PAT_1122

1122 Hamiltonian Cycle

The “Hamilton cycle problem” is to find a simple cycle that contains every vertex in a graph. Such a cycle is called a “Hamiltonian cycle”.

In this problem, you are supposed to tell if a given cycle is a Hamiltonian cycle.

Input Specification:

Each input file contains one test case. For each case, the first line contains 2 positive integers N (2<N≤200), the number of vertices, and M, the number of edges in an undirected graph. Then M lines follow, each describes an edge in the format Vertex1 Vertex2, where the vertices are numbered from 1 to N. The next line gives a positive integer K which is the number of queries, followed by K lines of queries, each in the format:

n V1 V2 … Vn

where n is the number of vertices in the list, and Vi’s are the vertices on a path.

Output Specification:

For each query, print in a line YES if the path does form a Hamiltonian cycle, or NO if not.

Sample Input:

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
6 10
6 2
3 4
1 5
2 5
3 1
4 1
1 6
6 3
1 2
4 5
6
7 5 1 4 3 6 2 5
6 5 1 4 3 6 2
9 6 2 1 6 3 4 5 2 6
4 1 2 5 1
7 6 1 3 4 5 2 6
7 6 1 2 5 4 3 1

Sample Output:

1
2
3
4
5
6
YES
NO
NO
NO
YES
NO

思路

要求判断给出的路径是否为哈密顿回路,那么哈密顿回路应该满足:

  1. 满足回路,即路径首尾相同
  2. 经过所有节点且经过一次
  3. 路径必须是连通的,即路径中相邻节点必须有路

程序

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	#include <iostream>
#include <cstdio>
#include <cstdlib>
#include <cstring>
#include <algorithm>
#include <set>
#include <queue>
#include <cmath>
#include <map>
#define INF 0x3f3f3f3f

using namespace std;

const int maxn = 205;
typedef long long ll;
int n,m,mapp[maxn][maxn],k,path[maxn*2];
bool used[maxn*2];

bool Judge(int z)
{
int num = 0;
for(int i = 0;i < z;i ++)
if(!used[path[i]])
used[path[i]] = true,num ++;
return num == n && z == n+1;
}
int main()
{
int x,y,z;
scanf("%d%d",&n,&m);
for(int i =0 ;i < m;i ++)
{
scanf("%d%d",&x,&y);
mapp[x][y] = mapp[y][x] = 1;
}
scanf("%d",&k);
for(int i =0 ;i < k;i ++)
{
scanf("%d",&z);
memset(used,false,sizeof(used));
for(int j = 0;j < z;j ++)
scanf("%d",path+j);
bool flag = true;
if(path[0] == path[z-1] && Judge(z))
{
for(int j = 1;j < z;j ++)
{
if(mapp[path[j]][path[j-1]] == 0)
{
flag = false;
break;
}
}
}
else
{
flag = false;
}
if(flag)
printf("YES\n");
else
printf("NO\n");
}
return 0;
}

参考资料

哈密顿回路 -百度百科

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