// BUG: unchecked code
struct virtual_tree {
    int n, root;
    vector<vector<pil>> e;
    LCA lca;
    vector<int> dfn;
    vector<ll> ps;
    virtual_tree(const vector<vector<pil>>& e, int root) : n(e.size() - 1), root(root), e(e), lca(vector<vector<int>> {{}, {}}), dfn(n + 1), ps(n + 1) {
        vector<vector<int>> ch(n + 1);
        for (int u = 0; u <= n; ++u) {
            for (auto&& [v, w] : e[u]) {
                ch[u].emplace_back(v);
            }
        }
        lca = LCA(ch);
        int t = 0;
        auto dfs = [&] (auto dfs, int v, int pa) -> void {
            dfn[v] = ++t;
            for (auto&& [u, w] : e[v]) {
                if (u == pa) continue;
                ps[u] = ps[v] + w;
                dfs(dfs, u, v);
            }
        };
        dfs(dfs, root, 0);
    }
    // returned vertex number starts from 0
    vector<vector<pil>> generate(vector<int> pivots) {
        int m = pivots.size();
        if (m == 0) return {};
        sort_by_key(pivots.begin(), pivots.end(), expr(dfn[v], int v));
        vector<int> a;
        for (int i = 0; i < m - 1; ++i) {
            int v = pivots[i];
            int u = lca.query(pivots[i], pivots[i + 1]);
            a.emplace_back(v);
            a.emplace_back(u);
        }
        a.emplace_back(pivots[m - 1]);
        sort_by_key(a.begin(), a.end(), expr(dfn[v], int v));
        m = unique(a.begin(), a.end()) - a.begin();
        unordered_map<int, int, safe_hash> mp;
        for (int i = 0;  i < m; ++i) {
            mp[a[i]] = i;
        }
        vector<vector<pil>> ret(m);
        for (int i = 0; i < m - 1; ++i) {
            int l = lca.query(a[i], a[i + 1]);
            edgew(ret, mp[l], i + 1, ps[a[i + 1]] - ps[l]);
        }
        return ret;
    }
};