【图论】最大流算法(FF,EK,dinic)_ff算法求最大流-优快云博客

本文链接：https://blog.youkuaiyun.com/Rainfoo/article/details/105523328

本文介绍了Dinic算法的最快速版本，包括在Acwing 2172题目中的应用，以及与FF和EK算法的对比。Dinic算法结合当前弧优化，能实现O(V^2*E)的时间复杂度。同时提到了基于DFS的FF算法和以BFS为主的EK算法，探讨了它们在解决最大流问题时的特点。

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Dinic算法+各种优化最快速版本Acwing.2172：

#include<bits/stdc++.h>
#define ll long long
#define rep(i,a,n) for(int i=a;i<=n;i++)
#define per(i,n,a) for(int i=n;i>=a;i--)
#define endl '\n'
#define mem(a) memset(a,0,sizeof(a))
#define IO ios::sync_with_stdio(false);cin.tie(0);
using namespace std;
const int INF=0x3f3f3f3f;
const int mod=1e9+7;
const int maxn=1e5+5;
int n,m,S,T;
int head[maxn],d[maxn],cur[maxn],tot;
struct E{
	int to,next,cap;
}edge[maxn<<1];
void add(int u,int v,int w){
	edge[tot].to=v;
	edge[tot].next=head[u];
	edge[tot].cap=w;
	head[u]=tot++;
}
bool bfs(){
	memset(d,-1,sizeof(d));	
	queue<int> q;
	q.push(S);d[S]=0;cur[S]=head[S];
	while(!q.empty()){
		int u=q.front();q.pop();
		for(int i=head[u];i!=-1;i=edge[i].next){
			int v=edge[i].to;
			if(d[v]==-1&&edge[i].cap){
				d[v]=d[u]+1;
				cur[v]=head[v];
				if(v==T) return true;
				q.push(v);
			}			
		}
	}
	return false;
}
int dfs(int u,int limit){
	if(u==T) return limit;
	int flow=0;
	for(int i=cur[u];i!=-1&&flow<limit;i=edge[i].next){
		cur[u]=i;
		int v=edge[i].to;
		if(d[v]==d[u]+1&&edge[i].cap){
			int delta=dfs(v,min(edge[i].cap,limit-flow));
			if(!delta) d[v]=-1;
			edge[i].cap-=delta;
			edge[i^1].cap+=delta;
			flow+=delta;
		}
	}
	return flow;
}
int dinic(){
	int maxFlow=0,delta;
	while(bfs()){
		while(delta=dfs(S,INF)){
			maxFlow+=delta;
		}
	}
	return maxFlow;  	
}
int main(){
	cin>>n>>m>>S>>T;
	memset(head,-1,sizeof(head));
	for(int i=1;i<=m;i++){
		int u,v,w;cin>>u>>v>>w;
		add(u,v,w);add(v,u,0);
	}
	cout<<dinic()<<endl;
}

模板题：洛谷P3376
算法一：Dinic算法+当前弧优化
O(V^2*E)

#include<bits/stdc++.h>
#define ll long long
#define rep(i,a,n) for(int i=a;i<=n;i++)
#define per(i,n,a) for(int i=n;i>=a;i--)
#define endl '\n'
#define mem(a) memset(a,0,sizeof(a))
#define IO ios::sync_with_stdio(false);cin.tie(0);
using namespace std;
const int INF=0x3f3f3f3f;
const int mod=1e9+7;
const int maxn=1e5+5;
int n,m,S,T;
int head[maxn],d[maxn],cur[maxn],tot;
struct E{
	int to,next,cap;
}edge[maxn<<1];
void add(int u,int v,int w){
	edge[tot].to=v;
	edge[tot].next=head[u];
	edge[tot].cap=w;
	head[u]=tot++;
}
bool bfs(){
	memset(d,0,sizeof(d));	
	queue<int> q;
	q.push(S);d[S]=1;
	while(!q.empty()){
		int u=q.front();q.pop();
		for(int i=head[u];i!=-1;i=edge[i].next){
			int v=edge[i].to;
			if(d[v]||edge[i].cap<=0) continue;
			q.push(v);d[v]=d[u]+1;
		}
	}
	for(int i=0;i<=n;i++) cur[i]=head[i];
	
	return d[T]!=0;
}
int dfs(int u,int flow){
	if(u==T) return flow;
	for(int i=cur[u];i!=-1;i=edge[i].next){
		cur[u]=i;
		int v=edge[i].to;
		if(d[v]!=d[u]+1||edge[i].cap<=0) continue;
		int delta=dfs(v,min(flow,edge[i].cap));
		if(delta<=0) continue;
		edge[i].cap-=delta;
		edge[i^1].cap+=delta;
		return delta;
	}
	return 0;
}
int get_maxFlow(){
	int maxFlow=0,delta;
	while(bfs()){
		while(delta=dfs(S,INF)){
			maxFlow+=delta;
		}
	}
	return maxFlow;  	
}
int main(){
	while(cin>>n>>m>>S>>T){
		memset(head,-1,sizeof(head));
		for(int i=1;i<=m;i++){
			int u,v,w;cin>>u>>v>>w;
			add(u,v,w);add(v,u,0);
		}
		cout<<get_maxFlow()<<endl;
	}
}

算法二：FF算法(dfs为主)

#include<bits/stdc++.h>
#define ll long long
#define rep(i,a,n) for(int i=a;i<=n;i++)
#define per(i,n,a) for(int i=n;i>=a;i--)
#define endl '\n'
#define mem(a) memset(a,0,sizeof(a))
#define IO ios::sync_with_stdio(false);cin.tie(0);
using namespace std;
const int INF=0x3f3f3f3f;
const int mod=1e9+7;
const int maxn=1e5+5;
int n,m,S,T;
int head[maxn],vis[maxn],tot;
struct E{
	int to,next,cap;
}edge[maxn<<1];
void add(int u,int v,int w){
	edge[tot].to=v;
	edge[tot].next=head[u];
	edge[tot].cap=w;
	head[u]=tot++;
}
int dfs(int u,int flow){
	if(u==T) return flow;
	vis[u]=1;
	for(int i=head[u];i!=-1;i=edge[i].next){
		int v=edge[i].to;
		if(edge[i].cap<=0||vis[v]) continue;
		int delta=dfs(v,min(flow,edge[i].cap));
		if(delta<=0) continue;
		edge[i].cap-=delta;
		edge[i^1].cap+=delta;
		return delta;
	}
	return 0;
}
int get_maxFlow(){
	int maxFlow=0,delta;
	while(delta=dfs(S,INF)){
		mem(vis);
		maxFlow+=delta;
	}
	return maxFlow;  	
}
int main(){
	while(cin>>n>>m>>S>>T){
		memset(head,-1,sizeof(head));
		for(int i=1;i<=m;i++){
			int u,v,w;cin>>u>>v>>w;
			add(u,v,w);add(v,u,0);
		}
		cout<<get_maxFlow()<<endl;
	}
}

算法三：EK算法(BFS为主)

#include<bits/stdc++.h>
#define ll long long
#define rep(i,a,n) for(int i=a;i<=n;i++)
#define per(i,n,a) for(int i=n;i>=a;i--)
#define endl '\n'
#define mem(a) memset(a,0,sizeof(a))
#define IO ios::sync_with_stdio(false);cin.tie(0);
using namespace std;
const int INF=0x3f3f3f3f;
const int mod=1e9+7;
const int maxn=1e5+5;
int n,m,S,T;
int head[maxn],pre[maxn],flow[maxn],Inq[maxn],tot;
struct E{
	int to,next,cap;
}edge[maxn<<1];
void add(int u,int v,int w){
	edge[tot].to=v;
	edge[tot].next=head[u];
	edge[tot].cap=w;
	head[u]=tot++;
}
bool bfs(){
	mem(Inq);
	queue<int> q;
	q.push(S);Inq[S]=1;flow[S]=INF;
	while(!q.empty()){
		int u=q.front();q.pop();
		for(int i=head[u];i!=-1;i=edge[i].next){
			int v=edge[i].to;
			if(Inq[v]||edge[i].cap<=0) continue;
			flow[v]=min(flow[u],edge[i].cap);
			pre[v]=i;
			q.push(v);Inq[v]=1;
			if(v==T) return 1;
		}
	}
	return 0;
}
int update(){
	int delta=flow[T],u=T;
	while(u!=S){
		int i=pre[u];
		edge[i].cap-=delta;
		edge[i^1].cap+=delta;
		u=edge[i^1].to;
	}
	return delta;
}
int get_maxFlow(){
	int maxFlow=0;
	while(bfs()) maxFlow+=update();
	return maxFlow;  	
}
int main(){
	while(cin>>n>>m>>S>>T){
		memset(head,-1,sizeof(head));
		memset(pre,0,sizeof(pre));
		for(int i=1;i<=m;i++){
			int u,v,w;cin>>u>>v>>w;
			add(u,v,w);add(v,u,0);
		}
		cout<<get_maxFlow()<<endl;
	}
}