#include<bits/stdc++.h>
#define fi first
#define se second
#define endl "\n"
#define ii pair<int, int>
using namespace std;

const int N = 1e5 + 10;

int p[N], sz[N], deg[N], a[N], b[N], order[N];

long long dist[20][N], f[N];

bool dd[N];

vector<pair<int, int> > adj[N];

struct line {
  long long m, c;
  long long eval(long long x) { return m * x + c; }
  float cut(line l) {
      return (float) (c - l.c) / (l.m - m);
  }
};

struct CHT {

vector<line> st;

void add(line cur) {
  if (!st.empty() && st.back().m == cur.m) {
    if (st.back().c >= cur.c) return;
    st.pop_back();
  }
  while (st.size() >= 2 && cur.cut(st.back()) <= st.back().cut(st.end()[-2])) st.pop_back();
  st.push_back(cur);
}

long long get(long long x) {
  if (st.empty()) return 1e18;
  int l = 0, r = st.size() - 1;
  while (l < r) {
    int mid = l + r >> 1;
    if (st[mid].eval(x) >= st[mid + 1].eval(x)) l = mid + 1;
    else r = mid;
  }
  return st[r].eval(x);
}

} cht[N];

int calc(int u, int par) {
  sz[u] = 1;
  for(auto &[v, w] : adj[u]) if (v != par && !dd[v]) sz[u] += calc(v, u);
  return sz[u];
}

int getcentr (int u, int par, int cnt) {
  for(auto &[v, w] : adj[u]) if (v != par && !dd[v] && sz[v] * 2 > cnt) {
    return getcentr (v, u, cnt);
  }
  return u;
}

void dfs(int u, int par, int j) {
  for(auto &[v, w] : adj[u]) if (v != par && !dd[v]) {
    dist[j][v] = dist[j][u] + w;
    dfs(v, u, j);
  }
}

void centroid(int u, int par) {
  int node = getcentr(u, u, calc(u, u));
  p[node] = par;
  dd[node] = 1;
  deg[node] = deg[par] + 1;
  dfs(node, node, deg[node]);
  for(auto &[v, w] : adj[node]) if (!dd[v]) centroid(v, node);
}

void add(int u) {
  int v = u;
  while (v) {
    cht[v].add({b[u], f[u] + dist[deg[v]][u] * b[u] + a[u]});
    v = p[v];
  }
}

long long get(int u) {
   int v = u;
   long long ret = 1e18;
   while (v) {
      ret = min(ret, cht[v].get(dist[deg[v]][u]));
      v = p[v];
   }
   return ret;
}

vector<long long> travel (vector<long long> A, vector<int> B, vector<int> U, vector<int> V, vector<int> W) {
  int n = A.size();
  for(int i = 1; i <= n; i++) a[i] = A[i - 1], b[i] = B[i - 1];
  for(int i = 0; i < n - 1; i++) {
    U[i]++;
    V[i]++;
    adj[U[i]].emplace_back(V[i], W[i]);
    adj[V[i]].emplace_back(U[i], W[i]);
  }

  centroid(1, 0);

  for(int i = 1; i <= n; i++) order[i] = i;
  sort(order + 1, order + n + 1, [] (const int &x, const int &y) {
       return b[x] > b[y];
       });

  int one = 0;
//  for(int i = 1; i <= n; i++) cout << order[i] << " "; cout << endl;
  for(int cur = 1; cur <= n; cur++) if (order[cur] == 1) one = cur;
  assert(one);
  add(1);
//  cout << get(2);
//  exit(0);
  for(int cur = one + 1; cur <= n; cur++) {
    int i = order[cur];
    f[i] = get(i);
//    cout << i << " " << f[i] << endl;
    add(i);
  }
//  exit(0);
  vector<long long> ret;
  for(int i = 2; i <= n; i++) {
    long long cost = get(i);
    ret.push_back(cost);
  }
  return ret;
}

#ifdef ngu

int main() {
  freopen ("task.inp", "r", stdin);
  freopen ("task.out", "w", stdout);

    int n;
    cin >> n;
    vector<long long> A(n);
    vector<int> B(n), U(n - 1), V(n - 1), W(n - 1);

    for(int i = 0; i < n; i++) cin >> A[i];
    for(int i = 0; i < n; i++) cin >> B[i];

    for(int i = 0; i < n - 1; i++) {
      cin >> U[i] >> V[i] >> W[i];
    }

    vector<long long> ans = travel(A, B, U, V, W);
    for(auto &x : ans) cout << x << " ";

//    for (long long x : travel(A, B, U, V, W)) {
//        cout << x << ' ';
//    }
    cout << '\n';
}
#endif // ngu