FF演算法(Ford-Fulkerson)
也稱標號法

司書寫Matlab可以直接用工具箱,所以只給了lingo程式

那麼就只記記思路:

我們如果手算,步驟是這樣的:
1.選定一條S到T的路,計算\(\delta_i\)(\(\delta_i\)是這條路中最小的邊權)
2.更新這條路上每條邊的權值(調整成參流)
比如:S\(\rightarrow\)A\(\rightarrow\)C\(\rightarrow\)T這條路,\(\delta\)=2,更新完之後SA為1,AC為0,CT為0
3.一二步重複執行,直到選的任一路的\(\delta\)都為0
4.這個網路的最大流就是\(\Sigma\)\(\delta_i\)

那麼用程式寫出來呢?因為我們開始任意選的路會選錯,所以不妨加條"後悔路"
也就是下圖這樣:

加入之後我們每次更新要注意把後悔路也更新了,更新方法是減\(\delta\).
怎麼理解呢?如果我們先選了S\(\rightarrow\)A\(\rightarrow\)C\(\rightarrow\)T這條路,CA就會被更新為2,TC被更新為2,這樣相當於有"退回"的可能了.可以選S\(\rightarrow\)B\(\rightarrow\)C\(\rightarrow\)A\(\rightarrow\)T這種道路.

附C++程式碼:

#define readl(a) scanf("%lld",&a)
#define reads(a) scanf("%s",&a)
#define readc(a) scanf("%c",&a)
#define pb push_back
#define mem(a) memset(a,o,sizeof(a))
#define Buff ios::sync_with_stdio(false)
typedef long long ll;
using name space std;
const int INF = 2e9+7;
const int N = 1e5+7;
const int M = 1e6+7;
const int base = 100;

struct edge//邊的結構體
{
	int to,nex,cap;//指向點,下一條邊,容量
	edge(int to=0,int nex=0,int cap=0):to(to),nex(nex),cap(cap){}
};
class FF
{
public:
	int S,T,n,m,cnt;
	int head[N],vis[N];
	edge e[M];
	void addedge(int u,int v,int cap)
	{
		e[++cnt]=edge(v,head[u],cap);
		head[u]=cnt;
	}
	
	void buildGraph(int _n,int _m,int _S,int _T)//建圖
	{
		n = _n;m = _m;
		S = _S;T = _T;
		mem(head);mem(vis);
		cnt = 1;
	
		for(int i=1;u,v,cap;i<=m;i++)
		{
			read(u);read(v);read(cap);
			addedge(u,v,cap);
			addedge(v,u,0);
		}
	}
	int dfs(int u.int flow)
	{
		if(u==T) return flow;
		vis[u] = 1;
		for(int i=head[u];i;i = e[i].nex)
		{
			int v = e[i].to;
			if(e[i].cap<=0||vis[v]) continue;
			int delta = dfs(v,min(flow,e[i].cap));
			if (delta<=0) continue;
			e[i].cap-=delta;
			e[i^1].cap+=delta;
			return maxFlow;
		}
	}
	int get_maxFlow()
	{
		int maxFlow = 0;
		while(delta = dfs(S,INF)) 
		{
			mem(vis);
			maxFlow += delta;
		}
		return maxFlow;
	}
};
FF ways;
signed main()
{
	int n,m, S, T;
	while(~scanf("%d %d %d %d",&n,&m,&S,&T))
	{
		ways.buildgraph(n,m,S,T);
		printf("%d\b",ways.get_maxFlow());
	}
	
}//源自b站某up