#include <cmath>

   /// Base graph class for UndirFullGraph.

   class UndirFullGraphBase {

     int NodeNum;

     int EdgeNum;

   public:

     typedef FullGraphBase Graph;

     class Node;

     class Edge;

   public:

     FullGraphBase() {}

     ///Creates a full graph with \c n nodes.

     void construct(int n) { NodeNum = n; EdgeNum = n * (n - 1) / 2; }

     ///

     //    FullGraphBase(const FullGraphBase &_g)

     //      : NodeNum(_g.nodeNum()), EdgeNum(NodeNum*NodeNum) { }

     typedef True NodeNumTag;

     typedef True EdgeNumTag;

     ///Number of nodes.

     int nodeNum() const { return NodeNum; }

     ///Number of edges.

     int edgeNum() const { return EdgeNum; }

     /// Maximum node ID.

     /// Maximum node ID.

     ///\sa id(Node)

     int maxId(Node = INVALID) const { return NodeNum-1; }

     /// Maximum edge ID.

     /// Maximum edge ID.

     ///\sa id(Edge)

     int maxId(Edge = INVALID) const { return EdgeNum-1; }

     Node tail(Edge e) const { 

       /// \todo we may do it faster

       return ((int)sqrt((double)(1 + 8 * e.id)) + 1) / 2; 

     }

     Node head(Edge e) const { 

       int tail = ((int)sqrt((double)(1 + 8 * e.id)) + 1) / 2;;

       return e.id - (tail) * (tail - 1) / 2; 

     }

     /// Node ID.

     /// The ID of a valid Node is a nonnegative integer not greater than

     /// \ref maxNodeId(). The range of the ID's is not surely continuous

     /// and the greatest node ID can be actually less then \ref maxNodeId().

     ///

     /// The ID of the \ref INVALID node is -1.

     ///\return The ID of the node \c v. 

     static int id(Node v) { return v.id; }

     /// Edge ID.

     /// The ID of a valid Edge is a nonnegative integer not greater than

     /// \ref maxEdgeId(). The range of the ID's is not surely continuous

     /// and the greatest edge ID can be actually less then \ref maxEdgeId().

     ///

     /// The ID of the \ref INVALID edge is -1.

     ///\return The ID of the edge \c e. 

     static int id(Edge e) { return e.id; }

     /// Finds an edge between two nodes.

     /// Finds an edge from node \c u to node \c v.

     ///

     /// If \c prev is \ref INVALID (this is the default value), then

     /// It finds the first edge from \c u to \c v. Otherwise it looks for

     /// the next edge from \c u to \c v after \c prev.

     /// \return The found edge or INVALID if there is no such an edge.

     Edge findEdge(Node u,Node v, Edge prev = INVALID) 

     {

       return prev.id == -1 ? Edge(*this, u.id, v.id) : INVALID;

     }

     class Node {

       friend class FullGraphBase;

     protected:

       int id;

       Node(int _id) { id = _id;}

     public:

       Node() {}

       Node (Invalid) { id = -1; }

       bool operator==(const Node node) const {return id == node.id;}

       bool operator!=(const Node node) const {return id != node.id;}

       bool operator<(const Node node) const {return id < node.id;}

     };

     class Edge {

       friend class FullGraphBase;

     protected:

       int id;  // NodeNum * head + tail;

       Edge(int _id) : id(_id) {}

       Edge(const FullGraphBase& _graph, int tail, int head) 

 	: id(_graph.NodeNum * head+tail) {}

     public:

       Edge() { }

       Edge (Invalid) { id = -1; }

       bool operator==(const Edge edge) const {return id == edge.id;}

       bool operator!=(const Edge edge) const {return id != edge.id;}

       bool operator<(const Edge edge) const {return id < edge.id;}

     };

     void first(Node& node) const {

       node.id = NodeNum-1;

     }

     static void next(Node& node) {

       --node.id;

     }

     void first(Edge& edge) const {

       edge.id = EdgeNum-1;

     }

     static void next(Edge& edge) {

       --edge.id;

     }

     void firstOut(Edge& edge, const Node& node) const {      

       edge.id = node.id != 0 ? node.id * (node.id - 1) / 2 : -1;

     }

     /// \todo with specialized iterators we can make faster iterating

     void nextOut(Edge& edge) const {

       int tail = ((int)sqrt((double)(1 + 8 * e.id)) + 1) / 2;;

       int head = e.id - (tail) * (tail - 1) / 2; 

       ++head;

       return head < tail ? tail * (tail - 1) / 2 + head : -1;

     }

     void firstIn(Edge& edge, const Node& node) const {

       edge.id = node.id * (node.id + 1) / 2 - 1;

     }

     void nextIn(Edge& edge) const {

       int tail = ((int)sqrt((double)(1 + 8 * e.id)) + 1) / 2;;

       int head = e.id - (tail) * (tail - 1) / 2; ++head;

       ++tail;

       return tail < nodeNum ? tail * (tail - 1) / 2 + head : -1;

     }

   };

   /// \todo UndirFullGraph from the UndirFullGraphBase