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:heavy_check_mark: verify/ds/dynamic_tree/yosupo_point_set_tree_path_composite_sum.test.cpp

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Code

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

#include "../../../utilities/fast_io.hpp"
#include <vector>

#include "../../../ds/dynamic_tree/rake_compress_link_cut_tree.hpp"
#include "../../../math/modint.hpp"

using mint = m1une::math::modint998244353;

template <class T>
struct PointSetTreePathCompositeSum {
    struct Point {
        T s;
        T x;

        static Point id() {
            return Point{T(0), T(0)};
        }

        Point inv() const {
            return Point{T(0) - s, T(0) - x};
        }
    };

    struct Path {
        T a;
        T b;
        T s;
        T x;
    };

    struct VertexValue {
        T x;
    };

    struct EdgeValue {
        T a;
        T b;
    };

    static Path make_vertex_path(const Point& d, const VertexValue& vertex) {
        return Path{T(1), T(0), d.s + vertex.x, d.x + T(1)};
    }

    static Path make_edge_path(const Point& d, const EdgeValue& edge) {
        return Path{edge.a, edge.b, d.s * edge.a + d.x * edge.b, d.x};
    }

    static Point to_point(const Path& path) {
        return Point{path.s, path.x};
    }

    static Point rake(const Point& a, const Point& b) {
        return Point{a.s + b.s, a.x + b.x};
    }

    static Path compress(const Path& parent_side, const Path& child_side) {
        return Path{
            parent_side.a * child_side.a,
            parent_side.a * child_side.b + parent_side.b,
            parent_side.s + parent_side.a * child_side.s + parent_side.b * child_side.x,
            parent_side.x + child_side.x
        };
    }
};

int main() {
    m1une::utilities::FastInput fast_input;
    m1une::utilities::FastOutput fast_output;

    using TreeDP = PointSetTreePathCompositeSum<mint>;
    using VertexValue = typename TreeDP::VertexValue;
    using EdgeValue = typename TreeDP::EdgeValue;
    using LCT = m1une::ds::RakeCompressLinkCutTree<TreeDP>;
    int n, q;
    fast_input >> n >> q;

    LCT lct;
    std::vector<int> vertex_id(n);
    for (int i = 0; i < n; i++) {
        mint a;
        fast_input >> a;
        vertex_id[i] = lct.add_vertex(VertexValue{a});
    }

    std::vector<int> edge_id(n - 1);
    for (int i = 0; i + 1 < n; i++) {
        int u, v;
        mint b, c;
        fast_input >> u >> v >> b >> c;
        edge_id[i] = lct.add_edge(vertex_id[u], vertex_id[v], EdgeValue{b, c});
    }

    for (int i = 0; i < q; i++) {
        int type;
        fast_input >> type;
        int root;
        if (type == 0) {
            int w;
            mint x;
            fast_input >> w >> x >> root;
            lct.set_vertex(vertex_id[w], VertexValue{x});
        } else {
            int e;
            mint y, z;
            fast_input >> e >> y >> z >> root;
            lct.set_edge(edge_id[e], EdgeValue{y, z});
        }
        fast_output << lct.component_prod(vertex_id[root]).s << '\n';
    }
}
#line 1 "verify/ds/dynamic_tree/yosupo_point_set_tree_path_composite_sum.test.cpp"
#define PROBLEM "https://judge.yosupo.jp/problem/point_set_tree_path_composite_sum"

#line 1 "utilities/fast_io.hpp"



#include <algorithm>
#include <array>
#include <cerrno>
#include <charconv>
#include <cstddef>
#include <cstdio>
#include <cstdlib>
#include <cstdint>
#include <cstring>
#include <iterator>
#include <string>
#include <sys/stat.h>
#include <type_traits>
#include <utility>
#include <unistd.h>
#include <vector>

namespace m1une {
namespace utilities {

struct FastOutput;

namespace internal {

// Shared with the convenience helpers in template.hpp.
inline FastOutput* standard_output_instance = nullptr;

// Detect std::begin(x), std::end(x).
template <class T, class = void>
struct is_range : std::false_type {};

template <class T>
struct is_range<T, std::void_t<
    decltype(std::begin(std::declval<T&>())),
    decltype(std::end(std::declval<T&>()))
>> : std::true_type {};

template <class T>
inline constexpr bool is_range_v = is_range<T>::value;

template <class T>
using range_reference_t = decltype(*std::begin(std::declval<T&>()));

template <class T>
using range_value_t = std::remove_cv_t<std::remove_reference_t<range_reference_t<T>>>;

template <class T, class = void>
struct range_stored_value {
    using type = range_value_t<T>;
};

template <class T>
struct range_stored_value<T, std::void_t<typename std::remove_cv_t<std::remove_reference_t<T>>::value_type>> {
    using type = typename std::remove_cv_t<std::remove_reference_t<T>>::value_type;
};

template <class T>
using range_stored_value_t = typename range_stored_value<T>::type;

// Treat strings and C strings as scalar output objects, not as ranges.
template <class T>
struct is_char_array : std::false_type {};

template <class T, std::size_t N>
struct is_char_array<T[N]>
    : std::bool_constant<std::is_same_v<std::remove_cv_t<T>, char>> {};

template <class T>
struct is_string_like
    : std::bool_constant<
          std::is_same_v<std::decay_t<T>, std::string>
          || std::is_same_v<std::decay_t<T>, const char*>
          || std::is_same_v<std::decay_t<T>, char*>
          || is_char_array<std::remove_reference_t<T>>::value
      > {};

template <class T>
inline constexpr bool is_string_like_v = is_string_like<T>::value;

// ModInt-like type: x.val() is printable, and x can be assigned from long long.
template <class T, class = void>
struct has_val_method : std::false_type {};

template <class T>
struct has_val_method<T, std::void_t<decltype(std::declval<const T&>().val())>>
    : std::true_type {};

template <class T>
inline constexpr bool has_val_method_v = has_val_method<T>::value;

template <class T, class = void>
struct has_static_mod_raw : std::false_type {};

template <class T>
struct has_static_mod_raw<
    T, std::void_t<decltype(T::mod()), decltype(T::raw(std::declval<uint32_t>()))>>
    : std::true_type {};

template <class T>
inline constexpr bool has_static_mod_raw_v = has_static_mod_raw<T>::value;

// libstdc++ before GCC 16 does not classify __int128 as an integral type in
// strict ISO modes such as -std=c++23. Keep the fast-I/O interface independent
// of that implementation detail.
template <class T>
inline constexpr bool is_integral_v =
    std::is_integral_v<T>
    || std::is_same_v<std::remove_cv_t<T>, __int128_t>
    || std::is_same_v<std::remove_cv_t<T>, __uint128_t>;

template <class T>
inline constexpr bool is_signed_v =
    std::is_signed_v<T>
    || std::is_same_v<std::remove_cv_t<T>, __int128_t>;

template <class T>
struct make_unsigned {
    using type = std::make_unsigned_t<T>;
};

template <>
struct make_unsigned<__int128_t> {
    using type = __uint128_t;
};

template <>
struct make_unsigned<__uint128_t> {
    using type = __uint128_t;
};

template <class T>
using make_unsigned_t = typename make_unsigned<std::remove_cv_t<T>>::type;

}  // namespace internal

struct FastInput {
    static constexpr int buffer_size = 1 << 20;

   private:
    std::FILE* _stream;
    char _buffer[buffer_size];
    int _position;
    int _length;
    int _file_descriptor;
    bool _streaming;

    bool refill() {
        _position = 0;
        if (_streaming) {
            ssize_t length;
            do {
                length = ::read(_file_descriptor, _buffer, buffer_size);
            } while (length < 0 && errno == EINTR);
            if (length <= 0) {
                _length = 0;
                return false;
            }
            _length = int(length);
        } else {
            _length = int(std::fread(_buffer, 1, buffer_size, _stream));
        }
        return _length != 0;
    }

    template <class T>
    bool read_integer_from_stream(T& value) {
        if (!skip_spaces()) return false;
        int c = read_char_raw();

        bool negative = false;
        if (c == '-') {
            negative = true;
            c = read_char_raw();
        }

        if constexpr (internal::is_signed_v<T>) {
            T result = 0;
            while ('0' <= c && c <= '9') {
                result = negative ? result * 10 - (c - '0')
                                  : result * 10 + (c - '0');
                c = read_char_raw();
            }
            value = result;
        } else {
            T result = 0;
            while ('0' <= c && c <= '9') {
                result = result * 10 + T(c - '0');
                c = read_char_raw();
            }
            value = negative ? T(0) - result : result;
        }
        return true;
    }

    bool prepare_number() {
        if (_length - _position >= 64) return true;
        const int remaining = _length - _position;
        if (remaining > 0) std::memmove(_buffer, _buffer + _position, remaining);
        const int added = int(std::fread(_buffer + remaining, 1, buffer_size - remaining, _stream));
        _position = 0;
        _length = remaining + added;
        if (_length < buffer_size) _buffer[_length] = '\0';
        return _length != 0;
    }

   public:
    explicit FastInput(std::FILE* stream = stdin)
        : _stream(stream),
          _position(0),
          _length(0),
          _file_descriptor(::fileno(stream)),
          _streaming([&] {
              struct stat status;
              return _file_descriptor >= 0
                     && ::fstat(_file_descriptor, &status) == 0
                     && !S_ISREG(status.st_mode);
          }()) {}

    FastInput(const FastInput&) = delete;
    FastInput& operator=(const FastInput&) = delete;

    int read_char_raw() {
        if (_position == _length && !refill()) return EOF;
        return _buffer[_position++];
    }

    bool skip_spaces() {
        int c = read_char_raw();
        while (c != EOF && c <= ' ') c = read_char_raw();
        if (c == EOF) return false;
        --_position;
        return true;
    }

    bool read(char& value) {
        if (!skip_spaces()) return false;
        value = char(read_char_raw());
        return true;
    }

    bool read(std::string& value) {
        if (!skip_spaces()) return false;
        value.clear();
        while (true) {
            const int begin = _position;
            while (_position < _length &&
                   static_cast<unsigned char>(_buffer[_position]) > ' ') {
                ++_position;
            }
            value.append(_buffer + begin, _position - begin);
            if (_position < _length) {
                ++_position;
                return true;
            }
            if (!refill()) return true;
        }
    }

    bool read(bool& value) {
        int x;
        if (!read(x)) return false;
        value = x != 0;
        return true;
    }

    template <class T>
    std::enable_if_t<
        internal::is_integral_v<T>
            && !std::is_same_v<std::remove_cv_t<T>, bool>
            && !std::is_same_v<std::remove_cv_t<T>, char>,
        bool
    >
    read(T& value) {
        if (_streaming) return read_integer_from_stream(value);
        if (!prepare_number()) return false;
        int c = static_cast<unsigned char>(_buffer[_position++]);
        while (c <= ' ') c = static_cast<unsigned char>(_buffer[_position++]);

        bool negative = false;
        if (c == '-') {
            negative = true;
            c = static_cast<unsigned char>(_buffer[_position++]);
        }

        if constexpr (internal::is_signed_v<T>) {
            T result = 0;
            while ('0' <= c && c <= '9') {
                const int first = c - '0';
                const int second = static_cast<unsigned char>(_buffer[_position]) - '0';
                if (0 <= second && second <= 9) {
                    result = negative ? result * 100 - (first * 10 + second)
                                      : result * 100 + (first * 10 + second);
                    ++_position;
                } else {
                    result = negative ? result * 10 - first : result * 10 + first;
                }
                c = static_cast<unsigned char>(_buffer[_position++]);
            }
            value = result;
        } else {
            T result = 0;
            while ('0' <= c && c <= '9') {
                const unsigned first = unsigned(c - '0');
                const int second = static_cast<unsigned char>(_buffer[_position]) - '0';
                if (0 <= second && second <= 9) {
                    result = result * 100 + T(first * 10 + unsigned(second));
                    ++_position;
                } else {
                    result = result * 10 + T(first);
                }
                c = static_cast<unsigned char>(_buffer[_position++]);
            }
            value = negative ? T(0) - result : result;
        }
        if (_position > _length) _position = _length;
        return true;
    }

    template <class T>
    std::enable_if_t<std::is_floating_point_v<T>, bool>
    read(T& value) {
        if (!skip_spaces()) return false;
        int c = read_char_raw();
        bool negative = false;
        if (c == '-' || c == '+') {
            negative = c == '-';
            c = read_char_raw();
        }

        long double result = 0;
        while ('0' <= c && c <= '9') {
            result = result * 10 + (c - '0');
            c = read_char_raw();
        }
        if (c == '.') {
            long double place = 0.1L;
            c = read_char_raw();
            while ('0' <= c && c <= '9') {
                result += (c - '0') * place;
                place *= 0.1L;
                c = read_char_raw();
            }
        }
        if (c == 'e' || c == 'E') {
            c = read_char_raw();
            bool exponent_negative = false;
            if (c == '-' || c == '+') {
                exponent_negative = c == '-';
                c = read_char_raw();
            }
            int exponent = 0;
            while ('0' <= c && c <= '9') {
                exponent = exponent * 10 + (c - '0');
                c = read_char_raw();
            }
            long double scale = 1;
            long double power = 10;
            while (exponent > 0) {
                if (exponent & 1) scale *= power;
                power *= power;
                exponent >>= 1;
            }
            result = exponent_negative ? result / scale : result * scale;
        }
        value = static_cast<T>(negative ? -result : result);
        return true;
    }

    template <class T>
    std::enable_if_t<
        internal::has_val_method_v<T>
            && !internal::is_integral_v<T>
            && !internal::is_range_v<T>,
        bool
    >
    read(T& value) {
        long long x;
        if (!read(x)) return false;
        if constexpr (internal::has_static_mod_raw_v<T>) {
            if (x >= 0 && uint64_t(x) < uint64_t(T::mod())) {
                value = T::raw(uint32_t(x));
            } else {
                value = T(x);
            }
        } else {
            value = T(x);
        }
        return true;
    }

    template <class First, class Second>
    bool read(std::pair<First, Second>& value) {
        if (!read(value.first)) return false;
        return read(value.second);
    }

    template <class Range>
    std::enable_if_t<
        internal::is_range_v<Range>
            && !internal::is_string_like_v<Range>,
        bool
    >
    read(Range& range) {
        using StoredValue = internal::range_stored_value_t<Range>;
        constexpr bool nested = internal::is_range_v<StoredValue>
                                && !internal::is_string_like_v<StoredValue>;

        for (auto&& value : range) {
            if constexpr (std::is_same_v<StoredValue, bool> && !nested) {
                bool x;
                if (!read(x)) return false;
                value = x;
            } else {
                if (!read(value)) return false;
            }
        }
        return true;
    }

    template <class First, class Second, class... Rest>
    bool read(First& first, Second& second, Rest&... rest) {
        if (!read(first)) return false;
        return read(second, rest...);
    }

    template <class T>
    FastInput& operator>>(T& value) {
        if (!read(value)) std::abort();
        return *this;
    }
};

struct FastOutput {
    static constexpr int buffer_size = 1 << 20;

   private:
    inline static const auto digit_quads = [] {
        std::array<char, 40000> result{};
        for (int i = 0; i < 10000; i++) {
            int value = i;
            for (int j = 3; j >= 0; j--) {
                result[4 * i + j] = char('0' + value % 10);
                value /= 10;
            }
        }
        return result;
    }();

    std::FILE* _stream;
    char _buffer[buffer_size];
    int _position;
    int _precision;
    std::chars_format _float_format;
    char _range_separator;
    std::string* _capture = nullptr;

    template <class T>
    std::string format_cell(const T& value) {
        std::string result;
        struct CaptureGuard {
            std::string*& target;
            std::string* previous;
            ~CaptureGuard() { target = previous; }
        } guard{_capture, _capture};
        _capture = &result;
        write(value);
        return result;
    }

    template <class Matrix>
    void write_aligned_matrix(const Matrix& matrix) {
        std::vector<std::vector<std::string>> rows;
        std::vector<std::size_t> widths;
        for (const auto& row : matrix) {
            auto& cells = rows.emplace_back();
            std::size_t column = 0;
            for (const auto& value : row) {
                cells.push_back(format_cell(value));
                if (column == widths.size()) widths.push_back(0);
                widths[column] = std::max(widths[column], cells.back().size());
                ++column;
            }
        }
        bool first = true;
        for (const auto& row : rows) {
            if (!first) write_char('\n');
            first = false;
            for (std::size_t column = 0; column < row.size(); ++column) {
                if (column != 0) write_char(_range_separator);
                for (std::size_t padding = row[column].size();
                     padding < widths[column]; ++padding) {
                    write_char(' ');
                }
                write(row[column]);
            }
        }
    }

   public:
    explicit FastOutput(std::FILE* stream = stdout)
        : _stream(stream),
          _position(0),
          _precision(6),
          _float_format(std::chars_format::general),
          _range_separator(' ') {
        if (_stream == stdout
            && internal::standard_output_instance == nullptr) {
            internal::standard_output_instance = this;
        }
    }

    FastOutput(const FastOutput&) = delete;
    FastOutput& operator=(const FastOutput&) = delete;

    ~FastOutput() {
        flush();
        if (internal::standard_output_instance == this) {
            internal::standard_output_instance = nullptr;
        }
    }

    void flush() {
        if (_position != 0) {
            std::fwrite(_buffer, 1, _position, _stream);
            _position = 0;
        }
        std::fflush(_stream);
    }

    void write_char(char c) {
        if (_capture != nullptr) {
            _capture->push_back(c);
            return;
        }
        if (_position == buffer_size) flush();
        _buffer[_position++] = c;
    }

    void write(const char* s) {
        while (*s != '\0') write_char(*s++);
    }

    void write(const std::string& s) {
        if (_capture != nullptr) {
            _capture->append(s);
            return;
        }
        std::size_t position = 0;
        while (position < s.size()) {
            if (_position == buffer_size) flush();
            const std::size_t copied =
                std::min<std::size_t>(buffer_size - _position, s.size() - position);
            std::memcpy(_buffer + _position, s.data() + position, copied);
            _position += int(copied);
            position += copied;
        }
    }

    void write(char c) {
        write_char(c);
    }

    void write(bool value) {
        write_char(value ? '1' : '0');
    }

    template <class T>
    std::enable_if_t<std::is_floating_point_v<T>>
    write(T value) {
        char digits[128];
        auto [end, error] = std::to_chars(
            digits,
            digits + sizeof(digits),
            value,
            _float_format,
            _precision
        );
        if (error != std::errc()) std::abort();
        for (const char* pointer = digits; pointer != end; pointer++) {
            write_char(*pointer);
        }
    }

    template <class T>
    std::enable_if_t<
        internal::is_integral_v<T>
            && !std::is_same_v<std::remove_cv_t<T>, bool>
            && !std::is_same_v<std::remove_cv_t<T>, char>
    >
    write(T value) {
        using Raw = std::remove_cv_t<T>;
        using Unsigned = internal::make_unsigned_t<Raw>;

        Unsigned magnitude;
        if constexpr (internal::is_signed_v<Raw>) {
            if (value < 0) {
                write_char('-');
                magnitude = Unsigned(0) - Unsigned(value);
            } else {
                magnitude = Unsigned(value);
            }
        } else {
            magnitude = value;
        }

        if (magnitude == 0) {
            write_char('0');
            return;
        }

        unsigned chunks[16];
        int count = 0;
        while (magnitude >= 10000) {
            const Unsigned quotient = magnitude / 10000;
            chunks[count++] = unsigned(magnitude - quotient * 10000);
            magnitude = quotient;
        }
        if (_capture == nullptr && _position > buffer_size - 64) flush();
        char captured[64];
        char* const begin = _capture != nullptr ? captured : _buffer + _position;
        char* destination = begin;
        const unsigned leading = unsigned(magnitude);
        const char* first = digit_quads.data() + 4 * leading;
        int skip = leading < 10 ? 3 : leading < 100 ? 2 : leading < 1000 ? 1 : 0;
        for (; skip < 4; skip++) *destination++ = first[skip];
        while (count--) {
            const char* digits = digit_quads.data() + 4 * chunks[count];
            std::memcpy(destination, digits, 4);
            destination += 4;
        }
        if (_capture != nullptr) {
            _capture->append(begin, destination - begin);
        } else {
            _position += int(destination - begin);
        }
    }

    template <class T>
    std::enable_if_t<
        internal::has_val_method_v<T>
            && !internal::is_integral_v<T>
            && !internal::is_range_v<T>
    >
    write(const T& value) {
        write(value.val());
    }

    template <class First, class Second>
    void write(const std::pair<First, Second>& value) {
        write(value.first);
        write_char(' ');
        write(value.second);
    }

    template <class Range>
    std::enable_if_t<
        internal::is_range_v<Range>
            && !internal::is_string_like_v<Range>
    >
    write(const Range& range) {
        using StoredValue = internal::range_stored_value_t<const Range>;
        constexpr bool nested = internal::is_range_v<StoredValue>
                                && !internal::is_string_like_v<StoredValue>;

        bool first = true;
        for (const auto& value : range) {
            if (!first) write_char(nested ? '\n' : _range_separator);
            first = false;
            if constexpr (std::is_same_v<StoredValue, bool> && !nested) {
                write(static_cast<bool>(value));
            } else {
                write(value);
            }
        }
    }

    template <class First, class... Rest>
    void print(const First& first, const Rest&... rest) {
        write(first);
        ((write_char(' '), write(rest)), ...);
    }

    void println() {
        write_char('\n');
    }

    void set_precision(int precision) {
        _precision = precision;
    }

    void set_fixed(int precision = 6) {
        _float_format = std::chars_format::fixed;
        _precision = precision;
    }

    void set_general(int precision = 6) {
        _float_format = std::chars_format::general;
        _precision = precision;
    }

    void set_range_separator(char separator) {
        _range_separator = separator;
    }

    template <class Matrix>
    void write_aligned(const Matrix& matrix) {
        using Row = internal::range_stored_value_t<const Matrix>;
        using Cell = internal::range_stored_value_t<const Row>;
        static_assert(internal::is_range_v<Row> && !internal::is_string_like_v<Row>,
                      "write_aligned requires a two-dimensional range");
        static_assert(!internal::is_range_v<Cell> || internal::is_string_like_v<Cell>,
                      "write_aligned requires scalar cells");
        write_aligned_matrix(matrix);
    }

    template <class Matrix>
    void println_aligned(const Matrix& matrix) {
        write_aligned(matrix);
        write_char('\n');
    }

    template <class... Args>
    void println(const Args&... args) {
        print(args...);
        write_char('\n');
    }

    template <class T>
    FastOutput& operator<<(const T& value) {
        write(value);
        return *this;
    }
};

}  // namespace utilities
}  // namespace m1une


#line 5 "verify/ds/dynamic_tree/yosupo_point_set_tree_path_composite_sum.test.cpp"

#line 1 "ds/dynamic_tree/rake_compress_link_cut_tree.hpp"



#include <cassert>
#line 6 "ds/dynamic_tree/rake_compress_link_cut_tree.hpp"
#include <variant>
#line 8 "ds/dynamic_tree/rake_compress_link_cut_tree.hpp"

namespace m1une {
namespace ds {

// Maintains a dynamic forest whose tree DP uses two different aggregate types.
// Point is a commutative group for virtual children, while Path is an ordered
// preferred-path cluster and does not need an inverse.
template <class TreeDPInfo>
struct RakeCompressLinkCutTree {
    using Point = typename TreeDPInfo::Point;
    using Path = typename TreeDPInfo::Path;
    using VertexValue = typename TreeDPInfo::VertexValue;
    using EdgeValue = typename TreeDPInfo::EdgeValue;
    using VertexId = int;
    using EdgeId = int;

   private:
    struct Node {
        int left = -1;
        int right = -1;
        int parent = -1;
        bool rev = false;
        std::variant<VertexValue, EdgeValue> value;
        Point virtual_prod;
        Path prod;
        Path rev_prod;

        explicit Node(const VertexValue& vertex_value)
            : value(std::in_place_index<0>, vertex_value),
              virtual_prod(Point::id()),
              prod(TreeDPInfo::make_vertex_path(virtual_prod, vertex_value)),
              rev_prod(prod) {}

        explicit Node(std::in_place_index_t<1>, const EdgeValue& edge_value)
            : value(std::in_place_index<1>, edge_value),
              virtual_prod(Point::id()),
              prod(TreeDPInfo::make_edge_path(virtual_prod, edge_value)),
              rev_prod(prod) {}
    };

    struct OriginalEdge {
        VertexId u = -1;
        VertexId v = -1;
        int node = -1;
        bool alive = false;
    };

    std::vector<Node> _nodes;
    std::vector<int> _vertex_nodes;
    std::vector<OriginalEdge> _edges;
    std::vector<int> _path_buffer;

    bool is_splay_root(int node) const {
        int parent = _nodes[node].parent;
        return parent == -1 || (_nodes[parent].left != node && _nodes[parent].right != node);
    }

    void update(int node) {
        Node& x = _nodes[node];
        Path self = x.value.index() == 0
            ? TreeDPInfo::make_vertex_path(x.virtual_prod, std::get<0>(x.value))
            : TreeDPInfo::make_edge_path(x.virtual_prod, std::get<1>(x.value));
        x.prod = self;
        x.rev_prod = self;

        if (x.left != -1) {
            x.prod = TreeDPInfo::compress(_nodes[x.left].prod, x.prod);
            x.rev_prod = TreeDPInfo::compress(x.rev_prod, _nodes[x.left].rev_prod);
        }
        if (x.right != -1) {
            x.prod = TreeDPInfo::compress(x.prod, _nodes[x.right].prod);
            x.rev_prod = TreeDPInfo::compress(_nodes[x.right].rev_prod, x.rev_prod);
        }
    }

    void add_virtual_child(int node, int child) {
        if (child == -1) return;
        Point contribution = TreeDPInfo::to_point(_nodes[child].prod);
        _nodes[node].virtual_prod = TreeDPInfo::rake(_nodes[node].virtual_prod, contribution);
    }

    void remove_virtual_child(int node, int child) {
        if (child == -1) return;
        Point contribution = TreeDPInfo::to_point(_nodes[child].prod);
        _nodes[node].virtual_prod = TreeDPInfo::rake(_nodes[node].virtual_prod, contribution.inv());
    }

    void apply_reverse(int node) {
        if (node == -1) return;
        Node& x = _nodes[node];
        std::swap(x.left, x.right);
        std::swap(x.prod, x.rev_prod);
        x.rev = !x.rev;
    }

    void push(int node) {
        if (node == -1 || !_nodes[node].rev) return;
        apply_reverse(_nodes[node].left);
        apply_reverse(_nodes[node].right);
        _nodes[node].rev = false;
    }

    void push_to(int node) {
        _path_buffer.clear();
        int cur = node;
        _path_buffer.push_back(cur);
        while (!is_splay_root(cur)) {
            cur = _nodes[cur].parent;
            _path_buffer.push_back(cur);
        }
        for (int i = int(_path_buffer.size()) - 1; i >= 0; i--) push(_path_buffer[i]);
    }

    void rotate(int node) {
        int parent = _nodes[node].parent;
        int grand = _nodes[parent].parent;
        bool is_right = _nodes[parent].right == node;
        int middle = is_right ? _nodes[node].left : _nodes[node].right;

        if (!is_splay_root(parent)) {
            if (_nodes[grand].left == parent) {
                _nodes[grand].left = node;
            } else {
                _nodes[grand].right = node;
            }
        }
        _nodes[node].parent = grand;

        if (is_right) {
            _nodes[node].left = parent;
            _nodes[parent].right = middle;
        } else {
            _nodes[node].right = parent;
            _nodes[parent].left = middle;
        }
        if (middle != -1) _nodes[middle].parent = parent;
        _nodes[parent].parent = node;

        update(parent);
        update(node);
    }

    void splay(int node) {
        push_to(node);
        while (!is_splay_root(node)) {
            int parent = _nodes[node].parent;
            int grand = _nodes[parent].parent;
            if (!is_splay_root(parent)) {
                bool zig_zig = (_nodes[parent].left == node) == (_nodes[grand].left == parent);
                rotate(zig_zig ? parent : node);
            }
            rotate(node);
        }
    }

    int access(int node) {
        int last = -1;
        for (int cur = node; cur != -1; cur = _nodes[cur].parent) {
            splay(cur);
            add_virtual_child(cur, _nodes[cur].right);
            remove_virtual_child(cur, last);
            _nodes[cur].right = last;
            if (last != -1) _nodes[last].parent = cur;
            update(cur);
            last = cur;
        }
        splay(node);
        return last;
    }

    void check_node(int node) const {
        assert(0 <= node && node < int(_nodes.size()));
    }

    void check_vertex(VertexId vertex) const {
        assert(0 <= vertex && vertex < int(_vertex_nodes.size()));
    }

    void check_edge(EdgeId edge_id) const {
        assert(0 <= edge_id && edge_id < int(_edges.size()));
    }

    int vertex_node(VertexId vertex) const {
        check_vertex(vertex);
        return _vertex_nodes[vertex];
    }

    int add_edge_node(const EdgeValue& edge_value) {
        _nodes.emplace_back(std::in_place_index<1>, edge_value);
        return int(_nodes.size()) - 1;
    }

    void set_vertex_node_value(int node, const VertexValue& vertex_value) {
        check_node(node);
        access(node);
        _nodes[node].value.template emplace<0>(vertex_value);
        update(node);
    }

    void set_edge_node_value(int node, const EdgeValue& edge_value) {
        check_node(node);
        access(node);
        _nodes[node].value.template emplace<1>(edge_value);
        update(node);
    }

    void evert_node(int node) {
        check_node(node);
        access(node);
        apply_reverse(node);
    }

    int component_root_node(int node) {
        check_node(node);
        access(node);
        int cur = node;
        push(cur);
        while (_nodes[cur].left != -1) {
            cur = _nodes[cur].left;
            push(cur);
        }
        splay(cur);
        return cur;
    }

    bool connected_nodes(int u, int v) {
        if (u == v) return true;
        return component_root_node(u) == component_root_node(v);
    }

    bool link_nodes(int u, int v) {
        check_node(u);
        check_node(v);
        if (u == v) return false;
        evert_node(u);
        if (component_root_node(v) == u) return false;
        access(v);
        _nodes[u].parent = v;
        add_virtual_child(v, u);
        update(v);
        return true;
    }

    bool cut_nodes(int u, int v) {
        check_node(u);
        check_node(v);
        if (u == v) return false;
        evert_node(u);
        access(v);
        if (_nodes[v].left != u || _nodes[u].right != -1) return false;
        _nodes[v].left = -1;
        _nodes[u].parent = -1;
        update(v);
        return true;
    }

   public:
    RakeCompressLinkCutTree() = default;

    explicit RakeCompressLinkCutTree(const std::vector<VertexValue>& values) {
        _nodes.reserve(values.size());
        _vertex_nodes.reserve(values.size());
        for (const VertexValue& value : values) add_vertex(value);
    }

    int vertex_count() const {
        return int(_vertex_nodes.size());
    }

    bool empty() const {
        return _vertex_nodes.empty();
    }

    VertexId add_vertex(const VertexValue& vertex_value) {
        VertexValue value = vertex_value;
        int node = int(_nodes.size());
        _nodes.emplace_back(value);
        _vertex_nodes.push_back(node);
        return int(_vertex_nodes.size()) - 1;
    }

    const VertexValue& get_vertex(VertexId vertex) const {
        return std::get<0>(_nodes[vertex_node(vertex)].value);
    }

    void set_vertex(VertexId vertex, const VertexValue& vertex_value) {
        set_vertex_node_value(vertex_node(vertex), vertex_value);
    }

    int edge_count() const {
        return int(_edges.size());
    }

    bool edge_alive(EdgeId edge_id) const {
        check_edge(edge_id);
        return _edges[edge_id].alive;
    }

    std::pair<VertexId, VertexId> edge_endpoints(EdgeId edge_id) const {
        check_edge(edge_id);
        return {_edges[edge_id].u, _edges[edge_id].v};
    }

    const EdgeValue& get_edge(EdgeId edge_id) const {
        check_edge(edge_id);
        return std::get<1>(_nodes[_edges[edge_id].node].value);
    }

    void set_edge(EdgeId edge_id, const EdgeValue& edge_value) {
        check_edge(edge_id);
        set_edge_node_value(_edges[edge_id].node, edge_value);
    }

    EdgeId add_edge(VertexId u, VertexId v, const EdgeValue& edge_value) {
        check_vertex(u);
        check_vertex(v);
        if (u == v || connected(u, v)) return -1;
        EdgeValue value = edge_value;
        int edge_id = int(_edges.size());
        int node = add_edge_node(value);
        _edges.push_back(OriginalEdge{u, v, node, true});
        bool ok1 = link_nodes(vertex_node(u), node);
        bool ok2 = link_nodes(node, vertex_node(v));
        assert(ok1 && ok2);
        return edge_id;
    }

    bool cut_edge(EdgeId edge_id) {
        check_edge(edge_id);
        OriginalEdge& edge = _edges[edge_id];
        if (!edge.alive) return false;
        bool ok1 = cut_nodes(vertex_node(edge.u), edge.node);
        bool ok2 = cut_nodes(edge.node, vertex_node(edge.v));
        if (ok1 && ok2) edge.alive = false;
        return ok1 && ok2;
    }

    void reroot(VertexId vertex) {
        evert_node(vertex_node(vertex));
    }

    bool connected(VertexId u, VertexId v) {
        check_vertex(u);
        check_vertex(v);
        return connected_nodes(vertex_node(u), vertex_node(v));
    }

    // Reroots the represented tree at vertex and returns its whole-tree cluster.
    Path component_prod(VertexId vertex) {
        int node = vertex_node(vertex);
        evert_node(node);
        return _nodes[node].prod;
    }

    Path query_component(VertexId vertex) {
        return component_prod(vertex);
    }
};

}  // namespace ds
}  // namespace m1une


#line 1 "math/modint.hpp"



#line 6 "math/modint.hpp"
#include <iostream>
#line 9 "math/modint.hpp"

namespace m1une {
namespace math {

template <uint32_t Modulus>
struct ModInt {
    static_assert(0 < Modulus, "Modulus must be positive");

   private:
    uint32_t _v;

   public:
    static constexpr uint32_t mod() {
        return Modulus;
    }

    static constexpr ModInt raw(uint32_t v) noexcept {
        ModInt x;
        x._v = v;
        return x;
    }

    constexpr ModInt() noexcept : _v(0) {}

    template <class Integer, std::enable_if_t<std::is_integral_v<Integer>, int> = 0>
    constexpr ModInt(Integer v) noexcept {
        if constexpr (std::is_signed_v<Integer>) {
            int64_t x = static_cast<int64_t>(v) % static_cast<int64_t>(Modulus);
            if (x < 0) x += Modulus;
            _v = static_cast<uint32_t>(x);
        } else {
            _v = static_cast<uint32_t>(static_cast<uint64_t>(v) % Modulus);
        }
    }

    constexpr uint32_t val() const noexcept {
        return _v;
    }

    constexpr ModInt& operator++() noexcept {
        _v++;
        if (_v == Modulus) _v = 0;
        return *this;
    }

    constexpr ModInt& operator--() noexcept {
        if (_v == 0) _v = Modulus;
        _v--;
        return *this;
    }

    constexpr ModInt operator++(int) noexcept {
        ModInt res = *this;
        ++*this;
        return res;
    }

    constexpr ModInt operator--(int) noexcept {
        ModInt res = *this;
        --*this;
        return res;
    }

    constexpr ModInt& operator+=(const ModInt& rhs) noexcept {
        _v += rhs._v;
        if (_v >= Modulus) _v -= Modulus;
        return *this;
    }

    constexpr ModInt& operator-=(const ModInt& rhs) noexcept {
        _v -= rhs._v;
        if (_v >= Modulus) _v += Modulus;
        return *this;
    }

    constexpr ModInt& operator*=(const ModInt& rhs) noexcept {
        uint64_t z = _v;
        z *= rhs._v;
        _v = static_cast<uint32_t>(z % Modulus);
        return *this;
    }

    constexpr ModInt& operator/=(const ModInt& rhs) noexcept {
        return *this *= rhs.inv();
    }

    constexpr ModInt operator+(const ModInt& rhs) const noexcept {
        return ModInt(*this) += rhs;
    }
    constexpr ModInt operator-(const ModInt& rhs) const noexcept {
        return ModInt(*this) -= rhs;
    }
    constexpr ModInt operator*(const ModInt& rhs) const noexcept {
        return ModInt(*this) *= rhs;
    }
    constexpr ModInt operator/(const ModInt& rhs) const noexcept {
        return ModInt(*this) /= rhs;
    }

    constexpr bool operator==(const ModInt& rhs) const noexcept {
        return _v == rhs._v;
    }
    constexpr bool operator!=(const ModInt& rhs) const noexcept {
        return _v != rhs._v;
    }

    constexpr ModInt pow(long long n) const noexcept {
        ModInt res = raw(1 % Modulus);
        ModInt x = n < 0 ? inv() : *this;
        uint64_t exponent = n < 0 ? uint64_t(-(n + 1)) + 1 : uint64_t(n);
        while (exponent > 0) {
            if (exponent & 1) res *= x;
            x *= x;
            exponent >>= 1;
        }
        return res;
    }

    constexpr ModInt inv() const noexcept {
        int64_t a = _v, b = Modulus, u = 1, v = 0;
        while (b) {
            int64_t t = a / b;
            a -= t * b;
            std::swap(a, b);
            u -= t * v;
            std::swap(u, v);
        }
        assert(a == 1);
        u %= Modulus;
        if (u < 0) u += Modulus;
        return raw(static_cast<uint32_t>(u));
    }

    friend std::ostream& operator<<(std::ostream& os, const ModInt& rhs) {
        return os << rhs._v;
    }

    friend std::istream& operator>>(std::istream& is, ModInt& rhs) {
        long long v;
        is >> v;
        rhs = ModInt(v);
        return is;
    }
};

using modint998244353 = ModInt<998244353>;
using modint1000000007 = ModInt<1000000007>;

template <int Id = 0>
struct DynamicModInt {
   private:
    uint32_t _v;
    inline static uint32_t _mod = 1;

   public:
    static uint32_t mod() noexcept {
        return _mod;
    }

    static void set_mod(uint32_t modulus) noexcept {
        assert(modulus > 0);
        assert(modulus <= uint32_t(1) << 31);
        _mod = modulus;
    }

    static DynamicModInt raw(uint32_t v) noexcept {
        assert(v < _mod);
        DynamicModInt x;
        x._v = v;
        return x;
    }

    DynamicModInt() noexcept : _v(0) {}

    template <class Integer, std::enable_if_t<std::is_integral_v<Integer>, int> = 0>
    DynamicModInt(Integer v) noexcept {
        if constexpr (std::is_signed_v<Integer>) {
            int64_t x = static_cast<int64_t>(v) % static_cast<int64_t>(_mod);
            if (x < 0) x += _mod;
            _v = static_cast<uint32_t>(x);
        } else {
            _v = static_cast<uint32_t>(static_cast<uint64_t>(v) % _mod);
        }
    }

    uint32_t val() const noexcept {
        return _v;
    }

    DynamicModInt& operator++() noexcept {
        _v++;
        if (_v == _mod) _v = 0;
        return *this;
    }

    DynamicModInt& operator--() noexcept {
        if (_v == 0) _v = _mod;
        _v--;
        return *this;
    }

    DynamicModInt operator++(int) noexcept {
        DynamicModInt result = *this;
        ++*this;
        return result;
    }

    DynamicModInt operator--(int) noexcept {
        DynamicModInt result = *this;
        --*this;
        return result;
    }

    DynamicModInt& operator+=(const DynamicModInt& rhs) noexcept {
        _v += rhs._v;
        if (_v >= _mod) _v -= _mod;
        return *this;
    }

    DynamicModInt& operator-=(const DynamicModInt& rhs) noexcept {
        _v -= rhs._v;
        if (_v >= _mod) _v += _mod;
        return *this;
    }

    DynamicModInt& operator*=(const DynamicModInt& rhs) noexcept {
        _v = static_cast<uint32_t>(uint64_t(_v) * rhs._v % _mod);
        return *this;
    }

    DynamicModInt& operator/=(const DynamicModInt& rhs) noexcept {
        return *this *= rhs.inv();
    }

    DynamicModInt operator+(const DynamicModInt& rhs) const noexcept {
        return DynamicModInt(*this) += rhs;
    }

    DynamicModInt operator-(const DynamicModInt& rhs) const noexcept {
        return DynamicModInt(*this) -= rhs;
    }

    DynamicModInt operator*(const DynamicModInt& rhs) const noexcept {
        return DynamicModInt(*this) *= rhs;
    }

    DynamicModInt operator/(const DynamicModInt& rhs) const noexcept {
        return DynamicModInt(*this) /= rhs;
    }

    bool operator==(const DynamicModInt& rhs) const noexcept {
        return _v == rhs._v;
    }

    bool operator!=(const DynamicModInt& rhs) const noexcept {
        return _v != rhs._v;
    }

    DynamicModInt pow(long long exponent) const noexcept {
        DynamicModInt result = raw(1 % _mod);
        DynamicModInt base = exponent < 0 ? inv() : *this;
        uint64_t magnitude =
            exponent < 0 ? uint64_t(-(exponent + 1)) + 1 : uint64_t(exponent);
        while (magnitude > 0) {
            if (magnitude & 1) result *= base;
            base *= base;
            magnitude >>= 1;
        }
        return result;
    }

    DynamicModInt inv() const noexcept {
        int64_t a = _v, b = _mod, u = 1, v = 0;
        while (b) {
            int64_t quotient = a / b;
            a -= quotient * b;
            std::swap(a, b);
            u -= quotient * v;
            std::swap(u, v);
        }
        assert(a == 1);
        u %= _mod;
        if (u < 0) u += _mod;
        return raw(static_cast<uint32_t>(u));
    }

    friend std::ostream& operator<<(std::ostream& os, const DynamicModInt& rhs) {
        return os << rhs._v;
    }

    friend std::istream& operator>>(std::istream& is, DynamicModInt& rhs) {
        long long value;
        is >> value;
        rhs = DynamicModInt(value);
        return is;
    }
};

}  // namespace math
}  // namespace m1une


#line 8 "verify/ds/dynamic_tree/yosupo_point_set_tree_path_composite_sum.test.cpp"

using mint = m1une::math::modint998244353;

template <class T>
struct PointSetTreePathCompositeSum {
    struct Point {
        T s;
        T x;

        static Point id() {
            return Point{T(0), T(0)};
        }

        Point inv() const {
            return Point{T(0) - s, T(0) - x};
        }
    };

    struct Path {
        T a;
        T b;
        T s;
        T x;
    };

    struct VertexValue {
        T x;
    };

    struct EdgeValue {
        T a;
        T b;
    };

    static Path make_vertex_path(const Point& d, const VertexValue& vertex) {
        return Path{T(1), T(0), d.s + vertex.x, d.x + T(1)};
    }

    static Path make_edge_path(const Point& d, const EdgeValue& edge) {
        return Path{edge.a, edge.b, d.s * edge.a + d.x * edge.b, d.x};
    }

    static Point to_point(const Path& path) {
        return Point{path.s, path.x};
    }

    static Point rake(const Point& a, const Point& b) {
        return Point{a.s + b.s, a.x + b.x};
    }

    static Path compress(const Path& parent_side, const Path& child_side) {
        return Path{
            parent_side.a * child_side.a,
            parent_side.a * child_side.b + parent_side.b,
            parent_side.s + parent_side.a * child_side.s + parent_side.b * child_side.x,
            parent_side.x + child_side.x
        };
    }
};

int main() {
    m1une::utilities::FastInput fast_input;
    m1une::utilities::FastOutput fast_output;

    using TreeDP = PointSetTreePathCompositeSum<mint>;
    using VertexValue = typename TreeDP::VertexValue;
    using EdgeValue = typename TreeDP::EdgeValue;
    using LCT = m1une::ds::RakeCompressLinkCutTree<TreeDP>;
    int n, q;
    fast_input >> n >> q;

    LCT lct;
    std::vector<int> vertex_id(n);
    for (int i = 0; i < n; i++) {
        mint a;
        fast_input >> a;
        vertex_id[i] = lct.add_vertex(VertexValue{a});
    }

    std::vector<int> edge_id(n - 1);
    for (int i = 0; i + 1 < n; i++) {
        int u, v;
        mint b, c;
        fast_input >> u >> v >> b >> c;
        edge_id[i] = lct.add_edge(vertex_id[u], vertex_id[v], EdgeValue{b, c});
    }

    for (int i = 0; i < q; i++) {
        int type;
        fast_input >> type;
        int root;
        if (type == 0) {
            int w;
            mint x;
            fast_input >> w >> x >> root;
            lct.set_vertex(vertex_id[w], VertexValue{x});
        } else {
            int e;
            mint y, z;
            fast_input >> e >> y >> z >> root;
            lct.set_edge(edge_id[e], EdgeValue{y, z});
        }
        fast_output << lct.component_prod(vertex_id[root]).s << '\n';
    }
}
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