POJ 1459--Power Network(三种方法求最大流)

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根据题意，新建一个超级节点，所有发电站的流量由此节点发出，新建一个超级汇点，所有节点使用的流量都流到此节点。然后采用最大流的各种方法求解。

不管用ford——fulkerson方法还是推送重贴标签，首先都不用管自环，显然一个最大流是不会包括自环的，自环只会减少流量。然后反向边也不需要如算法导论中加入另外的点进行处理。为了减少内存，直接用容量的减少代表流量的增加，正向边容量减少多少，正向流量就增加多少，同时增加相应的反向容量，意思就是说反向的能允许的流量增加，而如果接下来的操作反向容量减少，也就是有反向的流量，也即是正向流量的减少，不管正向反向流量都可增可少，但是注意的是实际上最大流只会有一个方向的流量，但是可以等效为上述操作。直至所有增广路径的残余流量都是0。-------------ford——fulkerson
push——relabel：此算法较难理解，初始化后若一个点有溢出，并且其所有能流入的点（因为在构图里面允许反向边，所以不用像算法导论里面允许一个反向的流量作为残余边来减少流量，而本算法在对反向容量增加时可以达到同样的效果。所以只要capacity[u][v]>0就说明还有残余流量）高度均大于等于它，就对此点更新高度。若此点高度等于其能流入的点高度加1，则对此边推入操作。因为在满足溢出，可流入条件下，推入和重贴标签是互斥的，可以优化。
Dinic：首先BFS分层，为了构造一个有向无环图，当BFS无法到达汇点时，已达最大流。然后对上图作DFS，类似于二分匹配，当DFS到最后一点，其不能向下迭代但是又不是汇点则将此点标记不再走。当到达汇点回溯时，只回溯到路径最小边(u,v)，因为显然从u点还可能向下找增广路径，可继续使用u点以前多余的流量，所以要对每点设置一个总增广值sumAug，记录此点能到达的所有增广路径使用的流量，然后再从此点回溯。

ford-fulkerson：

#include<iostream>#include<cstdio>#include<queue>#include<cstring>using namespace std;#define min(x,y) (x>y?y:x)#define INF   (0X7F7F)#define maxN  105short capacity[maxN][maxN];short nodeNum;short start;short dest;int maxFlow(){    short fa[maxN];    short minPathFlow[maxN];    queue<short> q;    short now,i,tmpFlow,son;    int maxFlow = 0;    while(1)    {        memset(fa,127,sizeof(short)*(nodeNum+2));        memset(minPathFlow,127,sizeof(short)*(nodeNum+2));        q.push(start);        while(!q.empty())              //寻找始终点最短路径        {            now = q.front();            q.pop();            for(i = 0;i <= nodeNum;i++)            {                tmpFlow = capacity[now][i];                if(fa[i] == INF&&tmpFlow > 0)                {                if(i == dest)                {                    minPathFlow[dest] = min(minPathFlow[now],tmpFlow);                    fa[dest] = now;           //并查集                    break;                }                else                {                    fa[i] = now;                    minPathFlow[i] = min(minPathFlow[now],tmpFlow);                    q.push(i);                }                }            }        }        if(minPathFlow[dest] != INF)          //更新流量        {            maxFlow += minPathFlow[dest];            son = dest;            while(son != start)            {                capacity[fa[son]][son] -= minPathFlow[dest];                capacity[son][fa[son]] += minPathFlow[dest];                son = fa[son];            }        }        else            break;    }    cout<<maxFlow<<endl;    return 0;}int main(){    char tmpCh;    short u,v,cmax,powerNum,consumerNum,lineNum;    while(scanf("%hd%hd%hd%hd",&nodeNum,&powerNum,&consumerNum,&lineNum)!=EOF)    {        memset(capacity,0,sizeof(short)*(maxN*maxN));        while(lineNum--)        {            cin>>tmpCh>>u>>tmpCh>>v>>tmpCh>>cmax;                capacity[u][v] = cmax;        }        start = nodeNum+1;        dest = nodeNum;        while(powerNum--)        {            cin>>tmpCh>>u>>tmpCh>>cmax;            capacity[start][u] = cmax;        }        while(consumerNum--)        {            cin>>tmpCh>>u>>tmpCh>>cmax;            capacity[u][dest] = cmax;        }        maxFlow();    }    return 0;}

push-relabel to front:

#include<iostream>#include<cstdio>#include<cstring>using namespace std;#define maxN 105#define min(x,y)   (x>y?y:x)#define INF  (0x7F7F)int capacity[maxN][maxN];short nodeNum;short start,dest;short height[maxN];int exceFlow[maxN];struct node{    short x;    node *next;    node *prev;}*head;bool push(short u,short v){    int minFlow = min(exceFlow[u],capacity[u][v]);    capacity[u][v] -= minFlow;    capacity[v][u] += minFlow;    exceFlow[u] -= minFlow;    exceFlow[v] += minFlow;    return true;}bool relabel(short u){    short i;    short minHeight = INF;    for(i = 0;i < nodeNum;i++)    {        if(capacity[u][i] > 0&&height[u] <= height[i])            minHeight = min(minHeight,height[i]);    }    height[u] = minHeight+1;    return true;}int discharge(short u){    short v = 0;    while(exceFlow[u] > 0)    {        if(v == nodeNum)        {            relabel(u);            v = 0;        }        else if(capacity[u][v] > 0&&height[u] == height[v]+1)        {            push(u,v);        }        else        {            v++;        }    }    return 1;}int moveToHead(node *tmp){    if(tmp->prev != NULL)    {        if(tmp->next != NULL)        {            tmp->prev->next = tmp->next;            tmp->next->prev = tmp->prev;        }        else        {            tmp->prev->next = NULL;        }        tmp->next = head;        head->prev = tmp;        head = tmp;        head->prev = NULL;    }    return 1;}int relabelToFront(){    short i,u;    bool IsMax = false;    short oldHeight;    bool IsRelabel;    /*****Initialize preflow******/    memset(height,0,sizeof(short)*(nodeNum));    height[start] = nodeNum;    memset(exceFlow,0,sizeof(int)*(nodeNum));    for(i = 0;i < nodeNum-2;i++)    {        if(capacity[start][i] > 0)        {            exceFlow[i] = capacity[start][i];            exceFlow[start] -= capacity[start][i];            capacity[i][start] = capacity[start][i];            capacity[start][i] = 0;        }    }    /*****Init List*********/    node *tmp;    head = NULL;    for(i = 0;i < nodeNum-2;i++)    {        tmp = new node;        tmp->x = i;        tmp->next = head;        if(head != NULL)            head->prev = tmp;        head = tmp;        head->prev = NULL;    }    tmp = head;    while(tmp != NULL)    {        u = tmp->x;        oldHeight = height[u];        discharge(u);        if(height[u] > oldHeight)        {            moveToHead(tmp);        }        tmp = tmp->next;    }    delete[] tmp;    delete[] head;    cout<<exceFlow[dest]<<endl;    return 1;}int main(){    short powerNum,consumerNum,lineNum;    int u,v,cmax;    char tmpCh;    while(cin>>nodeNum>>powerNum>>consumerNum>>lineNum)    {        memset(capacity,0,sizeof(int)*(maxN*maxN));        while(lineNum--)        {            cin>>tmpCh>>u>>tmpCh>>v>>tmpCh>>cmax;            if(u != v)            capacity[u][v] = cmax;        }        nodeNum += 2;        start = nodeNum-1;        dest = nodeNum-2;        while(powerNum--)        {            cin>>tmpCh>>u>>tmpCh>>cmax;            capacity[start][u] = cmax;        }        while(consumerNum--)        {            cin>>tmpCh>>u>>tmpCh>>cmax;            capacity[u][dest] = cmax;        }        relabelToFront();    }    return 0;}

Dinic:

#include<iostream>#include<cstdio>#include<cstring>using namespace std;#define maxN 105#define min(x,y)   (x>y?y:x)#define INF  (0x7F7F)int capacity[maxN][maxN];short nodeNum;short start,dest;short height[maxN];int maxFlow;int Aug(short u,int minFlow){    short i;    int tmpFlow,tmpAug;    int sumAug = 0;    if(u == dest)    {        maxFlow += minFlow;        return minFlow;           //返回增广路径中最小容量，能增加的最大流量    }    for(i = 1;i < nodeNum;i++)    {        if(capacity[u][i] > 0&&height[i] == height[u]+1)        {            tmpFlow = min(minFlow,capacity[u][i]);            tmpAug = Aug(i,tmpFlow);            sumAug += tmpAug;               //从u点出发可能有不同的增广路径，将所有路径的容量保存在sumAug            if(tmpAug)            {                capacity[u][i] -= tmpAug;                capacity[i][u] += tmpAug;                if(capacity[u][i] != 0)      //若流量减为0，即寻找的增广路径的最小容量在此点，则从此点父节点再往下增广，反之直接返回找到路径容量最小点                return sumAug;                else                {                    minFlow -= tmpAug;        //再往下增广时，能允许的流量减少                }            }            else            {                height[i] = INF;             //从i点已无增广路径，将此点锁定不被访问            }        }    }    return sumAug;}int BFS(){    short q[maxN];    short now,rear,i;    q[now = rear = 0] = start;            memset(height,127,sizeof(short)*(nodeNum));    height[start] = 0;    while(now <= rear)    {        for(i = 1;i < nodeNum;i++)        {            if(capacity[q[now]][i] > 0&&height[i] == INF)            {                height[i] = height[q[now]]+1;            //可流的两点高度相差一，高度相差一不一定可流                q[++rear] = i;                if(i == dest)                    return 1;            }        }        now++;    }    return 0;}int Dinic(){    maxFlow = 0;    while(BFS())    {        Aug(start,INF);    }    cout<<maxFlow<<endl;    return 1;}int main(){    short powerNum,consumerNum,lineNum;    int u,v,cmax;    char tmpCh;    while(cin>>nodeNum>>powerNum>>consumerNum>>lineNum)    {        memset(capacity,0,sizeof(int)*(maxN*maxN));        while(lineNum--)        {            cin>>tmpCh>>u>>tmpCh>>v>>tmpCh>>cmax;            if(u != v)            capacity[++u][++v] = cmax;        }        nodeNum += 2;        start = 0;        dest = nodeNum-1;        while(powerNum--)        {            cin>>tmpCh>>u>>tmpCh>>cmax;            capacity[start][++u] = cmax;        }        while(consumerNum--)        {            cin>>tmpCh>>u>>tmpCh>>cmax;            capacity[++u][dest] = cmax;        }        Dinic();    }    return 0;}

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