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:heavy_check_mark: graph/ds/rolling_hash_on_tree.hpp

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#include "random/base.hpp"
#include "graph/tree.hpp"
#include "mod/modint61.hpp"

// 木の辺に文字がちょうどひとつ書いてある (static)
template <typename TREE, bool EDGE>
struct Rolling_Hash_On_Tree {
  using mint = modint61;

  TREE& tree;
  int N;
  mint base;
  // edge に文字があると思って
  // dp1: [root,v]
  // dp2: [v,root]
  vc<int> dat;
  vc<mint> dp1, dp2;
  vc<mint> pow, ipow;

  template <typename F>
  Rolling_Hash_On_Tree(TREE& tree, F f, mint base_ = 0)
      : tree(tree), N(tree.N), base(base_) {
    if (base == mint(0)) base = RNG(mint::get_mod());
    build(f);
  }

  template <typename F>
  void build(F f) {
    dat.resize(N);
    if constexpr (EDGE) {
      FOR(i, N - 1) { dat[tree.e_to_v(i)] = f(i); }
    } else {
      FOR(i, N) { dat[i] = f(i); }
    }
    pow.resize(N + 1), ipow.resize(N + 1);
    pow[0] = 1, pow[1] = base;
    ipow[0] = 1, ipow[1] = base.inverse();
    FOR(i, 2, N + 1) pow[i] = pow[i - 1] * pow[1];
    FOR(i, 2, N + 1) ipow[i] = ipow[i - 1] * ipow[1];
    int root = tree.V[0];
    dp1.resize(N), dp2.resize(N);
    dp1[root] = dp2[root] = dat[0];
    FOR(i, 1, N) {
      int v = tree.V[i];
      int d = tree.depth[v], p = tree.parent[v];
      dp1[v] = base * dp1[p] + dat[v];
      dp2[v] = dp2[p] + pow[d] * dat[v];
    }
  }

  mint get(int a, int b) {
    int c = tree.lca(a, b);
    mint x1 = get_du(a, c), x2 = get_ud(c, b);
    int n2 = tree.depth[b] - tree.depth[c];
    if constexpr (!EDGE) { x1 = x1 * base + dat[c]; }
    return x1 * pow[n2] + x2;
  }

  int lcp(int s1, int t1, int s2, int t2) {
    return lcp_and_comp(s1, t1, s2, t2).fi;
  }

  // <=>
  char comp(int s1, int t1, int s2, int t2) {
    return lcp_and_comp(s1, t1, s2, t2).se;
  }

  pair<int, char> lcp_and_comp(int s1, int t1, int s2, int t2) {
    int lcp = 0;
    // heavy path の頂点列
    auto path1 = tree.get_path_decomposition(s1, t1, EDGE);
    auto path2 = tree.get_path_decomposition(s2, t2, EDGE);
    reverse(all(path1));
    reverse(all(path2));
    while (len(path1) && len(path2)) {
      int a, b, c, d;
      tie(a, b) = POP(path1), tie(c, d) = POP(path2);
      ll n1 = abs(a - b) + 1, n2 = abs(c - d) + 1;
      ll n = min(n1, n2);
      if (n < n1) {
        if (a <= b) { path1.eb(a + n, b), b = a + n - 1; }
        if (a > b) { path1.eb(a - n, b), b = a - n + 1; }
      }
      if (n < n2) {
        if (c <= d) { path2.eb(c + n, d), d = c + n - 1; }
        if (c > d) { path2.eb(c - n, d), d = c - n + 1; }
      }
      mint x1 = from_hld_pair(a, b), x2 = from_hld_pair(c, d);
      if (x1 == x2) {
        lcp += n;
        continue;
      }
      auto check = [&](ll n) -> bool {
        if (n == 0) return 1;
        mint x1 = (a <= b ? from_hld_pair(a, a + n - 1)
                          : from_hld_pair(a, a - n + 1));
        mint x2 = (c <= d ? from_hld_pair(c, c + n - 1)
                          : from_hld_pair(c, c - n + 1));
        return x1 == x2;
      };
      ll k = binary_search(check, 0, n);
      lcp += k;
      a = (a <= b ? a + k : a - k);
      c = (c <= d ? c + k : c - k);
      a = tree.V[a], c = tree.V[c];
      if (dat[a] < dat[c]) return {lcp, '<'};
      if (dat[a] == dat[c]) return {lcp, '='};
      if (dat[a] > dat[c]) return {lcp, '>'};
    }
    if (!path1.empty()) return {lcp, '>'};
    if (!path2.empty()) return {lcp, '<'};
    return {lcp, '='};
  }

private:
  mint get_ud(int a, int b) {
    return (a == -1 ? dp1[b]
                    : dp1[b] - dp1[a] * pow[tree.depth[b] - tree.depth[a]]);
  }
  mint get_du(int a, int b) {
    return (b == -1 ? dp2[a] : (dp2[a] - dp2[b]) * ipow[tree.depth[b] + 1]);
  }
  mint from_hld_pair(int a, int b) {
    if (a <= b) { return get_ud(tree.parent[tree.V[a]], tree.V[b]); }
    return get_du(tree.V[a], tree.parent[tree.V[b]]);
  }
};
#line 2 "random/base.hpp"

u64 RNG_64() {
  static u64 x_ = u64(chrono::duration_cast<chrono::nanoseconds>(chrono::high_resolution_clock::now().time_since_epoch()).count()) * 10150724397891781847ULL;
  x_ ^= x_ << 7;
  return x_ ^= x_ >> 9;
}

u64 RNG(u64 lim) { return RNG_64() % lim; }

ll RNG(ll l, ll r) { return l + RNG_64() % (r - l); }
#line 2 "graph/tree.hpp"

#line 2 "ds/hashmap.hpp"

// u64 -> Val

template <typename Val>
struct HashMap {
  // n は入れたいものの個数で ok

  HashMap(u32 n = 0) { build(n); }
  void build(u32 n) {
    u32 k = 8;
    while (k < n * 2) k *= 2;
    cap = k / 2, mask = k - 1;
    key.resize(k), val.resize(k), used.assign(k, 0);
  }

  // size を保ったまま. size=0 にするときは build すること.

  void clear() {
    used.assign(len(used), 0);
    cap = (mask + 1) / 2;
  }
  int size() { return len(used) / 2 - cap; }

  int index(const u64& k) {
    int i = 0;
    for (i = hash(k); used[i] && key[i] != k; i = (i + 1) & mask) {}
    return i;
  }

  Val& operator[](const u64& k) {
    if (cap == 0) extend();
    int i = index(k);
    if (!used[i]) { used[i] = 1, key[i] = k, val[i] = Val{}, --cap; }
    return val[i];
  }

  Val get(const u64& k, Val default_value) {
    int i = index(k);
    return (used[i] ? val[i] : default_value);
  }

  bool count(const u64& k) {
    int i = index(k);
    return used[i] && key[i] == k;
  }

  // f(key, val)

  template <typename F>
  void enumerate_all(F f) {
    FOR(i, len(used)) if (used[i]) f(key[i], val[i]);
  }

private:
  u32 cap, mask;
  vc<u64> key;
  vc<Val> val;
  vc<bool> used;

  u64 hash(u64 x) {
    static const u64 FIXED_RANDOM = std::chrono::steady_clock::now().time_since_epoch().count();
    x += FIXED_RANDOM;
    x = (x ^ (x >> 30)) * 0xbf58476d1ce4e5b9;
    x = (x ^ (x >> 27)) * 0x94d049bb133111eb;
    return (x ^ (x >> 31)) & mask;
  }

  void extend() {
    vc<pair<u64, Val>> dat;
    dat.reserve(len(used) / 2 - cap);
    FOR(i, len(used)) {
      if (used[i]) dat.eb(key[i], val[i]);
    }
    build(2 * len(dat));
    for (auto& [a, b]: dat) (*this)[a] = b;
  }
};
#line 3 "graph/base.hpp"

template <typename T>
struct Edge {
  int frm, to;
  T cost;
  int id;
};

template <typename T = int, bool directed = false>
struct Graph {
  static constexpr bool is_directed = directed;
  int N, M;
  using cost_type = T;
  using edge_type = Edge<T>;
  vector<edge_type> edges;
  vector<int> indptr;
  vector<edge_type> csr_edges;
  vc<int> vc_deg, vc_indeg, vc_outdeg;
  bool prepared;

  class OutgoingEdges {
  public:
    OutgoingEdges(const Graph* G, int l, int r) : G(G), l(l), r(r) {}

    const edge_type* begin() const {
      if (l == r) { return 0; }
      return &G->csr_edges[l];
    }

    const edge_type* end() const {
      if (l == r) { return 0; }
      return &G->csr_edges[r];
    }

  private:
    const Graph* G;
    int l, r;
  };

  bool is_prepared() { return prepared; }

  Graph() : N(0), M(0), prepared(0) {}
  Graph(int N) : N(N), M(0), prepared(0) {}

  void build(int n) {
    N = n, M = 0;
    prepared = 0;
    edges.clear();
    indptr.clear();
    csr_edges.clear();
    vc_deg.clear();
    vc_indeg.clear();
    vc_outdeg.clear();
  }

  void add(int frm, int to, T cost = 1, int i = -1) {
    assert(!prepared);
    assert(0 <= frm && 0 <= to && to < N);
    if (i == -1) i = M;
    auto e = edge_type({frm, to, cost, i});
    edges.eb(e);
    ++M;
  }

#ifdef FASTIO
  // wt, off
  void read_tree(bool wt = false, int off = 1) { read_graph(N - 1, wt, off); }

  void read_graph(int M, bool wt = false, int off = 1) {
    for (int m = 0; m < M; ++m) {
      INT(a, b);
      a -= off, b -= off;
      if (!wt) {
        add(a, b);
      } else {
        T c;
        read(c);
        add(a, b, c);
      }
    }
    build();
  }
#endif

  void build() {
    assert(!prepared);
    prepared = true;
    indptr.assign(N + 1, 0);
    for (auto&& e: edges) {
      indptr[e.frm + 1]++;
      if (!directed) indptr[e.to + 1]++;
    }
    for (int v = 0; v < N; ++v) { indptr[v + 1] += indptr[v]; }
    auto counter = indptr;
    csr_edges.resize(indptr.back() + 1);
    for (auto&& e: edges) {
      csr_edges[counter[e.frm]++] = e;
      if (!directed) csr_edges[counter[e.to]++] = edge_type({e.to, e.frm, e.cost, e.id});
    }
  }

  OutgoingEdges operator[](int v) const {
    assert(prepared);
    return {this, indptr[v], indptr[v + 1]};
  }

  vc<int> deg_array() {
    if (vc_deg.empty()) calc_deg();
    return vc_deg;
  }

  pair<vc<int>, vc<int>> deg_array_inout() {
    if (vc_indeg.empty()) calc_deg_inout();
    return {vc_indeg, vc_outdeg};
  }

  int deg(int v) {
    if (vc_deg.empty()) calc_deg();
    return vc_deg[v];
  }

  int in_deg(int v) {
    if (vc_indeg.empty()) calc_deg_inout();
    return vc_indeg[v];
  }

  int out_deg(int v) {
    if (vc_outdeg.empty()) calc_deg_inout();
    return vc_outdeg[v];
  }

#ifdef FASTIO
  void debug() {
#ifdef LOCAL
    print("Graph");
    if (!prepared) {
      print("frm to cost id");
      for (auto&& e: edges) print(e.frm, e.to, e.cost, e.id);
    } else {
      print("indptr", indptr);
      print("frm to cost id");
      FOR(v, N) for (auto&& e: (*this)[v]) print(e.frm, e.to, e.cost, e.id);
    }
#endif
  }
#endif

  vc<int> new_idx;
  vc<bool> used_e;

  // G における頂点 V[i] が、新しいグラフで i になるようにする
  // {G, es}
  // sum(deg(v)) の計算量になっていて、
  // 新しいグラフの n+m より大きい可能性があるので注意
  Graph<T, directed> rearrange(vc<int> V, bool keep_eid = 0) {
    if (len(new_idx) != N) new_idx.assign(N, -1);
    int n = len(V);
    FOR(i, n) new_idx[V[i]] = i;
    Graph<T, directed> G(n);
    vc<int> history;
    FOR(i, n) {
      for (auto&& e: (*this)[V[i]]) {
        if (len(used_e) <= e.id) used_e.resize(e.id + 1);
        if (used_e[e.id]) continue;
        int a = e.frm, b = e.to;
        if (new_idx[a] != -1 && new_idx[b] != -1) {
          history.eb(e.id);
          used_e[e.id] = 1;
          int eid = (keep_eid ? e.id : -1);
          G.add(new_idx[a], new_idx[b], e.cost, eid);
        }
      }
    }
    FOR(i, n) new_idx[V[i]] = -1;
    for (auto&& eid: history) used_e[eid] = 0;
    G.build();
    return G;
  }

  Graph<T, true> to_directed_tree(int root = -1) {
    if (root == -1) root = 0;
    assert(!is_directed && prepared && M == N - 1);
    Graph<T, true> G1(N);
    vc<int> par(N, -1);
    auto dfs = [&](auto& dfs, int v) -> void {
      for (auto& e: (*this)[v]) {
        if (e.to == par[v]) continue;
        par[e.to] = v, dfs(dfs, e.to);
      }
    };
    dfs(dfs, root);
    for (auto& e: edges) {
      int a = e.frm, b = e.to;
      if (par[a] == b) swap(a, b);
      assert(par[b] == a);
      G1.add(a, b, e.cost);
    }
    G1.build();
    return G1;
  }

  HashMap<int> MP_FOR_EID;

  int get_eid(u64 a, u64 b) {
    if (len(MP_FOR_EID) == 0) {
      MP_FOR_EID.build(N - 1);
      for (auto& e: edges) {
        u64 a = e.frm, b = e.to;
        u64 k = to_eid_key(a, b);
        MP_FOR_EID[k] = e.id;
      }
    }
    return MP_FOR_EID.get(to_eid_key(a, b), -1);
  }

  u64 to_eid_key(u64 a, u64 b) {
    if (!directed && a > b) swap(a, b);
    return N * a + b;
  }

private:
  void calc_deg() {
    assert(vc_deg.empty());
    vc_deg.resize(N);
    for (auto&& e: edges) vc_deg[e.frm]++, vc_deg[e.to]++;
  }

  void calc_deg_inout() {
    assert(vc_indeg.empty());
    vc_indeg.resize(N);
    vc_outdeg.resize(N);
    for (auto&& e: edges) { vc_indeg[e.to]++, vc_outdeg[e.frm]++; }
  }
};
#line 4 "graph/tree.hpp"

// HLD euler tour をとっていろいろ。

template <typename GT>
struct Tree {
  using Graph_type = GT;
  GT &G;
  using WT = typename GT::cost_type;
  int N;
  vector<int> LID, RID, head, V, parent, VtoE;
  vc<int> depth;
  vc<WT> depth_weighted;

  Tree(GT &G, int r = 0, bool hld = 1) : G(G) { build(r, hld); }

  void build(int r = 0, bool hld = 1) {
    if (r == -1) return; // build を遅延したいとき

    N = G.N;
    LID.assign(N, -1), RID.assign(N, -1), head.assign(N, r);
    V.assign(N, -1), parent.assign(N, -1), VtoE.assign(N, -1);
    depth.assign(N, -1), depth_weighted.assign(N, 0);
    assert(G.is_prepared());
    int t1 = 0;
    dfs_sz(r, -1, hld);
    dfs_hld(r, t1);
  }

  void dfs_sz(int v, int p, bool hld) {
    auto &sz = RID;
    parent[v] = p;
    depth[v] = (p == -1 ? 0 : depth[p] + 1);
    sz[v] = 1;
    int l = G.indptr[v], r = G.indptr[v + 1];
    auto &csr = G.csr_edges;
    // 使う辺があれば先頭にする

    for (int i = r - 2; i >= l; --i) {
      if (hld && depth[csr[i + 1].to] == -1) swap(csr[i], csr[i + 1]);
    }
    int hld_sz = 0;
    for (int i = l; i < r; ++i) {
      auto e = csr[i];
      if (depth[e.to] != -1) continue;
      depth_weighted[e.to] = depth_weighted[v] + e.cost;
      VtoE[e.to] = e.id;
      dfs_sz(e.to, v, hld);
      sz[v] += sz[e.to];
      if (hld && chmax(hld_sz, sz[e.to]) && l < i) { swap(csr[l], csr[i]); }
    }
  }

  void dfs_hld(int v, int &times) {
    LID[v] = times++;
    RID[v] += LID[v];
    V[LID[v]] = v;
    bool heavy = true;
    for (auto &&e: G[v]) {
      if (depth[e.to] <= depth[v]) continue;
      head[e.to] = (heavy ? head[v] : e.to);
      heavy = false;
      dfs_hld(e.to, times);
    }
  }

  vc<int> heavy_path_at(int v) {
    vc<int> P = {v};
    while (1) {
      int a = P.back();
      for (auto &&e: G[a]) {
        if (e.to != parent[a] && head[e.to] == v) {
          P.eb(e.to);
          break;
        }
      }
      if (P.back() == a) break;
    }
    return P;
  }

  int heavy_child(int v) {
    int k = LID[v] + 1;
    if (k == N) return -1;
    int w = V[k];
    return (parent[w] == v ? w : -1);
  }

  vc<int> memo_tail;

  int tail(int v) {
    if (memo_tail.empty()) {
      memo_tail.assign(N, -1);
      FOR_R(i, N) {
        int v = V[i];
        int w = heavy_child(v);
        memo_tail[v] = (w == -1 ? v : memo_tail[w]);
      }
    }
    return memo_tail[v];
  }

  int e_to_v(int eid) {
    auto e = G.edges[eid];
    return (parent[e.frm] == e.to ? e.frm : e.to);
  }
  int v_to_e(int v) { return VtoE[v]; }
  int get_eid(int u, int v) {
    if (parent[u] != v) swap(u, v);
    assert(parent[u] == v);
    return VtoE[u];
  }

  int ELID(int v) { return 2 * LID[v] - depth[v]; }
  int ERID(int v) { return 2 * RID[v] - depth[v] - 1; }

  // 目標地点へ進む個数が k

  int LA(int v, int k) {
    assert(k <= depth[v]);
    while (1) {
      int u = head[v];
      if (LID[v] - k >= LID[u]) return V[LID[v] - k];
      k -= LID[v] - LID[u] + 1;
      v = parent[u];
    }
  }
  int la(int u, int v) { return LA(u, v); }

  int LCA(int u, int v) {
    for (;; v = parent[head[v]]) {
      if (LID[u] > LID[v]) swap(u, v);
      if (head[u] == head[v]) return u;
    }
  }

  int meet(int a, int b, int c) { return LCA(a, b) ^ LCA(a, c) ^ LCA(b, c); }
  int lca(int u, int v) { return LCA(u, v); }

  int subtree_size(int v, int root = -1) {
    if (root == -1) return RID[v] - LID[v];
    if (v == root) return N;
    int x = jump(v, root, 1);
    if (in_subtree(v, x)) return RID[v] - LID[v];
    return N - RID[x] + LID[x];
  }

  int dist(int a, int b) {
    int c = LCA(a, b);
    return depth[a] + depth[b] - 2 * depth[c];
  }

  WT dist_weighted(int a, int b) {
    int c = LCA(a, b);
    return depth_weighted[a] + depth_weighted[b] - WT(2) * depth_weighted[c];
  }

  // a is in b

  bool in_subtree(int a, int b) { return LID[b] <= LID[a] && LID[a] < RID[b]; }

  int jump(int a, int b, ll k) {
    if (k == 1) {
      if (a == b) return -1;
      return (in_subtree(b, a) ? LA(b, depth[b] - depth[a] - 1) : parent[a]);
    }
    int c = LCA(a, b);
    int d_ac = depth[a] - depth[c];
    int d_bc = depth[b] - depth[c];
    if (k > d_ac + d_bc) return -1;
    if (k <= d_ac) return LA(a, k);
    return LA(b, d_ac + d_bc - k);
  }

  vc<int> collect_child(int v) {
    vc<int> res;
    for (auto &&e: G[v])
      if (e.to != parent[v]) res.eb(e.to);
    return res;
  }

  vc<int> collect_subtree(int v) { return {V.begin() + LID[v], V.begin() + RID[v]}; }

  vc<int> collect_light(int v) {
    vc<int> res;
    bool skip = true;
    for (auto &&e: G[v])
      if (e.to != parent[v]) {
        if (!skip) res.eb(e.to);
        skip = false;
      }
    return res;
  }

  vc<pair<int, int>> get_path_decomposition(int u, int v, bool edge) {
    // [始点, 終点] の"閉"区間列。

    vc<pair<int, int>> up, down;
    while (1) {
      if (head[u] == head[v]) break;
      if (LID[u] < LID[v]) {
        down.eb(LID[head[v]], LID[v]);
        v = parent[head[v]];
      } else {
        up.eb(LID[u], LID[head[u]]);
        u = parent[head[u]];
      }
    }
    if (LID[u] < LID[v]) down.eb(LID[u] + edge, LID[v]);
    elif (LID[v] + edge <= LID[u]) up.eb(LID[u], LID[v] + edge);
    reverse(all(down));
    up.insert(up.end(), all(down));
    return up;
  }

  // 辺の列の情報 (frm,to,str)

  // str = "heavy_up", "heavy_down", "light_up", "light_down"

  vc<tuple<int, int, string>> get_path_decomposition_detail(int u, int v) {
    vc<tuple<int, int, string>> up, down;
    while (1) {
      if (head[u] == head[v]) break;
      if (LID[u] < LID[v]) {
        if (v != head[v]) down.eb(head[v], v, "heavy_down"), v = head[v];
        down.eb(parent[v], v, "light_down"), v = parent[v];
      } else {
        if (u != head[u]) up.eb(u, head[u], "heavy_up"), u = head[u];
        up.eb(u, parent[u], "light_up"), u = parent[u];
      }
    }
    if (LID[u] < LID[v]) down.eb(u, v, "heavy_down");
    elif (LID[v] < LID[u]) up.eb(u, v, "heavy_up");
    reverse(all(down));
    concat(up, down);
    return up;
  }

  vc<int> restore_path(int u, int v) {
    vc<int> P;
    for (auto &&[a, b]: get_path_decomposition(u, v, 0)) {
      if (a <= b) {
        FOR(i, a, b + 1) P.eb(V[i]);
      } else {
        FOR_R(i, b, a + 1) P.eb(V[i]);
      }
    }
    return P;
  }

  // path [a,b] と [c,d] の交わり. 空ならば {-1,-1}.

  // https://codeforces.com/problemset/problem/500/G

  pair<int, int> path_intersection(int a, int b, int c, int d) {
    int ab = lca(a, b), ac = lca(a, c), ad = lca(a, d);
    int bc = lca(b, c), bd = lca(b, d), cd = lca(c, d);
    int x = ab ^ ac ^ bc, y = ab ^ ad ^ bd; // meet(a,b,c), meet(a,b,d)

    if (x != y) return {x, y};
    int z = ac ^ ad ^ cd;
    if (x != z) x = -1;
    return {x, x};
  }

  // uv path 上で check(v) を満たす最後の v

  // なければ （つまり check(v) が ng ）-1

  template <class F>
  int max_path(F check, int u, int v) {
    if (!check(u)) return -1;
    auto pd = get_path_decomposition(u, v, false);
    for (auto [a, b]: pd) {
      if (!check(V[a])) return u;
      if (check(V[b])) {
        u = V[b];
        continue;
      }
      int c = binary_search([&](int c) -> bool { return check(V[c]); }, a, b, 0);
      return V[c];
    }
    return u;
  }
};
#line 2 "mod/modint61.hpp"

struct modint61 {
  static constexpr u64 mod = (1ULL << 61) - 1;
  u64 val;
  constexpr modint61() : val(0ULL) {}
  constexpr modint61(u32 x) : val(x) {}
  constexpr modint61(u64 x) : val(x % mod) {}
  constexpr modint61(int x) : val((x < 0) ? (x + static_cast<ll>(mod)) : x) {}
  constexpr modint61(ll x) : val(((x %= static_cast<ll>(mod)) < 0) ? (x + static_cast<ll>(mod)) : x) {}
  static constexpr u64 get_mod() { return mod; }

  modint61 &operator+=(const modint61 &a) {
    val = ((val += a.val) >= mod) ? (val - mod) : val;
    return *this;
  }
  modint61 &operator-=(const modint61 &a) {
    val = ((val -= a.val) >= mod) ? (val + mod) : val;
    return *this;
  }
  modint61 &operator*=(const modint61 &a) {
    const unsigned __int128 y = static_cast<unsigned __int128>(val) * a.val;
    val = (y >> 61) + (y & mod);
    val = (val >= mod) ? (val - mod) : val;
    return *this;
  }
  modint61 operator-() const { return modint61(val ? mod - val : u64(0)); }
  modint61 &operator/=(const modint61 &a) { return (*this *= a.inverse()); }
  modint61 operator+(const modint61 &p) const { return modint61(*this) += p; }
  modint61 operator-(const modint61 &p) const { return modint61(*this) -= p; }
  modint61 operator*(const modint61 &p) const { return modint61(*this) *= p; }
  modint61 operator/(const modint61 &p) const { return modint61(*this) /= p; }
  bool operator<(const modint61 &other) const { return val < other.val; }
  bool operator==(const modint61 &p) const { return val == p.val; }
  bool operator!=(const modint61 &p) const { return val != p.val; }
  modint61 inverse() const {
    ll a = val, b = mod, u = 1, v = 0, t;
    while (b > 0) {
      t = a / b;
      swap(a -= t * b, b), swap(u -= t * v, v);
    }
    return modint61(u);
  }
  modint61 pow(ll n) const {
    assert(n >= 0);
    modint61 ret(1), mul(val);
    while (n > 0) {
      if (n & 1) ret *= mul;
      mul *= mul, n >>= 1;
    }
    return ret;
  }
};

#ifdef FASTIO
void rd(modint61 &x) {
  fastio::rd(x.val);
  assert(0 <= x.val && x.val < modint61::mod);
}

void wt(modint61 x) { fastio::wt(x.val); }
#endif
#line 4 "graph/ds/rolling_hash_on_tree.hpp"

// 木の辺に文字がちょうどひとつ書いてある (static)
template <typename TREE, bool EDGE>
struct Rolling_Hash_On_Tree {
  using mint = modint61;

  TREE& tree;
  int N;
  mint base;
  // edge に文字があると思って
  // dp1: [root,v]
  // dp2: [v,root]
  vc<int> dat;
  vc<mint> dp1, dp2;
  vc<mint> pow, ipow;

  template <typename F>
  Rolling_Hash_On_Tree(TREE& tree, F f, mint base_ = 0)
      : tree(tree), N(tree.N), base(base_) {
    if (base == mint(0)) base = RNG(mint::get_mod());
    build(f);
  }

  template <typename F>
  void build(F f) {
    dat.resize(N);
    if constexpr (EDGE) {
      FOR(i, N - 1) { dat[tree.e_to_v(i)] = f(i); }
    } else {
      FOR(i, N) { dat[i] = f(i); }
    }
    pow.resize(N + 1), ipow.resize(N + 1);
    pow[0] = 1, pow[1] = base;
    ipow[0] = 1, ipow[1] = base.inverse();
    FOR(i, 2, N + 1) pow[i] = pow[i - 1] * pow[1];
    FOR(i, 2, N + 1) ipow[i] = ipow[i - 1] * ipow[1];
    int root = tree.V[0];
    dp1.resize(N), dp2.resize(N);
    dp1[root] = dp2[root] = dat[0];
    FOR(i, 1, N) {
      int v = tree.V[i];
      int d = tree.depth[v], p = tree.parent[v];
      dp1[v] = base * dp1[p] + dat[v];
      dp2[v] = dp2[p] + pow[d] * dat[v];
    }
  }

  mint get(int a, int b) {
    int c = tree.lca(a, b);
    mint x1 = get_du(a, c), x2 = get_ud(c, b);
    int n2 = tree.depth[b] - tree.depth[c];
    if constexpr (!EDGE) { x1 = x1 * base + dat[c]; }
    return x1 * pow[n2] + x2;
  }

  int lcp(int s1, int t1, int s2, int t2) {
    return lcp_and_comp(s1, t1, s2, t2).fi;
  }

  // <=>
  char comp(int s1, int t1, int s2, int t2) {
    return lcp_and_comp(s1, t1, s2, t2).se;
  }

  pair<int, char> lcp_and_comp(int s1, int t1, int s2, int t2) {
    int lcp = 0;
    // heavy path の頂点列
    auto path1 = tree.get_path_decomposition(s1, t1, EDGE);
    auto path2 = tree.get_path_decomposition(s2, t2, EDGE);
    reverse(all(path1));
    reverse(all(path2));
    while (len(path1) && len(path2)) {
      int a, b, c, d;
      tie(a, b) = POP(path1), tie(c, d) = POP(path2);
      ll n1 = abs(a - b) + 1, n2 = abs(c - d) + 1;
      ll n = min(n1, n2);
      if (n < n1) {
        if (a <= b) { path1.eb(a + n, b), b = a + n - 1; }
        if (a > b) { path1.eb(a - n, b), b = a - n + 1; }
      }
      if (n < n2) {
        if (c <= d) { path2.eb(c + n, d), d = c + n - 1; }
        if (c > d) { path2.eb(c - n, d), d = c - n + 1; }
      }
      mint x1 = from_hld_pair(a, b), x2 = from_hld_pair(c, d);
      if (x1 == x2) {
        lcp += n;
        continue;
      }
      auto check = [&](ll n) -> bool {
        if (n == 0) return 1;
        mint x1 = (a <= b ? from_hld_pair(a, a + n - 1)
                          : from_hld_pair(a, a - n + 1));
        mint x2 = (c <= d ? from_hld_pair(c, c + n - 1)
                          : from_hld_pair(c, c - n + 1));
        return x1 == x2;
      };
      ll k = binary_search(check, 0, n);
      lcp += k;
      a = (a <= b ? a + k : a - k);
      c = (c <= d ? c + k : c - k);
      a = tree.V[a], c = tree.V[c];
      if (dat[a] < dat[c]) return {lcp, '<'};
      if (dat[a] == dat[c]) return {lcp, '='};
      if (dat[a] > dat[c]) return {lcp, '>'};
    }
    if (!path1.empty()) return {lcp, '>'};
    if (!path2.empty()) return {lcp, '<'};
    return {lcp, '='};
  }

private:
  mint get_ud(int a, int b) {
    return (a == -1 ? dp1[b]
                    : dp1[b] - dp1[a] * pow[tree.depth[b] - tree.depth[a]]);
  }
  mint get_du(int a, int b) {
    return (b == -1 ? dp2[a] : (dp2[a] - dp2[b]) * ipow[tree.depth[b] + 1]);
  }
  mint from_hld_pair(int a, int b) {
    if (a <= b) { return get_ud(tree.parent[tree.V[a]], tree.V[b]); }
    return get_du(tree.V[a], tree.parent[tree.V[b]]);
  }
};
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