#include "factories.h"
#include <bits/stdc++.h>
using namespace std;
typedef long long ll;
typedef long double ld;
typedef double db;
typedef string str;
typedef pair<int,int> pi;
typedef pair<ll,ll> pl;
typedef pair<db,db> pd;
typedef vector<int> vi;
typedef vector<ll> vl;
typedef vector<db> vd;
typedef vector<str> vs;
typedef vector<pi> vpi;
typedef vector<pl> vpl;
typedef vector<pd> vpd;
#define mp make_pair
#define f first
#define s second
#define sz(x) (int)x.size()
#define all(x) begin(x), end(x)
#define rall(x) (x).rbegin(), (x).rend()
#define rsz resize
#define ins insert
#define ft front()
#define bk back()
#define pf push_front
#define pb push_back
#define eb emplace_back
#define lb lower_bound
#define ub upper_bound
#define FOR(i,a,b) for (int i = (a); i < (b); ++i)
#define F0R(i,a) FOR(i,0,a)
#define ROF(i,a,b) for (int i = (b)-1; i >= (a); --i)
#define R0F(i,a) ROF(i,0,a)
#define trav(a,x) for (auto& a: x)
const int MOD = 1e9+7; // 998244353;
const int MX = 2e5+5;
const ll INF = 1e18;
const ld PI = acos((ld)-1);
const int xd[4] = {1,0,-1,0}, yd[4] = {0,1,0,-1};
template<class T> bool ckmin(T& a, const T& b) {
return b < a ? a = b, 1 : 0; }
template<class T> bool ckmax(T& a, const T& b) {
return a < b ? a = b, 1 : 0; }
int pct(int x) { return __builtin_popcount(x); }
int bit(int x) { return 31-__builtin_clz(x); } // floor(log2(x))
int cdiv(int a, int b) { return a/b+!(a<0||a%b == 0); } // division of a by b rounded up, assumes b > 0
// INPUT
template<class A> void re(complex<A>& c);
template<class A, class B> void re(pair<A,B>& p);
template<class A> void re(vector<A>& v);
template<class A, size_t SZ> void re(array<A,SZ>& a);
template<class T> void re(T& x) { cin >> x; }
void re(db& d) { str t; re(t); d = stod(t); }
void re(ld& d) { str t; re(t); d = stold(t); }
template<class H, class... T> void re(H& h, T&... t) { re(h); re(t...); }
template<class A> void re(complex<A>& c) { A a,b; re(a,b); c = {a,b}; }
template<class A, class B> void re(pair<A,B>& p) { re(p.f,p.s); }
template<class A> void re(vector<A>& x) { trav(a,x) re(a); }
template<class A, size_t SZ> void re(array<A,SZ>& x) { trav(a,x) re(a); }
// TO_STRING
#define ts to_string
template<class A, class B> str ts(pair<A,B> p);
template<class A> str ts(complex<A> c) { return ts(mp(c.real(),c.imag())); }
str ts(bool b) { return b ? "true" : "false"; }
str ts(char c) { str s = ""; s += c; return s; }
str ts(str s) { return s; }
str ts(const char* s) { return (str)s; }
str ts(vector<bool> v) {
bool fst = 1; str res = "{";
F0R(i,sz(v)) {
if (!fst) res += ", ";
fst = 0; res += ts(v[i]);
}
res += "}"; return res;
}
template<size_t SZ> str ts(bitset<SZ> b) {
str res = ""; F0R(i,SZ) res += char('0'+b[i]);
return res; }
template<class T> str ts(T v) {
bool fst = 1; str res = "{";
for (const auto& x: v) {
if (!fst) res += ", ";
fst = 0; res += ts(x);
}
res += "}"; return res;
}
template<class A, class B> str ts(pair<A,B> p) {
return "("+ts(p.f)+", "+ts(p.s)+")"; }
// OUTPUT
template<class A> void pr(A x) { cout << ts(x); }
template<class H, class... T> void pr(const H& h, const T&... t) {
pr(h); pr(t...); }
void ps() { pr("\n"); } // print w/ spaces
template<class H, class... T> void ps(const H& h, const T&... t) {
pr(h); if (sizeof...(t)) pr(" "); ps(t...); }
// DEBUG
void DBG() { cerr << "]" << endl; }
template<class H, class... T> void DBG(H h, T... t) {
cerr << to_string(h); if (sizeof...(t)) cerr << ", ";
DBG(t...); }
#ifdef LOCAL // compile with -DLOCAL
#define dbg(...) cerr << "[" << #__VA_ARGS__ << "]: [", DBG(__VA_ARGS__)
#else
#define dbg(...) 42
#endif
// FILE I/O
void setIn(string s) { freopen(s.c_str(),"r",stdin); }
void setOut(string s) { freopen(s.c_str(),"w",stdout); }
void unsyncIO() { ios_base::sync_with_stdio(0); cin.tie(0); }
void setIO(string s = "") {
unsyncIO();
// cin.exceptions(cin.failbit);
// throws exception when do smth illegal
// ex. try to read letter into int
if (sz(s)) { setIn(s+".in"), setOut(s+".out"); } // for USACO
}
mt19937 rng((uint32_t)chrono::steady_clock::now().time_since_epoch().count());
/**
* Description: 1D range minimum query. Can also do queries
* for any associative operation in $O(1)$ with D\&C
* Source: KACTL
* Verification:
* https://cses.fi/problemset/stats/1647/
* http://wcipeg.com/problem/ioi1223
* https://pastebin.com/ChpniVZL
* Memory: O(N\log N)
* Time: O(1)
*/
template<class T> struct RMQ { // floor(log_2(x))
int level(int x) { return 31-__builtin_clz(x); }
vector<T> v; vector<vi> jmp;
int comb(int a, int b) { // index of min
return v[a]==v[b]?min(a,b):(v[a]<v[b]?a:b); }
void init(const vector<T>& _v) {
v = _v; jmp = {vi(sz(v))}; iota(all(jmp[0]),0);
for (int j = 1; 1<<j <= sz(v); ++j) {
jmp.pb(vi(sz(v)-(1<<j)+1));
F0R(i,sz(jmp[j])) jmp[j][i] = comb(jmp[j-1][i],
jmp[j-1][i+(1<<(j-1))]);
}
}
int index(int l, int r) { // get index of min element
int d = level(r-l+1);
return comb(jmp[d][l],jmp[d][r-(1<<d)+1]); }
T query(int l, int r) { return v[index(l,r)]; }
};
/**
* Description: Euler Tour LCA w/ O(1) query. Compress takes a subset $S$
* of nodes and computes the minimal subtree that contains all the nodes
* pairwise LCA's and compressing edges. Returns a list of (par, orig\_index)
* representing a tree rooted at 0. The root points to itself.
* Source: USACO, Simon Lindholm (KACTL)
* Verification: USACO Debug the Bugs
* https://codeforces.com/contest/1320/problem/E
*/
template<int SZ> struct LCA {
int N, R = 1, depth[SZ], st[SZ];
ll dist[SZ];
vector<pair<int, ll>> adj[SZ]; vpi tmp; RMQ<pi> r;
void ae(int u, int v, ll d) { adj[u].pb(mp(v, d)), adj[v].pb(mp(u, d)); }
void dfs(int u, int prev, ll d = 0){
st[u] = sz(tmp), depth[u] = depth[prev]+1;
dist[u] = d;
tmp.pb({depth[u],u});
trav(v,adj[u]) if (v.f != prev)
dfs(v.f,u,d+v.s), tmp.pb({depth[u],u});
}
void init(int _N) { N = _N; dfs(R,0); r.init(tmp); }
int lca(int u, int v){
u = st[u], v = st[v]; if (u > v) swap(u,v);
return r.query(u,v).s; }
ll getDist(int u, int v) {
return dist[u]+dist[v]-2*dist[lca(u,v)]; }
vpi compress(vi S) {
static vi rev; rev.rsz(N+1);
auto cmp = [&](int a, int b) { return st[a] < st[b]; };
sort(all(S),cmp);
int m = sz(S)-1; F0R(i,m) S.pb(lca(S[i],S[i+1]));
sort(all(S),cmp); S.erase(unique(all(S)),end(S));
F0R(i,sz(S)) rev[S[i]] = i;
vpi ret = {pi(0,S[0])};
F0R(i,sz(S)-1) ret.eb(rev[lca(S[i],S[i+1])],S[i+1]);
return ret;
}
};
const int mx = 500005;
LCA<mx> tr;
int N;
ll bound;
bool isY[mx];
ll highest[mx][2];
int typ[mx];
void Init(int _N, int A[], int B[], int D[]) {
N = _N;
for(int i = 0; i <= N-2; i++){
int a = A[i];
int b = B[i];
a++;
b++;
ll d = D[i];
tr.ae(a, b, d);
}
tr.init(N+2);
bound = ll(N)*ll(log2(N)*0.5);
for(int i = 1; i <= N; i++){
typ[i] = 2;
highest[i][0] = INF;
highest[i][1] = INF;
}
//bound = 0;
/*for(int i = 1; i <= N; i++){
ps(i, tr.dist[i]);
}*/
}
ll ans;
vi children[mx];
void search(int node){
for(auto u: children[node]){
search(u);
for(int j = 0; j < 2; j++){
ckmin(highest[node][j], highest[u][j]);
}
}
if(typ[node] != 2){
int t = typ[node];
ckmin(highest[node][t], tr.getDist(node, 1));
}
}
long long Query(int S, int X[], int T, int Y[]) {
for(int i = 0; i < S; i++) X[i]++;
for(int i = 0; i < T; i++) Y[i]++;
vi nodes;
for(int i = 0; i < S; i++){
nodes.pb(X[i]);
typ[X[i]] = 0;
}
for(int i = 0; i < T; i++){
nodes.pb(Y[i]);
typ[Y[i]] = 1;
}
vpi par = tr.compress(nodes);
/*for(int i = 1; i <= N; i++){
ps(i, tr.dist[i]);
}
ps("nodes", nodes);
ps("par", par);*/
for(auto u: par){
if(par[u.f].s == u.s) continue;
children[par[u.f].s].pb(u.s);
}
/*for(int i = 1; i <= N; i++){
ps(i, children[i]);
}*/
search(par[0].s);
ans = INF;
for(auto x: par){
int u = x.s;
ckmin(ans, highest[u][0]+highest[u][1]-2*tr.getDist(1, u));
}
//reset arrays based on nodes
for(auto x: par){
int u = x.s;
typ[u] = 2;
children[u].clear();
highest[u][0] = highest[u][1] = INF;
}
return ans;
}
/*int main() {
setIO();
// you should actually read the stuff at the bottom
}*/
/* stuff you should look for
* int overflow, array bounds
* special cases (n=1?)
* do smth instead of nothing and stay organized
* WRITE STUFF DOWN
*/
