/* -*- C++ -*-

  *

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

  *

  * Copyright (C) 2003-2007

  * Egervary Jeno Kombinatorikus Optimalizalasi Kutatocsoport

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

  *

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

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

  * precise terms see the accompanying LICENSE file.

  *

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

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

  * purpose.

  *

  */

 #ifndef LEMON_DYNAMIC_TREE_H

 #define LEMON_DYNAMIC_TREE_H

 /// \ingroup auxdata

 /// \file

 /// \brief The dynamic tree data structure of Sleator and Tarjan.

 ///

 #include <vector>

 #include <limits>

 #include <lemon/tolerance.h>

 namespace lemon {

   /// \ingroup auxdata

   ///

   /// \brief The dynamic tree data structure of Sleator and Tarjan.

   ///

   /// This class provides an implementation of the dynamic tree data

   /// structure for maintaining a set of node-disjoint rooted

   /// trees. Each item has an associated value, and the item with

   /// minimum value can be find in \f$O(\log(n)\f$ on the path from a

   /// node to the its root, and the items on such path can be

   /// increased or decreased globally with a certain value in the same

   /// running time. We regard a tree edge as directed toward the root,

   /// that is from child to parent. Its structure can be modified by

   /// two basic operations: \e link(v,w) adds an edge between a root v

   /// and a node w in a different component; \e cut(v) removes the

   /// edge between v and its parent.

   /// 

   /// \param _Value The value type of the items.  

   /// \param _ItemIntMap Converts item type of node to integer.

   /// \param _Tolerance The tolerance class to handle computation

   /// problems.

   /// \param _enableSize If true then the data structre manatain the

   /// size of each tree. The feature is used in \ref GoldbergTarjan

   /// algorithm. The default value is true.

   ///

   /// \author Hamori Tamas

 #ifdef DOXYGEN

   template <typename _Value, typename _ItemIntMap, 

 	    typename _Tolerance, bool _enableSize>

 #else

   template <typename _Value, typename _ItemIntMap, 

 	    typename _Tolerance = lemon::Tolerance<_Value>,

 	    bool _enableSize = true>

 #endif

   class DynamicTree {

   public:

     /// \brief The integer map on the items.

     typedef _ItemIntMap ItemIntMap;

     /// \brief The item type of nodes.

     typedef typename ItemIntMap::Key Item;

     /// \brief The value type of the algorithms.

     typedef _Value Value;

     /// \brief The tolerance used by the algorithm.

     typedef _Tolerance Tolerance;

   private:

     class ItemData;

     std::vector<ItemData> _data;

     ItemIntMap &_iim;

     Value _max_value;

     Tolerance _tolerance;

   public:

     /// \brief The constructor of the class.

     ///

     /// \param iim The integer map on the items. 

     /// \param tolerance Tolerance class.

     DynamicTree(ItemIntMap &iim, const Tolerance& tolerance = Tolerance())

       : _iim(iim), _max_value(std::numeric_limits<Value>::max()), 

 	_tolerance(tolerance) {}

     ~DynamicTree() {}

     /// \brief Clears the data structure

     ///

     /// Clears the data structure

     void clear() {

       _data.clear();

     }

     /// \brief Sets the tolerance used by algorithm.

     ///

     /// Sets the tolerance used by algorithm.

     void tolerance(const Tolerance& tolerance) const {

       _tolerance = tolerance;

       return *this;

     } 

     /// \brief Returns the tolerance used by algorithm.

     ///

     /// Returns the tolerance used by algorithm.

     const Tolerance& tolerance() const {

       return tolerance;

     } 

     /// \brief Create a new tree containing a single node with cost zero.

     void makeTree(const Item &item) {

       _data[makePath(item)].successor = -1;

     }

     /// \brief Return the root of the tree containing node with itemtype

     /// \e item.

     Item findRoot(const Item &item) {

       return _data[findTail(expose(_iim[item]))].id;

     }

     /// \brief Return the the value of nodes in the tree containing

     /// node with itemtype \e item.

     int findSize(const Item &item) {

       return _data[expose(_iim[item])].size;

     }

     /// \brief Return the minimum cost containing node.

     /// 

     /// Return into \e d the minimum cost on the tree path from \e item

     /// to findRoot(item).  Return the last item (closest to its root)

     /// on this path of the minimum cost.

     Item findCost(const Item &item, Value& d){

       return _data[findPathCost(expose(_iim[item]),d)].id;

     }

     /// \brief Add \e x value to the cost of every node on the path from

     /// \e item to findRoot(item).

     void addCost(const Item &item, Value x){

       addPathCost(expose(_iim[item]), x);

     }

     /// \brief Combine the trees containing nodes \e item1 and \e item2

     /// by adding an edge from \e item1 \e item2.

     /// 

     /// This operation assumes that \e item1 is root and \e item2 is in

     /// a different tree.

     void link(const Item &item1, const Item &item2){

       int v = _iim[item1];

       int w = _iim[item2];

       int p = expose(w);

       join(-1, expose(v), p);

       _data[v].successor = -1;

       _data[v].size += _data[p].size;

     }    

     /// \brief Divide the tree containing node \e item into two trees by

     /// deleting the edge out of \e item.

     /// 

     /// This operation assumes that \e item is not a tree root.

     void cut(const Item &item) {

       int v = _iim[item];

       int p, q;

       expose(v);

       split(p, v, q);

       if (p != -1) {

 	_data[p].successor = v;

       }

       _data[v].size -= _data[q].size;

       if (q != -1) {

 	_data[q].successor = _data[v].successor;

       }

       _data[v].successor = -1;

     }

     ///\brief 

     Item parent(const Item &item){

       return _data[_iim[item].p].id;

     }

     ///\brief Return the upper bound of the costs.

     Value maxValue() const {

       return _max_value;

     }

   private:

     int makePath(const Item &item) {

       _iim.set(item, _data.size());

       ItemData v(item);

       _data.push_back(v);

       return _iim[item];

     }

     int findPath(int v){

       splay(v);

       return v;

     }

     int findPathCost(int p, Value &d){

       while ((_data[p].right != -1 && 

 	      !_tolerance.less(0, _data[_data[p].right].dmin)) || 

 	     (_data[p].left != -1 && _tolerance.less(0, _data[p].dcost))) {

 	if (_data[p].right != -1 && 

 	    !_tolerance.less(0, _data[_data[p].right].dmin)) {

 	  p = _data[p].right;

 	} else if (_data[p].left != -1 && 

 		   !_tolerance.less(0, _data[_data[p].left].dmin)){

 	  p = _data[p].left;

 	}

       }

       splay(p);

       d = _data[p].dmin;

       return p; 

     }

     int findTail(int p){

       while (_data[p].right != -1) {

 	p = _data[p].right;

       }

       splay(p);

       return p;

     }

     void addPathCost(int p, Value x){

       if (!_tolerance.less(x, _max_value)) {

 	_data[p].dmin = x;_data[p].dcost = x;

       } else if (!_tolerance.less(-x, _max_value)) {

 	_data[p].dmin = 0;

 	_data[p].dcost = 0;

       } else {

 	_data[p].dmin += x;

       }

     }

     void join(int p, int v, int q) {

       Value min = _max_value;

       Value pmin = _max_value;

       Value vmin = _data[v].dmin;

       Value qmin = _max_value;

       if (p != -1){

 	pmin = _data[p].dmin;

       }

       if (q != -1){

 	qmin = _data[q].dmin;

       }

       if (_tolerance.less(vmin, qmin)) {

 	if (_tolerance.less(vmin,pmin)) {

 	  min = vmin;

 	} else {

 	  min = pmin;

 	}

       } else if (_tolerance.less(qmin,pmin)) {

 	min = qmin;

       } else {

 	min = pmin;

       }

       if (p != -1){

 	_data[p].parent = v;

 	_data[p].dmin -= min;

       }

       if (q!=-1){

 	_data[q].parent = v;

 	if (_tolerance.less(_data[q].dmin,_max_value)) {

 	  _data[q].dmin -= min;

 	}

       }

       _data[v].left = p;

       _data[v].right = q;

       if (_tolerance.less(min,_max_value)) {

 	_data[v].dcost = _data[v].dmin - min;

       }

       _data[v].dmin = min;

     }

     void split(int &p, int v, int &q){

       splay(v);

       p = -1;

       if (_data[v].left != -1){

 	p = _data[v].left;

 	_data[p].dmin += _data[v].dmin;

 	_data[p].parent = -1;

 	_data[v].left = -1;

       }

       q = -1;

       if (_data[v].right != -1) {

 	q=_data[v].right;

 	if (_tolerance.less(_data[q].dmin, _max_value)) {

 	  _data[q].dmin += _data[v].dmin;

 	}

 	_data[q].parent = -1;

 	_data[v].right = -1;

       } 

       if (_tolerance.less(_data[v].dcost, _max_value)) {

 	_data[v].dmin += _data[v].dcost;

 	_data[v].dcost = 0;

       } else {

 	_data[v].dmin = _data[v].dcost;

       }

     }

     int expose(int v) {

       int p, q, r, w;

       p = -1;

       while (v != -1) {

 	w = _data[findPath(v)].successor;

 	split(q, v, r);

 	if (q != -1) {

 	  _data[q].successor = v;

 	}

 	join(p, v, r);

 	p = v;

 	v = w;

       }

       _data[p].successor = -1;

       return p;

     }

     void splay(int v) {

       while (_data[v].parent != -1) {

 	if (v == _data[_data[v].parent].left) {

 	  if (_data[_data[v].parent].parent == -1) {

 	    zig(v);

 	  } else {

 	    if (_data[v].parent == _data[_data[_data[v].parent].parent].left) {

 	      zig(_data[v].parent);

 	      zig(v);

 	    } else {

 	      zig(v);

 	      zag(v);

 	    }

 	  }

 	} else {

 	  if (_data[_data[v].parent].parent == -1) {

 	    zag(v);

 	  } else {

 	    if (_data[v].parent == _data[_data[_data[v].parent].parent].left) {

 	      zag(v);

 	      zig(v);

 	    } else {

 	      zag(_data[v].parent);

 	      zag(v);

 	    }

 	  }

 	}

       }

     }

     void zig(int v) {

       Value min = _data[_data[v].parent].dmin;

       int a = _data[v].parent;

       Value aa = _data[a].dcost;

       if (_tolerance.less(aa, _max_value)) { 

 	aa+= min;

       }

       int b = v;

       Value ab = min + _data[b].dmin;

       Value bb = _data[b].dcost;

       if (_tolerance.less(bb, _max_value)) { 

 	bb+= ab;

       }

       int c = -1;

       Value cc = _max_value;

       if (_data[a].right != -1) {

 	c = _data[a].right;

 	cc = _data[c].dmin;

 	if (_tolerance.less(cc, _max_value)) {

 	  cc+=min;

 	}

       }

       int d = -1;

       Value dd = _max_value;

       if (_data[v].left != -1){

 	d = _data[v].left;

 	dd = ab + _data[d].dmin;

       }

       int e = -1;

       Value ee = _max_value;

       if (_data[v].right != -1) {

 	e = _data[v].right;

 	ee = ab + _data[e].dmin;

       }

       Value min2;

       if (_tolerance.less(0, _data[b].dmin) || 

 	  (e != -1 && !_tolerance.less(0, _data[e].dmin))) {

 	min2 = min;

       } else {

 	if (_tolerance.less(aa, cc)) {

 	  if (_tolerance.less(aa, ee)) {

 	    min2 = aa;

 	  } else {

 	    min2 = ee;

 	  }

 	} else if (_tolerance.less(cc, ee)) {

 	  min2 = cc;

 	} else {

 	  min2 = ee;

 	}

       }

       _data[a].dcost = aa;

       if (_tolerance.less(aa, _max_value)) { 

 	_data[a].dcost -= min2;

       }

       _data[a].dmin = min2;

       if (_tolerance.less(min2,_max_value)) { 

 	_data[a].dmin -= min; 

       }

       _data[a].size -= _data[b].size;

       _data[b].dcost = bb;

       if (_tolerance.less(bb, _max_value)) { 

 	_data[b].dcost -= min;

       }

       _data[b].dmin = min;

       _data[b].size += _data[a].size;

       if (c != -1) {

 	_data[c].dmin = cc;

 	if (_tolerance.less(cc, _max_value)) {

 	  _data[c].dmin -= min2;

 	}

       }

       if (d != -1) {

 	_data[d].dmin = dd - min;

 	_data[a].size += _data[d].size;

 	_data[b].size -= _data[d].size;

       }

       if (e != -1) {

 	_data[e].dmin = ee - min2;

       }

       int w = _data[v].parent;

       _data[v].successor = _data[w].successor;

       _data[w].successor = -1;

       _data[v].parent = _data[w].parent;

       _data[w].parent = v;

       _data[w].left = _data[v].right;

       _data[v].right = w;

       if (_data[v].parent != -1){

 	if (_data[_data[v].parent].right == w) {

 	  _data[_data[v].parent].right = v;

 	} else {

 	  _data[_data[v].parent].left = v;

 	}

       }

       if (_data[w].left != -1){

 	_data[_data[w].left].parent = w;

       }

     }

     void zag(int v) {

       Value min = _data[_data[v].parent].dmin;

       int a = _data[v].parent;

       Value aa = _data[a].dcost;

       if (_tolerance.less(aa, _max_value)) { 

 	aa += min;

       }

       int b = v;

       Value ab = min + _data[b].dmin;

       Value bb = _data[b].dcost;

       if (_tolerance.less(bb, _max_value)) {

 	bb += ab;

       }

       int c = -1;

       Value cc = _max_value;

       if (_data[a].left != -1){

 	c = _data[a].left;

 	cc = min + _data[c].dmin;

       }

       int d = -1;

       Value dd = _max_value;

       if (_data[v].right!=-1) {

 	d = _data[v].right;

 	dd = _data[d].dmin;

 	if (_tolerance.less(dd, _max_value)) {

 	  dd += ab;

 	}

       }

       int e = -1;

       Value ee = _max_value;

       if (_data[v].left != -1){

 	e = _data[v].left;

 	ee = ab + _data[e].dmin;

       }

       Value min2;

       if (_tolerance.less(0, _data[b].dmin) || 

 	  (e != -1 && !_tolerance.less(0, _data[e].dmin))) {

 	min2 = min;

       } else {

 	if (_tolerance.less(aa, cc)) {

 	  if (_tolerance.less(aa, ee)) {

 	    min2 = aa;

 	  } else {

 	    min2 = ee;

 	  }

 	} else if (_tolerance.less(cc, ee)) {

 	  min2 = cc;

 	} else {

 	  min2 = ee;

 	}

       }

       _data[a].dcost = aa;

       if (_tolerance.less(aa, _max_value)) { 

 	_data[a].dcost -= min2;

       }

       _data[a].dmin = min2;

       if (_tolerance.less(min2, _max_value)) {

 	_data[a].dmin -= min;

       }

       _data[a].size -= _data[b].size;

       _data[b].dcost = bb;

       if (_tolerance.less(bb, _max_value)) { 

 	_data[b].dcost -= min;

       }

       _data[b].dmin = min;

       _data[b].size += _data[a].size;

       if (c != -1) {

 	_data[c].dmin = cc - min2;

       }

       if (d != -1) {

 	_data[d].dmin = dd;

 	_data[a].size += _data[d].size;

 	_data[b].size -= _data[d].size;

 	if (_tolerance.less(dd, _max_value)) {

 	  _data[d].dmin -= min;

 	}

       }

       if (e != -1) {

 	_data[e].dmin = ee - min2;

       }

       int w = _data[v].parent;

       _data[v].successor = _data[w].successor;

       _data[w].successor = -1;

       _data[v].parent = _data[w].parent;

       _data[w].parent = v;

       _data[w].right = _data[v].left;

       _data[v].left = w;

       if (_data[v].parent != -1){

 	if (_data[_data[v].parent].left == w) {

 	  _data[_data[v].parent].left = v;

 	} else {

 	  _data[_data[v].parent].right = v;

 	}

       }

       if (_data[w].right != -1){

 	_data[_data[w].right].parent = w;

       }

     }

   private:

     class ItemData {

     public:

       Item id;

       int size;

       int successor;

       int parent;

       int left;

       int right;

       Value dmin;

       Value dcost;

     public:

       ItemData(const Item &item)

 	: id(item), size(1), successor(), parent(-1), 

 	  left(-1), right(-1), dmin(0), dcost(0) {}

     };

   };

   template <typename _Value, typename _ItemIntMap, typename _Tolerance>

   class DynamicTree<_Value, _ItemIntMap, _Tolerance, false> {

   public:

     typedef _ItemIntMap ItemIntMap;

     typedef typename ItemIntMap::Key Item;

     typedef _Value Value;

     typedef _Tolerance Tolerance;

   private:

     class ItemData;

     std::vector<ItemData> _data;

     ItemIntMap &_iim;

     Value _max_value;

     Tolerance _tolerance;

   public:

     DynamicTree(ItemIntMap &iim, const Tolerance& tolerance = Tolerance())

       : _iim(iim), _max_value(std::numeric_limits<Value>::max()), 

 	_tolerance(tolerance) {}

     ~DynamicTree() {}

     void clear() {

       _data.clear();

     }

     void tolerance(const Tolerance& tolerance) const {

       _tolerance = tolerance;

       return *this;

     } 

     const Tolerance& tolerance() const {

       return tolerance;

     } 

     void makeTree(const Item &item) {

       _data[makePath(item)].successor = -1;

     }

     Item findRoot(const Item &item) {

       return _data[findTail(expose(_iim[item]))].id;

     }

     Item findCost(const Item &item, Value& d){

       return _data[findPathCost(expose(_iim[item]),d)].id;

     }

     void addCost(const Item &item, Value x){

       addPathCost(expose(_iim[item]), x);

     }

     void link(const Item &item1, const Item &item2){

       int v = _iim[item1];

       int w = _iim[item2];

       int p = expose(w);

       join(-1, expose(v), p);

       _data[v].successor = -1;

     }    

     void cut(const Item &item) {

       int v = _iim[item];

       int p, q;

       expose(v);

       split(p, v, q);

       if (p != -1) {

 	_data[p].successor = v;

       }

       if (q != -1) {

 	_data[q].successor = _data[v].successor;

       }

       _data[v].successor = -1;

     }

     Item parent(const Item &item){

       return _data[_iim[item].p].id;

     }

     Value maxValue() const {

       return _max_value;

     }

   private:

     int makePath(const Item &item) {

       _iim.set(item, _data.size());

       ItemData v(item);

       _data.push_back(v);

       return _iim[item];

     }

     int findPath(int v){

       splay(v);

       return v;

     }

     int findPathCost(int p, Value &d){

       while ((_data[p].right != -1 && 

 	      !_tolerance.less(0, _data[_data[p].right].dmin)) || 

 	     (_data[p].left != -1 && _tolerance.less(0, _data[p].dcost))) {

 	if (_data[p].right != -1 && 

 	    !_tolerance.less(0, _data[_data[p].right].dmin)) {

 	  p = _data[p].right;

 	} else if (_data[p].left != -1 && 

 		   !_tolerance.less(0, _data[_data[p].left].dmin)){

 	  p = _data[p].left;

 	}

       }

       splay(p);

       d = _data[p].dmin;

       return p; 

     }

     int findTail(int p){

       while (_data[p].right != -1) {

 	p = _data[p].right;

       }

       splay(p);

       return p;

     }

     void addPathCost(int p, Value x){

       if (!_tolerance.less(x, _max_value)) {

 	_data[p].dmin = x;_data[p].dcost = x;

       } else if (!_tolerance.less(-x, _max_value)) {

 	_data[p].dmin = 0;

 	_data[p].dcost = 0;

       } else {

 	_data[p].dmin += x;

       }

     }

     void join(int p, int v, int q) {

       Value min = _max_value;

       Value pmin = _max_value;

       Value vmin = _data[v].dmin;

       Value qmin = _max_value;

       if (p != -1){

 	pmin = _data[p].dmin;

       }

       if (q != -1){

 	qmin = _data[q].dmin;

       }

       if (_tolerance.less(vmin, qmin)) {

 	if (_tolerance.less(vmin,pmin)) {

 	  min = vmin;

 	} else {

 	  min = pmin;

 	}

       } else if (_tolerance.less(qmin,pmin)) {

 	min = qmin;

       } else {

 	min = pmin;

       }

       if (p != -1){

 	_data[p].parent = v;

 	_data[p].dmin -= min;

       }

       if (q!=-1){

 	_data[q].parent = v;

 	if (_tolerance.less(_data[q].dmin,_max_value)) {

 	  _data[q].dmin -= min;

 	}

       }

       _data[v].left = p;

       _data[v].right = q;

       if (_tolerance.less(min,_max_value)) {

 	_data[v].dcost = _data[v].dmin - min;

       }

       _data[v].dmin = min;

     }

     void split(int &p, int v, int &q){

       splay(v);

       p = -1;

       if (_data[v].left != -1){

 	p = _data[v].left;

 	_data[p].dmin += _data[v].dmin;

 	_data[p].parent = -1;

 	_data[v].left = -1;

       }

       q = -1;

       if (_data[v].right != -1) {

 	q=_data[v].right;

 	if (_tolerance.less(_data[q].dmin, _max_value)) {

 	  _data[q].dmin += _data[v].dmin;

 	}

 	_data[q].parent = -1;

 	_data[v].right = -1;

       } 

       if (_tolerance.less(_data[v].dcost, _max_value)) {

 	_data[v].dmin += _data[v].dcost;

 	_data[v].dcost = 0;

       } else {

 	_data[v].dmin = _data[v].dcost;

       }

     }

     int expose(int v) {

       int p, q, r, w;

       p = -1;

       while (v != -1) {

 	w = _data[findPath(v)].successor;

 	split(q, v, r);

 	if (q != -1) {

 	  _data[q].successor = v;

 	}

 	join(p, v, r);

 	p = v;

 	v = w;

       }

       _data[p].successor = -1;

       return p;

     }

     void splay(int v) {

       while (_data[v].parent != -1) {

 	if (v == _data[_data[v].parent].left) {

 	  if (_data[_data[v].parent].parent == -1) {

 	    zig(v);

 	  } else {

 	    if (_data[v].parent == _data[_data[_data[v].parent].parent].left) {

 	      zig(_data[v].parent);

 	      zig(v);

 	    } else {

 	      zig(v);

 	      zag(v);

 	    }

 	  }

 	} else {

 	  if (_data[_data[v].parent].parent == -1) {

 	    zag(v);

 	  } else {

 	    if (_data[v].parent == _data[_data[_data[v].parent].parent].left) {

 	      zag(v);

 	      zig(v);

 	    } else {

 	      zag(_data[v].parent);

 	      zag(v);

 	    }

 	  }

 	}

       }

     }

     void zig(int v) {

       Value min = _data[_data[v].parent].dmin;

       int a = _data[v].parent;

       Value aa = _data[a].dcost;

       if (_tolerance.less(aa, _max_value)) { 

 	aa+= min;

       }

       int b = v;

       Value ab = min + _data[b].dmin;

       Value bb = _data[b].dcost;

       if (_tolerance.less(bb, _max_value)) { 

 	bb+= ab;

       }

       int c = -1;

       Value cc = _max_value;

       if (_data[a].right != -1) {

 	c = _data[a].right;

 	cc = _data[c].dmin;

 	if (_tolerance.less(cc, _max_value)) {

 	  cc+=min;

 	}

       }

       int d = -1;

       Value dd = _max_value;

       if (_data[v].left != -1){

 	d = _data[v].left;

 	dd = ab + _data[d].dmin;

       }

       int e = -1;

       Value ee = _max_value;

       if (_data[v].right != -1) {

 	e = _data[v].right;

 	ee = ab + _data[e].dmin;

       }

       Value min2;

       if (_tolerance.less(0, _data[b].dmin) || 

 	  (e != -1 && !_tolerance.less(0, _data[e].dmin))) {

 	min2 = min;

       } else {

 	if (_tolerance.less(aa, cc)) {

 	  if (_tolerance.less(aa, ee)) {

 	    min2 = aa;

 	  } else {

 	    min2 = ee;

 	  }

 	} else if (_tolerance.less(cc, ee)) {

 	  min2 = cc;

 	} else {

 	  min2 = ee;

 	}

       }

       _data[a].dcost = aa;

       if (_tolerance.less(aa, _max_value)) { 

 	_data[a].dcost -= min2;

       }

       _data[a].dmin = min2;

       if (_tolerance.less(min2,_max_value)) { 

 	_data[a].dmin -= min; 

       }

       _data[b].dcost = bb;

       if (_tolerance.less(bb, _max_value)) { 

 	_data[b].dcost -= min;

       }

       _data[b].dmin = min;

       if (c != -1) {

 	_data[c].dmin = cc;

 	if (_tolerance.less(cc, _max_value)) {

 	  _data[c].dmin -= min2;

 	}

       }

       if (d != -1) {

 	_data[d].dmin = dd - min;

       }

       if (e != -1) {

 	_data[e].dmin = ee - min2;

       }

       int w = _data[v].parent;

       _data[v].successor = _data[w].successor;

       _data[w].successor = -1;

       _data[v].parent = _data[w].parent;

       _data[w].parent = v;

       _data[w].left = _data[v].right;

       _data[v].right = w;

       if (_data[v].parent != -1){

 	if (_data[_data[v].parent].right == w) {

 	  _data[_data[v].parent].right = v;

 	} else {

 	  _data[_data[v].parent].left = v;

 	}

       }

       if (_data[w].left != -1){

 	_data[_data[w].left].parent = w;

       }

     }

     void zag(int v) {

       Value min = _data[_data[v].parent].dmin;

       int a = _data[v].parent;

       Value aa = _data[a].dcost;

       if (_tolerance.less(aa, _max_value)) { 

 	aa += min;

       }

       int b = v;

       Value ab = min + _data[b].dmin;

       Value bb = _data[b].dcost;

       if (_tolerance.less(bb, _max_value)) {

 	bb += ab;

       }

       int c = -1;

       Value cc = _max_value;

       if (_data[a].left != -1){

 	c = _data[a].left;

 	cc = min + _data[c].dmin;

       }

       int d = -1;

       Value dd = _max_value;

       if (_data[v].right!=-1) {

 	d = _data[v].right;

 	dd = _data[d].dmin;

 	if (_tolerance.less(dd, _max_value)) {

 	  dd += ab;

 	}

       }

       int e = -1;

       Value ee = _max_value;

       if (_data[v].left != -1){

 	e = _data[v].left;

 	ee = ab + _data[e].dmin;

       }

       Value min2;

       if (_tolerance.less(0, _data[b].dmin) || 

 	  (e != -1 && !_tolerance.less(0, _data[e].dmin))) {

 	min2 = min;

       } else {

 	if (_tolerance.less(aa, cc)) {

 	  if (_tolerance.less(aa, ee)) {

 	    min2 = aa;

 	  } else {

 	    min2 = ee;

 	  }

 	} else if (_tolerance.less(cc, ee)) {

 	  min2 = cc;

 	} else {

 	  min2 = ee;

 	}

       }

       _data[a].dcost = aa;

       if (_tolerance.less(aa, _max_value)) { 

 	_data[a].dcost -= min2;

       }

       _data[a].dmin = min2;

       if (_tolerance.less(min2, _max_value)) {

 	_data[a].dmin -= min;

       }

       _data[b].dcost = bb;

       if (_tolerance.less(bb, _max_value)) { 

 	_data[b].dcost -= min;

       }

       _data[b].dmin = min;

       if (c != -1) {

 	_data[c].dmin = cc - min2;

       }

       if (d != -1) {

 	_data[d].dmin = dd;

 	if (_tolerance.less(dd, _max_value)) {

 	  _data[d].dmin -= min;

 	}

       }

       if (e != -1) {

 	_data[e].dmin = ee - min2;

       }

       int w = _data[v].parent;

       _data[v].successor = _data[w].successor;

       _data[w].successor = -1;

       _data[v].parent = _data[w].parent;

       _data[w].parent = v;

       _data[w].right = _data[v].left;

       _data[v].left = w;

       if (_data[v].parent != -1){

 	if (_data[_data[v].parent].left == w) {

 	  _data[_data[v].parent].left = v;

 	} else {

 	  _data[_data[v].parent].right = v;

 	}

       }

       if (_data[w].right != -1){

 	_data[_data[w].right].parent = w;

       }

     }

   private:

     class ItemData {

     public:

       Item id;

       int successor;

       int parent;

       int left;

       int right;

       Value dmin;

       Value dcost;

     public:

       ItemData(const Item &item)

 	: id(item), successor(), parent(-1), 

 	  left(-1), right(-1), dmin(0), dcost(0) {}

     };

   };

 }

 #endif