poj 1149 PIGS【最大流经典建图】

PIGS

Time Limit: 1000MS		Memory Limit: 10000K
Total Submissions: 18727		Accepted: 8508

Description

Mirko works on a pig farm that consists of M locked pig-houses and Mirko can't unlock any pighouse because he doesn't have the keys. Customers come to the farm one after another. Each of them has keys to some pig-houses and wants to buy a certain number of pigs. 
All data concerning customers planning to visit the farm on that particular day are available to Mirko early in the morning so that he can make a sales-plan in order to maximize the number of pigs sold. 
More precisely, the procedure is as following: the customer arrives, opens all pig-houses to which he has the key, Mirko sells a certain number of pigs from all the unlocked pig-houses to him, and, if Mirko wants, he can redistribute the remaining pigs across the unlocked pig-houses. 
An unlimited number of pigs can be placed in every pig-house. 
Write a program that will find the maximum number of pigs that he can sell on that day.

Input

The first line of input contains two integers M and N, 1 <= M <= 1000, 1 <= N <= 100, number of pighouses and number of customers. Pig houses are numbered from 1 to M and customers are numbered from 1 to N. 
The next line contains M integeres, for each pig-house initial number of pigs. The number of pigs in each pig-house is greater or equal to 0 and less or equal to 1000. 
The next N lines contains records about the customers in the following form ( record about the i-th customer is written in the (i+2)-th line): 
A K1 K2 ... KA B It means that this customer has key to the pig-houses marked with the numbers K1, K2, ..., KA (sorted nondecreasingly ) and that he wants to buy B pigs. Numbers A and B can be equal to 0.

Output

The first and only line of the output should contain the number of sold pigs.

Sample Input

3 3
3 1 10
2 1 2 2
2 1 3 3
1 2 6

Sample Output

7
 
网络流问题的难点还是在于建图
 
题意：一个工人在养猪场工作，他负责卖猪，现在有m个猪圈和n个顾客，每个猪圈的容量没有限制，但是工人没有猪圈的钥匙，而前来的顾客有这些猪圈中一些猪圈的钥匙，每个顾客来之后会打开所有他有钥匙的猪圈，然后买完自己想要买的猪的个数后，关闭所有他打开的猪圈，顾客总是一个接一个的来，当猪圈打开的时候，工人可以随意挪动打开的猪圈中的猪（只能在打开的猪圈中相互移动）问工人一天最多卖出去多少猪
 
输入：开头第一行两个整数m，n分别代表猪圈数和顾客数，接下来一行m个数分别表示对应的猪圈中猪的个数，在接下来n行每行第一个数a，表示接下来a个数意思是当前行（第i行）i顾客所持的猪圈的钥匙，最后一个数b代表这个顾客要买b头猪
 
题解：1、超级源点连接第一个到达j猪圈的顾客  权值为第j个猪圈的猪的头数，
2、因为每个同时打开的猪圈之间的猪可以相互走动，所以所有连接到j猪圈的顾客相互连接容量为无穷大
3、处理重边，比如说第一个猪圈的第一个顾客和第二个猪圈的第一个顾客都是cus1，第三个猪圈的第一个顾客是cus2，所以就有源点到cus1的重边，此时将这两条边合并即可，流量为二者流量和
4、每一个顾客连接超级汇点

#include<stdio.h>
#include<string.h>
#include<stack>
#include<queue>
#include<algorithm>
#include<vector>
#define MAXM 100100
#define MAX 10010
#define INF 0x7ffff
using namespace std;
vector<int>map[MAXM];
int n,m,a,b[MAX];
int ans,head[MAX];
int cur[MAX],dis[MAX];
int vis[MAX];
int pig[MAX];
struct node
{
	int from,to,cap,flow,next;
}edge[MAXM];
void init()
{
	ans=0;
	memset(head,-1,sizeof(head));
	for(int i=1;i<=m;i++)
	    map[i].clear();
}
void add(int u,int v,int w)
{
	int i,j;
	for(i=head[u];i!=-1;i=edge[i].next)
	{
		if(edge[i].to==v) //判断是否有重边
		    break;
	}
	if(i==-1)//i==-1表示当前这个点并没有边与其相连
	{
		edge[ans]={u,v,w,0,head[u]};
		head[u]=ans++;
		edge[ans]={v,u,0,0,head[v]};
		head[v]=ans++;
	}
	else//如果有重边  则将流量合并
	    if(w!=INF)
	        edge[i].cap+=w;
}
void input()
{
	int i,j,key;
	for(i=1;i<=m;i++)
		scanf("%d",&pig[i]);
    for(i=1;i<=n;i++)
    {
    	scanf("%d",&a);
    	while(a--)
    	{
    		scanf("%d",&key);
    		map[key].push_back(i);//存储先后到第可以个猪圈的顾客编号
    	}
    	scanf("%d",&b[i]);
    }
}
void getmap()
{
    int i,j;
    for(i=1;i<=m;i++)
	{
		int u=map[i][0];
    	add(0,u,pig[i]);//超级源点连接第一个到i猪圈的顾客
    	for(j=0;j<map[i].size()-1;j++)
    		add(map[i][j],map[i][j+1],INF);//所有到i猪圈的顾客按照顺序连接
    }
    for(i=1;i<=n;i++)
        add(i,n+1,b[i]);//所有顾客连接汇点
}
int bfs(int beg,int end)
{
    int i;
    memset(vis,0,sizeof(vis));
    memset(dis,-1,sizeof(dis));
    queue<int>q;
    while(!q.empty())
        q.pop();
    vis[beg]=1;
    dis[beg]=0;
    q.push(beg);
    while(!q.empty())
    {
        int u=q.front();
        q.pop();
        for(i=head[u];i!=-1;i=edge[i].next)//遍历所有的与u相连的边
        {
            node E=edge[i];
            if(!vis[E.to]&&E.cap>E.flow)//如果边未被访问且流量未满继续操作
            {
                dis[E.to]=dis[u]+1;//建立层次图
                vis[E.to]=1;//将当前点标记
                if(E.to==end)//如果当前点搜索到终点则停止搜索  返回1表示有从原点到达汇点的路径
                    return 1;
                q.push(E.to);//将当前点入队
            }
        }
    }
    return 0;//返回0表示未找到从源点到汇点的路径
}
int dfs(int x,int a,int end)//把找到的这条边上的所有当前流量加上a（a是这条路径中的最小残余流量）
{
    //int i;
    if(x==end||a==0)//如果搜索到终点或者最小的残余流量为0
        return a;
    int flow=0,f;
    for(int& i=cur[x];i!=-1;i=edge[i].next)//i从上次结束时的弧开始
    {
        node& E=edge[i];
        if(dis[E.to]==dis[x]+1&&(f=dfs(E.to,min(a,E.cap-E.flow),end))>0)//如果
        {//bfs中我们已经建立过层次图，现在如果 dis[E.to]==dis[x]+1表示是我们找到的路径
        //如果dfs>0表明最小的残余流量还有，我们要一直找到最小残余流量为0
            E.flow+=f;//正向边当前流量加上最小的残余流量
            edge[i^1].flow-=f;//反向边
            flow+=f;//总流量加上f
            a-=f;//最小可增流量减去f
            if(a==0)
                break;
        }
    }
    return flow;//所有边加上最小残余流量后的值
}
int Maxflow(int beg,int end)
{
    int flow=0;
    while(bfs(beg,end))//存在最短路径
    {
        memcpy(cur,head,sizeof(head));//复制数组
        flow+=dfs(beg,INF,end);
    }
    return flow;//最大流量
}
int main()
{
	while(scanf("%d%d",&m,&n)!=EOF)
	{
		init();
		input();
		getmap();
		printf("%d\n",Maxflow(0,n+1));
	}
	return 0;
}