目次
1126 Eulerian Path 
タイトルの説明
問題を解く構想.
プログラム
vectorにおけるresizeとreserve関数の違い
ブログ参照

1126 Eulerian Path 

タイトルの説明

In graph theory, an Eulerian path is a path in a graph which visits every edge exactly once. Similarly, an Eulerian circuit is an Eulerian path which starts and ends on the same vertex. They were first discussed by Leonhard Euler while solving the famous Seven Bridges of Konigsberg problem in 1736. It has been proven that connected graphs with all vertices of even degree have an Eulerian circuit, and such graphs are called Eulerian. If there are exactly two vertices of odd degree, all Eulerian paths start at one of them and end at the other. A graph that has an Eulerian path but not an Eulerian circuit is called semi-Eulerian. (Cited from https://en.wikipedia.org/wiki/Eulerian_path)
Given an undirected graph, you are supposed to tell if it is Eulerian, semi-Eulerian, or non-Eulerian.

Input Specification:

Each input file contains one test case. Each case starts with a line containing 2 numbers N (≤ 500), and M, which are the total number of vertices, and the number of edges, respectively. Then M lines follow, each describes an edge by giving the two ends of the edge (the vertices are numbered from 1 to N).

Output Specification:

For each test case, first print in a line the degrees of the vertices in ascending order of their indices. Then in the next line print your conclusion about the graph -- either  Eulerian ,  Semi-Eulerian , or  Non-Eulerian . Note that all the numbers in the first line must be separated by exactly 1 space, and there must be no extra space at the beginning or the end of the line.

Sample Input 1:

7 12
5 7
1 2
1 3
2 3
2 4
3 4
5 2
7 6
6 3
4 5
6 4
5 6

Sample Output 1:

2 4 4 4 4 4 2
Eulerian

Sample Input 2:

6 10
1 2
1 3
2 3
2 4
3 4
5 2
6 3
4 5
6 4
5 6

Sample Output 2:

2 4 4 4 3 3
Semi-Eulerian

Sample Input 3:

5 8
1 2
2 5
5 4
4 1
1 3
3 2
3 4
5 3

Sample Output 3:

3 3 4 3 3
Non-Eulerian

問題を解く構想.

オーラ図と半オーラ図:まず図が連通図であることを決定する
図Gにこのような経路が存在し、Gの各辺をちょうど1回通過するようにすれば、この経路をオーラ経路と呼ぶ.この経路が1つの回路であれば、オーラ戻り路となる.オーラ戻り路を有する図はオーラ図となり、オーラ経路を有するが、オーラ戻り路を有さない図は半オーラ図となる
無方向連通図では、度数が奇数の点の個数が0であればオーラ図、度数が奇数の点の個数が2であれば半オーラ図
有方向連通図の場合、各ノードの入度が出度に等しい場合にのみ、オーラ図
2つのノードを除いてのみ、残りのノードの入度が出度に等しく、この2つのノードが起点sの入度=出度-1を満たし、終了点tの出度=入度-1を満たす場合、半オーラ図

プログラム

#include 
#include 
#include 
#include 
#include 
#include 
#include 
#include 
#include 
#include 
#define INF 0x3f3f3f3f

using namespace std;
const int maxn = 550;
int degree[maxn],n,m,flag[maxn],cnt;
vector> grep;
void dfs(int pos)
{
    flag[pos] = 1;
    cnt++;
    for(int it : grep[pos])
        if(!flag[it])
            dfs(it);
}
int main()
{
    int x,y;
    scanf("%d%d",&n,&m);
    grep.resize(n+1);
    for(int i =0 ;i < m;i ++)
    {
        scanf("%d%d",&x,&y);
        degree[x] ++;degree[y]++;
        grep[x].push_back(y);
        grep[y].push_back(x);
    }
    int num = 0;
    bool flag = true;
    for(int i = 1;i <= n;i ++)
    {
        printf("%d%c",degree[i],i==n?'
':' ');
        if(degree[i] % 2 == 1)
            num++;
    }
    dfs(1);
    if(num == 0 && cnt == n)
        printf("Eulerian
");
    else if(num == 2 && cnt == n)
        printf("Semi-Eulerian
");
    else
        printf("Non-Eulerian
");
    return 0;
}

vectorにおけるresizeとreserve関数の違い

resizeは変化のsizeであり、reserveはcapacityである.
resizeのソースプログラムは次のとおりです.

  iterator erase(iterator first, iterator last) {              //          [first,last)
    iterator i = copy(last, finish, first);                    // 
    destroy(i, finish);                                        // 
    finish = finish - (last - first);                          // 
    return first;
  }
  void resize(size_type new_size, const T& x) {
    if (new_size < size())                                  // 
      erase(begin() + new_size, end());
    else
      insert(end(), new_size - size(), x);                  // end new_size-size() ， resize capacity 
  }
  void resize(size_type new_size) { resize(new_size, T()); }

reserveのソースプログラムは次のとおりです.

  void reserve(size_type n) {                                 // n( )， ， 
    if (capacity() < n) {
      const size_type old_size = size();                      //   
      iterator tmp = allocate_and_copy(n, start, finish);     // n ， strat finish 
      destroy(start, finish);                                 // 
      deallocate();                                           // 
      start = tmp;
      finish = tmp + old_size;
      end_of_storage = start + n;                            
    }
  }

ブログ参照

オーラ図とセミオーラ図