poj1734

Sightseeing trip

Time Limit: 1000MS		Memory Limit: 65536K
Total Submissions: 9078		Accepted: 3380		Special Judge

Description

There is a travel agency in Adelton town on Zanzibar island. It has decided to offer its clients, besides many other attractions, sightseeing the town. To earn as much as possible from this attraction, the agency has accepted a shrewd decision: it is necessary to find the shortest route which begins and ends at the same place. Your task is to write a program which finds such a route.

In the town there are N crossing points numbered from 1 to N and M two-way roads numbered from 1 to M. Two crossing points can be connected by multiple roads, but no road connects a crossing point with itself. Each sightseeing route is a sequence of road numbers y_1, ..., y_k, k>2. The road y_i (1<=i<=k-1) connects crossing points x_i and x_{i+1}, the road y_k connects crossing points x_k and x_1. All the numbers x_1,...,x_k should be different.The length of the sightseeing route is the sum of the lengths of all roads on the sightseeing route, i.e. L(y_1)+L(y_2)+...+L(y_k) where L(y_i) is the length of the road y_i (1<=i<=k). Your program has to find such a sightseeing route, the length of which is minimal, or to specify that it is not possible,because there is no sightseeing route in the town.

Input

The first line of input contains two positive integers: the number of crossing points N<=100 and the number of roads M<=10000. Each of the next M lines describes one road. It contains 3 positive integers: the number of its first crossing point, the number of the second one, and the length of the road (a positive integer less than 500).

Output

There is only one line in output. It contains either a string 'No solution.' in case there isn't any sightseeing route, or it contains the numbers of all crossing points on the shortest sightseeing route in the order how to pass them (i.e. the numbers x_1 to x_k from our definition of a sightseeing route), separated by single spaces. If there are multiple sightseeing routes of the minimal length, you can output any one of them.

Sample Input

Sample Output

1 3 5 2

Source

CEOI 1999

题目大意：给定图的N个点M条边，求出图中的最小环(无向图，有重边)。

解题思路：

int maxn=105;

int a[maxn][maxn],f[maxn][maxn];

a:邻接矩阵，存图

利用floyd算法；

f：记录任意两点间的最短距离，初值为a.

f^(k)[i][j]表示从顶点i到顶点j，中间顶点序号不大于k的最短路径长度。

f^(k)[i][j]=min(f^(k-1)[i][j],f^(k-1)[i][k]+f^(k-1)[k][j])

则最小环可以表示为a[i][k]+a[k][j]+f^(k-1)[i][j]

即表示从顶点i到顶点j，中间顶点序号不大于k-1的最短路径长度+i到k的边长+k到j的边长。（这样保证构成环，而没有重边）

#include<iostream>
#include<cstring>
using namespace std;
int n,m,ans=0x3f3f3f3f,s,t,temk=0x3f3f3f3f,cnt;
const int maxn=1000;
int a[maxn][maxn],d[maxn][maxn],f[maxn][maxn],path[maxn];
void dfs(int i,int j){
    if(f[i][j]==0){path[++cnt]=j;return;}
    dfs(f[i][j],j);
}
void floy(){    
    memset(path,0,sizeof(path));
    for(int i=1;i<=n;i++)
        for(int j=1;j<=n;j++)
            d[i][j]=a[i][j];
    for(int k=1;k<=n;k++){
        for(int i=1;i<k;i++)
            for(int j=i+1;j<k;j++)
                if((long long)a[i][k]+a[k][j]+d[i][j]<ans){//注意数据类型，3个连加，容易超Int 
                    ans=a[i][k]+a[k][j]+d[i][j];
                    s=i;t=j;
                    temk=k;
                    cnt=0;
                    path[++cnt]=s;    
                    dfs(s,t);//记录从s到t的中间节点，包含t,不含s. 
                    path[++cnt]=k;
                    
                }
            
        for(int i=1;i<=n;i++)
            for(int j=1;j<=n;j++)
                if(d[i][j]>d[i][k]+d[k][j]){
                    d[i][j]=d[i][k]+d[k][j];
                    f[i][j]=k;
            }
        }
        return ;
}
int main(){
    memset(a,0x3f,sizeof(a));
    memset(f,0,sizeof(f));
    cin>>n>>m;
    for(int i=1;i<=n;i++) a[i][i]=0;
    for(int i=1;i<=m;i++){
        int x,y,w;
        cin>>x>>y>>w;
        if(w<a[x][y]){
            a[x][y]=a[y][x]=w;
        }
    }
    floy();
    if(temk==0x3f3f3f3f)cout<<"No solution."<<endl;
    else {for(int i=1;i<=cnt;i++) cout<<path[i]<<' ';cout<<endl;}
    return 0;
}