#include "factories.h"
#include "bits/stdc++.h"
using namespace std;
#define fi first
#define se second
#define ll long long
#define dbg(v) cerr<<#v<<" = "<<v<<'\n'
#define vi vector<int>
#define vl vector <ll>
#define pii pair<int,int>
#define mp make_pair
#define db long double
#define pb push_back
#define all(s) s.begin(),s.end()
template < class P , class Q > ostream& operator<<(ostream& stream, pair < P , Q > v){ stream << "(" << v.fi << ',' << v.se << ")"; return stream;}
template < class T > ostream& operator<<(ostream& stream, const vector<T> v){ stream << "[ "; for (int i=0; i<(int)v.size(); i++) stream << v[i] << " "; stream << "]"; return stream;}
template < class T > T smin(T &a,T b) {if (a > b) a = b;return a;}
template < class T > T smax(T &a,T b) {if (a < b) a = b;return a;}
int n;
const int N = 2e6 + 5;
vector < pii > g[N];
int SZ[N];
void dfs_sz(int v = 0) {
SZ[v] = 1;
for (auto & u : g[v]) {
if (SZ[u.fi] != -1)
continue;
dfs_sz(u.fi);
SZ[v] += SZ[u.fi];
if (SZ[u.fi] > SZ[g[v][0].fi])
swap(u, g[v][0]);
}
}
int in[N];
int out[N];
int rin[N];
int nxt[N];
int pr[N];
int Time = 0;
ll D[N];
void dfs_hld(int v = 0) {
in[v] = ++Time;
rin[in[v]] = v;
for (auto u : g[v]) {
if (D[u.fi] != -1)
continue;
nxt[u.fi] = (u == g[v][0] ? nxt[v] : u.fi);
D[u.fi] = u.se + D[v];
pr[u.fi] = v;
dfs_hld(u.fi);
}
out[v] = Time;
}
ll t[N];
ll T[N];
ll lz[N];
int ok[N];
void upd_lz(int l,int r,int node) {
if (lz[node] != -1) {
if (l != r)
lz[node * 2] = lz[node * 2 + 1] = lz[node];
T[node] = t[node] + lz[node];
if (lz[node] == 0)
ok[node] = 0;
else
ok[node] = r - l + 1;
}
lz[node] = -1;
}
void build(int l,int r,int node) {
lz[node] = -1;
if (l == r)
return void(T[node] = t[node] = -2 * D[rin[l]]);
int m = (l + r) / 2;
build(l,m,node * 2);
build(m+1,r,node * 2 + 1);
T[node] = t[node] = min(t[node * 2],t[node * 2 + 1]);
}
void Upd(int l,int r,int node) {
if (l == r)
return;
int m = (l + r) / 2;
upd_lz(l,m,node * 2);
upd_lz(m + 1,r,node * 2 + 1);
T[node] = min(T[node * 2],T[node * 2 + 1]);
ok[node] = ok[node * 2] + ok[node * 2 + 1];
}
void update(int l,int r,int l1,int r1,ll v,int node) {
upd_lz(l,r,node);
if (l1 <= l && r <= r1)
lz[node] = v;
else {
int m = (l + r) / 2;
if (l1 <= m)
update(l,m,l1,r1,v,node * 2);
if (m+1<=r1)
update(m+1,r,l1,r1,v,node * 2 + 1);
Upd(l,r,node);
}
}
ll query(int l,int r,int l1,int r1,int node) {
upd_lz(l,r,node);
if (!ok[node])
return (ll)(1e18);
if (l1 <= l && r <= r1 && ok[node] == r - l + 1)
return T[node];
int m = (l + r) / 2;
ll ans = 1e18;
if (l1 <= m)
smin(ans,query(l,m,l1,r1,node * 2));
if (m+1<=r1)
smin(ans,query(m+1,r,l1,r1,node * 2 + 1));
Upd(l,r,node);
return ans;
}
void Init(int NN, int A[], int B[], int DD[]) {
n = NN;
for (int i = 0;i < n;++i) {
g[A[i]].pb(mp(B[i],DD[i]));
g[B[i]].pb(mp(A[i],DD[i]));
}
for (int i = 0;i < n;++i)
SZ[i] = D[i] = -1;
D[0] = 0;
dfs_sz();
dfs_hld();
build(1,n,1);
rin[0] = -1;
pr[0] = -1;
}
void upd(int node,ll v) {
while (node >= 0) {
update(1,n,in[nxt[node]],in[node],v,1);
node = pr[nxt[node]];
}
}
ll que(int node) {
ll ans = 1e18;
while (node >= 0) {
auto Q = query(1,n,in[nxt[node]],in[node],1);
smin(ans,Q);
node = pr[nxt[node]];
}
return ans;
}
long long Query(int S, int X[], int T, int Y[]) {
vi a,b;
for (int i = 0;i < S;++i)
a.pb(X[i]);
for (int i = 0;i < T;++i)
b.pb(Y[i]);
sort(all(a),[&](int x,int y) {
return D[x] > D[y];
});
for (auto it : a)
upd(it,D[it]);
ll mn1 = D[a[0]],mn2 = D[b[0]];
for (auto it : a)
smin(mn1,D[it]);
for (auto it : b)
smin(mn2,D[it]);
ll ans = mn1 + mn2;//vertex 0 isn't "covered"
for (auto it : b)
smin(ans,D[it] + que(it));
for (auto it : a)
upd(it,0ll);
return ans;
}
