tests/hld/vpath.test.cpp

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• Last update: 2025-03-21 23:17:51+09:00
• Problem: https://judge.yosupo.jp/problem/vertex_set_path_composite

Depends on

• Heavy-light decomposition (tools/hld.hpp)
• std::less by key (tools/less_by.hpp)
• $2^x$ (tools/pow2.hpp)

Code

// competitive-verifier: PROBLEM https://judge.yosupo.jp/problem/vertex_set_path_composite

#include <utility>
#include <iostream>
#include <vector>
#include "atcoder/modint.hpp"
#include "atcoder/segtree.hpp"
#include "tools/hld.hpp"

using ll = long long;
using mint = atcoder::modint998244353;

std::pair<mint, mint> op(const std::pair<mint, mint> e1, const std::pair<mint, mint> e2) {
  return ::std::make_pair(e1.first * e2.first, e1.first * e2.second + e1.second);
}
std::pair<mint, mint> po(const std::pair<mint, mint> e1, const std::pair<mint, mint> e2) {
  return op(e2, e1);
}
std::pair<mint, mint> e() {
  return ::std::make_pair(mint::raw(1), mint::raw(0));
}

int main() {
  std::cin.tie(nullptr);
  std::ios_base::sync_with_stdio(false);

  ll N, Q;
  std::cin >> N >> Q;
  std::vector<std::pair<mint, mint>> f;
  f.reserve(N);
  for (ll i = 0; i < N; ++i) {
    ll a, b;
    std::cin >> a >> b;
    f.emplace_back(mint::raw(a), mint::raw(b));
  }
  tools::hld hld(N);
  for (ll i = 0; i < N - 1; ++i) {
    ll u, v;
    std::cin >> u >> v;
    hld.add_edge(u, v);
  }

  hld.build(0);
  std::vector<std::pair<mint, mint>> g(N);
  for (ll i = 0; i < N; ++i) {
    g[hld.vid2dfs(i)] = f[i];
  }
  atcoder::segtree<std::pair<mint, mint>, op, e> segtree_to_root(g);
  atcoder::segtree<std::pair<mint, mint>, po, e> segtree_to_leaf(g);

  for (ll q = 0; q < Q; ++q) {
    ll t;
    std::cin >> t;
    if (t == 0) {
      ll p, c, d;
      std::cin >> p >> c >> d;
      segtree_to_root.set(hld.vid2dfs(p), std::make_pair(mint::raw(c), mint::raw(d)));
      segtree_to_leaf.set(hld.vid2dfs(p), std::make_pair(mint::raw(c), mint::raw(d)));
    } else {
      ll u, v, x;
      std::cin >> u >> v >> x;
      std::pair<mint, mint> prod = e();
      for (const auto& [from, to] : hld.vpath(u, v)) {
        if (from < to) {
          prod = op(segtree_to_leaf.prod(from, to), prod);
        } else {
          prod = op(segtree_to_root.prod(to, from), prod);
        }
      }
      std::cout << (prod.first * mint::raw(x) + prod.second).val() << '\n';
    }
  }

  return 0;
}

#line 1 "tests/hld/vpath.test.cpp"
// competitive-verifier: PROBLEM https://judge.yosupo.jp/problem/vertex_set_path_composite

#include <utility>
#include <iostream>
#include <vector>
#line 1 "lib/ac-library/atcoder/modint.hpp"



#include <cassert>
#include <numeric>
#include <type_traits>

#ifdef _MSC_VER
#include <intrin.h>
#endif

#line 1 "lib/ac-library/atcoder/internal_math.hpp"



#line 5 "lib/ac-library/atcoder/internal_math.hpp"

#ifdef _MSC_VER
#include <intrin.h>
#endif

namespace atcoder {

namespace internal {

// @param m `1 <= m`
// @return x mod m
constexpr long long safe_mod(long long x, long long m) {
    x %= m;
    if (x < 0) x += m;
    return x;
}

// Fast modular multiplication by barrett reduction
// Reference: https://en.wikipedia.org/wiki/Barrett_reduction
// NOTE: reconsider after Ice Lake
struct barrett {
    unsigned int _m;
    unsigned long long im;

    // @param m `1 <= m`
    explicit barrett(unsigned int m) : _m(m), im((unsigned long long)(-1) / m + 1) {}

    // @return m
    unsigned int umod() const { return _m; }

    // @param a `0 <= a < m`
    // @param b `0 <= b < m`
    // @return `a * b % m`
    unsigned int mul(unsigned int a, unsigned int b) const {
        // [1] m = 1
        // a = b = im = 0, so okay

        // [2] m >= 2
        // im = ceil(2^64 / m)
        // -> im * m = 2^64 + r (0 <= r < m)
        // let z = a*b = c*m + d (0 <= c, d < m)
        // a*b * im = (c*m + d) * im = c*(im*m) + d*im = c*2^64 + c*r + d*im
        // c*r + d*im < m * m + m * im < m * m + 2^64 + m <= 2^64 + m * (m + 1) < 2^64 * 2
        // ((ab * im) >> 64) == c or c + 1
        unsigned long long z = a;
        z *= b;
#ifdef _MSC_VER
        unsigned long long x;
        _umul128(z, im, &x);
#else
        unsigned long long x =
            (unsigned long long)(((unsigned __int128)(z)*im) >> 64);
#endif
        unsigned long long y = x * _m;
        return (unsigned int)(z - y + (z < y ? _m : 0));
    }
};

// @param n `0 <= n`
// @param m `1 <= m`
// @return `(x ** n) % m`
constexpr long long pow_mod_constexpr(long long x, long long n, int m) {
    if (m == 1) return 0;
    unsigned int _m = (unsigned int)(m);
    unsigned long long r = 1;
    unsigned long long y = safe_mod(x, m);
    while (n) {
        if (n & 1) r = (r * y) % _m;
        y = (y * y) % _m;
        n >>= 1;
    }
    return r;
}

// Reference:
// M. Forisek and J. Jancina,
// Fast Primality Testing for Integers That Fit into a Machine Word
// @param n `0 <= n`
constexpr bool is_prime_constexpr(int n) {
    if (n <= 1) return false;
    if (n == 2 || n == 7 || n == 61) return true;
    if (n % 2 == 0) return false;
    long long d = n - 1;
    while (d % 2 == 0) d /= 2;
    constexpr long long bases[3] = {2, 7, 61};
    for (long long a : bases) {
        long long t = d;
        long long y = pow_mod_constexpr(a, t, n);
        while (t != n - 1 && y != 1 && y != n - 1) {
            y = y * y % n;
            t <<= 1;
        }
        if (y != n - 1 && t % 2 == 0) {
            return false;
        }
    }
    return true;
}
template <int n> constexpr bool is_prime = is_prime_constexpr(n);

// @param b `1 <= b`
// @return pair(g, x) s.t. g = gcd(a, b), xa = g (mod b), 0 <= x < b/g
constexpr std::pair<long long, long long> inv_gcd(long long a, long long b) {
    a = safe_mod(a, b);
    if (a == 0) return {b, 0};

    // Contracts:
    // [1] s - m0 * a = 0 (mod b)
    // [2] t - m1 * a = 0 (mod b)
    // [3] s * |m1| + t * |m0| <= b
    long long s = b, t = a;
    long long m0 = 0, m1 = 1;

    while (t) {
        long long u = s / t;
        s -= t * u;
        m0 -= m1 * u;  // |m1 * u| <= |m1| * s <= b

        // [3]:
        // (s - t * u) * |m1| + t * |m0 - m1 * u|
        // <= s * |m1| - t * u * |m1| + t * (|m0| + |m1| * u)
        // = s * |m1| + t * |m0| <= b

        auto tmp = s;
        s = t;
        t = tmp;
        tmp = m0;
        m0 = m1;
        m1 = tmp;
    }
    // by [3]: |m0| <= b/g
    // by g != b: |m0| < b/g
    if (m0 < 0) m0 += b / s;
    return {s, m0};
}

// Compile time primitive root
// @param m must be prime
// @return primitive root (and minimum in now)
constexpr int primitive_root_constexpr(int m) {
    if (m == 2) return 1;
    if (m == 167772161) return 3;
    if (m == 469762049) return 3;
    if (m == 754974721) return 11;
    if (m == 998244353) return 3;
    int divs[20] = {};
    divs[0] = 2;
    int cnt = 1;
    int x = (m - 1) / 2;
    while (x % 2 == 0) x /= 2;
    for (int i = 3; (long long)(i)*i <= x; i += 2) {
        if (x % i == 0) {
            divs[cnt++] = i;
            while (x % i == 0) {
                x /= i;
            }
        }
    }
    if (x > 1) {
        divs[cnt++] = x;
    }
    for (int g = 2;; g++) {
        bool ok = true;
        for (int i = 0; i < cnt; i++) {
            if (pow_mod_constexpr(g, (m - 1) / divs[i], m) == 1) {
                ok = false;
                break;
            }
        }
        if (ok) return g;
    }
}
template <int m> constexpr int primitive_root = primitive_root_constexpr(m);

// @param n `n < 2^32`
// @param m `1 <= m < 2^32`
// @return sum_{i=0}^{n-1} floor((ai + b) / m) (mod 2^64)
unsigned long long floor_sum_unsigned(unsigned long long n,
                                      unsigned long long m,
                                      unsigned long long a,
                                      unsigned long long b) {
    unsigned long long ans = 0;
    while (true) {
        if (a >= m) {
            ans += n * (n - 1) / 2 * (a / m);
            a %= m;
        }
        if (b >= m) {
            ans += n * (b / m);
            b %= m;
        }

        unsigned long long y_max = a * n + b;
        if (y_max < m) break;
        // y_max < m * (n + 1)
        // floor(y_max / m) <= n
        n = (unsigned long long)(y_max / m);
        b = (unsigned long long)(y_max % m);
        std::swap(m, a);
    }
    return ans;
}

}  // namespace internal

}  // namespace atcoder


#line 1 "lib/ac-library/atcoder/internal_type_traits.hpp"



#line 7 "lib/ac-library/atcoder/internal_type_traits.hpp"

namespace atcoder {

namespace internal {

#ifndef _MSC_VER
template <class T>
using is_signed_int128 =
    typename std::conditional<std::is_same<T, __int128_t>::value ||
                                  std::is_same<T, __int128>::value,
                              std::true_type,
                              std::false_type>::type;

template <class T>
using is_unsigned_int128 =
    typename std::conditional<std::is_same<T, __uint128_t>::value ||
                                  std::is_same<T, unsigned __int128>::value,
                              std::true_type,
                              std::false_type>::type;

template <class T>
using make_unsigned_int128 =
    typename std::conditional<std::is_same<T, __int128_t>::value,
                              __uint128_t,
                              unsigned __int128>;

template <class T>
using is_integral = typename std::conditional<std::is_integral<T>::value ||
                                                  is_signed_int128<T>::value ||
                                                  is_unsigned_int128<T>::value,
                                              std::true_type,
                                              std::false_type>::type;

template <class T>
using is_signed_int = typename std::conditional<(is_integral<T>::value &&
                                                 std::is_signed<T>::value) ||
                                                    is_signed_int128<T>::value,
                                                std::true_type,
                                                std::false_type>::type;

template <class T>
using is_unsigned_int =
    typename std::conditional<(is_integral<T>::value &&
                               std::is_unsigned<T>::value) ||
                                  is_unsigned_int128<T>::value,
                              std::true_type,
                              std::false_type>::type;

template <class T>
using to_unsigned = typename std::conditional<
    is_signed_int128<T>::value,
    make_unsigned_int128<T>,
    typename std::conditional<std::is_signed<T>::value,
                              std::make_unsigned<T>,
                              std::common_type<T>>::type>::type;

#else

template <class T> using is_integral = typename std::is_integral<T>;

template <class T>
using is_signed_int =
    typename std::conditional<is_integral<T>::value && std::is_signed<T>::value,
                              std::true_type,
                              std::false_type>::type;

template <class T>
using is_unsigned_int =
    typename std::conditional<is_integral<T>::value &&
                                  std::is_unsigned<T>::value,
                              std::true_type,
                              std::false_type>::type;

template <class T>
using to_unsigned = typename std::conditional<is_signed_int<T>::value,
                                              std::make_unsigned<T>,
                                              std::common_type<T>>::type;

#endif

template <class T>
using is_signed_int_t = std::enable_if_t<is_signed_int<T>::value>;

template <class T>
using is_unsigned_int_t = std::enable_if_t<is_unsigned_int<T>::value>;

template <class T> using to_unsigned_t = typename to_unsigned<T>::type;

}  // namespace internal

}  // namespace atcoder


#line 14 "lib/ac-library/atcoder/modint.hpp"

namespace atcoder {

namespace internal {

struct modint_base {};
struct static_modint_base : modint_base {};

template <class T> using is_modint = std::is_base_of<modint_base, T>;
template <class T> using is_modint_t = std::enable_if_t<is_modint<T>::value>;

}  // namespace internal

template <int m, std::enable_if_t<(1 <= m)>* = nullptr>
struct static_modint : internal::static_modint_base {
    using mint = static_modint;

  public:
    static constexpr int mod() { return m; }
    static mint raw(int v) {
        mint x;
        x._v = v;
        return x;
    }

    static_modint() : _v(0) {}
    template <class T, internal::is_signed_int_t<T>* = nullptr>
    static_modint(T v) {
        long long x = (long long)(v % (long long)(umod()));
        if (x < 0) x += umod();
        _v = (unsigned int)(x);
    }
    template <class T, internal::is_unsigned_int_t<T>* = nullptr>
    static_modint(T v) {
        _v = (unsigned int)(v % umod());
    }

    unsigned int val() const { return _v; }

    mint& operator++() {
        _v++;
        if (_v == umod()) _v = 0;
        return *this;
    }
    mint& operator--() {
        if (_v == 0) _v = umod();
        _v--;
        return *this;
    }
    mint operator++(int) {
        mint result = *this;
        ++*this;
        return result;
    }
    mint operator--(int) {
        mint result = *this;
        --*this;
        return result;
    }

    mint& operator+=(const mint& rhs) {
        _v += rhs._v;
        if (_v >= umod()) _v -= umod();
        return *this;
    }
    mint& operator-=(const mint& rhs) {
        _v -= rhs._v;
        if (_v >= umod()) _v += umod();
        return *this;
    }
    mint& operator*=(const mint& rhs) {
        unsigned long long z = _v;
        z *= rhs._v;
        _v = (unsigned int)(z % umod());
        return *this;
    }
    mint& operator/=(const mint& rhs) { return *this = *this * rhs.inv(); }

    mint operator+() const { return *this; }
    mint operator-() const { return mint() - *this; }

    mint pow(long long n) const {
        assert(0 <= n);
        mint x = *this, r = 1;
        while (n) {
            if (n & 1) r *= x;
            x *= x;
            n >>= 1;
        }
        return r;
    }
    mint inv() const {
        if (prime) {
            assert(_v);
            return pow(umod() - 2);
        } else {
            auto eg = internal::inv_gcd(_v, m);
            assert(eg.first == 1);
            return eg.second;
        }
    }

    friend mint operator+(const mint& lhs, const mint& rhs) {
        return mint(lhs) += rhs;
    }
    friend mint operator-(const mint& lhs, const mint& rhs) {
        return mint(lhs) -= rhs;
    }
    friend mint operator*(const mint& lhs, const mint& rhs) {
        return mint(lhs) *= rhs;
    }
    friend mint operator/(const mint& lhs, const mint& rhs) {
        return mint(lhs) /= rhs;
    }
    friend bool operator==(const mint& lhs, const mint& rhs) {
        return lhs._v == rhs._v;
    }
    friend bool operator!=(const mint& lhs, const mint& rhs) {
        return lhs._v != rhs._v;
    }

  private:
    unsigned int _v;
    static constexpr unsigned int umod() { return m; }
    static constexpr bool prime = internal::is_prime<m>;
};

template <int id> struct dynamic_modint : internal::modint_base {
    using mint = dynamic_modint;

  public:
    static int mod() { return (int)(bt.umod()); }
    static void set_mod(int m) {
        assert(1 <= m);
        bt = internal::barrett(m);
    }
    static mint raw(int v) {
        mint x;
        x._v = v;
        return x;
    }

    dynamic_modint() : _v(0) {}
    template <class T, internal::is_signed_int_t<T>* = nullptr>
    dynamic_modint(T v) {
        long long x = (long long)(v % (long long)(mod()));
        if (x < 0) x += mod();
        _v = (unsigned int)(x);
    }
    template <class T, internal::is_unsigned_int_t<T>* = nullptr>
    dynamic_modint(T v) {
        _v = (unsigned int)(v % mod());
    }

    unsigned int val() const { return _v; }

    mint& operator++() {
        _v++;
        if (_v == umod()) _v = 0;
        return *this;
    }
    mint& operator--() {
        if (_v == 0) _v = umod();
        _v--;
        return *this;
    }
    mint operator++(int) {
        mint result = *this;
        ++*this;
        return result;
    }
    mint operator--(int) {
        mint result = *this;
        --*this;
        return result;
    }

    mint& operator+=(const mint& rhs) {
        _v += rhs._v;
        if (_v >= umod()) _v -= umod();
        return *this;
    }
    mint& operator-=(const mint& rhs) {
        _v += mod() - rhs._v;
        if (_v >= umod()) _v -= umod();
        return *this;
    }
    mint& operator*=(const mint& rhs) {
        _v = bt.mul(_v, rhs._v);
        return *this;
    }
    mint& operator/=(const mint& rhs) { return *this = *this * rhs.inv(); }

    mint operator+() const { return *this; }
    mint operator-() const { return mint() - *this; }

    mint pow(long long n) const {
        assert(0 <= n);
        mint x = *this, r = 1;
        while (n) {
            if (n & 1) r *= x;
            x *= x;
            n >>= 1;
        }
        return r;
    }
    mint inv() const {
        auto eg = internal::inv_gcd(_v, mod());
        assert(eg.first == 1);
        return eg.second;
    }

    friend mint operator+(const mint& lhs, const mint& rhs) {
        return mint(lhs) += rhs;
    }
    friend mint operator-(const mint& lhs, const mint& rhs) {
        return mint(lhs) -= rhs;
    }
    friend mint operator*(const mint& lhs, const mint& rhs) {
        return mint(lhs) *= rhs;
    }
    friend mint operator/(const mint& lhs, const mint& rhs) {
        return mint(lhs) /= rhs;
    }
    friend bool operator==(const mint& lhs, const mint& rhs) {
        return lhs._v == rhs._v;
    }
    friend bool operator!=(const mint& lhs, const mint& rhs) {
        return lhs._v != rhs._v;
    }

  private:
    unsigned int _v;
    static internal::barrett bt;
    static unsigned int umod() { return bt.umod(); }
};
template <int id> internal::barrett dynamic_modint<id>::bt(998244353);

using modint998244353 = static_modint<998244353>;
using modint1000000007 = static_modint<1000000007>;
using modint = dynamic_modint<-1>;

namespace internal {

template <class T>
using is_static_modint = std::is_base_of<internal::static_modint_base, T>;

template <class T>
using is_static_modint_t = std::enable_if_t<is_static_modint<T>::value>;

template <class> struct is_dynamic_modint : public std::false_type {};
template <int id>
struct is_dynamic_modint<dynamic_modint<id>> : public std::true_type {};

template <class T>
using is_dynamic_modint_t = std::enable_if_t<is_dynamic_modint<T>::value>;

}  // namespace internal

}  // namespace atcoder


#line 1 "lib/ac-library/atcoder/segtree.hpp"



#include <algorithm>
#line 6 "lib/ac-library/atcoder/segtree.hpp"
#include <functional>
#line 8 "lib/ac-library/atcoder/segtree.hpp"

#line 1 "lib/ac-library/atcoder/internal_bit.hpp"



#ifdef _MSC_VER
#include <intrin.h>
#endif

#if __cplusplus >= 202002L
#include <bit>
#endif

namespace atcoder {

namespace internal {

#if __cplusplus >= 202002L

using std::bit_ceil;

#else

// @return same with std::bit::bit_ceil
unsigned int bit_ceil(unsigned int n) {
    unsigned int x = 1;
    while (x < (unsigned int)(n)) x *= 2;
    return x;
}

#endif

// @param n `1 <= n`
// @return same with std::bit::countr_zero
int countr_zero(unsigned int n) {
#ifdef _MSC_VER
    unsigned long index;
    _BitScanForward(&index, n);
    return index;
#else
    return __builtin_ctz(n);
#endif
}

// @param n `1 <= n`
// @return same with std::bit::countr_zero
constexpr int countr_zero_constexpr(unsigned int n) {
    int x = 0;
    while (!(n & (1 << x))) x++;
    return x;
}

}  // namespace internal

}  // namespace atcoder


#line 10 "lib/ac-library/atcoder/segtree.hpp"

namespace atcoder {

#if __cplusplus >= 201703L

template <class S, auto op, auto e> struct segtree {
    static_assert(std::is_convertible_v<decltype(op), std::function<S(S, S)>>,
                  "op must work as S(S, S)");
    static_assert(std::is_convertible_v<decltype(e), std::function<S()>>,
                  "e must work as S()");

#else

template <class S, S (*op)(S, S), S (*e)()> struct segtree {

#endif

  public:
    segtree() : segtree(0) {}
    explicit segtree(int n) : segtree(std::vector<S>(n, e())) {}
    explicit segtree(const std::vector<S>& v) : _n(int(v.size())) {
        size = (int)internal::bit_ceil((unsigned int)(_n));
        log = internal::countr_zero((unsigned int)size);
        d = std::vector<S>(2 * size, e());
        for (int i = 0; i < _n; i++) d[size + i] = v[i];
        for (int i = size - 1; i >= 1; i--) {
            update(i);
        }
    }

    void set(int p, S x) {
        assert(0 <= p && p < _n);
        p += size;
        d[p] = x;
        for (int i = 1; i <= log; i++) update(p >> i);
    }

    S get(int p) const {
        assert(0 <= p && p < _n);
        return d[p + size];
    }

    S prod(int l, int r) const {
        assert(0 <= l && l <= r && r <= _n);
        S sml = e(), smr = e();
        l += size;
        r += size;

        while (l < r) {
            if (l & 1) sml = op(sml, d[l++]);
            if (r & 1) smr = op(d[--r], smr);
            l >>= 1;
            r >>= 1;
        }
        return op(sml, smr);
    }

    S all_prod() const { return d[1]; }

    template <bool (*f)(S)> int max_right(int l) const {
        return max_right(l, [](S x) { return f(x); });
    }
    template <class F> int max_right(int l, F f) const {
        assert(0 <= l && l <= _n);
        assert(f(e()));
        if (l == _n) return _n;
        l += size;
        S sm = e();
        do {
            while (l % 2 == 0) l >>= 1;
            if (!f(op(sm, d[l]))) {
                while (l < size) {
                    l = (2 * l);
                    if (f(op(sm, d[l]))) {
                        sm = op(sm, d[l]);
                        l++;
                    }
                }
                return l - size;
            }
            sm = op(sm, d[l]);
            l++;
        } while ((l & -l) != l);
        return _n;
    }

    template <bool (*f)(S)> int min_left(int r) const {
        return min_left(r, [](S x) { return f(x); });
    }
    template <class F> int min_left(int r, F f) const {
        assert(0 <= r && r <= _n);
        assert(f(e()));
        if (r == 0) return 0;
        r += size;
        S sm = e();
        do {
            r--;
            while (r > 1 && (r % 2)) r >>= 1;
            if (!f(op(d[r], sm))) {
                while (r < size) {
                    r = (2 * r + 1);
                    if (f(op(d[r], sm))) {
                        sm = op(d[r], sm);
                        r--;
                    }
                }
                return r + 1 - size;
            }
            sm = op(d[r], sm);
        } while ((r & -r) != r);
        return 0;
    }

  private:
    int _n, size, log;
    std::vector<S> d;

    void update(int k) { d[k] = op(d[2 * k], d[2 * k + 1]); }
};

}  // namespace atcoder


#line 1 "tools/hld.hpp"



#line 6 "tools/hld.hpp"
#include <iterator>
#include <limits>
#line 9 "tools/hld.hpp"
#include <ranges>
#include <stack>
#line 1 "lib/ac-library/atcoder/dsu.hpp"



#line 7 "lib/ac-library/atcoder/dsu.hpp"

namespace atcoder {

// Implement (union by size) + (path compression)
// Reference:
// Zvi Galil and Giuseppe F. Italiano,
// Data structures and algorithms for disjoint set union problems
struct dsu {
  public:
    dsu() : _n(0) {}
    explicit dsu(int n) : _n(n), parent_or_size(n, -1) {}

    int merge(int a, int b) {
        assert(0 <= a && a < _n);
        assert(0 <= b && b < _n);
        int x = leader(a), y = leader(b);
        if (x == y) return x;
        if (-parent_or_size[x] < -parent_or_size[y]) std::swap(x, y);
        parent_or_size[x] += parent_or_size[y];
        parent_or_size[y] = x;
        return x;
    }

    bool same(int a, int b) {
        assert(0 <= a && a < _n);
        assert(0 <= b && b < _n);
        return leader(a) == leader(b);
    }

    int leader(int a) {
        assert(0 <= a && a < _n);
        if (parent_or_size[a] < 0) return a;
        return parent_or_size[a] = leader(parent_or_size[a]);
    }

    int size(int a) {
        assert(0 <= a && a < _n);
        return -parent_or_size[leader(a)];
    }

    std::vector<std::vector<int>> groups() {
        std::vector<int> leader_buf(_n), group_size(_n);
        for (int i = 0; i < _n; i++) {
            leader_buf[i] = leader(i);
            group_size[leader_buf[i]]++;
        }
        std::vector<std::vector<int>> result(_n);
        for (int i = 0; i < _n; i++) {
            result[i].reserve(group_size[i]);
        }
        for (int i = 0; i < _n; i++) {
            result[leader_buf[i]].push_back(i);
        }
        result.erase(
            std::remove_if(result.begin(), result.end(),
                           [&](const std::vector<int>& v) { return v.empty(); }),
            result.end());
        return result;
    }

  private:
    int _n;
    // root node: -1 * component size
    // otherwise: parent
    std::vector<int> parent_or_size;
};

}  // namespace atcoder


#line 1 "tools/less_by.hpp"



namespace tools {

  template <class F>
  class less_by {
  private:
    F selector;

  public:
    less_by(const F& selector) : selector(selector) {
    }

    template <class T>
    bool operator()(const T& x, const T& y) const {
      return selector(x) < selector(y);
    }
  };
}


#line 1 "tools/pow2.hpp"



#line 5 "tools/pow2.hpp"
#include <cstddef>

namespace tools {

  template <typename T, typename ::std::enable_if<::std::is_unsigned<T>::value, ::std::nullptr_t>::type = nullptr>
  constexpr T pow2(const T x) {
    return static_cast<T>(1) << x;
  }

  template <typename T, typename ::std::enable_if<::std::is_signed<T>::value, ::std::nullptr_t>::type = nullptr>
  constexpr T pow2(const T x) {
    return static_cast<T>(static_cast<typename ::std::make_unsigned<T>::type>(1) << static_cast<typename ::std::make_unsigned<T>::type>(x));
  }
}


#line 16 "tools/hld.hpp"

namespace tools {
  class hld {
    bool m_built;
    ::std::vector<::std::vector<int>> m_graph;
    ::std::vector<int> m_edges;
    ::std::vector<int> m_parent;
    ::std::vector<int> m_depth;
    ::atcoder::dsu m_dsu;
    ::std::vector<int> m_out;
    ::std::vector<int> m_vid2dfs;
    ::std::vector<int> m_dfs2vid;
    ::std::vector<int> m_eid2dfs;
    ::std::vector<int> m_dfs2eid;
    ::std::vector<::std::vector<int>> m_ancestors;

  public:
    class vchildren_view : public ::std::ranges::view_interface<vchildren_view> {
      ::tools::hld const *m_parent;
      int m_v;

    public:
      class iterator {
      private:
        ::tools::hld const *m_parent;
        int m_v;
        int m_i;

      public:
        using difference_type = ::std::ptrdiff_t;
        using value_type = int;
        using reference = int;
        using pointer = int*;
        using iterator_category = ::std::input_iterator_tag;

        iterator() = default;
        iterator(::tools::hld const * const parent, const int v, const int i) :
          m_parent(parent),
          m_v(v),
          m_i(i) {
        }

        reference operator*() const {
          return this->m_parent->m_edges[this->m_parent->m_graph[this->m_v][this->m_i]] ^ this->m_v;
        }
        iterator& operator++() {
          ++this->m_i;
          return *this;
        }
        iterator operator++(int) {
          const auto self = *this;
          ++*this;
          return self;
        }
        friend bool operator==(const iterator& lhs, const iterator& rhs) {
          return lhs.m_parent == rhs.m_parent && lhs.m_v == rhs.m_v && lhs.m_i == rhs.m_i;
        }
        friend bool operator!=(const iterator& lhs, const iterator& rhs) {
          return !(lhs == rhs);
        }
      };

      vchildren_view() = default;
      vchildren_view(::tools::hld const * const parent, const int v) :
        m_parent(parent),
        m_v(v) {
      }

      iterator begin() const {
        return iterator(this->m_parent, this->m_v, 0);
      };
      iterator end() const {
        return iterator(this->m_parent, this->m_v, this->m_parent->m_graph[this->m_v].size());
      }
    };

    hld() = default;
    explicit hld(const int n) : m_built(false), m_graph(n) {
      assert(n >= 1);
    }

    int size() const {
      return this->m_graph.size();
    }

    void add_edge(const int u, const int v) {
      assert(!this->m_built);
      assert(0 <= u && u < this->size());
      assert(0 <= v && v < this->size());
      this->m_graph[u].push_back(this->m_edges.size());
      this->m_graph[v].push_back(this->m_edges.size());
      this->m_edges.push_back(u ^ v);
    }

    void build(const int root) {
      assert(!this->m_built);
      assert(0 <= root && root < this->size());
      assert(::std::ssize(this->m_edges) + 1 == this->size());

      this->m_parent.resize(this->size());
      this->m_depth.resize(this->size());
      this->m_dsu = ::atcoder::dsu(this->size());
      this->m_out.resize(this->size());
      this->m_vid2dfs.resize(this->size());
      this->m_dfs2vid.resize(this->size());
      this->m_eid2dfs.resize(this->m_edges.size());
      this->m_dfs2eid.resize(this->m_edges.size());

      ::std::vector<int> subtree_size(this->size());
      this->m_parent[root] = ::std::numeric_limits<int>::max();
      this->m_depth[root] = 0;
      ::std::stack<::std::pair<int, bool>> stack;
      stack.emplace(root, false);
      stack.emplace(root, true);
      while (!stack.empty()) {
        const auto [here, pre] = stack.top();
        stack.pop();
        if (pre) {
          for (const auto eid : this->m_graph[here]) {
            const auto next = this->m_edges[eid] ^ here;
            if (here == root || next != (this->m_edges[this->m_parent[here]] ^ here)) {
              this->m_parent[next] = eid;
              this->m_depth[next] = this->m_depth[here] + 1;
              stack.emplace(next, false);
              stack.emplace(next, true);
            }
          }
        } else {
          subtree_size[here] = 1;
          for (const auto eid : this->m_graph[here]) {
            const auto child = this->m_edges[eid] ^ here;
            if (here == root || child != (this->m_edges[this->m_parent[here]] ^ here)) {
              subtree_size[here] += subtree_size[child];
            }
          }
        }
      }

      for (int v = 0; v < this->size(); ++v) {
        if (v != root) {
          this->m_graph[v].erase(::std::ranges::find(this->m_graph[v], this->m_parent[v]));
        }
        if (this->m_graph[v].size() > 1) {
          ::std::iter_swap(
            this->m_graph[v].begin(),
            ::std::ranges::max_element(
              this->m_graph[v],
              ::tools::less_by([&](const int eid) { return subtree_size[this->m_edges[eid] ^ v]; })
            )
          );
        }
      }

      int dfs_order = 0;
      stack.emplace(root, false);
      stack.emplace(root, true);
      while (!stack.empty()) {
        const auto [here, pre] = stack.top();
        stack.pop();

        if (pre) {
          this->m_vid2dfs[here] = dfs_order;
          this->m_dfs2vid[dfs_order] = here;
          if (here != root) {
            this->m_eid2dfs[this->m_parent[here]] = dfs_order - 1;
            this->m_dfs2eid[dfs_order - 1] = this->m_parent[here];
          }
          ++dfs_order;

          if (!this->m_graph[here].empty()) {
            this->m_dsu.merge(here, this->m_edges[this->m_graph[here].front()] ^ here);
          }
          for (auto it = this->m_graph[here].rbegin(); it != this->m_graph[here].rend(); ++it) {
            stack.emplace(this->m_edges[*it] ^ here, false);
            stack.emplace(this->m_edges[*it] ^ here, true);
          }
        } else {
          this->m_out[here] = dfs_order;
        }
      }

      this->m_built = true;
    }

    int depth(const int v) const {
      assert(this->m_built);
      assert(0 <= v && v < this->size());
      return this->m_depth[v];
    }
    int vparent(const int v) const {
      assert(this->m_built);
      assert(0 <= v && v < this->size());
      assert(this->m_depth[v] > 0);
      return this->m_edges[this->m_parent[v]] ^ v;
    }
    int eparent(const int v) const {
      assert(this->m_built);
      assert(0 <= v && v < this->size());
      assert(this->m_depth[v] > 0);
      return this->m_parent[v];
    }
    int vancestor(const int v, const int k) {
      assert(this->m_built);
      assert(0 <= v && v < this->size());
      assert(0 <= k && k <= this->m_depth[v]);

      if (this->m_ancestors.empty()) {
        this->m_ancestors.resize(this->size());
        ::std::vector<int> targets(this->size());
        ::std::iota(targets.begin(), targets.end(), 0);
        targets.erase(::std::remove(targets.begin(), targets.end(), this->m_dfs2vid[0]), targets.end());
        for (const auto t : targets) {
          this->m_ancestors[t].push_back(this->vparent(t));
        }
        for (int g = 1; [&]() {
          targets.erase(::std::remove_if(targets.begin(), targets.end(), [&](const int t) {
            return this->m_depth[t] < ::tools::pow2(g);
          }), targets.end());
          return !targets.empty();
        }(); ++g) {
          for (const auto t : targets) {
            this->m_ancestors[t].push_back(this->m_ancestors[this->m_ancestors[t][g - 1]][g - 1]);
          }
        }
      }

      int res = v;
      for (int g = 0; ::tools::pow2(g) <= k; ++g) {
        if ((k >> g) & 1) {
          res = this->m_ancestors[res][g];
        }
      }

      return res;
    }
    ::tools::hld::vchildren_view vchildren(const int v) const & {
      assert(this->m_built);
      assert(0 <= v && v < this->size());
      return ::tools::hld::vchildren_view(this, v);
    }
    const ::std::vector<int>& echildren(const int v) const & {
      assert(this->m_built);
      assert(0 <= v && v < this->size());
      return this->m_graph[v];
    }

    int vid2dfs(const int v) const {
      assert(this->m_built);
      assert(0 <= v && v < this->size());
      return this->m_vid2dfs[v];
    }
    int dfs2vid(const int i) const {
      assert(this->m_built);
      assert(0 <= i && i < this->size());
      return this->m_dfs2vid[i];
    }
    int eid2dfs(const int e) const {
      assert(this->m_built);
      assert(0 <= e && e < this->size());
      return this->m_eid2dfs[e];
    }
    int dfs2eid(const int i) const {
      assert(this->m_built);
      assert(0 <= i && i < this->size());
      return this->m_dfs2eid[i];
    }

    int lca(int u, int v) {
      assert(this->m_built);
      assert(0 <= u && u < this->size());
      assert(0 <= v && v < this->size());

      while (!this->m_dsu.same(u, v)) {
        if (this->m_depth[this->m_dsu.leader(u)] >= this->m_depth[this->m_dsu.leader(v)]) {
          u = this->m_edges[this->m_parent[this->m_dsu.leader(u)]] ^ this->m_dsu.leader(u);
        } else {
          v = this->m_edges[this->m_parent[this->m_dsu.leader(v)]] ^ this->m_dsu.leader(v);
        }
      }
      if (this->m_depth[u] >= this->m_depth[v]) {
        return v;
      } else {
        return u;
      }
    }

    ::std::pair<int, int> vsubtree(const int v) const {
      assert(this->m_built);
      assert(0 <= v && v < this->size());
      return ::std::make_pair(this->m_vid2dfs[v], this->m_out[v]);
    }
    ::std::pair<int, int> esubtree(const int v) const {
      assert(this->m_built);
      assert(0 <= v && v < this->size());
      return ::std::make_pair(this->m_depth[v] == 0 ? 0 : this->m_eid2dfs[this->m_parent[v]] + 1, this->m_out[v] - 1);
    }

    ::std::vector<::std::pair<int, int>> vpath(int u, int v) {
      assert(this->m_built);
      assert(0 <= u && u < this->size());
      assert(0 <= v && v < this->size());

      ::std::vector<::std::pair<int, int>> head, tail;
      while (!this->m_dsu.same(u, v)) {
        if (this->m_depth[this->m_dsu.leader(u)] >= this->m_depth[this->m_dsu.leader(v)]) {
          head.emplace_back(this->m_vid2dfs[u] + 1, this->m_vid2dfs[this->m_dsu.leader(u)]);
          u = this->m_edges[this->m_parent[this->m_dsu.leader(u)]] ^ this->m_dsu.leader(u);
        } else {
          tail.emplace_back(this->m_vid2dfs[this->m_dsu.leader(v)], this->m_vid2dfs[v] + 1);
          v = this->m_edges[this->m_parent[this->m_dsu.leader(v)]] ^ this->m_dsu.leader(v);
        }
      }
      if (this->m_depth[u] >= this->m_depth[v]) {
        head.emplace_back(this->m_vid2dfs[u] + 1, this->m_vid2dfs[v]);
      } else {
        head.emplace_back(this->m_vid2dfs[u], this->m_vid2dfs[v] + 1);
      }

      ::std::copy(tail.rbegin(), tail.rend(), ::std::back_inserter(head));
      return head;
    }
    ::std::vector<::std::pair<int, int>> epath(int u, int v) {
      assert(this->m_built);
      assert(0 <= u && u < this->size());
      assert(0 <= v && v < this->size());

      ::std::vector<::std::pair<int, int>> head, tail;
      while (!this->m_dsu.same(u, v)) {
        if (this->m_depth[this->m_dsu.leader(u)] >= this->m_depth[this->m_dsu.leader(v)]) {
          head.emplace_back(this->m_eid2dfs[this->m_parent[u]] + 1, this->m_eid2dfs[this->m_parent[this->m_dsu.leader(u)]]);
          u = this->m_edges[this->m_parent[this->m_dsu.leader(u)]] ^ this->m_dsu.leader(u);
        } else {
          tail.emplace_back(this->m_eid2dfs[this->m_parent[this->m_dsu.leader(v)]], this->m_eid2dfs[this->m_parent[v]] + 1);
          v = this->m_edges[this->m_parent[this->m_dsu.leader(v)]] ^ this->m_dsu.leader(v);
        }
      }
      if (this->m_depth[u] > this->m_depth[v]) {
        head.emplace_back(this->m_eid2dfs[this->m_parent[u]] + 1, this->m_eid2dfs[this->m_graph[v].front()]);
      } else if (this->m_depth[u] < this->m_depth[v]) {
        head.emplace_back(this->m_eid2dfs[this->m_graph[u].front()], this->m_eid2dfs[this->m_parent[v]] + 1);
      }

      ::std::copy(tail.rbegin(), tail.rend(), ::std::back_inserter(head));
      return head;
    }
  };
}


#line 9 "tests/hld/vpath.test.cpp"

using ll = long long;
using mint = atcoder::modint998244353;

std::pair<mint, mint> op(const std::pair<mint, mint> e1, const std::pair<mint, mint> e2) {
  return ::std::make_pair(e1.first * e2.first, e1.first * e2.second + e1.second);
}
std::pair<mint, mint> po(const std::pair<mint, mint> e1, const std::pair<mint, mint> e2) {
  return op(e2, e1);
}
std::pair<mint, mint> e() {
  return ::std::make_pair(mint::raw(1), mint::raw(0));
}

int main() {
  std::cin.tie(nullptr);
  std::ios_base::sync_with_stdio(false);

  ll N, Q;
  std::cin >> N >> Q;
  std::vector<std::pair<mint, mint>> f;
  f.reserve(N);
  for (ll i = 0; i < N; ++i) {
    ll a, b;
    std::cin >> a >> b;
    f.emplace_back(mint::raw(a), mint::raw(b));
  }
  tools::hld hld(N);
  for (ll i = 0; i < N - 1; ++i) {
    ll u, v;
    std::cin >> u >> v;
    hld.add_edge(u, v);
  }

  hld.build(0);
  std::vector<std::pair<mint, mint>> g(N);
  for (ll i = 0; i < N; ++i) {
    g[hld.vid2dfs(i)] = f[i];
  }
  atcoder::segtree<std::pair<mint, mint>, op, e> segtree_to_root(g);
  atcoder::segtree<std::pair<mint, mint>, po, e> segtree_to_leaf(g);

  for (ll q = 0; q < Q; ++q) {
    ll t;
    std::cin >> t;
    if (t == 0) {
      ll p, c, d;
      std::cin >> p >> c >> d;
      segtree_to_root.set(hld.vid2dfs(p), std::make_pair(mint::raw(c), mint::raw(d)));
      segtree_to_leaf.set(hld.vid2dfs(p), std::make_pair(mint::raw(c), mint::raw(d)));
    } else {
      ll u, v, x;
      std::cin >> u >> v >> x;
      std::pair<mint, mint> prod = e();
      for (const auto& [from, to] : hld.vpath(u, v)) {
        if (from < to) {
          prod = op(segtree_to_leaf.prod(from, to), prod);
        } else {
          prod = op(segtree_to_root.prod(to, from), prod);
        }
      }
      std::cout << (prod.first * mint::raw(x) + prod.second).val() << '\n';
    }
  }

  return 0;
}

Test cases

Env	Name	Status	Elapsed	Memory
g++	almost_line_00	AC	232 ms	34 MB
g++	almost_line_01	AC	235 ms	33 MB
g++	example_00	AC	5 ms	4 MB
g++	example_01	AC	4 ms	4 MB
g++	line_00	AC	216 ms	34 MB
g++	line_01	AC	197 ms	34 MB
g++	long-path-decomposition_killer_00	AC	145 ms	33 MB
g++	max_random_00	AC	283 ms	33 MB
g++	max_random_01	AC	285 ms	33 MB
g++	max_random_02	AC	280 ms	33 MB
g++	random_00	AC	184 ms	21 MB
g++	random_01	AC	207 ms	28 MB
g++	random_02	AC	114 ms	11 MB
g++	small_00	AC	6 ms	4 MB
g++	small_01	AC	5 ms	4 MB
g++	small_02	AC	5 ms	4 MB
g++	small_03	AC	5 ms	4 MB
g++	small_04	AC	5 ms	4 MB
g++	worst_for_path_decomposition_00	AC	518 ms	33 MB
g++	worst_for_path_decomposition_01	AC	553 ms	33 MB

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