# HG changeset patch
# User Peter Kovacs <kpeter@inf.elte.hu>
# Date 2009-07-08 17:21:30
# Node ID 0747f332c4782fd4fd5e73d83f34fc6ee201b323
# Parent  9f529abcaebf13f19e61ba24fdd2c3631860af91

Improve and unify the documentation of heaps (#299)
and avoid a warning in SimpleBucketHeap::operator[].

diff --git a/lemon/bin_heap.h b/lemon/bin_heap.h
--- a/lemon/bin_heap.h
+++ b/lemon/bin_heap.h
@@ -21,7 +21,7 @@
 
 ///\ingroup auxdat
 ///\file
-///\brief Binary Heap implementation.
+///\brief Binary heap implementation.
 
 #include <vector>
 #include <utility>
@@ -29,45 +29,41 @@
 
 namespace lemon {
 
-  ///\ingroup auxdat
+  /// \ingroup auxdat
   ///
-  ///\brief A Binary Heap implementation.
+  /// \brief Binary heap data structure.
   ///
-  ///This class implements the \e binary \e heap data structure.
+  /// This class implements the \e binary \e heap data structure.
+  /// It fully conforms to the \ref concepts::Heap "heap concept".
   ///
-  ///A \e heap is a data structure for storing items with specified values
-  ///called \e priorities in such a way that finding the item with minimum
-  ///priority is efficient. \c CMP specifies the ordering of the priorities.
-  ///In a heap one can change the priority of an item, add or erase an
-  ///item, etc.
-  ///
-  ///\tparam PR Type of the priority of the items.
-  ///\tparam IM A read and writable item map with int values, used internally
-  ///to handle the cross references.
-  ///\tparam CMP A functor class for the ordering of the priorities.
-  ///The default is \c std::less<PR>.
-  ///
-  ///\sa FibHeap
-  ///\sa Dijkstra
+  /// \tparam PR Type of the priorities of the items.
+  /// \tparam IM A read-writable item map with \c int values, used
+  /// internally to handle the cross references.
+  /// \tparam CMP A functor class for comparing the priorities.
+  /// The default is \c std::less<PR>.
+#ifdef DOXYGEN
+  template <typename PR, typename IM, typename CMP>
+#else
   template <typename PR, typename IM, typename CMP = std::less<PR> >
+#endif
   class BinHeap {
+  public:
 
-  public:
-    ///\e
+    /// Type of the item-int map.
     typedef IM ItemIntMap;
-    ///\e
+    /// Type of the priorities.
     typedef PR Prio;
-    ///\e
+    /// Type of the items stored in the heap.
     typedef typename ItemIntMap::Key Item;
-    ///\e
+    /// Type of the item-priority pairs.
     typedef std::pair<Item,Prio> Pair;
-    ///\e
+    /// Functor type for comparing the priorities.
     typedef CMP Compare;
 
-    /// \brief Type to represent the items states.
+    /// \brief Type to represent the states of the items.
     ///
-    /// Each Item element have a state associated to it. It may be "in heap",
-    /// "pre heap" or "post heap". The latter two are indifferent from the
+    /// Each item has a state associated to it. It can be "in heap",
+    /// "pre-heap" or "post-heap". The latter two are indifferent from the
     /// heap's point of view, but may be useful to the user.
     ///
     /// The item-int map must be initialized in such way that it assigns
@@ -84,42 +80,43 @@
     ItemIntMap &_iim;
 
   public:
-    /// \brief The constructor.
+
+    /// \brief Constructor.
     ///
-    /// The constructor.
-    /// \param map should be given to the constructor, since it is used
-    /// internally to handle the cross references. The value of the map
-    /// must be \c PRE_HEAP (<tt>-1</tt>) for every item.
+    /// Constructor.
+    /// \param map A map that assigns \c int values to the items.
+    /// It is used internally to handle the cross references.
+    /// The assigned value must be \c PRE_HEAP (<tt>-1</tt>) for each item.
     explicit BinHeap(ItemIntMap &map) : _iim(map) {}
 
-    /// \brief The constructor.
+    /// \brief Constructor.
     ///
-    /// The constructor.
-    /// \param map should be given to the constructor, since it is used
-    /// internally to handle the cross references. The value of the map
-    /// should be PRE_HEAP (-1) for each element.
-    ///
-    /// \param comp The comparator function object.
+    /// Constructor.
+    /// \param map A map that assigns \c int values to the items.
+    /// It is used internally to handle the cross references.
+    /// The assigned value must be \c PRE_HEAP (<tt>-1</tt>) for each item.
+    /// \param comp The function object used for comparing the priorities.
     BinHeap(ItemIntMap &map, const Compare &comp)
       : _iim(map), _comp(comp) {}
 
 
-    /// The number of items stored in the heap.
+    /// \brief The number of items stored in the heap.
     ///
-    /// \brief Returns the number of items stored in the heap.
+    /// This function returns the number of items stored in the heap.
     int size() const { return _data.size(); }
 
-    /// \brief Checks if the heap stores no items.
+    /// \brief Check if the heap is empty.
     ///
-    /// Returns \c true if and only if the heap stores no items.
+    /// This function returns \c true if the heap is empty.
     bool empty() const { return _data.empty(); }
 
-    /// \brief Make empty this heap.
+    /// \brief Make the heap empty.
     ///
-    /// Make empty this heap. It does not change the cross reference map.
-    /// If you want to reuse what is not surely empty you should first clear
-    /// the heap and after that you should set the cross reference map for
-    /// each item to \c PRE_HEAP.
+    /// This functon makes the heap empty.
+    /// It does not change the cross reference map. If you want to reuse
+    /// a heap that is not surely empty, you should first clear it and
+    /// then you should set the cross reference map to \c PRE_HEAP
+    /// for each item.
     void clear() {
       _data.clear();
     }
@@ -171,44 +168,47 @@
     }
 
   public:
+
     /// \brief Insert a pair of item and priority into the heap.
     ///
-    /// Adds \c p.first to the heap with priority \c p.second.
+    /// This function inserts \c p.first to the heap with priority
+    /// \c p.second.
     /// \param p The pair to insert.
+    /// \pre \c p.first must not be stored in the heap.
     void push(const Pair &p) {
       int n = _data.size();
       _data.resize(n+1);
       bubble_up(n, p);
     }
 
-    /// \brief Insert an item into the heap with the given heap.
+    /// \brief Insert an item into the heap with the given priority.
     ///
-    /// Adds \c i to the heap with priority \c p.
+    /// This function inserts the given item into the heap with the
+    /// given priority.
     /// \param i The item to insert.
     /// \param p The priority of the item.
+    /// \pre \e i must not be stored in the heap.
     void push(const Item &i, const Prio &p) { push(Pair(i,p)); }
 
-    /// \brief Returns the item with minimum priority relative to \c Compare.
+    /// \brief Return the item having minimum priority.
     ///
-    /// This method returns the item with minimum priority relative to \c
-    /// Compare.
-    /// \pre The heap must be nonempty.
+    /// This function returns the item having minimum priority.
+    /// \pre The heap must be non-empty.
     Item top() const {
       return _data[0].first;
     }
 
-    /// \brief Returns the minimum priority relative to \c Compare.
+    /// \brief The minimum priority.
     ///
-    /// It returns the minimum priority relative to \c Compare.
-    /// \pre The heap must be nonempty.
+    /// This function returns the minimum priority.
+    /// \pre The heap must be non-empty.
     Prio prio() const {
       return _data[0].second;
     }
 
-    /// \brief Deletes the item with minimum priority relative to \c Compare.
+    /// \brief Remove the item having minimum priority.
     ///
-    /// This method deletes the item with minimum priority relative to \c
-    /// Compare from the heap.
+    /// This function removes the item having minimum priority.
     /// \pre The heap must be non-empty.
     void pop() {
       int n = _data.size()-1;
@@ -219,11 +219,12 @@
       _data.pop_back();
     }
 
-    /// \brief Deletes \c i from the heap.
+    /// \brief Remove the given item from the heap.
     ///
-    /// This method deletes item \c i from the heap.
-    /// \param i The item to erase.
-    /// \pre The item should be in the heap.
+    /// This function removes the given item from the heap if it is
+    /// already stored.
+    /// \param i The item to delete.
+    /// \pre \e i must be in the heap.
     void erase(const Item &i) {
       int h = _iim[i];
       int n = _data.size()-1;
@@ -236,22 +237,22 @@
       _data.pop_back();
     }
 
-
-    /// \brief Returns the priority of \c i.
+    /// \brief The priority of the given item.
     ///
-    /// This function returns the priority of item \c i.
+    /// This function returns the priority of the given item.
     /// \param i The item.
-    /// \pre \c i must be in the heap.
+    /// \pre \e i must be in the heap.
     Prio operator[](const Item &i) const {
       int idx = _iim[i];
       return _data[idx].second;
     }
 
-    /// \brief \c i gets to the heap with priority \c p independently
-    /// if \c i was already there.
+    /// \brief Set the priority of an item or insert it, if it is
+    /// not stored in the heap.
     ///
-    /// This method calls \ref push(\c i, \c p) if \c i is not stored
-    /// in the heap and sets the priority of \c i to \c p otherwise.
+    /// This method sets the priority of the given item if it is
+    /// already stored in the heap. Otherwise it inserts the given
+    /// item into the heap with the given priority.
     /// \param i The item.
     /// \param p The priority.
     void set(const Item &i, const Prio &p) {
@@ -267,37 +268,35 @@
       }
     }
 
-    /// \brief Decreases the priority of \c i to \c p.
+    /// \brief Decrease the priority of an item to the given value.
     ///
-    /// This method decreases the priority of item \c i to \c p.
+    /// This function decreases the priority of an item to the given value.
     /// \param i The item.
     /// \param p The priority.
-    /// \pre \c i must be stored in the heap with priority at least \c
-    /// p relative to \c Compare.
+    /// \pre \e i must be stored in the heap with priority at least \e p.
     void decrease(const Item &i, const Prio &p) {
       int idx = _iim[i];
       bubble_up(idx, Pair(i,p));
     }
 
-    /// \brief Increases the priority of \c i to \c p.
+    /// \brief Increase the priority of an item to the given value.
     ///
-    /// This method sets the priority of item \c i to \c p.
+    /// This function increases the priority of an item to the given value.
     /// \param i The item.
     /// \param p The priority.
-    /// \pre \c i must be stored in the heap with priority at most \c
-    /// p relative to \c Compare.
+    /// \pre \e i must be stored in the heap with priority at most \e p.
     void increase(const Item &i, const Prio &p) {
       int idx = _iim[i];
       bubble_down(idx, Pair(i,p), _data.size());
     }
 
-    /// \brief Returns if \c item is in, has already been in, or has
-    /// never been in the heap.
+    /// \brief Return the state of an item.
     ///
-    /// This method returns PRE_HEAP if \c item has never been in the
-    /// heap, IN_HEAP if it is in the heap at the moment, and POST_HEAP
-    /// otherwise. In the latter case it is possible that \c item will
-    /// get back to the heap again.
+    /// This method returns \c PRE_HEAP if the given item has never
+    /// been in the heap, \c IN_HEAP if it is in the heap at the moment,
+    /// and \c POST_HEAP otherwise.
+    /// In the latter case it is possible that the item will get back
+    /// to the heap again.
     /// \param i The item.
     State state(const Item &i) const {
       int s = _iim[i];
@@ -306,11 +305,11 @@
       return State(s);
     }
 
-    /// \brief Sets the state of the \c item in the heap.
+    /// \brief Set the state of an item in the heap.
     ///
-    /// Sets the state of the \c item in the heap. It can be used to
-    /// manually clear the heap when it is important to achive the
-    /// better time complexity.
+    /// This function sets the state of the given item in the heap.
+    /// It can be used to manually clear the heap when it is important
+    /// to achive better time complexity.
     /// \param i The item.
     /// \param st The state. It should not be \c IN_HEAP.
     void state(const Item& i, State st) {
@@ -327,12 +326,13 @@
       }
     }
 
-    /// \brief Replaces an item in the heap.
+    /// \brief Replace an item in the heap.
     ///
-    /// The \c i item is replaced with \c j item. The \c i item should
-    /// be in the heap, while the \c j should be out of the heap. The
-    /// \c i item will out of the heap and \c j will be in the heap
-    /// with the same prioriority as prevoiusly the \c i item.
+    /// This function replaces item \c i with item \c j.
+    /// Item \c i must be in the heap, while \c j must be out of the heap.
+    /// After calling this method, item \c i will be out of the
+    /// heap and \c j will be in the heap with the same prioriority
+    /// as item \c i had before.
     void replace(const Item& i, const Item& j) {
       int idx = _iim[i];
       _iim.set(i, _iim[j]);
diff --git a/lemon/bucket_heap.h b/lemon/bucket_heap.h
--- a/lemon/bucket_heap.h
+++ b/lemon/bucket_heap.h
@@ -21,7 +21,7 @@
 
 ///\ingroup auxdat
 ///\file
-///\brief Bucket Heap implementation.
+///\brief Bucket heap implementation.
 
 #include <vector>
 #include <utility>
@@ -55,33 +55,39 @@
 
   /// \ingroup auxdat
   ///
-  /// \brief A Bucket Heap implementation.
+  /// \brief Bucket heap data structure.
   ///
-  /// This class implements the \e bucket \e heap data structure. A \e heap
-  /// is a data structure for storing items with specified values called \e
-  /// priorities in such a way that finding the item with minimum priority is
-  /// efficient. The bucket heap is very simple implementation, it can store
-  /// only integer priorities and it stores for each priority in the
-  /// \f$ [0..C) \f$ range a list of items. So it should be used only when
-  /// the priorities are small. It is not intended to use as dijkstra heap.
+  /// This class implements the \e bucket \e heap data structure.
+  /// It practically conforms to the \ref concepts::Heap "heap concept",
+  /// but it has some limitations.
   ///
-  /// \param IM A read and write Item int map, used internally
-  /// to handle the cross references.
-  /// \param MIN If the given parameter is false then instead of the
-  /// minimum value the maximum can be retrivied with the top() and
-  /// prio() member functions.
+  /// The bucket heap is a very simple structure. It can store only
+  /// \c int priorities and it maintains a list of items for each priority
+  /// in the range <tt>[0..C)</tt>. So it should only be used when the
+  /// priorities are small. It is not intended to use as a Dijkstra heap.
+  ///
+  /// \tparam IM A read-writable item map with \c int values, used
+  /// internally to handle the cross references.
+  /// \tparam MIN Indicate if the heap is a \e min-heap or a \e max-heap.
+  /// The default is \e min-heap. If this parameter is set to \c false,
+  /// then the comparison is reversed, so the top(), prio() and pop()
+  /// functions deal with the item having maximum priority instead of the
+  /// minimum.
+  ///
+  /// \sa SimpleBucketHeap
   template <typename IM, bool MIN = true>
   class BucketHeap {
 
   public:
-    /// \e
-    typedef typename IM::Key Item;
-    /// \e
+
+    /// Type of the item-int map.
+    typedef IM ItemIntMap;
+    /// Type of the priorities.
     typedef int Prio;
-    /// \e
-    typedef std::pair<Item, Prio> Pair;
-    /// \e
-    typedef IM ItemIntMap;
+    /// Type of the items stored in the heap.
+    typedef typename ItemIntMap::Key Item;
+    /// Type of the item-priority pairs.
+    typedef std::pair<Item,Prio> Pair;
 
   private:
 
@@ -89,10 +95,10 @@
 
   public:
 
-    /// \brief Type to represent the items states.
+    /// \brief Type to represent the states of the items.
     ///
-    /// Each Item element have a state associated to it. It may be "in heap",
-    /// "pre heap" or "post heap". The latter two are indifferent from the
+    /// Each item has a state associated to it. It can be "in heap",
+    /// "pre-heap" or "post-heap". The latter two are indifferent from the
     /// heap's point of view, but may be useful to the user.
     ///
     /// The item-int map must be initialized in such way that it assigns
@@ -104,30 +110,32 @@
     };
 
   public:
-    /// \brief The constructor.
+
+    /// \brief Constructor.
     ///
-    /// The constructor.
-    /// \param map should be given to the constructor, since it is used
-    /// internally to handle the cross references. The value of the map
-    /// should be PRE_HEAP (-1) for each element.
+    /// Constructor.
+    /// \param map A map that assigns \c int values to the items.
+    /// It is used internally to handle the cross references.
+    /// The assigned value must be \c PRE_HEAP (<tt>-1</tt>) for each item.
     explicit BucketHeap(ItemIntMap &map) : _iim(map), _minimum(0) {}
 
-    /// The number of items stored in the heap.
+    /// \brief The number of items stored in the heap.
     ///
-    /// \brief Returns the number of items stored in the heap.
+    /// This function returns the number of items stored in the heap.
     int size() const { return _data.size(); }
 
-    /// \brief Checks if the heap stores no items.
+    /// \brief Check if the heap is empty.
     ///
-    /// Returns \c true if and only if the heap stores no items.
+    /// This function returns \c true if the heap is empty.
     bool empty() const { return _data.empty(); }
 
-    /// \brief Make empty this heap.
+    /// \brief Make the heap empty.
     ///
-    /// Make empty this heap. It does not change the cross reference
-    /// map.  If you want to reuse a heap what is not surely empty you
-    /// should first clear the heap and after that you should set the
-    /// cross reference map for each item to \c PRE_HEAP.
+    /// This functon makes the heap empty.
+    /// It does not change the cross reference map. If you want to reuse
+    /// a heap that is not surely empty, you should first clear it and
+    /// then you should set the cross reference map to \c PRE_HEAP
+    /// for each item.
     void clear() {
       _data.clear(); _first.clear(); _minimum = 0;
     }
@@ -174,19 +182,24 @@
     }
 
   public:
+
     /// \brief Insert a pair of item and priority into the heap.
     ///
-    /// Adds \c p.first to the heap with priority \c p.second.
+    /// This function inserts \c p.first to the heap with priority
+    /// \c p.second.
     /// \param p The pair to insert.
+    /// \pre \c p.first must not be stored in the heap.
     void push(const Pair& p) {
       push(p.first, p.second);
     }
 
     /// \brief Insert an item into the heap with the given priority.
     ///
-    /// Adds \c i to the heap with priority \c p.
+    /// This function inserts the given item into the heap with the
+    /// given priority.
     /// \param i The item to insert.
     /// \param p The priority of the item.
+    /// \pre \e i must not be stored in the heap.
     void push(const Item &i, const Prio &p) {
       int idx = _data.size();
       _iim[i] = idx;
@@ -197,10 +210,10 @@
       }
     }
 
-    /// \brief Returns the item with minimum priority.
+    /// \brief Return the item having minimum priority.
     ///
-    /// This method returns the item with minimum priority.
-    /// \pre The heap must be nonempty.
+    /// This function returns the item having minimum priority.
+    /// \pre The heap must be non-empty.
     Item top() const {
       while (_first[_minimum] == -1) {
         Direction::increase(_minimum);
@@ -208,10 +221,10 @@
       return _data[_first[_minimum]].item;
     }
 
-    /// \brief Returns the minimum priority.
+    /// \brief The minimum priority.
     ///
-    /// It returns the minimum priority.
-    /// \pre The heap must be nonempty.
+    /// This function returns the minimum priority.
+    /// \pre The heap must be non-empty.
     Prio prio() const {
       while (_first[_minimum] == -1) {
         Direction::increase(_minimum);
@@ -219,9 +232,9 @@
       return _minimum;
     }
 
-    /// \brief Deletes the item with minimum priority.
+    /// \brief Remove the item having minimum priority.
     ///
-    /// This method deletes the item with minimum priority from the heap.
+    /// This function removes the item having minimum priority.
     /// \pre The heap must be non-empty.
     void pop() {
       while (_first[_minimum] == -1) {
@@ -233,11 +246,12 @@
       relocate_last(idx);
     }
 
-    /// \brief Deletes \c i from the heap.
+    /// \brief Remove the given item from the heap.
     ///
-    /// This method deletes item \c i from the heap, if \c i was
-    /// already stored in the heap.
-    /// \param i The item to erase.
+    /// This function removes the given item from the heap if it is
+    /// already stored.
+    /// \param i The item to delete.
+    /// \pre \e i must be in the heap.
     void erase(const Item &i) {
       int idx = _iim[i];
       _iim[_data[idx].item] = -2;
@@ -245,22 +259,22 @@
       relocate_last(idx);
     }
 
-
-    /// \brief Returns the priority of \c i.
+    /// \brief The priority of the given item.
     ///
-    /// This function returns the priority of item \c i.
-    /// \pre \c i must be in the heap.
+    /// This function returns the priority of the given item.
     /// \param i The item.
+    /// \pre \e i must be in the heap.
     Prio operator[](const Item &i) const {
       int idx = _iim[i];
       return _data[idx].value;
     }
 
-    /// \brief \c i gets to the heap with priority \c p independently
-    /// if \c i was already there.
+    /// \brief Set the priority of an item or insert it, if it is
+    /// not stored in the heap.
     ///
-    /// This method calls \ref push(\c i, \c p) if \c i is not stored
-    /// in the heap and sets the priority of \c i to \c p otherwise.
+    /// This method sets the priority of the given item if it is
+    /// already stored in the heap. Otherwise it inserts the given
+    /// item into the heap with the given priority.
     /// \param i The item.
     /// \param p The priority.
     void set(const Item &i, const Prio &p) {
@@ -274,13 +288,12 @@
       }
     }
 
-    /// \brief Decreases the priority of \c i to \c p.
+    /// \brief Decrease the priority of an item to the given value.
     ///
-    /// This method decreases the priority of item \c i to \c p.
-    /// \pre \c i must be stored in the heap with priority at least \c
-    /// p relative to \c Compare.
+    /// This function decreases the priority of an item to the given value.
     /// \param i The item.
     /// \param p The priority.
+    /// \pre \e i must be stored in the heap with priority at least \e p.
     void decrease(const Item &i, const Prio &p) {
       int idx = _iim[i];
       unlace(idx);
@@ -291,13 +304,12 @@
       lace(idx);
     }
 
-    /// \brief Increases the priority of \c i to \c p.
+    /// \brief Increase the priority of an item to the given value.
     ///
-    /// This method sets the priority of item \c i to \c p.
-    /// \pre \c i must be stored in the heap with priority at most \c
-    /// p relative to \c Compare.
+    /// This function increases the priority of an item to the given value.
     /// \param i The item.
     /// \param p The priority.
+    /// \pre \e i must be stored in the heap with priority at most \e p.
     void increase(const Item &i, const Prio &p) {
       int idx = _iim[i];
       unlace(idx);
@@ -305,13 +317,13 @@
       lace(idx);
     }
 
-    /// \brief Returns if \c item is in, has already been in, or has
-    /// never been in the heap.
+    /// \brief Return the state of an item.
     ///
-    /// This method returns PRE_HEAP if \c item has never been in the
-    /// heap, IN_HEAP if it is in the heap at the moment, and POST_HEAP
-    /// otherwise. In the latter case it is possible that \c item will
-    /// get back to the heap again.
+    /// This method returns \c PRE_HEAP if the given item has never
+    /// been in the heap, \c IN_HEAP if it is in the heap at the moment,
+    /// and \c POST_HEAP otherwise.
+    /// In the latter case it is possible that the item will get back
+    /// to the heap again.
     /// \param i The item.
     State state(const Item &i) const {
       int idx = _iim[i];
@@ -319,11 +331,11 @@
       return State(idx);
     }
 
-    /// \brief Sets the state of the \c item in the heap.
+    /// \brief Set the state of an item in the heap.
     ///
-    /// Sets the state of the \c item in the heap. It can be used to
-    /// manually clear the heap when it is important to achive the
-    /// better time complexity.
+    /// This function sets the state of the given item in the heap.
+    /// It can be used to manually clear the heap when it is important
+    /// to achive better time complexity.
     /// \param i The item.
     /// \param st The state. It should not be \c IN_HEAP.
     void state(const Item& i, State st) {
@@ -361,31 +373,42 @@
 
   /// \ingroup auxdat
   ///
-  /// \brief A Simplified Bucket Heap implementation.
+  /// \brief Simplified bucket heap data structure.
   ///
   /// This class implements a simplified \e bucket \e heap data
-  /// structure.  It does not provide some functionality but it faster
-  /// and simplier data structure than the BucketHeap. The main
-  /// difference is that the BucketHeap stores for every key a double
-  /// linked list while this class stores just simple lists. In the
-  /// other way it does not support erasing each elements just the
-  /// minimal and it does not supports key increasing, decreasing.
+  /// structure. It does not provide some functionality, but it is
+  /// faster and simpler than BucketHeap. The main difference is
+  /// that BucketHeap stores a doubly-linked list for each key while
+  /// this class stores only simply-linked lists. It supports erasing
+  /// only for the item having minimum priority and it does not support
+  /// key increasing and decreasing.
   ///
-  /// \param IM A read and write Item int map, used internally
-  /// to handle the cross references.
-  /// \param MIN If the given parameter is false then instead of the
-  /// minimum value the maximum can be retrivied with the top() and
-  /// prio() member functions.
+  /// Note that this implementation does not conform to the
+  /// \ref concepts::Heap "heap concept" due to the lack of some 
+  /// functionality.
+  ///
+  /// \tparam IM A read-writable item map with \c int values, used
+  /// internally to handle the cross references.
+  /// \tparam MIN Indicate if the heap is a \e min-heap or a \e max-heap.
+  /// The default is \e min-heap. If this parameter is set to \c false,
+  /// then the comparison is reversed, so the top(), prio() and pop()
+  /// functions deal with the item having maximum priority instead of the
+  /// minimum.
   ///
   /// \sa BucketHeap
   template <typename IM, bool MIN = true >
   class SimpleBucketHeap {
 
   public:
-    typedef typename IM::Key Item;
+
+    /// Type of the item-int map.
+    typedef IM ItemIntMap;
+    /// Type of the priorities.
     typedef int Prio;
-    typedef std::pair<Item, Prio> Pair;
-    typedef IM ItemIntMap;
+    /// Type of the items stored in the heap.
+    typedef typename ItemIntMap::Key Item;
+    /// Type of the item-priority pairs.
+    typedef std::pair<Item,Prio> Pair;
 
   private:
 
@@ -393,10 +416,10 @@
 
   public:
 
-    /// \brief Type to represent the items states.
+    /// \brief Type to represent the states of the items.
     ///
-    /// Each Item element have a state associated to it. It may be "in heap",
-    /// "pre heap" or "post heap". The latter two are indifferent from the
+    /// Each item has a state associated to it. It can be "in heap",
+    /// "pre-heap" or "post-heap". The latter two are indifferent from the
     /// heap's point of view, but may be useful to the user.
     ///
     /// The item-int map must be initialized in such way that it assigns
@@ -409,48 +432,53 @@
 
   public:
 
-    /// \brief The constructor.
+    /// \brief Constructor.
     ///
-    /// The constructor.
-    /// \param map should be given to the constructor, since it is used
-    /// internally to handle the cross references. The value of the map
-    /// should be PRE_HEAP (-1) for each element.
+    /// Constructor.
+    /// \param map A map that assigns \c int values to the items.
+    /// It is used internally to handle the cross references.
+    /// The assigned value must be \c PRE_HEAP (<tt>-1</tt>) for each item.
     explicit SimpleBucketHeap(ItemIntMap &map)
       : _iim(map), _free(-1), _num(0), _minimum(0) {}
 
-    /// \brief Returns the number of items stored in the heap.
+    /// \brief The number of items stored in the heap.
     ///
-    /// The number of items stored in the heap.
+    /// This function returns the number of items stored in the heap.
     int size() const { return _num; }
 
-    /// \brief Checks if the heap stores no items.
+    /// \brief Check if the heap is empty.
     ///
-    /// Returns \c true if and only if the heap stores no items.
+    /// This function returns \c true if the heap is empty.
     bool empty() const { return _num == 0; }
 
-    /// \brief Make empty this heap.
+    /// \brief Make the heap empty.
     ///
-    /// Make empty this heap. It does not change the cross reference
-    /// map.  If you want to reuse a heap what is not surely empty you
-    /// should first clear the heap and after that you should set the
-    /// cross reference map for each item to \c PRE_HEAP.
+    /// This functon makes the heap empty.
+    /// It does not change the cross reference map. If you want to reuse
+    /// a heap that is not surely empty, you should first clear it and
+    /// then you should set the cross reference map to \c PRE_HEAP
+    /// for each item.
     void clear() {
       _data.clear(); _first.clear(); _free = -1; _num = 0; _minimum = 0;
     }
 
     /// \brief Insert a pair of item and priority into the heap.
     ///
-    /// Adds \c p.first to the heap with priority \c p.second.
+    /// This function inserts \c p.first to the heap with priority
+    /// \c p.second.
     /// \param p The pair to insert.
+    /// \pre \c p.first must not be stored in the heap.
     void push(const Pair& p) {
       push(p.first, p.second);
     }
 
     /// \brief Insert an item into the heap with the given priority.
     ///
-    /// Adds \c i to the heap with priority \c p.
+    /// This function inserts the given item into the heap with the
+    /// given priority.
     /// \param i The item to insert.
     /// \param p The priority of the item.
+    /// \pre \e i must not be stored in the heap.
     void push(const Item &i, const Prio &p) {
       int idx;
       if (_free == -1) {
@@ -471,10 +499,10 @@
       ++_num;
     }
 
-    /// \brief Returns the item with minimum priority.
+    /// \brief Return the item having minimum priority.
     ///
-    /// This method returns the item with minimum priority.
-    /// \pre The heap must be nonempty.
+    /// This function returns the item having minimum priority.
+    /// \pre The heap must be non-empty.
     Item top() const {
       while (_first[_minimum] == -1) {
         Direction::increase(_minimum);
@@ -482,10 +510,10 @@
       return _data[_first[_minimum]].item;
     }
 
-    /// \brief Returns the minimum priority.
+    /// \brief The minimum priority.
     ///
-    /// It returns the minimum priority.
-    /// \pre The heap must be nonempty.
+    /// This function returns the minimum priority.
+    /// \pre The heap must be non-empty.
     Prio prio() const {
       while (_first[_minimum] == -1) {
         Direction::increase(_minimum);
@@ -493,9 +521,9 @@
       return _minimum;
     }
 
-    /// \brief Deletes the item with minimum priority.
+    /// \brief Remove the item having minimum priority.
     ///
-    /// This method deletes the item with minimum priority from the heap.
+    /// This function removes the item having minimum priority.
     /// \pre The heap must be non-empty.
     void pop() {
       while (_first[_minimum] == -1) {
@@ -509,16 +537,15 @@
       --_num;
     }
 
-    /// \brief Returns the priority of \c i.
+    /// \brief The priority of the given item.
     ///
-    /// This function returns the priority of item \c i.
-    /// \warning This operator is not a constant time function
-    /// because it scans the whole data structure to find the proper
-    /// value.
-    /// \pre \c i must be in the heap.
+    /// This function returns the priority of the given item.
     /// \param i The item.
+    /// \pre \e i must be in the heap.
+    /// \warning This operator is not a constant time function because
+    /// it scans the whole data structure to find the proper value.
     Prio operator[](const Item &i) const {
-      for (int k = 0; k < _first.size(); ++k) {
+      for (int k = 0; k < int(_first.size()); ++k) {
         int idx = _first[k];
         while (idx != -1) {
           if (_data[idx].item == i) {
@@ -530,13 +557,13 @@
       return -1;
     }
 
-    /// \brief Returns if \c item is in, has already been in, or has
-    /// never been in the heap.
+    /// \brief Return the state of an item.
     ///
-    /// This method returns PRE_HEAP if \c item has never been in the
-    /// heap, IN_HEAP if it is in the heap at the moment, and POST_HEAP
-    /// otherwise. In the latter case it is possible that \c item will
-    /// get back to the heap again.
+    /// This method returns \c PRE_HEAP if the given item has never
+    /// been in the heap, \c IN_HEAP if it is in the heap at the moment,
+    /// and \c POST_HEAP otherwise.
+    /// In the latter case it is possible that the item will get back
+    /// to the heap again.
     /// \param i The item.
     State state(const Item &i) const {
       int idx = _iim[i];
diff --git a/lemon/concepts/heap.h b/lemon/concepts/heap.h
--- a/lemon/concepts/heap.h
+++ b/lemon/concepts/heap.h
@@ -16,13 +16,13 @@
  *
  */
 
+#ifndef LEMON_CONCEPTS_HEAP_H
+#define LEMON_CONCEPTS_HEAP_H
+
 ///\ingroup concept
 ///\file
 ///\brief The concept of heaps.
 
-#ifndef LEMON_CONCEPTS_HEAP_H
-#define LEMON_CONCEPTS_HEAP_H
-
 #include <lemon/core.h>
 #include <lemon/concept_check.h>
 
@@ -35,21 +35,27 @@
 
     /// \brief The heap concept.
     ///
-    /// Concept class describing the main interface of heaps. A \e heap
-    /// is a data structure for storing items with specified values called
-    /// \e priorities in such a way that finding the item with minimum
-    /// priority is efficient. In a heap one can change the priority of an
-    /// item, add or erase an item, etc.
+    /// This concept class describes the main interface of heaps.
+    /// The various heap structures are efficient
+    /// implementations of the abstract data type \e priority \e queue.
+    /// They store items with specified values called \e priorities
+    /// in such a way that finding and removing the item with minimum
+    /// priority are efficient. The basic operations are adding and
+    /// erasing items, changing the priority of an item, etc.
     ///
-    /// \tparam PR Type of the priority of the items.
-    /// \tparam IM A read and writable item map with int values, used
+    /// Heaps are crucial in several algorithms, such as Dijkstra and Prim.
+    /// Any class that conforms to this concept can be used easily in such
+    /// algorithms.
+    ///
+    /// \tparam PR Type of the priorities of the items.
+    /// \tparam IM A read-writable item map with \c int values, used
     /// internally to handle the cross references.
-    /// \tparam Comp A functor class for the ordering of the priorities.
+    /// \tparam CMP A functor class for comparing the priorities.
     /// The default is \c std::less<PR>.
 #ifdef DOXYGEN
-    template <typename PR, typename IM, typename Comp = std::less<PR> >
+    template <typename PR, typename IM, typename CMP>
 #else
-    template <typename PR, typename IM>
+    template <typename PR, typename IM, typename CMP = std::less<PR> >
 #endif
     class Heap {
     public:
@@ -64,109 +70,125 @@
       /// \brief Type to represent the states of the items.
       ///
       /// Each item has a state associated to it. It can be "in heap",
-      /// "pre heap" or "post heap". The later two are indifferent
-      /// from the point of view of the heap, but may be useful for
-      /// the user.
+      /// "pre-heap" or "post-heap". The latter two are indifferent from the
+      /// heap's point of view, but may be useful to the user.
       ///
       /// The item-int map must be initialized in such way that it assigns
       /// \c PRE_HEAP (<tt>-1</tt>) to any element to be put in the heap.
       enum State {
         IN_HEAP = 0,    ///< = 0. The "in heap" state constant.
-        PRE_HEAP = -1,  ///< = -1. The "pre heap" state constant.
-        POST_HEAP = -2  ///< = -2. The "post heap" state constant.
+        PRE_HEAP = -1,  ///< = -1. The "pre-heap" state constant.
+        POST_HEAP = -2  ///< = -2. The "post-heap" state constant.
       };
 
-      /// \brief The constructor.
+      /// \brief Constructor.
       ///
-      /// The constructor.
+      /// Constructor.
       /// \param map A map that assigns \c int values to keys of type
       /// \c Item. It is used internally by the heap implementations to
       /// handle the cross references. The assigned value must be
-      /// \c PRE_HEAP (<tt>-1</tt>) for every item.
+      /// \c PRE_HEAP (<tt>-1</tt>) for each item.
       explicit Heap(ItemIntMap &map) {}
 
+      /// \brief Constructor.
+      ///
+      /// Constructor.
+      /// \param map A map that assigns \c int values to keys of type
+      /// \c Item. It is used internally by the heap implementations to
+      /// handle the cross references. The assigned value must be
+      /// \c PRE_HEAP (<tt>-1</tt>) for each item.
+      /// \param comp The function object used for comparing the priorities.
+      explicit Heap(ItemIntMap &map, const CMP &comp) {}
+
       /// \brief The number of items stored in the heap.
       ///
-      /// Returns the number of items stored in the heap.
+      /// This function returns the number of items stored in the heap.
       int size() const { return 0; }
 
-      /// \brief Checks if the heap is empty.
+      /// \brief Check if the heap is empty.
       ///
-      /// Returns \c true if the heap is empty.
+      /// This function returns \c true if the heap is empty.
       bool empty() const { return false; }
 
-      /// \brief Makes the heap empty.
+      /// \brief Make the heap empty.
       ///
-      /// Makes the heap empty.
-      void clear();
+      /// This functon makes the heap empty.
+      /// It does not change the cross reference map. If you want to reuse
+      /// a heap that is not surely empty, you should first clear it and
+      /// then you should set the cross reference map to \c PRE_HEAP
+      /// for each item.
+      void clear() {}
 
-      /// \brief Inserts an item into the heap with the given priority.
+      /// \brief Insert an item into the heap with the given priority.
       ///
-      /// Inserts the given item into the heap with the given priority.
+      /// This function inserts the given item into the heap with the
+      /// given priority.
       /// \param i The item to insert.
       /// \param p The priority of the item.
+      /// \pre \e i must not be stored in the heap.
       void push(const Item &i, const Prio &p) {}
 
-      /// \brief Returns the item having minimum priority.
+      /// \brief Return the item having minimum priority.
       ///
-      /// Returns the item having minimum priority.
+      /// This function returns the item having minimum priority.
       /// \pre The heap must be non-empty.
       Item top() const {}
 
       /// \brief The minimum priority.
       ///
-      /// Returns the minimum priority.
+      /// This function returns the minimum priority.
       /// \pre The heap must be non-empty.
       Prio prio() const {}
 
-      /// \brief Removes the item having minimum priority.
+      /// \brief Remove the item having minimum priority.
       ///
-      /// Removes the item having minimum priority.
+      /// This function removes the item having minimum priority.
       /// \pre The heap must be non-empty.
       void pop() {}
 
-      /// \brief Removes an item from the heap.
+      /// \brief Remove the given item from the heap.
       ///
-      /// Removes the given item from the heap if it is already stored.
+      /// This function removes the given item from the heap if it is
+      /// already stored.
       /// \param i The item to delete.
+      /// \pre \e i must be in the heap.
       void erase(const Item &i) {}
 
-      /// \brief The priority of an item.
+      /// \brief The priority of the given item.
       ///
-      /// Returns the priority of the given item.
+      /// This function returns the priority of the given item.
       /// \param i The item.
-      /// \pre \c i must be in the heap.
+      /// \pre \e i must be in the heap.
       Prio operator[](const Item &i) const {}
 
-      /// \brief Sets the priority of an item or inserts it, if it is
+      /// \brief Set the priority of an item or insert it, if it is
       /// not stored in the heap.
       ///
       /// This method sets the priority of the given item if it is
-      /// already stored in the heap.
-      /// Otherwise it inserts the given item with the given priority.
+      /// already stored in the heap. Otherwise it inserts the given
+      /// item into the heap with the given priority.
       ///
       /// \param i The item.
       /// \param p The priority.
       void set(const Item &i, const Prio &p) {}
 
-      /// \brief Decreases the priority of an item to the given value.
+      /// \brief Decrease the priority of an item to the given value.
       ///
-      /// Decreases the priority of an item to the given value.
+      /// This function decreases the priority of an item to the given value.
       /// \param i The item.
       /// \param p The priority.
-      /// \pre \c i must be stored in the heap with priority at least \c p.
+      /// \pre \e i must be stored in the heap with priority at least \e p.
       void decrease(const Item &i, const Prio &p) {}
 
-      /// \brief Increases the priority of an item to the given value.
+      /// \brief Increase the priority of an item to the given value.
       ///
-      /// Increases the priority of an item to the given value.
+      /// This function increases the priority of an item to the given value.
       /// \param i The item.
       /// \param p The priority.
-      /// \pre \c i must be stored in the heap with priority at most \c p.
+      /// \pre \e i must be stored in the heap with priority at most \e p.
       void increase(const Item &i, const Prio &p) {}
 
-      /// \brief Returns if an item is in, has already been in, or has
-      /// never been in the heap.
+      /// \brief Return the state of an item.
       ///
       /// This method returns \c PRE_HEAP if the given item has never
       /// been in the heap, \c IN_HEAP if it is in the heap at the moment,
@@ -176,11 +198,11 @@
       /// \param i The item.
       State state(const Item &i) const {}
 
-      /// \brief Sets the state of an item in the heap.
+      /// \brief Set the state of an item in the heap.
       ///
-      /// Sets the state of the given item in the heap. It can be used
-      /// to manually clear the heap when it is important to achive the
-      /// better time complexity.
+      /// This function sets the state of the given item in the heap.
+      /// It can be used to manually clear the heap when it is important
+      /// to achive better time complexity.
       /// \param i The item.
       /// \param st The state. It should not be \c IN_HEAP.
       void state(const Item& i, State st) {}
diff --git a/lemon/fib_heap.h b/lemon/fib_heap.h
--- a/lemon/fib_heap.h
+++ b/lemon/fib_heap.h
@@ -21,9 +21,10 @@
 
 ///\file
 ///\ingroup auxdat
-///\brief Fibonacci Heap implementation.
+///\brief Fibonacci heap implementation.
 
 #include <vector>
+#include <utility>
 #include <functional>
 #include <lemon/math.h>
 
@@ -31,42 +32,37 @@
 
   /// \ingroup auxdat
   ///
-  ///\brief Fibonacci Heap.
+  /// \brief Fibonacci heap data structure.
   ///
-  ///This class implements the \e Fibonacci \e heap data structure. A \e heap
-  ///is a data structure for storing items with specified values called \e
-  ///priorities in such a way that finding the item with minimum priority is
-  ///efficient. \c CMP specifies the ordering of the priorities. In a heap
-  ///one can change the priority of an item, add or erase an item, etc.
+  /// This class implements the \e Fibonacci \e heap data structure.
+  /// It fully conforms to the \ref concepts::Heap "heap concept".
   ///
-  ///The methods \ref increase and \ref erase are not efficient in a Fibonacci
-  ///heap. In case of many calls to these operations, it is better to use a
-  ///\ref BinHeap "binary heap".
+  /// The methods \ref increase() and \ref erase() are not efficient in a
+  /// Fibonacci heap. In case of many calls of these operations, it is
+  /// better to use other heap structure, e.g. \ref BinHeap "binary heap".
   ///
-  ///\param PRIO Type of the priority of the items.
-  ///\param IM A read and writable Item int map, used internally
-  ///to handle the cross references.
-  ///\param CMP A class for the ordering of the priorities. The
-  ///default is \c std::less<PRIO>.
-  ///
-  ///\sa BinHeap
-  ///\sa Dijkstra
+  /// \tparam PR Type of the priorities of the items.
+  /// \tparam IM A read-writable item map with \c int values, used
+  /// internally to handle the cross references.
+  /// \tparam CMP A functor class for comparing the priorities.
+  /// The default is \c std::less<PR>.
 #ifdef DOXYGEN
-  template <typename PRIO, typename IM, typename CMP>
+  template <typename PR, typename IM, typename CMP>
 #else
-  template <typename PRIO, typename IM, typename CMP = std::less<PRIO> >
+  template <typename PR, typename IM, typename CMP = std::less<PR> >
 #endif
   class FibHeap {
   public:
-    ///\e
+
+    /// Type of the item-int map.
     typedef IM ItemIntMap;
-    ///\e
-    typedef PRIO Prio;
-    ///\e
+    /// Type of the priorities.
+    typedef PR Prio;
+    /// Type of the items stored in the heap.
     typedef typename ItemIntMap::Key Item;
-    ///\e
+    /// Type of the item-priority pairs.
     typedef std::pair<Item,Prio> Pair;
-    ///\e
+    /// Functor type for comparing the priorities.
     typedef CMP Compare;
 
   private:
@@ -80,10 +76,10 @@
 
   public:
 
-    /// \brief Type to represent the items states.
+    /// \brief Type to represent the states of the items.
     ///
-    /// Each Item element have a state associated to it. It may be "in heap",
-    /// "pre heap" or "post heap". The latter two are indifferent from the
+    /// Each item has a state associated to it. It can be "in heap",
+    /// "pre-heap" or "post-heap". The latter two are indifferent from the
     /// heap's point of view, but may be useful to the user.
     ///
     /// The item-int map must be initialized in such way that it assigns
@@ -94,60 +90,54 @@
       POST_HEAP = -2  ///< = -2.
     };
 
-    /// \brief The constructor
+    /// \brief Constructor.
     ///
-    /// \c map should be given to the constructor, since it is
-    ///   used internally to handle the cross references.
+    /// Constructor.
+    /// \param map A map that assigns \c int values to the items.
+    /// It is used internally to handle the cross references.
+    /// The assigned value must be \c PRE_HEAP (<tt>-1</tt>) for each item.
     explicit FibHeap(ItemIntMap &map)
       : _minimum(0), _iim(map), _num() {}
 
-    /// \brief The constructor
+    /// \brief Constructor.
     ///
-    /// \c map should be given to the constructor, since it is used
-    /// internally to handle the cross references. \c comp is an
-    /// object for ordering of the priorities.
+    /// Constructor.
+    /// \param map A map that assigns \c int values to the items.
+    /// It is used internally to handle the cross references.
+    /// The assigned value must be \c PRE_HEAP (<tt>-1</tt>) for each item.
+    /// \param comp The function object used for comparing the priorities.
     FibHeap(ItemIntMap &map, const Compare &comp)
       : _minimum(0), _iim(map), _comp(comp), _num() {}
 
     /// \brief The number of items stored in the heap.
     ///
-    /// Returns the number of items stored in the heap.
+    /// This function returns the number of items stored in the heap.
     int size() const { return _num; }
 
-    /// \brief Checks if the heap stores no items.
+    /// \brief Check if the heap is empty.
     ///
-    ///   Returns \c true if and only if the heap stores no items.
+    /// This function returns \c true if the heap is empty.
     bool empty() const { return _num==0; }
 
-    /// \brief Make empty this heap.
+    /// \brief Make the heap empty.
     ///
-    /// Make empty this heap. It does not change the cross reference
-    /// map.  If you want to reuse a heap what is not surely empty you
-    /// should first clear the heap and after that you should set the
-    /// cross reference map for each item to \c PRE_HEAP.
+    /// This functon makes the heap empty.
+    /// It does not change the cross reference map. If you want to reuse
+    /// a heap that is not surely empty, you should first clear it and
+    /// then you should set the cross reference map to \c PRE_HEAP
+    /// for each item.
     void clear() {
       _data.clear(); _minimum = 0; _num = 0;
     }
 
-    /// \brief \c item gets to the heap with priority \c value independently
-    /// if \c item was already there.
+    /// \brief Insert an item into the heap with the given priority.
     ///
-    /// This method calls \ref push(\c item, \c value) if \c item is not
-    /// stored in the heap and it calls \ref decrease(\c item, \c value) or
-    /// \ref increase(\c item, \c value) otherwise.
-    void set (const Item& item, const Prio& value) {
-      int i=_iim[item];
-      if ( i >= 0 && _data[i].in ) {
-        if ( _comp(value, _data[i].prio) ) decrease(item, value);
-        if ( _comp(_data[i].prio, value) ) increase(item, value);
-      } else push(item, value);
-    }
-
-    /// \brief Adds \c item to the heap with priority \c value.
-    ///
-    /// Adds \c item to the heap with priority \c value.
-    /// \pre \c item must not be stored in the heap.
-    void push (const Item& item, const Prio& value) {
+    /// This function inserts the given item into the heap with the
+    /// given priority.
+    /// \param item The item to insert.
+    /// \param prio The priority of the item.
+    /// \pre \e item must not be stored in the heap.
+    void push (const Item& item, const Prio& prio) {
       int i=_iim[item];
       if ( i < 0 ) {
         int s=_data.size();
@@ -168,40 +158,30 @@
         _data[i].right_neighbor=_data[_minimum].right_neighbor;
         _data[_minimum].right_neighbor=i;
         _data[i].left_neighbor=_minimum;
-        if ( _comp( value, _data[_minimum].prio) ) _minimum=i;
+        if ( _comp( prio, _data[_minimum].prio) ) _minimum=i;
       } else {
         _data[i].right_neighbor=_data[i].left_neighbor=i;
         _minimum=i;
       }
-      _data[i].prio=value;
+      _data[i].prio=prio;
       ++_num;
     }
 
-    /// \brief Returns the item with minimum priority relative to \c Compare.
+    /// \brief Return the item having minimum priority.
     ///
-    /// This method returns the item with minimum priority relative to \c
-    /// Compare.
-    /// \pre The heap must be nonempty.
+    /// This function returns the item having minimum priority.
+    /// \pre The heap must be non-empty.
     Item top() const { return _data[_minimum].name; }
 
-    /// \brief Returns the minimum priority relative to \c Compare.
+    /// \brief The minimum priority.
     ///
-    /// It returns the minimum priority relative to \c Compare.
-    /// \pre The heap must be nonempty.
-    const Prio& prio() const { return _data[_minimum].prio; }
+    /// This function returns the minimum priority.
+    /// \pre The heap must be non-empty.
+    Prio prio() const { return _data[_minimum].prio; }
 
-    /// \brief Returns the priority of \c item.
+    /// \brief Remove the item having minimum priority.
     ///
-    /// It returns the priority of \c item.
-    /// \pre \c item must be in the heap.
-    const Prio& operator[](const Item& item) const {
-      return _data[_iim[item]].prio;
-    }
-
-    /// \brief Deletes the item with minimum priority relative to \c Compare.
-    ///
-    /// This method deletes the item with minimum priority relative to \c
-    /// Compare from the heap.
+    /// This function removes the item having minimum priority.
     /// \pre The heap must be non-empty.
     void pop() {
       /*The first case is that there are only one root.*/
@@ -234,10 +214,12 @@
       --_num;
     }
 
-    /// \brief Deletes \c item from the heap.
+    /// \brief Remove the given item from the heap.
     ///
-    /// This method deletes \c item from the heap, if \c item was already
-    /// stored in the heap. It is quite inefficient in Fibonacci heaps.
+    /// This function removes the given item from the heap if it is
+    /// already stored.
+    /// \param item The item to delete.
+    /// \pre \e item must be in the heap.
     void erase (const Item& item) {
       int i=_iim[item];
 
@@ -252,43 +234,68 @@
       }
     }
 
-    /// \brief Decreases the priority of \c item to \c value.
+    /// \brief The priority of the given item.
     ///
-    /// This method decreases the priority of \c item to \c value.
-    /// \pre \c item must be stored in the heap with priority at least \c
-    ///   value relative to \c Compare.
-    void decrease (Item item, const Prio& value) {
+    /// This function returns the priority of the given item.
+    /// \param item The item.
+    /// \pre \e item must be in the heap.
+    Prio operator[](const Item& item) const {
+      return _data[_iim[item]].prio;
+    }
+
+    /// \brief Set the priority of an item or insert it, if it is
+    /// not stored in the heap.
+    ///
+    /// This method sets the priority of the given item if it is
+    /// already stored in the heap. Otherwise it inserts the given
+    /// item into the heap with the given priority.
+    /// \param item The item.
+    /// \param prio The priority.
+    void set (const Item& item, const Prio& prio) {
       int i=_iim[item];
-      _data[i].prio=value;
+      if ( i >= 0 && _data[i].in ) {
+        if ( _comp(prio, _data[i].prio) ) decrease(item, prio);
+        if ( _comp(_data[i].prio, prio) ) increase(item, prio);
+      } else push(item, prio);
+    }
+
+    /// \brief Decrease the priority of an item to the given value.
+    ///
+    /// This function decreases the priority of an item to the given value.
+    /// \param item The item.
+    /// \param prio The priority.
+    /// \pre \e item must be stored in the heap with priority at least \e prio.
+    void decrease (const Item& item, const Prio& prio) {
+      int i=_iim[item];
+      _data[i].prio=prio;
       int p=_data[i].parent;
 
-      if ( p!=-1 && _comp(value, _data[p].prio) ) {
+      if ( p!=-1 && _comp(prio, _data[p].prio) ) {
         cut(i,p);
         cascade(p);
       }
-      if ( _comp(value, _data[_minimum].prio) ) _minimum=i;
+      if ( _comp(prio, _data[_minimum].prio) ) _minimum=i;
     }
 
-    /// \brief Increases the priority of \c item to \c value.
+    /// \brief Increase the priority of an item to the given value.
     ///
-    /// This method sets the priority of \c item to \c value. Though
-    /// there is no precondition on the priority of \c item, this
-    /// method should be used only if it is indeed necessary to increase
-    /// (relative to \c Compare) the priority of \c item, because this
-    /// method is inefficient.
-    void increase (Item item, const Prio& value) {
+    /// This function increases the priority of an item to the given value.
+    /// \param item The item.
+    /// \param prio The priority.
+    /// \pre \e item must be stored in the heap with priority at most \e prio.
+    void increase (const Item& item, const Prio& prio) {
       erase(item);
-      push(item, value);
+      push(item, prio);
     }
 
-
-    /// \brief Returns if \c item is in, has already been in, or has never
-    /// been in the heap.
+    /// \brief Return the state of an item.
     ///
-    /// This method returns PRE_HEAP if \c item has never been in the
-    /// heap, IN_HEAP if it is in the heap at the moment, and POST_HEAP
-    /// otherwise. In the latter case it is possible that \c item will
-    /// get back to the heap again.
+    /// This method returns \c PRE_HEAP if the given item has never
+    /// been in the heap, \c IN_HEAP if it is in the heap at the moment,
+    /// and \c POST_HEAP otherwise.
+    /// In the latter case it is possible that the item will get back
+    /// to the heap again.
+    /// \param item The item.
     State state(const Item &item) const {
       int i=_iim[item];
       if( i>=0 ) {
@@ -298,11 +305,11 @@
       return State(i);
     }
 
-    /// \brief Sets the state of the \c item in the heap.
+    /// \brief Set the state of an item in the heap.
     ///
-    /// Sets the state of the \c item in the heap. It can be used to
-    /// manually clear the heap when it is important to achive the
-    /// better time _complexity.
+    /// This function sets the state of the given item in the heap.
+    /// It can be used to manually clear the heap when it is important
+    /// to achive better time complexity.
     /// \param i The item.
     /// \param st The state. It should not be \c IN_HEAP.
     void state(const Item& i, State st) {
diff --git a/lemon/radix_heap.h b/lemon/radix_heap.h
--- a/lemon/radix_heap.h
+++ b/lemon/radix_heap.h
@@ -21,7 +21,7 @@
 
 ///\ingroup auxdat
 ///\file
-///\brief Radix Heap implementation.
+///\brief Radix heap implementation.
 
 #include <vector>
 #include <lemon/error.h>
@@ -29,37 +29,35 @@
 namespace lemon {
 
 
-  /// \ingroup auxdata
+  /// \ingroup auxdat
   ///
-  /// \brief A Radix Heap implementation.
+  /// \brief Radix heap data structure.
   ///
-  /// This class implements the \e radix \e heap data structure. A \e heap
-  /// is a data structure for storing items with specified values called \e
-  /// priorities in such a way that finding the item with minimum priority is
-  /// efficient. This heap type can store only items with \e int priority.
-  /// In a heap one can change the priority of an item, add or erase an
-  /// item, but the priority cannot be decreased under the last removed
-  /// item's priority.
+  /// This class implements the \e radix \e heap data structure.
+  /// It practically conforms to the \ref concepts::Heap "heap concept",
+  /// but it has some limitations due its special implementation.
+  /// The type of the priorities must be \c int and the priority of an
+  /// item cannot be decreased under the priority of the last removed item.
   ///
-  /// \param IM A read and writable Item int map, used internally
-  /// to handle the cross references.
-  ///
-  /// \see BinHeap
-  /// \see Dijkstra
+  /// \tparam IM A read-writable item map with \c int values, used
+  /// internally to handle the cross references.
   template <typename IM>
   class RadixHeap {
 
   public:
-    typedef typename IM::Key Item;
+
+    /// Type of the item-int map.
+    typedef IM ItemIntMap;
+    /// Type of the priorities.
     typedef int Prio;
-    typedef IM ItemIntMap;
+    /// Type of the items stored in the heap.
+    typedef typename ItemIntMap::Key Item;
 
     /// \brief Exception thrown by RadixHeap.
     ///
-    /// This Exception is thrown when a smaller priority
-    /// is inserted into the \e RadixHeap then the last time erased.
+    /// This exception is thrown when an item is inserted into a
+    /// RadixHeap with a priority smaller than the last erased one.
     /// \see RadixHeap
-
     class UnderFlowPriorityError : public Exception {
     public:
       virtual const char* what() const throw() {
@@ -67,18 +65,18 @@
       }
     };
 
-    /// \brief Type to represent the items states.
+    /// \brief Type to represent the states of the items.
     ///
-    /// Each Item element have a state associated to it. It may be "in heap",
-    /// "pre heap" or "post heap". The latter two are indifferent from the
+    /// Each item has a state associated to it. It can be "in heap",
+    /// "pre-heap" or "post-heap". The latter two are indifferent from the
     /// heap's point of view, but may be useful to the user.
     ///
-    /// The ItemIntMap \e should be initialized in such way that it maps
-    /// PRE_HEAP (-1) to any element to be put in the heap...
+    /// The item-int map must be initialized in such way that it assigns
+    /// \c PRE_HEAP (<tt>-1</tt>) to any element to be put in the heap.
     enum State {
-      IN_HEAP = 0,
-      PRE_HEAP = -1,
-      POST_HEAP = -2
+      IN_HEAP = 0,    ///< = 0.
+      PRE_HEAP = -1,  ///< = -1.
+      POST_HEAP = -2  ///< = -2.
     };
 
   private:
@@ -101,47 +99,50 @@
 
     ItemIntMap &_iim;
 
+  public:
 
-  public:
-    /// \brief The constructor.
+    /// \brief Constructor.
     ///
-    /// The constructor.
-    ///
-    /// \param map It should be given to the constructor, since it is used
-    /// internally to handle the cross references. The value of the map
-    /// should be PRE_HEAP (-1) for each element.
-    ///
-    /// \param minimal The initial minimal value of the heap.
-    /// \param capacity It determines the initial capacity of the heap.
-    RadixHeap(ItemIntMap &map, int minimal = 0, int capacity = 0)
-      : _iim(map) {
-      boxes.push_back(RadixBox(minimal, 1));
-      boxes.push_back(RadixBox(minimal + 1, 1));
-      while (lower(boxes.size() - 1, capacity + minimal - 1)) {
+    /// Constructor.
+    /// \param map A map that assigns \c int values to the items.
+    /// It is used internally to handle the cross references.
+    /// The assigned value must be \c PRE_HEAP (<tt>-1</tt>) for each item.
+    /// \param minimum The initial minimum value of the heap.
+    /// \param capacity The initial capacity of the heap.
+    RadixHeap(ItemIntMap &map, int minimum = 0, int capacity = 0)
+      : _iim(map)
+    {
+      boxes.push_back(RadixBox(minimum, 1));
+      boxes.push_back(RadixBox(minimum + 1, 1));
+      while (lower(boxes.size() - 1, capacity + minimum - 1)) {
         extend();
       }
     }
 
-    /// The number of items stored in the heap.
+    /// \brief The number of items stored in the heap.
     ///
-    /// \brief Returns the number of items stored in the heap.
+    /// This function returns the number of items stored in the heap.
     int size() const { return data.size(); }
-    /// \brief Checks if the heap stores no items.
+
+    /// \brief Check if the heap is empty.
     ///
-    /// Returns \c true if and only if the heap stores no items.
+    /// This function returns \c true if the heap is empty.
     bool empty() const { return data.empty(); }
 
-    /// \brief Make empty this heap.
+    /// \brief Make the heap empty.
     ///
-    /// Make empty this heap. It does not change the cross reference
-    /// map.  If you want to reuse a heap what is not surely empty you
-    /// should first clear the heap and after that you should set the
-    /// cross reference map for each item to \c PRE_HEAP.
-    void clear(int minimal = 0, int capacity = 0) {
+    /// This functon makes the heap empty.
+    /// It does not change the cross reference map. If you want to reuse
+    /// a heap that is not surely empty, you should first clear it and
+    /// then you should set the cross reference map to \c PRE_HEAP
+    /// for each item.
+    /// \param minimum The minimum value of the heap.
+    /// \param capacity The capacity of the heap.
+    void clear(int minimum = 0, int capacity = 0) {
       data.clear(); boxes.clear();
-      boxes.push_back(RadixBox(minimal, 1));
-      boxes.push_back(RadixBox(minimal + 1, 1));
-      while (lower(boxes.size() - 1, capacity + minimal - 1)) {
+      boxes.push_back(RadixBox(minimum, 1));
+      boxes.push_back(RadixBox(minimum + 1, 1));
+      while (lower(boxes.size() - 1, capacity + minimum - 1)) {
         extend();
       }
     }
@@ -156,7 +157,7 @@
       return pr >= boxes[box].min + boxes[box].size;
     }
 
-    /// \brief Remove item from the box list.
+    // Remove item from the box list
     void remove(int index) {
       if (data[index].prev >= 0) {
         data[data[index].prev].next = data[index].next;
@@ -168,7 +169,7 @@
       }
     }
 
-    /// \brief Insert item into the box list.
+    // Insert item into the box list
     void insert(int box, int index) {
       if (boxes[box].first == -1) {
         boxes[box].first = index;
@@ -182,14 +183,14 @@
       data[index].box = box;
     }
 
-    /// \brief Add a new box to the box list.
+    // Add a new box to the box list
     void extend() {
       int min = boxes.back().min + boxes.back().size;
       int bs = 2 * boxes.back().size;
       boxes.push_back(RadixBox(min, bs));
     }
 
-    /// \brief Move an item up into the proper box.
+    // Move an item up into the proper box.
     void bubble_up(int index) {
       if (!lower(data[index].box, data[index].prio)) return;
       remove(index);
@@ -197,7 +198,7 @@
       insert(box, index);
     }
 
-    /// \brief Find up the proper box for the item with the given prio.
+    // Find up the proper box for the item with the given priority
     int findUp(int start, int pr) {
       while (lower(start, pr)) {
         if (++start == int(boxes.size())) {
@@ -207,7 +208,7 @@
       return start;
     }
 
-    /// \brief Move an item down into the proper box.
+    // Move an item down into the proper box
     void bubble_down(int index) {
       if (!upper(data[index].box, data[index].prio)) return;
       remove(index);
@@ -215,7 +216,7 @@
       insert(box, index);
     }
 
-    /// \brief Find up the proper box for the item with the given prio.
+    // Find down the proper box for the item with the given priority
     int findDown(int start, int pr) {
       while (upper(start, pr)) {
         if (--start < 0) throw UnderFlowPriorityError();
@@ -223,14 +224,14 @@
       return start;
     }
 
-    /// \brief Find the first not empty box.
+    // Find the first non-empty box
     int findFirst() {
       int first = 0;
       while (boxes[first].first == -1) ++first;
       return first;
     }
 
-    /// \brief Gives back the minimal prio of the box.
+    // Gives back the minimum priority of the given box
     int minValue(int box) {
       int min = data[boxes[box].first].prio;
       for (int k = boxes[box].first; k != -1; k = data[k].next) {
@@ -239,8 +240,7 @@
       return min;
     }
 
-    /// \brief Rearrange the items of the heap and makes the
-    /// first box not empty.
+    // Rearrange the items of the heap and make the first box non-empty
     void moveDown() {
       int box = findFirst();
       if (box == 0) return;
@@ -277,9 +277,12 @@
 
     /// \brief Insert an item into the heap with the given priority.
     ///
-    /// Adds \c i to the heap with priority \c p.
+    /// This function inserts the given item into the heap with the
+    /// given priority.
     /// \param i The item to insert.
     /// \param p The priority of the item.
+    /// \pre \e i must not be stored in the heap.
+    /// \warning This method may throw an \c UnderFlowPriorityException.
     void push(const Item &i, const Prio &p) {
       int n = data.size();
       _iim.set(i, n);
@@ -291,27 +294,27 @@
       insert(box, n);
     }
 
-    /// \brief Returns the item with minimum priority.
+    /// \brief Return the item having minimum priority.
     ///
-    /// This method returns the item with minimum priority.
-    /// \pre The heap must be nonempty.
+    /// This function returns the item having minimum priority.
+    /// \pre The heap must be non-empty.
     Item top() const {
       const_cast<RadixHeap<ItemIntMap>&>(*this).moveDown();
       return data[boxes[0].first].item;
     }
 
-    /// \brief Returns the minimum priority.
+    /// \brief The minimum priority.
     ///
-    /// It returns the minimum priority.
-    /// \pre The heap must be nonempty.
+    /// This function returns the minimum priority.
+    /// \pre The heap must be non-empty.
     Prio prio() const {
       const_cast<RadixHeap<ItemIntMap>&>(*this).moveDown();
       return data[boxes[0].first].prio;
      }
 
-    /// \brief Deletes the item with minimum priority.
+    /// \brief Remove the item having minimum priority.
     ///
-    /// This method deletes the item with minimum priority.
+    /// This function removes the item having minimum priority.
     /// \pre The heap must be non-empty.
     void pop() {
       moveDown();
@@ -321,11 +324,12 @@
       relocate_last(index);
     }
 
-    /// \brief Deletes \c i from the heap.
+    /// \brief Remove the given item from the heap.
     ///
-    /// This method deletes item \c i from the heap, if \c i was
-    /// already stored in the heap.
-    /// \param i The item to erase.
+    /// This function removes the given item from the heap if it is
+    /// already stored.
+    /// \param i The item to delete.
+    /// \pre \e i must be in the heap.
     void erase(const Item &i) {
       int index = _iim[i];
       _iim[i] = POST_HEAP;
@@ -333,24 +337,26 @@
       relocate_last(index);
    }
 
-    /// \brief Returns the priority of \c i.
+    /// \brief The priority of the given item.
     ///
-    /// This function returns the priority of item \c i.
-    /// \pre \c i must be in the heap.
+    /// This function returns the priority of the given item.
     /// \param i The item.
+    /// \pre \e i must be in the heap.
     Prio operator[](const Item &i) const {
       int idx = _iim[i];
       return data[idx].prio;
     }
 
-    /// \brief \c i gets to the heap with priority \c p independently
-    /// if \c i was already there.
+    /// \brief Set the priority of an item or insert it, if it is
+    /// not stored in the heap.
     ///
-    /// This method calls \ref push(\c i, \c p) if \c i is not stored
-    /// in the heap and sets the priority of \c i to \c p otherwise.
-    /// It may throw an \e UnderFlowPriorityException.
+    /// This method sets the priority of the given item if it is
+    /// already stored in the heap. Otherwise it inserts the given
+    /// item into the heap with the given priority.
     /// \param i The item.
     /// \param p The priority.
+    /// \pre \e i must be in the heap.
+    /// \warning This method may throw an \c UnderFlowPriorityException.
     void set(const Item &i, const Prio &p) {
       int idx = _iim[i];
       if( idx < 0 ) {
@@ -365,39 +371,38 @@
       }
     }
 
-
-    /// \brief Decreases the priority of \c i to \c p.
+    /// \brief Decrease the priority of an item to the given value.
     ///
-    /// This method decreases the priority of item \c i to \c p.
-    /// \pre \c i must be stored in the heap with priority at least \c p, and
-    /// \c should be greater or equal to the last removed item's priority.
+    /// This function decreases the priority of an item to the given value.
     /// \param i The item.
     /// \param p The priority.
+    /// \pre \e i must be stored in the heap with priority at least \e p.
+    /// \warning This method may throw an \c UnderFlowPriorityException.
     void decrease(const Item &i, const Prio &p) {
       int idx = _iim[i];
       data[idx].prio = p;
       bubble_down(idx);
     }
 
-    /// \brief Increases the priority of \c i to \c p.
+    /// \brief Increase the priority of an item to the given value.
     ///
-    /// This method sets the priority of item \c i to \c p.
-    /// \pre \c i must be stored in the heap with priority at most \c p
+    /// This function increases the priority of an item to the given value.
     /// \param i The item.
     /// \param p The priority.
+    /// \pre \e i must be stored in the heap with priority at most \e p.
     void increase(const Item &i, const Prio &p) {
       int idx = _iim[i];
       data[idx].prio = p;
       bubble_up(idx);
     }
 
-    /// \brief Returns if \c item is in, has already been in, or has
-    /// never been in the heap.
+    /// \brief Return the state of an item.
     ///
-    /// This method returns PRE_HEAP if \c item has never been in the
-    /// heap, IN_HEAP if it is in the heap at the moment, and POST_HEAP
-    /// otherwise. In the latter case it is possible that \c item will
-    /// get back to the heap again.
+    /// This method returns \c PRE_HEAP if the given item has never
+    /// been in the heap, \c IN_HEAP if it is in the heap at the moment,
+    /// and \c POST_HEAP otherwise.
+    /// In the latter case it is possible that the item will get back
+    /// to the heap again.
     /// \param i The item.
     State state(const Item &i) const {
       int s = _iim[i];
@@ -405,11 +410,11 @@
       return State(s);
     }
 
-    /// \brief Sets the state of the \c item in the heap.
+    /// \brief Set the state of an item in the heap.
     ///
-    /// Sets the state of the \c item in the heap. It can be used to
-    /// manually clear the heap when it is important to achive the
-    /// better time complexity.
+    /// This function sets the state of the given item in the heap.
+    /// It can be used to manually clear the heap when it is important
+    /// to achive better time complexity.
     /// \param i The item.
     /// \param st The state. It should not be \c IN_HEAP.
     void state(const Item& i, State st) {