Problem Portal

Portal1: Luogu

Description

For example, give a network diagram, its source and sink points, and find out the maximum flow of its network.

Input

The first line contains four positive integers \ (N,M,S,T \), which respectively represent the number of points, the number of directed edges, the source point serial number and the sink point serial number.

Next, each line of \ (M \) contains three positive integers \ (u_i,v_i,w_i \), indicating that the directed edge of \ (I \) starts from \ (w_i \) to \ (v_i \), and the edge weight is \ (w_i \) (that is, the maximum flow of the edge is \ (w_i \).

Output

A row containing a positive integer is the maximum flow of the network.

Sample Input

4 5 4 3
4 2 30
4 3 20
2 3 20
2 1 30
1 3 40

Sample Output

50

Hint

Data size:

Data for \ (30 \% \): \ (N \leq 10,M \leq 25 \);

Data for \ (70 \% \): \ (N \leq 200,M \leq 1000 \);

Data for \ (100 \% \): \ (N \leq 10000,M \leq 100000 \).

Example description:

There are \ (3 \) paths in the topic:

\(4 \to 2 \to 3 \), the route can pass the traffic of \ (20 \)

\(4 \to 3 \), traffic through \ (20 \)

\(4 \to 2 \to 1 \to 3 \), traffic that can pass through \ (10 \) (traffic that has consumed \ (20 \) before side \ (4 \to 2 \)

Therefore, the total flow is \ (20 + 20 + 10 = 50 \), and the output is \ (50 \).

Solution

Template problem, find the maximum flow.

Code

#include<iostream>
#include<algorithm>
#include<cstdio>
#include<cstring>
#include<cmath>
#include<queue>

using namespace std;

const int INF=0x3f3f3f3f, MAXN=10005, MAXM=200005;
int n, m, s, t, u, v, val, cnt, head[MAXN], dis[MAXN], pre[MAXN];
queue<int> Q;
struct node {
    int u, v, val, flow, nxt;
} edge[MAXM];
inline void addedge(int u, int v, int w) {//Forward star map
    edge[++cnt].u=u; edge[cnt].v=v; edge[cnt].val=w; edge[cnt].nxt=head[u]; head[u]=cnt;
}
inline bool Emonds_Karp() {
    memset(pre, 0, sizeof(pre));
    memset(dis, 0, sizeof(dis));//Initialization
    dis[s]=INF;
    pre[s]=0;
    Q.push(s);
    while(!Q.empty()) {//Similar to SPFA
        int x=Q.front();
        Q.pop();
        for (int i=head[x]; i; i=edge[i].nxt) {
            int v=edge[i].v;
            if (!dis[v] && edge[i].val>edge[i].flow) {
                Q.push(v);
                dis[v]=min(dis[x], edge[i].val-edge[i].flow);
                pre[v]=i;
            }
            if (dis[t]) break;
        }
    }
    return dis[t];
}
int main() {
    scanf("%d%d%d%d",&n, &m, &s, &t);
    cnt=1;
    for(int i=1; i<=m; i++) {
        scanf("%d%d%d",&u, &v, &val);
        addedge(u, v, val);
        addedge(v, u, 0);//Forward star map
    }
    int ans=0;
    while (Emonds_Karp()) {//Emonds Karp algorithm
        for (int i=t; i!=s; i=edge[pre[i]].u) {
            edge[pre[i]].flow+=dis[t];
            edge[pre[i] xor 1].val-=dis[t];
        }
        ans+=dis[t];
    }
    printf("%d\n",ans);
    return 0;
}

#include<iostream>
#include<algorithm>
#include<cstdio>
#include<cstring>
#include<cmath>
#include<queue>

using namespace std;

const int INF=0x3f3f3f3f, MAXN=10005, MAXM=200005;
struct node {
    int nxt, to, val;
} edge[MAXM];
queue<int> Q;
int n, m, s, t, u, v, cnt, val, dis[MAXN], head[MAXN];
inline void addedge(int u, int v, int w) {//Forward star map
    edge[++cnt].val=w; edge[cnt].to=v; edge[cnt].nxt=head[u]; head[u]=cnt;
}
inline bool bfs() {
    memset(dis, -1, sizeof(dis));
    dis[s]=0;
    Q.push(s);
    while (!Q.empty()) {
        int x=Q.front();
        Q.pop();
        for (int i=head[x]; ~i; i=edge[i].nxt) {
            int v=edge[i].to;
            if (edge[i].val && dis[v]==-1) {
                dis[v]=dis[x]+1;
                Q.push(v);
            }
        }
    }
    if (~dis[t]) return 1;
    return 0;
}
inline int dfs(int u, int flow) {
    if (u==t) return flow;
    int ret=flow;
    for (int i=head[u]; ~i; i=edge[i].nxt) {
        if (ret<=0) break;
        int v=edge[i].to;
        if (edge[i].val && dis[v]==dis[u]+1) {
            int x=dfs(v, min(edge[i].val, ret));
            ret-=x;
            edge[i].val-=x;
            edge[i xor 1].val+=x;
        }
    }
    return flow-ret;
}
inline int Dinic() {//Dinic algorithm
    int ret=0;
    while (bfs()) ret+=dfs(s, INF);
    return ret;
}
int main() {
    scanf("%d%d%d%d",&n, &m, &s, &t);
    memset(head, -1, sizeof(head));
    cnt=1;
    for (int i=1; i<=m; i++) {
        scanf("%d%d%d",&u, &v, &val);
        addedge(u, v, val);
        addedge(v, u, 0);//bilateral
    }
    printf("%d\n",Dinic());
    return 0;
}

#include<iostream>
#include<algorithm>
#include<cstdio>
#include<cstring>
#include<cmath>
#include<queue>

using namespace std;

const int INF=0x3f3f3f3f, MAXN=10005, MAXM=200005;
queue<int> Q;
int n, m, s, t, u, v, val, cnt, ans, head[MAXN], head1[MAXN], deep[MAXN], pre[MAXN], a[MAXN];
struct node {
    int v, w, nxt;
} edge[MAXM];
inline void addedge(int u, int v, int w) {//Forward star map
    edge[cnt].v=v; edge[cnt].w=w; edge[cnt].nxt=head[u]; head[u]=cnt++;
}
inline void bfs(int t) {
    for (int i=1; i<=n; i++)
        head1[i]=head[i];
    for (int i=1; i<=n; i++)
        deep[i]=n;
    deep[t]=0;
    Q.push(t);
    while (!Q.empty()) {
        int u=Q.front();
        Q.pop();
        for (int i=head[u]; ~i; i=edge[i].nxt)
            if (deep[edge[i].v]==n && edge[i^1].w) {
                deep[edge[i].v]=deep[u]+1;
                Q.push(edge[i].v);
            }
    }
}
int calc(int s,int t) {//Calculation
    int ans=INF, u=t;
    while (u!=s) {
        ans=min (ans,edge[pre[u]].w);
        u=edge[pre[u] xor 1].v;
    }
    u=t;
    while (u!=s) {
        edge[pre[u]].w-=ans;
        edge[pre[u] xor 1].w+=ans;
        u=edge[pre[u] xor 1].v;
    }
    return ans;
}
inline void ISAP(int s, int t) {//ISAP algorithm
    int u=s;
    bfs(t);
    for (int i=1; i<=n; i++)
        a[deep[i]]++;
    while (deep[s]<n) {
        if (u==t) {
            ans+=calc(s, t);
            u=s;
        }
        bool flag=0;
        for (int &i=head1[u]; ~i; i=edge[i].nxt)
            if (deep[u]==deep[edge[i].v]+1 && edge[i].w) {
                flag=1;
                u=edge[i].v;
                pre[edge[i].v]=i;
                break;
            }
        if (!flag) {
            int Min=n-1;
            for (int i=head[u]; ~i; i=edge[i].nxt)
                if (edge[i].w) Min=min(Min, deep[edge[i].v]);
            if ((--a[deep[u]])==0) break;
            a[deep[u]=Min+1]++;
            head1[u]=head[u];
            if (u!=s) u=edge[pre[u] xor 1].v;
        }
    }
}
int main() {
    memset(head, -1, sizeof (head));
    scanf("%d%d%d%d",&n, &m, &s, &t);
    for (int i=1; i<=m; i++) {
        scanf("%d%d%d",&u, &v, &val);
        addedge(u, v, val);
        addedge(v, u, 0);//bilateral
    }
    ISAP(s, t);
    printf ("%d\n",ans);
    return 0;
}