/* -*- 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.

 *

 */

///\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/bits/invalid.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;