diff -r 75f749682240 -r c80ef5912903 src/lemon/concept/graph.h --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/src/lemon/concept/graph.h Thu Nov 04 20:24:59 2004 +0000 @@ -0,0 +1,918 @@ +/* -*- C++ -*- + * src/lemon/concept/graph.h - Part of LEMON, a generic C++ optimization library + * + * Copyright (C) 2004 Egervary Jeno Kombinatorikus Optimalizalasi Kutatocsoport + * (Egervary Combinatorial Optimization Research Group, 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_CONCEPT_GRAPH_H +#define LEMON_CONCEPT_GRAPH_H + +///\ingroup concept +///\file +///\brief Declaration of Graph. + +#include +#include +#include +#include + +namespace lemon { + namespace concept { + + /// \addtogroup concept + /// @{ + +// /// An empty static graph class. + +// /// This class provides all the common features of a graph structure, +// /// however completely without implementations and real data structures +// /// behind the interface. +// /// All graph algorithms should compile with this class, but it will not +// /// run properly, of course. +// /// +// /// It can be used for checking the interface compatibility, +// /// or it can serve as a skeleton of a new graph structure. +// /// +// /// Also, you will find here the full documentation of a certain graph +// /// feature, the documentation of a real graph imlementation +// /// like @ref ListGraph or +// /// @ref SmartGraph will just refer to this structure. +// /// +// /// \todo A pages describing the concept of concept description would +// /// be nice. +// class StaticGraph +// { +// public: +// /// Defalult constructor. + +// /// Defalult constructor. +// /// +// StaticGraph() { } +// ///Copy consructor. + +// // ///\todo It is not clear, what we expect from a copy constructor. +// // ///E.g. How to assign the nodes/edges to each other? What about maps? +// // StaticGraph(const StaticGraph& g) { } + +// /// The base type of node iterators, +// /// or in other words, the trivial node iterator. + +// /// This is the base type of each node iterator, +// /// thus each kind of node iterator converts to this. +// /// More precisely each kind of node iterator should be inherited +// /// from the trivial node iterator. +// class Node { +// public: +// /// Default constructor + +// /// @warning The default constructor sets the iterator +// /// to an undefined value. +// Node() { } +// /// Copy constructor. + +// /// Copy constructor. +// /// +// Node(const Node&) { } + +// /// Invalid constructor \& conversion. + +// /// This constructor initializes the iterator to be invalid. +// /// \sa Invalid for more details. +// Node(Invalid) { } +// /// Equality operator + +// /// Two iterators are equal if and only if they point to the +// /// same object or both are invalid. +// bool operator==(Node) const { return true; } + +// /// Inequality operator + +// /// \sa operator==(Node n) +// /// +// bool operator!=(Node) const { return true; } + +// ///Comparison operator. + +// ///This is a strict ordering between the nodes. +// /// +// ///This ordering can be different from the order in which NodeIt +// ///goes through the nodes. +// ///\todo Possibly we don't need it. +// bool operator<(Node) const { return true; } +// }; + +// /// This iterator goes through each node. + +// /// This iterator goes through each node. +// /// Its usage is quite simple, for example you can count the number +// /// of nodes in graph \c g of type \c Graph like this: +// /// \code +// /// int count=0; +// /// for (Graph::NodeIt n(g); n!=INVALID ++n) ++count; +// /// \endcode +// class NodeIt : public Node { +// public: +// /// Default constructor + +// /// @warning The default constructor sets the iterator +// /// to an undefined value. +// NodeIt() { } +// /// Copy constructor. + +// /// Copy constructor. +// /// +// NodeIt(const NodeIt&) { } +// /// Invalid constructor \& conversion. + +// /// Initialize the iterator to be invalid. +// /// \sa Invalid for more details. +// NodeIt(Invalid) { } +// /// Sets the iterator to the first node. + +// /// Sets the iterator to the first node of \c g. +// /// +// NodeIt(const StaticGraph& g) { } +// /// Node -> NodeIt conversion. + +// /// Sets the iterator to the node of \c g pointed by the trivial +// /// iterator n. +// /// This feature necessitates that each time we +// /// iterate the edge-set, the iteration order is the same. +// NodeIt(const StaticGraph& g, const Node& n) { } +// /// Next node. + +// /// Assign the iterator to the next node. +// /// +// NodeIt& operator++() { return *this; } +// }; + + +// /// The base type of the edge iterators. + +// /// The base type of the edge iterators. +// /// +// class Edge { +// public: +// /// Default constructor + +// /// @warning The default constructor sets the iterator +// /// to an undefined value. +// Edge() { } +// /// Copy constructor. + +// /// Copy constructor. +// /// +// Edge(const Edge&) { } +// /// Initialize the iterator to be invalid. + +// /// Initialize the iterator to be invalid. +// /// +// Edge(Invalid) { } +// /// Equality operator + +// /// Two iterators are equal if and only if they point to the +// /// same object or both are invalid. +// bool operator==(Edge) const { return true; } +// /// Inequality operator + +// /// \sa operator==(Node n) +// /// +// bool operator!=(Edge) const { return true; } +// ///Comparison operator. + +// ///This is a strict ordering between the nodes. +// /// +// ///This ordering can be different from the order in which NodeIt +// ///goes through the nodes. +// ///\todo Possibly we don't need it. +// bool operator<(Edge) const { return true; } +// }; + +// /// This iterator goes trough the outgoing edges of a node. + +// /// This iterator goes trough the \e outgoing edges of a certain node +// /// of a graph. +// /// Its usage is quite simple, for example you can count the number +// /// of outgoing edges of a node \c n +// /// in graph \c g of type \c Graph as follows. +// /// \code +// /// int count=0; +// /// for (Graph::OutEdgeIt e(g, n); e!=INVALID; ++e) ++count; +// /// \endcode + +// class OutEdgeIt : public Edge { +// public: +// /// Default constructor + +// /// @warning The default constructor sets the iterator +// /// to an undefined value. +// OutEdgeIt() { } +// /// Copy constructor. + +// /// Copy constructor. +// /// +// OutEdgeIt(const OutEdgeIt&) { } +// /// Initialize the iterator to be invalid. + +// /// Initialize the iterator to be invalid. +// /// +// OutEdgeIt(Invalid) { } +// /// This constructor sets the iterator to first outgoing edge. + +// /// This constructor set the iterator to the first outgoing edge of +// /// node +// ///@param n the node +// ///@param g the graph +// OutEdgeIt(const StaticGraph& g, const Node& n) { } +// /// Edge -> OutEdgeIt conversion + +// /// Sets the iterator to the value of the trivial iterator \c e. +// /// This feature necessitates that each time we +// /// iterate the edge-set, the iteration order is the same. +// OutEdgeIt(const StaticGraph& g, const Edge& e) { } +// ///Next outgoing edge + +// /// Assign the iterator to the next +// /// outgoing edge of the corresponding node. +// OutEdgeIt& operator++() { return *this; } +// }; + +// /// This iterator goes trough the incoming edges of a node. + +// /// This iterator goes trough the \e incoming edges of a certain node +// /// of a graph. +// /// Its usage is quite simple, for example you can count the number +// /// of outgoing edges of a node \c n +// /// in graph \c g of type \c Graph as follows. +// /// \code +// /// int count=0; +// /// for(Graph::InEdgeIt e(g, n); e!=INVALID; ++e) ++count; +// /// \endcode + +// class InEdgeIt : public Edge { +// public: +// /// Default constructor + +// /// @warning The default constructor sets the iterator +// /// to an undefined value. +// InEdgeIt() { } +// /// Copy constructor. + +// /// Copy constructor. +// /// +// InEdgeIt(const InEdgeIt&) { } +// /// Initialize the iterator to be invalid. + +// /// Initialize the iterator to be invalid. +// /// +// InEdgeIt(Invalid) { } +// /// This constructor sets the iterator to first incoming edge. + +// /// This constructor set the iterator to the first incoming edge of +// /// node +// ///@param n the node +// ///@param g the graph +// InEdgeIt(const StaticGraph& g, const Node& n) { } +// /// Edge -> InEdgeIt conversion + +// /// Sets the iterator to the value of the trivial iterator \c e. +// /// This feature necessitates that each time we +// /// iterate the edge-set, the iteration order is the same. +// InEdgeIt(const StaticGraph& g, const Edge& n) { } +// /// Next incoming edge + +// /// Assign the iterator to the next inedge of the corresponding node. +// /// +// InEdgeIt& operator++() { return *this; } +// }; +// /// This iterator goes through each edge. + +// /// This iterator goes through each edge of a graph. +// /// Its usage is quite simple, for example you can count the number +// /// of edges in a graph \c g of type \c Graph as follows: +// /// \code +// /// int count=0; +// /// for(Graph::EdgeIt e(g); e!=INVALID; ++e) ++count; +// /// \endcode +// class EdgeIt : public Edge { +// public: +// /// Default constructor + +// /// @warning The default constructor sets the iterator +// /// to an undefined value. +// EdgeIt() { } +// /// Copy constructor. + +// /// Copy constructor. +// /// +// EdgeIt(const EdgeIt&) { } +// /// Initialize the iterator to be invalid. + +// /// Initialize the iterator to be invalid. +// /// +// EdgeIt(Invalid) { } +// /// This constructor sets the iterator to first edge. + +// /// This constructor set the iterator to the first edge of +// /// node +// ///@param g the graph +// EdgeIt(const StaticGraph& g) { } +// /// Edge -> EdgeIt conversion + +// /// Sets the iterator to the value of the trivial iterator \c e. +// /// This feature necessitates that each time we +// /// iterate the edge-set, the iteration order is the same. +// EdgeIt(const StaticGraph&, const Edge&) { } +// ///Next edge + +// /// Assign the iterator to the next +// /// edge of the corresponding node. +// EdgeIt& operator++() { return *this; } +// }; + +// /// First node of the graph. + +// /// \retval i the first node. +// /// \return the first node. +// /// +// NodeIt& first(NodeIt& i) const { return i; } + +// /// The first incoming edge. + +// /// The first incoming edge. +// /// +// InEdgeIt& first(InEdgeIt &i, Node) const { return i; } +// /// The first outgoing edge. + +// /// The first outgoing edge. +// /// +// OutEdgeIt& first(OutEdgeIt& i, Node) const { return i; } +// /// The first edge of the Graph. + +// /// The first edge of the Graph. +// /// +// EdgeIt& first(EdgeIt& i) const { return i; } + +// ///Gives back the head node of an edge. + +// ///Gives back the head node of an edge. +// /// +// Node head(Edge) const { return INVALID; } +// ///Gives back the tail node of an edge. + +// ///Gives back the tail node of an edge. +// /// +// Node tail(Edge) const { return INVALID; } + +// ///Gives back the \e id of a node. + +// ///\warning Not all graph structures provide this feature. +// /// +// ///\todo Should each graph provide \c id? +// int id(const Node&) const { return 0; } +// ///Gives back the \e id of an edge. + +// ///\warning Not all graph structures provide this feature. +// /// +// ///\todo Should each graph provide \c id? +// int id(const Edge&) const { return 0; } + +// ///\e + +// ///\todo Should it be in the concept? +// /// +// int nodeNum() const { return 0; } +// ///\e + +// ///\todo Should it be in the concept? +// /// +// int edgeNum() const { return 0; } + + +// ///Reference map of the nodes to type \c T. + +// /// \ingroup concept +// ///Reference map of the nodes to type \c T. +// /// \sa Reference +// /// \warning Making maps that can handle bool type (NodeMap) +// /// needs some extra attention! +// template class NodeMap : public ReferenceMap< Node, T > +// { +// public: + +// ///\e +// NodeMap(const StaticGraph&) { } +// ///\e +// NodeMap(const StaticGraph&, T) { } + +// ///Copy constructor +// template NodeMap(const NodeMap&) { } +// ///Assignment operator +// template NodeMap& operator=(const NodeMap&) +// { return *this; } +// }; + +// ///Reference map of the edges to type \c T. + +// /// \ingroup concept +// ///Reference map of the edges to type \c T. +// /// \sa Reference +// /// \warning Making maps that can handle bool type (EdgeMap) +// /// needs some extra attention! +// template class EdgeMap +// : public ReferenceMap +// { +// public: + +// ///\e +// EdgeMap(const StaticGraph&) { } +// ///\e +// EdgeMap(const StaticGraph&, T) { } + +// ///Copy constructor +// template EdgeMap(const EdgeMap&) { } +// ///Assignment operator +// template EdgeMap &operator=(const EdgeMap&) +// { return *this; } +// }; +// }; + +// struct DummyType { +// int value; +// DummyType() {} +// DummyType(int p) : value(p) {} +// DummyType& operator=(int p) { value = p; return *this;} +// }; + +// ///\brief Checks whether \c G meets the +// ///\ref lemon::concept::StaticGraph "StaticGraph" concept +// template void checkCompileStaticGraph(Graph &G) +// { +// typedef typename Graph::Node Node; +// typedef typename Graph::NodeIt NodeIt; +// typedef typename Graph::Edge Edge; +// typedef typename Graph::EdgeIt EdgeIt; +// typedef typename Graph::InEdgeIt InEdgeIt; +// typedef typename Graph::OutEdgeIt OutEdgeIt; + +// { +// Node i; Node j(i); Node k(INVALID); +// i=j; +// bool b; b=true; +// b=(i==INVALID); b=(i!=INVALID); +// b=(i==j); b=(i!=j); b=(iNodeIt conversion +// NodeIt ni(G,n); +// } +// { +// Edge i; Edge j(i); Edge k(INVALID); +// i=j; +// bool b; b=true; +// b=(i==INVALID); b=(i!=INVALID); +// b=(i==j); b=(i!=j); b=(iEdgeIt conversion +// EdgeIt ei(G,e); +// } +// { +// Node n; +// InEdgeIt i; InEdgeIt j(i); InEdgeIt k(INVALID); InEdgeIt l(G,n); +// i=j; +// j=G.first(i,n); +// j=++i; +// bool b; b=true; +// b=(i==INVALID); b=(i!=INVALID); +// Edge e(i); +// e=i; +// b=(i==j); b=(i!=j); b=(iInEdgeIt conversion +// InEdgeIt ei(G,e); +// } +// { +// Node n; +// OutEdgeIt i; OutEdgeIt j(i); OutEdgeIt k(INVALID); OutEdgeIt l(G,n); +// i=j; +// j=G.first(i,n); +// j=++i; +// bool b; b=true; +// b=(i==INVALID); b=(i!=INVALID); +// Edge e(i); +// e=i; +// b=(i==j); b=(i!=j); b=(iOutEdgeIt conversion +// OutEdgeIt ei(G,e); +// } +// { +// Node n,m; +// n=m=INVALID; +// Edge e; +// e=INVALID; +// n=G.tail(e); +// n=G.head(e); +// } +// // id tests +// { Node n; int i=G.id(n); i=i; } +// { Edge e; int i=G.id(e); i=i; } +// //NodeMap tests +// { +// Node k; +// typename Graph::template NodeMap m(G); +// //Const map +// typename Graph::template NodeMap const &cm = m; +// //Inicialize with default value +// typename Graph::template NodeMap mdef(G,12); +// //Copy +// typename Graph::template NodeMap mm(cm); +// //Copy from another type +// typename Graph::template NodeMap dm(cm); +// //Copy to more complex type +// typename Graph::template NodeMap em(cm); +// int v; +// v=m[k]; m[k]=v; m.set(k,v); +// v=cm[k]; + +// m=cm; +// dm=cm; //Copy from another type +// em=cm; //Copy to more complex type +// { +// //Check the typedef's +// typename Graph::template NodeMap::ValueType val; +// val=1; +// typename Graph::template NodeMap::KeyType key; +// key = typename Graph::NodeIt(G); +// } +// } +// { //bool NodeMap +// Node k; +// typename Graph::template NodeMap m(G); +// typename Graph::template NodeMap const &cm = m; //Const map +// //Inicialize with default value +// typename Graph::template NodeMap mdef(G,12); +// typename Graph::template NodeMap mm(cm); //Copy +// typename Graph::template NodeMap dm(cm); //Copy from another type +// bool v; +// v=m[k]; m[k]=v; m.set(k,v); +// v=cm[k]; + +// m=cm; +// dm=cm; //Copy from another type +// m=dm; //Copy to another type + +// { +// //Check the typedef's +// typename Graph::template NodeMap::ValueType val; +// val=true; +// typename Graph::template NodeMap::KeyType key; +// key= typename Graph::NodeIt(G); +// } +// } +// //EdgeMap tests +// { +// Edge k; +// typename Graph::template EdgeMap m(G); +// typename Graph::template EdgeMap const &cm = m; //Const map +// //Inicialize with default value +// typename Graph::template EdgeMap mdef(G,12); +// typename Graph::template EdgeMap mm(cm); //Copy +// typename Graph::template EdgeMap dm(cm);//Copy from another type +// int v; +// v=m[k]; m[k]=v; m.set(k,v); +// v=cm[k]; + +// m=cm; +// dm=cm; //Copy from another type +// { +// //Check the typedef's +// typename Graph::template EdgeMap::ValueType val; +// val=1; +// typename Graph::template EdgeMap::KeyType key; +// key= typename Graph::EdgeIt(G); +// } +// } +// { //bool EdgeMap +// Edge k; +// typename Graph::template EdgeMap m(G); +// typename Graph::template EdgeMap const &cm = m; //Const map +// //Inicialize with default value +// typename Graph::template EdgeMap mdef(G,12); +// typename Graph::template EdgeMap mm(cm); //Copy +// typename Graph::template EdgeMap dm(cm); //Copy from another type +// bool v; +// v=m[k]; m[k]=v; m.set(k,v); +// v=cm[k]; + +// m=cm; +// dm=cm; //Copy from another type +// m=dm; //Copy to another type +// { +// //Check the typedef's +// typename Graph::template EdgeMap::ValueType val; +// val=true; +// typename Graph::template EdgeMap::KeyType key; +// key= typename Graph::EdgeIt(G); +// } +// } +// } + +// /// An empty non-static graph class. + +// /// This class provides everything that \ref StaticGraph +// /// with additional functionality which enables to build a +// /// graph from scratch. +// class ExtendableGraph : public StaticGraph +// { +// public: +// /// Defalult constructor. + +// /// Defalult constructor. +// /// +// ExtendableGraph() { } +// ///Add a new node to the graph. + +// /// \return the new node. +// /// +// Node addNode() { return INVALID; } +// ///Add a new edge to the graph. + +// ///Add a new edge to the graph with tail node \c t +// ///and head node \c h. +// ///\return the new edge. +// Edge addEdge(Node h, Node t) { return INVALID; } + +// /// Resets the graph. + +// /// This function deletes all edges and nodes of the graph. +// /// It also frees the memory allocated to store them. +// /// \todo It might belong to \ref ErasableGraph. +// void clear() { } +// }; + + +// ///\brief Checks whether \c G meets the +// ///\ref lemon::concept::ExtendableGraph "ExtendableGraph" concept +// template void checkCompileExtendableGraph(Graph &G) +// { +// checkCompileStaticGraph(G); + +// typedef typename Graph::Node Node; +// typedef typename Graph::NodeIt NodeIt; +// typedef typename Graph::Edge Edge; +// typedef typename Graph::EdgeIt EdgeIt; +// typedef typename Graph::InEdgeIt InEdgeIt; +// typedef typename Graph::OutEdgeIt OutEdgeIt; + +// Node n,m; +// n=G.addNode(); +// m=G.addNode(); +// Edge e; +// e=G.addEdge(n,m); + +// // G.clear(); +// } + + +// /// An empty erasable graph class. + +// /// This class is an extension of \ref ExtendableGraph. It also makes it +// /// possible to erase edges or nodes. +// class ErasableGraph : public ExtendableGraph +// { +// public: +// /// Defalult constructor. + +// /// Defalult constructor. +// /// +// ErasableGraph() { } +// /// Deletes a node. + +// /// Deletes node \c n node. +// /// +// void erase(Node n) { } +// /// Deletes an edge. + +// /// Deletes edge \c e edge. +// /// +// void erase(Edge e) { } +// }; + +// template void checkCompileGraphEraseEdge(Graph &G) +// { +// typename Graph::Edge e; +// G.erase(e); +// } + +// template void checkCompileGraphEraseNode(Graph &G) +// { +// typename Graph::Node n; +// G.erase(n); +// } + +// ///\brief Checks whether \c G meets the +// ///\ref lemon::concept::EresableGraph "EresableGraph" concept +// template void checkCompileErasableGraph(Graph &G) +// { +// checkCompileExtendableGraph(G); +// checkCompileGraphEraseNode(G); +// checkCompileGraphEraseEdge(G); +// } + +// ///Checks whether a graph has findEdge() member function. + +// ///\todo findEdge() might be a global function. +// /// +// template void checkCompileGraphFindEdge(Graph &G) +// { +// typedef typename Graph::NodeIt Node; +// typedef typename Graph::NodeIt NodeIt; + +// G.findEdge(NodeIt(G),++NodeIt(G),G.findEdge(NodeIt(G),++NodeIt(G))); +// G.findEdge(Node(),Node(),G.findEdge(Node(),Node())); +// } + + + + /************* New GraphBase stuff **************/ + + + /// \bug The nodes and edges are not allowed to inherit from the + /// same baseclass. + + class BaseGraphItem { + public: + BaseGraphItem() {} + BaseGraphItem(Invalid) {} + + // We explicitely list these: + BaseGraphItem(BaseGraphItem const&) {} + BaseGraphItem& operator=(BaseGraphItem const&) { return *this; } + + bool operator==(BaseGraphItem) const { return false; } + bool operator!=(BaseGraphItem) const { return false; } + + // Technical requirement. Do we really need this? + bool operator<(BaseGraphItem) const { return false; } + }; + + + /// A minimal GraphBase concept + + /// This class describes a minimal concept which can be extended to a + /// full-featured graph with \ref GraphFactory. + class GraphBase { + public: + + GraphBase() {} + + + /// \bug Nodes and Edges are comparable each other + + class Node : public BaseGraphItem {}; + class Edge : public BaseGraphItem {}; + + // Graph operation + void firstNode(Node &n) const { } + void firstEdge(Edge &e) const { } + + void firstOutEdge(Edge &e, Node) const { } + void firstInEdge(Edge &e, Node) const { } + + void nextNode(Node &n) const { } + void nextEdge(Edge &e) const { } + + + // Question: isn't it reasonable if this methods have a Node + // parameter? Like this: + // Edge& nextOut(Edge &e, Node) const { return e; } + void nextOutEdge(Edge &e) const { } + void nextInEdge(Edge &e) const { } + + Node head(Edge) const { return Node(); } + Node tail(Edge) const { return Node(); } + + + // Do we need id, nodeNum, edgeNum and co. in this basic graphbase + // concept? + + + // Maps. + // + // We need a special slimer concept which does not provide maps (it + // wouldn't be strictly slimer, cause for map-factory id() & friends + // a required...) + + template + class NodeMap : public GraphMap {}; + + template + class EdgeMap : public GraphMap {}; + }; + + + + /**************** Concept checking classes ****************/ + + template + struct BaseGraphItemConcept { + void constraints() { + BGI i1; + BGI i2 = i1; + BGI i3 = INVALID; + + i1 = i3; + if( i1 == i3 ) { + if ( i2 != i3 && i3 < i2 ) + return; + } + } + }; + + + + class StaticGraph + : virtual public BaseGraphComponent, public IterableGraphComponent, public MappableGraphComponent { + public: + typedef BaseGraphComponent::Node Node; + typedef BaseGraphComponent::Edge Edge; + }; + + template + struct StaticGraphConcept { + void constraints() { + function_requires >(); + function_requires >(); + function_requires >(); + } + Graph& graph; + }; + + class ExtendableGraph + : virtual public BaseGraphComponent, public StaticGraph, public ExtendableGraphComponent, public ClearableGraphComponent { + public: + typedef BaseGraphComponent::Node Node; + typedef BaseGraphComponent::Edge Edge; + }; + + template + struct ExtendableGraphConcept { + void constraints() { + function_requires >(); + function_requires >(); + function_requires >(); + } + Graph& graph; + }; + + class ErasableGraph + : virtual public BaseGraphComponent, public ExtendableGraph, public ErasableGraphComponent { + public: + typedef BaseGraphComponent::Node Node; + typedef BaseGraphComponent::Edge Edge; + }; + + template + struct ErasableGraphConcept { + void constraints() { + function_requires >(); + function_requires >(); + } + Graph& graph; + }; + + // @} + } //namespace concept +} //namespace lemon + + + +#endif // LEMON_CONCEPT_GRAPH_H