cplib-cpp
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tree/test/vertex-set-path-composite.test.cpp
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- Last update: 2025-01-01 21:39:17+09:00
- Problem: https://judge.yosupo.jp/problem/vertex_set_path_composite
Depends on
- modint.hpp
- segmenttree/point-update-range-get_nonrecursive.hpp
- Heavy-light decomposition （HLD, 木の重軽分解）
(tree/heavy_light_decomposition.hpp)
Code
#define PROBLEM "https://judge.yosupo.jp/problem/vertex_set_path_composite"
#include "../../modint.hpp"
#include "../../segmenttree/point-update-range-get_nonrecursive.hpp"
#include "../heavy_light_decomposition.hpp"
#include <iostream>
#include <vector>
using namespace std;
using mint = ModInt<998244353>;
using P = pair<mint, mint>;
struct PointSetRangeComposite : public NonrecursiveSegmentTree<P, P, bool> {
using SegTree = NonrecursiveSegmentTree<P, P, bool>;
P merge_data(const P &vl, const P &vr) override {
return make_pair(vl.first * vr.first, vr.first * vl.second + vr.second);
};
P data2ret(const P &v, const bool &q) override { return v; }
P merge_ret(const P &vl, const P &vr) override { return merge_data(vl, vr); };
PointSetRangeComposite(const std::vector<P> &seq, P zero)
: SegTree::NonrecursiveSegmentTree() {
SegTree::initialize(seq, zero);
};
};
int main() {
cin.tie(nullptr), ios::sync_with_stdio(false);
int N, Q;
cin >> N >> Q;
vector<P> V(N);
for (auto &x : V) cin >> x.first >> x.second;
vector<pair<int, int>> edges(N - 1);
for (auto &[u, v] : edges) cin >> u >> v;
heavy_light_decomposition hld(N, edges);
hld.build();
vector<P> stinit = hld.segtree_rearrange(V);
PointSetRangeComposite segtree(stinit, P{1, 0});
reverse(stinit.begin(), stinit.end());
PointSetRangeComposite segtreeinv(stinit, P{1, 0});
while (Q--) {
int q, u, v, x;
cin >> q >> u >> v >> x;
if (q == 0) {
segtree.update(hld.subtree_begin[u], P{v, x});
segtreeinv.update(N - 1 - hld.subtree_begin[u], P{v, x});
} else {
mint ret = x;
hld.for_each_vertex_noncommutative(
u, v,
[&](int l, int r) -> void {
assert(0 <= l and l <= r and r < N);
P tmp = segtreeinv.get(N - 1 - r, N - 1 - l + 1);
ret = tmp.first * ret + tmp.second;
},
[&](int l, int r) -> void {
assert(0 <= l and l <= r and r < N);
P tmp = segtree.get(l, r + 1);
ret = tmp.first * ret + tmp.second;
});
cout << ret << '\n';
}
}
}#line 1 "tree/test/vertex-set-path-composite.test.cpp"
#define PROBLEM "https://judge.yosupo.jp/problem/vertex_set_path_composite"
#line 2 "modint.hpp"
#include <cassert>
#include <iostream>
#include <set>
#include <vector>
template <int md> struct ModInt {
using lint = long long;
constexpr static int mod() { return md; }
static int get_primitive_root() {
static int primitive_root = 0;
if (!primitive_root) {
primitive_root = [&]() {
std::set<int> fac;
int v = md - 1;
for (lint i = 2; i * i <= v; i++)
while (v % i == 0) fac.insert(i), v /= i;
if (v > 1) fac.insert(v);
for (int g = 1; g < md; g++) {
bool ok = true;
for (auto i : fac)
if (ModInt(g).pow((md - 1) / i) == 1) {
ok = false;
break;
}
if (ok) return g;
}
return -1;
}();
}
return primitive_root;
}
int val_;
int val() const noexcept { return val_; }
constexpr ModInt() : val_(0) {}
constexpr ModInt &_setval(lint v) { return val_ = (v >= md ? v - md : v), *this; }
constexpr ModInt(lint v) { _setval(v % md + md); }
constexpr explicit operator bool() const { return val_ != 0; }
constexpr ModInt operator+(const ModInt &x) const {
return ModInt()._setval((lint)val_ + x.val_);
}
constexpr ModInt operator-(const ModInt &x) const {
return ModInt()._setval((lint)val_ - x.val_ + md);
}
constexpr ModInt operator*(const ModInt &x) const {
return ModInt()._setval((lint)val_ * x.val_ % md);
}
constexpr ModInt operator/(const ModInt &x) const {
return ModInt()._setval((lint)val_ * x.inv().val() % md);
}
constexpr ModInt operator-() const { return ModInt()._setval(md - val_); }
constexpr ModInt &operator+=(const ModInt &x) { return *this = *this + x; }
constexpr ModInt &operator-=(const ModInt &x) { return *this = *this - x; }
constexpr ModInt &operator*=(const ModInt &x) { return *this = *this * x; }
constexpr ModInt &operator/=(const ModInt &x) { return *this = *this / x; }
friend constexpr ModInt operator+(lint a, const ModInt &x) { return ModInt(a) + x; }
friend constexpr ModInt operator-(lint a, const ModInt &x) { return ModInt(a) - x; }
friend constexpr ModInt operator*(lint a, const ModInt &x) { return ModInt(a) * x; }
friend constexpr ModInt operator/(lint a, const ModInt &x) { return ModInt(a) / x; }
constexpr bool operator==(const ModInt &x) const { return val_ == x.val_; }
constexpr bool operator!=(const ModInt &x) const { return val_ != x.val_; }
constexpr bool operator<(const ModInt &x) const {
return val_ < x.val_;
} // To use std::map<ModInt, T>
friend std::istream &operator>>(std::istream &is, ModInt &x) {
lint t;
return is >> t, x = ModInt(t), is;
}
constexpr friend std::ostream &operator<<(std::ostream &os, const ModInt &x) {
return os << x.val_;
}
constexpr ModInt pow(lint n) const {
ModInt ans = 1, tmp = *this;
while (n) {
if (n & 1) ans *= tmp;
tmp *= tmp, n >>= 1;
}
return ans;
}
static constexpr int cache_limit = std::min(md, 1 << 21);
static std::vector<ModInt> facs, facinvs, invs;
constexpr static void _precalculation(int N) {
const int l0 = facs.size();
if (N > md) N = md;
if (N <= l0) return;
facs.resize(N), facinvs.resize(N), invs.resize(N);
for (int i = l0; i < N; i++) facs[i] = facs[i - 1] * i;
facinvs[N - 1] = facs.back().pow(md - 2);
for (int i = N - 2; i >= l0; i--) facinvs[i] = facinvs[i + 1] * (i + 1);
for (int i = N - 1; i >= l0; i--) invs[i] = facinvs[i] * facs[i - 1];
}
constexpr ModInt inv() const {
if (this->val_ < cache_limit) {
if (facs.empty()) facs = {1}, facinvs = {1}, invs = {0};
while (this->val_ >= int(facs.size())) _precalculation(facs.size() * 2);
return invs[this->val_];
} else {
return this->pow(md - 2);
}
}
constexpr ModInt fac() const {
while (this->val_ >= int(facs.size())) _precalculation(facs.size() * 2);
return facs[this->val_];
}
constexpr ModInt facinv() const {
while (this->val_ >= int(facs.size())) _precalculation(facs.size() * 2);
return facinvs[this->val_];
}
constexpr ModInt doublefac() const {
lint k = (this->val_ + 1) / 2;
return (this->val_ & 1) ? ModInt(k * 2).fac() / (ModInt(2).pow(k) * ModInt(k).fac())
: ModInt(k).fac() * ModInt(2).pow(k);
}
constexpr ModInt nCr(int r) const {
if (r < 0 or this->val_ < r) return ModInt(0);
return this->fac() * (*this - r).facinv() * ModInt(r).facinv();
}
constexpr ModInt nPr(int r) const {
if (r < 0 or this->val_ < r) return ModInt(0);
return this->fac() * (*this - r).facinv();
}
static ModInt binom(int n, int r) {
static long long bruteforce_times = 0;
if (r < 0 or n < r) return ModInt(0);
if (n <= bruteforce_times or n < (int)facs.size()) return ModInt(n).nCr(r);
r = std::min(r, n - r);
ModInt ret = ModInt(r).facinv();
for (int i = 0; i < r; ++i) ret *= n - i;
bruteforce_times += r;
return ret;
}
// Multinomial coefficient, (k_1 + k_2 + ... + k_m)! / (k_1! k_2! ... k_m!)
// Complexity: O(sum(ks))
template <class Vec> static ModInt multinomial(const Vec &ks) {
ModInt ret{1};
int sum = 0;
for (int k : ks) {
assert(k >= 0);
ret *= ModInt(k).facinv(), sum += k;
}
return ret * ModInt(sum).fac();
}
// Catalan number, C_n = binom(2n, n) / (n + 1)
// C_0 = 1, C_1 = 1, C_2 = 2, C_3 = 5, C_4 = 14, ...
// https://oeis.org/A000108
// Complexity: O(n)
static ModInt catalan(int n) {
if (n < 0) return ModInt(0);
return ModInt(n * 2).fac() * ModInt(n + 1).facinv() * ModInt(n).facinv();
}
ModInt sqrt() const {
if (val_ == 0) return 0;
if (md == 2) return val_;
if (pow((md - 1) / 2) != 1) return 0;
ModInt b = 1;
while (b.pow((md - 1) / 2) == 1) b += 1;
int e = 0, m = md - 1;
while (m % 2 == 0) m >>= 1, e++;
ModInt x = pow((m - 1) / 2), y = (*this) * x * x;
x *= (*this);
ModInt z = b.pow(m);
while (y != 1) {
int j = 0;
ModInt t = y;
while (t != 1) j++, t *= t;
z = z.pow(1LL << (e - j - 1));
x *= z, z *= z, y *= z;
e = j;
}
return ModInt(std::min(x.val_, md - x.val_));
}
};
template <int md> std::vector<ModInt<md>> ModInt<md>::facs = {1};
template <int md> std::vector<ModInt<md>> ModInt<md>::facinvs = {1};
template <int md> std::vector<ModInt<md>> ModInt<md>::invs = {0};
using ModInt998244353 = ModInt<998244353>;
// using mint = ModInt<998244353>;
// using mint = ModInt<1000000007>;
#line 2 "segmenttree/point-update-range-get_nonrecursive.hpp"
#include <algorithm>
#line 4 "segmenttree/point-update-range-get_nonrecursive.hpp"
#include <functional>
#line 6 "segmenttree/point-update-range-get_nonrecursive.hpp"
#include <stack>
#line 8 "segmenttree/point-update-range-get_nonrecursive.hpp"
// CUT begin
// Nonrecursive Segment Tree (point-update, range-get)
// - Conditions for operations:
// - merge_data: [TDATA, TDATA] -> TDATA, e(x, y) == e(y, x)
// - data2ret: [TDATA, TQUERY] -> TRET
// - merge_ret: [TRET, TRET] -> TRET, g(defaultRET, x) == x, g(x, y) = g(y, x)
// - commutability f(e(x, y), q) == g(f(x, q), f(y, q))
template <typename TDATA, typename TRET, typename TQUERY> struct NonrecursiveSegmentTree {
int N;
TRET defaultRET;
virtual TDATA merge_data(const TDATA &, const TDATA &) = 0;
virtual TRET data2ret(const TDATA &, const TQUERY &) = 0;
virtual TRET merge_ret(const TRET &, const TRET &) = 0;
std::vector<TDATA> data;
inline TDATA &at(int i) { return data[i]; }
inline void _merge(int i) { at(i) = merge_data(at(i << 1), at((i << 1) + 1)); }
void initialize(const std::vector<TDATA> &seq, TRET RET_ZERO) {
N = seq.size();
defaultRET = RET_ZERO;
data = seq;
data.insert(data.end(), seq.begin(), seq.end());
for (int i = N - 1; i; i--) _merge(i);
}
NonrecursiveSegmentTree() = default;
void update(int pos, const TDATA &x) {
assert(pos >= 0 and pos < N);
at(pos + N) = x;
for (int i = pos + N; i > 1;) i >>= 1, _merge(i);
}
// [l, r), 0-indexed
TRET get(int l, int r, TQUERY query = NULL) {
assert(l >= 0 and r <= N);
TRET retl = defaultRET, retr = defaultRET;
l += N, r += N;
while (l < r) {
if (l & 1) retl = merge_ret(retl, data2ret(data[l++], query));
if (r & 1) retr = merge_ret(data2ret(data[--r], query), retr);
l >>= 1, r >>= 1;
}
return merge_ret(retl, retr);
}
// Calculate smallest r that satisfies condition(g(f(x_l, q), ..., f(x_{r - 1}, q)) == true
// Assumption: Monotonicity of g(x_l, ..., x_r) about r (l: fixed)
// Complexity: O(log N)
int binary_search(int l, std::function<bool(TRET)> condition, TQUERY query = NULL) {
std::stack<int> rs;
l += N;
int r = N * 2;
TRET retl = defaultRET;
if (condition(retl)) return l - N;
while (l < r) {
if (l & 1) {
TRET ret_tmp = merge_ret(retl, data2ret(data[l], query));
if (condition(ret_tmp)) {
while (l * 2 < N * 2) {
ret_tmp = merge_ret(retl, data2ret(data[l * 2], query));
if (condition(ret_tmp))
l *= 2;
else
retl = ret_tmp, l = l * 2 + 1;
}
return l - N;
}
l++;
retl = ret_tmp;
}
if (r & 1) rs.push(--r);
l >>= 1, r >>= 1;
}
while (!rs.empty()) {
l = rs.top();
rs.pop();
TRET ret_tmp = merge_ret(retl, data2ret(data[l], query));
if (condition(ret_tmp)) {
while (l * 2 < N * 2) {
ret_tmp = merge_ret(retl, data2ret(data[l * 2], query));
if (condition(ret_tmp))
l *= 2;
else
retl = ret_tmp, l = l * 2 + 1;
}
return l - N;
}
retl = ret_tmp;
}
return N;
}
template <typename T1, typename T2, typename T3>
friend std::ostream &operator<<(std::ostream &os, NonrecursiveSegmentTree<T1, T2, T3> s) {
os << "[SegmentTree (len: " << s.N << ')';
for (int i = 0; i < s.N; i++) os << s.at(i + s.N) << ',';
os << "]";
return os;
}
};
// Range Minimum Query
// - get: return min(x_l, ..., x_{r - 1})
template <typename T> struct RangeMinimumQuery : public NonrecursiveSegmentTree<T, T, bool> {
using SegTree = NonrecursiveSegmentTree<T, T, bool>;
T merge_data(const T &vl, const T &vr) override { return std::min(vl, vr); };
T data2ret(const T &v, const bool &q) override { return v; }
T merge_ret(const T &vl, const T &vr) override { return std::min(vl, vr); };
RangeMinimumQuery(const std::vector<T> &seq, T defaultmin)
: SegTree::NonrecursiveSegmentTree() {
SegTree::initialize(seq, defaultmin);
};
};
template <typename T> struct PointUpdateRangeSum : public NonrecursiveSegmentTree<T, T, bool> {
using SegTree = NonrecursiveSegmentTree<T, T, bool>;
T merge_data(const T &vl, const T &vr) override { return vl + vr; };
T data2ret(const T &v, const bool &q) override { return v; }
T merge_ret(const T &vl, const T &vr) override { return vl + vr; };
PointUpdateRangeSum(const std::vector<T> &seq, T zero) : SegTree::NonrecursiveSegmentTree() {
SegTree::initialize(seq, zero);
};
};
// Range Counting less than q Query
// - get: return (#{i | l <= i < r, x_i < q}, total sum of them).
template <typename T>
struct CountAndSumLessThan
: public NonrecursiveSegmentTree<std::vector<std::pair<T, T>>, std::pair<int, T>, T> {
using TDATA = std::vector<std::pair<T, T>>;
using TRET = std::pair<int, T>;
using TQUERY = T;
TDATA merge_data(const TDATA &vl, const TDATA &vr) override {
TDATA ret = vl;
ret.insert(ret.end(), vr.begin(), vr.end());
std::sort(ret.begin(), ret.end());
if (ret.size()) {
ret[0].second = ret[0].first;
for (size_t i = 1; i < ret.size(); i++)
ret[i].second = ret[i - 1].second + ret[i].first;
}
return ret;
}
TRET data2ret(const TDATA &vec, const TQUERY &q) override {
int i = std::lower_bound(vec.begin(), vec.end(), std::make_pair(q, q)) - vec.begin();
if (!i)
return std::make_pair(0, 0);
else
return std::make_pair(i, vec[i - 1].second);
}
TRET merge_ret(const TRET &l, const TRET &r) override {
return std::make_pair(l.first + r.first, l.second + r.second);
}
using SegTree = NonrecursiveSegmentTree<TDATA, TRET, TQUERY>;
CountAndSumLessThan(const std::vector<T> &seq) : SegTree::NonrecursiveSegmentTree() {
std::vector<TDATA> init;
for (auto x : seq) init.emplace_back(TDATA{std::pair<T, T>(x, x)});
SegTree::initialize(init, TRET(0, 0));
}
};
#line 5 "tree/heavy_light_decomposition.hpp"
#include <queue>
#line 7 "tree/heavy_light_decomposition.hpp"
#include <utility>
#line 9 "tree/heavy_light_decomposition.hpp"
// Heavy-light decomposition of trees (forest)
// Based on http://beet-aizu.hatenablog.com/entry/2017/12/12/235950
struct heavy_light_decomposition {
int V;
int k;
std::vector<std::vector<int>> e;
std::vector<int> par; // par[i] = parent of vertex i (Default: -1)
std::vector<int> depth; // depth[i] = distance between root and vertex i
std::vector<int> subtree_sz; // subtree_sz[i] = size of subtree whose root is i
std::vector<int> heavy_child; // heavy_child[i] = child of vertex i on heavy path (Default: -1)
std::vector<int> tree_id; // tree_id[i] = id of tree vertex i belongs to
// subtree_begin[i] = aligned id for vertex i (consecutive on heavy edges)
std::vector<int> subtree_begin, subtree_end;
std::vector<int> vis_order; // vis_order[subtree_begin[i]] = i
std::vector<int> head; // head[i] = id of vertex on heavy path of vertex i, nearest to root
heavy_light_decomposition(int n, const std::vector<std::pair<int, int>> &edges)
: V(n), k(0), e(n), par(n), depth(n), subtree_sz(n), heavy_child(n), tree_id(n, -1),
subtree_begin(n), subtree_end(n), vis_order(0), head(n) {
for (auto [u, v] : edges) {
assert(u != v);
e.at(u).emplace_back(v);
e.at(v).emplace_back(u);
}
}
void _build_dfs_1(int root) {
std::vector<std::pair<int, int>> st;
par[root] = -1;
depth[root] = 0;
st.emplace_back(root, 0);
while (!st.empty()) {
int now = st.back().first;
int &i = st.back().second;
if (i < (int)e[now].size()) {
int nxt = e[now][i++];
if (nxt == par[now]) continue;
par[nxt] = now;
depth[nxt] = depth[now] + 1;
st.emplace_back(nxt, 0);
} else {
st.pop_back();
int max_sub_sz = 0;
subtree_sz[now] = 1;
heavy_child[now] = -1;
for (auto nxt : e[now]) {
if (nxt == par[now]) continue;
subtree_sz[now] += subtree_sz[nxt];
if (max_sub_sz < subtree_sz[nxt]) {
max_sub_sz = subtree_sz[nxt], heavy_child[now] = nxt;
}
}
}
}
}
void _build_dfs_2(int now, int tree_id_now) {
std::vector<std::pair<int, bool>> st;
st.emplace_back(now, true);
head[now] = now;
while (!st.empty()) {
auto [now, mode] = st.back();
st.pop_back();
if (mode) {
tree_id[now] = tree_id_now;
subtree_begin[now] = k++;
vis_order.push_back(now);
st.emplace_back(now, false);
for (int nxt : e[now]) {
if (nxt == par[now] or nxt == heavy_child[now]) continue;
head[nxt] = nxt;
st.emplace_back(nxt, true);
}
if (heavy_child[now] != -1) {
head[heavy_child[now]] = head[now];
st.emplace_back(heavy_child[now], true);
}
} else {
subtree_end[now] = k;
}
}
}
void build(std::vector<int> roots = {0}) {
int tree_id_now = 0;
for (int r : roots) {
_build_dfs_1(r);
_build_dfs_2(r, tree_id_now++);
}
}
template <class T> std::vector<T> segtree_rearrange(const std::vector<T> &data) const {
assert(int(data.size()) == V);
std::vector<T> ret;
ret.reserve(vis_order.size());
for (int v : vis_order) ret.emplace_back(data[v]);
return ret;
}
void for_vertex(int u, const std::function<void(int)> &f) const { f(subtree_begin[u]); }
// query for vertices on path [u, v] (INCLUSIVE)
void
for_each_vertex(int u, int v, const std::function<void(int ancestor, int descendant)> &f) const {
while (true) {
if (subtree_begin[u] > subtree_begin[v]) std::swap(u, v);
f(std::max(subtree_begin[head[v]], subtree_begin[u]), subtree_begin[v]);
if (head[u] == head[v]) break;
v = par[head[v]];
}
}
void for_each_vertex_noncommutative(
int from, int to, const std::function<void(int ancestor, int descendant)> &fup,
const std::function<void(int ancestor, int descendant)> &fdown) const {
int u = from, v = to;
const int lca = lowest_common_ancestor(u, v), dlca = depth[lca];
while (u >= 0 and depth[u] > dlca) {
const int p = (depth[head[u]] > dlca ? head[u] : lca);
fup(subtree_begin[p] + (p == lca), subtree_begin[u]), u = par[p];
}
static std::vector<std::pair<int, int>> lrs;
int sz = 0;
while (v >= 0 and depth[v] >= dlca) {
const int p = (depth[head[v]] >= dlca ? head[v] : lca);
if (int(lrs.size()) == sz) lrs.emplace_back(0, 0);
lrs.at(sz++) = {p, v}, v = par.at(p);
}
while (sz--) fdown(subtree_begin[lrs.at(sz).first], subtree_begin[lrs.at(sz).second]);
}
// query for edges on path [u, v]
void for_each_edge(int u, int v, const std::function<void(int, int)> &f) const {
while (true) {
if (subtree_begin[u] > subtree_begin[v]) std::swap(u, v);
if (head[u] != head[v]) {
f(subtree_begin[head[v]], subtree_begin[v]);
v = par[head[v]];
} else {
if (u != v) f(subtree_begin[u] + 1, subtree_begin[v]);
break;
}
}
}
// query for vertices in subtree rooted at u
void for_subtree(int u, const std::function<void(int, int)> &f) const {
f(subtree_begin[u], subtree_end[u] - 1);
}
// lowest_common_ancestor: O(log V)
int lowest_common_ancestor(int u, int v) const {
assert(tree_id[u] == tree_id[v] and tree_id[u] >= 0);
while (true) {
if (subtree_begin[u] > subtree_begin[v]) std::swap(u, v);
if (head[u] == head[v]) return u;
v = par[head[v]];
}
}
int distance(int u, int v) const {
assert(tree_id[u] == tree_id[v] and tree_id[u] >= 0);
return depth[u] + depth[v] - 2 * depth[lowest_common_ancestor(u, v)];
}
// Level ancestor, O(log V)
// if k-th parent is out of range, return -1
int kth_parent(int v, int k) const {
if (k < 0) return -1;
while (v >= 0) {
int h = head.at(v), len = depth.at(v) - depth.at(h);
if (k <= len) return vis_order.at(subtree_begin.at(v) - k);
k -= len + 1, v = par.at(h);
}
return -1;
}
// Jump on tree, O(log V)
int s_to_t_by_k_steps(int s, int t, int k) const {
if (k < 0) return -1;
if (k == 0) return s;
int lca = lowest_common_ancestor(s, t);
if (k <= depth.at(s) - depth.at(lca)) return kth_parent(s, k);
return kth_parent(t, depth.at(s) + depth.at(t) - depth.at(lca) * 2 - k);
}
};
#line 7 "tree/test/vertex-set-path-composite.test.cpp"
using namespace std;
using mint = ModInt<998244353>;
using P = pair<mint, mint>;
struct PointSetRangeComposite : public NonrecursiveSegmentTree<P, P, bool> {
using SegTree = NonrecursiveSegmentTree<P, P, bool>;
P merge_data(const P &vl, const P &vr) override {
return make_pair(vl.first * vr.first, vr.first * vl.second + vr.second);
};
P data2ret(const P &v, const bool &q) override { return v; }
P merge_ret(const P &vl, const P &vr) override { return merge_data(vl, vr); };
PointSetRangeComposite(const std::vector<P> &seq, P zero)
: SegTree::NonrecursiveSegmentTree() {
SegTree::initialize(seq, zero);
};
};
int main() {
cin.tie(nullptr), ios::sync_with_stdio(false);
int N, Q;
cin >> N >> Q;
vector<P> V(N);
for (auto &x : V) cin >> x.first >> x.second;
vector<pair<int, int>> edges(N - 1);
for (auto &[u, v] : edges) cin >> u >> v;
heavy_light_decomposition hld(N, edges);
hld.build();
vector<P> stinit = hld.segtree_rearrange(V);
PointSetRangeComposite segtree(stinit, P{1, 0});
reverse(stinit.begin(), stinit.end());
PointSetRangeComposite segtreeinv(stinit, P{1, 0});
while (Q--) {
int q, u, v, x;
cin >> q >> u >> v >> x;
if (q == 0) {
segtree.update(hld.subtree_begin[u], P{v, x});
segtreeinv.update(N - 1 - hld.subtree_begin[u], P{v, x});
} else {
mint ret = x;
hld.for_each_vertex_noncommutative(
u, v,
[&](int l, int r) -> void {
assert(0 <= l and l <= r and r < N);
P tmp = segtreeinv.get(N - 1 - r, N - 1 - l + 1);
ret = tmp.first * ret + tmp.second;
},
[&](int l, int r) -> void {
assert(0 <= l and l <= r and r < N);
P tmp = segtree.get(l, r + 1);
ret = tmp.first * ret + tmp.second;
});
cout << ret << '\n';
}
}
}