/** \file
 * Defines the interface for the KDTree class.
 *
 * \author Martin F. Krafft <libkdtree@pobox.madduck.net>
 *
 * Paul Harris figured this stuff out (below)
 * Notes:
 * This is similar to a binary tree, but its not the same.
 * There are a few important differences:
 *
 *  * Each level is sorted by a different criteria (this is fundamental to the design).
 *
 *  * It is possible to have children IDENTICAL to its parent in BOTH branches
 *    This is different to a binary tree, where identical children are always to the right
 *    So, KDTree has the relationships:
 *    * The left branch is <= its parent (in binary tree, this relationship is a plain < )
 *    * The right branch is <= its parent (same as binary tree)
 *
 *    This is done for mostly for performance.
 *    Its a LOT easier to maintain a consistent tree if we use the <= relationship.
 *    Note that this relationship only makes a difference when searching for an exact
 *    item with find() or find_exact, other search, erase and insert functions don't notice
 *    the difference.
 *
 *    In the case of binary trees, you can safely assume that the next identical item
 *    will be the child leaf,
 *    but in the case of KDTree, the next identical item might
 *    be a long way down a subtree, because of the various different sort criteria.
 *
 *    So erase()ing a node from a KDTree could require serious and complicated
 *    tree rebalancing to maintain consistency... IF we required binary-tree-like relationships.
 *
 *    This has no effect on insert()s, a < test is good enough to keep consistency.
 *
 *    It has an effect on find() searches:
 *      * Instead of using compare(child,node) for a < relationship and following 1 branch,
 *        we must use !compare(node,child) for a <= relationship, and test BOTH branches, as
 *        we could potentially go down both branches.
 *
 *    It has no real effect on bounds-based searches (like find_nearest, find_within_range)
 *    as it compares vs a boundary and would follow both branches if required.
 *
 *    This has no real effect on erase()s, a < test is good enough to keep consistency.
 */

#ifndef INCLUDE_KDTREE_KDTREE_HPP
#define INCLUDE_KDTREE_KDTREE_HPP


//
//  This number is guarenteed to change with every release.
//
//  KDTREE_VERSION % 100 is the patch level
//  KDTREE_VERSION / 100 % 1000 is the minor version
//  KDTREE_VERSION / 100000 is the major version
#define KDTREE_VERSION 701
//
//  KDTREE_LIB_VERSION must be defined to be the same as KDTREE_VERSION
//  but as a *string* in the form "x_y[_z]" where x is the major version
//  number, y is the minor version number, and z is the patch level if not 0.
#define KDTREE_LIB_VERSION "0_7_1"


#include <vector>

#ifdef KDTREE_CHECK_PERFORMANCE_COUNTERS
#  include <map>
#endif
#include <algorithm>

#ifdef KDTREE_DEFINE_OSTREAM_OPERATORS
#  include <ostream>
#  include <stack>
#endif

#include <cmath>
#include <cstddef>
#include <cassert>

#include "function.hpp"
#include "allocator.hpp"
#include "iterator.hpp"
#include "node.hpp"
#include "region.hpp"

namespace KDTree
{

#ifdef KDTREE_CHECK_PERFORMANCE
   unsigned long long num_dist_calcs = 0;
#endif

  template <size_t const __K, typename _Val,
            typename _Acc = _Bracket_accessor<_Val>,
	    typename _Dist = squared_difference<typename _Acc::result_type,
						typename _Acc::result_type>,
            typename _Cmp = std::less<typename _Acc::result_type>,
            typename _Alloc = std::allocator<_Node<_Val> > >
    class KDTree : protected _Alloc_base<_Val, _Alloc>
    {
    protected:
      typedef _Alloc_base<_Val, _Alloc> _Base;
      typedef typename _Base::allocator_type allocator_type;

      typedef _Node_base* _Base_ptr;
      typedef _Node_base const* _Base_const_ptr;
      typedef _Node<_Val>* _Link_type;
      typedef _Node<_Val> const* _Link_const_type;

      typedef _Node_compare<_Val, _Acc, _Cmp> _Node_compare_;

    public:
      typedef _Region<__K, _Val, typename _Acc::result_type, _Acc, _Cmp>
        _Region_;
      typedef _Val value_type;
      typedef value_type* pointer;
      typedef value_type const* const_pointer;
      typedef value_type& reference;
      typedef value_type const& const_reference;
      typedef typename _Acc::result_type subvalue_type;
      typedef typename _Dist::distance_type distance_type;
      typedef size_t size_type;
      typedef ptrdiff_t difference_type;

      KDTree(_Acc const& __acc = _Acc(), _Dist const& __dist = _Dist(),
	     _Cmp const& __cmp = _Cmp(), const allocator_type& __a = allocator_type())
        : _Base(__a), _M_header(),
	  _M_count(0), _M_acc(__acc), _M_cmp(__cmp), _M_dist(__dist)
      {
         _M_empty_initialise();
      }

      KDTree(const KDTree& __x)
         : _Base(__x.get_allocator()), _M_header(), _M_count(0),
	   _M_acc(__x._M_acc), _M_cmp(__x._M_cmp), _M_dist(__x._M_dist)
      {
         _M_empty_initialise();
         // this is slow:
         // this->insert(begin(), __x.begin(), __x.end());
         // this->optimise();

         // this is much faster, as it skips a lot of useless work
         // do the optimisation before inserting
         // Needs to be stored in a vector first as _M_optimise()
         // sorts the data in the passed iterators directly.
         std::vector<value_type> temp;
         temp.reserve(__x.size());
         std::copy(__x.begin(),__x.end(),std::back_inserter(temp));
         _M_optimise(temp.begin(), temp.end(), 0);
      }

      template<typename _InputIterator>
        KDTree(_InputIterator __first, _InputIterator __last,
	       _Acc const& acc = _Acc(), _Dist const& __dist = _Dist(),
	       _Cmp const& __cmp = _Cmp(), const allocator_type& __a = allocator_type())
        : _Base(__a), _M_header(), _M_count(0),
	  _M_acc(acc), _M_cmp(__cmp), _M_dist(__dist)
      {
         _M_empty_initialise();
         // this is slow:
         // this->insert(begin(), __first, __last);
         // this->optimise();

         // this is much faster, as it skips a lot of useless work
         // do the optimisation before inserting
         // Needs to be stored in a vector first as _M_optimise()
         // sorts the data in the passed iterators directly.
         std::vector<value_type> temp;
         temp.reserve(std::distance(__first,__last));
         std::copy(__first,__last,std::back_inserter(temp));
         _M_optimise(temp.begin(), temp.end(), 0);

         // NOTE: this will BREAK users that are passing in
         // read-once data via the iterator...
         // We increment __first all the way to __last once within
         // the distance() call, and again within the copy() call.
         //
         // This should end up using some funky C++ concepts or 
         // type traits to check that the iterators can be used in this way...
      }


      // this will CLEAR the tree and fill it with the contents
      // of 'writable_vector'.  it will use the passed vector directly,
      // and will basically resort the vector many times over while
      // optimising the tree.
      //
      // Paul: I use this when I have already built up a vector of data
      // that I want to add, and I don't mind if its contents get shuffled
      // by the kdtree optimise routine.
      void efficient_replace_and_optimise( std::vector<value_type> & writable_vector )
      {
         this->clear();
         _M_optimise(writable_vector.begin(), writable_vector.end(), 0);
      }



      KDTree&
      operator=(const KDTree& __x)
      {
	if (this != &__x)
	  {
	    _M_acc = __x._M_acc;
	    _M_dist = __x._M_dist;
	    _M_cmp = __x._M_cmp;
         // this is slow:
         // this->insert(begin(), __x.begin(), __x.end());
         // this->optimise();

         // this is much faster, as it skips a lot of useless work
         // do the optimisation before inserting
         // Needs to be stored in a vector first as _M_optimise()
         // sorts the data in the passed iterators directly.
         std::vector<value_type> temp;
         temp.reserve(__x.size());
         std::copy(__x.begin(),__x.end(),std::back_inserter(temp));
         efficient_replace_and_optimise(temp);
	  }
	return *this;
      }

      ~KDTree()
      {
        this->clear();
      }

      allocator_type
      get_allocator() const
      {
        return _Base::get_allocator();
      }

      size_type
      size() const
      {
        return _M_count;
      }

      size_type
      max_size() const
      {
        return size_type(-1);
      }

      bool
      empty() const
      {
        return this->size() == 0;
      }

      void
      clear()
      {
        _M_erase_subtree(_M_get_root());
        _M_set_leftmost(&_M_header);
        _M_set_rightmost(&_M_header);
        _M_set_root(NULL);
        _M_count = 0;
      }

      /*! \brief Comparator for the values in the KDTree.

	The comparator shall not be modified, it could invalidate the tree.
	\return a copy of the comparator used by the KDTree.
       */
      _Cmp
      value_comp() const
      { return _M_cmp; }

      /*! \brief Accessor to the value's elements.

	This accessor shall not be modified, it could invalidate the tree.
	\return a copy of the accessor used by the KDTree.
       */
      _Acc
      value_acc() const
      { return _M_acc; }

      /*! \brief Distance calculator between 2 value's element.

	This functor can be modified. It's modification will only affect the
	behavior of the find and find_nearest functions.
	\return a reference to the distance calculator used by the KDTree.
       */
      const _Dist&
      value_distance() const
      { return _M_dist; }

      _Dist&
      value_distance()
      { return _M_dist; }

      // typedef _Iterator<_Val, reference, pointer> iterator;
      typedef _Iterator<_Val, const_reference, const_pointer> const_iterator;
      // No mutable iterator at this stage
      typedef const_iterator iterator;
      typedef std::reverse_iterator<const_iterator> const_reverse_iterator;
      typedef std::reverse_iterator<iterator> reverse_iterator;

      // Note: the static_cast in end() is invalid (_M_header is not convertable to a _Link_type), but
      // thats ok as it just means undefined behaviour if the user dereferences the end() iterator.

      const_iterator begin() const { return const_iterator(_M_get_leftmost()); }
      const_iterator end() const { return const_iterator(static_cast<_Link_const_type>(&_M_header)); }
      const_reverse_iterator rbegin() const { return const_reverse_iterator(end()); }
      const_reverse_iterator rend() const { return const_reverse_iterator(begin()); }

      iterator
      insert(iterator /* ignored */, const_reference __V)
      {
         return this->insert(__V);
      }

      iterator
      insert(const_reference __V)
      {
        if (!_M_get_root())
          {
            _Link_type __n = _M_new_node(__V, &_M_header);
            ++_M_count;
            _M_set_root(__n);
            _M_set_leftmost(__n);
            _M_set_rightmost(__n);
            return iterator(__n);
          }
        return _M_insert(_M_get_root(), __V, 0);
      }

      template <class _InputIterator>
      void insert(_InputIterator __first, _InputIterator __last) {
         for (; __first != __last; ++__first)
            this->insert(*__first);
      }

      void
      insert(iterator __pos, size_type __n, const value_type& __x)
      {
        for (; __n > 0; --__n)
          this->insert(__pos, __x);
      }

      template<typename _InputIterator>
      void
      insert(iterator __pos, _InputIterator __first, _InputIterator __last) {
         for (; __first != __last; ++__first)
            this->insert(__pos, *__first);
      }

      // Note: this uses the find() to location the item you want to erase.
      // find() compares by equivalence of location ONLY.  See the comments
      // above find_exact() for why you may not want this.
      //
      // If you want to erase ANY item that has the same location as __V,
      // then use this function.
      //
      // If you want to erase a PARTICULAR item, and not any other item
      // that might happen to have the same location, then you should use
      // erase_exact().
      void
      erase(const_reference __V) {
	const_iterator b =  this->find(__V);
        this->erase(b);
      }

      void
      erase_exact(const_reference __V) {
        this->erase(this->find_exact(__V));
      }

      // note: kept as const because its easier to const-cast it away
      void
      erase(const_iterator const& __IT)
      {
         assert(__IT != this->end());
        _Link_const_type target = __IT.get_raw_node();
        _Link_const_type n = target;
        size_type level = 0;
        while ((n = _S_parent(n)) != &_M_header)
           ++level;
        _M_erase( const_cast<_Link_type>(target), level );
        _M_delete_node( const_cast<_Link_type>(target) );
        --_M_count;
      }

/* this does not work since erasure changes sort order
      void
      erase(const_iterator __A, const_iterator const& __B)
      {
        if (0 && __A == this->begin() && __B == this->end())
          {
            this->clear();
          }
        else
          {
            while (__A != __B)
              this->erase(__A++);
          }
      }
*/

      // compares via equivalence
      // so if you are looking for any item with the same location,
      // according to the standard accessor comparisions,
      // then this is the function for you.
      template <class SearchVal>
      const_iterator
      find(SearchVal const& __V) const
      {
        if (!_M_get_root()) return this->end();
        return _M_find(_M_get_root(), __V, 0);
      }

      // compares via equality
      // if you are looking for a particular item in the tree,
      // and (for example) it has an ID that is checked via an == comparison
      // eg
      // struct Item
      // {
      //    size_type unique_id;
      //    bool operator==(Item const& a, Item const& b) { return a.unique_id == b.unique_id; }
      //    Location location;
      // };
      // Two items may be equivalent in location.  find() would return
      // either one of them.  But no two items have the same ID, so
      // find_exact() would always return the item with the same location AND id.
      //
      template <class SearchVal>
      const_iterator
      find_exact(SearchVal const& __V) const
      {
        if (!_M_get_root()) return this->end();
        return _M_find_exact(_M_get_root(), __V, 0);
      }

      // NOTE: see notes on find_within_range().
      size_type
        count_within_range(const_reference __V, subvalue_type const __R) const
        {
          if (!_M_get_root()) return 0;
          _Region_ __region(__V, __R, _M_acc, _M_cmp);
          return this->count_within_range(__region);
        }

      size_type
        count_within_range(_Region_ const& __REGION) const
        {
          if (!_M_get_root()) return 0;

          _Region_ __bounds(__REGION);
          return _M_count_within_range(_M_get_root(),
                               __REGION, __bounds, 0);
        }

      // NOTE: see notes on find_within_range().
      template <typename SearchVal, class Visitor>
        Visitor
        visit_within_range(SearchVal const& V, subvalue_type const R, Visitor visitor) const
        {
          if (!_M_get_root()) return visitor;
          _Region_ region(V, R, _M_acc, _M_cmp);
          return this->visit_within_range(region, visitor);
        }

      template <class Visitor>
        Visitor
        visit_within_range(_Region_ const& REGION, Visitor visitor) const
        {
          if (_M_get_root())
            {
              _Region_ bounds(REGION);
              return _M_visit_within_range(visitor, _M_get_root(), REGION, bounds, 0);
            }
          return visitor;
        }

      // NOTE: this will visit points based on 'Manhattan distance' aka city-block distance
      // aka taxicab metric. Meaning it will find all points within:
      //    max(x_dist,max(y_dist,z_dist));
      //  AND NOT than what you would expect: sqrt(x_dist*x_dist + y_dist*y_dist + z_dist*z_dist)
      //
      // This is because it converts the distance into a bounding-box 'region' and compares
      // against that.
      //
      // If you want the sqrt() behaviour, ask on the mailing list for different options.
      //
      template <typename SearchVal, typename _OutputIterator>
        _OutputIterator
        find_within_range(SearchVal const& val, subvalue_type const range,
                          _OutputIterator out) const
        {
          if (!_M_get_root()) return out;
          _Region_ region(val, range, _M_acc, _M_cmp);
          return this->find_within_range(region, out);
        }

      template <typename _OutputIterator>
        _OutputIterator
        find_within_range(_Region_ const& region,
                          _OutputIterator out) const
        {
          if (_M_get_root())
            {
              _Region_ bounds(region);
              out = _M_find_within_range(out, _M_get_root(),
                                   region, bounds, 0);
            }
          return out;
        }

      template <class SearchVal>
      std::pair<const_iterator, distance_type>
      find_nearest (SearchVal const& __val) const
      {
	if (_M_get_root())
	  {
	    std::pair<const _Node<_Val>*,
	      std::pair<size_type, typename _Acc::result_type> >
	      best = _S_node_nearest (__K, 0, __val,
				      _M_get_root(), &_M_header, _M_get_root(),
				      std::sqrt(_S_accumulate_node_distance
				      (__K, _M_dist, _M_acc, _M_get_root()->_M_value, __val)),
				      _M_cmp, _M_acc, _M_dist,
				      always_true<value_type>());
	    return std::pair<const_iterator, distance_type>
	      (best.first, best.second.second);
	  }
	  return std::pair<const_iterator, distance_type>(end(), 0);
      }

      template <class SearchVal>
      std::pair<const_iterator, distance_type>
      find_nearest (SearchVal const& __val, distance_type __max) const
      {
	if (_M_get_root())
	  {
        bool root_is_candidate = false;
	    const _Node<_Val>* node = _M_get_root();
       { // scope to ensure we don't use 'root_dist' anywhere else
	    distance_type root_dist = std::sqrt(_S_accumulate_node_distance
	      (__K, _M_dist, _M_acc, _M_get_root()->_M_value, __val));
	    if (root_dist <= __max)
	      {
            root_is_candidate = true;
            __max = root_dist;
	      }
       }
	    std::pair<const _Node<_Val>*,
	      std::pair<size_type, typename _Acc::result_type> >
	      best = _S_node_nearest (__K, 0, __val, _M_get_root(), &_M_header,
				      node, __max, _M_cmp, _M_acc, _M_dist,
				      always_true<value_type>());
       // make sure we didn't just get stuck with the root node...
       if (root_is_candidate || best.first != _M_get_root())
          return std::pair<const_iterator, distance_type>
            (best.first, best.second.second);
	  }
	  return std::pair<const_iterator, distance_type>(end(), __max);
      }

      template <class SearchVal, class _Predicate>
      std::pair<const_iterator, distance_type>
      find_nearest_if (SearchVal const& __val, distance_type __max,
		       _Predicate __p) const
      {
	if (_M_get_root())
	  {
        bool root_is_candidate = false;
	    const _Node<_Val>* node = _M_get_root();
	    if (__p(_M_get_root()->_M_value))
	      {
            { // scope to ensure we don't use root_dist anywhere else
	    distance_type root_dist = std::sqrt(_S_accumulate_node_distance
		  (__K, _M_dist, _M_acc, _M_get_root()->_M_value, __val));
		if (root_dist <= __max)
		  {
           root_is_candidate = true;
		    root_dist = __max;
		  }
            }
	      }
	    std::pair<const _Node<_Val>*,
	      std::pair<size_type, typename _Acc::result_type> >
	      best = _S_node_nearest (__K, 0, __val, _M_get_root(), &_M_header,
				      node, __max, _M_cmp, _M_acc, _M_dist, __p);
       // make sure we didn't just get stuck with the root node...
       if (root_is_candidate || best.first != _M_get_root())
          return std::pair<const_iterator, distance_type>
            (best.first, best.second.second);
	  }
	  return std::pair<const_iterator, distance_type>(end(), __max);
      }

      void
      optimise()
      {
        std::vector<value_type> __v(this->begin(),this->end());
        this->clear();
        _M_optimise(__v.begin(), __v.end(), 0);
      }

      void
      optimize()
      { // cater for people who cannot spell :)
        this->optimise();
      }

      void check_tree()
      {
         _M_check_node(_M_get_root(),0);
      }

    protected:

      void _M_check_children( _Link_const_type child, _Link_const_type parent, size_type const level, bool to_the_left )
      {
         assert(parent);
         if (child)
         {
	   _Node_compare_ compare(level % __K, _M_acc, _M_cmp);
            // REMEMBER! its a <= relationship for BOTH branches
            // for left-case (true), child<=node --> !(node<child)
            // for right-case (false), node<=child --> !(child<node)
            assert(!to_the_left || !compare(parent->_M_value,child->_M_value));  // check the left
            assert(to_the_left || !compare(child->_M_value,parent->_M_value));   // check the right
            // and recurse down the tree, checking everything
            _M_check_children(_S_left(child),parent,level,to_the_left);
            _M_check_children(_S_right(child),parent,level,to_the_left);
         }
      }

      void _M_check_node( _Link_const_type node, size_type const level )
      {
         if (node)
         {
            // (comparing on this level)
            // everything to the left of this node must be smaller than this
            _M_check_children( _S_left(node), node, level, true );
            // everything to the right of this node must be larger than this
            _M_check_children( _S_right(node), node, level, false );

            _M_check_node( _S_left(node), level+1 );
            _M_check_node( _S_right(node), level+1 );
         }
      }

      void _M_empty_initialise()
      {
        _M_set_leftmost(&_M_header);
        _M_set_rightmost(&_M_header);
	_M_header._M_parent = NULL;
        _M_set_root(NULL);
      }

      iterator
      _M_insert_left(_Link_type __N, const_reference __V)
      {
        _S_set_left(__N, _M_new_node(__V)); ++_M_count;
        _S_set_parent( _S_left(__N), __N );
        if (__N == _M_get_leftmost())
           _M_set_leftmost( _S_left(__N) );
        return iterator(_S_left(__N));
      }

      iterator
      _M_insert_right(_Link_type __N, const_reference __V)
      {
        _S_set_right(__N, _M_new_node(__V)); ++_M_count;
        _S_set_parent( _S_right(__N), __N );
        if (__N == _M_get_rightmost())
           _M_set_rightmost( _S_right(__N) );
        return iterator(_S_right(__N));
      }

      iterator
      _M_insert(_Link_type __N, const_reference __V,
             size_type const __L)
      {
        if (_Node_compare_(__L % __K, _M_acc, _M_cmp)(__V, __N->_M_value))
          {
            if (!_S_left(__N))
              return _M_insert_left(__N, __V);
            return _M_insert(_S_left(__N), __V, __L+1);
          }
        else
          {
            if (!_S_right(__N) || __N == _M_get_rightmost())
              return _M_insert_right(__N, __V);
            return _M_insert(_S_right(__N), __V, __L+1);
          }
      }

      _Link_type
      _M_erase(_Link_type dead_dad, size_type const level)
      {
         // find a new step_dad, he will become a drop-in replacement.
        _Link_type step_dad = _M_get_erase_replacement(dead_dad, level);

         // tell dead_dad's parent that his new child is step_dad
        if (dead_dad == _M_get_root())
           _M_set_root(step_dad);
        else if (_S_left(_S_parent(dead_dad)) == dead_dad)
            _S_set_left(_S_parent(dead_dad), step_dad);
        else
            _S_set_right(_S_parent(dead_dad), step_dad);

        // deal with the left and right edges of the tree...
        // if the dead_dad was at the edge, then substitude...
        // but if there IS no new dead, then left_most is the dead_dad's parent
         if (dead_dad == _M_get_leftmost())
           _M_set_leftmost( (step_dad ? step_dad : _S_parent(dead_dad)) );
         if (dead_dad == _M_get_rightmost())
           _M_set_rightmost( (step_dad ? step_dad : _S_parent(dead_dad)) );

        if (step_dad)
          {
             // step_dad gets dead_dad's parent
            _S_set_parent(step_dad, _S_parent(dead_dad));

            // first tell the children that step_dad is their new dad
            if (_S_left(dead_dad))
               _S_set_parent(_S_left(dead_dad), step_dad);
            if (_S_right(dead_dad))
               _S_set_parent(_S_right(dead_dad), step_dad);

            // step_dad gets dead_dad's children
            _S_set_left(step_dad, _S_left(dead_dad));
            _S_set_right(step_dad, _S_right(dead_dad));
          }

        return step_dad;
      }



      _Link_type
      _M_get_erase_replacement(_Link_type node, size_type const level)
      {
         // if 'node' is null, then we can't do any better
        if (_S_is_leaf(node))
           return NULL;

        std::pair<_Link_type,size_type> candidate;
        // if there is nothing to the left, find a candidate on the right tree
        if (!_S_left(node))
          candidate = _M_get_j_min( std::pair<_Link_type,size_type>(_S_right(node),level), level+1);
        // ditto for the right
        else if ((!_S_right(node)))
          candidate = _M_get_j_max( std::pair<_Link_type,size_type>(_S_left(node),level), level+1);
        // we have both children ...
        else
         {
            // we need to do a little more work in order to find a good candidate
            // this is actually a technique used to choose a node from either the
            // left or right branch RANDOMLY, so that the tree has a greater change of
            // staying balanced.
            // If this were a true binary tree, we would always hunt down the right branch.
            // See top for notes.
	   _Node_compare_ compare(level % __K, _M_acc, _M_cmp);
            // compare the children based on this level's criteria...
            // (this gives virtually random results)
            if (compare(_S_right(node)->_M_value, _S_left(node)->_M_value))
               // the right is smaller, get our replacement from the SMALLEST on the right
               candidate = _M_get_j_min(std::pair<_Link_type,size_type>(_S_right(node),level), level+1);
            else
               candidate = _M_get_j_max( std::pair<_Link_type,size_type>(_S_left(node),level), level+1);
         }

        // we have a candidate replacement by now.
        // remove it from the tree, but don't delete it.
        // it must be disconnected before it can be reconnected.
        _Link_type parent = _S_parent(candidate.first);
        if (_S_left(parent) == candidate.first)
           _S_set_left(parent, _M_erase(candidate.first, candidate.second));
        else
           _S_set_right(parent, _M_erase(candidate.first, candidate.second));

        return candidate.first;
      }



      std::pair<_Link_type,size_type>
      _M_get_j_min( std::pair<_Link_type,size_type> const node, size_type const level)
      {
         typedef std::pair<_Link_type,size_type> Result;
        if (_S_is_leaf(node.first))
            return Result(node.first,level);

        _Node_compare_ compare(node.second % __K, _M_acc, _M_cmp);
        Result candidate = node;
        if (_S_left(node.first))
          {
            Result left = _M_get_j_min(Result(_S_left(node.first), node.second), level+1);
            if (compare(left.first->_M_value, candidate.first->_M_value))
                candidate = left;
          }
        if (_S_right(node.first))
          {
            Result right = _M_get_j_min( Result(_S_right(node.first),node.second), level+1);
            if (compare(right.first->_M_value, candidate.first->_M_value))
                candidate = right;
          }
        if (candidate.first == node.first)
           return Result(candidate.first,level);

        return candidate;
      }



      std::pair<_Link_type,size_type>
      _M_get_j_max( std::pair<_Link_type,size_type> const node, size_type const level)
      {
         typedef std::pair<_Link_type,size_type> Result;

        if (_S_is_leaf(node.first))
            return Result(node.first,level);

        _Node_compare_ compare(node.second % __K, _M_acc, _M_cmp);
        Result candidate = node;
        if (_S_left(node.first))
          {
            Result left = _M_get_j_max( Result(_S_left(node.first),node.second), level+1);
            if (compare(candidate.first->_M_value, left.first->_M_value))
                candidate = left;
          }
        if (_S_right(node.first))
          {
            Result right = _M_get_j_max(Result(_S_right(node.first),node.second), level+1);
            if (compare(candidate.first->_M_value, right.first->_M_value))
                candidate = right;
          }

        if (candidate.first == node.first)
           return Result(candidate.first,level);

        return candidate;
      }


      void
      _M_erase_subtree(_Link_type __n)
      {
        while (__n)
          {
            _M_erase_subtree(_S_right(__n));
            _Link_type __t = _S_left(__n);
            _M_delete_node(__n);
            __n = __t;
          }
      }

      const_iterator
      _M_find(_Link_const_type node, const_reference value, size_type const level) const
      {
         // be aware! This is very different to normal binary searches, because of the <=
         // relationship used. See top for notes.
         // Basically we have to check ALL branches, as we may have an identical node
         // in different branches.
          const_iterator found = this->end();

	  _Node_compare_ compare(level % __K, _M_acc, _M_cmp);
        if (!compare(node->_M_value,value))   // note, this is a <= test
          {
           // this line is the only difference between _M_find_exact() and _M_find()
            if (_M_matches_node(node, value, level))
              return const_iterator(node);   // return right away
            if (_S_left(node))
               found = _M_find(_S_left(node), value, level+1);
          }
        if ( _S_right(node) && found == this->end() && !compare(value,node->_M_value))   // note, this is a <= test
            found = _M_find(_S_right(node), value, level+1);
        return found;
      }

      const_iterator
      _M_find_exact(_Link_const_type node, const_reference value, size_type const level) const
      {
         // be aware! This is very different to normal binary searches, because of the <=
         // relationship used. See top for notes.
         // Basically we have to check ALL branches, as we may have an identical node
         // in different branches.
          const_iterator found = this->end();

	  _Node_compare_ compare(level % __K, _M_acc, _M_cmp);
        if (!compare(node->_M_value,value))  // note, this is a <= test
        {
           // this line is the only difference between _M_find_exact() and _M_find()
            if (value == *const_iterator(node))
              return const_iterator(node);   // return right away
           if (_S_left(node))
            found = _M_find_exact(_S_left(node), value, level+1);
        }

        // note: no else!  items that are identical can be down both branches
        if ( _S_right(node) && found == this->end() && !compare(value,node->_M_value))   // note, this is a <= test
            found = _M_find_exact(_S_right(node), value, level+1);
        return found;
      }

      bool
      _M_matches_node_in_d(_Link_const_type __N, const_reference __V,
                           size_type const __L) const
      {
        _Node_compare_ compare(__L % __K, _M_acc, _M_cmp);
        return !(compare(__N->_M_value, __V) || compare(__V, __N->_M_value));
      }

      bool
      _M_matches_node_in_other_ds(_Link_const_type __N, const_reference __V,
                                  size_type const __L = 0) const
      {
        size_type __i = __L;
        while ((__i = (__i + 1) % __K) != __L % __K)
          if (!_M_matches_node_in_d(__N, __V, __i)) return false;
        return true;
      }

      bool
      _M_matches_node(_Link_const_type __N, const_reference __V,
                      size_type __L = 0) const
      {
        return _M_matches_node_in_d(__N, __V, __L)
          && _M_matches_node_in_other_ds(__N, __V, __L);
      }

      size_type
        _M_count_within_range(_Link_const_type __N, _Region_ const& __REGION,
                             _Region_ const& __BOUNDS,
                             size_type const __L) const
        {
           size_type count = 0;
          if (__REGION.encloses(_S_value(__N)))
            {
               ++count;
            }
          if (_S_left(__N))
            {
              _Region_ __bounds(__BOUNDS);
              __bounds.set_high_bound(_S_value(__N), __L);
              if (__REGION.intersects_with(__bounds))
                count += _M_count_within_range(_S_left(__N),
                                     __REGION, __bounds, __L+1);
            }
          if (_S_right(__N))
            {
              _Region_ __bounds(__BOUNDS);
              __bounds.set_low_bound(_S_value(__N), __L);
              if (__REGION.intersects_with(__bounds))
                count += _M_count_within_range(_S_right(__N),
                                     __REGION, __bounds, __L+1);
            }

          return count;
        }


      template <class Visitor>
        Visitor
        _M_visit_within_range(Visitor visitor,
                             _Link_const_type N, _Region_ const& REGION,
                             _Region_ const& BOUNDS,
                             size_type const L) const
        {
          if (REGION.encloses(_S_value(N)))
            {
              visitor(_S_value(N));
            }
          if (_S_left(N))
            {
              _Region_ bounds(BOUNDS);
              bounds.set_high_bound(_S_value(N), L);
              if (REGION.intersects_with(bounds))
                visitor = _M_visit_within_range(visitor, _S_left(N),
                                     REGION, bounds, L+1);
            }
          if (_S_right(N))
            {
              _Region_ bounds(BOUNDS);
              bounds.set_low_bound(_S_value(N), L);
              if (REGION.intersects_with(bounds))
                visitor = _M_visit_within_range(visitor, _S_right(N),
                                     REGION, bounds, L+1);
            }

          return visitor;
        }



      template <typename _OutputIterator>
        _OutputIterator
        _M_find_within_range(_OutputIterator out,
                             _Link_const_type __N, _Region_ const& __REGION,
                             _Region_ const& __BOUNDS,
                             size_type const __L) const
        {
          if (__REGION.encloses(_S_value(__N)))
            {
              *out++ = _S_value(__N);
            }
          if (_S_left(__N))
            {
              _Region_ __bounds(__BOUNDS);
              __bounds.set_high_bound(_S_value(__N), __L);
              if (__REGION.intersects_with(__bounds))
                out = _M_find_within_range(out, _S_left(__N),
                                     __REGION, __bounds, __L+1);
            }
          if (_S_right(__N))
            {
              _Region_ __bounds(__BOUNDS);
              __bounds.set_low_bound(_S_value(__N), __L);
              if (__REGION.intersects_with(__bounds))
                out = _M_find_within_range(out, _S_right(__N),
                                     __REGION, __bounds, __L+1);
            }

          return out;
        }


      template <typename _Iter>
        void
        _M_optimise(_Iter const& __A, _Iter const& __B,
                    size_type const __L)
      {
        if (__A == __B) return;
        _Node_compare_ compare(__L % __K, _M_acc, _M_cmp);
        _Iter __m = __A + (__B - __A) / 2;
        std::nth_element(__A, __m, __B, compare);
        this->insert(*__m);
        if (__m != __A) _M_optimise(__A, __m, __L+1);
        if (++__m != __B) _M_optimise(__m, __B, __L+1);
      }

      _Link_const_type
      _M_get_root() const
      {
         return const_cast<_Link_const_type>(_M_root);
      }

      _Link_type
      _M_get_root()
      {
         return _M_root;
      }

      void _M_set_root(_Link_type n)
      {
         _M_root = n;
      }

      _Link_const_type
      _M_get_leftmost() const
      {
        return static_cast<_Link_type>(_M_header._M_left);
      }

      void
      _M_set_leftmost( _Node_base * a )
      {
         _M_header._M_left = a;
      }

      _Link_const_type
      _M_get_rightmost() const
      {
        return static_cast<_Link_type>( _M_header._M_right );
      }

      void
      _M_set_rightmost( _Node_base * a )
      {
         _M_header._M_right = a;
      }

      static _Link_type
      _S_parent(_Base_ptr N)
      {
        return static_cast<_Link_type>( N->_M_parent );
      }

      static _Link_const_type
      _S_parent(_Base_const_ptr N)
      {
        return static_cast<_Link_const_type>( N->_M_parent );
      }

      static void
      _S_set_parent(_Base_ptr N, _Base_ptr p)
      {
        N->_M_parent = p;
      }

      static void
      _S_set_left(_Base_ptr N, _Base_ptr l)
      {
        N->_M_left = l;
      }

      static _Link_type
      _S_left(_Base_ptr N)
      {
        return static_cast<_Link_type>( N->_M_left );
      }

      static _Link_const_type
      _S_left(_Base_const_ptr N)
      {
        return static_cast<_Link_const_type>( N->_M_left );
      }

      static void
      _S_set_right(_Base_ptr N, _Base_ptr r)
      {
        N->_M_right = r;
      }

      static _Link_type
      _S_right(_Base_ptr N)
      {
        return static_cast<_Link_type>( N->_M_right );
      }

      static _Link_const_type
      _S_right(_Base_const_ptr N)
      {
        return static_cast<_Link_const_type>( N->_M_right );
      }

      static bool
      _S_is_leaf(_Base_const_ptr N)
      {
        return !_S_left(N) && !_S_right(N);
      }

      static const_reference
      _S_value(_Link_const_type N)
      {
        return N->_M_value;
      }

      static const_reference
      _S_value(_Base_const_ptr N)
      {
        return static_cast<_Link_const_type>(N)->_M_value;
      }

      static _Link_const_type
      _S_minimum(_Link_const_type __X)
      {
        return static_cast<_Link_const_type> ( _Node_base::_S_minimum(__X) );
      }

      static _Link_const_type
      _S_maximum(_Link_const_type __X)
      {
        return static_cast<_Link_const_type>( _Node_base::_S_maximum(__X) );
      }


      _Link_type
      _M_new_node(const_reference __V, //  = value_type(),
                  _Base_ptr const __PARENT = NULL,
                  _Base_ptr const __LEFT = NULL,
                  _Base_ptr const __RIGHT = NULL)
      {
         typename _Base::NoLeakAlloc noleak(this);
         _Link_type new_node = noleak.get();
         _Base::_M_construct_node(new_node, __V, __PARENT, __LEFT, __RIGHT);
         noleak.disconnect();
         return new_node;
      }

      /* WHAT was this for?
      _Link_type
      _M_clone_node(_Link_const_type __X)
      {
        _Link_type __ret = _M_allocate_node(__X->_M_value);
        // TODO
        return __ret;
      }
      */

      void
      _M_delete_node(_Link_type __p)
      {
        _Base::_M_destroy_node(__p);
        _Base::_M_deallocate_node(__p);
      }

      _Link_type _M_root;
      _Node_base _M_header;
      size_type _M_count;
      _Acc _M_acc;
      _Cmp _M_cmp;
      _Dist _M_dist;

#ifdef KDTREE_DEFINE_OSTREAM_OPERATORS
      friend std::ostream&
      operator<<(std::ostream& o,
		 KDTree<__K, _Val, _Acc, _Dist, _Cmp, _Alloc> const& tree)
    {
      o << "meta node:   " << tree._M_header << std::endl;
      o << "root node:   " << tree._M_root << std::endl;

      if (tree.empty())
        return o << "[empty " << __K << "d-tree " << &tree << "]";

      o << "nodes total: " << tree.size() << std::endl;
      o << "dimensions:  " << __K << std::endl;

      typedef KDTree<__K, _Val, _Acc, _Dist, _Cmp, _Alloc> _Tree;
      typedef typename _Tree::_Link_type _Link_type;

      std::stack<_Link_const_type> s;
      s.push(tree._M_get_root());

      while (!s.empty())
        {
          _Link_const_type n = s.top();
          s.pop();
          o << *n << std::endl;
          if (_Tree::_S_left(n)) s.push(_Tree::_S_left(n));
          if (_Tree::_S_right(n)) s.push(_Tree::_S_right(n));
        }

      return o;
    }
#endif

  };


} // namespace KDTree

#endif // include guard

/* COPYRIGHT --
 *
 * This file is part of libkdtree++, a C++ template KD-Tree sorting container.
 * libkdtree++ is (c) 2004-2007 Martin F. Krafft <libkdtree@pobox.madduck.net>
 * and Sylvain Bougerel <sylvain.bougerel.devel@gmail.com> distributed under the
 * terms of the Artistic License 2.0. See the ./COPYING file in the source tree
 * root for more information.
 * Parts of this file are (c) 2004-2007 Paul Harris <paulharris@computer.org>.
 *
 * THIS PACKAGE IS PROVIDED "AS IS" AND WITHOUT ANY EXPRESS OR IMPLIED
 * WARRANTIES, INCLUDING, WITHOUT LIMITATION, THE IMPLIED WARRANTIES
 * OF MERCHANTIBILITY AND FITNESS FOR A PARTICULAR PURPOSE.
 */