上下界网络流学习笔记

1、无源汇可行流

对每条边(from,to,cap)改造

cap=upper-lower

inflow[to]+=lower

outflow[from]+=lower

然后对每个点Addedge(s,i,inflow[i]),Addedge(i,t,ouflow[i])

#include<iostream>
#include<cstdio>
#include<cstring>
#include<vector>
#include<queue>
using namespace std;
const int maxn=30000;
const int oo=1000000000;

int n,m;

int inflow[maxn],outflow[maxn];

struct Edge{
	int from,to,cap,flow;
};
vector<int>G[maxn];
vector<Edge>edges;
void Addedge(int x,int y,int z){
	Edge e;
	e.from=x;e.to=y;e.cap=z;e.flow=0;
	edges.push_back(e);
	e.from=y;e.to=x;e.cap=0;e.flow=0;
	edges.push_back(e);
	int c=edges.size();
	G[x].push_back(c-2);
	G[y].push_back(c-1);
}

int s,t;
queue<int>q;
int vis[maxn];
int d[maxn];
int Bfs(){
	memset(vis,0,sizeof(vis));
	vis[s]=1;d[s]=0;q.push(s);
	while(!q.empty()){
		int x=q.front();q.pop();
		for(int i=0;i<G[x].size();++i){
			Edge e=edges[G[x][i]];
			if((e.cap>e.flow)&&(!vis[e.to])){
				vis[e.to]=1;
				d[e.to]=d[x]+1;
				q.push(e.to);
			}
		}
	}
	return vis[t];
}

int Dfs(int x,int a){
	if((x==t)||(a==0))return a;
	
	int nowflow=0,f=0;
	for(int i=0;i<G[x].size();++i){
		Edge e=edges[G[x][i]];
		if((d[x]+1==d[e.to])&&((f=Dfs(e.to,min(a,e.cap-e.flow)))>0)){
			nowflow+=f;
			a-=f;
			edges[G[x][i]].flow+=f;
			edges[G[x][i]^1].flow-=f;
			if(a==0)break;
		}
	}
	return nowflow;
}

int Maxflow(){
	int flow=0;
	while(Bfs())flow+=Dfs(s,oo);
	return flow;
}

int Lo[maxn];
int main(){
	scanf("%d%d",&n,&m);
	
	for(int i=1;i<=m;++i){
		int x,y,l,h;
		scanf("%d%d%d%d",&x,&y,&l,&h);
		Addedge(x,y,h-l);
		inflow[y]+=l;
		outflow[x]+=l;
		Lo[i*2-2]=l;
	}
	
	s=n+1;t=s+1;
	for(int i=1;i<=n;++i)Addedge(s,i,inflow[i]);
	for(int i=1;i<=n;++i)Addedge(i,t,outflow[i]);
	
	Maxflow();
	int fla=1;
	for(int i=0;i<G[s].size();++i){
		Edge e=edges[G[s][i]];
		if(e.cap!=e.flow)fla=0;
	}
	
	if(fla){
		printf("YES
");
		for(int i=0;i<m+m;i+=2){
			printf("%d
",edges[i].flow+Lo[i]);
		}
	}else{
		printf("NO
");
	}
	
	return 0;
}

　　

 2、

有源汇可行流

Addedge(t,s,+oo)

然后转化为无源汇可行流

3、

有源汇最大流

先求出可行流

然后跑s->t的最大流就为答案

#include<iostream>
#include<cstdio>
#include<cstring>
#include<vector>
#include<queue>
using namespace std;
const int maxn=100009;
const int oo=1000000000;

int ss,tt;
int n,m;

int inflow[maxn],outflow[maxn],suminflow=0;

struct Edge{
	int from,to,cap,flow;
};
vector<int>G[maxn];
vector<Edge>edges;
void Addedge(int x,int y,int z){
	Edge e;
	e.from=x;e.to=y;e.cap=z;e.flow=0;
	edges.push_back(e);
	e.from=y;e.to=x;e.cap=0;e.flow=0;
	edges.push_back(e);
	int c=edges.size();
	G[x].push_back(c-2);
	G[y].push_back(c-1);
}

int s,t;
queue<int>q;
int vis[maxn];
int d[maxn];
int Bfs(){
	memset(vis,0,sizeof(vis));
	vis[s]=1;d[s]=0;q.push(s);
	while(!q.empty()){
		int x=q.front();q.pop();
		for(int i=0;i<G[x].size();++i){
			Edge e=edges[G[x][i]];
			if((e.cap>e.flow)&&(!vis[e.to])){
				vis[e.to]=1;
				d[e.to]=d[x]+1;
				q.push(e.to);
			}
		}
	}
	return vis[t];
}

int Dfs(int x,int a){
	if((x==t)||(a==0))return a;
	
	int nowflow=0,f=0;
	for(int i=0;i<G[x].size();++i){
		Edge e=edges[G[x][i]];
		if((d[x]+1==d[e.to])&&((f=Dfs(e.to,min(a,e.cap-e.flow)))>0)){
			nowflow+=f;
			a-=f;
			edges[G[x][i]].flow+=f;
			edges[G[x][i]^1].flow-=f;
			if(a==0)break;
		}
	}
	return nowflow;
}

int Maxflow(){
	int flow=0;
	while(Bfs())flow+=Dfs(s,oo);
	return flow;
}

int main(){
	scanf("%d%d%d%d",&n,&m,&ss,&tt);
	s=n+1;t=s+1;
	for(int i=1;i<=m;++i){
		int x,y,lo,hi;
		scanf("%d%d%d%d",&x,&y,&lo,&hi);
		Addedge(x,y,hi-lo);
		inflow[y]+=lo;
		outflow[x]+=lo;
		suminflow+=lo;
	}
	for(int i=1;i<=n;++i){
		Addedge(s,i,inflow[i]);
		Addedge(i,t,outflow[i]);
	}
	Addedge(tt,ss,oo);
	
	if(Maxflow()!=suminflow){
		printf("please go home to sleep
");
	}else{
		s=ss;t=tt;
		printf("%d
",Maxflow());
	}
	return 0;
}

　　4、有源汇上下界最小流

先跑可行流

然后把t->s的边删掉

在跑t->s的最大流

用t->s的边的流量- t->s的最大流

#include<iostream>
#include<cstdio>
#include<cstring>
#include<vector>
#include<queue>
using namespace std;
const int maxn=50009;
const int oo=1000000000;

int n,m,ss,tt;
int inflow[maxn],outflow[maxn],suminflow;

struct Edge{
	int from,to,cap,flow;
};
vector<int>G[maxn];
vector<Edge>edges;
void Addedge(int x,int y,int z){
	Edge e;
	e.from=x;e.to=y;e.cap=z;e.flow=0;
	edges.push_back(e);
	e.from=y;e.to=x;e.cap=0;e.flow=0;
	edges.push_back(e);
	int c=edges.size();
	G[x].push_back(c-2);
	G[y].push_back(c-1);
}

int s,t;
queue<int>q;
int vis[maxn];
int d[maxn];
int Bfs(){
	memset(vis,0,sizeof(vis));
	vis[s]=1;d[s]=0;q.push(s);
	while(!q.empty()){
		int x=q.front();q.pop();
		for(int i=0;i<G[x].size();++i){
			Edge e=edges[G[x][i]];
			if((e.cap>e.flow)&&(!vis[e.to])){
				vis[e.to]=1;
				d[e.to]=d[x]+1;
				q.push(e.to);
			}
		}
	}
	return vis[t];
}

int Dfs(int x,int a){
	if((x==t)||(a==0))return a;
	
	int nowflow=0,f=0;
	for(int i=0;i<G[x].size();++i){
		Edge e=edges[G[x][i]];
		if((d[x]+1==d[e.to])&&((f=Dfs(e.to,min(a,e.cap-e.flow)))>0)){
			nowflow+=f;
			a-=f;
			edges[G[x][i]].flow+=f;
			edges[G[x][i]^1].flow-=f;
			if(a==0)break;
		}
	}
	return nowflow;
}

int Maxflow(){
	int flow=0;
	while(Bfs())flow+=Dfs(s,oo);
	return flow;
}

int main(){
	scanf("%d%d%d%d",&n,&m,&ss,&tt);
	for(int i=1;i<=m;++i){
		int x,y,lo,hi;
		scanf("%d%d%d%d",&x,&y,&lo,&hi);
		inflow[y]+=lo;
		outflow[x]+=lo;
		Addedge(x,y,hi-lo);
		suminflow+=lo;
	}
	s=n+1;t=s+1;
	Addedge(tt,ss,oo);
	for(int i=1;i<=n;++i){
		Addedge(s,i,inflow[i]);
		Addedge(i,t,outflow[i]);
	}
	
	if(Maxflow()!=suminflow){
		printf("please go home to sleep
");
	}else{
		int tm;
		s=ss;t=tt;
		for(int i=0;i<G[t].size();++i){
			Edge e=edges[G[t][i]];
			if(e.to==s&&e.cap==oo){
				tm=edges[G[t][i]].flow;
				edges[G[t][i]].flow=edges[G[t][i]].cap=0;
				edges[G[t][i]^1].flow=edges[G[t][i]^1].cap=0;
			}
		}
		swap(s,t);
		printf("%d
",tm-Maxflow());
	}
	return 0;
}