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#define PROBLEM "https://judge.yosupo.jp/problem/vertex_set_path_composite" #include <iostream> #include <vector> #include "../../data_structure/segtree.hpp" #include "../../graph/base.hpp" #include "../../modint/modint.hpp" #include "../../tree/heavy_light_decomposition.hpp" using mint = ebi::modint998244353; using i64 = std::int64_t; struct S { mint c, d; }; S op(S a, S b) { return {b.c * a.c, b.c * a.d + b.d}; } S e() { return {1, 0}; } int main() { int n, q; std::cin >> n >> q; ebi::Graph<int> g(n); std::vector<S> fx(n); for (int i = 0; i < n; i++) { int a, b; std::cin >> a >> b; fx[i] = {a, b}; } g.read_tree(0); ebi::heavy_light_decomposition hld(g); ebi::segtree<S, op, e> seg1(n); ebi::segtree<S, op, e> seg2(n); for (int i = 0; i < n; i++) { int idx = hld.idx(i); seg1.set(idx, fx[i]); seg2.set(n - 1 - idx, fx[i]); } S ans = e(); auto f = [&](int l, int r) -> void { if (l <= r) { ans = op(ans, seg1.prod(l, r)); } else { ans = op(ans, seg2.prod(n - l, n - r)); } }; while (q--) { int t; std::cin >> t; if (t == 0) { int p, c, d; std::cin >> p >> c >> d; int idx = hld.idx(p); seg1.set(idx, {c, d}); seg2.set(n - 1 - idx, {c, d}); } else { int u, v, x; std::cin >> u >> v >> x; ans = e(); hld.path_noncommutative_query(u, v, true, f); std::cout << (ans.c * x + ans.d).val() << '\n'; } } }
#line 1 "test/data_structure/Vertex_Set_Path_Compositie.test.cpp" #define PROBLEM "https://judge.yosupo.jp/problem/vertex_set_path_composite" #include <iostream> #include <vector> #line 2 "data_structure/segtree.hpp" #include <cassert> #line 5 "data_structure/segtree.hpp" namespace ebi { template <class S, S (*op)(S, S), S (*e)()> struct segtree { private: int n; int sz; std::vector<S> data; void update(int i) { data[i] = op(data[2 * i], data[2 * i + 1]); } public: segtree(int n_) : segtree(std::vector<S>(n_, e())) {} segtree(const std::vector<S> &v) : n((int)v.size()), sz(1) { while (sz < n) sz *= 2; data = std::vector<S>(2 * sz, e()); for (int i = 0; i < n; i++) { data[sz + i] = v[i]; } for (int i = sz - 1; i >= 1; i--) update(i); } void set(int p, S x) { assert(0 <= p && p < n); p += sz; data[p] = x; while (p > 1) { p >>= 1; update(p); } } S get(int p) const { assert(0 <= p && p < n); return data[p + sz]; } S prod(int l, int r) const { assert(0 <= l && l <= r && r <= n); S sml = e(), smr = e(); l += sz; r += sz; while (l < r) { if (l & 1) sml = op(sml, data[l++]); if (r & 1) smr = op(data[--r], smr); l >>= 1; r >>= 1; } return op(sml, smr); } S all_prod() const { return data[1]; } 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 += sz; S sm = e(); do { while (l % 2 == 0) l >>= 1; if (!f(op(sm, data[l]))) { while (l < sz) { l = 2 * l; if (f(op(sm, data[l]))) { sm = op(sm, data[l]); l++; } } return l - sz; } sm = op(sm, data[l]); l++; } while ((l & -l) != l); return n; } 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 += sz; S sm = e(); do { r--; while (r > 1 && (r % 2)) r >>= 1; if (!f(op(data[r], sm))) { while (r < sz) { r = 2 * r + 1; if (f(op(data[r], sm))) { sm = op(data[r], sm); r--; } } return r + 1 - sz; } sm = op(data[r], sm); } while ((r & -r) != r); return 0; } S operator[](int p) const { return data[sz + p]; } }; } // namespace ebi #line 2 "graph/base.hpp" #line 5 "graph/base.hpp" #include <ranges> #line 7 "graph/base.hpp" #line 2 "data_structure/simple_csr.hpp" #line 4 "data_structure/simple_csr.hpp" #include <utility> #line 6 "data_structure/simple_csr.hpp" namespace ebi { template <class E> struct simple_csr { simple_csr() = default; simple_csr(int n, const std::vector<std::pair<int, E>>& elements) : start(n + 1, 0), elist(elements.size()) { for (auto e : elements) { start[e.first + 1]++; } for (auto i : std::views::iota(0, n)) { start[i + 1] += start[i]; } auto counter = start; for (auto [i, e] : elements) { elist[counter[i]++] = e; } } simple_csr(const std::vector<std::vector<E>>& es) : start(es.size() + 1, 0) { int n = es.size(); for (auto i : std::views::iota(0, n)) { start[i + 1] = (int)es[i].size() + start[i]; } elist.resize(start.back()); for (auto i : std::views::iota(0, n)) { std::copy(es[i].begin(), es[i].end(), elist.begin() + start[i]); } } int size() const { return (int)start.size() - 1; } const auto operator[](int i) const { return std::ranges::subrange(elist.begin() + start[i], elist.begin() + start[i + 1]); } auto operator[](int i) { return std::ranges::subrange(elist.begin() + start[i], elist.begin() + start[i + 1]); } const auto operator()(int i, int l, int r) const { return std::ranges::subrange(elist.begin() + start[i] + l, elist.begin() + start[i + 1] + r); } auto operator()(int i, int l, int r) { return std::ranges::subrange(elist.begin() + start[i] + l, elist.begin() + start[i + 1] + r); } private: std::vector<int> start; std::vector<E> elist; }; } // namespace ebi #line 9 "graph/base.hpp" namespace ebi { template <class T> struct Edge { int from, to; T cost; int id; }; template <class E> struct Graph { using cost_type = E; using edge_type = Edge<cost_type>; Graph(int n_) : n(n_) {} Graph() = default; void add_edge(int u, int v, cost_type c) { buff.emplace_back(u, edge_type{u, v, c, m}); edges.emplace_back(edge_type{u, v, c, m++}); } void add_undirected_edge(int u, int v, cost_type c) { buff.emplace_back(u, edge_type{u, v, c, m}); buff.emplace_back(v, edge_type{v, u, c, m}); edges.emplace_back(edge_type{u, v, c, m}); m++; } void read_tree(int offset = 1, bool is_weighted = false) { read_graph(n - 1, offset, false, is_weighted); } void read_parents(int offset = 1) { for (auto i : std::views::iota(1, n)) { int p; std::cin >> p; p -= offset; add_undirected_edge(p, i, 1); } build(); } void read_graph(int e, int offset = 1, bool is_directed = false, bool is_weighted = false) { for (int i = 0; i < e; i++) { int u, v; std::cin >> u >> v; u -= offset; v -= offset; if (is_weighted) { cost_type c; std::cin >> c; if (is_directed) { add_edge(u, v, c); } else { add_undirected_edge(u, v, c); } } else { if (is_directed) { add_edge(u, v, 1); } else { add_undirected_edge(u, v, 1); } } } build(); } void build() { assert(!prepared); csr = simple_csr<edge_type>(n, buff); buff.clear(); prepared = true; } int size() const { return n; } int node_number() const { return n; } int edge_number() const { return m; } edge_type get_edge(int i) const { return edges[i]; } std::vector<edge_type> get_edges() const { return edges; } const auto operator[](int i) const { return csr[i]; } auto operator[](int i) { return csr[i]; } private: int n, m = 0; std::vector<std::pair<int,edge_type>> buff; std::vector<edge_type> edges; simple_csr<edge_type> csr; bool prepared = false; }; } // namespace ebi #line 2 "modint/modint.hpp" #line 5 "modint/modint.hpp" #line 2 "modint/base.hpp" #include <concepts> #line 6 "modint/base.hpp" namespace ebi { template <class T> concept Modint = requires(T a, T b) { a + b; a - b; a * b; a / b; a.inv(); a.val(); a.pow(std::declval<long long>()); T::mod(); }; template <Modint mint> std::istream &operator>>(std::istream &os, mint &a) { long long x; os >> x; a = x; return os; } template <Modint mint> std::ostream &operator<<(std::ostream &os, const mint &a) { return os << a.val(); } } // namespace ebi #line 7 "modint/modint.hpp" namespace ebi { template <int m> struct static_modint { private: using modint = static_modint; public: static constexpr int mod() { return m; } static constexpr modint raw(int v) { modint x; x._v = v; return x; } constexpr static_modint() : _v(0) {} constexpr static_modint(long long v) { v %= (long long)umod(); if (v < 0) v += (long long)umod(); _v = (unsigned int)v; } constexpr unsigned int val() const { return _v; } constexpr unsigned int value() const { return val(); } constexpr modint &operator++() { _v++; if (_v == umod()) _v = 0; return *this; } constexpr modint &operator--() { if (_v == 0) _v = umod(); _v--; return *this; } constexpr modint operator++(int) { modint res = *this; ++*this; return res; } constexpr modint operator--(int) { modint res = *this; --*this; return res; } constexpr modint &operator+=(const modint &rhs) { _v += rhs._v; if (_v >= umod()) _v -= umod(); return *this; } constexpr modint &operator-=(const modint &rhs) { _v -= rhs._v; if (_v >= umod()) _v += umod(); return *this; } constexpr modint &operator*=(const modint &rhs) { unsigned long long x = _v; x *= rhs._v; _v = (unsigned int)(x % (unsigned long long)umod()); return *this; } constexpr modint &operator/=(const modint &rhs) { return *this = *this * rhs.inv(); } constexpr modint operator+() const { return *this; } constexpr modint operator-() const { return modint() - *this; } constexpr modint pow(long long n) const { assert(0 <= n); modint x = *this, res = 1; while (n) { if (n & 1) res *= x; x *= x; n >>= 1; } return res; } constexpr modint inv() const { assert(_v); return pow(umod() - 2); } friend modint operator+(const modint &lhs, const modint &rhs) { return modint(lhs) += rhs; } friend modint operator-(const modint &lhs, const modint &rhs) { return modint(lhs) -= rhs; } friend modint operator*(const modint &lhs, const modint &rhs) { return modint(lhs) *= rhs; } friend modint operator/(const modint &lhs, const modint &rhs) { return modint(lhs) /= rhs; } friend bool operator==(const modint &lhs, const modint &rhs) { return lhs.val() == rhs.val(); } friend bool operator!=(const modint &lhs, const modint &rhs) { return !(lhs == rhs); } private: unsigned int _v = 0; static constexpr unsigned int umod() { return m; } }; using modint998244353 = static_modint<998244353>; using modint1000000007 = static_modint<1000000007>; } // namespace ebi #line 2 "tree/heavy_light_decomposition.hpp" #include <algorithm> #line 6 "tree/heavy_light_decomposition.hpp" #line 8 "tree/heavy_light_decomposition.hpp" namespace ebi { template <class T> struct heavy_light_decomposition { private: void dfs_sz(int v, Graph<T> &g) { for (auto &e : g[v]) { if (e.to == par[v]) continue; par[e.to] = v; depth_[e.to] = depth_[v] + 1; dist[e.to] = dist[v] + e.cost; dfs_sz(e.to, g); sz[v] += sz[e.to]; if (sz[e.to] > sz[g[v][0].to] || g[v][0].to == par[v]) std::swap(e, g[v][0]); } } void dfs_hld(int v, const Graph<T> &g) { in[v] = num++; rev[in[v]] = v; for (auto e : g[v]) { if (e.to == par[v]) continue; nxt[e.to] = (e.to == g[v][0].to ? nxt[v] : e.to); dfs_hld(e.to, g); } out[v] = num; } // [u, v) パスの取得 (v は u の祖先) std::vector<std::pair<int, int>> ascend(int u, int v) const { std::vector<std::pair<int, int>> res; while (nxt[u] != nxt[v]) { res.emplace_back(in[u], in[nxt[u]]); u = par[nxt[u]]; } if (u != v) res.emplace_back(in[u], in[v] + 1); return res; } // (u, v] パスの取得 (u は v の祖先) std::vector<std::pair<int, int>> descend(int u, int v) const { if (u == v) return {}; if (nxt[u] == nxt[v]) return {{in[u] + 1, in[v]}}; auto res = descend(u, par[nxt[v]]); res.emplace_back(in[nxt[v]], in[v]); return res; } public: heavy_light_decomposition(Graph<T> gh, int root = 0) : n(gh.size()), sz(n, 1), in(n), out(n), nxt(n), par(n, -1), depth_(n, 0), rev(n), dist(n, 0) { nxt[root] = root; dfs_sz(root, gh); dfs_hld(root, gh); } int idx(int u) const { return in[u]; } int rev_idx(int i) const { return rev[i]; } int la(int v, int k) const { while (1) { int u = nxt[v]; if (in[u] <= in[v] - k) return rev[in[v] - k]; k -= in[v] - in[u] + 1; v = par[u]; } } int lca(int u, int v) const { while (nxt[u] != nxt[v]) { if (in[u] < in[v]) std::swap(u, v); u = par[nxt[u]]; } return depth_[u] < depth_[v] ? u : v; } int jump(int s, int t, int i) const { if (i == 0) return s; int l = lca(s, t); int d = depth_[s] + depth_[t] - depth_[l] * 2; if (d < i) return -1; if (depth_[s] - depth_[l] >= i) return la(s, i); i = d - i; return la(t, i); } std::vector<int> path(int s, int t) const { int l = lca(s, t); std::vector<int> a, b; for (; s != l; s = par[s]) a.emplace_back(s); for (; t != l; t = par[t]) b.emplace_back(t); a.emplace_back(l); std::reverse(b.begin(), b.end()); a.insert(a.end(), b.begin(), b.end()); return a; } int root_of_heavy_path(int u) const { return nxt[u]; } int parent(int u) const { return par[u]; } T distance(int u, int v) const { return dist[u] + dist[v] - 2 * dist[lca(u, v)]; } T distance_from_root(int v) const { return dist[v]; } T depth(int v) const { return depth_[v]; } bool at_path(int u, int v, int s) const { return distance(u, v) == distance(u, s) + distance(s, v); } template <class F> void path_noncommutative_query(int u, int v, bool vertex, const F &f) const { int l = lca(u, v); for (auto [a, b] : ascend(u, l)) f(a + 1, b); if (vertex) f(in[l], in[l] + 1); for (auto [a, b] : descend(l, v)) f(a, b + 1); } std::vector<std::pair<int, int>> path_sections(int u, int v, bool vertex) const { int l = lca(u, v); std::vector<std::pair<int, int>> sections; for (auto [a, b] : ascend(u, l)) sections.emplace_back(a + 1, b); if (vertex) sections.emplace_back(in[l], in[l] + 1); for (auto [a, b] : descend(l, v)) sections.emplace_back(a, b + 1); return sections; } template <class F> int max_path(int u, int v, bool vertex, F binary_search) const { int prev = -1; int l = lca(u, v); for (auto [a, b] : ascend(u, l)) { a++; int m = binary_search(a, b); if (m == b) { prev = rev[b]; } else { return (m == a ? prev : rev[m]); } } if (vertex) { int m = binary_search(in[l], in[l] + 1); if (m == in[l]) { return prev; } else { prev = l; } } for (auto [a, b] : descend(l, v)) { b++; int m = binary_search(a, b); if (m == b) { prev = rev[b - 1]; } else { return m == a ? prev : rev[m - 1]; } } return v; } template <class F> void subtree_query(int u, bool vertex, const F &f) { f(in[u] + int(!vertex), out[u]); } const std::vector<int> &dfs_order() const { return rev; } std::vector<std::pair<int, int>> lca_based_auxiliary_tree_dfs_order( std::vector<int> vs) const; std::pair<std::vector<int>, Graph<T>> lca_based_auxiliary_tree( std::vector<int> vs) const; private: int n; std::vector<int> sz, in, out, nxt, par, depth_, rev; std::vector<T> dist; int num = 0; }; } // namespace ebi #line 10 "test/data_structure/Vertex_Set_Path_Compositie.test.cpp" using mint = ebi::modint998244353; using i64 = std::int64_t; struct S { mint c, d; }; S op(S a, S b) { return {b.c * a.c, b.c * a.d + b.d}; } S e() { return {1, 0}; } int main() { int n, q; std::cin >> n >> q; ebi::Graph<int> g(n); std::vector<S> fx(n); for (int i = 0; i < n; i++) { int a, b; std::cin >> a >> b; fx[i] = {a, b}; } g.read_tree(0); ebi::heavy_light_decomposition hld(g); ebi::segtree<S, op, e> seg1(n); ebi::segtree<S, op, e> seg2(n); for (int i = 0; i < n; i++) { int idx = hld.idx(i); seg1.set(idx, fx[i]); seg2.set(n - 1 - idx, fx[i]); } S ans = e(); auto f = [&](int l, int r) -> void { if (l <= r) { ans = op(ans, seg1.prod(l, r)); } else { ans = op(ans, seg2.prod(n - l, n - r)); } }; while (q--) { int t; std::cin >> t; if (t == 0) { int p, c, d; std::cin >> p >> c >> d; int idx = hld.idx(p); seg1.set(idx, {c, d}); seg2.set(n - 1 - idx, {c, d}); } else { int u, v, x; std::cin >> u >> v >> x; ans = e(); hld.path_noncommutative_query(u, v, true, f); std::cout << (ans.c * x + ans.d).val() << '\n'; } } }