最大流模板

ISAP+BFS初始化+栈优化

跑的飞快，但比较长

点数最大大概\(10^5\),边数最大大概\(4 \times 10^5\)

#include <bits/stdc++.h>
using namespace std;
const int inf = 0x3f3f3f3f;
const int N = 60050;
struct node {
    int v, cap, flow, nxt;
    node () {}
    node (int v, int cap, int flow, int nxt): v(v), cap(cap), flow(flow), nxt(nxt) {}
} edge[N * 10];
int head[N], tot;
void init() {
    memset(head, -1, sizeof(head));
    tot = 0;
}
int dep[N];
void adde(int u, int v, int w) {
    edge[tot] = node(v, w, 0, head[u]);
    head[u] = tot++;
    edge[tot] = node(u, 0, 0, head[v]);
    head[v] = tot++;
}
int gap[N], cur[N];
void bfs(int s, int t) {
    memset(dep, -1, sizeof(dep));
    memset(gap, 0, sizeof(gap));
    gap[0] = 1;
    queue<int> q;
    dep[t] = 0;
    q.push(t);
    while (!q.empty()) {
        int u = q.front(); q.pop();
        for (int i = head[u]; ~i; i = edge[i].nxt) {
            int v = edge[i].v;
            if (dep[v] != -1) continue;
            q.push(v);
            dep[v] = dep[u] + 1;
            gap[dep[v]]++;
        }
    }
}
int S[N];
int sap(int s, int t, int n) {
    bfs(s, t);
    memcpy(cur, head, sizeof(head));
    int top = 0;
    int u = s;
    int ans = 0;
    while (dep[s] < n) {
        if (u == t) {
            int minn = inf;
            int inser;
            for (int i = 0; i < top; i++) {
                if (minn > edge[S[i]].cap - edge[S[i]].flow) {
                    minn = edge[S[i]].cap - edge[S[i]].flow;
                    inser = i;
                }
            }
            for (int i = 0; i < top; i++) {
                edge[S[i]].flow += minn;
                edge[S[i] ^ 1].flow -= minn;
            }
            ans += minn;
            top = inser;
            u = edge[S[top] ^ 1].v;
            continue;
        }
        bool flag = false;
        int v;
        for (int i = cur[u]; ~i; i = edge[i].nxt) {
            v = edge[i].v;
            if (edge[i].cap - edge[i].flow && dep[v] + 1 == dep[u]) {
                flag = true;
                cur[u] = i;
                break;
            }
        }
        if (flag) {
            S[top++] = cur[u];
            u = v;
            continue;
        }
        int minn = n;
        for (int i = head[u]; ~i; i = edge[i].nxt) {
            if (edge[i].cap - edge[i].flow && dep[edge[i].v] < minn) {
                minn = dep[edge[i].v];
                cur[u] = i;
            }
        }
        gap[dep[u]]--;
        if (!gap[dep[u]]) return ans;
        dep[u] = minn + 1;
        gap[dep[u]]++;
        if (u != s) u = edge[S[--top] ^ 1].v;
    }
    return ans;
}

普通ISAP 邻接表形式

跑的也挺快的，大概所需时间是dinic的2/3，无递归，可以完美替代dinic

点数最大大概\(10^5\),边数最大大概\(4 \times 10^5\)

#include <bits/stdc++.h>
using namespace std;
const int inf = 0x3f3f3f3f;
const int N = 20050;
struct node {
    int v, cap, nxt;
    node () {}
    node (int v, int cap, int nxt): v(v), cap(cap), nxt(nxt) {}
} edge[N * 100];
int head[N], tot;
void init() {
    memset(head, -1, sizeof(head));
    tot = 0;
}
int dep[N];
void adde(int u, int v, int w) {
    edge[tot] = node(v, w, head[u]);
    head[u] = tot++;
    edge[tot] = node(u, 0, head[v]);
    head[v] = tot++;
}
int cur[N];
int gap[N];
int pre[N];
int sap(int s, int t, int n) {
    memset(dep, 0, sizeof(dep));
    memset(gap, 0, sizeof(gap));
    memcpy(cur, head, sizeof(head));
    memset(pre, 0, sizeof(pre));
    gap[0] = n;
    int u = s;
    pre[u] = -1;
    int ans = 0;
    while (dep[s] < n) {
        if (u == t) {
            int minn = inf;
            for (int i = pre[u]; ~i; i = pre[edge[i ^ 1].v]) {
                if (minn > edge[i].cap) minn = edge[i].cap;
            }
            for (int i = pre[u]; ~i; i = pre[edge[i ^ 1].v]) {
                edge[i].cap -= minn;
                edge[i ^ 1].cap += minn;
            }
            u = s;
            ans += minn;
            continue;
        }
        bool flag = false;
        int v;
        for (int i = cur[u]; ~i; i = edge[i].nxt) {
            v = edge[i].v;
            if (edge[i].cap && dep[v] + 1 == dep[u]) {
                flag = true;
                cur[u] = pre[v] = i;
                break;
            }
        }
        if (flag) {
            u = v;
            continue;
        }
        int minn = n;
        for (int i = head[u]; ~i; i = edge[i].nxt) {
            if (edge[i].cap && dep[edge[i].v] < minn) {
                minn = dep[edge[i].v];
                cur[u] = i;
            }
        }
        gap[dep[u]]--;
        if (!gap[dep[u]]) return ans;
        dep[u] = minn + 1;
        gap[dep[u]]++;
        if (u != s) u = edge[pre[u] ^ 1].v;
    }
    return ans;
}

还有一种递归写法，挺慢的，和dinic差不多，还不如dinic好写，没啥用。

Dinic

大部分时间够用，但普通ISAP只比dinic长10行，代码量相差不大。

#include <bits/stdc++.h>
using namespace std;
const int inf = 0x3f3f3f3f;
const int N = 60050;
struct node {
    int u, v, cap, nxt;
    node () {}
    node (int u, int v, int cap, int nxt): u(v), v(v), cap(cap), nxt(nxt) {}
} edge[N * 10];
int head[N], tot;
void init() {
    memset(head, -1, sizeof(head));
    tot = 0;
}
int dep[N];
void adde(int u, int v, int w) {
    edge[tot] = node(u, v, w, head[u]);
    head[u] = tot++;
    edge[tot] = node(v, u, 0, head[v]);
    head[v] = tot++;
}
bool bfs(int s, int t) {
    queue<int> q;
    memset(dep, -1, sizeof(dep));
    dep[s] = 0;
    q.push(s);
    while (!q.empty()) {
        int u = q.front(); q.pop();
        for (int i = head[u]; ~i; i = edge[i].nxt) {
            int v = edge[i].v;
            if (edge[i].cap > 0 && dep[v] == -1) {
                dep[v] = dep[u] + 1;
                q.push(v);
            }
        }
    }
    return dep[t] > 0;
}
int dfs(int u, int t, int f) {//f是到这点的残余流量，used是这点已经流出的流量
    if (u == t) return f;
    int w, used = 0;
    for (int i = head[u]; ~i; i = edge[i].nxt) {
        int v = edge[i].v;
        int cap = edge[i].cap;
        if (cap && dep[v] == dep[u] + 1) {
            w = f - used;
            w = dfs(v, t, min(w, cap));
            edge[i].cap -= w;
            edge[i ^ 1].cap += w;
            used += w;
            if (used == f) return f;
        }
    }
    if (!used) dep[u] = -1;
    return used;
}
int dinic(int s, int t) {
    int maxflow = 0;
    while (bfs(s, t)) {
       maxflow += dfs(s, t, inf);
    }
    return maxflow;
}

最高标号预流推进(HLPP)

我觉得可能用不到把，复杂度是\(n^2 \sqrt{m}\)，但是常数比较大，和isap跑的差不多快

给个别人的板子，自己没写

#include <bits/stdc++.h>
using namespace std;
inline int Read(){
    int x=0;
    char c=getchar();
    while(c>'9'||c<'0')c=getchar();
    while(c>='0'&&c<='9')x=x*10+c-'0',c=getchar();
    return x;
}
const int inf=1<<30;
int top=1,head[10100];
int n,m,s,t;
int e[10100],h[10100],cnth[20100];//每个点对应的余流，高度；每个高度有多少个点 
struct cmp{
    inline bool operator () (int a,int b) const{
        return h[a]<h[b];
    }
};
struct Node{
    int v;
    int val;
    int next;
}node[400100];
inline void addedge(int u,int v,int val){
    node[++top].v=v;
    node[top].val=val;
    node[top].next=head[u];
    head[u]=top;
}
inline void add(int u,int v,int val){
    addedge(u,v,val);
    addedge(v,u,0);
}
int inque[11000];
void bfs(){
    memset(h,0x3f,sizeof(h));
    h[t]=0;
    queue<int>qu;
    qu.push(t);
    while(!qu.empty()){
        int u=qu.front();
        qu.pop();
        inque[u]=0;
        for(int i=head[u];i;i=node[i].next){
            int d=node[i].v;
            if(node[i^1].val&&h[d]>h[u]+1){//反向跑 
                h[d]=h[u]+1;
                if(inque[d]==0){
                    qu.push(d);
                    inque[d]=1;
                }
            }
        }
    }
    return;
}
priority_queue<int,vector<int>,cmp>q;
inline void push_(int u){
    for(int i=head[u];i;i=node[i].next){
        int d=node[i].v;
        if(node[i].val&&h[d]+1==h[u]){//可以推流 
            int mi=min(node[i].val,e[u]);
            node[i].val-=mi;
            node[i^1].val+=mi;
            e[u]-=mi;
            e[d]+=mi;
            if(inque[d]==0&&d!=t&&d!=s){
                q.push(d);
                inque[d]=1;
            }
            if(e[u]==0)break;//已经推完了 
        }
    }
}//推流 
inline void relabel(int u){
    h[u]=inf;
    for(int i=head[u];i;i=node[i].next){
        int d=node[i].v;
        if(node[i].val&&h[d]+1<h[u]){
            h[u]=h[d]+1;
        }
    }
}//把u的高度更改为与u相邻的最低的点的高度加1 
int hlpp(){
    register int i;
    bfs();
    if(h[s]==0x3f3f3f3f)return 0;//s与t不连通 
    h[s]=n;
    for(i=1;i<=n;i++)if(h[i]<0x3f3f3f3f)cnth[h[i]]++;//统计各个高度的点数，注意不要让下标越界
    for(i=head[s];i;i=node[i].next){
        int d=node[i].v;    
        int mi=node[i].val;
        if(mi){
            e[s]-=mi;
            e[d]+=mi;
            node[i].val-=mi;
            node[i^1].val+=mi;
            if(d!=t&&inque[d]==0&&d!=s){
            q.push(d);
            inque[d]=1; 
            }
        }
    }//从s向周围点推流
    while(!q.empty()){
        int u=q.top();
        inque[u]=0;
        q.pop();
        push_(u);
        if(e[u]){//还有余流 
            cnth[h[u]]--;
            if(cnth[h[u]]==0){
                for(int i=1;i<=n;i++){
                    if(i!=s&&i!=t&&h[i]>h[u]&&h[i]<n+1){
                        h[i]=n+1;//标记无法到达 
                    }
                }
            }//gap优化 
            relabel(u);
            cnth[h[u]]++;
            q.push(u);
            inque[u]=1;
        }
    }
    return e[t];
} 
int main(){
    n=Read(),m=Read(),s=Read(),t=Read();
    register int i;
    int u,v,val;
    for(i=1;i<=m;i++)u=Read(),v=Read(),val=Read(),add(u,v,val);
    printf("%d",hlpp());
    return 0;
}