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:heavy_check_mark: verify/graph/incremental_scc.test.cpp

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Code

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

#include "../../graph/incremental_scc.hpp"

#include <cassert>
#include <cstdint>
#include <utility>
#include <vector>

#include "../../ds/dsu/dsu.hpp"
#include "../../graph/scc.hpp"
#include "../../math/modint.hpp"
#include "../../utilities/fast_io.hpp"

namespace {

using Graph = m1une::graph::Graph<>;

void validate(const Graph& graph, const std::vector<std::pair<int, int>>& edges) {
    const int n = graph.size();
    const int m = graph.edge_count();
    assert(int(edges.size()) == m);
    const std::vector<int> actual = m1une::graph::incremental_scc(graph);
    assert(int(actual.size()) == m);

    std::vector<int> expected(m, m + 1);
    m1une::ds::Dsu replay(n);
    for (int time = 1; time <= m; time++) {
        Graph prefix(n);
        for (int edge_id = 0; edge_id < time; edge_id++) {
            if (!graph.is_edge_alive(edge_id)) continue;
            prefix.add_directed_edge(
                edges[edge_id].first,
                edges[edge_id].second
            );
        }
        const auto components =
            m1une::graph::strongly_connected_components(prefix);
        for (int edge_id = 0; edge_id < m; edge_id++) {
            if (edge_id >= time || !graph.is_edge_alive(edge_id)) continue;
            if (expected[edge_id] == m + 1 && components.same(
                    edges[edge_id].first,
                    edges[edge_id].second
                )) {
                expected[edge_id] = time;
            }
            if (actual[edge_id] == time) {
                replay.merge(edges[edge_id].first, edges[edge_id].second);
            }
        }
        for (int first = 0; first < n; first++) {
            for (int second = 0; second < n; second++) {
                assert(replay.same(first, second) ==
                       components.same(first, second));
            }
        }
    }
    assert(actual == expected);
}

void test_fixed() {
    Graph empty(3);
    validate(empty, std::vector<std::pair<int, int>>());

    Graph graph(4);
    std::vector<std::pair<int, int>> edges;
    edges.emplace_back(0, 1);
    graph.add_directed_edge(0, 1);
    edges.emplace_back(1, 2);
    graph.add_directed_edge(1, 2);
    edges.emplace_back(2, 0);
    graph.add_directed_edge(2, 0);
    edges.emplace_back(2, 3);
    graph.add_directed_edge(2, 3);
    edges.emplace_back(3, 3);
    graph.add_directed_edge(3, 3);
    validate(graph, edges);
    const std::vector<int> time = m1une::graph::incremental_scc(graph);
    assert(time[0] == 3);
    assert(time[1] == 3);
    assert(time[2] == 3);
    assert(time[3] == 6);
    assert(time[4] == 5);

    graph.erase_edge(2);
    validate(graph, edges);
}

void test_randomized() {
    std::uint64_t state = 20260718;
    auto random = [&state]() {
        state ^= state << 7;
        state ^= state >> 9;
        return state;
    };

    for (int trial = 0; trial < 500; trial++) {
        const int n = 1 + int(random() % 8);
        const int m = int(random() % 17);
        Graph graph(n);
        std::vector<std::pair<int, int>> edges;
        edges.reserve(m);
        for (int edge_id = 0; edge_id < m; edge_id++) {
            const int from = int(random() % n);
            const int to = int(random() % n);
            edges.emplace_back(from, to);
            const int id = graph.add_directed_edge(from, to);
            if (random() % 7 == 0) graph.erase_edge(id);
        }
        validate(graph, edges);
    }
}

}  // namespace

int main() {
    test_fixed();
    test_randomized();

    using Mint = m1une::math::modint998244353;
    m1une::utilities::FastInput fast_input;
    m1une::utilities::FastOutput fast_output;
    int vertex_count, edge_count;
    fast_input >> vertex_count >> edge_count;

    std::vector<Mint> weight(vertex_count);
    for (Mint& value : weight) {
        int input;
        fast_input >> input;
        value = Mint(input);
    }

    Graph graph(vertex_count);
    std::vector<std::pair<int, int>> edges;
    edges.reserve(edge_count);
    for (int edge_id = 0; edge_id < edge_count; edge_id++) {
        int from, to;
        fast_input >> from >> to;
        edges.emplace_back(from, to);
        graph.add_directed_edge(from, to);
    }

    const std::vector<int> merge_time =
        m1une::graph::incremental_scc(graph);
    std::vector<std::vector<int>> edges_by_time(edge_count + 1);
    for (int edge_id = 0; edge_id < edge_count; edge_id++) {
        if (merge_time[edge_id] <= edge_count) {
            edges_by_time[merge_time[edge_id]].push_back(edge_id);
        }
    }

    m1une::ds::Dsu components(vertex_count);
    Mint answer = 0;
    for (int time = 1; time <= edge_count; time++) {
        for (int edge_id : edges_by_time[time]) {
            const int first = components.leader(edges[edge_id].first);
            const int second = components.leader(edges[edge_id].second);
            if (first == second) continue;
            answer += weight[first] * weight[second];
            components.merge(first, second, [&](int leader, int absorbed) {
                weight[leader] += weight[absorbed];
            });
        }
        fast_output << answer.val() << '\n';
    }
}
#line 1 "verify/graph/incremental_scc.test.cpp"
#define PROBLEM "https://judge.yosupo.jp/problem/incremental_scc"

#line 1 "graph/incremental_scc.hpp"



#include <algorithm>
#include <cassert>
#include <cstddef>
#include <utility>
#include <vector>

#line 1 "graph/graph.hpp"



#include <array>
#line 8 "graph/graph.hpp"

namespace m1une {
namespace graph {

template <class T = int>
struct Edge {
    using cost_type = T;

    int from;
    int to;
    T cost;
    int id;
    bool alive;

    Edge() : from(-1), to(-1), cost(T()), id(-1), alive(true) {}
    Edge(int from_, int to_, T cost_ = T(1), int id_ = -1, bool alive_ = true)
        : from(from_), to(to_), cost(cost_), id(id_), alive(alive_) {}

    int other(int v) const {
        assert(v == from || v == to);
        return from ^ to ^ v;
    }
};

template <class T = int>
struct Graph {
    using edge_type = Edge<T>;
    using cost_type = T;

   private:
    struct EdgePositions {
        std::array<std::pair<int, int>, 2> value{};
        int size = 0;

        void push_back(std::pair<int, int> position) {
            assert(size < 2);
            value[size++] = position;
        }
    };

    int _n;
    int _edge_count;
    std::vector<std::vector<edge_type>> _g;
    std::vector<EdgePositions> _edge_positions;

   public:
    Graph() : _n(0), _edge_count(0) {}
    explicit Graph(int n) : _n(n), _edge_count(0), _g(n) {
        assert(0 <= n);
    }

    int size() const {
        return _n;
    }

    bool empty() const {
        return _n == 0;
    }

    int edge_count() const {
        return _edge_count;
    }

    int add_vertex() {
        _g.emplace_back();
        return _n++;
    }

    int add_directed_edge(int from, int to, T cost = T(1)) {
        assert(0 <= from && from < _n);
        assert(0 <= to && to < _n);
        int id = _edge_count++;
        int idx = int(_g[from].size());
        _g[from].push_back(edge_type(from, to, cost, id));
        _edge_positions.emplace_back();
        _edge_positions.back().push_back({from, idx});
        return id;
    }

    int add_edge(int u, int v, T cost = T(1)) {
        assert(0 <= u && u < _n);
        assert(0 <= v && v < _n);
        int id = _edge_count++;
        int u_idx = int(_g[u].size());
        _g[u].push_back(edge_type(u, v, cost, id));
        int v_idx = int(_g[v].size());
        _g[v].push_back(edge_type(v, u, cost, id));
        _edge_positions.emplace_back();
        _edge_positions.back().push_back({u, u_idx});
        _edge_positions.back().push_back({v, v_idx});
        return id;
    }

    void set_edge_alive(int id, bool alive) {
        assert(0 <= id && id < _edge_count);
        for (int i = 0; i < _edge_positions[id].size; ++i) {
            auto [v, idx] = _edge_positions[id].value[i];
            _g[v][idx].alive = alive;
        }
    }

    void erase_edge(int id) {
        set_edge_alive(id, false);
    }

    void revive_edge(int id) {
        set_edge_alive(id, true);
    }

    bool is_edge_alive(int id) const {
        assert(0 <= id && id < _edge_count);
        assert(_edge_positions[id].size != 0);
        auto [v, idx] = _edge_positions[id].value[0];
        return _g[v][idx].alive;
    }

    const std::vector<edge_type>& operator[](int v) const {
        assert(0 <= v && v < _n);
        return _g[v];
    }

    std::vector<edge_type>& operator[](int v) {
        assert(0 <= v && v < _n);
        return _g[v];
    }

    const std::vector<std::vector<edge_type>>& adjacency() const {
        return _g;
    }

    std::vector<std::vector<edge_type>>& adjacency() {
        return _g;
    }

    std::vector<edge_type> edges(bool include_inactive = false) const {
        std::vector<edge_type> result;
        result.reserve(_edge_count);
        std::vector<char> used(_edge_count, false);
        for (int v = 0; v < _n; v++) {
            for (const auto& e : _g[v]) {
                if (!include_inactive && !e.alive) continue;
                if (0 <= e.id && e.id < _edge_count) {
                    if (used[e.id]) continue;
                    used[e.id] = true;
                }
                result.push_back(e);
            }
        }
        return result;
    }

    Graph reversed() const {
        Graph result(_n);
        result._edge_count = _edge_count;
        result._edge_positions.assign(_edge_count, {});
        for (int v = 0; v < _n; v++) {
            for (const auto& e : _g[v]) {
                int idx = int(result._g[e.to].size());
                result._g[e.to].push_back(edge_type(e.to, e.from, e.cost, e.id, e.alive));
                if (0 <= e.id && e.id < _edge_count) result._edge_positions[e.id].push_back({e.to, idx});
            }
        }
        return result;
    }
};

}  // namespace graph
}  // namespace m1une


#line 11 "graph/incremental_scc.hpp"

namespace m1une {
namespace graph {

namespace incremental_scc_detail {

struct EdgeEvent {
    int id;
    int from;
    int to;
};

inline std::vector<int> component_ids(
    int vertex_count,
    const std::vector<EdgeEvent>& edges,
    int time
) {
    std::vector<int> begin(vertex_count + 1, 0);
    std::vector<int> reverse_begin(vertex_count + 1, 0);
    int edge_count = 0;
    for (const EdgeEvent& edge : edges) {
        if (edge.id >= time) continue;
        begin[edge.from + 1]++;
        reverse_begin[edge.to + 1]++;
        edge_count++;
    }
    for (int vertex = 0; vertex < vertex_count; vertex++) {
        begin[vertex + 1] += begin[vertex];
        reverse_begin[vertex + 1] += reverse_begin[vertex];
    }

    std::vector<int> adjacency(edge_count);
    std::vector<int> reverse_adjacency(edge_count);
    std::vector<int> cursor = begin;
    std::vector<int> reverse_cursor = reverse_begin;
    for (const EdgeEvent& edge : edges) {
        if (edge.id >= time) continue;
        adjacency[cursor[edge.from]++] = edge.to;
        reverse_adjacency[reverse_cursor[edge.to]++] = edge.from;
    }
    std::vector<int>().swap(cursor);
    std::vector<int>().swap(reverse_cursor);

    std::vector<char> visited(vertex_count, false);
    std::vector<int> next_position(begin.begin(), begin.end() - 1);
    std::vector<int> order;
    order.reserve(vertex_count);
    std::vector<int> stack;
    for (int start = 0; start < vertex_count; start++) {
        if (visited[start]) continue;
        visited[start] = true;
        stack.push_back(start);
        while (!stack.empty()) {
            const int vertex = stack.back();
            int& position = next_position[vertex];
            if (position < begin[vertex + 1]) {
                const int to = adjacency[position++];
                if (!visited[to]) {
                    visited[to] = true;
                    stack.push_back(to);
                }
            } else {
                order.push_back(vertex);
                stack.pop_back();
            }
        }
    }

    std::vector<int> component(vertex_count, -1);
    int component_count = 0;
    for (auto iterator = order.rbegin(); iterator != order.rend(); ++iterator) {
        const int start = *iterator;
        if (component[start] != -1) continue;
        component[start] = component_count;
        stack.push_back(start);
        while (!stack.empty()) {
            const int vertex = stack.back();
            stack.pop_back();
            for (int position = reverse_begin[vertex];
                 position < reverse_begin[vertex + 1]; position++) {
                const int to = reverse_adjacency[position];
                if (component[to] != -1) continue;
                component[to] = component_count;
                stack.push_back(to);
            }
        }
        component_count++;
    }
    return component;
}

}  // namespace incremental_scc_detail

// For every directed edge e, returns the first time t after e is inserted such
// that its endpoints are in the same SCC. At time t, edges with IDs less than
// t have been inserted. edge_count() + 1 means this never happens.
template <class T>
std::vector<int> incremental_scc(const Graph<T>& graph) {
    using incremental_scc_detail::EdgeEvent;
    using incremental_scc_detail::component_ids;

    const int vertex_count = graph.size();
    const int edge_count = graph.edge_count();
    const int never = edge_count + 1;
    std::vector<int> merge_time(edge_count, never);
    if (edge_count == 0) return merge_time;

    std::vector<EdgeEvent> edges_by_id(edge_count);
    std::vector<char> initialized(edge_count, false);
    for (int vertex = 0; vertex < vertex_count; vertex++) {
        for (const Edge<T>& edge : graph[vertex]) {
            assert(0 <= edge.id && edge.id < edge_count);
            assert(!initialized[edge.id]);
            if (initialized[edge.id]) continue;
            initialized[edge.id] = true;
            edges_by_id[edge.id] = EdgeEvent{edge.id, edge.from, edge.to};
        }
    }

    std::vector<EdgeEvent> events;
    events.reserve(edge_count);
    for (int edge_id = 0; edge_id < edge_count; edge_id++) {
        assert(initialized[edge_id]);
        if (graph.is_edge_alive(edge_id)) {
            events.push_back(edges_by_id[edge_id]);
        }
    }
    std::vector<EdgeEvent>().swap(edges_by_id);
    std::vector<char>().swap(initialized);

    std::vector<int> new_index(vertex_count, -1);
    auto divide = [&](
        auto&& self,
        std::vector<EdgeEvent> current,
        int left,
        int right
    ) -> void {
        if (current.empty() || right == left + 1) return;
        const int middle = left + (right - left) / 2;

        std::vector<int> touched;
        touched.reserve(std::min(
            std::size_t(vertex_count),
            current.size() * 2
        ));
        int compressed_count = 0;
        for (const EdgeEvent& edge : current) {
            if (new_index[edge.from] == -1) {
                new_index[edge.from] = compressed_count++;
                touched.push_back(edge.from);
            }
            if (new_index[edge.to] == -1) {
                new_index[edge.to] = compressed_count++;
                touched.push_back(edge.to);
            }
        }
        for (EdgeEvent& edge : current) {
            edge.from = new_index[edge.from];
            edge.to = new_index[edge.to];
        }
        for (int vertex : touched) new_index[vertex] = -1;

        std::vector<EdgeEvent> earlier;
        std::vector<EdgeEvent> later;
        earlier.reserve(current.size() / 2);
        later.reserve(current.size() / 2);
        {
            std::vector<int> component =
                component_ids(compressed_count, current, middle);
            for (const EdgeEvent& edge : current) {
                const int from_component = component[edge.from];
                const int to_component = component[edge.to];
                if (edge.id < middle &&
                    from_component == to_component) {
                    merge_time[edge.id] =
                        std::min(merge_time[edge.id], middle);
                    earlier.push_back(edge);
                } else {
                    later.push_back(EdgeEvent{
                        edge.id,
                        from_component,
                        to_component
                    });
                }
            }
        }

        std::vector<EdgeEvent>().swap(current);
        self(self, std::move(earlier), left, middle);
        self(self, std::move(later), middle, right);
    };
    divide(divide, std::move(events), 0, edge_count + 1);
    return merge_time;
}

}  // namespace graph
}  // namespace m1une


#line 4 "verify/graph/incremental_scc.test.cpp"

#line 6 "verify/graph/incremental_scc.test.cpp"
#include <cstdint>
#line 9 "verify/graph/incremental_scc.test.cpp"

#line 1 "ds/dsu/dsu.hpp"



#line 5 "ds/dsu/dsu.hpp"
#include <numeric>
#line 8 "ds/dsu/dsu.hpp"

namespace m1une {
namespace ds {

struct Dsu {
   private:
    int _n;
    // parent_or_size[i] is the parent of i if it's >= 0.
    // If it's < 0, then i is a root and -parent_or_size[i] is the size of the group.
    std::vector<int> parent_or_size;

    // Returns {new leader, absorbed leader}. The absorbed leader is -1 when
    // both vertices already belong to the same component.
    std::pair<int, int> merge_leaders(int a, int b) {
        int x = leader(a), y = leader(b);
        if (x == y) return {x, -1};
        if (-parent_or_size[x] < -parent_or_size[y]) std::swap(x, y);
        parent_or_size[x] += parent_or_size[y];
        parent_or_size[y] = x;
        return {x, y};
    }

   public:
    Dsu() : _n(0) {}
    explicit Dsu(int n) : _n(n), parent_or_size(n, -1) {}

    // Merges the group containing 'a' with the group containing 'b'.
    // Returns the leader of the merged group.
    int merge(int a, int b) {
        return merge_leaders(a, b).first;
    }

    // Invokes callback(new_leader, absorbed_leader) after an actual merge.
    // Returns the leader of the merged group.
    template <class Callback>
    int merge(int a, int b, Callback&& callback) {
        std::pair<int, int> merged = merge_leaders(a, b);
        if (merged.second != -1) callback(merged.first, merged.second);
        return merged.first;
    }

    // Returns true if 'a' and 'b' belong to the same group.
    bool same(int a, int b) {
        return leader(a) == leader(b);
    }

    // Returns the leader (representative) of the group containing 'a'.
    int leader(int a) {
        if (parent_or_size[a] < 0) return a;
        // Path compression
        return parent_or_size[a] = leader(parent_or_size[a]);
    }

    // Returns the size of the group containing 'a'.
    int size(int a) {
        return -parent_or_size[leader(a)];
    }

    // Returns a list of all groups, where each group is a vector of its elements.
    std::vector<std::vector<int>> groups() {
        std::vector<int> leader_buf(_n), group_size(_n);
        for (int i = 0; i < _n; i++) {
            leader_buf[i] = leader(i);
            group_size[leader_buf[i]]++;
        }
        std::vector<std::vector<int>> result(_n);
        for (int i = 0; i < _n; i++) {
            result[i].reserve(group_size[i]);
        }
        for (int i = 0; i < _n; i++) {
            result[leader_buf[i]].push_back(i);
        }
        result.erase(std::remove_if(result.begin(), result.end(), [&](const std::vector<int>& v) { return v.empty(); }),
                     result.end());
        return result;
    }
};

}  // namespace ds
}  // namespace m1une


#line 1 "graph/scc.hpp"



#line 9 "graph/scc.hpp"

#line 11 "graph/scc.hpp"

namespace m1une {
namespace graph {

struct SccResult {
    int count;
    std::vector<int> comp;
    std::vector<std::vector<int>> groups;

    bool same(int u, int v) const {
        assert(0 <= u && u < int(comp.size()));
        assert(0 <= v && v < int(comp.size()));
        return comp[u] == comp[v];
    }

    template <class T>
    Graph<int> dag(const Graph<T>& g) const {
        std::vector<std::pair<int, int>> edges;
        for (int v = 0; v < g.size(); v++) {
            for (const auto& e : g[v]) {
                if (!e.alive) continue;
                int a = comp[e.from], b = comp[e.to];
                if (a != b) edges.emplace_back(a, b);
            }
        }
        std::sort(edges.begin(), edges.end());
        edges.erase(std::unique(edges.begin(), edges.end()), edges.end());

        Graph<int> result(count);
        for (auto [a, b] : edges) result.add_directed_edge(a, b);
        return result;
    }
};

template <class T>
SccResult strongly_connected_components(const Graph<T>& g) {
    const int n = g.size();
    std::vector<std::vector<int>> reverse_graph(n);
    for (int vertex = 0; vertex < n; vertex++) {
        for (const auto& edge : g[vertex]) {
            if (edge.alive) reverse_graph[edge.to].push_back(vertex);
        }
    }

    std::vector<char> seen(n, false);
    std::vector<int> order;
    order.reserve(n);
    std::vector<std::pair<int, std::size_t>> dfs_stack;
    for (int start = 0; start < n; start++) {
        if (seen[start]) continue;
        seen[start] = true;
        dfs_stack.emplace_back(start, 0);
        while (!dfs_stack.empty()) {
            int vertex = dfs_stack.back().first;
            std::size_t& edge_index = dfs_stack.back().second;
            while (edge_index < g[vertex].size() &&
                   !g[vertex][edge_index].alive) {
                edge_index++;
            }
            if (edge_index == g[vertex].size()) {
                order.push_back(vertex);
                dfs_stack.pop_back();
                continue;
            }
            const int to = g[vertex][edge_index++].to;
            if (!seen[to]) {
                seen[to] = true;
                dfs_stack.emplace_back(to, 0);
            }
        }
    }

    std::vector<int> comp(n, -1);
    std::vector<std::vector<int>> groups;
    std::vector<int> stack;
    for (auto iterator = order.rbegin(); iterator != order.rend(); ++iterator) {
        const int start = *iterator;
        if (comp[start] != -1) continue;
        const int component = int(groups.size());
        groups.emplace_back();
        comp[start] = component;
        stack.push_back(start);
        while (!stack.empty()) {
            const int vertex = stack.back();
            stack.pop_back();
            groups.back().push_back(vertex);
            for (int to : reverse_graph[vertex]) {
                if (comp[to] != -1) continue;
                comp[to] = component;
                stack.push_back(to);
            }
        }
    }

    return SccResult{int(groups.size()), std::move(comp), std::move(groups)};
}

}  // namespace graph
}  // namespace m1une


#line 1 "math/modint.hpp"



#line 6 "math/modint.hpp"
#include <iostream>
#include <type_traits>
#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 1 "utilities/fast_io.hpp"



#line 6 "utilities/fast_io.hpp"
#include <cerrno>
#include <charconv>
#line 9 "utilities/fast_io.hpp"
#include <cstdio>
#include <cstdlib>
#line 12 "utilities/fast_io.hpp"
#include <cstring>
#include <iterator>
#include <string>
#include <sys/stat.h>
#line 18 "utilities/fast_io.hpp"
#include <unistd.h>
#line 20 "utilities/fast_io.hpp"

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 14 "verify/graph/incremental_scc.test.cpp"

namespace {

using Graph = m1une::graph::Graph<>;

void validate(const Graph& graph, const std::vector<std::pair<int, int>>& edges) {
    const int n = graph.size();
    const int m = graph.edge_count();
    assert(int(edges.size()) == m);
    const std::vector<int> actual = m1une::graph::incremental_scc(graph);
    assert(int(actual.size()) == m);

    std::vector<int> expected(m, m + 1);
    m1une::ds::Dsu replay(n);
    for (int time = 1; time <= m; time++) {
        Graph prefix(n);
        for (int edge_id = 0; edge_id < time; edge_id++) {
            if (!graph.is_edge_alive(edge_id)) continue;
            prefix.add_directed_edge(
                edges[edge_id].first,
                edges[edge_id].second
            );
        }
        const auto components =
            m1une::graph::strongly_connected_components(prefix);
        for (int edge_id = 0; edge_id < m; edge_id++) {
            if (edge_id >= time || !graph.is_edge_alive(edge_id)) continue;
            if (expected[edge_id] == m + 1 && components.same(
                    edges[edge_id].first,
                    edges[edge_id].second
                )) {
                expected[edge_id] = time;
            }
            if (actual[edge_id] == time) {
                replay.merge(edges[edge_id].first, edges[edge_id].second);
            }
        }
        for (int first = 0; first < n; first++) {
            for (int second = 0; second < n; second++) {
                assert(replay.same(first, second) ==
                       components.same(first, second));
            }
        }
    }
    assert(actual == expected);
}

void test_fixed() {
    Graph empty(3);
    validate(empty, std::vector<std::pair<int, int>>());

    Graph graph(4);
    std::vector<std::pair<int, int>> edges;
    edges.emplace_back(0, 1);
    graph.add_directed_edge(0, 1);
    edges.emplace_back(1, 2);
    graph.add_directed_edge(1, 2);
    edges.emplace_back(2, 0);
    graph.add_directed_edge(2, 0);
    edges.emplace_back(2, 3);
    graph.add_directed_edge(2, 3);
    edges.emplace_back(3, 3);
    graph.add_directed_edge(3, 3);
    validate(graph, edges);
    const std::vector<int> time = m1une::graph::incremental_scc(graph);
    assert(time[0] == 3);
    assert(time[1] == 3);
    assert(time[2] == 3);
    assert(time[3] == 6);
    assert(time[4] == 5);

    graph.erase_edge(2);
    validate(graph, edges);
}

void test_randomized() {
    std::uint64_t state = 20260718;
    auto random = [&state]() {
        state ^= state << 7;
        state ^= state >> 9;
        return state;
    };

    for (int trial = 0; trial < 500; trial++) {
        const int n = 1 + int(random() % 8);
        const int m = int(random() % 17);
        Graph graph(n);
        std::vector<std::pair<int, int>> edges;
        edges.reserve(m);
        for (int edge_id = 0; edge_id < m; edge_id++) {
            const int from = int(random() % n);
            const int to = int(random() % n);
            edges.emplace_back(from, to);
            const int id = graph.add_directed_edge(from, to);
            if (random() % 7 == 0) graph.erase_edge(id);
        }
        validate(graph, edges);
    }
}

}  // namespace

int main() {
    test_fixed();
    test_randomized();

    using Mint = m1une::math::modint998244353;
    m1une::utilities::FastInput fast_input;
    m1une::utilities::FastOutput fast_output;
    int vertex_count, edge_count;
    fast_input >> vertex_count >> edge_count;

    std::vector<Mint> weight(vertex_count);
    for (Mint& value : weight) {
        int input;
        fast_input >> input;
        value = Mint(input);
    }

    Graph graph(vertex_count);
    std::vector<std::pair<int, int>> edges;
    edges.reserve(edge_count);
    for (int edge_id = 0; edge_id < edge_count; edge_id++) {
        int from, to;
        fast_input >> from >> to;
        edges.emplace_back(from, to);
        graph.add_directed_edge(from, to);
    }

    const std::vector<int> merge_time =
        m1une::graph::incremental_scc(graph);
    std::vector<std::vector<int>> edges_by_time(edge_count + 1);
    for (int edge_id = 0; edge_id < edge_count; edge_id++) {
        if (merge_time[edge_id] <= edge_count) {
            edges_by_time[merge_time[edge_id]].push_back(edge_id);
        }
    }

    m1une::ds::Dsu components(vertex_count);
    Mint answer = 0;
    for (int time = 1; time <= edge_count; time++) {
        for (int edge_id : edges_by_time[time]) {
            const int first = components.leader(edges[edge_id].first);
            const int second = components.leader(edges[edge_id].second);
            if (first == second) continue;
            answer += weight[first] * weight[second];
            components.merge(first, second, [&](int leader, int absorbed) {
                weight[leader] += weight[absorbed];
            });
        }
        fast_output << answer.val() << '\n';
    }
}
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