item will
   345     /// to the heap again.
   338     /// get back to the heap again.
   346     /// \param item The item.
   339     State state(const Item &item) const {
   347     State state(const Item &item) const {
   340       int i=iimap[item];
   348       int i=_iim[item];
   341       if( i>=0 ) {
   349       if( i>=0 ) {
   342         if ( container[i].in ) i=0;
   350         if ( _data[i].in ) i=0;
   343         else i=-2;
   351         else i=-2;
   344       }
   352       }
   345       return State(i);
   353       return State(i);
   346     }
   354     }
   347 
   355 
   348     /// \brief Sets the state of the \c item in the heap.
   356     /// \brief Set the state of an item in the heap.
   349     ///
   357     ///
   350     /// Sets the state of the \c item in the heap. It can be used to
   358     /// This function sets the state of the given item in the heap.
   351     /// manually clear the heap when it is important to achive the
   359     /// It can be used to manually clear the heap when it is important
   352     /// better time complexity.
   360     /// to achive better time complexity.
   353     /// \param i The item.
   361     /// \param i The item.
   354     /// \param st The state. It should not be \c IN_HEAP.
   362     /// \param st The state. It should not be \c IN_HEAP.
   355     void state(const Item& i, State st) {
   363     void state(const Item& i, State st) {
   356       switch (st) {
   364       switch (st) {
   357       case POST_HEAP:
   365       case POST_HEAP:
   358       case PRE_HEAP:
   366       case PRE_HEAP:
   359         if (state(i) == IN_HEAP) {
   367         if (state(i) == IN_HEAP) {
   360           erase(i);
   368           erase(i);
   361         }
   369         }
   362         iimap[i] = st;
   370         _iim[i] = st;
   363         break;
   371         break;
   364       case IN_HEAP:
   372       case IN_HEAP:
   365         break;
   373         break;
   366       }
   374       }
   367     }
   375     }
   368 
   376 
   369   private:
   377   private:
   370     int find_min() {
   378     int findMin() {
   371       int min_loc=-1, min_val;
   379       int min_loc=-1, min_val;
   372       int x=head;
   380       int x=_head;
   373       if( x!=-1 ) {
   381       if( x!=-1 ) {
   374         min_val=container[x].prio;
   382         min_val=_data[x].prio;
   375         min_loc=x;
   383         min_loc=x;
   376         x=container[x].right_neighbor;
   384         x=_data[x].right_neighbor;
   377 
   385 
   378         while( x!=-1 ) {
   386         while( x!=-1 ) {
   379           if( comp( container[x].prio,min_val ) ) {
   387           if( _comp( _data[x].prio,min_val ) ) {
   380             min_val=container[x].prio;
   388             min_val=_data[x].prio;
   381             min_loc=x;
   389             min_loc=x;
   382           }
   390           }
   383           x=container[x].right_neighbor;
   391           x=_data[x].right_neighbor;
   384         }
   392         }
   385       }
   393       }
   386       return min_loc;
   394       return min_loc;
   387     }
   395     }
   388 
   396 
   389     void merge(int a) {
   397     void merge(int a) {
   390       interleave(a);
   398       interleave(a);
   391 
   399 
   392       int x=head;
   400       int x=_head;
   393       if( -1!=x ) {
   401       if( -1!=x ) {
   394         int x_prev=-1, x_next=container[x].right_neighbor;
   402         int x_prev=-1, x_next=_data[x].right_neighbor;
   395         while( -1!=x_next ) {
   403         while( -1!=x_next ) {
   396           if( container[x].degree!=container[x_next].degree || ( -1!=container[x_next].right_neighbor && container[container[x_next].right_neighbor].degree==container[x].degree ) ) {
   404           if( _data[x].degree!=_data[x_next].degree || ( -1!=_data[x_next].right_neighbor && _data[_data[x_next].right_neighbor].degree==_data[x].degree ) ) {
   397             x_prev=x;
   405             x_prev=x;
   398             x=x_next;
   406             x=x_next;
   399           }
   407           }
   400           else {
   408           else {
   401             if( comp(container[x].prio,container[x_next].prio) ) {
   409             if( _comp(_data[x].prio,_data[x_next].prio) ) {
   402               container[x].right_neighbor=container[x_next].right_neighbor;
   410               _data[x].right_neighbor=_data[x_next].right_neighbor;
   403               fuse(x_next,x);
   411               fuse(x_next,x);
   404             }
   412             }
   405             else {
   413             else {
   406               if( -1==x_prev ) { head=x_next; }
   414               if( -1==x_prev ) { _head=x_next; }
   407               else {
   415               else {
   408                 container[x_prev].right_neighbor=x_next;
   416                 _data[x_prev].right_neighbor=x_next;
   409               }
   417               }
   410               fuse(x,x_next);
   418               fuse(x,x_next);
   411               x=x_next;
   419               x=x_next;
   412             }
   420             }
   413           }
   421           }
   414           x_next=container[x].right_neighbor;
   422           x_next=_data[x].right_neighbor;
   415         }
   423         }
   416       }
   424       }
   417     }
   425     }
   418 
   426 
   419     void interleave(int a) {
   427     void interleave(int a) {
   420       int other=-1, head_other=-1;
   428       int other=-1, head_other=-1;
   421 
   429 
   422       while( -1!=a || -1!=head ) {
   430       while( -1!=a || -1!=_head ) {
   423         if( -1==a ) {
   431         if( -1==a ) {
   424           if( -1==head_other ) {
   432           if( -1==head_other ) {
   425             head_other=head;
   433             head_other=_head;
   426           }
   434           }
   427           else {
   435           else {
   428             container[other].right_neighbor=head;
   436             _data[other].right_neighbor=_head;
   429           }
   437           }
   430           head=-1;
   438           _head=-1;
   431         }
   439         }
   432         else if( -1==head ) {
   440         else if( -1==_head ) {
   433           if( -1==head_other ) {
   441           if( -1==head_other ) {
   434             head_other=a;
   442             head_other=a;
   435           }
   443           }
   436           else {
   444           else {
   437             container[other].right_neighbor=a;
   445             _data[other].right_neighbor=a;
   438           }
   446           }
   439           a=-1;
   447           a=-1;
   440         }
   448         }
   441         else {
   449         else {
   442           if( container[a].degree<container[head].degree ) {
   450           if( _data[a].degree<_data[_head].degree ) {
   443             if( -1==head_other ) {
   451             if( -1==head_other ) {
   444               head_other=a;
   452               head_other=a;
   445             }
   453             }
   446             else {
   454             else {
   447               container[other].right_neighbor=a;
   455               _data[other].right_neighbor=a;
   448             }
   456             }
   449             other=a;
   457             other=a;
   450             a=container[a].right_neighbor;
   458             a=_data[a].right_neighbor;
   451           }
   459           }
   452           else {
   460           else {
   453             if( -1==head_other ) {
   461             if( -1==head_other ) {
   454               head_other=head;
   462               head_other=_head;
   455             }
   463             }
   456             else {
   464             else {
   457               container[other].right_neighbor=head;
   465               _data[other].right_neighbor=_head;
   458             }
   466             }
   459             other=head;
   467             other=_head;
   460             head=container[head].right_neighbor;
   468             _head=_data[_head].right_neighbor;
   461           }
   469           }
   462         }
   470         }
   463       }
   471       }
   464       head=head_other;
   472       _head=head_other;
   465     }
   473     }
   466 
   474 
   467     // Lacing a under b
   475     // Lacing a under b
   468     void fuse(int a, int b) {
   476     void fuse(int a, int b) {
   469       container[a].parent=b;
   477       _data[a].parent=b;
   470       container[a].right_neighbor=container[b].child;
   478       _data[a].right_neighbor=_data[b].child;
   471       container[b].child=a;
   479       _data[b].child=a;
   472 
   480 
   473       ++container[b].degree;
   481       ++_data[b].degree;
   474     }
   482     }
   475 
   483 
   476     // It is invoked only if a has siblings.
   484     // It is invoked only if a has siblings.
   477     void unlace(int a) {
   485     void unlace(int a) {
   478       int neighb=container[a].right_neighbor;
   486       int neighb=_data[a].right_neighbor;
   479       int other=head;
   487       int other=_head;
   480 
   488 
   481       while( container[other].right_neighbor!=a )
   489       while( _data[other].right_neighbor!=a )
   482         other=container[other].right_neighbor;
   490         other=_data[other].right_neighbor;
   483       container[other].right_neighbor=neighb;
   491       _data[other].right_neighbor=neighb;
   484     }
   492     }
   485 
   493 
   486   private:
   494   private:
   487 
   495 
   488     class store {
   496     class store {