cpl

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:heavy_check_mark: graph/test/vertex_set_path_composite.test.cpp

Depends on

Code

#define PROBLEM "https://judge.yosupo.jp/problem/vertex_set_path_composite"

#define FAST_IO

#include "../../template/template.hpp"
#include "../../number_theory/mod_int.hpp"
#include "../../graph/graph.hpp"
#include "../../graph/heavy_light_decomposition.hpp"
#include "../../data_structure/segment_tree.hpp"

using Mint = ModInt<mod998244353>;

struct Linear {
    Mint a, b;
    Linear() : a(Mint(1)), b(Mint(0)) {}
    Linear(Mint _a, Mint _b) : a(_a), b(_b) {}
    Mint operator()(Mint x) const {
        return a * x + b;
    }
};

Linear composite(const Linear &f, const Linear &g) {
    return Linear(f.a * g.a, f.a * g.b + f.b);
}

struct Ops {
    using Value = Linear;
    static Value id() {
        return Linear();
    }
    static Value op(const Value &f, const Value &g) {
        return composite(f, g);
    }
};

int main() {
    i32 n, q;
    cin >> n >> q;
    Vec<Linear> fs(n);
    REP(i, n) {
        cin >> fs[i].a >> fs[i].b;
    }
    Graph<i32> g(n);
    REP(e, n - 1) {
        i32 u, v;
        cin >> u >> v;
        g.add_undirected_edge(u, v);
    }
    HeavyLightDecomposition hld(g);
    SegmentTree<Reversible<Ops>> seg(n, [&](i32 i) -> typename Reversible<Ops>::Value {
        Linear f = fs[hld.time_to_vertex(i)];
        return pair<Linear, Linear>(f, f);
    });
    REP(qi, q) {
        i32 type;
        cin >> type;
        if (type == 0) {
            i32 p;
            Linear f;
            cin >> p >> f.a >> f.b;
            seg.update(hld.in_time(p), pair<Linear, Linear>(f, f));
        } else {
            i32 u, v;
            Mint x;
            cin >> u >> v >> x;
            Linear f;
            for (auto [l, r] : hld.path(u, v, false)) {
                if (l < r) {
                    f = composite(seg.prod(l, r + 1).second, f);
                } else {
                    f = composite(seg.prod(r, l + 1).first, f);
                }
            }
            cout << f(x) << '\n';
        }
    }
}
#line 1 "graph/test/vertex_set_path_composite.test.cpp"
#define PROBLEM "https://judge.yosupo.jp/problem/vertex_set_path_composite"

#define FAST_IO

#line 1 "template/template.hpp"
#include <algorithm>
#include <array>
#include <bitset>
#include <cassert>
#include <cmath>
#include <iomanip>
#include <iostream>
#include <list>
#include <map>
#include <numeric>
#include <queue>
#include <random>
#include <set>
#include <stack>
#include <string>
#include <tuple>
#include <unordered_map>
#include <unordered_set>
#include <utility>
#include <vector>

#define OVERRIDE(a, b, c, d, ...) d
#define REP2(i, n) for (i32 i = 0; i < (i32) (n); ++i)
#define REP3(i, m, n) for (i32 i = (i32) (m); i < (i32) (n); ++i)
#define REP(...) OVERRIDE(__VA_ARGS__, REP3, REP2)(__VA_ARGS__)
#define PER(i, n) for (i32 i = (i32) (n) - 1; i >= 0; --i)
#define ALL(x) begin(x), end(x)

using namespace std;

using u32 = unsigned int;
using u64 = unsigned long long;
using u128 = __uint128_t;
using i32 = signed int;
using i64 = signed long long;
using i128 = __int128_t;
using f64 = double;
using f80 = long double;

template <typename T>
using Vec = vector<T>;

template <typename T>
bool chmin(T &x, const T &y) {
    if (x > y) {
        x = y;
        return true;
    }
    return false;
}
template <typename T>
bool chmax(T &x, const T &y) {
    if (x < y) {
        x = y;
        return true;
    }
    return false;
}

istream &operator>>(istream &is, i128 &x) {
    i64 v;
    is >> v;
    x = v;
    return is;
}
ostream &operator<<(ostream &os, i128 x) {
    os << (i64) x;
    return os;
}
istream &operator>>(istream &is, u128 &x) {
    u64 v;
    is >> v;
    x = v;
    return is;
}
ostream &operator<<(ostream &os, u128 x) {
    os << (u64) x;
    return os;
}

[[maybe_unused]] constexpr i32 INF = 1000000100;
[[maybe_unused]] constexpr i64 INF64 = 3000000000000000100;
struct SetUpIO {
    SetUpIO() {
#ifdef FAST_IO
        ios::sync_with_stdio(false);
        cin.tie(nullptr);
#endif
        cout << fixed << setprecision(15);
    }
} set_up_io;
#line 2 "number_theory/mod_int.hpp"

#line 5 "number_theory/mod_int.hpp"
#include <type_traits>

#line 2 "number_theory/utils.hpp"

constexpr bool is_prime(unsigned n) {
    if (n == 0 || n == 1) {
        return false;
    }
    for (unsigned i = 2; i * i <= n; ++i) {
        if (n % i == 0) {
            return false;
        }
    }
    return true;
}

constexpr unsigned mod_pow(unsigned x, unsigned y, unsigned mod) {
    unsigned ret = 1, self = x;
    while (y != 0) {
        if (y & 1) {
            ret = (unsigned) ((unsigned long long) ret * self % mod);
        }
        self = (unsigned) ((unsigned long long) self * self % mod);
        y /= 2;
    }
    return ret;
}

template <unsigned mod>
constexpr unsigned primitive_root() {
    static_assert(is_prime(mod), "`mod` must be a prime number.");
    if (mod == 2) {
        return 1;
    }

    unsigned primes[32] = {};
    int it = 0;
    {
        unsigned m = mod - 1;
        for (unsigned i = 2; i * i <= m; ++i) {
            if (m % i == 0) {
                primes[it++] = i;
                while (m % i == 0) {
                    m /= i;
                }
            }
        }
        if (m != 1) {
            primes[it++] = m;
        }
    }
    for (unsigned i = 2; i < mod; ++i) {
        bool ok = true;
        for (int j = 0; j < it; ++j) {
            if (mod_pow(i, (mod - 1) / primes[j], mod) == 1) {
                ok = false;
                break;
            }
        }
        if (ok)
            return i;
    }
    return 0;
}

// y >= 1
template <typename T>
constexpr T safe_mod(T x, T y) {
    x %= y;
    if (x < 0) {
        x += y;
    }
    return x;
}

// y != 0
template <typename T>
constexpr T floor_div(T x, T y) {
    if (y < 0) {
        x *= -1;
        y *= -1;
    }
    if (x >= 0) {
        return x / y;
    } else {
        return -((-x + y - 1) / y);
    }
}

// y != 0
template <typename T>
constexpr T ceil_div(T x, T y) {
    if (y < 0) {
        x *= -1;
        y *= -1;
    }
    if (x >= 0) {
        return (x + y - 1) / y;
    } else {
        return -(-x / y);
    }
}
#line 8 "number_theory/mod_int.hpp"

template <unsigned mod>
class ModInt {
    static_assert(mod != 0, "`mod` must not be equal to 0.");
    static_assert(
        mod < (1u << 31),
        "`mod` must be less than (1u << 31) = 2147483648.");

    unsigned val;

public:
    static constexpr unsigned get_mod() {
        return mod;
    }
    
    constexpr ModInt() : val(0) {}
    template <typename T, std::enable_if_t<std::is_signed_v<T>> * = nullptr>
    constexpr ModInt(T x) : val((unsigned) ((long long) x % (long long) mod + (x < 0 ? mod : 0))) {}
    template <typename T, std::enable_if_t<std::is_unsigned_v<T>> * = nullptr>
    constexpr ModInt(T x) : val((unsigned) (x % mod)) {}

    static constexpr ModInt raw(unsigned x) {
        ModInt<mod> ret;
        ret.val = x;
        return ret;
    }

    constexpr unsigned get_val() const {
        return val;
    }

    constexpr ModInt operator+() const {
        return *this;
    }
    constexpr ModInt operator-() const {
        return ModInt<mod>(0u) - *this;
    }

    constexpr ModInt &operator+=(const ModInt &rhs) {
        val += rhs.val;
        if (val >= mod)
            val -= mod;
        return *this;
    }
    constexpr ModInt &operator-=(const ModInt &rhs) {
        if (val < rhs.val)
            val += mod;
        val -= rhs.val;
        return *this;
    }
    constexpr ModInt &operator*=(const ModInt &rhs) {
        val = (unsigned long long)val * rhs.val % mod;
        return *this;
    }
    constexpr ModInt &operator/=(const ModInt &rhs) {
        val = (unsigned long long)val * rhs.inv().val % mod;
        return *this;
    }

    friend constexpr ModInt operator+(const ModInt &lhs, const ModInt &rhs) {
        return ModInt<mod>(lhs) += rhs;
    }
    friend constexpr ModInt operator-(const ModInt &lhs, const ModInt &rhs) {
        return ModInt<mod>(lhs) -= rhs;
    }
    friend constexpr ModInt operator*(const ModInt &lhs, const ModInt &rhs) {
        return ModInt<mod>(lhs) *= rhs;
    }
    friend constexpr ModInt operator/(const ModInt &lhs, const ModInt &rhs) {
        return ModInt<mod>(lhs) /= rhs;
    }

    constexpr ModInt pow(unsigned long long x) const {
        ModInt<mod> ret = ModInt<mod>::raw(1);
        ModInt<mod> self = *this;
        while (x != 0) {
            if (x & 1)
                ret *= self;
            self *= self;
            x >>= 1;
        }
        return ret;
    }
    constexpr ModInt inv() const {
        static_assert(is_prime(mod), "`mod` must be a prime number.");
        assert(val != 0);
        return this->pow(mod - 2);
    }

    friend std::istream &operator>>(std::istream &is, ModInt<mod> &x) {
        long long val;
        is >> val;
        x.val = val % mod + (val < 0 ? mod : 0);
        return is;
    }

    friend std::ostream &operator<<(std::ostream &os, const ModInt<mod> &x) {
        os << x.val;
        return os;
    }

    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.val != rhs.val;
    }
};

[[maybe_unused]] constexpr unsigned mod998244353 = 998244353;
[[maybe_unused]] constexpr unsigned mod1000000007 = 1000000007;

#line 2 "graph/graph.hpp"

#line 7 "graph/graph.hpp"

template <typename Edge>
class Graph {
    std::vector<std::vector<Edge>> edges;

public:
    Graph() : edges() {}
    Graph(int v) : edges(v) {
        assert(v >= 0);
    }
    
    std::vector<int> add_vertices(int n) {
        int v = (int) edges.size();
        std::vector<int> idx(n);
        std::iota(idx.begin(), idx.end(), v);
        edges.resize(edges.size() + n);
        return idx;
    }

    template <typename... T>
    void add_directed_edge(int from, int to, T &&...val) {
        assert(from >= 0 && from < (int) edges.size());
        assert(to >= 0 && to < (int) edges.size());
        edges[from].emplace_back(Edge(to, std::forward<T>(val)...));
    }

    template <typename... T>
    void add_undirected_edge(int u, int v, const T &...val) {
        assert(u >= 0 && u < (int) edges.size());
        assert(v >= 0 && v < (int) edges.size());
        edges[u].emplace_back(Edge(v, val...));
        edges[v].emplace_back(Edge(u, val...));
    }

    int size() const {
        return (int) edges.size();
    }

    const std::vector<Edge> &operator[](int v) const {
        assert(v >= 0 && v < (int) edges.size());
        return edges[v];
    }

    std::vector<Edge> &operator[](int v) {
        assert(v >= 0 && v < (int) edges.size());
        return edges[v];
    }
};

struct UnweightedEdge {
    int to;

    UnweightedEdge(int t) : to(t) {}
    
    explicit operator int() const {
        return to;
    }

    using Weight = int;
    Weight weight() const {
        return 1;
    }
};

template <typename T>
struct WeightedEdge {
    int to;
    T wt;

    WeightedEdge(int t, const T &w) : to(t), wt(w) {}

    explicit operator int() const {
        return to;
    }

    using Weight = T;
    Weight weight() const {
        return wt;
    }
};

#line 2 "graph/heavy_light_decomposition.hpp"

#line 7 "graph/heavy_light_decomposition.hpp"

class HeavyLightDecomposition {
    std::vector<int> siz;
    std::vector<int> par;
    std::vector<int> hea;
    std::vector<int> in;
    std::vector<int> out;
    std::vector<int> dep;
    std::vector<int> rev;

    template <typename G>
    void dfs1(G &g, int v) {
        if (!g[v].empty() && (int) g[v][0] == par[v]) {
            std::swap(g[v][0], g[v].back());
        }
        for (auto &e : g[v]) {
            int u = (int)e;
            if (u != par[v]) {
                par[u] = v;
                dfs1(g, u);
                siz[v] += siz[u];
                if (siz[u] > siz[(int) g[v][0]]) {
                    std::swap(g[v][0], e);
                }
            }
        }
    }

    template <typename G>
    void dfs2(const G &g, int v, int &time) {
        in[v] = time;
        rev[time++] = v;
        for (auto &e : g[v]) {
            int u = (int)e;
            if (u == par[v]) {
                continue;
            }
            if (u == (int) g[v][0]) {
                hea[u] = hea[v];
            } else {
                hea[u] = u;
            }
            dep[u] = dep[v] + 1;
            dfs2(g, u, time);
        }
        out[v] = time;
    }

public:
    template <typename G>
    HeavyLightDecomposition(G &g, int root = 0) :
        siz(g.size(), 1),
        par(g.size(), root),
        hea(g.size(), root),
        in(g.size(), 0),
        out(g.size(), 0),
        dep(g.size(), 0),
        rev(g.size(), 0) {
        assert(root >= 0 && root < (int) g.size());
        dfs1(g, root);
        int time = 0;
        dfs2(g, root, time);
    }

    int subtree_size(int v) const {
        assert(v >= 0 && v < (int) siz.size());
        return siz[v];
    }

    int parent(int v) const {
        assert(v >= 0 && v < (int) par.size());
        return par[v];
    }

    int in_time(int v) const {
        assert(v >= 0 && v < (int) in.size());
        return in[v];
    }

    int out_time(int v) const {
        assert(v >= 0 && v < (int) out.size());
        return out[v];
    }

    int depth(int v) const {
        assert(v >= 0 && v < (int) dep.size());
        return dep[v];
    }

    int time_to_vertex(int t) const {
        assert(t >= 0 && t < (int) rev.size());
        return rev[t];
    }
    
    int la(int v, int k) const {
        assert(v >= 0 && v < (int) dep.size());
        assert(k >= 0);
        if (k > dep[v]) {
            return -1;
        }
        while (true) {
            int u = hea[v];
            if (in[u] + k <= in[v]) {
                return rev[in[v] - k];
            }
            k -= in[v] - in[u] + 1;
            v = par[u];
        }
        return 0;
    }
    
    int forward(int v, int dst) const {
        assert(v >= 0 && v < (int) dep.size());
        assert(dst >= 0 && dst < (int) dep.size());
        assert(v != dst);
        int l = lca(v, dst);
        if (l == v) {
            return la(dst, dist(v, dst) - 1);
        } else {
            return par[v];
        }
    }

    int lca(int u, int v) const {
        assert(u >= 0 && u < (int) dep.size());
        assert(v >= 0 && v < (int) dep.size());
        while (u != v) {
            if (in[u] > in[v]) {
                std::swap(u, v);
            }
            if (hea[u] == hea[v]) {
                v = u;
            } else {
                v = par[hea[v]];
            }
        }
        return u;
    }

    int dist(int u, int v) const {
        assert(u >= 0 && u < (int) dep.size());
        assert(v >= 0 && v < (int) dep.size());
        return dep[u] + dep[v] - 2 * dep[lca(u, v)];
    }

    std::vector<std::pair<int, int>> path(int u, int v, bool edge) const {
        assert(u >= 0 && u < (int) dep.size());
        assert(v >= 0 && v < (int) dep.size());
        std::vector<std::pair<int, int>> fromu, fromv;
        bool rev = false;
        while (true) {
            if (u == v && edge) {
                break;
            }
            if (in[u] > in[v]) {
                std::swap(u, v);
                std::swap(fromu, fromv);
                rev ^= true;
            }
            if (hea[u] == hea[v]) {
                fromv.emplace_back(in[v], in[u] + (int)edge);
                v = u;
                break;
            } else {
                fromv.emplace_back(in[v], in[hea[v]]);
                v = par[hea[v]];
            }
        }
        if (rev) {
            std::swap(fromu, fromv);
        }
        std::reverse(fromv.begin(), fromv.end());
        fromu.reserve(fromv.size());
        for (auto [x, y] : fromv) {
            fromu.emplace_back(y, x);
        }
        return fromu;
    }
    
    int jump(int u, int v, int k) const {
        assert(u >= 0 && u < (int) dep.size());
        assert(v >= 0 && v < (int) dep.size());
        assert(k >= 0);
        int l = lca(u, v);
        int dis = dep[u] + dep[v] - 2 * dep[l];
        if (k > dis) {
            return -1;
        }
        if (k <= dep[u] - dep[l]) {
            return la(u, k);
        } else {
            return la(v, dis - k);
        }
    }
    
    int meet(int u, int v, int w) const {
        return lca(u, v) ^ lca(v, w) ^ lca(w, u);
    }
};

#line 2 "data_structure/segment_tree.hpp"

#line 6 "data_structure/segment_tree.hpp"

#line 2 "data_structure/operations.hpp"

#include <limits>
#line 5 "data_structure/operations.hpp"

template <typename T>
struct Add {
    using Value = T;
    static Value id() {
        return T(0);
    }
    static Value op(const Value &lhs, const Value &rhs) {
        return lhs + rhs;
    }
    static Value inv(const Value &x) {
        return -x;
    }
};

template <typename T>
struct Mul {
    using Value = T;
    static Value id() {
        return Value(1);
    }
    static Value op(const Value &lhs, const Value &rhs) {
        return lhs * rhs;
    }
    static Value inv(const Value &x) {
        return Value(1) / x;
    }
};

template <typename T>
struct Min {
    using Value = T;
    static Value id() {
        return std::numeric_limits<T>::max();
    }
    static Value op(const Value &lhs, const Value &rhs) {
        return std::min(lhs, rhs);
    }
};

template <typename T>
struct Max {
    using Value = T;
    static Value id() {
        return std::numeric_limits<Value>::min();
    }
    static Value op(const Value &lhs, const Value &rhs) {
        return std::max(lhs, rhs);
    }
};

template <typename T>
struct Xor {
    using Value = T;
    static Value id() {
        return T(0);
    }
    static Value op(const Value &lhs, const Value &rhs) {
        return lhs ^ rhs;
    }
    static Value inv(const Value &x) {
        return x;
    }
};

template <typename Monoid>
struct Reversible {
    using Value = std::pair<typename Monoid::Value, typename Monoid::Value>;
    static Value id() {
        return Value(Monoid::id(), Monoid::id());
    }
    static Value op(const Value &v1, const Value &v2) {
        return Value(
            Monoid::op(v1.first, v2.first),
            Monoid::op(v2.second, v1.second));
    }
};

#line 8 "data_structure/segment_tree.hpp"

template <typename Monoid>
class SegmentTree {
public:
    using Value = typename Monoid::Value;

private:
    int old_length;
    int length;
    std::vector<Value> node;

    static int ceil2(int n) {
        int l = 1;
        while (l < n) {
            l <<= 1;
        }
        return l;
    }

public:
    SegmentTree(int n) :
        old_length(n),
        length(ceil2(old_length)),
        node(length << 1, Monoid::id()) {
        assert(n >= 0);
    }

    SegmentTree(const std::vector<Value> &v) :
        old_length((int) v.size()),
        length(ceil2(old_length)),
        node(length << 1, Monoid::id()) {
        for (int i = 0; i < old_length; ++i) {
            node[i + length] = v[i];
        }
        for (int i = length - 1; i > 0; --i) {
            node[i] = Monoid::op(node[i << 1], node[i << 1 | 1]);
        }
    }

    template <typename F>
    SegmentTree(int n, const F &f) :
        old_length(n), length(ceil2(n)), node(length << 1, Monoid::id()) {
        assert(n >= 0);
        for (int i = 0; i < old_length; ++i) {
            node[i + length] = f(i);
        }
        for (int i = length - 1; i > 0; --i) {
            node[i] = Monoid::op(node[i << 1], node[i << 1 | 1]);
        }
    }

    const Value &operator[](int idx) const {
        assert(idx >= 0 && idx < old_length);
        return node[idx + length];
    }

    void update(int idx, Value val) {
        assert(idx >= 0 && idx < old_length);
        idx += length;
        node[idx] = std::move(val);
        while (idx != 1) {
            idx >>= 1;
            node[idx] = Monoid::op(node[idx << 1], node[idx << 1 | 1]);
        }
    }

    Value prod(int l, int r) const {
        assert(l >= 0 && l <= r && r <= old_length);
        Value prodl = Monoid::id();
        Value prodr = Monoid::id();
        l += length;
        r += length;
        while (l != r) {
            if (l & 1) {
                prodl = Monoid::op(prodl, node[l++]);
            }
            if (r & 1) {
                prodr = Monoid::op(node[--r], prodr);
            }
            l >>= 1;
            r >>= 1;
        }
        return Monoid::op(prodl, prodr);
    }
    
    Value all_prod() const {
        return node[1];
    }
};

#line 10 "graph/test/vertex_set_path_composite.test.cpp"

using Mint = ModInt<mod998244353>;

struct Linear {
    Mint a, b;
    Linear() : a(Mint(1)), b(Mint(0)) {}
    Linear(Mint _a, Mint _b) : a(_a), b(_b) {}
    Mint operator()(Mint x) const {
        return a * x + b;
    }
};

Linear composite(const Linear &f, const Linear &g) {
    return Linear(f.a * g.a, f.a * g.b + f.b);
}

struct Ops {
    using Value = Linear;
    static Value id() {
        return Linear();
    }
    static Value op(const Value &f, const Value &g) {
        return composite(f, g);
    }
};

int main() {
    i32 n, q;
    cin >> n >> q;
    Vec<Linear> fs(n);
    REP(i, n) {
        cin >> fs[i].a >> fs[i].b;
    }
    Graph<i32> g(n);
    REP(e, n - 1) {
        i32 u, v;
        cin >> u >> v;
        g.add_undirected_edge(u, v);
    }
    HeavyLightDecomposition hld(g);
    SegmentTree<Reversible<Ops>> seg(n, [&](i32 i) -> typename Reversible<Ops>::Value {
        Linear f = fs[hld.time_to_vertex(i)];
        return pair<Linear, Linear>(f, f);
    });
    REP(qi, q) {
        i32 type;
        cin >> type;
        if (type == 0) {
            i32 p;
            Linear f;
            cin >> p >> f.a >> f.b;
            seg.update(hld.in_time(p), pair<Linear, Linear>(f, f));
        } else {
            i32 u, v;
            Mint x;
            cin >> u >> v >> x;
            Linear f;
            for (auto [l, r] : hld.path(u, v, false)) {
                if (l < r) {
                    f = composite(seg.prod(l, r + 1).second, f);
                } else {
                    f = composite(seg.prod(r, l + 1).first, f);
                }
            }
            cout << f(x) << '\n';
        }
    }
}
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