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				/* -- mode: C++; indent-tabs-mode: nil; --
				*
				* This file is a part of LEMON, a generic C++ optimization library.
				*
				* Copyright (C) 2003-2013
				* 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_UNION_FIND_H
				#define LEMON_UNION_FIND_H

				//!\ingroup auxdat
				//!\file
				//!\brief Union-Find data structures.
				//!

				#include <vector>
				#include <list>
				#include <utility>
				#include <algorithm>
				#include <functional>

				#include <lemon/core.h>

				namespace lemon {

				/// \ingroup auxdat
				///
				/// \brief A \e Union-Find data structure implementation
				///
				/// The class implements the \e Union-Find data structure.
				/// The union operation uses rank heuristic, while
				/// the find operation uses path compression.
				/// This is a very simple but efficient implementation, providing
				/// only four methods: join (union), find, insert and size.
				/// For more features, see the \ref UnionFindEnum class.
				///
				/// It is primarily used in Kruskal algorithm for finding minimal
				/// cost spanning tree in a graph.
				/// \sa kruskal()
				///
				/// \pre You need to add all the elements by the \ref insert()
				/// method.
				template <typename IM>
				class UnionFind {
				public:

				///\e
				typedef IM ItemIntMap;
				///\e
				typedef typename ItemIntMap::Key Item;

				private:
				// If the items vector stores negative value for an item then
				// that item is root item and it has -items[it] component size.
				// Else the items[it] contains the index of the parent.
				std::vector<int> items;
				ItemIntMap& index;

				bool rep(int idx) const {
				return items[idx] < 0;
				}

				int repIndex(int idx) const {
				int k = idx;
				while (!rep(k)) {
				k = items[k] ;
				}
				while (idx != k) {
				int next = items[idx];
				const_cast<int&>(items[idx]) = k;
				idx = next;
				}
				return k;
				}

				public:

				/// \brief Constructor
				///
				/// Constructor of the UnionFind class. You should give an item to
				/// integer map which will be used from the data structure. If you
				/// modify directly this map that may cause segmentation fault,
				/// invalid data structure, or infinite loop when you use again
				/// the union-find.
				UnionFind(ItemIntMap& m) : index(m) {}

				/// \brief Returns the index of the element's component.
				///
				/// The method returns the index of the element's component.
				/// This is an integer between zero and the number of inserted elements.
				///
				int find(const Item& a) {
				return repIndex(index[a]);
				}

				/// \brief Clears the union-find data structure
				///
				/// Erase each item from the data structure.
				void clear() {
				items.clear();
				}

				/// \brief Inserts a new element into the structure.
				///
				/// This method inserts a new element into the data structure.
				///
				/// The method returns the index of the new component.
				int insert(const Item& a) {
				int n = items.size();
				items.push_back(-1);
				index.set(a,n);
				return n;
				}

				/// \brief Joining the components of element \e a and element \e b.
				///
				/// This is the \e union operation of the Union-Find structure.
				/// Joins the component of element \e a and component of
				/// element \e b. If \e a and \e b are in the same component then
				/// it returns false otherwise it returns true.
				bool join(const Item& a, const Item& b) {
				int ka = repIndex(index[a]);
				int kb = repIndex(index[b]);

				if ( ka == kb )
				return false;

				if (items[ka] < items[kb]) {
				items[ka] += items[kb];
				items[kb] = ka;
				} else {
				items[kb] += items[ka];
				items[ka] = kb;
				}
				return true;
				}

				/// \brief Returns the size of the component of element \e a.
				///
				/// Returns the size of the component of element \e a.
				int size(const Item& a) {
				int k = repIndex(index[a]);
				return - items[k];
				}

				};

				/// \ingroup auxdat
				///
				/// \brief A \e Union-Find data structure implementation which
				/// is able to enumerate the components.
				///
				/// The class implements a \e Union-Find data structure
				/// which is able to enumerate the components and the items in
				/// a component. If you don't need this feature then perhaps it's
				/// better to use the \ref UnionFind class which is more efficient.
				///
				/// The union operation uses rank heuristic, while
				/// the find operation uses path compression.
				///
				/// \pre You need to add all the elements by the \ref insert()
				/// method.
				///
				template <typename IM>
				class UnionFindEnum {
				public:

				///\e
				typedef IM ItemIntMap;
				///\e
				typedef typename ItemIntMap::Key Item;

				private:

				ItemIntMap& index;

				// If the parent stores negative value for an item then that item
				// is root item and it has ~(items[it].parent) component id. Else
				// the items[it].parent contains the index of the parent.
				//
				// The \c next and \c prev provides the double-linked
				// cyclic list of one component's items.
				struct ItemT {
				int parent;
				Item item;

				int next, prev;
				};

				std::vector<ItemT> items;
				int firstFreeItem;

				struct ClassT {
				int size;
				int firstItem;
				int next, prev;
				};

				std::vector<ClassT> classes;
				int firstClass, firstFreeClass;

				int newClass() {
				if (firstFreeClass == -1) {
				int cdx = classes.size();
				classes.push_back(ClassT());
				return cdx;
				} else {
				int cdx = firstFreeClass;
				firstFreeClass = classes[firstFreeClass].next;
				return cdx;
				}
				}

				int newItem() {
				if (firstFreeItem == -1) {
				int idx = items.size();
				items.push_back(ItemT());
				return idx;
				} else {
				int idx = firstFreeItem;
				firstFreeItem = items[firstFreeItem].next;
				return idx;
				}
				}


				bool rep(int idx) const {
				return items[idx].parent < 0;
				}

				int repIndex(int idx) const {
				int k = idx;
				while (!rep(k)) {
				k = items[k].parent;
				}
				while (idx != k) {
				int next = items[idx].parent;
				const_cast<int&>(items[idx].parent) = k;
				idx = next;
				}
				return k;
				}

				int classIndex(int idx) const {
				return ~(items[repIndex(idx)].parent);
				}

				void singletonItem(int idx) {
				items[idx].next = idx;
				items[idx].prev = idx;
				}

				void laceItem(int idx, int rdx) {
				items[idx].prev = rdx;
				items[idx].next = items[rdx].next;
				items[items[rdx].next].prev = idx;
				items[rdx].next = idx;
				}

				void unlaceItem(int idx) {
				items[items[idx].prev].next = items[idx].next;
				items[items[idx].next].prev = items[idx].prev;

				items[idx].next = firstFreeItem;
				firstFreeItem = idx;
				}

				void spliceItems(int ak, int bk) {
				items[items[ak].prev].next = bk;
				items[items[bk].prev].next = ak;
				int tmp = items[ak].prev;
				items[ak].prev = items[bk].prev;
				items[bk].prev = tmp;

				}

				void laceClass(int cls) {
				if (firstClass != -1) {
				classes[firstClass].prev = cls;
				}
				classes[cls].next = firstClass;
				classes[cls].prev = -1;
				firstClass = cls;
				}

				void unlaceClass(int cls) {
				if (classes[cls].prev != -1) {
				classes[classes[cls].prev].next = classes[cls].next;
				} else {
				firstClass = classes[cls].next;
				}
				if (classes[cls].next != -1) {
				classes[classes[cls].next].prev = classes[cls].prev;
				}

				classes[cls].next = firstFreeClass;
				firstFreeClass = cls;
				}

				public:

				UnionFindEnum(ItemIntMap& _index)
				: index(_index), items(), firstFreeItem(-1),
				firstClass(-1), firstFreeClass(-1) {}

				/// \brief Inserts the given element into a new component.
				///
				/// This method creates a new component consisting only of the
				/// given element.
				///
				int insert(const Item& item) {
				int idx = newItem();

				index.set(item, idx);

				singletonItem(idx);
				items[idx].item = item;

				int cdx = newClass();

				items[idx].parent = ~cdx;

				laceClass(cdx);
				classes[cdx].size = 1;
				classes[cdx].firstItem = idx;

				firstClass = cdx;

				return cdx;
				}

				/// \brief Inserts the given element into the component of the others.
				///
				/// This methods inserts the element \e a into the component of the
				/// element \e comp.
				void insert(const Item& item, int cls) {
				int rdx = classes[cls].firstItem;
				int idx = newItem();

				index.set(item, idx);

				laceItem(idx, rdx);

				items[idx].item = item;
				items[idx].parent = rdx;

				++classes[~(items[rdx].parent)].size;
				}

				/// \brief Clears the union-find data structure
				///
				/// Erase each item from the data structure.
				void clear() {
				items.clear();
				firstClass = -1;
				firstFreeItem = -1;
				}

				/// \brief Finds the component of the given element.
				///
				/// The method returns the component id of the given element.
				int find(const Item &item) const {
				return ~(items[repIndex(index[item])].parent);
				}

				/// \brief Joining the component of element \e a and element \e b.
				///
				/// This is the \e union operation of the Union-Find structure.
				/// Joins the component of element \e a and component of
				/// element \e b. If \e a and \e b are in the same component then
				/// returns -1 else returns the remaining class.
				int join(const Item& a, const Item& b) {

				int ak = repIndex(index[a]);
				int bk = repIndex(index[b]);

				if (ak == bk) {
				return -1;
				}

				int acx = ~(items[ak].parent);
				int bcx = ~(items[bk].parent);

				int rcx;

				if (classes[acx].size > classes[bcx].size) {
				classes[acx].size += classes[bcx].size;
				items[bk].parent = ak;
				unlaceClass(bcx);
				rcx = acx;
				} else {
				classes[bcx].size += classes[acx].size;
				items[ak].parent = bk;
				unlaceClass(acx);
				rcx = bcx;
				}
				spliceItems(ak, bk);

				return rcx;
				}

				/// \brief Returns the size of the class.
				///
				/// Returns the size of the class.
				int size(int cls) const {
				return classes[cls].size;
				}

				/// \brief Splits up the component.
				///
				/// Splitting the component into singleton components (component
				/// of size one).
				void split(int cls) {
				int fdx = classes[cls].firstItem;
				int idx = items[fdx].next;
				while (idx != fdx) {
				int next = items[idx].next;

				singletonItem(idx);

				int cdx = newClass();
				items[idx].parent = ~cdx;

				laceClass(cdx);
				classes[cdx].size = 1;
				classes[cdx].firstItem = idx;

				idx = next;
				}

				items[idx].prev = idx;
				items[idx].next = idx;

				classes[~(items[idx].parent)].size = 1;

				}

				/// \brief Removes the given element from the structure.
				///
				/// Removes the element from its component and if the component becomes
				/// empty then removes that component from the component list.
				///
				/// \warning It is an error to remove an element which is not in
				/// the structure.
				/// \warning This running time of this operation is proportional to the
				/// number of the items in this class.
				void erase(const Item& item) {
				int idx = index[item];
				int fdx = items[idx].next;

				int cdx = classIndex(idx);
				if (idx == fdx) {
				unlaceClass(cdx);
				items[idx].next = firstFreeItem;
				firstFreeItem = idx;
				return;
				} else {
				classes[cdx].firstItem = fdx;
				--classes[cdx].size;
				items[fdx].parent = ~cdx;

				unlaceItem(idx);
				idx = items[fdx].next;
				while (idx != fdx) {
				items[idx].parent = fdx;
				idx = items[idx].next;
				}

				}

				}

				/// \brief Gives back a representant item of the component.
				///
				/// Gives back a representant item of the component.
				Item item(int cls) const {
				return items[classes[cls].firstItem].item;
				}

				/// \brief Removes the component of the given element from the structure.
				///
				/// Removes the component of the given element from the structure.
				///
				/// \warning It is an error to give an element which is not in the
				/// structure.
				void eraseClass(int cls) {
				int fdx = classes[cls].firstItem;
				unlaceClass(cls);
				items[items[fdx].prev].next = firstFreeItem;
				firstFreeItem = fdx;
				}

				/// \brief LEMON style iterator for the representant items.
				///
				/// ClassIt is a lemon style iterator for the components. It iterates
				/// on the ids of the classes.
				class ClassIt {
				public:
				/// \brief Constructor of the iterator
				///
				/// Constructor of the iterator
				ClassIt(const UnionFindEnum& ufe) : unionFind(&ufe) {
				cdx = unionFind->firstClass;
				}

				/// \brief Constructor to get invalid iterator
				///
				/// Constructor to get invalid iterator
				ClassIt(Invalid) : unionFind(0), cdx(-1) {}

				/// \brief Increment operator
				///
				/// It steps to the next representant item.
				ClassIt& operator++() {
				cdx = unionFind->classes[cdx].next;
				return *this;
				}

				/// \brief Conversion operator
				///
				/// It converts the iterator to the current representant item.
				operator int() const {
				return cdx;
				}

				/// \brief Equality operator
				///
				/// Equality operator
				bool operator==(const ClassIt& i) {
				return i.cdx == cdx;
				}

				/// \brief Inequality operator
				///
				/// Inequality operator
				bool operator!=(const ClassIt& i) {
				return i.cdx != cdx;
				}

				private:
				const UnionFindEnum* unionFind;
				int cdx;
				};

				/// \brief LEMON style iterator for the items of a component.
				///
				/// ClassIt is a lemon style iterator for the components. It iterates
				/// on the items of a class. By example if you want to iterate on
				/// each items of each classes then you may write the next code.
				///\code
				/// for (ClassIt cit(ufe); cit != INVALID; ++cit) {
				/// std::cout << "Class: ";
				/// for (ItemIt iit(ufe, cit); iit != INVALID; ++iit) {
				/// std::cout << toString(iit) << ' ' << std::endl;
				/// }
				/// std::cout << std::endl;
				/// }
				///\endcode
				class ItemIt {
				public:
				/// \brief Constructor of the iterator
				///
				/// Constructor of the iterator. The iterator iterates
				/// on the class of the \c item.
				ItemIt(const UnionFindEnum& ufe, int cls) : unionFind(&ufe) {
				fdx = idx = unionFind->classes[cls].firstItem;
				}

				/// \brief Constructor to get invalid iterator
				///
				/// Constructor to get invalid iterator
				ItemIt(Invalid) : unionFind(0), idx(-1) {}

				/// \brief Increment operator
				///
				/// It steps to the next item in the class.
				ItemIt& operator++() {
				idx = unionFind->items[idx].next;
				if (idx == fdx) idx = -1;
				return *this;
				}

				/// \brief Conversion operator
				///
				/// It converts the iterator to the current item.
				operator const Item&() const {
				return unionFind->items[idx].item;
				}

				/// \brief Equality operator
				///
				/// Equality operator
				bool operator==(const ItemIt& i) {
				return i.idx == idx;
				}

				/// \brief Inequality operator
				///
				/// Inequality operator
				bool operator!=(const ItemIt& i) {
				return i.idx != idx;
				}

				private:
				const UnionFindEnum* unionFind;
				int idx, fdx;
				};

				};

				/// \ingroup auxdat
				///
				/// \brief A \e Extend-Find data structure implementation which
				/// is able to enumerate the components.
				///
				/// The class implements an \e Extend-Find data structure which is
				/// able to enumerate the components and the items in a
				/// component. The data structure is a simplification of the
				/// Union-Find structure, and it does not allow to merge two components.
				///
				/// \pre You need to add all the elements by the \ref insert()
				/// method.
				template <typename IM>
				class ExtendFindEnum {
				public:

				///\e
				typedef IM ItemIntMap;
				///\e
				typedef typename ItemIntMap::Key Item;

				private:

				ItemIntMap& index;

				struct ItemT {
				int cls;
				Item item;
				int next, prev;
				};

				std::vector<ItemT> items;
				int firstFreeItem;

				struct ClassT {
				int firstItem;
				int next, prev;
				};

				std::vector<ClassT> classes;

				int firstClass, firstFreeClass;

				int newClass() {
				if (firstFreeClass != -1) {
				int cdx = firstFreeClass;
				firstFreeClass = classes[cdx].next;
				return cdx;
				} else {
				classes.push_back(ClassT());
				return classes.size() - 1;
				}
				}

				int newItem() {
				if (firstFreeItem != -1) {
				int idx = firstFreeItem;
				firstFreeItem = items[idx].next;
				return idx;
				} else {
				items.push_back(ItemT());
				return items.size() - 1;
				}
				}

				public:

				/// \brief Constructor
				ExtendFindEnum(ItemIntMap& _index)
				: index(_index), items(), firstFreeItem(-1),
				classes(), firstClass(-1), firstFreeClass(-1) {}

				/// \brief Inserts the given element into a new component.
				///
				/// This method creates a new component consisting only of the
				/// given element.
				int insert(const Item& item) {
				int cdx = newClass();
				classes[cdx].prev = -1;
				classes[cdx].next = firstClass;
				if (firstClass != -1) {
				classes[firstClass].prev = cdx;
				}
				firstClass = cdx;

				int idx = newItem();
				items[idx].item = item;
				items[idx].cls = cdx;
				items[idx].prev = idx;
				items[idx].next = idx;

				classes[cdx].firstItem = idx;

				index.set(item, idx);

				return cdx;
				}

				/// \brief Inserts the given element into the given component.
				///
				/// This methods inserts the element \e item a into the \e cls class.
				void insert(const Item& item, int cls) {
				int idx = newItem();
				int rdx = classes[cls].firstItem;
				items[idx].item = item;
				items[idx].cls = cls;

				items[idx].prev = rdx;
				items[idx].next = items[rdx].next;
				items[items[rdx].next].prev = idx;
				items[rdx].next = idx;

				index.set(item, idx);
				}

				/// \brief Clears the union-find data structure
				///
				/// Erase each item from the data structure.
				void clear() {
				items.clear();
				classes.clear();
				firstClass = firstFreeClass = firstFreeItem = -1;
				}

				/// \brief Gives back the class of the \e item.
				///
				/// Gives back the class of the \e item.
				int find(const Item &item) const {
				return items[index[item]].cls;
				}

				/// \brief Gives back a representant item of the component.
				///
				/// Gives back a representant item of the component.
				Item item(int cls) const {
				return items[classes[cls].firstItem].item;
				}

				/// \brief Removes the given element from the structure.
				///
				/// Removes the element from its component and if the component becomes
				/// empty then removes that component from the component list.
				///
				/// \warning It is an error to remove an element which is not in
				/// the structure.
				void erase(const Item &item) {
				int idx = index[item];
				int cdx = items[idx].cls;

				if (idx == items[idx].next) {
				if (classes[cdx].prev != -1) {
				classes[classes[cdx].prev].next = classes[cdx].next;
				} else {
				firstClass = classes[cdx].next;
				}
				if (classes[cdx].next != -1) {
				classes[classes[cdx].next].prev = classes[cdx].prev;
				}
				classes[cdx].next = firstFreeClass;
				firstFreeClass = cdx;
				} else {
				classes[cdx].firstItem = items[idx].next;
				items[items[idx].next].prev = items[idx].prev;
				items[items[idx].prev].next = items[idx].next;
				}
				items[idx].next = firstFreeItem;
				firstFreeItem = idx;

				}


				/// \brief Removes the component of the given element from the structure.
				///
				/// Removes the component of the given element from the structure.
				///
				/// \warning It is an error to give an element which is not in the
				/// structure.
				void eraseClass(int cdx) {
				int idx = classes[cdx].firstItem;
				items[items[idx].prev].next = firstFreeItem;
				firstFreeItem = idx;

				if (classes[cdx].prev != -1) {
				classes[classes[cdx].prev].next = classes[cdx].next;
				} else {
				firstClass = classes[cdx].next;
				}
				if (classes[cdx].next != -1) {
				classes[classes[cdx].next].prev = classes[cdx].prev;
				}
				classes[cdx].next = firstFreeClass;
				firstFreeClass = cdx;
				}

				/// \brief LEMON style iterator for the classes.
				///
				/// ClassIt is a lemon style iterator for the components. It iterates
				/// on the ids of classes.
				class ClassIt {
				public:
				/// \brief Constructor of the iterator
				///
				/// Constructor of the iterator
				ClassIt(const ExtendFindEnum& ufe) : extendFind(&ufe) {
				cdx = extendFind->firstClass;
				}

				/// \brief Constructor to get invalid iterator
				///
				/// Constructor to get invalid iterator
				ClassIt(Invalid) : extendFind(0), cdx(-1) {}

				/// \brief Increment operator
				///
				/// It steps to the next representant item.
				ClassIt& operator++() {
				cdx = extendFind->classes[cdx].next;
				return *this;
				}

				/// \brief Conversion operator
				///
				/// It converts the iterator to the current class id.
				operator int() const {
				return cdx;
				}

				/// \brief Equality operator
				///
				/// Equality operator
				bool operator==(const ClassIt& i) {
				return i.cdx == cdx;
				}

				/// \brief Inequality operator
				///
				/// Inequality operator
				bool operator!=(const ClassIt& i) {
				return i.cdx != cdx;
				}

				private:
				const ExtendFindEnum* extendFind;
				int cdx;
				};

				/// \brief LEMON style iterator for the items of a component.
				///
				/// ClassIt is a lemon style iterator for the components. It iterates
				/// on the items of a class. By example if you want to iterate on
				/// each items of each classes then you may write the next code.
				///\code
				/// for (ClassIt cit(ufe); cit != INVALID; ++cit) {
				/// std::cout << "Class: ";
				/// for (ItemIt iit(ufe, cit); iit != INVALID; ++iit) {
				/// std::cout << toString(iit) << ' ' << std::endl;
				/// }
				/// std::cout << std::endl;
				/// }
				///\endcode
				class ItemIt {
				public:
				/// \brief Constructor of the iterator
				///
				/// Constructor of the iterator. The iterator iterates
				/// on the class of the \c item.
				ItemIt(const ExtendFindEnum& ufe, int cls) : extendFind(&ufe) {
				fdx = idx = extendFind->classes[cls].firstItem;
				}

				/// \brief Constructor to get invalid iterator
				///
				/// Constructor to get invalid iterator
				ItemIt(Invalid) : extendFind(0), idx(-1) {}

				/// \brief Increment operator
				///
				/// It steps to the next item in the class.
				ItemIt& operator++() {
				idx = extendFind->items[idx].next;
				if (fdx == idx) idx = -1;
				return *this;
				}

				/// \brief Conversion operator
				///
				/// It converts the iterator to the current item.
				operator const Item&() const {
				return extendFind->items[idx].item;
				}

				/// \brief Equality operator
				///
				/// Equality operator
				bool operator==(const ItemIt& i) {
				return i.idx == idx;
				}

				/// \brief Inequality operator
				///
				/// Inequality operator
				bool operator!=(const ItemIt& i) {
				return i.idx != idx;
				}

				private:
				const ExtendFindEnum* extendFind;
				int idx, fdx;
				};

				};

				/// \ingroup auxdat
				///
				/// \brief A \e Union-Find data structure implementation which
				/// is able to store a priority for each item and retrieve the minimum of
				/// each class.
				///
				/// A \e Union-Find data structure implementation which is able to
				/// store a priority for each item and retrieve the minimum of each
				/// class. In addition, it supports the joining and splitting the
				/// components. If you don't need this feature then you makes
				/// better to use the \ref UnionFind class which is more efficient.
				///
				/// The union-find data strcuture based on a (2, 16)-tree with a
				/// tournament minimum selection on the internal nodes. The insert
				/// operation takes O(1), the find, set, decrease and increase takes
				/// O(log(n)), where n is the number of nodes in the current
				/// component. The complexity of join and split is O(log(n)*k),
				/// where n is the sum of the number of the nodes and k is the
				/// number of joined components or the number of the components
				/// after the split.
				///
				/// \pre You need to add all the elements by the \ref insert()
				/// method.
				template <typename V, typename IM, typename Comp = std::less<V> >
				class HeapUnionFind {
				public:

				///\e
				typedef V Value;
				///\e
				typedef typename IM::Key Item;
				///\e
				typedef IM ItemIntMap;
				///\e
				typedef Comp Compare;

				private:

				static const int cmax = 16;

				ItemIntMap& index;

				struct ClassNode {
				int parent;
				int depth;

				int left, right;
				int next, prev;
				};

				int first_class;
				int first_free_class;
				std::vector<ClassNode> classes;

				int newClass() {
				if (first_free_class < 0) {
				int id = classes.size();
				classes.push_back(ClassNode());
				return id;
				} else {
				int id = first_free_class;
				first_free_class = classes[id].next;
				return id;
				}
				}

				void deleteClass(int id) {
				classes[id].next = first_free_class;
				first_free_class = id;
				}

				struct ItemNode {
				int parent;
				Item item;
				Value prio;
				int next, prev;
				int left, right;
				int size;
				};

				int first_free_node;
				std::vector<ItemNode> nodes;

				int newNode() {
				if (first_free_node < 0) {
				int id = nodes.size();
				nodes.push_back(ItemNode());
				return id;
				} else {
				int id = first_free_node;
				first_free_node = nodes[id].next;
				return id;
				}
				}

				void deleteNode(int id) {
				nodes[id].next = first_free_node;
				first_free_node = id;
				}

				Comp comp;

				int findClass(int id) const {
				int kd = id;
				while (kd >= 0) {
				kd = nodes[kd].parent;
				}
				return ~kd;
				}

				int leftNode(int id) const {
				int kd = ~(classes[id].parent);
				for (int i = 0; i < classes[id].depth; ++i) {
				kd = nodes[kd].left;
				}
				return kd;
				}

				int nextNode(int id) const {
				int depth = 0;
				while (id >= 0 && nodes[id].next == -1) {
				id = nodes[id].parent;
				++depth;
				}
				if (id < 0) {
				return -1;
				}
				id = nodes[id].next;
				while (depth--) {
				id = nodes[id].left;
				}
				return id;
				}


				void setPrio(int id) {
				int jd = nodes[id].left;
				nodes[id].prio = nodes[jd].prio;
				nodes[id].item = nodes[jd].item;
				jd = nodes[jd].next;
				while (jd != -1) {
				if (comp(nodes[jd].prio, nodes[id].prio)) {
				nodes[id].prio = nodes[jd].prio;
				nodes[id].item = nodes[jd].item;
				}
				jd = nodes[jd].next;
				}
				}

				void push(int id, int jd) {
				nodes[id].size = 1;
				nodes[id].left = nodes[id].right = jd;
				nodes[jd].next = nodes[jd].prev = -1;
				nodes[jd].parent = id;
				}

				void pushAfter(int id, int jd) {
				int kd = nodes[id].parent;
				if (nodes[id].next != -1) {
				nodes[nodes[id].next].prev = jd;
				if (kd >= 0) {
				nodes[kd].size += 1;
				}
				} else {
				if (kd >= 0) {
				nodes[kd].right = jd;
				nodes[kd].size += 1;
				}
				}
				nodes[jd].next = nodes[id].next;
				nodes[jd].prev = id;
				nodes[id].next = jd;
				nodes[jd].parent = kd;
				}

				void pushRight(int id, int jd) {
				nodes[id].size += 1;
				nodes[jd].prev = nodes[id].right;
				nodes[jd].next = -1;
				nodes[nodes[id].right].next = jd;
				nodes[id].right = jd;
				nodes[jd].parent = id;
				}

				void popRight(int id) {
				nodes[id].size -= 1;
				int jd = nodes[id].right;
				nodes[nodes[jd].prev].next = -1;
				nodes[id].right = nodes[jd].prev;
				}

				void splice(int id, int jd) {
				nodes[id].size += nodes[jd].size;
				nodes[nodes[id].right].next = nodes[jd].left;
				nodes[nodes[jd].left].prev = nodes[id].right;
				int kd = nodes[jd].left;
				while (kd != -1) {
				nodes[kd].parent = id;
				kd = nodes[kd].next;
				}
				nodes[id].right = nodes[jd].right;
				}

				void split(int id, int jd) {
				int kd = nodes[id].parent;
				nodes[kd].right = nodes[id].prev;
				nodes[nodes[id].prev].next = -1;

				nodes[jd].left = id;
				nodes[id].prev = -1;
				int num = 0;
				while (id != -1) {
				nodes[id].parent = jd;
				nodes[jd].right = id;
				id = nodes[id].next;
				++num;
				}
				nodes[kd].size -= num;
				nodes[jd].size = num;
				}

				void pushLeft(int id, int jd) {
				nodes[id].size += 1;
				nodes[jd].next = nodes[id].left;
				nodes[jd].prev = -1;
				nodes[nodes[id].left].prev = jd;
				nodes[id].left = jd;
				nodes[jd].parent = id;
				}

				void popLeft(int id) {
				nodes[id].size -= 1;
				int jd = nodes[id].left;
				nodes[nodes[jd].next].prev = -1;
				nodes[id].left = nodes[jd].next;
				}

				void repairLeft(int id) {
				int jd = ~(classes[id].parent);
				while (nodes[jd].left != -1) {
				int kd = nodes[jd].left;
				if (nodes[jd].size == 1) {
				if (nodes[jd].parent < 0) {
				classes[id].parent = ~kd;
				classes[id].depth -= 1;
				nodes[kd].parent = ~id;
				deleteNode(jd);
				jd = kd;
				} else {
				int pd = nodes[jd].parent;
				if (nodes[nodes[jd].next].size < cmax) {
				pushLeft(nodes[jd].next, nodes[jd].left);
				if (less(jd, nodes[jd].next) \|\|
				nodes[jd].item == nodes[pd].item) {
				nodes[nodes[jd].next].prio = nodes[jd].prio;
				nodes[nodes[jd].next].item = nodes[jd].item;
				}
				popLeft(pd);
				deleteNode(jd);
				jd = pd;
				} else {
				int ld = nodes[nodes[jd].next].left;
				popLeft(nodes[jd].next);
				pushRight(jd, ld);
				if (less(ld, nodes[jd].left) \|\|
				nodes[ld].item == nodes[pd].item) {
				nodes[jd].item = nodes[ld].item;
				nodes[jd].prio = nodes[ld].prio;
				}
				if (nodes[nodes[jd].next].item == nodes[ld].item) {
				setPrio(nodes[jd].next);
				}
				jd = nodes[jd].left;
				}
				}
				} else {
				jd = nodes[jd].left;
				}
				}
				}

				void repairRight(int id) {
				int jd = ~(classes[id].parent);
				while (nodes[jd].right != -1) {
				int kd = nodes[jd].right;
				if (nodes[jd].size == 1) {
				if (nodes[jd].parent < 0) {
				classes[id].parent = ~kd;
				classes[id].depth -= 1;
				nodes[kd].parent = ~id;
				deleteNode(jd);
				jd = kd;
				} else {
				int pd = nodes[jd].parent;
				if (nodes[nodes[jd].prev].size < cmax) {
				pushRight(nodes[jd].prev, nodes[jd].right);
				if (less(jd, nodes[jd].prev) \|\|
				nodes[jd].item == nodes[pd].item) {
				nodes[nodes[jd].prev].prio = nodes[jd].prio;
				nodes[nodes[jd].prev].item = nodes[jd].item;
				}
				popRight(pd);
				deleteNode(jd);
				jd = pd;
				} else {
				int ld = nodes[nodes[jd].prev].right;
				popRight(nodes[jd].prev);
				pushLeft(jd, ld);
				if (less(ld, nodes[jd].right) \|\|
				nodes[ld].item == nodes[pd].item) {
				nodes[jd].item = nodes[ld].item;
				nodes[jd].prio = nodes[ld].prio;
				}
				if (nodes[nodes[jd].prev].item == nodes[ld].item) {
				setPrio(nodes[jd].prev);
				}
				jd = nodes[jd].right;
				}
				}
				} else {
				jd = nodes[jd].right;
				}
				}
				}


				bool less(int id, int jd) const {
				return comp(nodes[id].prio, nodes[jd].prio);
				}

				public:

				/// \brief Returns true when the given class is alive.
				///
				/// Returns true when the given class is alive, ie. the class is
				/// not nested into other class.
				bool alive(int cls) const {
				return classes[cls].parent < 0;
				}

				/// \brief Returns true when the given class is trivial.
				///
				/// Returns true when the given class is trivial, ie. the class
				/// contains just one item directly.
				bool trivial(int cls) const {
				return classes[cls].left == -1;
				}

				/// \brief Constructs the union-find.
				///
				/// Constructs the union-find.
				/// \brief _index The index map of the union-find. The data
				/// structure uses internally for store references.
				HeapUnionFind(ItemIntMap& _index)
				: index(_index), first_class(-1),
				first_free_class(-1), first_free_node(-1) {}

				/// \brief Clears the union-find data structure
				///
				/// Erase each item from the data structure.
				void clear() {
				nodes.clear();
				classes.clear();
				first_free_node = first_free_class = first_class = -1;
				}

				/// \brief Insert a new node into a new component.
				///
				/// Insert a new node into a new component.
				/// \param item The item of the new node.
				/// \param prio The priority of the new node.
				/// \return The class id of the one-item-heap.
				int insert(const Item& item, const Value& prio) {
				int id = newNode();
				nodes[id].item = item;
				nodes[id].prio = prio;
				nodes[id].size = 0;

				nodes[id].prev = -1;
				nodes[id].next = -1;

				nodes[id].left = -1;
				nodes[id].right = -1;

				nodes[id].item = item;
				index[item] = id;

				int class_id = newClass();
				classes[class_id].parent = ~id;
				classes[class_id].depth = 0;

				classes[class_id].left = -1;
				classes[class_id].right = -1;

				if (first_class != -1) {
				classes[first_class].prev = class_id;
				}
				classes[class_id].next = first_class;
				classes[class_id].prev = -1;
				first_class = class_id;

				nodes[id].parent = ~class_id;

				return class_id;
				}

				/// \brief The class of the item.
				///
				/// \return The alive class id of the item, which is not nested into
				/// other classes.
				///
				/// The time complexity is O(log(n)).
				int find(const Item& item) const {
				return findClass(index[item]);
				}

				/// \brief Joins the classes.
				///
				/// The current function joins the given classes. The parameter is
				/// an STL range which should be contains valid class ids. The
				/// time complexity is O(log(n)*k) where n is the overall number
				/// of the joined nodes and k is the number of classes.
				/// \return The class of the joined classes.
				/// \pre The range should contain at least two class ids.
				template <typename Iterator>
				int join(Iterator begin, Iterator end) {
				std::vector<int> cs;
				for (Iterator it = begin; it != end; ++it) {
				cs.push_back(*it);
				}

				int class_id = newClass();
				{ // creation union-find

				if (first_class != -1) {
				classes[first_class].prev = class_id;
				}
				classes[class_id].next = first_class;
				classes[class_id].prev = -1;
				first_class = class_id;

				classes[class_id].depth = classes[cs[0]].depth;
				classes[class_id].parent = classes[cs[0]].parent;
				nodes[~(classes[class_id].parent)].parent = ~class_id;

				int l = cs[0];

				classes[class_id].left = l;
				classes[class_id].right = l;

				if (classes[l].next != -1) {
				classes[classes[l].next].prev = classes[l].prev;
				}
				classes[classes[l].prev].next = classes[l].next;

				classes[l].prev = -1;
				classes[l].next = -1;

				classes[l].depth = leftNode(l);
				classes[l].parent = class_id;

				}

				{ // merging of heap
				int l = class_id;
				for (int ci = 1; ci < int(cs.size()); ++ci) {
				int r = cs[ci];
				int rln = leftNode(r);
				if (classes[l].depth > classes[r].depth) {
				int id = ~(classes[l].parent);
				for (int i = classes[r].depth + 1; i < classes[l].depth; ++i) {
				id = nodes[id].right;
				}
				while (id >= 0 && nodes[id].size == cmax) {
				int new_id = newNode();
				int right_id = nodes[id].right;

				popRight(id);
				if (nodes[id].item == nodes[right_id].item) {
				setPrio(id);
				}
				push(new_id, right_id);
				pushRight(new_id, ~(classes[r].parent));

				if (less(~classes[r].parent, right_id)) {
				nodes[new_id].item = nodes[~classes[r].parent].item;
				nodes[new_id].prio = nodes[~classes[r].parent].prio;
				} else {
				nodes[new_id].item = nodes[right_id].item;
				nodes[new_id].prio = nodes[right_id].prio;
				}

				id = nodes[id].parent;
				classes[r].parent = ~new_id;
				}
				if (id < 0) {
				int new_parent = newNode();
				nodes[new_parent].next = -1;
				nodes[new_parent].prev = -1;
				nodes[new_parent].parent = ~l;

				push(new_parent, ~(classes[l].parent));
				pushRight(new_parent, ~(classes[r].parent));
				setPrio(new_parent);

				classes[l].parent = ~new_parent;
				classes[l].depth += 1;
				} else {
				pushRight(id, ~(classes[r].parent));
				while (id >= 0 && less(~(classes[r].parent), id)) {
				nodes[id].prio = nodes[~(classes[r].parent)].prio;
				nodes[id].item = nodes[~(classes[r].parent)].item;
				id = nodes[id].parent;
				}
				}
				} else if (classes[r].depth > classes[l].depth) {
				int id = ~(classes[r].parent);
				for (int i = classes[l].depth + 1; i < classes[r].depth; ++i) {
				id = nodes[id].left;
				}
				while (id >= 0 && nodes[id].size == cmax) {
				int new_id = newNode();
				int left_id = nodes[id].left;

				popLeft(id);
				if (nodes[id].prio == nodes[left_id].prio) {
				setPrio(id);
				}
				push(new_id, left_id);
				pushLeft(new_id, ~(classes[l].parent));

				if (less(~classes[l].parent, left_id)) {
				nodes[new_id].item = nodes[~classes[l].parent].item;
				nodes[new_id].prio = nodes[~classes[l].parent].prio;
				} else {
				nodes[new_id].item = nodes[left_id].item;
				nodes[new_id].prio = nodes[left_id].prio;
				}

				id = nodes[id].parent;
				classes[l].parent = ~new_id;

				}
				if (id < 0) {
				int new_parent = newNode();
				nodes[new_parent].next = -1;
				nodes[new_parent].prev = -1;
				nodes[new_parent].parent = ~l;

				push(new_parent, ~(classes[r].parent));
				pushLeft(new_parent, ~(classes[l].parent));
				setPrio(new_parent);

				classes[r].parent = ~new_parent;
				classes[r].depth += 1;
				} else {
				pushLeft(id, ~(classes[l].parent));
				while (id >= 0 && less(~(classes[l].parent), id)) {
				nodes[id].prio = nodes[~(classes[l].parent)].prio;
				nodes[id].item = nodes[~(classes[l].parent)].item;
				id = nodes[id].parent;
				}
				}
				nodes[~(classes[r].parent)].parent = ~l;
				classes[l].parent = classes[r].parent;
				classes[l].depth = classes[r].depth;
				} else {
				if (classes[l].depth != 0 &&
				nodes[~(classes[l].parent)].size +
				nodes[~(classes[r].parent)].size <= cmax) {
				splice(~(classes[l].parent), ~(classes[r].parent));
				deleteNode(~(classes[r].parent));
				if (less(~(classes[r].parent), ~(classes[l].parent))) {
				nodes[~(classes[l].parent)].prio =
				nodes[~(classes[r].parent)].prio;
				nodes[~(classes[l].parent)].item =
				nodes[~(classes[r].parent)].item;
				}
				} else {
				int new_parent = newNode();
				nodes[new_parent].next = nodes[new_parent].prev = -1;
				push(new_parent, ~(classes[l].parent));
				pushRight(new_parent, ~(classes[r].parent));
				setPrio(new_parent);

				classes[l].parent = ~new_parent;
				classes[l].depth += 1;
				nodes[new_parent].parent = ~l;
				}
				}
				if (classes[r].next != -1) {
				classes[classes[r].next].prev = classes[r].prev;
				}
				classes[classes[r].prev].next = classes[r].next;

				classes[r].prev = classes[l].right;
				classes[classes[l].right].next = r;
				classes[l].right = r;
				classes[r].parent = l;

				classes[r].next = -1;
				classes[r].depth = rln;
				}
				}
				return class_id;
				}

				/// \brief Split the class to subclasses.
				///
				/// The current function splits the given class. The join, which
				/// made the current class, stored a reference to the
				/// subclasses. The \c splitClass() member restores the classes
				/// and creates the heaps. The parameter is an STL output iterator
				/// which will be filled with the subclass ids. The time
				/// complexity is O(log(n)*k) where n is the overall number of
				/// nodes in the splitted classes and k is the number of the
				/// classes.
				template <typename Iterator>
				void split(int cls, Iterator out) {
				std::vector<int> cs;
				{ // splitting union-find
				int id = cls;
				int l = classes[id].left;

				classes[l].parent = classes[id].parent;
				classes[l].depth = classes[id].depth;

				nodes[~(classes[l].parent)].parent = ~l;

				*out++ = l;

				while (l != -1) {
				cs.push_back(l);
				l = classes[l].next;
				}

				classes[classes[id].right].next = first_class;
				classes[first_class].prev = classes[id].right;
				first_class = classes[id].left;

				if (classes[id].next != -1) {
				classes[classes[id].next].prev = classes[id].prev;
				}
				classes[classes[id].prev].next = classes[id].next;

				deleteClass(id);
				}

				{
				for (int i = 1; i < int(cs.size()); ++i) {
				int l = classes[cs[i]].depth;
				while (nodes[nodes[l].parent].left == l) {
				l = nodes[l].parent;
				}
				int r = l;
				while (nodes[l].parent >= 0) {
				l = nodes[l].parent;
				int new_node = newNode();

				nodes[new_node].prev = -1;
				nodes[new_node].next = -1;

				split(r, new_node);
				pushAfter(l, new_node);
				setPrio(l);
				setPrio(new_node);
				r = new_node;
				}
				classes[cs[i]].parent = ~r;
				classes[cs[i]].depth = classes[~(nodes[l].parent)].depth;
				nodes[r].parent = ~cs[i];

				nodes[l].next = -1;
				nodes[r].prev = -1;

				repairRight(~(nodes[l].parent));
				repairLeft(cs[i]);

				*out++ = cs[i];
				}
				}
				}

				/// \brief Gives back the priority of the current item.
				///
				/// Gives back the priority of the current item.
				const Value& operator[](const Item& item) const {
				return nodes[index[item]].prio;
				}

				/// \brief Sets the priority of the current item.
				///
				/// Sets the priority of the current item.
				void set(const Item& item, const Value& prio) {
				if (comp(prio, nodes[index[item]].prio)) {
				decrease(item, prio);
				} else if (!comp(prio, nodes[index[item]].prio)) {
				increase(item, prio);
				}
				}

				/// \brief Increase the priority of the current item.
				///
				/// Increase the priority of the current item.
				void increase(const Item& item, const Value& prio) {
				int id = index[item];
				int kd = nodes[id].parent;
				nodes[id].prio = prio;
				while (kd >= 0 && nodes[kd].item == item) {
				setPrio(kd);
				kd = nodes[kd].parent;
				}
				}

				/// \brief Increase the priority of the current item.
				///
				/// Increase the priority of the current item.
				void decrease(const Item& item, const Value& prio) {
				int id = index[item];
				int kd = nodes[id].parent;
				nodes[id].prio = prio;
				while (kd >= 0 && less(id, kd)) {
				nodes[kd].prio = prio;
				nodes[kd].item = item;
				kd = nodes[kd].parent;
				}
				}

				/// \brief Gives back the minimum priority of the class.
				///
				/// Gives back the minimum priority of the class.
				const Value& classPrio(int cls) const {
				return nodes[~(classes[cls].parent)].prio;
				}

				/// \brief Gives back the minimum priority item of the class.
				///
				/// \return Gives back the minimum priority item of the class.
				const Item& classTop(int cls) const {
				return nodes[~(classes[cls].parent)].item;
				}

				/// \brief Gives back a representant item of the class.
				///
				/// Gives back a representant item of the class.
				/// The representant is indpendent from the priorities of the
				/// items.
				const Item& classRep(int id) const {
				int parent = classes[id].parent;
				return nodes[parent >= 0 ? classes[id].depth : leftNode(id)].item;
				}

				/// \brief LEMON style iterator for the items of a class.
				///
				/// ClassIt is a lemon style iterator for the components. It iterates
				/// on the items of a class. By example if you want to iterate on
				/// each items of each classes then you may write the next code.
				///\code
				/// for (ClassIt cit(huf); cit != INVALID; ++cit) {
				/// std::cout << "Class: ";
				/// for (ItemIt iit(huf, cit); iit != INVALID; ++iit) {
				/// std::cout << toString(iit) << ' ' << std::endl;
				/// }
				/// std::cout << std::endl;
				/// }
				///\endcode
				class ItemIt {
				private:

				const HeapUnionFind* _huf;
				int _id, _lid;

				public:

				/// \brief Default constructor
				///
				/// Default constructor
				ItemIt() {}

				ItemIt(const HeapUnionFind& huf, int cls) : _huf(&huf) {
				int id = cls;
				int parent = _huf->classes[id].parent;
				if (parent >= 0) {
				_id = _huf->classes[id].depth;
				if (_huf->classes[id].next != -1) {
				_lid = _huf->classes[_huf->classes[id].next].depth;
				} else {
				_lid = -1;
				}
				} else {
				_id = _huf->leftNode(id);
				_lid = -1;
				}
				}

				/// \brief Increment operator
				///
				/// It steps to the next item in the class.
				ItemIt& operator++() {
				_id = _huf->nextNode(_id);
				return *this;
				}

				/// \brief Conversion operator
				///
				/// It converts the iterator to the current item.
				operator const Item&() const {
				return _huf->nodes[_id].item;
				}

				/// \brief Equality operator
				///
				/// Equality operator
				bool operator==(const ItemIt& i) {
				return i._id == _id;
				}

				/// \brief Inequality operator
				///
				/// Inequality operator
				bool operator!=(const ItemIt& i) {
				return i._id != _id;
				}

				/// \brief Equality operator
				///
				/// Equality operator
				bool operator==(Invalid) {
				return _id == _lid;
				}

				/// \brief Inequality operator
				///
				/// Inequality operator
				bool operator!=(Invalid) {
				return _id != _lid;
				}

				};

				/// \brief Class iterator
				///
				/// The iterator stores
				class ClassIt {
				private:

				const HeapUnionFind* _huf;
				int _id;

				public:

				ClassIt(const HeapUnionFind& huf)
				: _huf(&huf), _id(huf.first_class) {}

				ClassIt(const HeapUnionFind& huf, int cls)
				: _huf(&huf), _id(huf.classes[cls].left) {}

				ClassIt(Invalid) : _huf(0), _id(-1) {}

				const ClassIt& operator++() {
				_id = _huf->classes[_id].next;
				return *this;
				}

				/// \brief Equality operator
				///
				/// Equality operator
				bool operator==(const ClassIt& i) {
				return i._id == _id;
				}

				/// \brief Inequality operator
				///
				/// Inequality operator
				bool operator!=(const ClassIt& i) {
				return i._id != _id;
				}

				operator int() const {
				return _id;
				}

				};

				};

				//! @}

				} //namespace lemon

				#endif //LEMON_UNION_FIND_H