POJ 1273 Drainage Ditches （最大流 ：Dinic | EK）

Drainage Ditches

Time Limit : 2000/1000ms (Java/Other)   Memory Limit : 65536/32768K (Java/Other)

Total Submission(s) : 51   Accepted Submission(s) : 33

Problem Description

Every time it rains on Farmer John's fields, a pond forms over Bessie's favorite clover patch. This means that the clover is covered by water for awhile and takes quite a long time to regrow. Thus, Farmer John has built a set of drainage ditches so that Bessie's clover patch is never covered in water. Instead, the water is drained to a nearby stream. Being an ace engineer, Farmer John has also installed regulators at the beginning of each ditch, so he can control at what rate water flows into that ditch. 
Farmer John knows not only how many gallons of water each ditch can transport per minute but also the exact layout of the ditches, which feed out of the pond and into each other and stream in a potentially complex network. 
Given all this information, determine the maximum rate at which water can be transported out of the pond and into the stream. For any given ditch, water flows in only one direction, but there might be a way that water can flow in a circle. 

 

Input

The input includes several cases. For each case, the first line contains two space-separated integers, N (0 <= N <= 200) and M (2 <= M <= 200). N is the number of ditches that Farmer John has dug. M is the number of intersections points for those ditches. Intersection 1 is the pond. Intersection point M is the stream. Each of the following N lines contains three integers, Si, Ei, and Ci. Si and Ei (1 <= Si, Ei <= M) designate the intersections between which this ditch flows. Water will flow through this ditch from Si to Ei. Ci (0 <= Ci <= 10,000,000) is the maximum rate at which water will flow through the ditch.

 

Output

For each case, output a single integer, the maximum rate at which water may emptied from the pond. 

 

Sample Input

5 4
1 2 40
1 4 20
2 4 20
2 3 30
3 4 10

 

Sample Output

50

 

Source

USACO 93

 

 Dinic算法详见     http://www.cnblogs.com/acSzz/archive/2012/09/13/2683820.html

 

求最大流的本质，就是不停的寻找增广路径。直到找不到增广路径为止。
对于这个一般性的过程，Dinic算法的优化如下：

（1）
Dinic算法首先对图进行一次BFS，然后在BFS生成的层次图中进行多次DFS。
层次图的意思就是，只有在BFS树中深度相差1的节点才是连接的。
这就切断了原有的图中的许多不必要的连接。很牛逼！
这是需要证明的，估计证明也很复杂。

（2）
除此之外，每次DFS完后，会找到路径中容量最小的一条边。
在这条边之前的路径的容量是大于等于这条边的容量的。
那么从这条边之前的点，可能引发出别的增广路径。
比如说 S -> b -> c -> d -> T 是一条增广路径，容量最小的边是 b -> c。
可能存在一条 S -> b -> e -> f -> g -> T 这样的增广路径。
这样的话，在找到第一条增广路径后，只需要回溯到 b 点，就可以继续找下去了。
这样做的好处是，避免了找到一条路径就从头开始寻找另外一条的开销。
也就是再次从 S 寻找到 b 的开销。
这个过程看似复杂，但是代码实现起来很优雅，因为它的本质就是回溯！

（3）
在同一次 DFS 中。如果从一个点引发不出任何的增广路径，就将这个点在层次图中抹去。Di

 Dinic :

#include<iostream>
#include<cstdio>
#include<cstring>
#include<queue>

using namespace std;

const int VM=220;
const int INF=0x3f3f3f3f;

int n,m,src,des;
int map[VM][VM],dep[VM];    //dep[i]表示当前点到起点src的层数

int BFS(){      // 重新建图（按层数建图）
    queue<int> q;
    while(!q.empty())
        q.pop();
    memset(dep,-1,sizeof(dep));
    dep[src]=0;
    q.push(src);
    while(!q.empty()){
        int u=q.front();
        q.pop();
        for(int v=1;v<=m;v++)
            if(map[u][v]>0 && dep[v]==-1){      // 如果可以到达且还没有访问
                dep[v]=dep[u]+1;
                q.push(v);
            }
    }
    return dep[des]!=-1;
}

int DFS(int u,int minx){    // 查找路径上的最小的流量
    if(u==des)
        return minx;
    int tmp;
    for(int v=1;v<=m;v++)
        if(map[u][v]>0 && dep[v]==dep[u]+1 && (tmp=DFS(v,min(minx,map[u][v])))){
            map[u][v]-=tmp;     //正向减少
            map[v][u]+=tmp;     //反向增加
            return tmp;
        }
    return 0;
}

int Dinic(){
    int ans=0,tmp;
    while(BFS()){
        while(1){
            tmp=DFS(1,INF);
            if(tmp==0)
                break;
            ans+=tmp;
        }
    }
    return ans;
}

int main(){

    //freopen("input.txt","r",stdin);

    while(~scanf("%d%d",&n,&m)){
        memset(map,0,sizeof(map));
        int u,v,w;
        for(int i=0;i<n;i++){
            scanf("%d%d%d",&u,&v,&w);
            map[u][v]+=w;   //防止有重边
        }
        src=1,  des=m;
        printf("%d\n",Dinic());
    }
    return 0;
}

EK :  

#include<iostream>
#include<cstdio>
#include<cstring>
#include<queue>

using namespace std;

const int VM=220;
const int INF=0x3f3f3f3f;

int n,m,max_flow;   //max_flow是最大流
int map[VM][VM],flow[VM][VM];   // map[i][j]是每条边的容量,flow[i][j]是每条边的流量 
int res[VM],pre[VM];    //res[]是每个点的剩余流量,pre[]是每个点的父亲

int EK(int src,int des){
    max_flow=0;
    queue<int> q;
    while(!q.empty())
        q.pop();
    memset(flow,0,sizeof(flow));    //最开始每条边的流量都是0
    while(1){
        memset(res,0,sizeof(res));  //残余流量得变0,一开始所有点都没流入对吧
        res[src]=INF;   //源点嘛,剩余流量无限是必须的...
        q.push(src);    //从源点开始进行BFS找增广路
        int u,v;
        while(!q.empty()){
            u=q.front();
            q.pop();
            for(v=1;v<=m;v++)       //遍历所有点,找可行边
                if(!res[v] && map[u][v]>flow[u][v]){    //该点剩余流量为0 且 容量大于流量,也就是找到了新的结点 
                    pre[v]=u;   //找到新结点,父节点得记录一下吧
                    q.push(v);
                    res[v]=min(res[u],map[u][v]-flow[u][v]);    //如果u的剩余流量能填满uv就填满,不能的话就把u这点的流量全部流向uv 
                }
        }
        if(res[des]==0)     //如果当前已经是最大流,汇点没有残余流量
            return max_flow;
        for(u=des;u!=src;u=pre[u]){     //如果还能增广,那么回溯,从汇点往回更新每条走过的边的流量
            flow[pre[u]][u]+=res[des];  //更新正向流量   （注意这里更新的是流量，而不是容量）
            flow[u][pre[u]]-=res[des];  //更新反向流量
        }
        max_flow+=res[des];
    }
}

int main(){

    //freopen("input.txt","r",stdin);

    while(~scanf("%d%d",&n,&m)){
        memset(map,0,sizeof(map));
        memset(pre,0,sizeof(pre));
        int u,v,w;
        while(n--){
            scanf("%d%d%d",&u,&v,&w);
            map[u][v]+=w;   //有重边
        }
        printf("%d\n",EK(1,m));
    }
    return 0;
}