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

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Code

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

#include "../../graph/directed_mst.hpp"

#include <cassert>
#include "../../utilities/fast_io.hpp"
#include <optional>
#include <queue>
#include <random>
#include <vector>

namespace {

struct TestEdge {
    int from;
    int to;
    long long cost;
    bool alive;
};

std::optional<long long> brute_directed_mst(
    int n,
    int root,
    const std::vector<TestEdge>& edges
) {
    std::vector<std::vector<int>> incoming(n);
    for (int index = 0; index < int(edges.size()); index++) {
        if (edges[index].alive && edges[index].from != edges[index].to) {
            incoming[edges[index].to].push_back(index);
        }
    }
    for (int vertex = 0; vertex < n; vertex++) {
        if (vertex != root && incoming[vertex].empty()) return std::nullopt;
    }

    std::vector<int> vertices;
    for (int vertex = 0; vertex < n; vertex++) {
        if (vertex != root) vertices.push_back(vertex);
    }

    std::optional<long long> answer;
    std::vector<int> chosen(n, -1);
    auto search = [&](auto&& self, int index, long long cost) -> void {
        if (index == int(vertices.size())) {
            std::vector<std::vector<int>> tree(n);
            for (int vertex : vertices) {
                tree[edges[chosen[vertex]].from].push_back(vertex);
            }
            std::vector<char> reached(n, false);
            std::queue<int> queue;
            reached[root] = true;
            queue.push(root);
            while (!queue.empty()) {
                int vertex = queue.front();
                queue.pop();
                for (int to : tree[vertex]) {
                    if (reached[to]) continue;
                    reached[to] = true;
                    queue.push(to);
                }
            }
            for (char value : reached) {
                if (!value) return;
            }
            if (!answer || cost < *answer) answer = cost;
            return;
        }

        int vertex = vertices[index];
        for (int edge_index : incoming[vertex]) {
            chosen[vertex] = edge_index;
            self(self, index + 1, cost + edges[edge_index].cost);
        }
    };
    search(search, 0, 0);
    return answer;
}

std::optional<long long> brute_rootless_directed_mst(
    int n,
    const std::vector<TestEdge>& edges
) {
    std::optional<long long> answer;
    for (int root = 0; root < n; root++) {
        auto candidate = brute_directed_mst(n, root, edges);
        if (candidate && (!answer || *candidate < *answer)) answer = candidate;
    }
    return answer;
}

void validate_result(
    const m1une::graph::DirectedMinimumSpanningTree<long long>& result,
    int n,
    int root,
    const std::vector<TestEdge>& edges
) {
    assert(result.root == root);
    assert(int(result.parent.size()) == n);
    assert(int(result.parent_edge.size()) == n);
    assert(int(result.edges.size()) == n - 1);
    assert(result.parent[root] == root);
    assert(result.parent_edge[root] == -1);

    long long total = 0;
    std::vector<std::vector<int>> tree(n);
    for (int vertex = 0; vertex < n; vertex++) {
        if (vertex == root) continue;
        int edge_id = result.parent_edge[vertex];
        assert(0 <= edge_id && edge_id < int(edges.size()));
        const auto& edge = edges[edge_id];
        assert(edge.alive);
        assert(edge.to == vertex);
        assert(edge.from == result.parent[vertex]);
        total += edge.cost;
        tree[edge.from].push_back(vertex);
    }
    assert(total == result.cost);

    std::vector<char> reached(n, false);
    std::queue<int> queue;
    reached[root] = true;
    queue.push(root);
    while (!queue.empty()) {
        int vertex = queue.front();
        queue.pop();
        for (int to : tree[vertex]) {
            assert(!reached[to]);
            reached[to] = true;
            queue.push(to);
        }
    }
    for (char value : reached) assert(value);
}

void randomized_test() {
    std::mt19937 random(20260711);
    for (int test = 0; test < 500; test++) {
        int n = std::uniform_int_distribution<int>(1, 5)(random);
        int root = std::uniform_int_distribution<int>(0, n - 1)(random);
        int m = std::uniform_int_distribution<int>(0, 10)(random);

        m1une::graph::Graph<long long> graph(n);
        std::vector<TestEdge> edges;
        for (int index = 0; index < m; index++) {
            int from = std::uniform_int_distribution<int>(0, n - 1)(random);
            int to = std::uniform_int_distribution<int>(0, n - 1)(random);
            long long cost = std::uniform_int_distribution<int>(-5, 10)(random);
            int id = graph.add_directed_edge(from, to, cost);
            bool alive = std::uniform_int_distribution<int>(0, 8)(random) != 0;
            graph.set_edge_alive(id, alive);
            edges.push_back(TestEdge{from, to, cost, alive});
        }

        auto expected = brute_directed_mst(n, root, edges);
        auto actual = m1une::graph::directed_mst(graph, root);
        assert(expected.has_value() == actual.has_value());
        if (actual) {
            assert(actual->cost == *expected);
            validate_result(*actual, n, root, edges);
        }

        auto rootless_expected = brute_rootless_directed_mst(n, edges);
        auto rootless_actual = m1une::graph::directed_mst(graph);
        assert(rootless_expected.has_value() == rootless_actual.has_value());
        if (rootless_actual) {
            assert(rootless_actual->cost == *rootless_expected);
            validate_result(
                *rootless_actual,
                n,
                rootless_actual->root,
                edges
            );
        }
    }
}

}  // namespace

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

    randomized_test();
    int n, m, root;
    fast_input >> n >> m >> root;
    m1une::graph::Graph<long long> graph(n);
    for (int index = 0; index < m; index++) {
        int from, to;
        long long cost;
        fast_input >> from >> to >> cost;
        graph.add_directed_edge(from, to, cost);
    }

    auto answer = m1une::graph::directed_mst(graph, root);
    assert(answer.has_value());
    fast_output << answer->cost << '\n';
    for (int vertex = 0; vertex < n; vertex++) {
        if (vertex) fast_output << ' ';
        fast_output << answer->parent[vertex];
    }
    fast_output << '\n';
}
#line 1 "verify/graph/directed_mst.test.cpp"
#define PROBLEM "https://judge.yosupo.jp/problem/directedmst"

#line 1 "graph/directed_mst.hpp"



#include <cassert>
#include <optional>
#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 10 "graph/directed_mst.hpp"

namespace m1une {
namespace graph {

template <class T>
struct DirectedMinimumSpanningTree {
    T cost;
    std::vector<int> parent;
    std::vector<int> parent_edge;
    std::vector<Edge<T>> edges;
    int root;
};

namespace internal {

template <class T>
struct DirectedMstEdge {
    int from = -1;
    int to = -1;
    T cost = T(0);
    int id = -1;
};

template <class T>
struct DirectedMstHeapPool {
    using StoredEdge = DirectedMstEdge<T>;

    struct Node {
        StoredEdge edge;
        T offset = T(0);
        int child = -1;
        int sibling = -1;
    };

    struct Heap {
        int root = -1;
        int size = 0;
    };

    std::vector<Node> nodes;

    explicit DirectedMstHeapPool(int capacity = 0) {
        nodes.reserve(capacity);
    }

    T key(int node) const {
        return nodes[node].edge.cost + nodes[node].offset;
    }

    int meld_roots(int first, int second) {
        if (first == -1) return second;
        if (second == -1) return first;
        if (key(second) < key(first)) std::swap(first, second);
        nodes[second].offset -= nodes[first].offset;
        nodes[second].sibling = nodes[first].child;
        nodes[first].child = second;
        return first;
    }

    void push(Heap& heap, const StoredEdge& edge) {
        const int node = int(nodes.size());
        nodes.push_back(Node{edge, T(0), -1, -1});
        heap.root = meld_roots(heap.root, node);
        heap.size++;
    }

    void meld(Heap& destination, Heap& source) {
        destination.root = meld_roots(destination.root, source.root);
        destination.size += source.size;
        source.root = -1;
        source.size = 0;
    }

    const StoredEdge& top(const Heap& heap) const {
        assert(heap.root != -1);
        return nodes[heap.root].edge;
    }

    T top_key(const Heap& heap) const {
        assert(heap.root != -1);
        return key(heap.root);
    }

    void add_all(Heap& heap, const T& delta) {
        assert(heap.root != -1);
        nodes[heap.root].offset += delta;
    }

    void pop(Heap& heap) {
        assert(heap.root != -1 && heap.size > 0);
        const int old_root = heap.root;
        int child = nodes[old_root].child;
        std::vector<int> pairs;
        while (child != -1) {
            int first = child;
            child = nodes[first].sibling;
            nodes[first].sibling = -1;
            nodes[first].offset += nodes[old_root].offset;

            if (child != -1) {
                int second = child;
                child = nodes[second].sibling;
                nodes[second].sibling = -1;
                nodes[second].offset += nodes[old_root].offset;
                first = meld_roots(first, second);
            }
            pairs.push_back(first);
        }

        heap.root = -1;
        for (auto it = pairs.rbegin(); it != pairs.rend(); ++it) {
            heap.root = meld_roots(*it, heap.root);
        }
        heap.size--;
    }
};

struct DirectedMstDsu {
    std::vector<int> parent;

    explicit DirectedMstDsu(int n) : parent(n, -1) {}

    int leader(int vertex) {
        int root = vertex;
        while (parent[root] != -1) root = parent[root];
        while (vertex != root) {
            int next = parent[vertex];
            parent[vertex] = root;
            vertex = next;
        }
        return root;
    }
};

template <class T>
struct DirectedMstRootlessCost {
    int artificial_edges;
    T original_cost;

    DirectedMstRootlessCost() : artificial_edges(0), original_cost(T(0)) {}
    explicit DirectedMstRootlessCost(int zero)
        : artificial_edges(zero), original_cost(T(0)) {
        assert(zero == 0);
    }
    DirectedMstRootlessCost(int artificial_edges_, const T& original_cost_)
        : artificial_edges(artificial_edges_), original_cost(original_cost_) {}

    DirectedMstRootlessCost& operator+=(const DirectedMstRootlessCost& other) {
        artificial_edges += other.artificial_edges;
        original_cost += other.original_cost;
        return *this;
    }

    DirectedMstRootlessCost& operator-=(const DirectedMstRootlessCost& other) {
        artificial_edges -= other.artificial_edges;
        original_cost -= other.original_cost;
        return *this;
    }

    friend DirectedMstRootlessCost operator+(
        DirectedMstRootlessCost first,
        const DirectedMstRootlessCost& second
    ) {
        return first += second;
    }

    friend DirectedMstRootlessCost operator-(
        DirectedMstRootlessCost first,
        const DirectedMstRootlessCost& second
    ) {
        return first -= second;
    }

    friend bool operator<(
        const DirectedMstRootlessCost& first,
        const DirectedMstRootlessCost& second
    ) {
        if (first.artificial_edges != second.artificial_edges) {
            return first.artificial_edges < second.artificial_edges;
        }
        return first.original_cost < second.original_cost;
    }
};

}  // namespace internal

// Returns a minimum-cost spanning arborescence rooted at root, or nullopt when
// some vertex is unreachable from the root using active directed edges.
template <class T>
std::optional<DirectedMinimumSpanningTree<T>> directed_mst(
    const Graph<T>& graph,
    int root
) {
    const int n = graph.size();
    assert(0 <= root && root < n);
    const int maximum_node_count = 2 * n;

    int active_edge_count = 0;
#ifndef NDEBUG
    std::vector<int> incidence(graph.edge_count(), 0);
#endif
    for (int vertex = 0; vertex < n; vertex++) {
        for (const Edge<T>& edge : graph[vertex]) {
            if (!edge.alive) continue;
            assert(0 <= edge.id && edge.id < graph.edge_count());
#ifndef NDEBUG
            incidence[edge.id]++;
#endif
            active_edge_count++;
        }
    }
#ifndef NDEBUG
    for (int count : incidence) {
        if (count != 0) assert(count == 1);
    }
#endif

    using StoredEdge = internal::DirectedMstEdge<T>;
    using HeapPool = internal::DirectedMstHeapPool<T>;
    HeapPool pool(active_edge_count);
    std::vector<typename HeapPool::Heap> heaps(maximum_node_count);
    for (int vertex = 0; vertex < n; vertex++) {
        for (const Edge<T>& edge : graph[vertex]) {
            if (!edge.alive) continue;
            pool.push(heaps[edge.to], StoredEdge{edge.from, edge.to, edge.cost, edge.id});
        }
    }

    internal::DirectedMstDsu dsu(maximum_node_count);
    std::vector<int> contraction_parent(maximum_node_count, -1);
    std::vector<int> visited(maximum_node_count, 0);
    std::vector<StoredEdge> selected(maximum_node_count);
    int node_count = n;
    int visit_token = 1;
    visited[root] = 1;

    for (int start = 0; start < n; start++) {
        if (visited[start] != 0) continue;
        visit_token++;
        int component = start;
        while (visited[component] == 0 || visited[component] == visit_token) {
            if (visited[component] == visit_token) {
                if (node_count == maximum_node_count) return std::nullopt;
                const int contracted = node_count++;
                int current = component;
                do {
                    const T reduction = T(0) - pool.top_key(heaps[current]);
                    pool.add_all(heaps[current], reduction);
                    pool.meld(heaps[contracted], heaps[current]);
                    contraction_parent[current] = contracted;
                    dsu.parent[current] = contracted;
                    current = dsu.leader(selected[current].from);
                } while (current != contracted);
                component = contracted;
            }

            assert(visited[component] == 0);
            visited[component] = visit_token;
            while (heaps[component].size > 0 &&
                   dsu.leader(pool.top(heaps[component]).from) == component) {
                pool.pop(heaps[component]);
            }
            if (heaps[component].size == 0) return std::nullopt;
            selected[component] = pool.top(heaps[component]);
            component = dsu.leader(selected[component].from);
        }
    }

    DirectedMinimumSpanningTree<T> result;
    result.cost = T(0);
    result.parent.assign(n, -1);
    result.parent_edge.assign(n, -1);
    result.root = root;
    result.parent[root] = root;

    std::vector<char> expanded(node_count, false);
    std::vector<StoredEdge> chosen(n);
    for (int component = node_count - 1; component >= 0; component--) {
        if (component == root || expanded[component]) continue;
        const StoredEdge& edge = selected[component];
        if (edge.id == -1) return std::nullopt;
        int vertex = edge.to;
        while (vertex != -1 && !expanded[vertex]) {
            expanded[vertex] = true;
            vertex = contraction_parent[vertex];
        }
        result.cost += edge.cost;
        result.parent[edge.to] = edge.from;
        result.parent_edge[edge.to] = edge.id;
        chosen[edge.to] = edge;
    }

    result.edges.reserve(n - 1);
    for (int vertex = 0; vertex < n; vertex++) {
        if (vertex == root) continue;
        if (result.parent[vertex] == -1) return std::nullopt;
        const StoredEdge& edge = chosen[vertex];
        result.edges.emplace_back(edge.from, edge.to, edge.cost, edge.id, true);
    }
    return result;
}

// Chooses the root that gives a minimum-cost spanning arborescence.
template <class T>
std::optional<DirectedMinimumSpanningTree<T>> directed_mst(
    const Graph<T>& graph
) {
    const int n = graph.size();
    if (n == 0) return std::nullopt;

    using Cost = internal::DirectedMstRootlessCost<T>;
    Graph<Cost> augmented(n + 1);
    std::vector<int> original_edge_id;
    original_edge_id.reserve(graph.edge_count() + n);

#ifndef NDEBUG
    std::vector<int> incidence(graph.edge_count(), 0);
#endif
    for (int vertex = 0; vertex < n; vertex++) {
        for (const Edge<T>& edge : graph[vertex]) {
            if (!edge.alive) continue;
#ifndef NDEBUG
            assert(0 <= edge.id && edge.id < graph.edge_count());
            incidence[edge.id]++;
#endif
            augmented.add_directed_edge(
                edge.from,
                edge.to,
                Cost(0, edge.cost)
            );
            original_edge_id.push_back(edge.id);
        }
    }
#ifndef NDEBUG
    for (int count : incidence) {
        if (count != 0) assert(count == 1);
    }
#endif

    const int artificial_root = n;
    for (int vertex = 0; vertex < n; vertex++) {
        augmented.add_directed_edge(
            artificial_root,
            vertex,
            Cost(1, T(0))
        );
        original_edge_id.push_back(-1);
    }

    auto augmented_result = directed_mst(augmented, artificial_root);
    if (!augmented_result || augmented_result->cost.artificial_edges != 1) {
        return std::nullopt;
    }

    DirectedMinimumSpanningTree<T> result;
    result.cost = augmented_result->cost.original_cost;
    result.parent.assign(n, -1);
    result.parent_edge.assign(n, -1);
    result.root = -1;
    result.edges.reserve(n - 1);

    for (int vertex = 0; vertex < n; vertex++) {
        int augmented_edge_id = augmented_result->parent_edge[vertex];
        assert(0 <= augmented_edge_id &&
               augmented_edge_id < int(original_edge_id.size()));
        int edge_id = original_edge_id[augmented_edge_id];
        if (edge_id == -1) {
            assert(result.root == -1);
            result.root = vertex;
            result.parent[vertex] = vertex;
            continue;
        }

        result.parent[vertex] = augmented_result->parent[vertex];
        result.parent_edge[vertex] = edge_id;
        result.edges.emplace_back(
            result.parent[vertex],
            vertex,
            augmented_result->edges[vertex].cost.original_cost,
            edge_id,
            true
        );
    }
    assert(result.root != -1);
    return result;
}

}  // namespace graph
}  // namespace m1une


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

#line 1 "utilities/fast_io.hpp"



#include <algorithm>
#line 6 "utilities/fast_io.hpp"
#include <cerrno>
#include <charconv>
#include <cstddef>
#include <cstdio>
#include <cstdlib>
#include <cstdint>
#include <cstring>
#include <iterator>
#include <string>
#include <sys/stat.h>
#include <type_traits>
#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 8 "verify/graph/directed_mst.test.cpp"
#include <queue>
#include <random>
#line 11 "verify/graph/directed_mst.test.cpp"

namespace {

struct TestEdge {
    int from;
    int to;
    long long cost;
    bool alive;
};

std::optional<long long> brute_directed_mst(
    int n,
    int root,
    const std::vector<TestEdge>& edges
) {
    std::vector<std::vector<int>> incoming(n);
    for (int index = 0; index < int(edges.size()); index++) {
        if (edges[index].alive && edges[index].from != edges[index].to) {
            incoming[edges[index].to].push_back(index);
        }
    }
    for (int vertex = 0; vertex < n; vertex++) {
        if (vertex != root && incoming[vertex].empty()) return std::nullopt;
    }

    std::vector<int> vertices;
    for (int vertex = 0; vertex < n; vertex++) {
        if (vertex != root) vertices.push_back(vertex);
    }

    std::optional<long long> answer;
    std::vector<int> chosen(n, -1);
    auto search = [&](auto&& self, int index, long long cost) -> void {
        if (index == int(vertices.size())) {
            std::vector<std::vector<int>> tree(n);
            for (int vertex : vertices) {
                tree[edges[chosen[vertex]].from].push_back(vertex);
            }
            std::vector<char> reached(n, false);
            std::queue<int> queue;
            reached[root] = true;
            queue.push(root);
            while (!queue.empty()) {
                int vertex = queue.front();
                queue.pop();
                for (int to : tree[vertex]) {
                    if (reached[to]) continue;
                    reached[to] = true;
                    queue.push(to);
                }
            }
            for (char value : reached) {
                if (!value) return;
            }
            if (!answer || cost < *answer) answer = cost;
            return;
        }

        int vertex = vertices[index];
        for (int edge_index : incoming[vertex]) {
            chosen[vertex] = edge_index;
            self(self, index + 1, cost + edges[edge_index].cost);
        }
    };
    search(search, 0, 0);
    return answer;
}

std::optional<long long> brute_rootless_directed_mst(
    int n,
    const std::vector<TestEdge>& edges
) {
    std::optional<long long> answer;
    for (int root = 0; root < n; root++) {
        auto candidate = brute_directed_mst(n, root, edges);
        if (candidate && (!answer || *candidate < *answer)) answer = candidate;
    }
    return answer;
}

void validate_result(
    const m1une::graph::DirectedMinimumSpanningTree<long long>& result,
    int n,
    int root,
    const std::vector<TestEdge>& edges
) {
    assert(result.root == root);
    assert(int(result.parent.size()) == n);
    assert(int(result.parent_edge.size()) == n);
    assert(int(result.edges.size()) == n - 1);
    assert(result.parent[root] == root);
    assert(result.parent_edge[root] == -1);

    long long total = 0;
    std::vector<std::vector<int>> tree(n);
    for (int vertex = 0; vertex < n; vertex++) {
        if (vertex == root) continue;
        int edge_id = result.parent_edge[vertex];
        assert(0 <= edge_id && edge_id < int(edges.size()));
        const auto& edge = edges[edge_id];
        assert(edge.alive);
        assert(edge.to == vertex);
        assert(edge.from == result.parent[vertex]);
        total += edge.cost;
        tree[edge.from].push_back(vertex);
    }
    assert(total == result.cost);

    std::vector<char> reached(n, false);
    std::queue<int> queue;
    reached[root] = true;
    queue.push(root);
    while (!queue.empty()) {
        int vertex = queue.front();
        queue.pop();
        for (int to : tree[vertex]) {
            assert(!reached[to]);
            reached[to] = true;
            queue.push(to);
        }
    }
    for (char value : reached) assert(value);
}

void randomized_test() {
    std::mt19937 random(20260711);
    for (int test = 0; test < 500; test++) {
        int n = std::uniform_int_distribution<int>(1, 5)(random);
        int root = std::uniform_int_distribution<int>(0, n - 1)(random);
        int m = std::uniform_int_distribution<int>(0, 10)(random);

        m1une::graph::Graph<long long> graph(n);
        std::vector<TestEdge> edges;
        for (int index = 0; index < m; index++) {
            int from = std::uniform_int_distribution<int>(0, n - 1)(random);
            int to = std::uniform_int_distribution<int>(0, n - 1)(random);
            long long cost = std::uniform_int_distribution<int>(-5, 10)(random);
            int id = graph.add_directed_edge(from, to, cost);
            bool alive = std::uniform_int_distribution<int>(0, 8)(random) != 0;
            graph.set_edge_alive(id, alive);
            edges.push_back(TestEdge{from, to, cost, alive});
        }

        auto expected = brute_directed_mst(n, root, edges);
        auto actual = m1une::graph::directed_mst(graph, root);
        assert(expected.has_value() == actual.has_value());
        if (actual) {
            assert(actual->cost == *expected);
            validate_result(*actual, n, root, edges);
        }

        auto rootless_expected = brute_rootless_directed_mst(n, edges);
        auto rootless_actual = m1une::graph::directed_mst(graph);
        assert(rootless_expected.has_value() == rootless_actual.has_value());
        if (rootless_actual) {
            assert(rootless_actual->cost == *rootless_expected);
            validate_result(
                *rootless_actual,
                n,
                rootless_actual->root,
                edges
            );
        }
    }
}

}  // namespace

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

    randomized_test();
    int n, m, root;
    fast_input >> n >> m >> root;
    m1une::graph::Graph<long long> graph(n);
    for (int index = 0; index < m; index++) {
        int from, to;
        long long cost;
        fast_input >> from >> to >> cost;
        graph.add_directed_edge(from, to, cost);
    }

    auto answer = m1une::graph::directed_mst(graph, root);
    assert(answer.has_value());
    fast_output << answer->cost << '\n';
    for (int vertex = 0; vertex < n; vertex++) {
        if (vertex) fast_output << ' ';
        fast_output << answer->parent[vertex];
    }
    fast_output << '\n';
}
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