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#define PROBLEM "https://judge.yosupo.jp/problem/vertex_set_path_composite" #include <bits/stdc++.h> using namespace std; #include "../SegTree.h" #include "../../Math/modint.h" #include "../HeavyLight_adamant.h" using modular = ModInt<998244353>; // SegTree ops struct F { modular a, b; }; F op(const F& l, const F& r) { return F{ l.a*r.a, r.a*l.b + r.b }; } struct Node { F forward, backward; }; Node op(Node l, Node r) { return Node { op(l.forward, r.forward), op(r.backward, l.backward) }; } Node e() { return Node { F{1, 0}, F{1, 0} }; } #define REP(i, a) for (int i = 0, _##i = (a); i < _##i; ++i) int32_t main() { ios::sync_with_stdio(0); cin.tie(0); int n, q; cin >> n >> q; vector<F> fs(n); REP(i,n) { int a, b; cin >> a >> b; fs[i] = {a, b}; } vector<vector<int>> g(n); REP(i,n-1) { int u, v; cin >> u >> v; g[u].push_back(v); g[v].push_back(u); } HLD hld(g, 0); vector<Node> nodes; REP(i,n) { auto f = fs[hld.order[i]]; nodes.push_back({f, f}); } SegTree<Node, op, e> tree(nodes); while (q--) { int typ; cin >> typ; if (typ == 0) { int p, a, b; cin >> p >> a >> b; tree.set(hld.in[p], {{a, b}, {a, b}}); } else { int start, end, x; cin >> start >> end >> x; auto segments = hld.getSegments(start, end); F res {1, 0}; for (auto [u, v] : segments) { if (hld.in[u] <= hld.in[v]) { res = op(res, tree.prod(hld.in[u], hld.in[v] + 1).forward); } else { res = op(res, tree.prod(hld.in[v], hld.in[u] + 1).backward); } } cout << (res.a * modular(x) + res.b) << '\n'; } } return 0; }
#line 1 "DataStructure/test/hld_vertexsetpathcomposite.test.cpp" #define PROBLEM "https://judge.yosupo.jp/problem/vertex_set_path_composite" #include <bits/stdc++.h> using namespace std; #line 1 "DataStructure/SegTree.h" // SegTree, copied from AtCoder library {{{ // AtCoder doc: https://atcoder.github.io/ac-library/master/document_en/segtree.html // // Notes: // - Index of elements from 0 -> n-1 // - Range queries are [l, r-1] // // Tested: // - (binary search) https://atcoder.jp/contests/practice2/tasks/practice2_j // - https://oj.vnoi.info/problem/gss // - https://oj.vnoi.info/problem/nklineup // - (max_right & min_left for delete position queries) https://oj.vnoi.info/problem/segtree_itstr // - https://judge.yosupo.jp/problem/point_add_range_sum // - https://judge.yosupo.jp/problem/point_set_range_composite int ceil_pow2(int n) { int x = 0; while ((1U << x) < (unsigned int)(n)) x++; return x; } template< class T, // data type for nodes T (*op) (T, T), // operator to combine 2 nodes T (*e)() // identity element > struct SegTree { SegTree() : SegTree(0) {} explicit SegTree(int n) : SegTree(vector<T> (n, e())) {} explicit SegTree(const vector<T>& v) : _n((int) v.size()) { log = ceil_pow2(_n); size = 1<<log; d = vector<T> (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); } } // 0 <= p < n void set(int p, T x) { assert(0 <= p && p < _n); p += size; d[p] = x; for (int i = 1; i <= log; i++) update(p >> i); } // 0 <= p < n T get(int p) const { assert(0 <= p && p < _n); return d[p + size]; } // Get product in range [l, r-1] // 0 <= l <= r <= n // For empty segment (l == r) -> return e() T prod(int l, int r) const { assert(0 <= l && l <= r && r <= _n); T 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); } T all_prod() const { return d[1]; } // Binary search on SegTree to find largest r: // f(op(a[l] .. a[r-1])) = true (assuming empty array is always true) // f(op(a[l] .. a[r])) = false (assuming op(..., a[n]), which is out of bound, is always false) template <bool (*f)(T)> int max_right(int l) const { return max_right(l, [](T 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; T 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; } // Binary search on SegTree to find smallest l: // f(op(a[l] .. a[r-1])) = true (assuming empty array is always true) // f(op(a[l-1] .. a[r-1])) = false (assuming op(a[-1], ..), which is out of bound, is always false) template <bool (*f)(T)> int min_left(int r) const { return min_left(r, [](T 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; T 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; vector<T> d; void update(int k) { d[k] = op(d[2*k], d[2*k+1]); } }; // }}} // SegTree examples {{{ // Examples: Commonly used SegTree ops: max / min / sum struct MaxSegTreeOp { static int op(int x, int y) { return max(x, y); } static int e() { return INT_MIN; } }; struct MinSegTreeOp { static int op(int x, int y) { return min(x, y); } static int e() { return INT_MAX; } }; struct SumSegTreeOp { static long long op(long long x, long long y) { return x + y; } static long long e() { return 0; } }; // using STMax = SegTree<int, MaxSegTreeOp::op, MaxSegTreeOp::e>; // using STMin = SegTree<int, MinSegTreeOp::op, MinSegTreeOp::e>; // using STSum = SegTree<int, SumSegTreeOp::op, SumSegTreeOp::e>; // }}} #line 1 "Math/modint.h" // ModInt {{{ template<int MD> struct ModInt { using ll = long long; int x; constexpr ModInt() : x(0) {} constexpr ModInt(ll v) { _set(v % MD + MD); } constexpr static int mod() { return MD; } constexpr explicit operator bool() const { return x != 0; } constexpr ModInt operator + (const ModInt& a) const { return ModInt()._set((ll) x + a.x); } constexpr ModInt operator - (const ModInt& a) const { return ModInt()._set((ll) x - a.x + MD); } constexpr ModInt operator * (const ModInt& a) const { return ModInt()._set((ll) x * a.x % MD); } constexpr ModInt operator / (const ModInt& a) const { return ModInt()._set((ll) x * a.inv().x % MD); } constexpr ModInt operator - () const { return ModInt()._set(MD - x); } constexpr ModInt& operator += (const ModInt& a) { return *this = *this + a; } constexpr ModInt& operator -= (const ModInt& a) { return *this = *this - a; } constexpr ModInt& operator *= (const ModInt& a) { return *this = *this * a; } constexpr ModInt& operator /= (const ModInt& a) { return *this = *this / a; } friend constexpr ModInt operator + (ll a, const ModInt& b) { return ModInt()._set(a % MD + b.x); } friend constexpr ModInt operator - (ll a, const ModInt& b) { return ModInt()._set(a % MD - b.x + MD); } friend constexpr ModInt operator * (ll a, const ModInt& b) { return ModInt()._set(a % MD * b.x % MD); } friend constexpr ModInt operator / (ll a, const ModInt& b) { return ModInt()._set(a % MD * b.inv().x % MD); } constexpr bool operator == (const ModInt& a) const { return x == a.x; } constexpr bool operator != (const ModInt& a) const { return x != a.x; } friend std::istream& operator >> (std::istream& is, ModInt& other) { ll val; is >> val; other = ModInt(val); return is; } constexpr friend std::ostream& operator << (std::ostream& os, const ModInt& other) { return os << other.x; } constexpr ModInt pow(ll k) const { ModInt ans = 1, tmp = x; while (k) { if (k & 1) ans *= tmp; tmp *= tmp; k >>= 1; } return ans; } constexpr ModInt inv() const { if (x < 1000111) { _precalc(1000111); return invs[x]; } int a = x, b = MD, ax = 1, bx = 0; while (b) { int q = a/b, t = a%b; a = b; b = t; t = ax - bx*q; ax = bx; bx = t; } assert(a == 1); if (ax < 0) ax += MD; return ax; } static std::vector<ModInt> factorials, inv_factorials, invs; constexpr static void _precalc(int n) { if (factorials.empty()) { factorials = {1}; inv_factorials = {1}; invs = {0}; } if (n > MD) n = MD; int old_sz = factorials.size(); if (n <= old_sz) return; factorials.resize(n); inv_factorials.resize(n); invs.resize(n); for (int i = old_sz; i < n; ++i) factorials[i] = factorials[i-1] * i; inv_factorials[n-1] = factorials.back().pow(MD - 2); for (int i = n - 2; i >= old_sz; --i) inv_factorials[i] = inv_factorials[i+1] * (i+1); for (int i = n-1; i >= old_sz; --i) invs[i] = inv_factorials[i] * factorials[i-1]; } static int get_primitive_root() { static int primitive_root = 0; if (!primitive_root) { primitive_root = [&]() { std::set<int> fac; int v = MD - 1; for (ll 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; } static ModInt C(int n, int k) { _precalc(n + 1); return factorials[n] * inv_factorials[k] * inv_factorials[n-k]; } private: // Internal, DO NOT USE. // val must be in [0, 2*MD) constexpr inline __attribute__((always_inline)) ModInt& _set(ll v) { x = v >= MD ? v - MD : v; return *this; } }; template <int MD> std::vector<ModInt<MD>> ModInt<MD>::factorials = {1}; template <int MD> std::vector<ModInt<MD>> ModInt<MD>::inv_factorials = {1}; template <int MD> std::vector<ModInt<MD>> ModInt<MD>::invs = {0}; // }}} #line 1 "DataStructure/HeavyLight_adamant.h" // Index from 0 // Best used with SegTree.h // // Usage: // HLD hld(g, root); // // build segment tree. Note that we must use hld.order[i] // vector<T> nodes; // for (int i = 0; i < n; i++) // nodes.push_back(initial_value[hld.order[i]]) // SegTree<S, op, e> st(nodes); // // // Update single vertex // st.set(hld.in[u], new_value) // // // Update path // hld.apply_path(from, to, is_edge, [&] (int l, int r) { // st.apply(l, r+1, F); // }); // // // Query path // hld.prod_path_commutative<S, op, e> (from, to, is_edge, [&] (int l, int r) { // return st.prod(l, r+1); // }); // // Tested: // - (vertex, path) https://judge.yosupo.jp/problem/vertex_add_path_sum // - (vertex, path, non-commutative) https://judge.yosupo.jp/problem/vertex_set_path_composite // - (vertex, subtree) https://judge.yosupo.jp/problem/vertex_add_subtree_sum // - (vertex, path, non-commutative, 1-index) https://oj.vnoi.info/problem/icpc21_mt_l // - (vertex, path) https://oj.vnoi.info/problem/qtree3 // // - (edge, path) https://oj.vnoi.info/problem/qtreex // - (edge, path) https://oj.vnoi.info/problem/lubenica // - (edge, path) https://oj.vnoi.info/problem/pwalk // - (edge, path, lazy) https://oj.vnoi.info/problem/kbuild // - (edge, path, lazy) https://oj.vnoi.info/problem/onbridge // // - (lca) https://oj.vnoi.info/problem/fselect // - (kth_parent) https://cses.fi/problemset/task/1687 // HeavyLight {{{ struct HLD { HLD(const vector<vector<int>>& _g, int root) : n(_g.size()), g(_g), parent(n), depth(n), sz(n), dfs_number(0), nxt(n), in(n), out(n), order(n) { assert(0 <= root && root < n); // init parent, depth, sz // also move most heavy child of u to g[u][0] depth[root] = 0; dfs_sz(root, -1); // init nxt, in, out nxt[root] = root; dfs_hld(root); } int lca(int u, int v) const { assert(0 <= u && u < n); assert(0 <= v && v < n); while (true) { if (in[u] > in[v]) swap(u, v); // in[u] <= in[v] if (nxt[u] == nxt[v]) return u; v = parent[nxt[v]]; } } // return k-th parent // if no such parent -> return -1 int kth_parent(int u, int k) const { assert(0 <= u && u < n); if (depth[u] < k) return -1; while (true) { int v = nxt[u]; if (in[u] - k >= in[v]) return order[in[u] - k]; k -= in[u] - in[v] + 1; u = parent[v]; } } // return k-th vertex on path from u -> v (0 <= k) // if k > distance -> return -1 int kth_vertex_on_path(int u, int v, int k) const { assert(0 <= u && u < n); assert(0 <= v && v < n); int l = lca(u, v); int ul = depth[u] - depth[l]; if (k <= ul) return kth_parent(u, k); k -= ul; int vl = depth[v] - depth[l]; if (k <= vl) return kth_parent(v, vl - k); return -1; } int dist(int u, int v) const { assert(0 <= u && u < n); assert(0 <= v && v < n); int l = lca(u, v); return depth[u] + depth[v] - 2*depth[l]; } // apply f on vertices on path [u, v] // edge = true -> apply on edge // // f(l, r) should update segment tree [l, r] INCLUSIVE void apply_path(int u, int v, bool edge, const function<void(int, int)> &f) { assert(0 <= u && u < n); assert(0 <= v && v < n); if (u == v && edge) return; while (true) { if (in[u] > in[v]) swap(u, v); // in[u] <= in[v] if (nxt[u] == nxt[v]) break; f(in[nxt[v]], in[v]); v = parent[nxt[v]]; } if (u == v && edge) return; f(in[u] + edge, in[v]); } // get prod of path u -> v // edge = true -> get on edges // // f(l, r) should query segment tree [l, r] INCLUSIVE // f must be commutative. For non-commutative, use getSegments below template<class S, S (*op) (S, S), S (*e)()> S prod_path_commutative( int u, int v, bool edge, const function<S(int, int)>& f) const { assert(0 <= u && u < n); assert(0 <= v && v < n); if (u == v && edge) { return e(); } S su = e(), sv = e(); while (true) { if (in[u] > in[v]) { swap(u, v); swap(su, sv); } if (nxt[u] == nxt[v]) break; sv = op(sv, f(in[nxt[v]], in[v])); v = parent[nxt[v]]; } if (u == v && edge) { return op(su, sv); } else { return op(su, op(sv, f(in[u] + edge, in[v]))); } } // f(l, r) modify seg_tree [l, r] INCLUSIVE void apply_subtree(int u, bool edge, const function<void(int, int)>& f) { assert(0 <= u && u < n); f(in[u] + edge, out[u] - 1); } // f(l, r) queries seg_tree [l, r] INCLUSIVE template<class S> S prod_subtree_commutative(int u, bool edge, const function<S(S, S)>& f) { assert(0 <= u && u < n); return f(in[u] + edge, out[u] - 1); } // Useful when functions are non-commutative // Return all segments on path from u -> v // For this problem, the order (u -> v is different from v -> u) vector< pair<int,int> > getSegments(int u, int v) const { assert(0 <= u && u < n); assert(0 <= v && v < n); vector< pair<int,int> > upFromU, upFromV; int fu = nxt[u], fv = nxt[v]; while (fu != fv) { // u and v are on different chains if (depth[fu] >= depth[fv]) { // move u up upFromU.push_back({u, fu}); u = parent[fu]; fu = nxt[u]; } else { // move v up upFromV.push_back({fv, v}); v = parent[fv]; fv = nxt[v]; } } upFromU.push_back({u, v}); reverse(upFromV.begin(), upFromV.end()); upFromU.insert(upFromU.end(), upFromV.begin(), upFromV.end()); return upFromU; } // return true if u is ancestor bool isAncestor(int u, int v) const { return in[u] <= in[v] && out[v] <= out[u]; } // private: int n; vector<vector<int>> g; vector<int> parent; // par[u] = parent of u. par[root] = -1 vector<int> depth; // depth[u] = distance from root -> u vector<int> sz; // sz[u] = size of subtree rooted at u int dfs_number; vector<int> nxt; // nxt[u] = vertex on heavy path of u, nearest to root vector<int> in, out; // subtree(u) is in range [in[u], out[u]-1] vector<int> order; // euler tour void dfs_sz(int u, int fu) { parent[u] = fu; sz[u] = 1; // remove parent from adjacency list auto it = std::find(g[u].begin(), g[u].end(), fu); if (it != g[u].end()) g[u].erase(it); for (int& v : g[u]) { depth[v] = depth[u] + 1; dfs_sz(v, u); sz[u] += sz[v]; if (sz[v] > sz[g[u][0]]) swap(v, g[u][0]); } } void dfs_hld(int u) { order[dfs_number] = u; in[u] = dfs_number++; for (int v : g[u]) { nxt[v] = (v == g[u][0] ? nxt[u] : v); dfs_hld(v); } out[u] = dfs_number; } }; // }}} #line 9 "DataStructure/test/hld_vertexsetpathcomposite.test.cpp" using modular = ModInt<998244353>; // SegTree ops struct F { modular a, b; }; F op(const F& l, const F& r) { return F{ l.a*r.a, r.a*l.b + r.b }; } struct Node { F forward, backward; }; Node op(Node l, Node r) { return Node { op(l.forward, r.forward), op(r.backward, l.backward) }; } Node e() { return Node { F{1, 0}, F{1, 0} }; } #define REP(i, a) for (int i = 0, _##i = (a); i < _##i; ++i) int32_t main() { ios::sync_with_stdio(0); cin.tie(0); int n, q; cin >> n >> q; vector<F> fs(n); REP(i,n) { int a, b; cin >> a >> b; fs[i] = {a, b}; } vector<vector<int>> g(n); REP(i,n-1) { int u, v; cin >> u >> v; g[u].push_back(v); g[v].push_back(u); } HLD hld(g, 0); vector<Node> nodes; REP(i,n) { auto f = fs[hld.order[i]]; nodes.push_back({f, f}); } SegTree<Node, op, e> tree(nodes); while (q--) { int typ; cin >> typ; if (typ == 0) { int p, a, b; cin >> p >> a >> b; tree.set(hld.in[p], {{a, b}, {a, b}}); } else { int start, end, x; cin >> start >> end >> x; auto segments = hld.getSegments(start, end); F res {1, 0}; for (auto [u, v] : segments) { if (hld.in[u] <= hld.in[v]) { res = op(res, tree.prod(hld.in[u], hld.in[v] + 1).forward); } else { res = op(res, tree.prod(hld.in[v], hld.in[u] + 1).backward); } } cout << (res.a * modular(x) + res.b) << '\n'; } } return 0; }