问题描述
The competitors of Bubble Cup X gathered after the competition and discussed what is the best way to get to know the host country and its cities.
After exploring the map of Serbia for a while, the competitors came up with the following facts: the country has V cities which are indexed with numbers from 1 to V, and there are E bi-directional roads that connect the cites. Each road has a weight (the time needed to cross that road). There are N teams at the Bubble Cup and the competitors came up with the following plan: each of the N teams will start their journey in one of the V cities, and some of the teams share the starting position.
They want to find the shortest time T, such that every team can move in these T minutes, and the number of different cities they end up in is at least K (because they will only get to know the cities they end up in). A team doesn't have to be on the move all the time, if they like it in a particular city, they can stay there and wait for the time to pass.
Please help the competitors to determine the shortest time T so it's possible for them to end up in at least K different cities or print -1 if that is impossible no matter how they move.
Note that there can exist multiple roads between some cities.
输入格式
The first line contains four integers: V, E, N and K (1 ≤ V ≤ 600, 1 ≤ E ≤ 20000, 1 ≤ N ≤ min(V, 200), 1 ≤ K ≤ N), number of cities, number of roads, number of teams and the smallest number of different cities they need to end up in, respectively.
The second line contains N integers, the cities where the teams start their journey.
Next E lines contain information about the roads in following format: Ai Bi Ti (1 ≤ Ai, Bi ≤ V, 1 ≤ Ti ≤ 10000), which means that there is a road connecting cities Ai and Bi, and you need Ti minutes to cross that road.
输出格式
Output a single integer that represents the minimal time the teams can move for, such that they end up in at least K different cities or output -1 if there is no solution.
If the solution exists, result will be no greater than 1731311.
样例输入
6 7 5 4
5 5 2 2 5
1 3 3
1 5 2
1 6 5
2 5 4
2 6 7
3 4 11
3 5 3
样例输出
3
样例解释
Three teams start from city 5, and two teams start from city 2. If they agree to move for 3 minutes, one possible situation would be the following: Two teams in city 2, one team in city 5, one team in city 3 , and one team in city 1. And we see that there are four different cities the teams end their journey at.
题目大意
给定一个 v个点 e条边的带权无向图，在图上有 n个人，第 i个人位于点 xi，一个人通过一条边需要花费这条边的边权的时间。
现在每个人可以自由地走。求最短多少时间后满足结束后有人的节点数 ≥ m
解析
观察到最后的答案就是走过的最长时间。那么，这就变成了一个最大值最小的问题，可以用二分答案解决。
二分需要的时间mid。因为最后是至少m做城市有人，所以不妨当做是用m个人去匹配m座城市，那么就变成了一个二分图匹配问题。对于每个人，向他所在的城市在mid时间内可以到达的城市连边，这可以用Floyd求出两两最短路得到。然后二分图匹配，如果匹配数大于等于m说明可行。
代码
#include <iostream>
#include <cstdio>
#include <cstring>
#define N 1202
using namespace std;
int v,e,n,m,i,j,k,dis[N][N],g[N][N],pos[N],match[N];
bool vis[N];
int read()
{
char c=getchar();
int w=0;
while(c<'0'||c>'9') c=getchar();
while(c<='9'&&c>='0'){
w=w*10+c-'0';
c=getchar();
}
return w;
}
bool dfs(int x)
{
for(int y=1;y<=v;y++){
if(g[x][y]&&!vis[y]){
vis[y]=1;
if(!match[y]||dfs(match[y])){
match[y]=x;
return 1;
}
}
}
return 0;
}
int hungary()
{
memset(match,0,sizeof(match));
int ans=0;
for(int i=1;i<=n;i++){
memset(vis,0,sizeof(vis));
if(dfs(i)) ans++;
}
return ans;
}
bool check(int x)
{
memset(g,0,sizeof(g));
for(int i=1;i<=n;i++){
for(int j=1;j<=v;j++){
if(dis[pos[i]][j]<=x) g[i][j]=1;
}
}
int ans=hungary();
return (ans>=m);
}
int main()
{
v=read();e=read();n=read();m=read();
for(i=1;i<=n;i++) pos[i]=read();
memset(dis,0x3f,sizeof(dis));
for(i=1;i<=v;i++) dis[i][i]=0;
for(i=1;i<=e;i++){
int u=read(),v=read(),w=read();
dis[u][v]=dis[v][u]=min(dis[u][v],w);
}
for(k=1;k<=v;k++){
for(i=1;i<=v;i++){
for(j=1;j<=v;j++) dis[i][j]=min(dis[i][j],dis[i][k]+dis[k][j]);
}
}
int l=0,r=1731311,mid,ans=-1;
while(l<=r){
mid=(l+r)/2;
if(check(mid)){
ans=mid;
r=mid-1;
}
else l=mid+1;
}
printf("%d
",ans);
return 0;
}