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最新推荐文章于 2024-04-20 18:03:27 发布

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最新推荐文章于 2024-04-20 18:03:27 发布

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分类专栏：图论

本文链接：https://blog.youkuaiyun.com/weixin_46220615/article/details/113832080

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30 篇文章

订阅专栏

$一 .$ 网络流简述

$二 .$ 最大流

P3376 【模板】网络最大流

$E K :$

#include <set>
#include <map>
#include <list>
#include <cmath>
#include <stack>
#include <queue>
#include <string>
#include <bitset>
#include <vector>
#include<cstring>
#include <stdio.h>
#include <iostream>
#include <algorithm>
using namespace std;
typedef long long ll;
#define INF 0x3f3f3f3f
#define lowbit(x)  ((x)&-(x))
typedef unsigned long long ull;
#define freedom() ios::sync_with_stdio(false);
inline int read(){int s=0,w=1;char ch=getchar();
    while(ch<'0'||ch>'9'){if(ch=='-')w=-1;ch=getchar();}
    while(ch>='0'&&ch<='9') s=s*10+ch-'0',ch=getchar();return s*w;}
const int maxn = 5e3+5;
int head[maxn],tot;
struct Edge{
    int to,next;
    ll val;
}e[maxn<<2];
void add_edge(int u,int v,int w)
{
    e[tot].to=v;
    e[tot].next=head[u];
    e[tot].val=w;
    head[u]=tot++;
}
bool vis[maxn];
int dis[maxn]; // 一条增广路的最大可行流量
int pre[maxn]; // 记录边的前驱
bool bfs(int s,int t)
{
    memset(vis,0,sizeof(vis));
    queue<int>q;
    q.push(s),vis[s]=true;
    dis[s]=INF;
    while (!q.empty())
    {
        int x=q.front(); q.pop();
        for(int i=head[x];i!=-1;i=e[i].next)
        {
            int y=e[i].to,w=e[i].val;
            if(vis[y]||w==0) continue;
            dis[y]=min(dis[x],w);
            pre[y]=i;
            if(y==t) return true;
            q.push(y); vis[y]=true;
        }
    }
    return false;
}
ll EK(int s,int t)
{
    ll ans=0;
    while (bfs(s,t))
    {
        ans+=dis[t];
        for(int i=t;i!=s;i=e[pre[i]^1].to)
            e[pre[i]].val-=dis[t],e[pre[i]^1].val+=dis[t]; // 正向边+，反向边-。
    }
    return ans;
}
int main()
{
    int n=read(),m=read(),s=read(),t=read();
    memset(head,-1,sizeof(head));
    for(int i=1;i<=m;i++)
    {
        int u=read(),v=read(),w=read();
        add_edge(u,v,w);
        add_edge(v,u,0);
    }
    printf("%lld\n",EK(s,t));
    return 0;
}

$D i n i c :$

#include <set>
#include <map>
#include <list>
#include <cmath>
#include <stack>
#include <queue>
#include <string>
#include <bitset>
#include <vector>
#include<cstring>
#include <stdio.h>
#include <iostream>
#include <algorithm>
using namespace std;
typedef long long ll;
#define INF 0x3f3f3f3f
#define lowbit(x)  ((x)&-(x))
typedef unsigned long long ull;
#define freedom() ios::sync_with_stdio(false);
inline int read(){int s=0,w=1;char ch=getchar();
    while(ch<'0'||ch>'9'){if(ch=='-')w=-1;ch=getchar();}
    while(ch>='0'&&ch<='9') s=s*10+ch-'0',ch=getchar();return s*w;}
const int maxn = 5e3+5;
int head[maxn],tot;
struct Edge{
    int to,next;
    ll val;
}e[maxn<<2];
void add_edge(int u,int v,int w)
{
    e[tot].to=v;
    e[tot].next=head[u];
    e[tot].val=w;
    head[u]=tot++;
}
bool vis[maxn];
int deep[maxn]; // 记录每个点的层
bool bfs(int s,int t) // 对图进行分层
{
    memset(deep,0,sizeof(deep));
    queue<int>q;
    q.push(s);
    deep[s]=1;
    while (!q.empty())
    {
        int x=q.front(); q.pop();
        for(int i=head[x];i!=-1;i=e[i].next)
        {
            int y=e[i].to;
            ll w=e[i].val;

            if(deep[y]||!w) continue;

            deep[y]=deep[x]+1;
            q.push(y);
        }
    }
    return deep[t];
}
ll dfs(int x,int t,ll In)
{
// In:流入该点的流量，Out:流出该点的流量    
    if(x==t) return In;
    ll Out=0;
    for(int i=head[x];i!=-1;i=e[i].next)
    {
        int y=e[i].to;
        if(e[i].val&&deep[y]==deep[x]+1)
        {
            ll res=dfs(y,t,min(e[i].val,In));

            e[i].val-=res;
            e[i^1].val+=res;
            In-=res;
            Out+=res;
        }
    }
    if(In!=Out) deep[x]=0; 
    return Out;
}
ll Dinic(int s,int t)
{
    ll ans=0;
    while (bfs(s,t))
        ans+=dfs(s,t,1e18);
    return ans;
}
int main()
{
    int n=read(),m=read(),s=read(),t=read();
    memset(head,-1,sizeof(head));
    for(int i=1;i<=m;i++)
    {
        int u=read(),v=read(),w=read();
        add_edge(u,v,w);
        add_edge(v,u,0);
    }
    printf("%lld\n",Dinic(s,t));
    return 0;
}

$三 .$ 费用流

P3381 【模板】最小费用最大流

#include <set>
#include <map>
#include <list>
#include <cmath>
#include <stack>
#include <queue>
#include <string>
#include <bitset>
#include <vector>
#include<cstring>
#include <stdio.h>
#include <iostream>
#include <algorithm>
using namespace std;
typedef long long ll;
//#define INF 0x3f3f3f3f
typedef unsigned long long ull;
inline int read(){int s=0,w=1;char ch=getchar();
    while(ch<'0'||ch>'9'){if(ch=='-')w=-1;ch=getchar();}
    while(ch>='0'&&ch<='9') s=s*10+ch-'0',ch=getchar();return s*w;}
const int maxn = 1e5+5;
const ll INF=1e18;
int tot,head[maxn];
struct Edge{
    int to,next;
    int val,w;
}edge[maxn];
void add_edge(int u,int v,int w,int c)
{
    edge[tot].to=v;
    edge[tot].next=head[u];
    edge[tot].val=w;
    edge[tot].w=c;
    head[u]=tot++;
}
ll dis[maxn],di[maxn];
int pre[maxn];
bool vis[maxn];
bool spfa(int s,int t)
{
    memset(vis,false,sizeof(vis));
    for(int i=0;i<1e5;i++) dis[i]=INF;
    queue<int>q;
    q.push(s);
    vis[s]=true;
    dis[s]=0;
    di[s]=INF;
    while (!q.empty())
    {
        int x=q.front(); q.pop();
        vis[x]=false;
        for(int i=head[x];i!=-1;i=edge[i].next)
        {
            if(edge[i].val==0) continue;
            int y=edge[i].to;
            if(dis[y]>dis[x]+edge[i].w)
            {
                dis[y]=dis[x]+edge[i].w;
                di[y]=min(di[x],(ll)edge[i].val);
                pre[y]=i;
                if(!vis[y]) q.push(y),vis[y]=true;
            }
        }
    }
    if(dis[t]==INF) return false;
    return true;
}
ll EK(int s,int t)
{
    ll Flow=0,W=0;
    while (spfa(s,t))
    {
        Flow+=di[t];
        for(int i=t;i!=s;i=edge[pre[i]^1].to)
            edge[pre[i]].val-=di[t],edge[pre[i]^1].val+=di[t];

        W+=di[t]*dis[t];
    }
    printf("%lld ",Flow);
    return W;
}
int main()
{
    int n=read(),m=read(),s=read(),t=read();
    memset(head,-1,sizeof(head));
    for(int i=1;i<=m;i++)
    {
        int u=read(),v=read(),w=read(),c=read();
        add_edge(u,v,w,c);
        add_edge(v,u,0,-c);
    }
    printf("%lld\n",EK(s,t));
    return 0;
}

最小割&最小割集

2883: 病毒侵染

$分析：使\prod\limits_{i=1}^{k}{w}_{i}最小 ,取对数后为\sum\limits_{i=1}^{k}{w}_{i}$
$这样就转化成了求最小割集$

#include <set>
#include <map>
#include <list>
#include <cmath>
#include <stack>
#include <queue>
#include <string>
#include <bitset>
#include <vector>
#include<cstring>
#include <stdio.h>
#include <iostream>
#include <algorithm>
using namespace std;
typedef long long ll;
#define INF 0x3f3f3f3f
typedef unsigned long long ull;
inline int read(){int s=0,w=1;char ch=getchar();
    while(ch<'0'||ch>'9'){if(ch=='-')w=-1;ch=getchar();}
    while(ch>='0'&&ch<='9') s=s*10+ch-'0',ch=getchar();return s*w;}
const int maxn = 1e5+5;
int tot,head[maxn];
struct Edge{
    int u,to,next;
    double val;
    int ww;
}edge[maxn];
void add_edge(int u,int v,int w)
{
    double val=log(w);
    edge[tot]={u,v,head[u],val,w};
    head[u]=tot++;
    edge[tot]={v,u,head[v],val,w};
    head[v]=tot++;
}
int dep[maxn];
bool bfs(int s,int t)
{
    memset(dep,0,sizeof(dep));
    queue<int>q;
    q.push(s);
    dep[s]=1;
    while (!q.empty())
    {
        int x=q.front(); q.pop();
        for(int i=head[x];i!=-1;i=edge[i].next)
        {
            int y=edge[i].to;
            if(dep[y]||edge[i].val==0) continue;
            dep[y]=dep[x]+1;
            q.push(y);
        }
    }
    return dep[t];
}
double dfs(int s,int t,double In)
{
    if(s==t) return In;
    double Out=0;
    for(int i=head[s];i!=-1;i=edge[i].next)
    {
        int y=edge[i].to;
        if(edge[i].val==0||dep[y]!=dep[s]+1) continue;
        double res=dfs(y,t,min(edge[i].val,In));
 
        edge[i].val-=res;
        edge[i^1].val+=res;
 
        In-=res;
        Out+=res;
    }
    if(In!=Out) dep[s]=0;
    return Out;
}
double Dinic(int s,int t)
{
    double res=0;
    while (bfs(s,t))
        res+=dfs(s,t,1e9);
    return res;
}
int Cs[maxn];
void ColS(int x)
{
    Cs[x]=1;
    for(int i=head[x];i!=-1;i=edge[i].next)
    {
        int y=edge[i].to;
        if(!Cs[y]&&edge[i].val) ColS(y);
    }
}
int main()
{
    int n=read(),m=read();
    memset(head,-1,sizeof(head));
    for(int i=1;i<=m;i++)
    {
        int u=read(),v=read(),w=read();
        add_edge(u,v,w);
    }
    double Flow_max=Dinic(1,n);
 //   printf("%lld\n",Flow_max);
    ColS(1);
    ll mod=1e9+7;
    ll ans=1;
    bool flag=false;
    for(int i=0;i<=tot;i+=2)
        if(Cs[edge[i].u]+Cs[edge[i].to]==1) flag=true,ans=ans*edge[i].ww%mod;
    if(!flag) puts("0");
    else printf("%lld\n",ans);
    return 0;
}
/*
6 7
1 2 3
2 3 2
3 6 3
1 4 3
4 5 2
2 5 3
5 6 3
 
6
 */