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

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Code

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

#include "../../graph/bipartite.hpp"

#include <algorithm>
#include <cassert>
#include <cstdint>
#include "../../utilities/fast_io.hpp"
#include <utility>
#include <vector>

namespace {

void check(
    int left_size,
    int right_size,
    const std::vector<std::pair<int, int>>& edges
) {
    std::vector<int> left_degree(left_size, 0);
    std::vector<int> right_degree(right_size, 0);
    int maximum_degree = 0;
    for (auto [left, right] : edges) {
        left_degree[left]++;
        right_degree[right]++;
        maximum_degree = std::max(maximum_degree, left_degree[left]);
        maximum_degree = std::max(maximum_degree, right_degree[right]);
    }

    auto result = m1une::graph::bipartite_edge_coloring(left_size, right_size, edges);
    assert(result.color_count == maximum_degree);
    assert(result.color.size() == edges.size());
    std::vector<std::vector<char>> left_used(
        left_size,
        std::vector<char>(maximum_degree, false)
    );
    std::vector<std::vector<char>> right_used(
        right_size,
        std::vector<char>(maximum_degree, false)
    );
    for (int edge = 0; edge < int(edges.size()); edge++) {
        auto [left, right] = edges[edge];
        int color = result.color[edge];
        assert(0 <= color && color < maximum_degree);
        assert(!left_used[left][color]);
        assert(!right_used[right][color]);
        left_used[left][color] = true;
        right_used[right][color] = true;
    }
}

void test_fixed() {
    check(0, 0, std::vector<std::pair<int, int>>());

    std::vector<std::pair<int, int>> star;
    for (int right = 0; right < 20; right++) star.emplace_back(0, right);
    check(1, 20, star);

    std::vector<std::pair<int, int>> parallel;
    for (int edge = 0; edge < 15; edge++) parallel.emplace_back(0, 0);
    check(1, 1, parallel);

    std::vector<std::pair<int, int>> complete;
    for (int left = 0; left < 7; left++) {
        for (int right = 0; right < 7; right++) complete.emplace_back(left, right);
    }
    check(7, 7, complete);
}

void test_randomized() {
    std::uint64_t state = 393393;
    auto random = [&state]() {
        state ^= state << 7;
        state ^= state >> 9;
        return state;
    };

    for (int trial = 0; trial < 3000; trial++) {
        int left_size = 1 + int(random() % 10);
        int right_size = 1 + int(random() % 10);
        int edge_count = int(random() % 50);
        std::vector<std::pair<int, int>> edges;
        edges.reserve(edge_count);
        for (int edge = 0; edge < edge_count; edge++) {
            edges.emplace_back(
                int(random() % left_size),
                int(random() % right_size)
            );
        }
        check(left_size, right_size, edges);
    }
}

}  // namespace

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

    test_fixed();
    test_randomized();
    int left_size, right_size, edge_count;
    fast_input >> left_size >> right_size >> edge_count;
    std::vector<std::pair<int, int>> edges(edge_count);
    for (auto& [left, right] : edges) fast_input >> left >> right;

    auto result = m1une::graph::bipartite_edge_coloring(left_size, right_size, edges);
    fast_output << result.color_count << '\n';
    for (int color : result.color) fast_output << color << '\n';
}
#line 1 "verify/graph/bipartite_edge_coloring.test.cpp"
#define PROBLEM "https://judge.yosupo.jp/problem/bipartite_edge_coloring"

#line 1 "graph/bipartite.hpp"



#include <algorithm>
#include <cassert>
#include <cstddef>
#include <cstdint>
#include <limits>
#include <optional>
#include <queue>
#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 15 "graph/bipartite.hpp"

namespace m1une {
namespace graph {

struct BipartiteResult {
    bool is_bipartite;
    std::vector<int> color;
    std::vector<int> left_vertices;
    std::vector<int> right_vertices;
    std::vector<int> left_id;
    std::vector<int> right_id;
};

template <class T>
BipartiteResult bipartite(const Graph<T>& g) {
    int n = g.size();
    BipartiteResult result;
    result.is_bipartite = true;
    result.color.assign(n, -1);
    result.left_id.assign(n, -1);
    result.right_id.assign(n, -1);

    std::vector<std::vector<int>> adjacency(n);
    for (const auto& e : g.edges()) {
        adjacency[e.from].push_back(e.to);
        adjacency[e.to].push_back(e.from);
    }

    std::queue<int> que;
    for (int s = 0; s < n; s++) {
        if (result.color[s] != -1) continue;
        result.color[s] = 0;
        que.push(s);
        while (!que.empty()) {
            int v = que.front();
            que.pop();
            for (int to : adjacency[v]) {
                if (result.color[to] == -1) {
                    result.color[to] = result.color[v] ^ 1;
                    que.push(to);
                } else if (result.color[to] == result.color[v]) {
                    result.is_bipartite = false;
                    return result;
                }
            }
        }
    }

    for (int v = 0; v < n; v++) {
        if (result.color[v] == 0) {
            result.left_id[v] = int(result.left_vertices.size());
            result.left_vertices.push_back(v);
        } else {
            result.right_id[v] = int(result.right_vertices.size());
            result.right_vertices.push_back(v);
        }
    }

    return result;
}

template <class T>
bool is_bipartite(const Graph<T>& g) {
    return bipartite(g).is_bipartite;
}

struct BipartiteVertexSet {
    std::vector<int> left;
    std::vector<int> right;

    int size() const {
        return int(left.size() + right.size());
    }
};

struct BipartiteMatching {
    struct Edge {
        int left;
        int right;
        int id;
        bool alive;
    };

    struct Pair {
        int left;
        int right;
        int edge_id;
    };

   private:
    int _left_size;
    int _right_size;
    std::vector<Edge> _edges;
    std::vector<std::vector<int>> _adj;
    std::vector<std::vector<int>> _radj;
    std::vector<int> _left_match;
    std::vector<int> _right_match;
    std::vector<int> _left_match_edge;
    std::vector<int> _right_match_edge;
    bool _calculated;

    void invalidate() {
        _calculated = false;
    }

    void ensure_matching() {
        if (!_calculated) max_matching();
    }

   public:
    BipartiteMatching() : BipartiteMatching(0, 0) {}

    BipartiteMatching(int left_size, int right_size)
        : _left_size(left_size),
          _right_size(right_size),
          _adj(left_size),
          _radj(right_size),
          _left_match(left_size, -1),
          _right_match(right_size, -1),
          _left_match_edge(left_size, -1),
          _right_match_edge(right_size, -1),
          _calculated(false) {
        assert(0 <= left_size);
        assert(0 <= right_size);
    }

    int left_size() const {
        return _left_size;
    }

    int right_size() const {
        return _right_size;
    }

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

    int add_edge(int left, int right) {
        assert(0 <= left && left < _left_size);
        assert(0 <= right && right < _right_size);
        int id = int(_edges.size());
        _edges.push_back(Edge{left, right, id, true});
        _adj[left].push_back(id);
        _radj[right].push_back(id);
        invalidate();
        return id;
    }

    Edge get_edge(int i) const {
        assert(0 <= i && i < int(_edges.size()));
        return _edges[i];
    }

    std::vector<Edge> edges(bool include_inactive = false) const {
        std::vector<Edge> result;
        result.reserve(_edges.size());
        for (const auto& e : _edges) {
            if (include_inactive || e.alive) result.push_back(e);
        }
        return result;
    }

    void set_edge_alive(int id, bool alive) {
        assert(0 <= id && id < int(_edges.size()));
        _edges[id].alive = alive;
        invalidate();
    }

    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 < int(_edges.size()));
        return _edges[id].alive;
    }

    int max_matching() {
        _left_match.assign(_left_size, -1);
        _right_match.assign(_right_size, -1);
        _left_match_edge.assign(_left_size, -1);
        _right_match_edge.assign(_right_size, -1);

        std::vector<int> dist(_left_size);
        auto bfs = [&]() -> bool {
            std::queue<int> que;
            bool found = false;
            for (int l = 0; l < _left_size; l++) {
                if (_left_match[l] == -1) {
                    dist[l] = 0;
                    que.push(l);
                } else {
                    dist[l] = -1;
                }
            }

            while (!que.empty()) {
                int l = que.front();
                que.pop();
                for (int id : _adj[l]) {
                    const auto& e = _edges[id];
                    if (!e.alive) continue;
                    int next_left = _right_match[e.right];
                    if (next_left == -1) {
                        found = true;
                    } else if (dist[next_left] == -1) {
                        dist[next_left] = dist[l] + 1;
                        que.push(next_left);
                    }
                }
            }
            return found;
        };

        auto dfs = [&](auto self, int l) -> bool {
            for (int id : _adj[l]) {
                const auto& e = _edges[id];
                if (!e.alive) continue;
                int next_left = _right_match[e.right];
                if (next_left != -1 && (dist[next_left] != dist[l] + 1 || !self(self, next_left))) {
                    continue;
                }
                _left_match[l] = e.right;
                _right_match[e.right] = l;
                _left_match_edge[l] = id;
                _right_match_edge[e.right] = id;
                return true;
            }
            dist[l] = -1;
            return false;
        };

        int result = 0;
        while (bfs()) {
            for (int l = 0; l < _left_size; l++) {
                if (_left_match[l] == -1 && dfs(dfs, l)) result++;
            }
        }

        _calculated = true;
        return result;
    }

    int matching_size() {
        ensure_matching();
        int result = 0;
        for (int right : _left_match) {
            if (right != -1) result++;
        }
        return result;
    }

    std::vector<int> left_match() {
        ensure_matching();
        return _left_match;
    }

    std::vector<int> right_match() {
        ensure_matching();
        return _right_match;
    }

    std::vector<Pair> matching() {
        ensure_matching();
        std::vector<Pair> result;
        for (int l = 0; l < _left_size; l++) {
            if (_left_match[l] != -1) result.push_back(Pair{l, _left_match[l], _left_match_edge[l]});
        }
        return result;
    }

    BipartiteVertexSet minimum_vertex_cover() {
        ensure_matching();

        std::vector<char> visited_left(_left_size, false), visited_right(_right_size, false);
        std::queue<int> que;
        for (int l = 0; l < _left_size; l++) {
            if (_left_match[l] == -1) {
                visited_left[l] = true;
                que.push(l);
            }
        }

        while (!que.empty()) {
            int l = que.front();
            que.pop();
            for (int id : _adj[l]) {
                const auto& e = _edges[id];
                if (!e.alive || _left_match_edge[l] == id || visited_right[e.right]) continue;
                visited_right[e.right] = true;
                int next_left = _right_match[e.right];
                if (next_left != -1 && !visited_left[next_left]) {
                    visited_left[next_left] = true;
                    que.push(next_left);
                }
            }
        }

        BipartiteVertexSet result;
        for (int l = 0; l < _left_size; l++) {
            if (!visited_left[l]) result.left.push_back(l);
        }
        for (int r = 0; r < _right_size; r++) {
            if (visited_right[r]) result.right.push_back(r);
        }
        return result;
    }

    BipartiteVertexSet maximum_independent_set() {
        auto cover = minimum_vertex_cover();
        std::vector<char> in_left_cover(_left_size, false), in_right_cover(_right_size, false);
        for (int l : cover.left) in_left_cover[l] = true;
        for (int r : cover.right) in_right_cover[r] = true;

        BipartiteVertexSet result;
        for (int l = 0; l < _left_size; l++) {
            if (!in_left_cover[l]) result.left.push_back(l);
        }
        for (int r = 0; r < _right_size; r++) {
            if (!in_right_cover[r]) result.right.push_back(r);
        }
        return result;
    }

    std::optional<std::vector<int>> minimum_edge_cover() {
        ensure_matching();

        std::vector<int> result;
        std::vector<char> covered_left(_left_size, false), covered_right(_right_size, false);
        std::vector<char> used_edge(_edges.size(), false);

        auto use_edge = [&](int id) {
            if (used_edge[id]) return;
            used_edge[id] = true;
            result.push_back(id);
            covered_left[_edges[id].left] = true;
            covered_right[_edges[id].right] = true;
        };

        for (int l = 0; l < _left_size; l++) {
            if (_left_match_edge[l] != -1) use_edge(_left_match_edge[l]);
        }

        for (int l = 0; l < _left_size; l++) {
            if (covered_left[l]) continue;
            int id = -1;
            for (int edge_id : _adj[l]) {
                if (_edges[edge_id].alive) {
                    id = edge_id;
                    break;
                }
            }
            if (id == -1) return std::nullopt;
            use_edge(id);
        }

        for (int r = 0; r < _right_size; r++) {
            if (covered_right[r]) continue;
            int id = -1;
            for (int edge_id : _radj[r]) {
                if (_edges[edge_id].alive) {
                    id = edge_id;
                    break;
                }
            }
            if (id == -1) return std::nullopt;
            use_edge(id);
        }

        return result;
    }
};

struct BipartiteMatchingGraph {
    BipartiteResult parts;
    BipartiteMatching matching;
    std::vector<int> original_edge_id;

    int left_vertex(int left) const {
        assert(0 <= left && left < int(parts.left_vertices.size()));
        return parts.left_vertices[left];
    }

    int right_vertex(int right) const {
        assert(0 <= right && right < int(parts.right_vertices.size()));
        return parts.right_vertices[right];
    }

    int original_edge(int edge_id) const {
        assert(0 <= edge_id && edge_id < int(original_edge_id.size()));
        return original_edge_id[edge_id];
    }
};

template <class T>
std::optional<BipartiteMatchingGraph> make_bipartite_matching(const Graph<T>& g) {
    auto parts = bipartite(g);
    if (!parts.is_bipartite) return std::nullopt;

    BipartiteMatchingGraph result;
    result.parts = parts;
    result.matching = BipartiteMatching(int(parts.left_vertices.size()), int(parts.right_vertices.size()));

    for (const auto& e : g.edges()) {
        int left, right;
        if (parts.color[e.from] == 0) {
            left = parts.left_id[e.from];
            right = parts.right_id[e.to];
        } else {
            left = parts.left_id[e.to];
            right = parts.right_id[e.from];
        }
        int id = result.matching.add_edge(left, right);
        if (int(result.original_edge_id.size()) <= id) result.original_edge_id.resize(id + 1);
        result.original_edge_id[id] = e.id;
    }

    return result;
}

struct BipartiteEdgeColoringResult {
    int color_count;
    std::vector<int> color;
};

namespace detail {

struct BipartiteEdgeColoringGroups {
    int count;
    std::vector<int> group;
};

inline BipartiteEdgeColoringGroups group_vertices(
    const std::vector<int>& degree,
    int maximum_degree
) {
    BipartiteEdgeColoringGroups result;
    result.count = 0;
    result.group.assign(degree.size(), -1);
    int current_degree = 0;
    for (int vertex = 0; vertex < int(degree.size()); vertex++) {
        if (degree[vertex] == 0) continue;
        if (result.count == 0 || current_degree + degree[vertex] > maximum_degree) {
            result.count++;
            current_degree = 0;
        }
        result.group[vertex] = result.count - 1;
        current_degree += degree[vertex];
    }
    return result;
}

class BipartiteEdgeColoringSolver {
   private:
    int _side_size;
    int _original_edge_count;
    std::vector<int> _left;
    std::vector<int> _right;
    std::vector<int> _color;
    std::vector<int> _used_stamp;
    int _stamp;

    int other_endpoint(int vertex, int edge) const {
        if (vertex < _side_size) return _side_size + _right[edge];
        return _left[edge];
    }

    std::vector<int> perfect_matching(const std::vector<int>& edge_ids) const {
        std::vector<std::vector<int>> adjacency(_side_size);
        for (int edge : edge_ids) adjacency[_left[edge]].push_back(edge);

        std::vector<int> right_match(_side_size, -1);
        std::vector<int> left_match_edge(_side_size, -1);
        int matching_size = 0;
        for (int left = 0; left < _side_size; left++) {
            for (int edge : adjacency[left]) {
                int right = _right[edge];
                if (right_match[right] != -1) continue;
                right_match[right] = left;
                left_match_edge[left] = edge;
                matching_size++;
                break;
            }
        }

        std::vector<int> distance(_side_size);
        std::vector<int> next_edge(_side_size);
        std::vector<int> left_stack;
        std::vector<int> path_edges;
        left_stack.reserve(_side_size);
        path_edges.reserve(_side_size);

        while (matching_size < _side_size) {
            std::queue<int> queue;
            std::fill(distance.begin(), distance.end(), -1);
            for (int left = 0; left < _side_size; left++) {
                if (left_match_edge[left] != -1) continue;
                distance[left] = 0;
                queue.push(left);
            }

            bool reachable_free_right = false;
            while (!queue.empty()) {
                int left = queue.front();
                queue.pop();
                for (int edge : adjacency[left]) {
                    int next_left = right_match[_right[edge]];
                    if (next_left == -1) {
                        reachable_free_right = true;
                    } else if (distance[next_left] == -1) {
                        distance[next_left] = distance[left] + 1;
                        queue.push(next_left);
                    }
                }
            }
            assert(reachable_free_right);

            std::fill(next_edge.begin(), next_edge.end(), 0);
            int augmented = 0;
            for (int root = 0; root < _side_size; root++) {
                if (left_match_edge[root] != -1 || distance[root] == -1) continue;
                left_stack.clear();
                path_edges.clear();
                left_stack.push_back(root);
                bool found = false;

                while (!left_stack.empty() && !found) {
                    int left = left_stack.back();
                    bool advanced = false;
                    while (next_edge[left] < int(adjacency[left].size())) {
                        int edge = adjacency[left][next_edge[left]++];
                        int right = _right[edge];
                        int next_left = right_match[right];
                        if (next_left == -1) {
                            left_match_edge[left] = edge;
                            right_match[right] = left;
                            for (int index = int(path_edges.size()) - 1; index >= 0; index--) {
                                int path_edge = path_edges[index];
                                int path_left = left_stack[index];
                                left_match_edge[path_left] = path_edge;
                                right_match[_right[path_edge]] = path_left;
                            }
                            found = true;
                            break;
                        }
                        if (distance[next_left] != distance[left] + 1) continue;
                        path_edges.push_back(edge);
                        left_stack.push_back(next_left);
                        advanced = true;
                        break;
                    }
                    if (found || advanced) continue;
                    distance[left] = -1;
                    left_stack.pop_back();
                    if (path_edges.size() == left_stack.size() && !path_edges.empty()) {
                        path_edges.pop_back();
                    }
                }
                if (found) augmented++;
            }
            assert(augmented > 0);
            matching_size += augmented;
        }

        return left_match_edge;
    }

    std::pair<std::vector<int>, std::vector<int>> split_even(
        const std::vector<int>& edge_ids
    ) {
        std::vector<std::vector<int>> incidence(std::size_t(2) * _side_size);
        for (int edge : edge_ids) {
            incidence[_left[edge]].push_back(edge);
            incidence[_side_size + _right[edge]].push_back(edge);
        }

        _stamp++;
        assert(_stamp > 0);
        std::vector<int> next_edge(std::size_t(2) * _side_size, 0);
        std::vector<int> first;
        std::vector<int> second;
        first.reserve(edge_ids.size() / 2);
        second.reserve(edge_ids.size() / 2);

        for (int start = 0; start < 2 * _side_size; start++) {
            while (true) {
                while (next_edge[start] < int(incidence[start].size()) &&
                       _used_stamp[incidence[start][next_edge[start]]] == _stamp) {
                    next_edge[start]++;
                }
                if (next_edge[start] == int(incidence[start].size())) break;

                int vertex = start;
                bool parity = false;
                do {
                    while (next_edge[vertex] < int(incidence[vertex].size()) &&
                           _used_stamp[incidence[vertex][next_edge[vertex]]] == _stamp) {
                        next_edge[vertex]++;
                    }
                    assert(next_edge[vertex] < int(incidence[vertex].size()));
                    int edge = incidence[vertex][next_edge[vertex]++];
                    _used_stamp[edge] = _stamp;
                    if (!parity) {
                        first.push_back(edge);
                    } else {
                        second.push_back(edge);
                    }
                    parity = !parity;
                    vertex = other_endpoint(vertex, edge);
                } while (vertex != start);
                assert(!parity);
            }
        }
        assert(first.size() == second.size());
        return {std::move(first), std::move(second)};
    }

    void color_regular(const std::vector<int>& edge_ids, int degree, int offset) {
        assert(std::size_t(_side_size) * std::size_t(degree) == edge_ids.size());
        if (degree == 0) return;
        if (degree == 1) {
            for (int edge : edge_ids) {
                if (edge < _original_edge_count) _color[edge] = offset;
            }
            return;
        }

        if (degree % 2 == 1) {
            std::vector<int> matching = perfect_matching(edge_ids);
            _stamp++;
            assert(_stamp > 0);
            for (int edge : matching) {
                _used_stamp[edge] = _stamp;
                if (edge < _original_edge_count) _color[edge] = offset;
            }
            std::vector<int> remaining;
            remaining.reserve(edge_ids.size() - matching.size());
            for (int edge : edge_ids) {
                if (_used_stamp[edge] != _stamp) remaining.push_back(edge);
            }
            color_regular(remaining, degree - 1, offset + 1);
            return;
        }

        auto [first, second] = split_even(edge_ids);
        color_regular(first, degree / 2, offset);
        color_regular(second, degree / 2, offset + degree / 2);
    }

   public:
    BipartiteEdgeColoringSolver(
        int side_size,
        int original_edge_count,
        std::vector<int> left,
        std::vector<int> right
    )
        : _side_size(side_size),
          _original_edge_count(original_edge_count),
          _left(std::move(left)),
          _right(std::move(right)),
          _color(original_edge_count, -1),
          _used_stamp(_left.size(), 0),
          _stamp(0) {}

    std::vector<int> solve(int degree) {
        std::vector<int> edge_ids(_left.size());
        for (int edge = 0; edge < int(edge_ids.size()); edge++) edge_ids[edge] = edge;
        color_regular(edge_ids, degree, 0);
        for (int color : _color) assert(0 <= color && color < degree);
        return _color;
    }
};

}  // namespace detail

// Returns an optimal edge coloring of a bipartite multigraph.
inline BipartiteEdgeColoringResult bipartite_edge_coloring(
    int left_size,
    int right_size,
    const std::vector<std::pair<int, int>>& edges
) {
    assert(left_size >= 0);
    assert(right_size >= 0);
    assert(edges.size() <= std::size_t(std::numeric_limits<int>::max()));

    std::vector<int> left_degree(left_size, 0);
    std::vector<int> right_degree(right_size, 0);
    int maximum_degree = 0;
    for (auto [left, right] : edges) {
        assert(0 <= left && left < left_size);
        assert(0 <= right && right < right_size);
        left_degree[left]++;
        right_degree[right]++;
        maximum_degree = std::max(maximum_degree, left_degree[left]);
        maximum_degree = std::max(maximum_degree, right_degree[right]);
    }

    BipartiteEdgeColoringResult result;
    result.color_count = maximum_degree;
    if (edges.empty()) return result;

    detail::BipartiteEdgeColoringGroups left_groups =
        detail::group_vertices(left_degree, maximum_degree);
    detail::BipartiteEdgeColoringGroups right_groups =
        detail::group_vertices(right_degree, maximum_degree);
    int side_size = std::max(left_groups.count, right_groups.count);

    std::vector<int> contracted_left;
    std::vector<int> contracted_right;
    contracted_left.reserve(std::size_t(3) * edges.size());
    contracted_right.reserve(std::size_t(3) * edges.size());
    std::vector<int> contracted_left_degree(side_size, 0);
    std::vector<int> contracted_right_degree(side_size, 0);
    for (auto [left, right] : edges) {
        int contracted_left_vertex = left_groups.group[left];
        int contracted_right_vertex = right_groups.group[right];
        contracted_left.push_back(contracted_left_vertex);
        contracted_right.push_back(contracted_right_vertex);
        contracted_left_degree[contracted_left_vertex]++;
        contracted_right_degree[contracted_right_vertex]++;
    }

    int left = 0;
    int right = 0;
    while (true) {
        while (left < side_size && contracted_left_degree[left] == maximum_degree) left++;
        while (right < side_size && contracted_right_degree[right] == maximum_degree) right++;
        if (left == side_size || right == side_size) break;
        contracted_left.push_back(left);
        contracted_right.push_back(right);
        contracted_left_degree[left]++;
        contracted_right_degree[right]++;
    }
    assert(left == side_size && right == side_size);
    assert(contracted_left.size() == std::size_t(side_size) * std::size_t(maximum_degree));

    detail::BipartiteEdgeColoringSolver solver(
        side_size,
        int(edges.size()),
        std::move(contracted_left),
        std::move(contracted_right)
    );
    result.color = solver.solve(maximum_degree);
    return result;
}

}  // namespace graph
}  // namespace m1une


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

#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>
#include <type_traits>
#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 11 "verify/graph/bipartite_edge_coloring.test.cpp"

namespace {

void check(
    int left_size,
    int right_size,
    const std::vector<std::pair<int, int>>& edges
) {
    std::vector<int> left_degree(left_size, 0);
    std::vector<int> right_degree(right_size, 0);
    int maximum_degree = 0;
    for (auto [left, right] : edges) {
        left_degree[left]++;
        right_degree[right]++;
        maximum_degree = std::max(maximum_degree, left_degree[left]);
        maximum_degree = std::max(maximum_degree, right_degree[right]);
    }

    auto result = m1une::graph::bipartite_edge_coloring(left_size, right_size, edges);
    assert(result.color_count == maximum_degree);
    assert(result.color.size() == edges.size());
    std::vector<std::vector<char>> left_used(
        left_size,
        std::vector<char>(maximum_degree, false)
    );
    std::vector<std::vector<char>> right_used(
        right_size,
        std::vector<char>(maximum_degree, false)
    );
    for (int edge = 0; edge < int(edges.size()); edge++) {
        auto [left, right] = edges[edge];
        int color = result.color[edge];
        assert(0 <= color && color < maximum_degree);
        assert(!left_used[left][color]);
        assert(!right_used[right][color]);
        left_used[left][color] = true;
        right_used[right][color] = true;
    }
}

void test_fixed() {
    check(0, 0, std::vector<std::pair<int, int>>());

    std::vector<std::pair<int, int>> star;
    for (int right = 0; right < 20; right++) star.emplace_back(0, right);
    check(1, 20, star);

    std::vector<std::pair<int, int>> parallel;
    for (int edge = 0; edge < 15; edge++) parallel.emplace_back(0, 0);
    check(1, 1, parallel);

    std::vector<std::pair<int, int>> complete;
    for (int left = 0; left < 7; left++) {
        for (int right = 0; right < 7; right++) complete.emplace_back(left, right);
    }
    check(7, 7, complete);
}

void test_randomized() {
    std::uint64_t state = 393393;
    auto random = [&state]() {
        state ^= state << 7;
        state ^= state >> 9;
        return state;
    };

    for (int trial = 0; trial < 3000; trial++) {
        int left_size = 1 + int(random() % 10);
        int right_size = 1 + int(random() % 10);
        int edge_count = int(random() % 50);
        std::vector<std::pair<int, int>> edges;
        edges.reserve(edge_count);
        for (int edge = 0; edge < edge_count; edge++) {
            edges.emplace_back(
                int(random() % left_size),
                int(random() % right_size)
            );
        }
        check(left_size, right_size, edges);
    }
}

}  // namespace

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

    test_fixed();
    test_randomized();
    int left_size, right_size, edge_count;
    fast_input >> left_size >> right_size >> edge_count;
    std::vector<std::pair<int, int>> edges(edge_count);
    for (auto& [left, right] : edges) fast_input >> left >> right;

    auto result = m1une::graph::bipartite_edge_coloring(left_size, right_size, edges);
    fast_output << result.color_count << '\n';
    for (int color : result.color) fast_output << color << '\n';
}
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