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

  *

  * 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_MAPS_H

 #define LEMON_MAPS_H

 #include <iterator>

 #include <functional>

 #include <vector>

 #include <lemon/core.h>

 ///\file

 ///\ingroup maps

 ///\brief Miscellaneous property maps

 #include <map>

 namespace lemon {

   /// \addtogroup maps

   /// @{

   /// Base class of maps.

   /// Base class of maps. It provides the necessary type definitions

   /// required by the map %concepts.

   template<typename K, typename V>

   class MapBase {

   public:

     /// \biref The key type of the map.

     typedef K Key;

     /// \brief The value type of the map.

     /// (The type of objects associated with the keys).

     typedef V Value;

   };

   /// Null map. (a.k.a. DoNothingMap)

   /// This map can be used if you have to provide a map only for

   /// its type definitions, or if you have to provide a writable map,

   /// but data written to it is not required (i.e. it will be sent to

   /// <tt>/dev/null</tt>).

   /// It conforms the \ref concepts::ReadWriteMap "ReadWriteMap" concept.

   ///

   /// \sa ConstMap

   template<typename K, typename V>

   class NullMap : public MapBase<K, V> {

   public:

     typedef MapBase<K, V> Parent;

     typedef typename Parent::Key Key;

     typedef typename Parent::Value Value;

     /// Gives back a default constructed element.

     Value operator[](const Key&) const { return Value(); }

     /// Absorbs the value.

     void set(const Key&, const Value&) {}

   };

   /// Returns a \c NullMap class

   /// This function just returns a \c NullMap class.

   /// \relates NullMap

   template <typename K, typename V>

   NullMap<K, V> nullMap() {

     return NullMap<K, V>();

   }

   /// Constant map.

   /// This \ref concepts::ReadMap "readable map" assigns a specified

   /// value to each key.

   ///

   /// In other aspects it is equivalent to \c NullMap.

   /// So it conforms the \ref concepts::ReadWriteMap "ReadWriteMap"

   /// concept, but it absorbs the data written to it.

   ///

   /// The simplest way of using this map is through the constMap()

   /// function.

   ///

   /// \sa NullMap

   /// \sa IdentityMap

   template<typename K, typename V>

   class ConstMap : public MapBase<K, V> {

   private:

     V _value;

   public:

     typedef MapBase<K, V> Parent;

     typedef typename Parent::Key Key;

     typedef typename Parent::Value Value;

     /// Default constructor

     /// Default constructor.

     /// The value of the map will be default constructed.

     ConstMap() {}

     /// Constructor with specified initial value

     /// Constructor with specified initial value.

     /// \param v The initial value of the map.

     ConstMap(const Value &v) : _value(v) {}

     /// Gives back the specified value.

     Value operator[](const Key&) const { return _value; }

     /// Absorbs the value.

     void set(const Key&, const Value&) {}

     /// Sets the value that is assigned to each key.

     void setAll(const Value &v) {

       _value = v;

     }

     template<typename V1>

     ConstMap(const ConstMap<K, V1> &, const Value &v) : _value(v) {}

   };

   /// Returns a \c ConstMap class

   /// This function just returns a \c ConstMap class.

   /// \relates ConstMap

   template<typename K, typename V>

   inline ConstMap<K, V> constMap(const V &v) {

     return ConstMap<K, V>(v);

   }

   template<typename K, typename V>

   inline ConstMap<K, V> constMap() {

     return ConstMap<K, V>();

   }

   template<typename T, T v>

   struct Const {};

   /// Constant map with inlined constant value.

   /// This \ref concepts::ReadMap "readable map" assigns a specified

   /// value to each key.

   ///

   /// In other aspects it is equivalent to \c NullMap.

   /// So it conforms the \ref concepts::ReadWriteMap "ReadWriteMap"

   /// concept, but it absorbs the data written to it.

   ///

   /// The simplest way of using this map is through the constMap()

   /// function.

   ///

   /// \sa NullMap

   /// \sa IdentityMap

   template<typename K, typename V, V v>

   class ConstMap<K, Const<V, v> > : public MapBase<K, V> {

   public:

     typedef MapBase<K, V> Parent;

     typedef typename Parent::Key Key;

     typedef typename Parent::Value Value;

     /// Constructor.

     ConstMap() {}

     /// Gives back the specified value.

     Value operator[](const Key&) const { return v; }

     /// Absorbs the value.

     void set(const Key&, const Value&) {}

   };

   /// Returns a \c ConstMap class with inlined constant value

   /// This function just returns a \c ConstMap class with inlined

   /// constant value.

   /// \relates ConstMap

   template<typename K, typename V, V v>

   inline ConstMap<K, Const<V, v> > constMap() {

     return ConstMap<K, Const<V, v> >();

   }

   /// Identity map.

   /// This \ref concepts::ReadMap "read-only map" gives back the given

   /// key as value without any modification.

   ///

   /// \sa ConstMap

   template <typename T>

   class IdentityMap : public MapBase<T, T> {

   public:

     typedef MapBase<T, T> Parent;

     typedef typename Parent::Key Key;

     typedef typename Parent::Value Value;

     /// Gives back the given value without any modification.

     Value operator[](const Key &k) const {

       return k;

     }

   };

   /// Returns an \c IdentityMap class

   /// This function just returns an \c IdentityMap class.

   /// \relates IdentityMap

   template<typename T>

   inline IdentityMap<T> identityMap() {

     return IdentityMap<T>();

   }

   /// \brief Map for storing values for integer keys from the range

   /// <tt>[0..size-1]</tt>.

   ///

   /// This map is essentially a wrapper for \c std::vector. It assigns

   /// values to integer keys from the range <tt>[0..size-1]</tt>.

   /// It can be used with some data structures, for example

   /// \c UnionFind, \c BinHeap, when the used items are small

   /// integers. This map conforms the \ref concepts::ReferenceMap

   /// "ReferenceMap" concept.

   ///

   /// The simplest way of using this map is through the rangeMap()

   /// function.

   template <typename V>

   class RangeMap : public MapBase<int, V> {

     template <typename V1>

     friend class RangeMap;

   private:

     typedef std::vector<V> Vector;

     Vector _vector;

   public:

     typedef MapBase<int, V> Parent;

     /// Key type

     typedef typename Parent::Key Key;

     /// Value type

     typedef typename Parent::Value Value;

     /// Reference type

     typedef typename Vector::reference Reference;

     /// Const reference type

     typedef typename Vector::const_reference ConstReference;

     typedef True ReferenceMapTag;

   public:

     /// Constructor with specified default value.

     RangeMap(int size = 0, const Value &value = Value())

       : _vector(size, value) {}

     /// Constructs the map from an appropriate \c std::vector.

     template <typename V1>

     RangeMap(const std::vector<V1>& vector)

       : _vector(vector.begin(), vector.end()) {}

     /// Constructs the map from another \c RangeMap.

     template <typename V1>

     RangeMap(const RangeMap<V1> &c)

       : _vector(c._vector.begin(), c._vector.end()) {}

     /// Returns the size of the map.

     int size() {

       return _vector.size();

     }

     /// Resizes the map.

     /// Resizes the underlying \c std::vector container, so changes the

     /// keyset of the map.

     /// \param size The new size of the map. The new keyset will be the

     /// range <tt>[0..size-1]</tt>.

     /// \param value The default value to assign to the new keys.

     void resize(int size, const Value &value = Value()) {

       _vector.resize(size, value);

     }

   private:

     RangeMap& operator=(const RangeMap&);

   public:

     ///\e

     Reference operator[](const Key &k) {

       return _vector[k];

     }

     ///\e

     ConstReference operator[](const Key &k) const {

       return _vector[k];

     }

     ///\e

     void set(const Key &k, const Value &v) {

       _vector[k] = v;

     }

   };

   /// Returns a \c RangeMap class

   /// This function just returns a \c RangeMap class.

   /// \relates RangeMap

   template<typename V>

   inline RangeMap<V> rangeMap(int size = 0, const V &value = V()) {

     return RangeMap<V>(size, value);

   }

   /// \brief Returns a \c RangeMap class created from an appropriate

   /// \c std::vector

   /// This function just returns a \c RangeMap class created from an

   /// appropriate \c std::vector.

   /// \relates RangeMap

   template<typename V>

   inline RangeMap<V> rangeMap(const std::vector<V> &vector) {

     return RangeMap<V>(vector);

   }

   /// Map type based on \c std::map

   /// This map is essentially a wrapper for \c std::map with addition

   /// that you can specify a default value for the keys that are not

   /// stored actually. This value can be different from the default

   /// contructed value (i.e. \c %Value()).

   /// This type conforms the \ref concepts::ReferenceMap "ReferenceMap"

   /// concept.

   ///

   /// This map is useful if a default value should be assigned to most of

   /// the keys and different values should be assigned only to a few

   /// keys (i.e. the map is "sparse").

   /// The name of this type also refers to this important usage.

   ///

   /// Apart form that this map can be used in many other cases since it

   /// is based on \c std::map, which is a general associative container.

   /// However keep in mind that it is usually not as efficient as other

   /// maps.

   ///

   /// The simplest way of using this map is through the sparseMap()

   /// function.

   template <typename K, typename V, typename Compare = std::less<K> >

   class SparseMap : public MapBase<K, V> {

     template <typename K1, typename V1, typename C1>

     friend class SparseMap;

   public:

     typedef MapBase<K, V> Parent;

     /// Key type

     typedef typename Parent::Key Key;

     /// Value type

     typedef typename Parent::Value Value;

     /// Reference type

     typedef Value& Reference;

     /// Const reference type

     typedef const Value& ConstReference;

     typedef True ReferenceMapTag;

   private:

     typedef std::map<K, V, Compare> Map;

     Map _map;

     Value _value;

   public:

     /// \brief Constructor with specified default value.

     SparseMap(const Value &value = Value()) : _value(value) {}

     /// \brief Constructs the map from an appropriate \c std::map, and

     /// explicitly specifies a default value.

     template <typename V1, typename Comp1>

     SparseMap(const std::map<Key, V1, Comp1> &map,

               const Value &value = Value())

       : _map(map.begin(), map.end()), _value(value) {}

     /// \brief Constructs the map from another \c SparseMap.

     template<typename V1, typename Comp1>

     SparseMap(const SparseMap<Key, V1, Comp1> &c)

       : _map(c._map.begin(), c._map.end()), _value(c._value) {}

   private:

     SparseMap& operator=(const SparseMap&);

   public:

     ///\e

     Reference operator[](const Key &k) {

       typename Map::iterator it = _map.lower_bound(k);

       if (it != _map.end() && !_map.key_comp()(k, it->first))

         return it->second;

       else

         return _map.insert(it, std::make_pair(k, _value))->second;

     }

     ///\e

     ConstReference operator[](const Key &k) const {

       typename Map::const_iterator it = _map.find(k);

       if (it != _map.end())

         return it->second;

       else

         return _value;

     }

     ///\e

     void set(const Key &k, const Value &v) {

       typename Map::iterator it = _map.lower_bound(k);

       if (it != _map.end() && !_map.key_comp()(k, it->first))

         it->second = v;

       else

         _map.insert(it, std::make_pair(k, v));

     }

     ///\e

     void setAll(const Value &v) {

       _value = v;

       _map.clear();

     }

   };

   /// Returns a \c SparseMap class

   /// This function just returns a \c SparseMap class with specified

   /// default value.

   /// \relates SparseMap

   template<typename K, typename V, typename Compare>

   inline SparseMap<K, V, Compare> sparseMap(const V& value = V()) {

     return SparseMap<K, V, Compare>(value);

   }

   template<typename K, typename V>

   inline SparseMap<K, V, std::less<K> > sparseMap(const V& value = V()) {

     return SparseMap<K, V, std::less<K> >(value);

   }

   /// \brief Returns a \c SparseMap class created from an appropriate

   /// \c std::map

   /// This function just returns a \c SparseMap class created from an

   /// appropriate \c std::map.

   /// \relates SparseMap

   template<typename K, typename V, typename Compare>

   inline SparseMap<K, V, Compare>

     sparseMap(const std::map<K, V, Compare> &map, const V& value = V())

   {

     return SparseMap<K, V, Compare>(map, value);

   }

   /// @}

   /// \addtogroup map_adaptors

   /// @{

   /// Composition of two maps

   /// This \ref concepts::ReadMap "read-only map" returns the

   /// composition of two given maps. That is to say, if \c m1 is of

   /// type \c M1 and \c m2 is of \c M2, then for

   /// \code

   ///   ComposeMap<M1, M2> cm(m1,m2);

   /// \endcode

   /// <tt>cm[x]</tt> will be equal to <tt>m1[m2[x]]</tt>.

   ///

   /// The \c Key type of the map is inherited from \c M2 and the

   /// \c Value type is from \c M1.

   /// \c M2::Value must be convertible to \c M1::Key.

   ///

   /// The simplest way of using this map is through the composeMap()

   /// function.

   ///

   /// \sa CombineMap

   template <typename M1, typename M2>

   class ComposeMap : public MapBase<typename M2::Key, typename M1::Value> {

     const M1 &_m1;

     const M2 &_m2;

   public:

     typedef MapBase<typename M2::Key, typename M1::Value> Parent;

     typedef typename Parent::Key Key;

     typedef typename Parent::Value Value;

     /// Constructor

     ComposeMap(const M1 &m1, const M2 &m2) : _m1(m1), _m2(m2) {}

     /// \e

     typename MapTraits<M1>::ConstReturnValue

     operator[](const Key &k) const { return _m1[_m2[k]]; }

   };

   /// Returns a \c ComposeMap class

   /// This function just returns a \c ComposeMap class.

   ///

   /// If \c m1 and \c m2 are maps and the \c Value type of \c m2 is

   /// convertible to the \c Key of \c m1, then <tt>composeMap(m1,m2)[x]</tt>

   /// will be equal to <tt>m1[m2[x]]</tt>.

   ///

   /// \relates ComposeMap

   template <typename M1, typename M2>

   inline ComposeMap<M1, M2> composeMap(const M1 &m1, const M2 &m2) {

     return ComposeMap<M1, M2>(m1, m2);

   }

   /// Combination of two maps using an STL (binary) functor.

   /// This \ref concepts::ReadMap "read-only map" takes two maps and a

   /// binary functor and returns the combination of the two given maps

   /// using the functor.

   /// That is to say, if \c m1 is of type \c M1 and \c m2 is of \c M2

   /// and \c f is of \c F, then for

   /// \code

   ///   CombineMap<M1,M2,F,V> cm(m1,m2,f);

   /// \endcode

   /// <tt>cm[x]</tt> will be equal to <tt>f(m1[x],m2[x])</tt>.

   ///

   /// The \c Key type of the map is inherited from \c M1 (\c M1::Key

   /// must be convertible to \c M2::Key) and the \c Value type is \c V.

   /// \c M2::Value and \c M1::Value must be convertible to the

   /// corresponding input parameter of \c F and the return type of \c F

   /// must be convertible to \c V.

   ///

   /// The simplest way of using this map is through the combineMap()

   /// function.

   ///

   /// \sa ComposeMap

   template<typename M1, typename M2, typename F,

            typename V = typename F::result_type>

   class CombineMap : public MapBase<typename M1::Key, V> {

     const M1 &_m1;

     const M2 &_m2;

     F _f;

   public:

     typedef MapBase<typename M1::Key, V> Parent;

     typedef typename Parent::Key Key;

     typedef typename Parent::Value Value;

     /// Constructor

     CombineMap(const M1 &m1, const M2 &m2, const F &f = F())

       : _m1(m1), _m2(m2), _f(f) {}

     /// \e

     Value operator[](const Key &k) const { return _f(_m1[k],_m2[k]); }

   };

   /// Returns a \c CombineMap class

   /// This function just returns a \c CombineMap class.

   ///

   /// For example, if \c m1 and \c m2 are both maps with \c double

   /// values, then

   /// \code

   ///   combineMap(m1,m2,std::plus<double>())

   /// \endcode

   /// is equivalent to

   /// \code

   ///   addMap(m1,m2)

   /// \endcode

   ///

   /// This function is specialized for adaptable binary function

   /// classes and C++ functions.

   ///

   /// \relates CombineMap

   template<typename M1, typename M2, typename F, typename V>

   inline CombineMap<M1, M2, F, V>

   combineMap(const M1 &m1, const M2 &m2, const F &f) {

     return CombineMap<M1, M2, F, V>(m1,m2,f);

   }

   template<typename M1, typename M2, typename F>

   inline CombineMap<M1, M2, F, typename F::result_type>

   combineMap(const M1 &m1, const M2 &m2, const F &f) {

     return combineMap<M1, M2, F, typename F::result_type>(m1,m2,f);

   }

   template<typename M1, typename M2, typename K1, typename K2, typename V>

   inline CombineMap<M1, M2, V (*)(K1, K2), V>

   combineMap(const M1 &m1, const M2 &m2, V (*f)(K1, K2)) {

     return combineMap<M1, M2, V (*)(K1, K2), V>(m1,m2,f);

   }

   /// Converts an STL style (unary) functor to a map

   /// This \ref concepts::ReadMap "read-only map" returns the value

   /// of a given functor. Actually, it just wraps the functor and

   /// provides the \c Key and \c Value typedefs.

   ///

   /// Template parameters \c K and \c V will become its \c Key and

   /// \c Value. In most cases they have to be given explicitly because

   /// a functor typically does not provide \c argument_type and

   /// \c result_type typedefs.

   /// Parameter \c F is the type of the used functor.

   ///

   /// The simplest way of using this map is through the functorToMap()

   /// function.

   ///

   /// \sa MapToFunctor

   template<typename F,

            typename K = typename F::argument_type,

            typename V = typename F::result_type>

   class FunctorToMap : public MapBase<K, V> {

     F _f;

   public:

     typedef MapBase<K, V> Parent;

     typedef typename Parent::Key Key;

     typedef typename Parent::Value Value;

     /// Constructor

     FunctorToMap(const F &f = F()) : _f(f) {}

     /// \e

     Value operator[](const Key &k) const { return _f(k); }

   };

   /// Returns a \c FunctorToMap class

   /// This function just returns a \c FunctorToMap class.

   ///

   /// This function is specialized for adaptable binary function

   /// classes and C++ functions.

   ///

   /// \relates FunctorToMap

   template<typename K, typename V, typename F>

   inline FunctorToMap<F, K, V> functorToMap(const F &f) {

     return FunctorToMap<F, K, V>(f);

   }

   template <typename F>

   inline FunctorToMap<F, typename F::argument_type, typename F::result_type>

     functorToMap(const F &f)

   {

     return FunctorToMap<F, typename F::argument_type,

       typename F::result_type>(f);

   }

   template <typename K, typename V>

   inline FunctorToMap<V (*)(K), K, V> functorToMap(V (*f)(K)) {

     return FunctorToMap<V (*)(K), K, V>(f);

   }

   /// Converts a map to an STL style (unary) functor

   /// This class converts a map to an STL style (unary) functor.

   /// That is it provides an <tt>operator()</tt> to read its values.

   ///

   /// For the sake of convenience it also works as a usual

   /// \ref concepts::ReadMap "readable map", i.e. <tt>operator[]</tt>

   /// and the \c Key and \c Value typedefs also exist.

   ///

   /// The simplest way of using this map is through the mapToFunctor()

   /// function.

   ///

   ///\sa FunctorToMap

   template <typename M>

   class MapToFunctor : public MapBase<typename M::Key, typename M::Value> {

     const M &_m;

   public:

     typedef MapBase<typename M::Key, typename M::Value> Parent;

     typedef typename Parent::Key Key;

     typedef typename Parent::Value Value;

     typedef typename Parent::Key argument_type;

     typedef typename Parent::Value result_type;

     /// Constructor

     MapToFunctor(const M &m) : _m(m) {}

     /// \e

     Value operator()(const Key &k) const { return _m[k]; }

     /// \e

     Value operator[](const Key &k) const { return _m[k]; }

   };

   /// Returns a \c MapToFunctor class

   /// This function just returns a \c MapToFunctor class.

   /// \relates MapToFunctor

   template<typename M>

   inline MapToFunctor<M> mapToFunctor(const M &m) {

     return MapToFunctor<M>(m);

   }

   /// \brief Map adaptor to convert the \c Value type of a map to

   /// another type using the default conversion.

   /// Map adaptor to convert the \c Value type of a \ref concepts::ReadMap

   /// "readable map" to another type using the default conversion.

   /// The \c Key type of it is inherited from \c M and the \c Value

   /// type is \c V.

   /// This type conforms the \ref concepts::ReadMap "ReadMap" concept.

   ///

   /// The simplest way of using this map is through the convertMap()

   /// function.

   template <typename M, typename V>

   class ConvertMap : public MapBase<typename M::Key, V> {

     const M &_m;

   public:

     typedef MapBase<typename M::Key, V> Parent;

     typedef typename Parent::Key Key;

     typedef typename Parent::Value Value;

     /// Constructor

     /// Constructor.

     /// \param m The underlying map.

     ConvertMap(const M &m) : _m(m) {}

     /// \e

     Value operator[](const Key &k) const { return _m[k]; }

   };

   /// Returns a \c ConvertMap class

   /// This function just returns a \c ConvertMap class.

   /// \relates ConvertMap

   template<typename V, typename M>

   inline ConvertMap<M, V> convertMap(const M &map) {

     return ConvertMap<M, V>(map);

   }

   /// Applies all map setting operations to two maps

   /// This map has two \ref concepts::WriteMap "writable map" parameters

   /// and each write request will be passed to both of them.

   /// If \c M1 is also \ref concepts::ReadMap "readable", then the read

   /// operations will return the corresponding values of \c M1.

   ///

   /// The \c Key and \c Value types are inherited from \c M1.

   /// The \c Key and \c Value of \c M2 must be convertible from those

   /// of \c M1.

   ///

   /// The simplest way of using this map is through the forkMap()

   /// function.

   template<typename  M1, typename M2>

   class ForkMap : public MapBase<typename M1::Key, typename M1::Value> {

     M1 &_m1;

     M2 &_m2;

   public:

     typedef MapBase<typename M1::Key, typename M1::Value> Parent;

     typedef typename Parent::Key Key;

     typedef typename Parent::Value Value;

     /// Constructor

     ForkMap(M1 &m1, M2 &m2) : _m1(m1), _m2(m2) {}

     /// Returns the value associated with the given key in the first map.

     Value operator[](const Key &k) const { return _m1[k]; }

     /// Sets the value associated with the given key in both maps.

     void set(const Key &k, const Value &v) { _m1.set(k,v); _m2.set(k,v); }

   };

   /// Returns a \c ForkMap class

   /// This function just returns a \c ForkMap class.

   /// \relates ForkMap

   template <typename M1, typename M2>

   inline ForkMap<M1,M2> forkMap(M1 &m1, M2 &m2) {

     return ForkMap<M1,M2>(m1,m2);

   }

   /// Sum of two maps

   /// This \ref concepts::ReadMap "read-only map" returns the sum

   /// of the values of the two given maps.

   /// Its \c Key and \c Value types are inherited from \c M1.

   /// The \c Key and \c Value of \c M2 must be convertible to those of

   /// \c M1.

   ///

   /// If \c m1 is of type \c M1 and \c m2 is of \c M2, then for

   /// \code

   ///   AddMap<M1,M2> am(m1,m2);

   /// \endcode

   /// <tt>am[x]</tt> will be equal to <tt>m1[x]+m2[x]</tt>.

   ///

   /// The simplest way of using this map is through the addMap()

   /// function.

   ///

   /// \sa SubMap, MulMap, DivMap

   /// \sa ShiftMap, ShiftWriteMap

   template<typename M1, typename M2>

   class AddMap : public MapBase<typename M1::Key, typename M1::Value> {

     const M1 &_m1;

     const M2 &_m2;

   public:

     typedef MapBase<typename M1::Key, typename M1::Value> Parent;

     typedef typename Parent::Key Key;

     typedef typename Parent::Value Value;

     /// Constructor

     AddMap(const M1 &m1, const M2 &m2) : _m1(m1), _m2(m2) {}

     /// \e

     Value operator[](const Key &k) const { return _m1[k]+_m2[k]; }

   };

   /// Returns an \c AddMap class

   /// This function just returns an \c AddMap class.

   ///

   /// For example, if \c m1 and \c m2 are both maps with \c double

   /// values, then <tt>addMap(m1,m2)[x]</tt> will be equal to

   /// <tt>m1[x]+m2[x]</tt>.

   ///

   /// \relates AddMap

   template<typename M1, typename M2>

   inline AddMap<M1, M2> addMap(const M1 &m1, const M2 &m2) {

     return AddMap<M1, M2>(m1,m2);

   }

   /// Difference of two maps

   /// This \ref concepts::ReadMap "read-only map" returns the difference

   /// of the values of the two given maps.

   /// Its \c Key and \c Value types are inherited from \c M1.

   /// The \c Key and \c Value of \c M2 must be convertible to those of

   /// \c M1.

   ///

   /// If \c m1 is of type \c M1 and \c m2 is of \c M2, then for

   /// \code

   ///   SubMap<M1,M2> sm(m1,m2);

   /// \endcode

   /// <tt>sm[x]</tt> will be equal to <tt>m1[x]-m2[x]</tt>.

   ///

   /// The simplest way of using this map is through the subMap()

   /// function.

   ///

   /// \sa AddMap, MulMap, DivMap

   template<typename M1, typename M2>

   class SubMap : public MapBase<typename M1::Key, typename M1::Value> {

     const M1 &_m1;

     const M2 &_m2;

   public:

     typedef MapBase<typename M1::Key, typename M1::Value> Parent;

     typedef typename Parent::Key Key;

     typedef typename Parent::Value Value;

     /// Constructor

     SubMap(const M1 &m1, const M2 &m2) : _m1(m1), _m2(m2) {}

     /// \e

     Value operator[](const Key &k) const { return _m1[k]-_m2[k]; }

   };

   /// Returns a \c SubMap class

   /// This function just returns a \c SubMap class.

   ///

   /// For example, if \c m1 and \c m2 are both maps with \c double

   /// values, then <tt>subMap(m1,m2)[x]</tt> will be equal to

   /// <tt>m1[x]-m2[x]</tt>.

   ///

   /// \relates SubMap

   template<typename M1, typename M2>

   inline SubMap<M1, M2> subMap(const M1 &m1, const M2 &m2) {

     return SubMap<M1, M2>(m1,m2);

   }

   /// Product of two maps

   /// This \ref concepts::ReadMap "read-only map" returns the product

   /// of the values of the two given maps.

   /// Its \c Key and \c Value types are inherited from \c M1.

   /// The \c Key and \c Value of \c M2 must be convertible to those of

   /// \c M1.

   ///

   /// If \c m1 is of type \c M1 and \c m2 is of \c M2, then for

   /// \code

   ///   MulMap<M1,M2> mm(m1,m2);

   /// \endcode

   /// <tt>mm[x]</tt> will be equal to <tt>m1[x]*m2[x]</tt>.

   ///

   /// The simplest way of using this map is through the mulMap()

   /// function.

   ///

   /// \sa AddMap, SubMap, DivMap

   /// \sa ScaleMap, ScaleWriteMap

   template<typename M1, typename M2>

   class MulMap : public MapBase<typename M1::Key, typename M1::Value> {

     const M1 &_m1;

     const M2 &_m2;

   public:

     typedef MapBase<typename M1::Key, typename M1::Value> Parent;

     typedef typename Parent::Key Key;

     typedef typename Parent::Value Value;

     /// Constructor

     MulMap(const M1 &m1,const M2 &m2) : _m1(m1), _m2(m2) {}

     /// \e

     Value operator[](const Key &k) const { return _m1[k]*_m2[k]; }

   };

   /// Returns a \c MulMap class

   /// This function just returns a \c MulMap class.

   ///

   /// For example, if \c m1 and \c m2 are both maps with \c double

   /// values, then <tt>mulMap(m1,m2)[x]</tt> will be equal to

   /// <tt>m1[x]*m2[x]</tt>.

   ///

   /// \relates MulMap

   template<typename M1, typename M2>

   inline MulMap<M1, M2> mulMap(const M1 &m1,const M2 &m2) {

     return MulMap<M1, M2>(m1,m2);

   }

   /// Quotient of two maps

   /// This \ref concepts::ReadMap "read-only map" returns the quotient

   /// of the values of the two given maps.

   /// Its \c Key and \c Value types are inherited from \c M1.

   /// The \c Key and \c Value of \c M2 must be convertible to those of

   /// \c M1.

   ///

   /// If \c m1 is of type \c M1 and \c m2 is of \c M2, then for

   /// \code

   ///   DivMap<M1,M2> dm(m1,m2);

   /// \endcode

   /// <tt>dm[x]</tt> will be equal to <tt>m1[x]/m2[x]</tt>.

   ///

   /// The simplest way of using this map is through the divMap()

   /// function.

   ///

   /// \sa AddMap, SubMap, MulMap

   template<typename M1, typename M2>

   class DivMap : public MapBase<typename M1::Key, typename M1::Value> {

     const M1 &_m1;

     const M2 &_m2;

   public:

     typedef MapBase<typename M1::Key, typename M1::Value> Parent;

     typedef typename Parent::Key Key;

     typedef typename Parent::Value Value;

     /// Constructor

     DivMap(const M1 &m1,const M2 &m2) : _m1(m1), _m2(m2) {}

     /// \e

     Value operator[](const Key &k) const { return _m1[k]/_m2[k]; }

   };

   /// Returns a \c DivMap class

   /// This function just returns a \c DivMap class.

   ///

   /// For example, if \c m1 and \c m2 are both maps with \c double

   /// values, then <tt>divMap(m1,m2)[x]</tt> will be equal to

   /// <tt>m1[x]/m2[x]</tt>.

   ///

   /// \relates DivMap

   template<typename M1, typename M2>

   inline DivMap<M1, M2> divMap(const M1 &m1,const M2 &m2) {

     return DivMap<M1, M2>(m1,m2);

   }

   /// Shifts a map with a constant.

   /// This \ref concepts::ReadMap "read-only map" returns the sum of

   /// the given map and a constant value (i.e. it shifts the map with

   /// the constant). Its \c Key and \c Value are inherited from \c M.

   ///

   /// Actually,

   /// \code

   ///   ShiftMap<M> sh(m,v);

   /// \endcode

   /// is equivalent to

   /// \code

   ///   ConstMap<M::Key, M::Value> cm(v);

   ///   AddMap<M, ConstMap<M::Key, M::Value> > sh(m,cm);

   /// \endcode

   ///

   /// The simplest way of using this map is through the shiftMap()

   /// function.

   ///

   /// \sa ShiftWriteMap

   template<typename M, typename C = typename M::Value>

   class ShiftMap : public MapBase<typename M::Key, typename M::Value> {

     const M &_m;

     C _v;

   public:

     typedef MapBase<typename M::Key, typename M::Value> Parent;

     typedef typename Parent::Key Key;

     typedef typename Parent::Value Value;

     /// Constructor

     /// Constructor.

     /// \param m The undelying map.

     /// \param v The constant value.

     ShiftMap(const M &m, const C &v) : _m(m), _v(v) {}

     /// \e

     Value operator[](const Key &k) const { return _m[k]+_v; }

   };

   /// Shifts a map with a constant (read-write version).

   /// This \ref concepts::ReadWriteMap "read-write map" returns the sum

   /// of the given map and a constant value (i.e. it shifts the map with

   /// the constant). Its \c Key and \c Value are inherited from \c M.

   /// It makes also possible to write the map.

   ///

   /// The simplest way of using this map is through the shiftWriteMap()

   /// function.

   ///

   /// \sa ShiftMap

   template<typename M, typename C = typename M::Value>

   class ShiftWriteMap : public MapBase<typename M::Key, typename M::Value> {

     M &_m;

     C _v;

   public:

     typedef MapBase<typename M::Key, typename M::Value> Parent;

     typedef typename Parent::Key Key;

     typedef typename Parent::Value Value;

     /// Constructor

     /// Constructor.

     /// \param m The undelying map.

     /// \param v The constant value.

     ShiftWriteMap(M &m, const C &v) : _m(m), _v(v) {}

     /// \e

     Value operator[](const Key &k) const { return _m[k]+_v; }

     /// \e

     void set(const Key &k, const Value &v) { _m.set(k, v-_v); }

   };

   /// Returns a \c ShiftMap class

   /// This function just returns a \c ShiftMap class.

   ///

   /// For example, if \c m is a map with \c double values and \c v is

   /// \c double, then <tt>shiftMap(m,v)[x]</tt> will be equal to

   /// <tt>m[x]+v</tt>.

   ///

   /// \relates ShiftMap

   template<typename M, typename C>

   inline ShiftMap<M, C> shiftMap(const M &m, const C &v) {

     return ShiftMap<M, C>(m,v);

   }

   /// Returns a \c ShiftWriteMap class

   /// This function just returns a \c ShiftWriteMap class.

   ///

   /// For example, if \c m is a map with \c double values and \c v is

   /// \c double, then <tt>shiftWriteMap(m,v)[x]</tt> will be equal to

   /// <tt>m[x]+v</tt>.

   /// Moreover it makes also possible to write the map.

   ///

   /// \relates ShiftWriteMap

   template<typename M, typename C>

   inline ShiftWriteMap<M, C> shiftWriteMap(M &m, const C &v) {

     return ShiftWriteMap<M, C>(m,v);

   }

   /// Scales a map with a constant.

   /// This \ref concepts::ReadMap "read-only map" returns the value of

   /// the given map multiplied from the left side with a constant value.

   /// Its \c Key and \c Value are inherited from \c M.

   ///

   /// Actually,

   /// \code

   ///   ScaleMap<M> sc(m,v);

   /// \endcode

   /// is equivalent to

   /// \code

   ///   ConstMap<M::Key, M::Value> cm(v);

   ///   MulMap<ConstMap<M::Key, M::Value>, M> sc(cm,m);

   /// \endcode

   ///

   /// The simplest way of using this map is through the scaleMap()

   /// function.

   ///

   /// \sa ScaleWriteMap

   template<typename M, typename C = typename M::Value>

   class ScaleMap : public MapBase<typename M::Key, typename M::Value> {

     const M &_m;

     C _v;

   public:

     typedef MapBase<typename M::Key, typename M::Value> Parent;

     typedef typename Parent::Key Key;

     typedef typename Parent::Value Value;

     /// Constructor

     /// Constructor.

     /// \param m The undelying map.

     /// \param v The constant value.

     ScaleMap(const M &m, const C &v) : _m(m), _v(v) {}

     /// \e

     Value operator[](const Key &k) const { return _v*_m[k]; }

   };

   /// Scales a map with a constant (read-write version).

   /// This \ref concepts::ReadWriteMap "read-write map" returns the value of

   /// the given map multiplied from the left side with a constant value.

   /// Its \c Key and \c Value are inherited from \c M.

   /// It can also be used as write map if the \c / operator is defined

   /// between \c Value and \c C and the given multiplier is not zero.

   ///

   /// The simplest way of using this map is through the scaleWriteMap()

   /// function.

   ///

   /// \sa ScaleMap

   template<typename M, typename C = typename M::Value>

   class ScaleWriteMap : public MapBase<typename M::Key, typename M::Value> {

     M &_m;

     C _v;

   public:

     typedef MapBase<typename M::Key, typename M::Value> Parent;

     typedef typename Parent::Key Key;

     typedef typename Parent::Value Value;

     /// Constructor

     /// Constructor.

     /// \param m The undelying map.

     /// \param v The constant value.

     ScaleWriteMap(M &m, const C &v) : _m(m), _v(v) {}

     /// \e

     Value operator[](const Key &k) const { return _v*_m[k]; }

     /// \e

     void set(const Key &k, const Value &v) { _m.set(k, v/_v); }

   };

   /// Returns a \c ScaleMap class

   /// This function just returns a \c ScaleMap class.

   ///

   /// For example, if \c m is a map with \c double values and \c v is

   /// \c double, then <tt>scaleMap(m,v)[x]</tt> will be equal to

   /// <tt>v*m[x]</tt>.

   ///

   /// \relates ScaleMap

   template<typename M, typename C>

   inline ScaleMap<M, C> scaleMap(const M &m, const C &v) {

     return ScaleMap<M, C>(m,v);

   }

   /// Returns a \c ScaleWriteMap class

   /// This function just returns a \c ScaleWriteMap class.

   ///

   /// For example, if \c m is a map with \c double values and \c v is

   /// \c double, then <tt>scaleWriteMap(m,v)[x]</tt> will be equal to

   /// <tt>v*m[x]</tt>.

   /// Moreover it makes also possible to write the map.

   ///

   /// \relates ScaleWriteMap

   template<typename M, typename C>

   inline ScaleWriteMap<M, C> scaleWriteMap(M &m, const C &v) {

     return ScaleWriteMap<M, C>(m,v);

   }

   /// Negative of a map

   /// This \ref concepts::ReadMap "read-only map" returns the negative

   /// of the values of the given map (using the unary \c - operator).

   /// Its \c Key and \c Value are inherited from \c M.

   ///

   /// If M::Value is \c int, \c double etc., then

   /// \code

   ///   NegMap<M> neg(m);

   /// \endcode

   /// is equivalent to

   /// \code

   ///   ScaleMap<M> neg(m,-1);

   /// \endcode

   ///

   /// The simplest way of using this map is through the negMap()

   /// function.

   ///

   /// \sa NegWriteMap

   template<typename M>

   class NegMap : public MapBase<typename M::Key, typename M::Value> {

     const M& _m;

   public:

     typedef MapBase<typename M::Key, typename M::Value> Parent;

     typedef typename Parent::Key Key;

     typedef typename Parent::Value Value;

     /// Constructor

     NegMap(const M &m) : _m(m) {}

     /// \e

     Value operator[](const Key &k) const { return -_m[k]; }

   };

   /// Negative of a map (read-write version)

   /// This \ref concepts::ReadWriteMap "read-write map" returns the

   /// negative of the values of the given map (using the unary \c -

   /// operator).

   /// Its \c Key and \c Value are inherited from \c M.

   /// It makes also possible to write the map.

   ///

   /// If M::Value is \c int, \c double etc., then

   /// \code

   ///   NegWriteMap<M> neg(m);

   /// \endcode

   /// is equivalent to

   /// \code

   ///   ScaleWriteMap<M> neg(m,-1);

   /// \endcode

   ///

   /// The simplest way of using this map is through the negWriteMap()

   /// function.

   ///

   /// \sa NegMap

   template<typename M>

   class NegWriteMap : public MapBase<typename M::Key, typename M::Value> {

     M &_m;

   public:

     typedef MapBase<typename M::Key, typename M::Value> Parent;

     typedef typename Parent::Key Key;

     typedef typename Parent::Value Value;

     /// Constructor

     NegWriteMap(M &m) : _m(m) {}

     /// \e

     Value operator[](const Key &k) const { return -_m[k]; }

     /// \e

     void set(const Key &k, const Value &v) { _m.set(k, -v); }

   };

   /// Returns a \c NegMap class

   /// This function just returns a \c NegMap class.

   ///

   /// For example, if \c m is a map with \c double values, then

   /// <tt>negMap(m)[x]</tt> will be equal to <tt>-m[x]</tt>.

   ///

   /// \relates NegMap

   template <typename M>

   inline NegMap<M> negMap(const M &m) {

     return NegMap<M>(m);

   }

   /// Returns a \c NegWriteMap class

   /// This function just returns a \c NegWriteMap class.

   ///

   /// For example, if \c m is a map with \c double values, then

   /// <tt>negWriteMap(m)[x]</tt> will be equal to <tt>-m[x]</tt>.

   /// Moreover it makes also possible to write the map.

   ///

   /// \relates NegWriteMap

   template <typename M>

   inline NegWriteMap<M> negWriteMap(M &m) {

     return NegWriteMap<M>(m);

   }

   /// Absolute value of a map

   /// This \ref concepts::ReadMap "read-only map" returns the absolute

   /// value of the values of the given map.

   /// Its \c Key and \c Value are inherited from \c M.

   /// \c Value must be comparable to \c 0 and the unary \c -

   /// operator must be defined for it, of course.

   ///

   /// The simplest way of using this map is through the absMap()

   /// function.

   template<typename M>

   class AbsMap : public MapBase<typename M::Key, typename M::Value> {

     const M &_m;

   public:

     typedef MapBase<typename M::Key, typename M::Value> Parent;

     typedef typename Parent::Key Key;

     typedef typename Parent::Value Value;

     /// Constructor

     AbsMap(const M &m) : _m(m) {}

     /// \e

     Value operator[](const Key &k) const {

       Value tmp = _m[k];

       return tmp >= 0 ? tmp : -tmp;

     }

   };

   /// Returns an \c AbsMap class

   /// This function just returns an \c AbsMap class.

   ///

   /// For example, if \c m is a map with \c double values, then

   /// <tt>absMap(m)[x]</tt> will be equal to <tt>m[x]</tt> if

   /// it is positive or zero and <tt>-m[x]</tt> if <tt>m[x]</tt> is

   /// negative.

   ///

   /// \relates AbsMap

   template<typename M>

   inline AbsMap<M> absMap(const M &m) {

     return AbsMap<M>(m);

   }

   /// @}

   // Logical maps and map adaptors:

   /// \addtogroup maps

   /// @{

   /// Constant \c true map.

   /// This \ref concepts::ReadMap "read-only map" assigns \c true to

   /// each key.

   ///

   /// Note that

   /// \code

   ///   TrueMap<K> tm;

   /// \endcode

   /// is equivalent to

   /// \code

   ///   ConstMap<K,bool> tm(true);

   /// \endcode

   ///

   /// \sa FalseMap

   /// \sa ConstMap

   template <typename K>

   class TrueMap : public MapBase<K, bool> {

   public:

     typedef MapBase<K, bool> Parent;

     typedef typename Parent::Key Key;

     typedef typename Parent::Value Value;

     /// Gives back \c true.

     Value operator[](const Key&) const { return true; }

   };

   /// Returns a \c TrueMap class

   /// This function just returns a \c TrueMap class.

   /// \relates TrueMap

   template<typename K>

   inline TrueMap<K> trueMap() {

     return TrueMap<K>();

   }

   /// Constant \c false map.

   /// This \ref concepts::ReadMap "read-only map" assigns \c false to

   /// each key.

   ///

   /// Note that

   /// \code

   ///   FalseMap<K> fm;

   /// \endcode

   /// is equivalent to

   /// \code

   ///   ConstMap<K,bool> fm(false);

   /// \endcode

   ///

   /// \sa TrueMap

   /// \sa ConstMap

   template <typename K>

   class FalseMap : public MapBase<K, bool> {

   public:

     typedef MapBase<K, bool> Parent;

     typedef typename Parent::Key Key;

     typedef typename Parent::Value Value;

     /// Gives back \c false.

     Value operator[](const Key&) const { return false; }

   };

   /// Returns a \c FalseMap class

   /// This function just returns a \c FalseMap class.

   /// \relates FalseMap

   template<typename K>

   inline FalseMap<K> falseMap() {

     return FalseMap<K>();

   }

   /// @}

   /// \addtogroup map_adaptors

   /// @{

   /// Logical 'and' of two maps

   /// This \ref concepts::ReadMap "read-only map" returns the logical

   /// 'and' of the values of the two given maps.

   /// Its \c Key type is inherited from \c M1 and its \c Value type is

   /// \c bool. \c M2::Key must be convertible to \c M1::Key.

   ///

   /// If \c m1 is of type \c M1 and \c m2 is of \c M2, then for

   /// \code

   ///   AndMap<M1,M2> am(m1,m2);

   /// \endcode

   /// <tt>am[x]</tt> will be equal to <tt>m1[x]&&m2[x]</tt>.

   ///

   /// The simplest way of using this map is through the andMap()

   /// function.

   ///

   /// \sa OrMap

   /// \sa NotMap, NotWriteMap

   template<typename M1, typename M2>

   class AndMap : public MapBase<typename M1::Key, bool> {

     const M1 &_m1;

     const M2 &_m2;

   public:

     typedef MapBase<typename M1::Key, bool> Parent;

     typedef typename Parent::Key Key;

     typedef typename Parent::Value Value;

     /// Constructor

     AndMap(const M1 &m1, const M2 &m2) : _m1(m1), _m2(m2) {}

     /// \e

     Value operator[](const Key &k) const { return _m1[k]&&_m2[k]; }

   };

   /// Returns an \c AndMap class

   /// This function just returns an \c AndMap class.

   ///

   /// For example, if \c m1 and \c m2 are both maps with \c bool values,

   /// then <tt>andMap(m1,m2)[x]</tt> will be equal to

   /// <tt>m1[x]&&m2[x]</tt>.

   ///

   /// \relates AndMap

   template<typename M1, typename M2>

   inline AndMap<M1, M2> andMap(const M1 &m1, const M2 &m2) {

     return AndMap<M1, M2>(m1,m2);

   }

   /// Logical 'or' of two maps

   /// This \ref concepts::ReadMap "read-only map" returns the logical

   /// 'or' of the values of the two given maps.

   /// Its \c Key type is inherited from \c M1 and its \c Value type is

   /// \c bool. \c M2::Key must be convertible to \c M1::Key.

   ///

   /// If \c m1 is of type \c M1 and \c m2 is of \c M2, then for

   /// \code

   ///   OrMap<M1,M2> om(m1,m2);

   /// \endcode

   /// <tt>om[x]</tt> will be equal to <tt>m1[x]||m2[x]</tt>.

   ///

   /// The simplest way of using this map is through the orMap()

   /// function.

   ///

   /// \sa AndMap

   /// \sa NotMap, NotWriteMap

   template<typename M1, typename M2>

   class OrMap : public MapBase<typename M1::Key, bool> {

     const M1 &_m1;

     const M2 &_m2;

   public:

     typedef MapBase<typename M1::Key, bool> Parent;

     typedef typename Parent::Key Key;

     typedef typename Parent::Value Value;

     /// Constructor

     OrMap(const M1 &m1, const M2 &m2) : _m1(m1), _m2(m2) {}

     /// \e

     Value operator[](const Key &k) const { return _m1[k]||_m2[k]; }

   };

   /// Returns an \c OrMap class

   /// This function just returns an \c OrMap class.

   ///

   /// For example, if \c m1 and \c m2 are both maps with \c bool values,

   /// then <tt>orMap(m1,m2)[x]</tt> will be equal to

   /// <tt>m1[x]||m2[x]</tt>.

   ///

   /// \relates OrMap

   template<typename M1, typename M2>

   inline OrMap<M1, M2> orMap(const M1 &m1, const M2 &m2) {

     return OrMap<M1, M2>(m1,m2);

   }

   /// Logical 'not' of a map

   /// This \ref concepts::ReadMap "read-only map" returns the logical

   /// negation of the values of the given map.

   /// Its \c Key is inherited from \c M and its \c Value is \c bool.

   ///

   /// The simplest way of using this map is through the notMap()

   /// function.

   ///

   /// \sa NotWriteMap

   template <typename M>

   class NotMap : public MapBase<typename M::Key, bool> {

     const M &_m;

   public:

     typedef MapBase<typename M::Key, bool> Parent;

     typedef typename Parent::Key Key;

     typedef typename Parent::Value Value;

     /// Constructor

     NotMap(const M &m) : _m(m) {}

     /// \e

     Value operator[](const Key &k) const { return !_m[k]; }

   };

   /// Logical 'not' of a map (read-write version)

   /// This \ref concepts::ReadWriteMap "read-write map" returns the

   /// logical negation of the values of the given map.

   /// Its \c Key is inherited from \c M and its \c Value is \c bool.

   /// It makes also possible to write the map. When a value is set,

   /// the opposite value is set to the original map.

   ///

   /// The simplest way of using this map is through the notWriteMap()

   /// function.

   ///

   /// \sa NotMap

   template <typename M>

   class NotWriteMap : public MapBase<typename M::Key, bool> {

     M &_m;

   public:

     typedef MapBase<typename M::Key, bool> Parent;

     typedef typename Parent::Key Key;

     typedef typename Parent::Value Value;

     /// Constructor

     NotWriteMap(M &m) : _m(m) {}

     /// \e

     Value operator[](const Key &k) const { return !_m[k]; }

     /// \e

     void set(const Key &k, bool v) { _m.set(k, !v); }

   };

   /// Returns a \c NotMap class

   /// This function just returns a \c NotMap class.

   ///

   /// For example, if \c m is a map with \c bool values, then

   /// <tt>notMap(m)[x]</tt> will be equal to <tt>!m[x]</tt>.

   ///

   /// \relates NotMap

   template <typename M>

   inline NotMap<M> notMap(const M &m) {

     return NotMap<M>(m);

   }

   /// Returns a \c NotWriteMap class

   /// This function just returns a \c NotWriteMap class.

   ///

   /// For example, if \c m is a map with \c bool values, then

   /// <tt>notWriteMap(m)[x]</tt> will be equal to <tt>!m[x]</tt>.

   /// Moreover it makes also possible to write the map.

   ///

   /// \relates NotWriteMap

   template <typename M>

   inline NotWriteMap<M> notWriteMap(M &m) {

     return NotWriteMap<M>(m);

   }

   /// Combination of two maps using the \c == operator

   /// This \ref concepts::ReadMap "read-only map" assigns \c true to

   /// the keys for which the corresponding values of the two maps are

   /// equal.

   /// Its \c Key type is inherited from \c M1 and its \c Value type is

   /// \c bool. \c M2::Key must be convertible to \c M1::Key.

   ///

   /// If \c m1 is of type \c M1 and \c m2 is of \c M2, then for

   /// \code

   ///   EqualMap<M1,M2> em(m1,m2);

   /// \endcode

   /// <tt>em[x]</tt> will be equal to <tt>m1[x]==m2[x]</tt>.

   ///

   /// The simplest way of using this map is through the equalMap()

   /// function.

   ///

   /// \sa LessMap

   template<typename M1, typename M2>

   class EqualMap : public MapBase<typename M1::Key, bool> {

     const M1 &_m1;

     const M2 &_m2;

   public:

     typedef MapBase<typename M1::Key, bool> Parent;

     typedef typename Parent::Key Key;

     typedef typename Parent::Value Value;

     /// Constructor

     EqualMap(const M1 &m1, const M2 &m2) : _m1(m1), _m2(m2) {}

     /// \e

     Value operator[](const Key &k) const { return _m1[k]==_m2[k]; }

   };

   /// Returns an \c EqualMap class

   /// This function just returns an \c EqualMap class.

   ///

   /// For example, if \c m1 and \c m2 are maps with keys and values of

   /// the same type, then <tt>equalMap(m1,m2)[x]</tt> will be equal to

   /// <tt>m1[x]==m2[x]</tt>.

   ///

   /// \relates EqualMap

   template<typename M1, typename M2>

   inline EqualMap<M1, M2> equalMap(const M1 &m1, const M2 &m2) {

     return EqualMap<M1, M2>(m1,m2);

   }

   /// Combination of two maps using the \c < operator

   /// This \ref concepts::ReadMap "read-only map" assigns \c true to

   /// the keys for which the corresponding value of the first map is

   /// less then the value of the second map.

   /// Its \c Key type is inherited from \c M1 and its \c Value type is

   /// \c bool. \c M2::Key must be convertible to \c M1::Key.

   ///

   /// If \c m1 is of type \c M1 and \c m2 is of \c M2, then for

   /// \code

   ///   LessMap<M1,M2> lm(m1,m2);

   /// \endcode

   /// <tt>lm[x]</tt> will be equal to <tt>m1[x]<m2[x]</tt>.

   ///

   /// The simplest way of using this map is through the lessMap()

   /// function.

   ///

   /// \sa EqualMap

   template<typename M1, typename M2>

   class LessMap : public MapBase<typename M1::Key, bool> {

     const M1 &_m1;

     const M2 &_m2;

   public:

     typedef MapBase<typename M1::Key, bool> Parent;

     typedef typename Parent::Key Key;

     typedef typename Parent::Value Value;

     /// Constructor

     LessMap(const M1 &m1, const M2 &m2) : _m1(m1), _m2(m2) {}

     /// \e

     Value operator[](const Key &k) const { return _m1[k]<_m2[k]; }

   };

   /// Returns an \c LessMap class

   /// This function just returns an \c LessMap class.

   ///

   /// For example, if \c m1 and \c m2 are maps with keys and values of

   /// the same type, then <tt>lessMap(m1,m2)[x]</tt> will be equal to

   /// <tt>m1[x]<m2[x]</tt>.

   ///

   /// \relates LessMap

   template<typename M1, typename M2>

   inline LessMap<M1, M2> lessMap(const M1 &m1, const M2 &m2) {

     return LessMap<M1, M2>(m1,m2);

   }

   namespace _maps_bits {

     template <typename _Iterator, typename Enable = void>

     struct IteratorTraits {

       typedef typename std::iterator_traits<_Iterator>::value_type Value;

     };

     template <typename _Iterator>

     struct IteratorTraits<_Iterator,

       typename exists<typename _Iterator::container_type>::type>

     {

       typedef typename _Iterator::container_type::value_type Value;

     };

   }

   /// \brief Writable bool map for logging each \c true assigned element

   ///

   /// A \ref concepts::WriteMap "writable" bool map for logging

   /// each \c true assigned element, i.e it copies subsequently each

   /// keys set to \c true to the given iterator.

   /// The most important usage of it is storing certain nodes or arcs

   /// that were marked \c true by an algorithm.

   ///

   /// There are several algorithms that provide solutions through bool

   /// maps and most of them assign \c true at most once for each key.

   /// In these cases it is a natural request to store each \c true

   /// assigned elements (in order of the assignment), which can be

   /// easily done with LoggerBoolMap.

   ///

   /// The simplest way of using this map is through the loggerBoolMap()

   /// function.

   ///

   /// \tparam It The type of the iterator.

   /// \tparam Ke The key type of the map. The default value set

   /// according to the iterator type should work in most cases.

   ///

   /// \note The container of the iterator must contain enough space

   /// for the elements or the iterator should be an inserter iterator.

 #ifdef DOXYGEN

   template <typename It, typename Ke>

 #else

   template <typename It,

             typename Ke=typename _maps_bits::IteratorTraits<It>::Value>

 #endif

   class LoggerBoolMap {

   public:

     typedef It Iterator;

     typedef Ke Key;

     typedef bool Value;

     /// Constructor

     LoggerBoolMap(Iterator it)

       : _begin(it), _end(it) {}

     /// Gives back the given iterator set for the first key

     Iterator begin() const {

       return _begin;

     }

     /// Gives back the the 'after the last' iterator

     Iterator end() const {

       return _end;

     }

     /// The set function of the map

     void set(const Key& key, Value value) {

       if (value) {

         *_end++ = key;

       }

     }

   private:

     Iterator _begin;

     Iterator _end;

   };

   /// Returns a \c LoggerBoolMap class

   /// This function just returns a \c LoggerBoolMap class.

   ///

   /// The most important usage of it is storing certain nodes or arcs

   /// that were marked \c true by an algorithm.

   /// For example it makes easier to store the nodes in the processing

   /// order of Dfs algorithm, as the following examples show.

   /// \code

   ///   std::vector<Node> v;

   ///   dfs(g,s).processedMap(loggerBoolMap(std::back_inserter(v))).run();

   /// \endcode

   /// \code

   ///   std::vector<Node> v(countNodes(g));

   ///   dfs(g,s).processedMap(loggerBoolMap(v.begin())).run();

   /// \endcode

   ///

   /// \note The container of the iterator must contain enough space

   /// for the elements or the iterator should be an inserter iterator.

   ///

   /// \note LoggerBoolMap is just \ref concepts::WriteMap "writable", so

   /// it cannot be used when a readable map is needed, for example as

   /// \c ReachedMap for \c Bfs, \c Dfs and \c Dijkstra algorithms.

   ///

   /// \relates LoggerBoolMap

   template<typename Iterator>

   inline LoggerBoolMap<Iterator> loggerBoolMap(Iterator it) {

     return LoggerBoolMap<Iterator>(it);

   }

   /// Provides an immutable and unique id for each item in the graph.

   /// The IdMap class provides a unique and immutable id for each item of the

   /// same type (e.g. node) in the graph. This id is <ul><li>\b unique:

   /// different items (nodes) get different ids <li>\b immutable: the id of an

   /// item (node) does not change (even if you delete other nodes).  </ul>

   /// Through this map you get access (i.e. can read) the inner id values of

   /// the items stored in the graph. This map can be inverted with its member

   /// class \c InverseMap or with the \c operator() member.

   ///

   template <typename _Graph, typename _Item>

   class IdMap {

   public:

     typedef _Graph Graph;

     typedef int Value;

     typedef _Item Item;

     typedef _Item Key;

     /// \brief Constructor.

     ///

     /// Constructor of the map.

     explicit IdMap(const Graph& graph) : _graph(&graph) {}

     /// \brief Gives back the \e id of the item.

     ///

     /// Gives back the immutable and unique \e id of the item.

     int operator[](const Item& item) const { return _graph->id(item);}

     /// \brief Gives back the item by its id.

     ///

     /// Gives back the item by its id.

     Item operator()(int id) { return _graph->fromId(id, Item()); }

   private:

     const Graph* _graph;

   public:

     /// \brief The class represents the inverse of its owner (IdMap).

     ///

     /// The class represents the inverse of its owner (IdMap).

     /// \see inverse()

     class InverseMap {

     public:

       /// \brief Constructor.

       ///

       /// Constructor for creating an id-to-item map.

       explicit InverseMap(const Graph& graph) : _graph(&graph) {}

       /// \brief Constructor.

       ///

       /// Constructor for creating an id-to-item map.

       explicit InverseMap(const IdMap& map) : _graph(map._graph) {}

       /// \brief Gives back the given item from its id.

       ///

       /// Gives back the given item from its id.

       ///

       Item operator[](int id) const { return _graph->fromId(id, Item());}

     private:

       const Graph* _graph;

     };

     /// \brief Gives back the inverse of the map.

     ///

     /// Gives back the inverse of the IdMap.

     InverseMap inverse() const { return InverseMap(*_graph);}

   };

   /// \brief General invertable graph-map type.

   /// This type provides simple invertable graph-maps.

   /// The InvertableMap wraps an arbitrary ReadWriteMap

   /// and if a key is set to a new value then store it

   /// in the inverse map.

   ///

   /// The values of the map can be accessed

   /// with stl compatible forward iterator.

   ///

   /// \tparam _Graph The graph type.

   /// \tparam _Item The item type of the graph.

   /// \tparam _Value The value type of the map.

   ///

   /// \see IterableValueMap

   template <typename _Graph, typename _Item, typename _Value>

   class InvertableMap

     : protected ItemSetTraits<_Graph, _Item>::template Map<_Value>::Type {

   private:

     typedef typename ItemSetTraits<_Graph, _Item>::

     template Map<_Value>::Type Map;

     typedef _Graph Graph;

     typedef std::map<_Value, _Item> Container;

     Container _inv_map;

   public:

     /// The key type of InvertableMap (Node, Arc, Edge).

     typedef typename Map::Key Key;

     /// The value type of the InvertableMap.

     typedef typename Map::Value Value;

     /// \brief Constructor.

     ///

     /// Construct a new InvertableMap for the graph.

     ///

     explicit InvertableMap(const Graph& graph) : Map(graph) {}

     /// \brief Forward iterator for values.

     ///

     /// This iterator is an stl compatible forward

     /// iterator on the values of the map. The values can

     /// be accessed in the [beginValue, endValue) range.

     ///

     class ValueIterator

       : public std::iterator<std::forward_iterator_tag, Value> {

       friend class InvertableMap;

     private:

       ValueIterator(typename Container::const_iterator _it)

         : it(_it) {}

     public:

       ValueIterator() {}

       ValueIterator& operator++() { ++it; return *this; }

       ValueIterator operator++(int) {

         ValueIterator tmp(*this);

         operator++();

         return tmp;

       }

       const Value& operator*() const { return it->first; }

       const Value* operator->() const { return &(it->first); }

       bool operator==(ValueIterator jt) const { return it == jt.it; }

       bool operator!=(ValueIterator jt) const { return it != jt.it; }

     private:

       typename Container::const_iterator it;

     };

     /// \brief Returns an iterator to the first value.

     ///

     /// Returns an stl compatible iterator to the

     /// first value of the map. The values of the

     /// map can be accessed in the [beginValue, endValue)

     /// range.

     ValueIterator beginValue() const {

       return ValueIterator(_inv_map.begin());

     }

     /// \brief Returns an iterator after the last value.

     ///

     /// Returns an stl compatible iterator after the

     /// last value of the map. The values of the

     /// map can be accessed in the [beginValue, endValue)

     /// range.

     ValueIterator endValue() const {

       return ValueIterator(_inv_map.end());

     }

     /// \brief The setter function of the map.

     ///

     /// Sets the mapped value.

     void set(const Key& key, const Value& val) {

       Value oldval = Map::operator[](key);

       typename Container::iterator it = _inv_map.find(oldval);

       if (it != _inv_map.end() && it->second == key) {

         _inv_map.erase(it);

       }

       _inv_map.insert(make_pair(val, key));

       Map::set(key, val);

     }

     /// \brief The getter function of the map.

     ///

     /// It gives back the value associated with the key.

     typename MapTraits<Map>::ConstReturnValue

     operator[](const Key& key) const {

       return Map::operator[](key);

     }

     /// \brief Gives back the item by its value.

     ///

     /// Gives back the item by its value.

     Key operator()(const Value& key) const {

       typename Container::const_iterator it = _inv_map.find(key);

       return it != _inv_map.end() ? it->second : INVALID;

     }

   protected:

     /// \brief Erase the key from the map.

     ///

     /// Erase the key to the map. It is called by the

     /// \c AlterationNotifier.

     virtual void erase(const Key& key) {

       Value val = Map::operator[](key);

       typename Container::iterator it = _inv_map.find(val);

       if (it != _inv_map.end() && it->second == key) {

         _inv_map.erase(it);

       }

       Map::erase(key);

     }

     /// \brief Erase more keys from the map.

     ///

     /// Erase more keys from the map. It is called by the

     /// \c AlterationNotifier.

     virtual void erase(const std::vector<Key>& keys) {

       for (int i = 0; i < int(keys.size()); ++i) {

         Value val = Map::operator[](keys[i]);

         typename Container::iterator it = _inv_map.find(val);

         if (it != _inv_map.end() && it->second == keys[i]) {

           _inv_map.erase(it);

         }

       }

       Map::erase(keys);

     }

     /// \brief Clear the keys from the map and inverse map.

     ///

     /// Clear the keys from the map and inverse map. It is called by the

     /// \c AlterationNotifier.

     virtual void clear() {

       _inv_map.clear();

       Map::clear();

     }

   public:

     /// \brief The inverse map type.

     ///

     /// The inverse of this map. The subscript operator of the map

     /// gives back always the item what was last assigned to the value.

     class InverseMap {

     public:

       /// \brief Constructor of the InverseMap.

       ///

       /// Constructor of the InverseMap.

       explicit InverseMap(const InvertableMap& inverted)

         : _inverted(inverted) {}

       /// The value type of the InverseMap.

       typedef typename InvertableMap::Key Value;

       /// The key type of the InverseMap.

       typedef typename InvertableMap::Value Key;

       /// \brief Subscript operator.

       ///

       /// Subscript operator. It gives back always the item

       /// what was last assigned to the value.

       Value operator[](const Key& key) const {

         return _inverted(key);

       }

     private:

       const InvertableMap& _inverted;

     };

     /// \brief It gives back the just readable inverse map.

     ///

     /// It gives back the just readable inverse map.

     InverseMap inverse() const {

       return InverseMap(*this);

     }

   };

   /// \brief Provides a mutable, continuous and unique descriptor for each

   /// item in the graph.

   ///

   /// The DescriptorMap class provides a unique and continuous (but mutable)

   /// descriptor (id) for each item of the same type (e.g. node) in the

   /// graph. This id is <ul><li>\b unique: different items (nodes) get

   /// different ids <li>\b continuous: the range of the ids is the set of

   /// integers between 0 and \c n-1, where \c n is the number of the items of

   /// this type (e.g. nodes) (so the id of a node can change if you delete an

   /// other node, i.e. this id is mutable).  </ul> This map can be inverted

   /// with its member class \c InverseMap, or with the \c operator() member.

   ///

   /// \tparam _Graph The graph class the \c DescriptorMap belongs to.

   /// \tparam _Item The Item is the Key of the Map. It may be Node, Arc or

   /// Edge.

   template <typename _Graph, typename _Item>

   class DescriptorMap

     : protected ItemSetTraits<_Graph, _Item>::template Map<int>::Type {

     typedef _Item Item;

     typedef typename ItemSetTraits<_Graph, _Item>::template Map<int>::Type Map;

   public:

     /// The graph class of DescriptorMap.

     typedef _Graph Graph;

     /// The key type of DescriptorMap (Node, Arc, Edge).

     typedef typename Map::Key Key;

     /// The value type of DescriptorMap.

     typedef typename Map::Value Value;

     /// \brief Constructor.

     ///

     /// Constructor for descriptor map.

     explicit DescriptorMap(const Graph& _graph) : Map(_graph) {

       Item it;

       const typename Map::Notifier* nf = Map::notifier();

       for (nf->first(it); it != INVALID; nf->next(it)) {

         Map::set(it, _inv_map.size());

         _inv_map.push_back(it);

       }

     }

   protected:

     /// \brief Add a new key to the map.

     ///

     /// Add a new key to the map. It is called by the

     /// \c AlterationNotifier.

     virtual void add(const Item& item) {

       Map::add(item);

       Map::set(item, _inv_map.size());

       _inv_map.push_back(item);

     }

     /// \brief Add more new keys to the map.

     ///

     /// Add more new keys to the map. It is called by the

     /// \c AlterationNotifier.

     virtual void add(const std::vector<Item>& items) {

       Map::add(items);

       for (int i = 0; i < int(items.size()); ++i) {

         Map::set(items[i], _inv_map.size());

         _inv_map.push_back(items[i]);

       }

     }

     /// \brief Erase the key from the map.

     ///

     /// Erase the key from the map. It is called by the

     /// \c AlterationNotifier.

     virtual void erase(const Item& item) {

       Map::set(_inv_map.back(), Map::operator[](item));

       _inv_map[Map::operator[](item)] = _inv_map.back();

       _inv_map.pop_back();

       Map::erase(item);

     }

     /// \brief Erase more keys from the map.

     ///

     /// Erase more keys from the map. It is called by the

     /// \c AlterationNotifier.

     virtual void erase(const std::vector<Item>& items) {

       for (int i = 0; i < int(items.size()); ++i) {

         Map::set(_inv_map.back(), Map::operator[](items[i]));

         _inv_map[Map::operator[](items[i])] = _inv_map.back();

         _inv_map.pop_back();

       }

       Map::erase(items);

     }

     /// \brief Build the unique map.

     ///

     /// Build the unique map. It is called by the

     /// \c AlterationNotifier.

     virtual void build() {

       Map::build();

       Item it;

       const typename Map::Notifier* nf = Map::notifier();

       for (nf->first(it); it != INVALID; nf->next(it)) {

         Map::set(it, _inv_map.size());

         _inv_map.push_back(it);

       }

     }

     /// \brief Clear the keys from the map.

     ///

     /// Clear the keys from the map. It is called by the

     /// \c AlterationNotifier.

     virtual void clear() {

       _inv_map.clear();

       Map::clear();

     }

   public:

     /// \brief Returns the maximal value plus one.

     ///

     /// Returns the maximal value plus one in the map.

     unsigned int size() const {

       return _inv_map.size();

     }

     /// \brief Swaps the position of the two items in the map.

     ///

     /// Swaps the position of the two items in the map.

     void swap(const Item& p, const Item& q) {

       int pi = Map::operator[](p);

       int qi = Map::operator[](q);

       Map::set(p, qi);

       _inv_map[qi] = p;

       Map::set(q, pi);

       _inv_map[pi] = q;

     }

     /// \brief Gives back the \e descriptor of the item.

     ///

     /// Gives back the mutable and unique \e descriptor of the map.

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

       return Map::operator[](item);

     }

     /// \brief Gives back the item by its descriptor.

     ///

     /// Gives back th item by its descriptor.

     Item operator()(int id) const {

       return _inv_map[id];

     }

   private:

     typedef std::vector<Item> Container;

     Container _inv_map;

   public:

     /// \brief The inverse map type of DescriptorMap.

     ///

     /// The inverse map type of DescriptorMap.

     class InverseMap {

     public:

       /// \brief Constructor of the InverseMap.

       ///

       /// Constructor of the InverseMap.

       explicit InverseMap(const DescriptorMap& inverted)

         : _inverted(inverted) {}

       /// The value type of the InverseMap.

       typedef typename DescriptorMap::Key Value;

       /// The key type of the InverseMap.

       typedef typename DescriptorMap::Value Key;

       /// \brief Subscript operator.

       ///

       /// Subscript operator. It gives back the item

       /// that the descriptor belongs to currently.

       Value operator[](const Key& key) const {

         return _inverted(key);

       }

       /// \brief Size of the map.

       ///

       /// Returns the size of the map.

       unsigned int size() const {

         return _inverted.size();

       }

     private:

       const DescriptorMap& _inverted;

     };

     /// \brief Gives back the inverse of the map.

     ///

     /// Gives back the inverse of the map.

     const InverseMap inverse() const {

       return InverseMap(*this);

     }

   };

   /// \brief Returns the source of the given arc.

   ///

   /// The SourceMap gives back the source Node of the given arc.

   /// \see TargetMap

   template <typename Digraph>

   class SourceMap {

   public:

     typedef typename Digraph::Node Value;

     typedef typename Digraph::Arc Key;

     /// \brief Constructor

     ///

     /// Constructor

     /// \param _digraph The digraph that the map belongs to.

     explicit SourceMap(const Digraph& digraph) : _digraph(digraph) {}

     /// \brief The subscript operator.

     ///

     /// The subscript operator.

     /// \param arc The arc

     /// \return The source of the arc

     Value operator[](const Key& arc) const {

       return _digraph.source(arc);

     }

   private:

     const Digraph& _digraph;

   };

   /// \brief Returns a \c SourceMap class.

   ///

   /// This function just returns an \c SourceMap class.

   /// \relates SourceMap

   template <typename Digraph>

   inline SourceMap<Digraph> sourceMap(const Digraph& digraph) {

     return SourceMap<Digraph>(digraph);

   }

   /// \brief Returns the target of the given arc.

   ///

   /// The TargetMap gives back the target Node of the given arc.

   /// \see SourceMap

   template <typename Digraph>

   class TargetMap {

   public:

     typedef typename Digraph::Node Value;

     typedef typename Digraph::Arc Key;

     /// \brief Constructor

     ///

     /// Constructor

     /// \param _digraph The digraph that the map belongs to.

     explicit TargetMap(const Digraph& digraph) : _digraph(digraph) {}

     /// \brief The subscript operator.

     ///

     /// The subscript operator.

     /// \param e The arc

     /// \return The target of the arc

     Value operator[](const Key& e) const {

       return _digraph.target(e);

     }

   private:

     const Digraph& _digraph;

   };

   /// \brief Returns a \c TargetMap class.

   ///

   /// This function just returns a \c TargetMap class.

   /// \relates TargetMap

   template <typename Digraph>

   inline TargetMap<Digraph> targetMap(const Digraph& digraph) {

     return TargetMap<Digraph>(digraph);

   }

   /// \brief Returns the "forward" directed arc view of an edge.

   ///

   /// Returns the "forward" directed arc view of an edge.

   /// \see BackwardMap

   template <typename Graph>

   class ForwardMap {

   public:

     typedef typename Graph::Arc Value;

     typedef typename Graph::Edge Key;

     /// \brief Constructor

     ///

     /// Constructor

     /// \param _graph The graph that the map belongs to.

     explicit ForwardMap(const Graph& graph) : _graph(graph) {}

     /// \brief The subscript operator.

     ///

     /// The subscript operator.

     /// \param key An edge

     /// \return The "forward" directed arc view of edge

     Value operator[](const Key& key) const {

       return _graph.direct(key, true);

     }

   private:

     const Graph& _graph;

   };

   /// \brief Returns a \c ForwardMap class.

   ///

   /// This function just returns an \c ForwardMap class.

   /// \relates ForwardMap

   template <typename Graph>

   inline ForwardMap<Graph> forwardMap(const Graph& graph) {

     return ForwardMap<Graph>(graph);

   }

   /// \brief Returns the "backward" directed arc view of an edge.

   ///

   /// Returns the "backward" directed arc view of an edge.

   /// \see ForwardMap

   template <typename Graph>

   class BackwardMap {

   public:

     typedef typename Graph::Arc Value;

     typedef typename Graph::Edge Key;

     /// \brief Constructor

     ///

     /// Constructor

     /// \param _graph The graph that the map belongs to.

     explicit BackwardMap(const Graph& graph) : _graph(graph) {}

     /// \brief The subscript operator.

     ///

     /// The subscript operator.

     /// \param key An edge

     /// \return The "backward" directed arc view of edge

     Value operator[](const Key& key) const {

       return _graph.direct(key, false);

     }

   private:

     const Graph& _graph;

   };

   /// \brief Returns a \c BackwardMap class

   /// This function just returns a \c BackwardMap class.

   /// \relates BackwardMap

   template <typename Graph>

   inline BackwardMap<Graph> backwardMap(const Graph& graph) {

     return BackwardMap<Graph>(graph);

   }

   /// \brief Potential difference map

   ///

   /// If there is an potential map on the nodes then we

   /// can get an arc map as we get the substraction of the

   /// values of the target and source.

   template <typename Digraph, typename NodeMap>

   class PotentialDifferenceMap {

   public:

     typedef typename Digraph::Arc Key;

     typedef typename NodeMap::Value Value;

     /// \brief Constructor

     ///

     /// Contructor of the map

     explicit PotentialDifferenceMap(const Digraph& digraph,

                                     const NodeMap& potential)

       : _digraph(digraph), _potential(potential) {}

     /// \brief Const subscription operator

     ///

     /// Const subscription operator

     Value operator[](const Key& arc) const {

       return _potential[_digraph.target(arc)] -

         _potential[_digraph.source(arc)];

     }

   private:

     const Digraph& _digraph;

     const NodeMap& _potential;

   };

   /// \brief Returns a PotentialDifferenceMap.

   ///

   /// This function just returns a PotentialDifferenceMap.

   /// \relates PotentialDifferenceMap

   template <typename Digraph, typename NodeMap>

   PotentialDifferenceMap<Digraph, NodeMap>

   potentialDifferenceMap(const Digraph& digraph, const NodeMap& potential) {

     return PotentialDifferenceMap<Digraph, NodeMap>(digraph, potential);

   }

   /// \brief Map of the node in-degrees.

   ///

   /// This map returns the in-degree of a node. Once it is constructed,

   /// the degrees are stored in a standard NodeMap, so each query is done

   /// in constant time. On the other hand, the values are updated automatically

   /// whenever the digraph changes.

   ///

   /// \warning Besides addNode() and addArc(), a digraph structure may provide

   /// alternative ways to modify the digraph. The correct behavior of InDegMap

   /// is not guarantied if these additional features are used. For example

   /// the functions \ref ListDigraph::changeSource() "changeSource()",

   /// \ref ListDigraph::changeTarget() "changeTarget()" and

   /// \ref ListDigraph::reverseArc() "reverseArc()"

   /// of \ref ListDigraph will \e not update the degree values correctly.

   ///

   /// \sa OutDegMap

   template <typename _Digraph>

   class InDegMap

     : protected ItemSetTraits<_Digraph, typename _Digraph::Arc>

       ::ItemNotifier::ObserverBase {

   public:

     typedef _Digraph Digraph;

     typedef int Value;

     typedef typename Digraph::Node Key;

     typedef typename ItemSetTraits<Digraph, typename Digraph::Arc>

     ::ItemNotifier::ObserverBase Parent;

   private:

     class AutoNodeMap

       : public ItemSetTraits<Digraph, Key>::template Map<int>::Type {

     public:

       typedef typename ItemSetTraits<Digraph, Key>::

       template Map<int>::Type Parent;

       AutoNodeMap(const Digraph& digraph) : Parent(digraph, 0) {}

       virtual void add(const Key& key) {

         Parent::add(key);

         Parent::set(key, 0);

       }

       virtual void add(const std::vector<Key>& keys) {

         Parent::add(keys);

         for (int i = 0; i < int(keys.size()); ++i) {

           Parent::set(keys[i], 0);

         }

       }

       virtual void build() {

         Parent::build();

         Key it;

         typename Parent::Notifier* nf = Parent::notifier();

         for (nf->first(it); it != INVALID; nf->next(it)) {

           Parent::set(it, 0);

         }

       }

     };

   public:

     /// \brief Constructor.

     ///

     /// Constructor for creating in-degree map.

     explicit InDegMap(const Digraph& digraph)

       : _digraph(digraph), _deg(digraph) {

       Parent::attach(_digraph.notifier(typename Digraph::Arc()));

       for(typename Digraph::NodeIt it(_digraph); it != INVALID; ++it) {

         _deg[it] = countInArcs(_digraph, it);

       }

     }

     /// Gives back the in-degree of a Node.

     int operator[](const Key& key) const {

       return _deg[key];

     }

   protected:

     typedef typename Digraph::Arc Arc;

     virtual void add(const Arc& arc) {

       ++_deg[_digraph.target(arc)];

     }

     virtual void add(const std::vector<Arc>& arcs) {

       for (int i = 0; i < int(arcs.size()); ++i) {

         ++_deg[_digraph.target(arcs[i])];

       }

     }

     virtual void erase(const Arc& arc) {

       --_deg[_digraph.target(arc)];

     }

     virtual void erase(const std::vector<Arc>& arcs) {

       for (int i = 0; i < int(arcs.size()); ++i) {

         --_deg[_digraph.target(arcs[i])];

       }

     }

     virtual void build() {

       for(typename Digraph::NodeIt it(_digraph); it != INVALID; ++it) {

         _deg[it] = countInArcs(_digraph, it);

       }

     }

     virtual void clear() {

       for(typename Digraph::NodeIt it(_digraph); it != INVALID; ++it) {

         _deg[it] = 0;

       }

     }

   private:

     const Digraph& _digraph;

     AutoNodeMap _deg;

   };

   /// \brief Map of the node out-degrees.

   ///

   /// This map returns the out-degree of a node. Once it is constructed,

   /// the degrees are stored in a standard NodeMap, so each query is done

   /// in constant time. On the other hand, the values are updated automatically

   /// whenever the digraph changes.

   ///

   /// \warning Besides addNode() and addArc(), a digraph structure may provide

   /// alternative ways to modify the digraph. The correct behavior of OutDegMap

   /// is not guarantied if these additional features are used. For example

   /// the functions \ref ListDigraph::changeSource() "changeSource()",

   /// \ref ListDigraph::changeTarget() "changeTarget()" and

   /// \ref ListDigraph::reverseArc() "reverseArc()"

   /// of \ref ListDigraph will \e not update the degree values correctly.

   ///

   /// \sa InDegMap

   template <typename _Digraph>

   class OutDegMap

     : protected ItemSetTraits<_Digraph, typename _Digraph::Arc>

       ::ItemNotifier::ObserverBase {

   public:

     typedef _Digraph Digraph;

     typedef int Value;

     typedef typename Digraph::Node Key;

     typedef typename ItemSetTraits<Digraph, typename Digraph::Arc>

     ::ItemNotifier::ObserverBase Parent;

   private:

     class AutoNodeMap

       : public ItemSetTraits<Digraph, Key>::template Map<int>::Type {

     public:

       typedef typename ItemSetTraits<Digraph, Key>::

       template Map<int>::Type Parent;

       AutoNodeMap(const Digraph& digraph) : Parent(digraph, 0) {}

       virtual void add(const Key& key) {

         Parent::add(key);

         Parent::set(key, 0);

       }

       virtual void add(const std::vector<Key>& keys) {

         Parent::add(keys);

         for (int i = 0; i < int(keys.size()); ++i) {

           Parent::set(keys[i], 0);

         }

       }

       virtual void build() {

         Parent::build();

         Key it;

         typename Parent::Notifier* nf = Parent::notifier();

         for (nf->first(it); it != INVALID; nf->next(it)) {

           Parent::set(it, 0);

         }

       }

     };

   public:

     /// \brief Constructor.

     ///

     /// Constructor for creating out-degree map.

     explicit OutDegMap(const Digraph& digraph)

       : _digraph(digraph), _deg(digraph) {

       Parent::attach(_digraph.notifier(typename Digraph::Arc()));

       for(typename Digraph::NodeIt it(_digraph); it != INVALID; ++it) {

         _deg[it] = countOutArcs(_digraph, it);

       }

     }