URAL 1614. National Project “Trams” （图论大YY）

1614. National Project “Trams”

Time limit: 0.5 second
Memory limit: 64 MB

President has declared the development of tram service a priority national project. As a part of this project, Yekaterinburg will receive enough budget funds to carry out a complete reconstruction of the city's tram network.

There are 2 n tram stops in Yekaterinburg. In the course of reconstruction, the stops will be left in their places and no new stops will be built, but new tram railways will be laid so that it will be possible to go by tram from every tram stop to any other tram stop without any intermediate stops.

Having studied messages at the tram forum, the city authorities found out that citizens would be satisfied with the reconstruction only if for every pair of tram stops there would be a tram route connecting these stops without any intermediate stops. It is clear that the network of  n(2 n − 1) routes consisting of only two stops each satisfies this requirement. However, Yekaterinburg Mayor wants exactly  n routes to be left after the reconstruction, and each tram must make exactly 2 n stops (including the final ones) on its route. Trams must go along these routes in both directions. Suggest a plan of reconstruction that will satisfy both citizens and Mayor.

Input

The only input line contains the integer  n,  1 ≤  n ≤ 100.

Output

Output  n lines describing tram routes. Each route is a sequence of integers in the range from 1 to 2 n separated by a space. A route may go through a stop several times. If the problem has several solutions, you may output any of them. If there is no solution, output one line containing the number −1.

Sample

input	output
3	1 6 2 1 3 4 2 3 6 5 4 6 5 1 4 2 5 3

Problem Author: Alexander Ipatov (idea — Magaz Asanov)
Problem Source: The 12th Urals Collegiate Programing Championship, March 29, 2008

Tags:  tricky problem   ( hide tags for unsolved problems )

Difficulty: 731     Printable version     Submit solution     Discussion (4)
My submissions     All submissions (1014)     All accepted submissions (481)    Solutions rating (383)

题意:有N=2*n阶完全图，找n条路径，每条路径有2*n-1条边，正好n条路组成了完全图

思路：YY，每个顶点的入度为N-1奇数，所以他们要分别做端点一次，分别做中间点n次

所以按端点对称构造路径，比如N=6，分别以（1，4）（2，6）（3，5）作对称点，这样3条路径的形状都一样

没条路再经过其他点各一次即可，比如构造出：1->2->6->3->5->4这种交错形状的路径，构造方式不止一种

#include<cstdio>
#include<iostream>
#include<iostream>
#include<cstring>
using namespace std;
int path[110];  //构造一条形状相同的路径
int main()
{
    int n;
    while(~scanf("%d",&n))
    {
        int t=2*n+2,cnt=0,l=1;
        //对称构造
        while(cnt!=2*n)
        {
            path[++cnt]=((t-l)-1)%(2*n)+1;
            path[++cnt]=++l;
        }
        //输出
        for(int i=1;i<=n;i++)
            for(int j=1;j<=2*n;j++)
            {
                printf("%d%c",(path[j]+i-2)%(2*n)+1,j==2*n? '\n':' ');
            }
    }
    return 0;
}

大神的构造方法：

#include <cstdio>
#include <cstring>
#include <iostream>
using namespace std;

int N, M;

int main() {
    int i, j, k;
    cin>>N;
    M = N*2;
    for (i = 0; i < N; ++i) {
        int now = i;
        printf("%d", now+1);
        for (j = 1; j < M; ++j) {
            now = (now+j)%M;
            printf(" %d", now+1);
        }
        puts("");
    }
    return 0;
}