• 上下界的网络流模板

    上下界网络流问题对于每一条边、都有流量上下限的限制

    而普通的网络流就只有上限限制

    下面分别给出几种经典上下界网络流问题的模板

    参考博文Ⅰ、参考博文Ⅱ

    1、无源汇的上下界可行流

    实际也就是能否找出一个循环流、使得每个点的流入总流量 == 流出总流量

    对于原图的每一条边在网络流中容量应当为 （上界 - 下界）

    而后计算每个点流入流量的下界总和记为 in 、流出流量的下界总和记为 out

    抽象出超级源汇 ss 与 tt

    对于原图中的每一个点

    如果 in - out > 0 则 ss 与这个点连边、容量为 in - out

    如果 in - out < 0 则 这个点与 tt 连边、容量为 out - in

    最后如果 ss 的出边都满流的话、说明可行

    给出建图伪代码

    for each v in Graph

         if( in[v] - out[v] ) connect ss -> v、cap = in[v] - out[v]、FlowSum += in[v] - out[v]

         else if( in[v] - out[v] ) connect v -> tt、cap = out[v] - in[v]

    最后如果最大流 == FlowSum 则说明可行、否则不行

    LOJ 115 上下界可行流模板题

    #include<bits/stdc++.h>
#define LL long long
#define ULL unsigned long long

#define scl(i) scanf("%lld", &i)
#define scll(i, j) scanf("%lld %lld", &i, &j)
#define sclll(i, j, k) scanf("%lld %lld %lld", &i, &j, &k)
#define scllll(i, j, k, l) scanf("%lld %lld %lld %lld", &i, &j, &k, &l)

#define scs(i) scanf("%s", i)
#define sci(i) scanf("%d", &i)
#define scd(i) scanf("%lf", &i)
#define scIl(i) scanf("%I64d", &i)
#define scii(i, j) scanf("%d %d", &i, &j)
#define scdd(i, j) scanf("%lf %lf", &i, &j)
#define scIll(i, j) scanf("%I64d %I64d", &i, &j)
#define sciii(i, j, k) scanf("%d %d %d", &i, &j, &k)
#define scddd(i, j, k) scanf("%lf %lf %lf", &i, &j, &k)
#define scIlll(i, j, k) scanf("%I64d %I64d %I64d", &i, &j, &k)
#define sciiii(i, j, k, l) scanf("%d %d %d %d", &i, &j, &k, &l)
#define scdddd(i, j, k, l) scanf("%lf %lf %lf %lf", &i, &j, &k, &l)
#define scIllll(i, j, k, l) scanf("%I64d %I64d %I64d %I64d", &i, &j, &k, &l)

#define lson l, m, rt<<1
#define rson m+1, r, rt<<1|1
#define lowbit(i) (i & (-i))
#define mem(i, j) memset(i, j, sizeof(i))

#define fir first
#define sec second
#define VI vector<int>
#define ins(i) insert(i)
#define pb(i) push_back(i)
#define pii pair<int, int>
#define VL vector<long long>
#define mk(i, j) make_pair(i, j)
#define all(i) i.begin(), i.end()
#define pll pair<long long, long long>

#define _TIME 0
#define _INPUT 0
#define _OUTPUT 0
clock_t START, END;
void __stTIME();
void __enTIME();
void __IOPUT();
using namespace std;

const int maxn = 2000;
const int maxm = (10405) << 1;
struct Edge{
    int x,y,nxt;
    LL cap;
    Edge(){}
    Edge(int a,int b,LL c,LL d)
    { x=a,y=b,cap=c,nxt=d; }
};

struct Dinic{
    static const int N=maxn, M=maxm;
    static const LL INF=0x7fffffff;
    Edge e[M];
    int n,S,T,fst[N],cur[N],EdgeCnt;
    int q[N],dis[N],head,tail;
    LL MaxFlow;

    void Clear(int _n){
        n=_n,EdgeCnt=1;
        memset(fst,0,sizeof fst);
    }
    void AddEdge(int a,int b,LL c){
        e[++EdgeCnt]=Edge(a,b,c,fst[a]),fst[a]=EdgeCnt;
        e[++EdgeCnt]=Edge(b,a,0,fst[b]),fst[b]=EdgeCnt;
    }
    void init(){
        for (int i=1;i<=n;i++)
            cur[i]=fst[i];
    }
    void init(int _S,int _T){
        S=_S,T=_T,MaxFlow=0,init();
    }
    int bfs(){
        memset(dis,0,sizeof dis);
        head=tail=0;
        q[++tail]=T,dis[T]=1;
        while (head<tail)
            for (int x=q[++head],i=fst[x];i;i=e[i].nxt)
                if (!dis[e[i].y]&&e[i^1].cap){
                    if (e[i].y==T)
                        return 1;
                    dis[q[++tail]=e[i].y]=dis[x]+1;
                }
        return (bool)dis[S];
    }
    LL dfs(int x,LL Flow){
        if (x==T||!Flow)
            return Flow;
        LL now=Flow;
        for (int &i=cur[x];i;i=e[i].nxt){
            int y=e[i].y;
            if (dis[x]==dis[y]+1&&e[i].cap){
                LL d=dfs(y,min(now,e[i].cap));
                e[i].cap-=d,e[i^1].cap+=d,now-=d;
                if(now==0) break;
            }
        }
        return Flow-now;
    }
    LL GetMaxFlow(int _S,int _T){
        init(_S,_T);
        while (bfs()) init(),MaxFlow+=dfs(S,INF);
        return MaxFlow;
    }
}DC;

LL in[maxn], out[maxn], low[maxm];

int main(void){__stTIME();__IOPUT();

    int n, m;

    scii(n, m);

    DC.Clear(n+2);

    int ss = n+1, tt = n+2;

    for(int i=1; i<=m; i++){
        int u, v;
        LL upper;
        scii(u, v);
        scll(low[i], upper);
        out[u] += low[i];
        in[v] += low[i];
        DC.AddEdge(u, v, upper - low[i]);
    }

    LL FlowSum = 0;
    for(int i=1; i<=n; i++){
        if(in[i] - out[i] > 0) DC.AddEdge(ss, i, in[i] - out[i]), FlowSum += in[i] - out[i];
        else if(in[i] - out[i] < 0) DC.AddEdge(i, tt, out[i] - in[i]);
    }

    if(DC.GetMaxFlow(ss, tt) < FlowSum) puts("NO");
    else{
        puts("YES");
        for(int i=1; i<=m; i++) printf("%lld
", low[i] + DC.e[i<<1 | 1].cap);
    }




__enTIME();return 0;}


void __stTIME()
{
    #if _TIME
        START = clock();
    #endif
}

void __enTIME()
{
    #if _TIME
        END = clock();
        cerr<<"execute time = "<<(double)(END-START)/CLOCKS_PER_SEC<<endl;
    #endif
}

void __IOPUT()
{
    #if _INPUT
        freopen("in.txt", "r", stdin);
    #endif
    #if _OUTPUT
        freopen("out.txt", "w", stdout);
    #endif
}
    View Code

    2、有源汇的上下界可行流

    只要在无源汇的网络流基础上添加多一条边即可

    添加从汇到源的一条边、上界为 INF、下界为 0

    其余操作和无源汇的操作一样

    3、有源汇的上下界最大流

    上述可行流算法跑出来的并不一定是最大流

    方法就是先跑一遍有源汇的上下界可行流

    如果可行流跑出来的结果为可行则

    保持这个网络图不要变

    再跑一次从源到汇的最大流即为答案

    LOJ 116 有源汇的上下界最大流模板题

    #include<bits/stdc++.h>
#define LL long long
#define ULL unsigned long long

#define scl(i) scanf("%lld", &i)
#define scll(i, j) scanf("%lld %lld", &i, &j)
#define sclll(i, j, k) scanf("%lld %lld %lld", &i, &j, &k)
#define scllll(i, j, k, l) scanf("%lld %lld %lld %lld", &i, &j, &k, &l)

#define scs(i) scanf("%s", i)
#define sci(i) scanf("%d", &i)
#define scd(i) scanf("%lf", &i)
#define scIl(i) scanf("%I64d", &i)
#define scii(i, j) scanf("%d %d", &i, &j)
#define scdd(i, j) scanf("%lf %lf", &i, &j)
#define scIll(i, j) scanf("%I64d %I64d", &i, &j)
#define sciii(i, j, k) scanf("%d %d %d", &i, &j, &k)
#define scddd(i, j, k) scanf("%lf %lf %lf", &i, &j, &k)
#define scIlll(i, j, k) scanf("%I64d %I64d %I64d", &i, &j, &k)
#define sciiii(i, j, k, l) scanf("%d %d %d %d", &i, &j, &k, &l)
#define scdddd(i, j, k, l) scanf("%lf %lf %lf %lf", &i, &j, &k, &l)
#define scIllll(i, j, k, l) scanf("%I64d %I64d %I64d %I64d", &i, &j, &k, &l)

#define lson l, m, rt<<1
#define rson m+1, r, rt<<1|1
#define lowbit(i) (i & (-i))
#define mem(i, j) memset(i, j, sizeof(i))

#define fir first
#define sec second
#define VI vector<int>
#define ins(i) insert(i)
#define pb(i) push_back(i)
#define pii pair<int, int>
#define VL vector<long long>
#define mk(i, j) make_pair(i, j)
#define all(i) i.begin(), i.end()
#define pll pair<long long, long long>

#define _TIME 0
#define _INPUT 0
#define _OUTPUT 0
clock_t START, END;
void __stTIME();
void __enTIME();
void __IOPUT();
using namespace std;

const int maxn = 2000;
const int maxm = (10405) << 1;
const LL INF=0x7fffffff;

struct Edge{
    int x,y,nxt;
    LL cap;
    Edge(){}
    Edge(int a,int b,LL c,LL d)
    { x=a,y=b,cap=c,nxt=d; }
};

struct Dinic{
    static const int N=maxn, M=maxm;
    Edge e[M];
    int n,S,T,fst[N],cur[N],EdgeCnt;
    int q[N],dis[N],head,tail;
    LL MaxFlow;

    void Clear(int _n){
        n=_n,EdgeCnt=1;
        memset(fst,0,sizeof fst);
    }
    void AddEdge(int a,int b,LL c){
        e[++EdgeCnt]=Edge(a,b,c,fst[a]),fst[a]=EdgeCnt;
        e[++EdgeCnt]=Edge(b,a,0,fst[b]),fst[b]=EdgeCnt;
    }
    void init(){
        for (int i=1;i<=n;i++)
            cur[i]=fst[i];
    }
    void init(int _S,int _T){
        S=_S,T=_T,MaxFlow=0,init();
    }
    int bfs(){
        memset(dis,0,sizeof dis);
        head=tail=0;
        q[++tail]=T,dis[T]=1;
        while (head<tail)
            for (int x=q[++head],i=fst[x];i;i=e[i].nxt)
                if (!dis[e[i].y]&&e[i^1].cap){
                    if (e[i].y==T)
                        return 1;
                    dis[q[++tail]=e[i].y]=dis[x]+1;
                }
        return (bool)dis[S];
    }
    LL dfs(int x,LL Flow){
        if (x==T||!Flow)
            return Flow;
        LL now=Flow;
        for (int &i=cur[x];i;i=e[i].nxt){
            int y=e[i].y;
            if (dis[x]==dis[y]+1&&e[i].cap){
                LL d=dfs(y,min(now,e[i].cap));
                e[i].cap-=d,e[i^1].cap+=d,now-=d;
                if(now==0) break;
            }
        }
        return Flow-now;
    }
    LL GetMaxFlow(int _S,int _T){
        init(_S,_T);
        while (bfs()) init(),MaxFlow+=dfs(S,INF);
        return MaxFlow;
    }
}DC;

LL in[maxn], out[maxn], low[maxm];

int main(void){__stTIME();__IOPUT();

    int n, m, s, t;

    sciiii(n, m, s, t);

    DC.Clear(n+2);

    int ss = n+1, tt = n+2;

    for(int i=1; i<=m; i++){
        int u, v;
        LL upper;
        scii(u, v);
        scll(low[i], upper);
        out[u] += low[i];
        in[v] += low[i];
        DC.AddEdge(u, v, upper - low[i]);
    }DC.AddEdge(t, s, INF);

    LL FlowSum = 0;
    for(int i=1; i<=n; i++){
        if(in[i] - out[i] > 0) DC.AddEdge(ss, i, in[i] - out[i]), FlowSum += in[i] - out[i];
        else if(in[i] - out[i] < 0) DC.AddEdge(i, tt, out[i] - in[i]);
    }

    if(DC.GetMaxFlow(ss, tt) < FlowSum) puts("please go home to sleep");
    else printf("%lld
", DC.GetMaxFlow(s, t));




__enTIME();return 0;}


void __stTIME()
{
    #if _TIME
        START = clock();
    #endif
}

void __enTIME()
{
    #if _TIME
        END = clock();
        cerr<<"execute time = "<<(double)(END-START)/CLOCKS_PER_SEC<<endl;
    #endif
}

void __IOPUT()
{
    #if _INPUT
        freopen("in.txt", "r", stdin);
    #endif
    #if _OUTPUT
        freopen("out.txt", "w", stdout);
    #endif
}
    View Code

    4、有源汇的上下界最小流

    跑源汇的上下界可行流一次记为 F1

    然后添加汇到源的一条边、上界为 INF、下界为 0

    添加边后再跑一次上下界可行流一次记为 F2

    若 F1 + F2 < ( 超级源点 ss 所有出边的流量之和 )

    即满流的情况、则说明可行

    最小流就是刚刚添加的从汇到源的那条边的流量

    ( 不过有点慢、要快的可以上 LOJ 提交记录里面找找快速的代码是如何实现的 )

    LOJ 117 有源汇的上下界最小流模板

    #include<bits/stdc++.h>
#define LL long long
#define ULL unsigned long long

#define scl(i) scanf("%lld", &i)
#define scll(i, j) scanf("%lld %lld", &i, &j)
#define sclll(i, j, k) scanf("%lld %lld %lld", &i, &j, &k)
#define scllll(i, j, k, l) scanf("%lld %lld %lld %lld", &i, &j, &k, &l)

#define scs(i) scanf("%s", i)
#define sci(i) scanf("%d", &i)
#define scd(i) scanf("%lf", &i)
#define scIl(i) scanf("%I64d", &i)
#define scii(i, j) scanf("%d %d", &i, &j)
#define scdd(i, j) scanf("%lf %lf", &i, &j)
#define scIll(i, j) scanf("%I64d %I64d", &i, &j)
#define sciii(i, j, k) scanf("%d %d %d", &i, &j, &k)
#define scddd(i, j, k) scanf("%lf %lf %lf", &i, &j, &k)
#define scIlll(i, j, k) scanf("%I64d %I64d %I64d", &i, &j, &k)
#define sciiii(i, j, k, l) scanf("%d %d %d %d", &i, &j, &k, &l)
#define scdddd(i, j, k, l) scanf("%lf %lf %lf %lf", &i, &j, &k, &l)
#define scIllll(i, j, k, l) scanf("%I64d %I64d %I64d %I64d", &i, &j, &k, &l)

#define lson l, m, rt<<1
#define rson m+1, r, rt<<1|1
#define lowbit(i) (i & (-i))
#define mem(i, j) memset(i, j, sizeof(i))

#define fir first
#define sec second
#define VI vector<int>
#define ins(i) insert(i)
#define pb(i) push_back(i)
#define pii pair<int, int>
#define VL vector<long long>
#define mk(i, j) make_pair(i, j)
#define all(i) i.begin(), i.end()
#define pll pair<long long, long long>

#define _TIME 0
#define _INPUT 0
#define _OUTPUT 0
clock_t START, END;
void __stTIME();
void __enTIME();
void __IOPUT();
using namespace std;

const int maxn = 55000 + 10;
const int maxm = (225003 + 1000)<<1;
const LL INF=0x7fffffff;

struct Edge{
    int x,y,nxt;
    LL cap;
    Edge(){}
    Edge(int a,int b,LL c,LL d)
    { x=a,y=b,cap=c,nxt=d; }
};

struct Dinic{
    static const int N=maxn, M=maxm;
    Edge e[M];
    int n,S,T,fst[N],cur[N],EdgeCnt;
    int q[N],dis[N],head,tail;
    LL MaxFlow;

    void Clear(int _n){
        n=_n,EdgeCnt=1;
        memset(fst,0,sizeof fst);
    }
    void AddEdge(int a,int b,LL c){
        e[++EdgeCnt]=Edge(a,b,c,fst[a]),fst[a]=EdgeCnt;
        e[++EdgeCnt]=Edge(b,a,0,fst[b]),fst[b]=EdgeCnt;
    }
    void init(){
        for (int i=1;i<=n;i++)
            cur[i]=fst[i];
    }
    void init(int _S,int _T){
        S=_S,T=_T,MaxFlow=0,init();
    }
    int bfs(){
        memset(dis,0,sizeof dis);
        head=tail=0;
        q[++tail]=T,dis[T]=1;
        while (head<tail)
            for (int x=q[++head],i=fst[x];i;i=e[i].nxt)
                if (!dis[e[i].y]&&e[i^1].cap){
                    if (e[i].y==T)
                        return 1;
                    dis[q[++tail]=e[i].y]=dis[x]+1;
                }
        return (bool)dis[S];
    }
    LL dfs(int x,LL Flow){
        if (x==T||!Flow)
            return Flow;
        LL now=Flow;
        for (int &i=cur[x];i;i=e[i].nxt){
            int y=e[i].y;
            if (dis[x]==dis[y]+1&&e[i].cap){
                LL d=dfs(y,min(now,e[i].cap));
                e[i].cap-=d,e[i^1].cap+=d,now-=d;
                if(now==0) break;
            }
        }
        return Flow-now;
    }
    LL GetMaxFlow(int _S,int _T){
        init(_S,_T);
        while (bfs()) init(),MaxFlow+=dfs(S,INF);
        return MaxFlow;
    }
}DC;

LL in[maxn], out[maxn], low[maxm];

int main(void){__stTIME();__IOPUT();

    int n, m, s, t;

    sciiii(n, m, s, t);

    DC.Clear(n+2);

    int ss = n+1, tt = n+2;

    for(int i=1; i<=m; i++){
        int u, v;
        LL upper;
        scii(u, v);
        scll(low[i], upper);
        out[u] += low[i];
        in[v] += low[i];
        DC.AddEdge(u, v, upper - low[i]);
    }

    LL FlowSum = 0;
    for(int i=1; i<=n; i++){
        if(in[i] - out[i] > 0) DC.AddEdge(ss, i, in[i] - out[i]), FlowSum += in[i] - out[i];
        else if(in[i] - out[i] < 0) DC.AddEdge(i, tt, out[i] - in[i]);
    }

    LL F1 = DC.GetMaxFlow(ss, tt);
    int id = (DC.EdgeCnt + 2)>>1;
    DC.AddEdge(t, s, INF);
    LL F2 = DC.GetMaxFlow(ss, tt);

    if(F1+F2 < FlowSum) puts("please go home to sleep");
    else printf("%lld
", DC.e[id << 1 | 1].cap);




__enTIME();return 0;}


void __stTIME()
{
    #if _TIME
        START = clock();
    #endif
}

void __enTIME()
{
    #if _TIME
        END = clock();
        cerr<<"execute time = "<<(double)(END-START)/CLOCKS_PER_SEC<<endl;
    #endif
}

void __IOPUT()
{
    #if _INPUT
        freopen("in.txt", "r", stdin);
    #endif
    #if _OUTPUT
        freopen("out.txt", "w", stdout);
    #endif
}
    View Code

    5、有上下界的费用流

    不好意思、不会......
  • 相关阅读:
    JavaScript监听、设置全部ajax访问属性获取返回值（状态码）
    如何在Vue项目中使用vw实现移动端适配
    git 常用命令金字教程
    移动web开发之像素和DPR详解
    小程序-微信开发者工具使用
    小程序开发框架:Taro（一）基础篇
    vscode代码自动补全失效
    JS、C#编码解码
    PHP网页缓存技术
    ajaxfileupload回到json带<pre>
  • 原文地址：https://www.cnblogs.com/qwertiLH/p/9648323.html
Copyright © 2020-2023  润新知