有上下界的网络流 loj115 loj116 loj 117

参考文章

无源汇有上下界的可行流

有源汇有上下界的最大流

有源汇有上下界的最小流

无源汇有上下界可行流

以 loj115 为例。

剥离出必要边与自由边。

#include <iostream>
#include <cstring>
#include <cstdio>
#include <queue>
using namespace std;
int n, m, ss, tt, hea[205], uu, vv, ww, xx, cnt, tot, maxFlow, lev[205];
const int oo=0x3f3f3f3f;
queue<int> d;
struct Edge{
	int too, nxt, val, lim;
}edge[100005];
void add_edge(int fro, int too, int val, int lim){
	edge[cnt].nxt = hea[fro];
	edge[cnt].too = too;
	edge[cnt].lim = lim;
	edge[cnt].val = val;
	hea[fro] = cnt++;
}
void addEdge(int fro, int too, int val, int lim){
	add_edge(fro, too, val, lim);
	add_edge(too, fro, 0, lim);
}
bool bfs(){
	memset(lev, 0, sizeof(lev));
	lev[ss] = 1;
	d.push(ss);
	while(!d.empty()){
		int x=d.front();
		d.pop();
		for(int i=hea[x]; i!=-1; i=edge[i].nxt){
			int t=edge[i].too;
			if(!lev[t] && edge[i].val>0){
				lev[t] = lev[x] + 1;
				d.push(t);
			}
		}
	}
	return lev[tt]!=0;
}
int dfs(int x, int lim){
	if(x==tt)	return lim;
	int addFlow=0;
	for(int i=hea[x]; i!=-1 && addFlow<lim; i=edge[i].nxt){
		int t=edge[i].too;
		if(lev[t]==lev[x]+1 && edge[i].val>0){
			int tmp=dfs(t, min(lim-addFlow, edge[i].val));
			edge[i].val -= tmp;
			edge[i^1].val += tmp;
			addFlow += tmp;
		}
	}
	return addFlow;
}
void dinic(){
	while(bfs())	maxFlow += dfs(ss, oo);
}
int main(){
	memset(hea, -1, sizeof(hea));
	cin>>n>>m;
	ss = 0; tt = n + 1;
	for(int i=1; i<=m; i++){
		scanf("%d %d %d %d", &uu, &vv, &ww, &xx);
		addEdge(uu, vv, xx-ww, ww);
		addEdge(ss, vv, ww, ww);
		addEdge(uu, tt, ww, ww);//有一种优化是统计每个点流入下界与流出下界的差，然后根据差的正负决定是连ss还是tt
		tot += ww;
	}
	dinic();
	if(maxFlow<tot)	printf("NO\n");
	else{
		printf("YES\n");
		for(int i=0; i<cnt; i+=6)
			printf("%d\n", edge[i].lim+edge[i^1].val);
	}
	return 0;
}

有源汇有上下界最大流

以 loj116 为例。

#include <iostream>
#include <cstring>
#include <cstdio>
#include <queue>
using namespace std;
struct Edge{
    int too, nxt, val, lim;
}edge[60005];
int n, m, ss, tt, sss, ttt, hea[255], cnt, lev[255], uu, vv, ww, xx;
int maxFlow, tot;
const int oo=0x3f3f3f3f;
queue<int> d;
void add_edge(int fro, int too, int val, int lim){
    edge[cnt].nxt = hea[fro];
    edge[cnt].too = too;
    edge[cnt].val = val;
    edge[cnt].lim = lim;
    hea[fro] = cnt++;
}
void addEdge(int fro, int too, int val, int lim){
    add_edge(fro, too, val-lim, lim);
    add_edge(too, fro, 0, lim);
    add_edge(sss, too, lim, lim);
    add_edge(too, sss, 0, lim);
    add_edge(fro, ttt, lim, lim);
    add_edge(ttt, fro, 0, lim);
}
bool bfs(int fro, int too){
    memset(lev, 0, sizeof(lev));
    lev[fro] = 1;
    d.push(fro);
    while(!d.empty()){
        int x=d.front();
        d.pop();
        for(int i=hea[x]; i!=-1; i=edge[i].nxt){
            int t=edge[i].too;
            if(!lev[t] && edge[i].val>0){
                lev[t] = lev[x] + 1;
                d.push(t);
            }
        }
    }
    return lev[too]!=0;
}
int dfs(int fro, int too, int lim){
    if(fro==too)    return lim;
    int addFlow=0;
    for(int i=hea[fro]; i!=-1 && addFlow<lim; i=edge[i].nxt){
        int t=edge[i].too;
        if(lev[t]==lev[fro]+1 && edge[i].val>0){
            int tmp=dfs(t, too, min(lim-addFlow, edge[i].val));
            edge[i].val -= tmp;
            edge[i^1].val += tmp;
            addFlow += tmp;
        }
    }
    return addFlow;
}
void dinic(int fro, int too){
    maxFlow = 0;
    while(bfs(fro, too))    maxFlow += dfs(fro, too, oo);
}
int main(){
    cin>>n>>m>>ss>>tt;
    memset(hea, -1, sizeof(hea));
    sss = 0; ttt = n + 1;
    for(int i=1; i<=m; i++){
        scanf("%d %d %d %d", &uu, &vv, &ww, &xx);
        addEdge(uu, vv, xx, ww);
        tot += ww;
    }
    addEdge(tt, ss, oo, 0);
    dinic(sss, ttt);
    if(maxFlow<tot) printf("please go home to sleep\n");
    else{
        dinic(ss, tt);
        cout<<maxFlow<<endl;
    }
    return 0;
}

有源汇有上下界最小流

以 loj117 为例

#include <iostream>
#include <cstring>
#include <cstdio>
#include <queue>
using namespace std;
typedef long long ll;
int uu, vv, n, m, ss, tt, sss, ttt, hea[50105], cnt, lev[50105], cur[50105];
ll du[50105], ww, xx, tot, minFlow;
const ll oo=0x3f3f3f3f3f3f3f3f;
queue<int> d;
struct Edge{
	int too, nxt;
	ll val;
}edge[400005];
void add_edge(int fro, int too, ll val){
	edge[cnt].nxt = hea[fro];
	edge[cnt].too = too;
	edge[cnt].val = val;
	hea[fro] = cnt++;
}
void addEdge(int fro, int too, ll val){
	add_edge(fro, too, val);
	add_edge(too, fro, 0);
}
bool bfs(){
	memset(lev, 0, sizeof(lev));
	lev[sss] = 1;
	d.push(sss);
	while(!d.empty()){
		int x=d.front();
		d.pop();
		for(int i=hea[x]; i!=-1; i=edge[i].nxt){
			int t=edge[i].too;
			if(!lev[t] && edge[i].val>0){
				lev[t] = lev[x] + 1;
				d.push(t);
			}
		}
	}
	return lev[ttt]!=0;
}
ll dfs(int x, ll lim){
	if(x==ttt)	return lim;
	ll addFlow=0;
	for(int &i=cur[x]; i!=-1; i=edge[i].nxt){
		int t=edge[i].too;
		if(lev[t]==lev[x]+1 && edge[i].val>0){
			ll tmp=dfs(t, min(lim-addFlow, edge[i].val));
			edge[i].val -= tmp;
			edge[i^1].val += tmp;
			addFlow += tmp;
			if(addFlow==lim)	break;//注意，当前弧优化的addFlow的限制一定要在这里写。
		}
	}
	return addFlow;
}
void dinic(){
	while(bfs()){
		for(int i=sss; i<=ttt; i++)	cur[i] = hea[i];
		minFlow += dfs(sss, oo);
	}
}
int main(){
	memset(hea, -1, sizeof(hea));
	cin>>n>>m>>ss>>tt;
	sss = 0; ttt = n + 1;
	for(int i=1; i<=m; i++){
		scanf("%d %d %lld %lld", &uu, &vv, &ww, &xx);
		addEdge(uu, vv, xx-ww);
		du[uu] -= ww;
		du[vv] += ww;
	}
	for(int i=1; i<=n; i++){
		if(du[i]>0)	tot += du[i], addEdge(sss, i, du[i]);
		else if(du[i]<0)	addEdge(i, ttt, -du[i]);
	}
	dinic();
	addEdge(tt, ss, oo);
	dinic();
	if(minFlow<tot)	printf("please go home to sleep\n");
	else	printf("%lld\n", edge[cnt-1].val);
	return 0;
}