/* -*- 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.

  *

  */

 ///\ingroup paths

 ///\file

 ///\brief Classes for representing paths in digraphs.

 ///

 #ifndef LEMON_PATH_H

 #define LEMON_PATH_H

 #include <vector>

 #include <algorithm>

 #include <lemon/error.h>

 #include <lemon/core.h>

 #include <lemon/concepts/path.h>

 namespace lemon {

   /// \addtogroup paths

   /// @{

   /// \brief A structure for representing directed paths in a digraph.

   ///

   /// A structure for representing directed path in a digraph.

   /// \tparam _Digraph The digraph type in which the path is.

   ///

   /// In a sense, the path can be treated as a list of arcs. The

   /// lemon path type stores just this list. As a consequence, it

   /// cannot enumerate the nodes of the path and the source node of

   /// a zero length path is undefined.

   ///

   /// This implementation is a back and front insertable and erasable

   /// path type. It can be indexed in O(1) time. The front and back

   /// insertion and erase is done in O(1) (amortized) time. The

   /// implementation uses two vectors for storing the front and back

   /// insertions.

   template <typename _Digraph>

   class Path {

   public:

     typedef _Digraph Digraph;

     typedef typename Digraph::Arc Arc;

     /// \brief Default constructor

     ///

     /// Default constructor

     Path() {}

     /// \brief Template copy constructor

     ///

     /// This constuctor initializes the path from any other path type.

     /// It simply makes a copy of the given path.

     template <typename CPath>

     Path(const CPath& cpath) {

       copyPath(*this, cpath);

     }

     /// \brief Template copy assignment

     ///

     /// This operator makes a copy of a path of any other type.

     template <typename CPath>

     Path& operator=(const CPath& cpath) {

       copyPath(*this, cpath);

       return *this;

     }

     /// \brief Lemon style iterator for path arcs

     ///

     /// This class is used to iterate on the arcs of the paths.

     class ArcIt {

       friend class Path;

     public:

       /// \brief Default constructor

       ArcIt() {}

       /// \brief Invalid constructor

       ArcIt(Invalid) : path(0), idx(-1) {}

       /// \brief Initializate the iterator to the first arc of path

       ArcIt(const Path &_path)

         : path(&_path), idx(_path.empty() ? -1 : 0) {}

     private:

       ArcIt(const Path &_path, int _idx)

         : path(&_path), idx(_idx) {}

     public:

       /// \brief Conversion to Arc

       operator const Arc&() const {

         return path->nth(idx);

       }

       /// \brief Next arc

       ArcIt& operator++() {

         ++idx;

         if (idx >= path->length()) idx = -1;

         return *this;

       }

       /// \brief Comparison operator

       bool operator==(const ArcIt& e) const { return idx==e.idx; }

       /// \brief Comparison operator

       bool operator!=(const ArcIt& e) const { return idx!=e.idx; }

       /// \brief Comparison operator

       bool operator<(const ArcIt& e) const { return idx<e.idx; }

     private:

       const Path *path;

       int idx;

     };

     /// \brief Length of the path.

     int length() const { return head.size() + tail.size(); }

     /// \brief Return whether the path is empty.

     bool empty() const { return head.empty() && tail.empty(); }

     /// \brief Reset the path to an empty one.

     void clear() { head.clear(); tail.clear(); }

     /// \brief The nth arc.

     ///

     /// \pre n is in the [0..length() - 1] range

     const Arc& nth(int n) const {

       return n < int(head.size()) ? *(head.rbegin() + n) :

         *(tail.begin() + (n - head.size()));

     }

     /// \brief Initialize arc iterator to point to the nth arc

     ///

     /// \pre n is in the [0..length() - 1] range

     ArcIt nthIt(int n) const {

       return ArcIt(*this, n);

     }

     /// \brief The first arc of the path

     const Arc& front() const {

       return head.empty() ? tail.front() : head.back();

     }

     /// \brief Add a new arc before the current path

     void addFront(const Arc& arc) {

       head.push_back(arc);

     }

     /// \brief Erase the first arc of the path

     void eraseFront() {

       if (!head.empty()) {

         head.pop_back();

       } else {

         head.clear();

         int halfsize = tail.size() / 2;

         head.resize(halfsize);

         std::copy(tail.begin() + 1, tail.begin() + halfsize + 1,

                   head.rbegin());

         std::copy(tail.begin() + halfsize + 1, tail.end(), tail.begin());

         tail.resize(tail.size() - halfsize - 1);

       }

     }

     /// \brief The last arc of the path

     const Arc& back() const {

       return tail.empty() ? head.front() : tail.back();

     }

     /// \brief Add a new arc behind the current path

     void addBack(const Arc& arc) {

       tail.push_back(arc);

     }

     /// \brief Erase the last arc of the path

     void eraseBack() {

       if (!tail.empty()) {

         tail.pop_back();

       } else {

         int halfsize = head.size() / 2;

         tail.resize(halfsize);

         std::copy(head.begin() + 1, head.begin() + halfsize + 1,

                   tail.rbegin());

         std::copy(head.begin() + halfsize + 1, head.end(), head.begin());

         head.resize(head.size() - halfsize - 1);

       }

     }

     typedef True BuildTag;

     template <typename CPath>

     void build(const CPath& path) {

       int len = path.length();

       tail.reserve(len);

       for (typename CPath::ArcIt it(path); it != INVALID; ++it) {

         tail.push_back(it);

       }

     }

     template <typename CPath>

     void buildRev(const CPath& path) {

       int len = path.length();

       head.reserve(len);

       for (typename CPath::RevArcIt it(path); it != INVALID; ++it) {

         head.push_back(it);

       }

     }

   protected:

     typedef std::vector<Arc> Container;

     Container head, tail;

   };

   /// \brief A structure for representing directed paths in a digraph.

   ///

   /// A structure for representing directed path in a digraph.

   /// \tparam _Digraph The digraph type in which the path is.

   ///

   /// In a sense, the path can be treated as a list of arcs. The

   /// lemon path type stores just this list. As a consequence it

   /// cannot enumerate the nodes in the path and the zero length paths

   /// cannot store the source.

   ///

   /// This implementation is a just back insertable and erasable path

   /// type. It can be indexed in O(1) time. The back insertion and

   /// erasure is amortized O(1) time. This implementation is faster

   /// then the \c Path type because it use just one vector for the

   /// arcs.

   template <typename _Digraph>

   class SimplePath {

   public:

     typedef _Digraph Digraph;

     typedef typename Digraph::Arc Arc;

     /// \brief Default constructor

     ///

     /// Default constructor

     SimplePath() {}

     /// \brief Template copy constructor

     ///

     /// This path can be initialized with any other path type. It just

     /// makes a copy of the given path.

     template <typename CPath>

     SimplePath(const CPath& cpath) {

       copyPath(*this, cpath);

     }

     /// \brief Template copy assignment

     ///

     /// This path can be initialized with any other path type. It just

     /// makes a copy of the given path.

     template <typename CPath>

     SimplePath& operator=(const CPath& cpath) {

       copyPath(*this, cpath);

       return *this;

     }

     /// \brief Iterator class to iterate on the arcs of the paths

     ///

     /// This class is used to iterate on the arcs of the paths

     ///

     /// Of course it converts to Digraph::Arc

     class ArcIt {

       friend class SimplePath;

     public:

       /// Default constructor

       ArcIt() {}

       /// Invalid constructor

       ArcIt(Invalid) : path(0), idx(-1) {}

       /// \brief Initializate the constructor to the first arc of path

       ArcIt(const SimplePath &_path)

         : path(&_path), idx(_path.empty() ? -1 : 0) {}

     private:

       /// Constructor with starting point

       ArcIt(const SimplePath &_path, int _idx)

         : idx(_idx), path(&_path) {}

     public:

       ///Conversion to Digraph::Arc

       operator const Arc&() const {

         return path->nth(idx);

       }

       /// Next arc

       ArcIt& operator++() {

         ++idx;

         if (idx >= path->length()) idx = -1;

         return *this;

       }

       /// Comparison operator

       bool operator==(const ArcIt& e) const { return idx==e.idx; }

       /// Comparison operator

       bool operator!=(const ArcIt& e) const { return idx!=e.idx; }

       /// Comparison operator

       bool operator<(const ArcIt& e) const { return idx<e.idx; }

     private:

       const SimplePath *path;

       int idx;

     };

     /// \brief Length of the path.

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

     /// \brief Return true if the path is empty.

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

     /// \brief Reset the path to an empty one.

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

     /// \brief The nth arc.

     ///

     /// \pre n is in the [0..length() - 1] range

     const Arc& nth(int n) const {

       return data[n];

     }

     /// \brief  Initializes arc iterator to point to the nth arc.

     ArcIt nthIt(int n) const {

       return ArcIt(*this, n);

     }

     /// \brief The first arc of the path.

     const Arc& front() const {

       return data.front();

     }

     /// \brief The last arc of the path.

     const Arc& back() const {

       return data.back();

     }

     /// \brief Add a new arc behind the current path.

     void addBack(const Arc& arc) {

       data.push_back(arc);

     }

     /// \brief Erase the last arc of the path

     void eraseBack() {

       data.pop_back();

     }

     typedef True BuildTag;

     template <typename CPath>

     void build(const CPath& path) {

       int len = path.length();

       data.resize(len);

       int index = 0;

       for (typename CPath::ArcIt it(path); it != INVALID; ++it) {

         data[index] = it;;

         ++index;

       }

     }

     template <typename CPath>

     void buildRev(const CPath& path) {

       int len = path.length();

       data.resize(len);

       int index = len;

       for (typename CPath::RevArcIt it(path); it != INVALID; ++it) {

         --index;

         data[index] = it;;

       }

     }

   protected:

     typedef std::vector<Arc> Container;

     Container data;

   };

   /// \brief A structure for representing directed paths in a digraph.

   ///

   /// A structure for representing directed path in a digraph.

   /// \tparam _Digraph The digraph type in which the path is.

   ///

   /// In a sense, the path can be treated as a list of arcs. The

   /// lemon path type stores just this list. As a consequence it

   /// cannot enumerate the nodes in the path and the zero length paths

   /// cannot store the source.

   ///

   /// This implementation is a back and front insertable and erasable

   /// path type. It can be indexed in O(k) time, where k is the rank

   /// of the arc in the path. The length can be computed in O(n)

   /// time. The front and back insertion and erasure is O(1) time

   /// and it can be splited and spliced in O(1) time.

   template <typename _Digraph>

   class ListPath {

   public:

     typedef _Digraph Digraph;

     typedef typename Digraph::Arc Arc;

   protected:

     // the std::list<> is incompatible

     // hard to create invalid iterator

     struct Node {

       Arc arc;

       Node *next, *prev;

     };

     Node *first, *last;

     std::allocator<Node> alloc;

   public:

     /// \brief Default constructor

     ///

     /// Default constructor

     ListPath() : first(0), last(0) {}

     /// \brief Template copy constructor

     ///

     /// This path can be initialized with any other path type. It just

     /// makes a copy of the given path.

     template <typename CPath>

     ListPath(const CPath& cpath) : first(0), last(0) {

       copyPath(*this, cpath);

     }

     /// \brief Destructor of the path

     ///

     /// Destructor of the path

     ~ListPath() {

       clear();

     }

     /// \brief Template copy assignment

     ///

     /// This path can be initialized with any other path type. It just

     /// makes a copy of the given path.

     template <typename CPath>

     ListPath& operator=(const CPath& cpath) {

       copyPath(*this, cpath);

       return *this;

     }

     /// \brief Iterator class to iterate on the arcs of the paths

     ///

     /// This class is used to iterate on the arcs of the paths

     ///

     /// Of course it converts to Digraph::Arc

     class ArcIt {

       friend class ListPath;

     public:

       /// Default constructor

       ArcIt() {}

       /// Invalid constructor

       ArcIt(Invalid) : path(0), node(0) {}

       /// \brief Initializate the constructor to the first arc of path

       ArcIt(const ListPath &_path)

         : path(&_path), node(_path.first) {}

     protected:

       ArcIt(const ListPath &_path, Node *_node)

         : path(&_path), node(_node) {}

     public:

       ///Conversion to Digraph::Arc

       operator const Arc&() const {

         return node->arc;

       }

       /// Next arc

       ArcIt& operator++() {

         node = node->next;

         return *this;

       }

       /// Comparison operator

       bool operator==(const ArcIt& e) const { return node==e.node; }

       /// Comparison operator

       bool operator!=(const ArcIt& e) const { return node!=e.node; }

       /// Comparison operator

       bool operator<(const ArcIt& e) const { return node<e.node; }

     private:

       const ListPath *path;

       Node *node;

     };

     /// \brief The nth arc.

     ///

     /// This function looks for the nth arc in O(n) time.

     /// \pre n is in the [0..length() - 1] range

     const Arc& nth(int n) const {

       Node *node = first;

       for (int i = 0; i < n; ++i) {

         node = node->next;

       }

       return node->arc;

     }

     /// \brief Initializes arc iterator to point to the nth arc.

     ArcIt nthIt(int n) const {

       Node *node = first;

       for (int i = 0; i < n; ++i) {

         node = node->next;

       }

       return ArcIt(*this, node);

     }

     /// \brief Length of the path.

     int length() const {

       int len = 0;

       Node *node = first;

       while (node != 0) {

         node = node->next;

         ++len;

       }

       return len;

     }

     /// \brief Return true if the path is empty.

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

     /// \brief Reset the path to an empty one.

     void clear() {

       while (first != 0) {

         last = first->next;

         alloc.destroy(first);

         alloc.deallocate(first, 1);

         first = last;

       }

     }

     /// \brief The first arc of the path

     const Arc& front() const {

       return first->arc;

     }

     /// \brief Add a new arc before the current path

     void addFront(const Arc& arc) {

       Node *node = alloc.allocate(1);

       alloc.construct(node, Node());

       node->prev = 0;

       node->next = first;

       node->arc = arc;

       if (first) {

         first->prev = node;

         first = node;

       } else {

         first = last = node;

       }

     }

     /// \brief Erase the first arc of the path

     void eraseFront() {

       Node *node = first;

       first = first->next;

       if (first) {

         first->prev = 0;

       } else {

         last = 0;

       }

       alloc.destroy(node);

       alloc.deallocate(node, 1);

     }

     /// \brief The last arc of the path.

     const Arc& back() const {

       return last->arc;

     }

     /// \brief Add a new arc behind the current path.

     void addBack(const Arc& arc) {

       Node *node = alloc.allocate(1);

       alloc.construct(node, Node());

       node->next = 0;

       node->prev = last;

       node->arc = arc;

       if (last) {

         last->next = node;

         last = node;

       } else {

         last = first = node;

       }

     }

     /// \brief Erase the last arc of the path

     void eraseBack() {

       Node *node = last;

       last = last->prev;

       if (last) {

         last->next = 0;

       } else {

         first = 0;

       }

       alloc.destroy(node);

       alloc.deallocate(node, 1);

     }

     /// \brief Splice a path to the back of the current path.

     ///

     /// It splices \c tpath to the back of the current path and \c

     /// tpath becomes empty. The time complexity of this function is

     /// O(1).

     void spliceBack(ListPath& tpath) {

       if (first) {

         if (tpath.first) {

           last->next = tpath.first;

           tpath.first->prev = last;

           last = tpath.last;

         }

       } else {

         first = tpath.first;

         last = tpath.last;

       }

       tpath.first = tpath.last = 0;

     }

     /// \brief Splice a path to the front of the current path.

     ///

     /// It splices \c tpath before the current path and \c tpath

     /// becomes empty. The time complexity of this function

     /// is O(1).

     void spliceFront(ListPath& tpath) {

       if (first) {

         if (tpath.first) {

           first->prev = tpath.last;

           tpath.last->next = first;

           first = tpath.first;

         }

       } else {

         first = tpath.first;

         last = tpath.last;

       }

       tpath.first = tpath.last = 0;

     }

     /// \brief Splice a path into the current path.

     ///

     /// It splices the \c tpath into the current path before the

     /// position of \c it iterator and \c tpath becomes empty. The

     /// time complexity of this function is O(1). If the \c it is

     /// \c INVALID then it will splice behind the current path.

     void splice(ArcIt it, ListPath& tpath) {

       if (it.node) {

         if (tpath.first) {

           tpath.first->prev = it.node->prev;

           if (it.node->prev) {

             it.node->prev->next = tpath.first;

           } else {

             first = tpath.first;

           }

           it.node->prev = tpath.last;

           tpath.last->next = it.node;

         }

       } else {

         if (first) {

           if (tpath.first) {

             last->next = tpath.first;

             tpath.first->prev = last;

             last = tpath.last;

           }

         } else {

           first = tpath.first;

           last = tpath.last;

         }

       }

       tpath.first = tpath.last = 0;

     }

     /// \brief Split the current path.

     ///

     /// It splits the current path into two parts. The part before

     /// the iterator \c it will remain in the current path and the part

     /// starting with

     /// \c it will put into \c tpath. If \c tpath have arcs

     /// before the operation they are removed first.  The time

     /// complexity of this function is O(1) plus the the time of emtying

     /// \c tpath. If \c it is \c INVALID then it just clears \c tpath

     void split(ArcIt it, ListPath& tpath) {

       tpath.clear();

       if (it.node) {

         tpath.first = it.node;

         tpath.last = last;

         if (it.node->prev) {

           last = it.node->prev;

           last->next = 0;

         } else {

           first = last = 0;

         }

         it.node->prev = 0;

       }

     }

     typedef True BuildTag;

     template <typename CPath>

     void build(const CPath& path) {

       for (typename CPath::ArcIt it(path); it != INVALID; ++it) {

         addBack(it);

       }

     }

     template <typename CPath>

     void buildRev(const CPath& path) {

       for (typename CPath::RevArcIt it(path); it != INVALID; ++it) {

         addFront(it);

       }

     }

   };

   /// \brief A structure for representing directed paths in a digraph.

   ///

   /// A structure for representing directed path in a digraph.

   /// \tparam _Digraph The digraph type in which the path is.

   ///

   /// In a sense, the path can be treated as a list of arcs. The

   /// lemon path type stores just this list. As a consequence it

   /// cannot enumerate the nodes in the path and the source node of

   /// a zero length path is undefined.

   ///

   /// This implementation is completly static, i.e. it can be copy constucted

   /// or copy assigned from another path, but otherwise it cannot be

   /// modified.

   ///

   /// Being the the most memory efficient path type in LEMON,

   /// it is intented to be

   /// used when you want to store a large number of paths.

   template <typename _Digraph>

   class StaticPath {

   public:

     typedef _Digraph Digraph;

     typedef typename Digraph::Arc Arc;

     /// \brief Default constructor

     ///

     /// Default constructor

     StaticPath() : len(0), arcs(0) {}

     /// \brief Template copy constructor

     ///

     /// This path can be initialized from any other path type.

     template <typename CPath>

     StaticPath(const CPath& cpath) : arcs(0) {

       copyPath(*this, cpath);

     }

     /// \brief Destructor of the path

     ///

     /// Destructor of the path

     ~StaticPath() {

       if (arcs) delete[] arcs;

     }

     /// \brief Template copy assignment

     ///

     /// This path can be made equal to any other path type. It simply

     /// makes a copy of the given path.

     template <typename CPath>

     StaticPath& operator=(const CPath& cpath) {

       copyPath(*this, cpath);

       return *this;

     }

     /// \brief Iterator class to iterate on the arcs of the paths

     ///

     /// This class is used to iterate on the arcs of the paths

     ///

     /// Of course it converts to Digraph::Arc

     class ArcIt {

       friend class StaticPath;

     public:

       /// Default constructor

       ArcIt() {}

       /// Invalid constructor

       ArcIt(Invalid) : path(0), idx(-1) {}

       /// Initializate the constructor to the first arc of path

       ArcIt(const StaticPath &_path)

         : path(&_path), idx(_path.empty() ? -1 : 0) {}

     private:

       /// Constructor with starting point

       ArcIt(const StaticPath &_path, int _idx)

         : idx(_idx), path(&_path) {}

     public:

       ///Conversion to Digraph::Arc

       operator const Arc&() const {

         return path->nth(idx);

       }

       /// Next arc

       ArcIt& operator++() {

         ++idx;

         if (idx >= path->length()) idx = -1;

         return *this;

       }

       /// Comparison operator

       bool operator==(const ArcIt& e) const { return idx==e.idx; }

       /// Comparison operator

       bool operator!=(const ArcIt& e) const { return idx!=e.idx; }

       /// Comparison operator

       bool operator<(const ArcIt& e) const { return idx<e.idx; }

     private:

       const StaticPath *path;

       int idx;

     };

     /// \brief The nth arc.

     ///

     /// \pre n is in the [0..length() - 1] range

     const Arc& nth(int n) const {

       return arcs[n];

     }

     /// \brief The arc iterator pointing to the nth arc.

     ArcIt nthIt(int n) const {

       return ArcIt(*this, n);

     }

     /// \brief The length of the path.

     int length() const { return len; }

     /// \brief Return true when the path is empty.

     int empty() const { return len == 0; }

     /// \break Erase all arcs in the digraph.

     void clear() {

       len = 0;

       if (arcs) delete[] arcs;

       arcs = 0;

     }

     /// \brief The first arc of the path.

     const Arc& front() const {

       return arcs[0];

     }

     /// \brief The last arc of the path.

     const Arc& back() const {

       return arcs[len - 1];

     }

     typedef True BuildTag;

     template <typename CPath>

     void build(const CPath& path) {

       len = path.length();

       arcs = new Arc[len];

       int index = 0;

       for (typename CPath::ArcIt it(path); it != INVALID; ++it) {

         arcs[index] = it;

         ++index;

       }

     }

     template <typename CPath>

     void buildRev(const CPath& path) {

       len = path.length();

       arcs = new Arc[len];

       int index = len;

       for (typename CPath::RevArcIt it(path); it != INVALID; ++it) {

         --index;

         arcs[index] = it;

       }

     }

   private:

     int len;

     Arc* arcs;

   };

   ///////////////////////////////////////////////////////////////////////

   // Additional utilities

   ///////////////////////////////////////////////////////////////////////

   namespace _path_bits {

     template <typename Path, typename Enable = void>

     struct RevPathTagIndicator {

       static const bool value = false;

     };

     template <typename Path>

     struct RevPathTagIndicator<

       Path,

       typename enable_if<typename Path::RevPathTag, void>::type

       > {

       static const bool value = true;

     };

     template <typename Path, typename Enable = void>

     struct BuildTagIndicator {

       static const bool value = false;

     };

     template <typename Path>

     struct BuildTagIndicator<

       Path,

       typename enable_if<typename Path::BuildTag, void>::type

     > {

       static const bool value = true;

     };

     template <typename Target, typename Source,

               bool buildEnable = BuildTagIndicator<Target>::value,

               bool revEnable = RevPathTagIndicator<Source>::value>

     struct PathCopySelector {

       static void copy(Target& target, const Source& source) {

         target.clear();

         for (typename Source::ArcIt it(source); it != INVALID; ++it) {

           target.addBack(it);

         }

       }

     };

     template <typename Target, typename Source>

     struct PathCopySelector<Target, Source, false, true> {

       static void copy(Target& target, const Source& source) {

         target.clear();

         for (typename Source::RevArcIt it(source); it != INVALID; ++it) {

           target.addFront(it);

         }

       }

     };

     template <typename Target, typename Source>

     struct PathCopySelector<Target, Source, true, false> {

       static void copy(Target& target, const Source& source) {

         target.clear();

         target.build(source);

       }

     };

     template <typename Target, typename Source>

     struct PathCopySelector<Target, Source, true, true> {

       static void copy(Target& target, const Source& source) {

         target.clear();

         target.buildRev(source);

       }

     };

   }

   /// \brief Make a copy of a path.

   ///

   ///  This function makes a copy of a path.

   template <typename Target, typename Source>

   void copyPath(Target& target, const Source& source) {

     checkConcept<concepts::PathDumper<typename Source::Digraph>, Source>();

     _path_bits::PathCopySelector<Target, Source>::copy(target, source);

   }

   /// \brief Check the consistency of a path.

   ///

   /// This function checks that the target of each arc is the same

   /// as the source of the next one.

   ///

   template <typename Digraph, typename Path>

   bool checkPath(const Digraph& digraph, const Path& path) {

     typename Path::ArcIt it(path);

     if (it == INVALID) return true;

     typename Digraph::Node node = digraph.target(it);

     ++it;

     while (it != INVALID) {

       if (digraph.source(it) != node) return false;

       node = digraph.target(it);

       ++it;

     }

     return true;

   }

   /// \brief The source of a path

   ///

   /// This function returns the source of the given path.

   template <typename Digraph, typename Path>

   typename Digraph::Node pathSource(const Digraph& digraph, const Path& path) {

     return digraph.source(path.front());

   }

   /// \brief The target of a path

   ///

   /// This function returns the target of the given path.

   template <typename Digraph, typename Path>

   typename Digraph::Node pathTarget(const Digraph& digraph, const Path& path) {

     return digraph.target(path.back());

   }

   /// \brief Class which helps to iterate through the nodes of a path

   ///

   /// In a sense, the path can be treated as a list of arcs. The

   /// lemon path type stores only this list. As a consequence, it

   /// cannot enumerate the nodes in the path and the zero length paths

   /// cannot have a source node.

   ///

   /// This class implements the node iterator of a path structure. To

   /// provide this feature, the underlying digraph should be passed to

   /// the constructor of the iterator.

   template <typename Path>

   class PathNodeIt {

   private:

     const typename Path::Digraph *_digraph;

     typename Path::ArcIt _it;

     typename Path::Digraph::Node _nd;

   public:

     typedef typename Path::Digraph Digraph;

     typedef typename Digraph::Node Node;

     /// Default constructor

     PathNodeIt() {}

     /// Invalid constructor

     PathNodeIt(Invalid)

       : _digraph(0), _it(INVALID), _nd(INVALID) {}

     /// Constructor

     PathNodeIt(const Digraph& digraph, const Path& path)

       : _digraph(&digraph), _it(path) {

       _nd = (_it != INVALID ? _digraph->source(_it) : INVALID);

     }

     /// Constructor

     PathNodeIt(const Digraph& digraph, const Path& path, const Node& src)

       : _digraph(&digraph), _it(path), _nd(src) {}

     ///Conversion to Digraph::Node

     operator Node() const {

       return _nd;

     }

     /// Next node

     PathNodeIt& operator++() {

       if (_it == INVALID) _nd = INVALID;

       else {

         _nd = _digraph->target(_it);

         ++_it;

       }

       return *this;

     }

     /// Comparison operator

     bool operator==(const PathNodeIt& n) const {

       return _it == n._it && _nd == n._nd;

     }

     /// Comparison operator

     bool operator!=(const PathNodeIt& n) const {

       return _it != n._it || _nd != n._nd;

     }

     /// Comparison operator

     bool operator<(const PathNodeIt& n) const {

       return (_it < n._it && _nd != INVALID);

     }

   };

   ///@}

 } // namespace lemon

 #endif // LEMON_PATH_H