Eulerian tour

Eulerian tour

问题描述:
      给定一个无向图G,一条路径经过图G的每一条边,且仅经过一次,这条路径称为欧拉路径(Eulerian Tour),如果欧拉路径的起始顶点和终点是同一顶点,则称为欧拉回路(Eulerian circuit).
    算法:
    无向图G存在欧拉路径的充要条件:图G是连通的,且至多除两个点外(可以为0个,连接图不可能有且仅有一个顶点的度为奇数)其它所有顶点的度为偶数.
    无向图G存在欧拉回路的充要条件:图G是连通的且所有顶点的度为偶数;
    算法描述:
    

 1  tour: 数组,用于存储欧拉路径,反序输出即为欧拉路径
 2  pos:  int       
 3 
   find_eulerian_circuit()      
 4  {             
 5     pos = 0 ;            
 6     find_circuit( 1 );      
 7  }     
 8 
    find_eulerian_tour()     
 9  {            
10    find a vertex i ,the degree of which  is  odd           
11    pos = 0 ;            
12    find_circuit(i);    
13  }      
14 
    find_circuit(vertex i)      
15  {         
16      while (exist j,(i,j)  is  the edge of G)          
17     {               
18        remove edge(i,j);                
19         find_circuit(j);           
20      }            
21     tour[pos ++ ] = i;      
22  } 
23 

 

 USACO  3.2 Riding the fence,就是一个求欧拉路径的问题.
 问题描述:    

Farmer John owns a large number of fences that must be repaired annually. He traverses the fences by riding a horse along each and every one of them (and nowhere else) and fixing the broken parts.

Farmer John is as lazy as the next farmer and hates to ride the same fence twice. Your program must read in a description of a network of fences and tell Farmer John a path to traverse each fence length exactly once, if possible. Farmer J can, if he wishes, start and finish at any fence intersection.

Every fence connects two fence intersections, which are numbered inclusively from 1 through 500 (though some farms have far fewer than 500 intersections). Any number of fences (>=1) can meet at a fence intersection. It is always possible to ride from any fence to any other fence (i.e., all fences are "connected").

Your program must output the path of intersections that, if interpreted as a base 500 number, would have the smallest magnitude.

There will always be at least one solution for each set of input data supplied to your program for testing.

PROGRAM NAME: fence

INPUT FORMAT

Line 1:	The number of fences, F (1 <= F <= 1024)
Line 2..F+1:	A pair of integers (1 <= i,j <= 500) that tell which pair of intersections this fence connects.

SAMPLE INPUT (file fence.in)

9

1 2

2 3

3 4

4 2

4 5

2 5

5 6

5 7

4 6

OUTPUT FORMAT

The output consists of F+1 lines, each containing a single integer. Print the number of the starting intersection on the first line, the next intersection's number on the next line, and so on, until the final intersection on the last line. There might be many possible answers to any given input set, but only one is ordered correctly.

SAMPLE OUTPUT (file fence.out)

1

2

3

4

2

5

4

6

5

7

   解答:简单的欧拉路径问题,图采用邻接表存储,附原码

  

/*
ID: kuramaw1
PROG: fence
LANG: C++
*/

#include  < fstream >

using  std::ifstream;
using  std::ofstream;
using  std::endl;

#ifdef _DEBUG
#include  < iostream >
using  std::cout;
#endif

#define   MAX_V 500
#define   MAX_EDGE 1025

#define   MAX(a,b) ((a)>(b)?(a):(b))

struct  grapha
{
     struct  node
    {
         short  v;
        node  *  next;
        node( const   short  _v =- 1 ):v(_v),next(NULL)
        {

        }
        
    };

     struct  ver
    {
        node  *  r;
         short  d; // degree
        ver():d( 0 )
        {
            r = new  node();

        }
         ~ ver()
        {
            node  *  n = r;
             while (n != NULL)
            {
                node  *  t = n;
                n = n -> next;
                delete t;
            }
        }
        inline  void  add_neighbor( const   short   & v)
        {
            node  *  t = new  node(v);
            node  *  p = r;
            node  *  n = p -> next;
             while (n != NULL  &&  v > n -> v)
            {
                p = n;
                n = n -> next;
            }
            p -> next = t;
            t -> next = n;
            d ++ ;
        }
        inline  void   remove_neighbor( const   short   & v)
        {
            node  *  p = r;
            node  *  n = p -> next;
             while (n != NULL  &&  v != n -> v)
            {
                p = n;
                n = n -> next;
            }
             if (n != NULL)
            {
                p -> next = n -> next;
                delete n;
                d -- ;
            }

        }
    };

    ver v[MAX_V];
     short  n;
     short   *  tour;
     short  pos;

    grapha():n( 0 ),tour(NULL)
    {

    }

     void  add_edge( const   short    & _u, const   short   & _v)
    {
        v[_u - 1 ].add_neighbor(_v - 1 );
        v[_v - 1 ].add_neighbor(_u - 1 );
         short  t = MAX(_u,_v);
         if (t > n)
            n = t;
    }

     void  find_tour( const   short   & s)
    {
             while (v[s].d > 0 )
            {
                 short  j = v[s].r -> next -> v;
                v[s].remove_neighbor(j);
                v[j].remove_neighbor(s);
                find_tour(j);
            }
            tour[pos ++ ] = s + 1 ;

    }

     void  Eulerian_tour( short   *  _tour)
    {
        tour = _tour;
        pos = 0 ;
         bool  b = false ;
         for ( int  i = 0 ;i < n;i ++ )
          if (v[i].d  %   2 != 0 )
         {
             find_tour(i);
             b = true ;
              break ;
         }
        if ( ! b)
           find_tour( 0 );

    }


};

grapha g;
short  tour[MAX_EDGE];
short  f;
int  main()
{
    ifstream  in ( " fence.in " );
     in >> f;
     for ( short  i = 0 ;i < f;i ++ )
    {
         short  u,v;
         in >> u >> v;
        g.add_edge(u,v);
    }

     // do
    g.Eulerian_tour(tour);



     // out
    ofstream  out ( " fence.out " );
     for ( int  i = f;i >= 0 ;i -- )
      out << tour[i] << endl;
     out .close();
}