POJ2391：Ombrophobic Bovines(最大流+Floyd+二分)

Ombrophobic Bovines

Time Limit: 1000MSMemory Limit: 65536K

Total Submissions: 21660Accepted: 4658

题目链接：http://poj.org/problem?id=2391

Description:

FJ's cows really hate getting wet so much that the mere thought of getting caught in the rain makes them shake in their hooves. They have decided to put a rain siren on the farm to let them know when rain is approaching. They intend to create a rain evacuation plan so that all the cows can get to shelter before the rain begins. Weather forecasting is not always correct, though. In order to minimize false alarms, they want to sound the siren as late as possible while still giving enough time for all the cows to get to some shelter. 

The farm has F (1 <= F <= 200) fields on which the cows graze. A set of P (1 <= P <= 1500) paths connects them. The paths are wide, so that any number of cows can traverse a path in either direction. 

Some of the farm's fields have rain shelters under which the cows can shield themselves. These shelters are of limited size, so a single shelter might not be able to hold all the cows. Fields are small compared to the paths and require no time for cows to traverse. 

Compute the minimum amount of time before rain starts that the siren must be sounded so that every cow can get to some shelter.

Input:

* Line 1: Two space-separated integers: F and P 

* Lines 2..F+1: Two space-separated integers that describe a field. The first integer (range: 0..1000) is the number of cows in that field. The second integer (range: 0..1000) is the number of cows the shelter in that field can hold. Line i+1 describes field i. 

* Lines F+2..F+P+1: Three space-separated integers that describe a path. The first and second integers (both range 1..F) tell the fields connected by the path. The third integer (range: 1..1,000,000,000) is how long any cow takes to traverse it.

Output:

* Line 1: The minimum amount of time required for all cows to get under a shelter, presuming they plan their routes optimally. If it not possible for the all the cows to get under a shelter, output "-1".

Sample Input:

3 4
7 2
0 4
2 6
1 2 40
3 2 70
2 3 90
1 3 120

Sample Output:

110

题意：

给出n个点，m条无向路径，现在每个点都有一定数量的牛，然后我们知道每个点都能承装的最多的牛，以及经过一条路径需要的时间。

现在每头牛都要迁徙，问最少需要多少时间所有的牛都可以成功跑进点中，满足题中给出的条件。

 

题解：

考虑网络流，将点拆开，每两个点之间连一条边，容量为这两个点之间的花费最小时间。最小花费由floyd易求。

源点连点，容量为这个点有多少头牛；汇点连点，容量为这个点最多能容纳多少头牛。

显然时间越多牛就越可能全部到达，但我们这里要求的最大时间最小，可以跑个最小费用最大流，但时间复杂度有点高。

我们就考虑二分时间，然后利用二分的时间来限制前往哪些点，之后跑最大流就好了。

 

代码如下：

#include <cstdio>
#include <cstring>
#include <algorithm>
#include <iostream>
#include <queue>
#define s 0
#define t 2*n+1
#define INF 1e9
using namespace std;
typedef long long ll;
const int N = 505, M = 1505;
int n,m,tot,need;
int lim[N],has[N],head[N],d[N];
ll mp[N][N];
struct Edge{
    int v,next,c;
}e[(N*N)<<1];
void adde(int u,int v,int c){
    e[tot].v=v;e[tot].c=c;e[tot].next=head[u];head[u]=tot++;
    e[tot].v=u;e[tot].c=0;e[tot].next=head[v];head[v]=tot++;
}
bool bfs(int S,int T){
    memset(d,0,sizeof(d));d[S]=1;
    queue <int > q;q.push(S);
    while(!q.empty()){
        int u=q.front();q.pop();
        for(int i=head[u];i!=-1;i=e[i].next){
            int v=e[i].v;
            if(!d[v] && e[i].c>0){
                d[v]=d[u]+1;
                q.push(v);
            }
        }
    }
    return d[T]!=0;
}
int dfs(int S,int a){
    int flow=0,f;
    if(S==t || a==0) return a;
    for(int i=head[S];i!=-1;i=e[i].next){
        int v=e[i].v;
        if(d[v]!=d[S]+1) continue ;
        f=dfs(v,min(a,e[i].c));
        if(f){
            e[i].c-=f;
            e[i^1].c+=f;
            flow+=f;
            a-=f;
            if(a==0) break;
        }
    }
    if(!flow) d[S]=-1;
    return flow;
}
int Dinic(){
    int max_flow=0;
    while(bfs(0,t)) max_flow+=dfs(0,INF);
    return max_flow;
}
bool check(ll x){
    memset(head,-1,sizeof(head));tot=0;
    for(int i=1;i<=n;i++) adde(s,i,has[i]);
    for(int i=n+1;i<=2*n;i++) adde(i,t,lim[i-n]);
    for(int i=1;i<=n;i++)
        for(int j=1;j<=n;j++)
            if(mp[i][j]<=x) adde(i,j+n,INF);
    int flow = Dinic();
    for(int i=head[s];i!=-1;i=e[i].next){
        int now = e[i].c;
        if(now>0) return false;
    }
    return true ;
}
int main(){
    scanf("%d%d",&n,&m);
    for(int i=1;i<=n;i++) scanf("%d%d",&has[i],&lim[i]);
    for(int i=1;i<=n;i++)
        for(int j=1;j<=n;j++)
            mp[i][j]=i==j ? 0 : 1e18;
    for(int i=1;i<=m;i++){
        int u,v;ll w;
        scanf("%d%d%I64d",&u,&v,&w);
        ll tmp = mp[u][v];
        mp[u][v]=mp[v][u]=min(tmp,w);
    }
    for(int k=1;k<=n;k++)
        for(int i=1;i<=n;i++)
            for(int j=1;j<=n;j++)
                if(mp[i][j]>mp[i][k]+mp[k][j] && mp[i][k]!=1e18 && mp[k][j]!=1e18)
                    mp[i][j]=mp[i][k]+mp[k][j];
    ll l = 0,r = 1e18,mid;
    while(l<r){
        mid = (l+r)>>1;
        if(check(mid)) r=mid;
        else l=mid+1;
    }
    if(r==1e18) cout<<-1;
    else cout<<l<<endl;
    return 0;
}