洛谷-P3376 【模板】網路最大流
阿新 • • 發佈:2022-05-31
P3376 【模板】網路最大流
網路流
FF 演算法
直接用 dfs 求增廣路
時間複雜度 \(O(ef)\),\(e\) 為邊,\(f\) 為最大流
#include <iostream> #include <cstdio> using namespace std; typedef long long ll; const int maxn = 1e5 + 10; const ll inf = 1e17 + 10; int head[maxn], nex[maxn], tp = 2, to[maxn], vis[maxn]; ll vol[maxn]; int n, m, s, t; void add(int u, int v, int w) { to[tp] = v; vol[tp] = w; nex[tp] = head[u]; head[u] = tp; tp++; } ll dfs(int now, ll flow) { if(t == now) return flow; vis[now] = 1; for(int i=head[now]; i; i=nex[i]) { int v = to[i]; if(vol[i] <= 0 || vis[v] == 1) continue; ll f = dfs(v, min(flow, vol[i])); if(f != -1) { vol[i] -= f; vol[i ^ 1] += f; return f; } } return -1; } ll FF() { ll ans = 0, now = 0; while((now = dfs(s, inf)) != -1) { ans += now; for(int i=0; i<=n; i++) vis[i] = 0; } return ans; } int main() { ios::sync_with_stdio(false); cin.tie(0); cout.tie(0); cin >> n >> m >> s >> t; while(m--) { int l, r, w; cin >> l >> r >> w; add(l, r, w); add(r, l, 0); } cout << FF() << endl; return 0; }
EK 演算法
改用 bfs 求增廣路
時間複雜度 \(O(ve^2)\),\(v\) 為點數,\(e\) 為邊數
#include <iostream> #include <cstdio> #include <queue> using namespace std; typedef long long ll; const int maxn = 1e5 + 10; const ll inf = 1e17 + 10; int nex[maxn], head[maxn], to[maxn], tp = 1, last[maxn]; ll vol[maxn], flow[maxn]; int n, m, s, t; inline void add(int u, int v, int w) { tp++; nex[tp] = head[u]; head[u] = tp; vol[tp] = w; to[tp] = v; } bool bfs() { queue<int>q; q.push(s); last[s] = 0; while(q.size()) { int now = q.front(); q.pop(); if(now == t) break; for(int i=head[now]; i; i=nex[i]) { int v = to[i]; if(last[v] != -1 || vol[i] <= 0) continue; last[v] = i ^ 1; flow[v] = min(vol[i], flow[now]); q.push(v); } } return last[t] != -1; } ll EK() { ll ans = 0; flow[s] = inf; for(int i=0; i<=n; i++) last[i] = -1; while(bfs()) { for(int i=last[t]; i; i=last[to[i]]) { vol[i] += flow[t]; vol[i ^ 1] -= flow[t]; } ans += flow[t]; flow[s] = inf; for(int i=0; i<=n; i++) last[i] = -1; } return ans; } int main() { ios::sync_with_stdio(false); cin.tie(0); cout.tie(0); cin >> n >> m >> s >> t; while(m--) { int l, r, w; cin >> l >> r >> w; add(l, r, w); add(r, l, 0); } cout << EK() << endl; return 0; }
Dinic 演算法
用 bfs 分層之後,用 dfs 多次求該分層的所有增廣路
時間複雜度 \(O(v^2e)\),\(v\) 為點數,\(e\) 為邊數
#include <iostream> #include <cstdio> #include <queue> using namespace std; typedef long long ll; const int maxn = 1e5 + 10; const ll inf = 1e17 + 10; int head[maxn], nex[maxn], to[maxn], tp = 1, dep[maxn]; ll vol[maxn]; int n, m, s, t; void add(int l, int r, int w) { tp++; nex[tp] = head[l]; vol[tp] = w; to[tp] = r; head[l] = tp; } bool bfs() { queue<int>q; for(int i=0; i<=n; i++) dep[i] = -1; q.push(s); dep[s] = 0; while(q.size()) { int now = q.front(); q.pop(); for(int i=head[now]; i; i=nex[i]) { int u = to[i]; if(dep[u] != -1 || vol[i] <= 0) continue; dep[u] = dep[now] + 1; q.push(u); } } return dep[t] != -1; } ll dfs(int now, ll flow) { if(now == t) return flow; ll ans = 0; for(int i=head[now]; i && flow; i=nex[i]) { int u = to[i]; if(vol[i] <= 0 || dep[u] != dep[now] + 1) continue; ll f = dfs(u, min(flow, vol[i])); vol[i] -= f; vol[i ^ 1] += f; ans += f; flow -= f; } return ans; } ll dinic() { ll ans = 0; while(bfs()) ans += dfs(s, inf); return ans; } int main() { ios::sync_with_stdio(false); cin.tie(0); cout.tie(0); cin >> n >> m >> s >> t; while(m--) { int l, r, w; cin >> l >> r >> w; add(l, r, w); add(r, l, 0); } cout << dinic() << endl; return 0; }
Dinic + 弧優化
在一次 dfs 求增廣路的圖中,每流完一條邊,可以保證不會再被使用,所以可以在本次 dfs 中刪除該邊
時間複雜度不變,但是速度提高非常大
#include <iostream>
#include <cstdio>
#include <queue>
using namespace std;
typedef long long ll;
const int maxn = 1e5 + 10;
const ll inf = 1e17 + 10;
int head[maxn], nex[maxn], to[maxn], tp = 1, dep[maxn];
int cur[maxn];
ll vol[maxn];
int n, m, s, t;
void add(int l, int r, int w)
{
tp++;
nex[tp] = head[l];
vol[tp] = w;
to[tp] = r;
head[l] = tp;
}
bool bfs()
{
queue<int>q;
for(int i=0; i<=n; i++) dep[i] = -1;
for(int i=0; i<=n; i++) cur[i] = head[i];
q.push(s);
dep[s] = 0;
while(q.size())
{
int now = q.front();
q.pop();
for(int i=head[now]; i; i=nex[i])
{
int u = to[i];
if(dep[u] != -1 || vol[i] <= 0) continue;
dep[u] = dep[now] + 1;
q.push(u);
}
}
return dep[t] != -1;
}
ll dfs(int now, ll flow)
{
if(now == t)
return flow;
ll ans = 0;
for(int i=cur[now]; i && flow; i=nex[i])
{
cur[now] = i;
int u = to[i];
if(vol[i] <= 0 || dep[u] != dep[now] + 1) continue;
ll f = dfs(u, min(flow, vol[i]));
vol[i] -= f;
vol[i ^ 1] += f;
ans += f;
flow -= f;
}
return ans;
}
ll dinic()
{
ll ans = 0;
while(bfs())
ans += dfs(s, inf);
return ans;
}
int main()
{
ios::sync_with_stdio(false);
cin.tie(0);
cout.tie(0);
cin >> n >> m >> s >> t;
while(m--)
{
int l, r, w;
cin >> l >> r >> w;
add(l, r, w);
add(r, l, 0);
}
cout << dinic() << endl;
return 0;
}