/* -*- C++ -*-

 *

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

 *

 * Copyright (C) 2003-2008

 * Egervary Jeno Kombinatorikus Optimalizalasi Kutatocsoport

 * (Egervary Research Group on Combinatorial Optimization, EGRES).

 *

 * Permission to use, modify and distribute this software is granted

 * provided that this copyright notice appears in all copies. For

 * precise terms see the accompanying LICENSE file.

 *

 * This software is provided "AS IS" with no warranty of any kind,

 * express or implied, and with no claim as to its suitability for any

 * purpose.

 *

 */

#ifndef LEMON_MAPS_H

#define LEMON_MAPS_H

#include <iterator>

#include <functional>

#include <vector>

#include <lemon/bits/utility.h>

#include <lemon/bits/traits.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 \ref NullMap class

  /// This function just returns a \ref 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 \ref 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 is 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 \ref ConstMap class

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

  /// \relates ConstMap

  template<typename K, typename V>

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

    return ConstMap<K, V>(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 \ref 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 \ref ConstMap class with inlined constant value

  /// This function just returns a \ref 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 \ref IdentityMap class

  /// This function just returns an \ref 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

  /// \ref UnionFind, \ref 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 \ref 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 \ref RangeMap class

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

  /// \c std::vector

  /// This function just returns a \ref 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 \ref 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 \ref SparseMap class

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

  /// \c std::map

  /// This function just returns a \ref 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

  ///

  /// \todo Check the requirements.

  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 \ref ComposeMap class

  /// This function just returns a \ref 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

  ///

  /// \todo Check the requirements.

  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 \ref CombineMap class

  /// This function just returns a \ref 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> {

    const 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 \ref FunctorToMap class

  /// This function just returns a \ref 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 \ref MapToFunctor class

  /// This function just returns a \ref 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 \ref ConvertMap class

  /// This function just returns a \ref 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 \ref ForkMap class

  /// This function just returns a \ref 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 \ref AddMap class

  /// This function just returns an \ref 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 \ref SubMap class

  /// This function just returns a \ref 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 \ref MulMap class

  /// This function just returns a \ref 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 \ref DivMap class

  /// This function just returns a \ref 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;