Library

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:heavy_check_mark: test/tree/Point_Set_Tree_Path_Composition_Sum_Fixed_Root.test.cpp

Depends on

Code

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

#include "../../modint/modint.hpp"
#include "../../template/template.hpp"
#include "../../tree/dp_on_static_top_tree.hpp"

namespace ebi {

using mint = modint998244353;

struct Path {
    mint a, b, s, c;
};

struct Point {
    mint s, c;
};

void main_() {
    int n, q;
    std::cin >> n >> q;
    Graph<int> g(2 * n - 1);
    std::vector<Path> a(2 * n - 1);
    rep(i, 0, n) {
        int x;
        std::cin >> x;
        a[i] = {1, 0, x, 1};
    }
    rep(i, 0, n - 1) {
        int u, v, b, c;
        std::cin >> u >> v >> b >> c;
        g.add_undirected_edge(u, i + n, 1);
        g.add_undirected_edge(v, i + n, 1);
        a[i + n] = {b, c, 0, 0};
    }
    g.build();
    auto compress = [&](Path p, Path c) -> Path {
        return {p.a * c.a, p.b + p.a * c.b, p.s + p.a * c.s + p.b * c.c,
                p.c + c.c};
    };
    auto rake = [&](Point a, Point b) -> Point {
        return {a.s + b.s, a.c + b.c};
    };
    auto add_edge = [&](Path a) -> Point { return {a.s, a.c}; };
    auto add_vertex = [&](Point a, Path v) -> Path {
        return {v.a, v.b, v.s + v.a * a.s + v.b * a.c, a.c + v.c};
    };
    dp_on_static_top_tree<int, Path, Point, decltype(compress), decltype(rake),
                          decltype(add_edge), decltype(add_vertex)>
        dp(g, 0, a, compress, rake, add_edge, add_vertex);
    while (q--) {
        int t;
        std::cin >> t;
        if (t == 0) {
            int w, x;
            std::cin >> w >> x;
            dp.set(w, {1, 0, x, 1});
        } else {
            int e, y, z;
            std::cin >> e >> y >> z;
            dp.set(e + n, {y, z, 0, 0});
        }
        std::cout << dp.get().s << '\n';
    }
}

}  // namespace ebi

int main() {
    ebi::fast_io();
    int t = 1;
    // std::cin >> t;
    while (t--) {
        ebi::main_();
    }
    return 0;
}
#line 1 "test/tree/Point_Set_Tree_Path_Composition_Sum_Fixed_Root.test.cpp"
#define PROBLEM                        \
    "https://judge.yosupo.jp/problem/point_set_tree_path_composite_sum_fixed_root"

#line 2 "modint/modint.hpp"

#include <cassert>

#include <iostream>


#line 2 "modint/base.hpp"

#include <concepts>
#line 5 "modint/base.hpp"
#include <utility>

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) {}

    template <std::signed_integral T> constexpr static_modint(T v) {
        long long x = (long long)(v % (long long)(umod()));
        if (x < 0) x += umod();
        _v = (unsigned int)(x);
    }

    template <std::unsigned_integral T> constexpr static_modint(T v) {
        _v = (unsigned int)(v % umod());
    }

    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 1 "template/template.hpp"
#include <bits/stdc++.h>

#define rep(i, a, n) for (int i = (int)(a); i < (int)(n); i++)
#define rrep(i, a, n) for (int i = ((int)(n)-1); i >= (int)(a); i--)
#define Rep(i, a, n) for (i64 i = (i64)(a); i < (i64)(n); i++)
#define RRep(i, a, n) for (i64 i = ((i64)(n)-i64(1)); i >= (i64)(a); i--)
#define all(v) (v).begin(), (v).end()
#define rall(v) (v).rbegin(), (v).rend()

#line 2 "template/debug_template.hpp"

#line 4 "template/debug_template.hpp"

namespace ebi {

#ifdef LOCAL
#define debug(...)                                                      \
    std::cerr << "LINE: " << __LINE__ << "  [" << #__VA_ARGS__ << "]:", \
        debug_out(__VA_ARGS__)
#else
#define debug(...)
#endif

void debug_out() {
    std::cerr << std::endl;
}

template <typename Head, typename... Tail> void debug_out(Head h, Tail... t) {
    std::cerr << " " << h;
    if (sizeof...(t) > 0) std::cerr << " :";
    debug_out(t...);
}

}  // namespace ebi
#line 2 "template/int_alias.hpp"

#line 4 "template/int_alias.hpp"

namespace ebi {

using ld = long double;
using std::size_t;
using i8 = std::int8_t;
using u8 = std::uint8_t;
using i16 = std::int16_t;
using u16 = std::uint16_t;
using i32 = std::int32_t;
using u32 = std::uint32_t;
using i64 = std::int64_t;
using u64 = std::uint64_t;
using i128 = __int128_t;
using u128 = __uint128_t;

}  // namespace ebi
#line 2 "template/io.hpp"

#line 5 "template/io.hpp"
#include <optional>
#line 7 "template/io.hpp"

#line 9 "template/io.hpp"

namespace ebi {

template <typename T1, typename T2>
std::ostream &operator<<(std::ostream &os, const std::pair<T1, T2> &pa) {
    return os << pa.first << " " << pa.second;
}

template <typename T1, typename T2>
std::istream &operator>>(std::istream &os, std::pair<T1, T2> &pa) {
    return os >> pa.first >> pa.second;
}

template <typename T>
std::ostream &operator<<(std::ostream &os, const std::vector<T> &vec) {
    for (std::size_t i = 0; i < vec.size(); i++)
        os << vec[i] << (i + 1 == vec.size() ? "" : " ");
    return os;
}

template <typename T>
std::istream &operator>>(std::istream &os, std::vector<T> &vec) {
    for (T &e : vec) std::cin >> e;
    return os;
}

template <typename T>
std::ostream &operator<<(std::ostream &os, const std::optional<T> &opt) {
    if (opt) {
        os << opt.value();
    } else {
        os << "invalid value";
    }
    return os;
}

void fast_io() {
    std::cout << std::fixed << std::setprecision(15);
    std::cin.tie(nullptr);
    std::ios::sync_with_stdio(false);
}

}  // namespace ebi
#line 2 "template/utility.hpp"

#line 5 "template/utility.hpp"

#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 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) {
        assert(!prepared && u < n && v < n);
        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) {
        assert(!prepared && u < n && v < n);
        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 {
        assert(prepared);
        return edges[i];
    }

    std::vector<edge_type> get_edges() const {
        assert(prepared);
        return edges;
    }

    const auto operator[](int i) const {
        assert(prepared);
        return csr[i];
    }
    auto operator[](int i) {
        assert(prepared);
        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 8 "template/utility.hpp"

namespace ebi {

template <class T> inline bool chmin(T &a, T b) {
    if (a > b) {
        a = b;
        return true;
    }
    return false;
}

template <class T> inline bool chmax(T &a, T b) {
    if (a < b) {
        a = b;
        return true;
    }
    return false;
}

template <class T> T safe_ceil(T a, T b) {
    if (a % b == 0)
        return a / b;
    else if (a >= 0)
        return (a / b) + 1;
    else
        return -((-a) / b);
}

template <class T> T safe_floor(T a, T b) {
    if (a % b == 0)
        return a / b;
    else if (a >= 0)
        return a / b;
    else
        return -((-a) / b) - 1;
}

constexpr i64 LNF = std::numeric_limits<i64>::max() / 4;

constexpr int INF = std::numeric_limits<int>::max() / 2;

const std::vector<int> dy = {1, 0, -1, 0, 1, 1, -1, -1};
const std::vector<int> dx = {0, 1, 0, -1, 1, -1, 1, -1};

}  // namespace ebi
#line 2 "tree/dp_on_static_top_tree.hpp"

#line 2 "tree/static_top_tree.hpp"

#line 4 "tree/static_top_tree.hpp"

namespace ebi {

enum Type { Vertex, Compress, Rake, AddEdge, AddVertex };

template <class T> struct static_top_tree {
  private:
    struct Node {
        int par = -1, lch = -1, rch = -1;
        Type ty = Type::Vertex;
    };

    void dfs_sz(int v) {
        for (auto &e : g[v]) {
            if (e.to == par[v]) continue;
            par[e.to] = v;
            dfs_sz(e.to);
            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]);
            }
        }
    }

    int new_node(int k, int l, int r, Type t) {
        if (k == -1) {
            k = (int)stt.size();
            stt.emplace_back(-1, l, r, t);
        } else {
            stt[k].lch = l;
            stt[k].rch = r;
            stt[k].ty = t;
        }
        if (l != -1) stt[l].par = k;
        if (r != -1) stt[r].par = k;
        return k;
    }

    std::pair<int, int> merge(const std::vector<std::pair<int, int>> &a,
                              Type t) {
        if (a.size() == 1) {
            return a[0];
        }
        int sum = 0;
        for (auto [v_, s] : a) sum += s;
        std::vector<std::pair<int, int>> b, c;
        for (auto [i, s] : a) {
            if (sum > s)
                b.emplace_back(i, s);
            else
                c.emplace_back(i, s);
            sum -= 2 * s;
        }
        auto [i, si] = merge(b, t);
        auto [j, sj] = merge(c, t);
        return {new_node(-1, i, j, t), si + sj};
    }

    std::pair<int, int> compress(int v) {
        std::vector<std::pair<int, int>> path{add_vertex(v)};
        while (g[v][0].to != par[v]) {
            path.emplace_back(add_vertex(v = g[v][0].to));
        }
        return merge(path, Type::Compress);
    }

    std::pair<int, int> rake(int v) {
        std::vector<std::pair<int, int>> ch;
        for (int i = 1; i < (int)g[v].size(); i++) {
            if (g[v][i].to == par[v]) continue;
            ch.emplace_back(add_edge(g[v][i].to));
        }
        return ch.empty() ? std::pair<int, int>{-1, 0} : merge(ch, Type::Rake);
    }

    std::pair<int, int> add_edge(int v) {
        auto [i, si] = compress(v);
        return {new_node(-1, i, -1, Type::AddEdge), si};
    }

    std::pair<int, int> add_vertex(int v) {
        auto [i, si] = rake(v);
        return {new_node(v, i, -1, i == -1 ? Type::Vertex : Type::AddVertex),
                si + 1};
    }

  public:
    static_top_tree(Graph<T> g_, int root = 0)
        : n(g_.size()), g(g_), par(n, -1), sz(n, 1), stt(n) {
        if (n == 1) {
            stt_root = 0;
            return;
        }
        dfs_sz(root);
        stt_root = compress(root).first;
    }

    int node_num() const {
        return (int)stt.size();
    }

    int parent(int v) const {
        return stt[v].par;
    }

    std::pair<int, int> child(int v) const {
        return {stt[v].lch, stt[v].rch};
    }

    int left_child(int v) const {
        return stt[v].lch;
    }

    int right_child(int v) const {
        return stt[v].rch;
    }

    Type type(int v) const {
        return stt[v].ty;
    }

    int root() const {
        return stt_root;
    }

  private:
    int n;
    Graph<T> g;
    std::vector<int> par, sz;
    std::vector<Node> stt;
    int stt_root;
};

}  // namespace ebi
#line 4 "tree/dp_on_static_top_tree.hpp"

namespace ebi {

template <class T, class Path, class Point, class Compress, class Rake,
          class Add_edge, class Add_vertex>
struct dp_on_static_top_tree {
  private:
    void dfs(int v) {
        auto [lch, rch] = stt.child(v);
        if (lch != -1) dfs(lch);
        if (rch != -1) dfs(rch);
        update_(v);
    }

    void update_(int v) {
        if (stt.type(v) == Type::Vertex) {
            path[v] = vertex[v];
        } else if (stt.type(v) == Type::Compress) {
            path[v] =
                compress(path[stt.left_child(v)], path[stt.right_child(v)]);
        } else if (stt.type(v) == Type::Rake) {
            point[v] =
                rake(point[stt.left_child(v)], point[stt.right_child(v)]);
        } else if (stt.type(v) == Type::AddEdge) {
            point[v] = add_edge(path[stt.left_child(v)]);
        } else if (stt.type(v) == Type::AddVertex) {
            path[v] = add_vertex(point[stt.left_child(v)], vertex[v]);
        }
    }

    void update(int v) {
        while (v != -1) {
            update_(v);
            v = stt.parent(v);
        }
    }

  public:
    dp_on_static_top_tree(const Graph<T> &g, int root,
                          const std::vector<Path> &vertex_,
                          const Compress &compress_, const Rake &rake_,
                          const Add_edge &add_edge_,
                          const Add_vertex &add_vertex_)
        : stt(g, root),
          n(stt.node_num()),
          path(n),
          point(n),
          vertex(vertex_),
          compress(compress_),
          rake(rake_),
          add_edge(add_edge_),
          add_vertex(add_vertex_) {
        dfs(stt.root());
    }

    Path get() const {
        return path[stt.root()];
    }

    void set(int v, Path x) {
        vertex[v] = x;
        update(v);
    }

  private:
    static_top_tree<T> stt;

    int n;
    std::vector<Path> path;
    std::vector<Point> point;
    std::vector<Path> vertex;
    const Compress compress;
    const Rake rake;
    const Add_edge add_edge;
    const Add_vertex add_vertex;
};

}  // namespace ebi
#line 7 "test/tree/Point_Set_Tree_Path_Composition_Sum_Fixed_Root.test.cpp"

namespace ebi {

using mint = modint998244353;

struct Path {
    mint a, b, s, c;
};

struct Point {
    mint s, c;
};

void main_() {
    int n, q;
    std::cin >> n >> q;
    Graph<int> g(2 * n - 1);
    std::vector<Path> a(2 * n - 1);
    rep(i, 0, n) {
        int x;
        std::cin >> x;
        a[i] = {1, 0, x, 1};
    }
    rep(i, 0, n - 1) {
        int u, v, b, c;
        std::cin >> u >> v >> b >> c;
        g.add_undirected_edge(u, i + n, 1);
        g.add_undirected_edge(v, i + n, 1);
        a[i + n] = {b, c, 0, 0};
    }
    g.build();
    auto compress = [&](Path p, Path c) -> Path {
        return {p.a * c.a, p.b + p.a * c.b, p.s + p.a * c.s + p.b * c.c,
                p.c + c.c};
    };
    auto rake = [&](Point a, Point b) -> Point {
        return {a.s + b.s, a.c + b.c};
    };
    auto add_edge = [&](Path a) -> Point { return {a.s, a.c}; };
    auto add_vertex = [&](Point a, Path v) -> Path {
        return {v.a, v.b, v.s + v.a * a.s + v.b * a.c, a.c + v.c};
    };
    dp_on_static_top_tree<int, Path, Point, decltype(compress), decltype(rake),
                          decltype(add_edge), decltype(add_vertex)>
        dp(g, 0, a, compress, rake, add_edge, add_vertex);
    while (q--) {
        int t;
        std::cin >> t;
        if (t == 0) {
            int w, x;
            std::cin >> w >> x;
            dp.set(w, {1, 0, x, 1});
        } else {
            int e, y, z;
            std::cin >> e >> y >> z;
            dp.set(e + n, {y, z, 0, 0});
        }
        std::cout << dp.get().s << '\n';
    }
}

}  // namespace ebi

int main() {
    ebi::fast_io();
    int t = 1;
    // std::cin >> t;
    while (t--) {
        ebi::main_();
    }
    return 0;
}
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