URAL1004 Sightseeing Trip(Folyd求最小环，打印路径)

题目：

1004. Sightseeing Trip

Time limit: 0.5 second
Memory limit: 64 MB

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 y1, …, yk, k > 2. The road yi (1 ≤ i ≤ k − 1) connects crossing pointsxi and xi+1, the road yk connects crossing points xk and x1. All the numbers x1, …, xk 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(y1) + L(y2) + … + L(yk) where L(yi) is the length of the road yi (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

Input contains T tests (1 ≤ T ≤ 5). The first line of each test contains two integers: the number of crossing points N and the number of roads M (3 ≤ N ≤ 100; 3 ≤ M ≤ N · (N − 1)). Each of the next M lines describes one road. It contains 3 integers: the number of its first crossing point a, the number of the second one b, and the length of the road l (1 ≤ a, b ≤ N; a ≠ b;1 ≤ l ≤ 300). Input is ended with a “−1” line.

Output

Each line of output is an answer. 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 x1 to xk 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

inputoutput

5 71 4 11 3 3003 1 101 2 162 3 1002 5 155 3 204 31 2 101 3 201 4 30-1

1 3 5 2No solution.

Problem Source: Central European Olympiad in Informatics 1999

思路：

题意给了一张无向图，求的是这个图的最小环（就是从某个点出发再走回它自己的最短路），并且把路径打印出来。

所以我们求出一个最短路+次短路的和就是所求的路径.我们以环权值最小为标准（最短路权值+次短路权值），记录从i~j的路径以及能松弛i和j点的k中代价最小的那个,一直更新路径，如果遇见更短的路径那么就放弃之前记录的重新记录.floyd是按照结点的顺序更新最短路的，所以我们在更新最短路之前先找到一个连接点k，当前的点k肯定不存在于已存在的最短路dis[i][j]的路径上，因为我们还没用这个k去更新最短路，这样一个环就找到了

map[]来存储原图

dis[]来存储松弛后的图

path[]记录路径

vis[][]标记两个点之间经过哪个点中转过

ans来存储当前最小的环的值，如果遇到更小的，那么就重新记录路径

代码：

#include <cstdio>#include <cstring>#include <cstdlib>#include <cmath>#include <string>#include <iostream>#include <stack>#include <queue>#include <vector>#include <algorithm>#define mem(a,b) memset(a,b,sizeof(a))#define N 110#define inf 9999999#define ll long longusing namespace std;int n,m,ans,cnt;int map[N][N],dis[N][N];int vis[N][N],path[N];void init(){  for(int i=1; i<=n; i++)      for(int j=1; j<=n; j++)      {          dis[i][j]=inf;          vis[i][j]=0;      }}void record(int i,int j)//递归记录路径{  if(vis[i][j])  {      record(i,vis[i][j]);      record(vis[i][j],j);  }  else      path[cnt++]=j;}void floyd(){  ans=inf;  for(int k=1; k<=n; k++)  {      //找出最小环并且记录路径      for(int i=1; i<k; i++)          for(int j=i+1; j<k; j++)              if(ans>dis[i][j]+map[i][k]+map[k][j])//如果最短路加上次短路权值可以成环，就记录              {                  ans=dis[i][j]+map[i][k]+map[k][j];                  cnt=0;                  path[cnt++]=i;                  record(i,j);//回溯记录路径                  path[cnt++]=k;              }      //正常的folyd      for(int i=1; i<=n; i++)          for(int j=1; j<=n; j++)              if(dis[i][j]>dis[i][k]+dis[k][j])              {                  dis[i][j]=dis[i][k]+dis[k][j];                  vis[i][j]=k;              }  }}int main(){  int u,v,w;  while(~scanf("%d",&n)&&n!=-1)  {      scanf("%d",&m);      init();      for(int i=1; i<=m; i++)      {          scanf("%d%d%d",&u,&v,&w);          if(w<dis[u][v])              dis[u][v]=dis[v][u]=w;      }      for(int i=1; i<=n; i++)          for(int j=1; j<=n; j++)              map[i][j]=dis[i][j];      floyd();      if(ans==inf)          puts("No solution.");      else      {          printf("%d",path[0]);          for(int i=1; i<cnt; i++)              printf(" %d",path[i]);          puts("");      }  }  return 0;}