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#define PROBLEM "https://judge.yosupo.jp/problem/dynamic_tree_vertex_set_path_composite" #include <bits/stdc++.h> using namespace std; #include "../../Math/modint.h" using modular = ModInt<998244353>; #define PATH_QUERIES_ONLY struct T { modular a, b; T() : a(1), b(0) {} T(modular _a, modular _b) : a(_a), b(_b) {} // return f(g()) T operator + (const T& g) const { return T { a * g.a, a * g.b + b, }; } T operator += (const T& g) { b = a * g.b + b; a = a * g.a; return *this; } }; #include "../LinkCutTree.h" #define FOR(i, a, b) for (int i = (a), _##i = (b); i <= _##i; ++i) #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; LinkCut tree(n); FOR(i,1,n) { modular a, b; cin >> a >> b; tree.set(i, T{a, b}); } REP(i,n-1) { int u, v; cin >> u >> v; ++u; ++v; tree.link(u, v); } while (q--) { int typ; cin >> typ; if (typ == 0) { // remove (u, v), add (w, x) int u, v, w, x; cin >> u >> v >> w >> x; ++u; ++v; ++w; ++x; tree.cut(u, v); tree.link(w, x); } else if (typ == 1) { // set f(p) = cx + d int p; cin >> p; ++p; modular c, d; cin >> c >> d; tree.set(p, T{c, d}); } else if (typ == 2) { // get path (u, v) and apply f(x) int u, v; cin >> u >> v; ++u; ++v; modular x; cin >> x; auto f = tree.getPath(u, v); cout << f.a * x + f.b << '\n'; } } return 0; }
#line 1 "DataStructure/test/link_cut_tree_vertexsetpathcomposite.test.cpp" #define PROBLEM "https://judge.yosupo.jp/problem/dynamic_tree_vertex_set_path_composite" #include <bits/stdc++.h> using namespace std; #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 7 "DataStructure/test/link_cut_tree_vertexsetpathcomposite.test.cpp" using modular = ModInt<998244353>; #define PATH_QUERIES_ONLY struct T { modular a, b; T() : a(1), b(0) {} T(modular _a, modular _b) : a(_a), b(_b) {} // return f(g()) T operator + (const T& g) const { return T { a * g.a, a * g.b + b, }; } T operator += (const T& g) { b = a * g.b + b; a = a * g.a; return *this; } }; #line 1 "DataStructure/LinkCutTree.h" // copied from https://codeforces.com/blog/entry/75885 // - Index from 1 // - T needs to support + operation // For subtree queries -> requires - operation // --> see this comment for how to handle it: https://codeforces.com/blog/entry/67637?#comment-650424 // - Not using template here, since inheritance becomes very ugly // - Doesn't support lazy update (so no subtree updates) // - For query on *edge* weights (instead of vertex weights) // --> for each edge, add a new node in LinkCut tree. // See https://oj.vnoi.info/problem/icpc22_mn_b for example // // Tested: // - https://judge.yosupo.jp/problem/dynamic_tree_subtree_add_subtree_sum // - https://judge.yosupo.jp/problem/dynamic_tree_vertex_set_path_composite // - https://judge.yosupo.jp/problem/dynamic_tree_vertex_add_subtree_sum // - (edge weights) https://oj.vnoi.info/problem/icpc22_mn_b // - (link, cut, connected) https://www.spoj.com/problems/DYNACON1/ // Add this for path queries only // #define PATH_QUERIES_ONLY // TODO: Specify T // using T = long long; // Link Cut Tree {{{ // SplayTree {{{ struct SplayTree { // can we replace SplayTreeById and use this only? struct Node { array<int, 2> child = {0, 0}; int parent = 0; // Path aggregates // - path[0] = go from left -> right // - path[1] = go from right -> left array<T, 2> path; // default to T constructor T self; // Subtree aggregates T sub, vir; bool reverse = false; }; vector<Node> nodes; SplayTree(int n) : nodes(n + 1) {} void splay(int x) { for (pushDown(x); ~getDirection(x); ) { int y = nodes[x].parent; int z = nodes[y].parent; pushDown(z); pushDown(y); pushDown(x); int dx = getDirection(x); int dy = getDirection(y); if (~dy) rotate(dx != dy ? x : y); rotate(x); } } // private: // Return t where nodes[parent(x)].child[t] == x int getDirection(int x) { int p = nodes[x].parent; if (!p) return -1; return nodes[p].child[0] == x ? 0 : nodes[p].child[1] == x ? 1 : -1; } /** * Before: * z * | * y * / * x * \ * xchild * * After: * z * | * x * \ * y * / * xchild */ void rotate(int x) { int y = nodes[x].parent, dx = getDirection(x); int z = nodes[y].parent, dy = getDirection(y); setChild(y, nodes[x].child[!dx], dx); setChild(x, y, !dx); if (~dy) setChild(z, x, dy); nodes[x].parent = z; } void pushDown(int x) { if (!x) return; if (nodes[x].reverse) { auto [l, r] = nodes[x].child; nodes[l].reverse ^= 1; nodes[r].reverse ^= 1; swap(nodes[x].child[0], nodes[x].child[1]); swap(nodes[x].path[0], nodes[x].path[1]); nodes[x].reverse = false; } } void pushUp(int x) { auto [l, r] = nodes[x].child; pushDown(l); pushDown(r); nodes[x].path[0] = nodes[l].path[0] + nodes[x].self + nodes[r].path[0]; nodes[x].path[1] = nodes[r].path[1] + nodes[x].self + nodes[l].path[1]; nodes[x].sub = nodes[x].vir + nodes[l].sub + nodes[r].sub + nodes[x].self; } void setChild(int x, int y, int dir) { nodes[x].child[dir] = y; nodes[y].parent = x; pushUp(x); } }; // }}} struct LinkCut : SplayTree { LinkCut(int n) : SplayTree(n) {} bool is_connected(int u, int v) { return LCA(u, v) > 0; } void link(int u, int v) { reroot(u); access(v); nodes[v].vir = nodes[v].vir + nodes[u].sub; nodes[u].parent = v; pushUp(v); } void cut(int u, int v) { reroot(u); access(v); nodes[v].child[0] = nodes[u].parent = 0; pushUp(v); } // Returns 0 if u and v are not connected int LCA(int u, int v) { if (u == v) return u; access(u); int ret = access(v); return nodes[u].parent ? ret : 0; } T getPath(int u, int v) { reroot(u); access(v); return nodes[v].path[1]; } void set(int u, T val) { access(u); nodes[u].self = val; pushUp(u); } T get(int u) { return nodes[u].self; } // Get aggregate of subtree(u). v is parent of u. There must exist edge(v, u) (?) T getSubtree(int u, int v) { reroot(v); access(u); return nodes[u].vir + nodes[u].self; } // private: void reroot(int x) { access(x); nodes[x].reverse ^= 1; pushDown(x); } int access(int x) { int u = x, v = 0; for (; u; v = u, u = nodes[u].parent) { splay(u); int& ov = nodes[u].child[1]; nodes[u].vir = nodes[u].vir + nodes[ov].sub; #ifndef PATH_QUERIES_ONLY // T requires subtract for subtree queries nodes[u].vir -= nodes[v].sub; #endif ov = v; pushUp(u); } return splay(x), v; } }; // }}} // Example for custom type: // https://judge.yosupo.jp/problem/dynamic_tree_vertex_set_path_composite {{{ // Since T doesn't support subtract -> comment out line // nodes[u].vir -= nodes[v].sub /** struct T { modular a, b; T() : a(1), b(0) {} T(modular _a, modular _b) : a(_a), b(_b) {} // return f(g()) T operator + (const T& g) const { return T { a * g.a, a * g.b + b, }; } T operator += (const T& g) { b = a * g.b + b; a = a * g.a; return *this; } }; */ // }}} #line 31 "DataStructure/test/link_cut_tree_vertexsetpathcomposite.test.cpp" #define FOR(i, a, b) for (int i = (a), _##i = (b); i <= _##i; ++i) #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; LinkCut tree(n); FOR(i,1,n) { modular a, b; cin >> a >> b; tree.set(i, T{a, b}); } REP(i,n-1) { int u, v; cin >> u >> v; ++u; ++v; tree.link(u, v); } while (q--) { int typ; cin >> typ; if (typ == 0) { // remove (u, v), add (w, x) int u, v, w, x; cin >> u >> v >> w >> x; ++u; ++v; ++w; ++x; tree.cut(u, v); tree.link(w, x); } else if (typ == 1) { // set f(p) = cx + d int p; cin >> p; ++p; modular c, d; cin >> c >> d; tree.set(p, T{c, d}); } else if (typ == 2) { // get path (u, v) and apply f(x) int u, v; cin >> u >> v; ++u; ++v; modular x; cin >> x; auto f = tree.getPath(u, v); cout << f.a * x + f.b << '\n'; } } return 0; }