Library

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:heavy_check_mark: Directed MST
(graph/directed_mst.hpp)

説明

根と有向グラフを与えて、有向最小全域木を構築する。返り値 $(x, p)$ について、 $x$ は木の重みの総和、 $p_{i}$ は頂点 $i$ の親である。 $p_{root} = root$ である。 $O(M\log {N})$

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Code

#pragma once

#include <cassert>
#include <numeric>
#include <optional>
#include <ranges>
#include <utility>
#include <vector>

#include "../data_structure/skew_heap.hpp"
#include "../graph/base.hpp"

namespace ebi {

namespace internal {

struct union_find_not_merge_tech {
  public:
    union_find_not_merge_tech(int n_) : n(n_), data(n, -1) {}

    int leader(int u) {
        if (data[u] < 0) return u;
        return data[u] = leader(data[u]);
    }

    bool merge(int u, int v) {
        u = leader(u);
        v = leader(v);
        if (u == v) return false;
        data[u] += data[v];
        data[v] = u;
        return true;
    }

    bool same(int u, int v) {
        return leader(u) == leader(v);
    }

    int size(int u) {
        u = leader(u);
        return -data[u];
    }

  private:
    int n;
    std::vector<int> data;
};

}  // namespace internal

template <class T>
std::optional<std::pair<T, std::vector<int>>> directed_mst(int root,
                                                           const Graph<T> &g) {
    using Heap = skew_heap<T, Edge<T>, std::greater<T>>;
    int n = g.node_number();
    std::vector<int> used(2 * n, 0);
    std::vector<int> par_node(2 * n, -1);
    std::vector<int> best_id(2 * n, -1);
    std::vector<T> best_cost(2 * n);
    internal::union_find_not_merge_tech uf(2 * n);
    std::vector<Heap> heap(2 * n);
    int nxt = n;
    used[root] = 2;
    for (auto e : g.get_edges()) {
        heap[e.to].push(e.cost, e);
    }
    T sum = 0;
    for (auto s : std::views::iota(0, n)) {
        if (used[s]) continue;
        std::vector<int> path = {s};
        while (1) {
            int a = path.back();
            if (heap[a].empty()) return std::nullopt;
            auto [c, e] = heap[a].top();
            heap[a].pop();
            int v = uf.leader(e.from);
            if (a == v) continue;
            used[a] = 1;
            best_id[a] = e.id;
            best_cost[a] = c;
            sum += c;
            if (!used[v]) {
                path.emplace_back(v);
                continue;
            } else if (used[v] == 1) {
                int w = -1;
                int u = nxt++;
                while (w != v) {
                    w = path.back();
                    path.pop_back();
                    T sub = best_cost[w];
                    heap[w].add(-sub);
                    heap[u].meld(heap[w]);
                    par_node[w] = u;
                    uf.merge(u, w);
                    used[w] = 2;
                }
                path.emplace_back(u);
            } else
                break;
        }
        for (auto v : path) used[v] = 2;
    }
    std::vector<int> par(n, -1);
    std::vector<bool> done(nxt, false);
    done[root] = true;
    par[root] = root;
    T ret = 0;
    for (auto i : std::views::iota(0, nxt) | std::views::reverse) {
        if (done[i]) continue;
        int id = best_id[i];
        auto e = g.get_edge(id);
        par[e.to] = e.from;
        ret += e.cost;
        int x = e.to;
        while (x != -1 && !done[x]) {
            done[x] = true;
            x = par_node[x];
        }
    }
    assert(sum == ret);
    return std::pair<T, std::vector<int>>{sum, par};
}

}  // namespace ebi
#line 2 "graph/directed_mst.hpp"

#include <cassert>
#include <numeric>
#include <optional>
#include <ranges>
#include <utility>
#include <vector>

#line 2 "data_structure/skew_heap.hpp"

#line 4 "data_structure/skew_heap.hpp"
#include <functional>
#include <memory>
#line 7 "data_structure/skew_heap.hpp"

namespace ebi {

template <class Key, class T, class Compare = std::less<Key>> struct skew_heap {
  private:
    using value_type = Key;
    using Self = skew_heap<Key, T, Compare>;

    struct Node;
    using iterator = std::shared_ptr<Node>;

    struct Node {
        Node(value_type x_, T info_) : x(x_), info(info_) {}

        void propagate() {
            if (lhs) lhs->lazy += lazy;
            if (rhs) rhs->lazy += lazy;
            x += lazy;
            lazy = value_type();
        }

        value_type x;
        T info;
        value_type lazy = value_type();
        iterator lhs = nullptr, rhs = nullptr;
    };

    iterator internal_meld(iterator lhs, iterator rhs) {
        if (lhs == nullptr) return rhs;
        if (rhs == nullptr) return lhs;
        lhs->propagate();
        rhs->propagate();
        if (Compare()(lhs->x, rhs->x)) {
            std::swap(lhs, rhs);
        }
        lhs->rhs = internal_meld(lhs->rhs, rhs);
        std::swap(lhs->lhs, lhs->rhs);
        return lhs;
    }

  public:
    skew_heap() = default;

    void push(value_type x, T info) {
        root = internal_meld(root, std::make_shared<Node>(x, info));
        sz++;
    }

    void pop() {
        assert(!empty());
        root = internal_meld(root->lhs, root->rhs);
        sz--;
    }

    void meld(Self &heap) {
        root = internal_meld(root, heap.root);
        sz += heap.sz;
    }

    void add(value_type lazy) {
        if (root == nullptr) return;
        root->lazy += lazy;
        root->propagate();
    }

    bool empty() const {
        return root == nullptr;
    }

    std::pair<value_type, T> top() const {
        return {root->x, root->info};
    }

    int size() const {
        return sz;
    }

  private:
    iterator root = nullptr;
    int sz = 0;
};

}  // namespace ebi
#line 2 "graph/base.hpp"

#line 4 "graph/base.hpp"
#include <iostream>
#line 7 "graph/base.hpp"

#line 2 "data_structure/simple_csr.hpp"

#line 6 "data_structure/simple_csr.hpp"

namespace ebi {

template <class E> struct simple_csr {
    simple_csr() = default;

    simple_csr(int n, const std::vector<std::pair<int, E>>& elements)
        : start(n + 1, 0), elist(elements.size()) {
        for (auto e : elements) {
            start[e.first + 1]++;
        }
        for (auto i : std::views::iota(0, n)) {
            start[i + 1] += start[i];
        }
        auto counter = start;
        for (auto [i, e] : elements) {
            elist[counter[i]++] = e;
        }
    }

    simple_csr(const std::vector<std::vector<E>>& es)
        : start(es.size() + 1, 0) {
        int n = es.size();
        for (auto i : std::views::iota(0, n)) {
            start[i + 1] = (int)es[i].size() + start[i];
        }
        elist.resize(start.back());
        for (auto i : std::views::iota(0, n)) {
            std::copy(es[i].begin(), es[i].end(), elist.begin() + start[i]);
        }
    }

    int size() const {
        return (int)start.size() - 1;
    }

    const auto operator[](int i) const {
        return std::ranges::subrange(elist.begin() + start[i],
                                     elist.begin() + start[i + 1]);
    }
    auto operator[](int i) {
        return std::ranges::subrange(elist.begin() + start[i],
                                     elist.begin() + start[i + 1]);
    }

    const auto operator()(int i, int l, int r) const {
        return std::ranges::subrange(elist.begin() + start[i] + l,
                                     elist.begin() + start[i + 1] + r);
    }
    auto operator()(int i, int l, int r) {
        return std::ranges::subrange(elist.begin() + start[i] + l,
                                     elist.begin() + start[i + 1] + r);
    }

  private:
    std::vector<int> start;
    std::vector<E> elist;
};

}  // namespace ebi
#line 9 "graph/base.hpp"

namespace ebi {

template <class T> struct Edge {
    int from, to;
    T cost;
    int id;
};

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

    Graph(int n_) : n(n_) {}

    Graph() = default;

    void add_edge(int u, int v, cost_type c) {
        assert(!prepared && u < n && v < n);
        buff.emplace_back(u, edge_type{u, v, c, m});
        edges.emplace_back(edge_type{u, v, c, m++});
    }

    void add_undirected_edge(int u, int v, cost_type c) {
        assert(!prepared && u < n && v < n);
        buff.emplace_back(u, edge_type{u, v, c, m});
        buff.emplace_back(v, edge_type{v, u, c, m});
        edges.emplace_back(edge_type{u, v, c, m});
        m++;
    }

    void read_tree(int offset = 1, bool is_weighted = false) {
        read_graph(n - 1, offset, false, is_weighted);
    }

    void read_parents(int offset = 1) {
        for (auto i : std::views::iota(1, n)) {
            int p;
            std::cin >> p;
            p -= offset;
            add_undirected_edge(p, i, 1);
        }
        build();
    }

    void read_graph(int e, int offset = 1, bool is_directed = false,
                    bool is_weighted = false) {
        for (int i = 0; i < e; i++) {
            int u, v;
            std::cin >> u >> v;
            u -= offset;
            v -= offset;
            if (is_weighted) {
                cost_type c;
                std::cin >> c;
                if (is_directed) {
                    add_edge(u, v, c);
                } else {
                    add_undirected_edge(u, v, c);
                }
            } else {
                if (is_directed) {
                    add_edge(u, v, 1);
                } else {
                    add_undirected_edge(u, v, 1);
                }
            }
        }
        build();
    }

    void build() {
        assert(!prepared);
        csr = simple_csr<edge_type>(n, buff);
        buff.clear();
        prepared = true;
    }

    int size() const {
        return n;
    }

    int node_number() const {
        return n;
    }

    int edge_number() const {
        return m;
    }

    edge_type get_edge(int i) const {
        assert(prepared);
        return edges[i];
    }

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

    const auto operator[](int i) const {
        assert(prepared);
        return csr[i];
    }
    auto operator[](int i) {
        assert(prepared);
        return csr[i];
    }

  private:
    int n, m = 0;

    std::vector<std::pair<int, edge_type>> buff;

    std::vector<edge_type> edges;
    simple_csr<edge_type> csr;
    bool prepared = false;
};

}  // namespace ebi
#line 12 "graph/directed_mst.hpp"

namespace ebi {

namespace internal {

struct union_find_not_merge_tech {
  public:
    union_find_not_merge_tech(int n_) : n(n_), data(n, -1) {}

    int leader(int u) {
        if (data[u] < 0) return u;
        return data[u] = leader(data[u]);
    }

    bool merge(int u, int v) {
        u = leader(u);
        v = leader(v);
        if (u == v) return false;
        data[u] += data[v];
        data[v] = u;
        return true;
    }

    bool same(int u, int v) {
        return leader(u) == leader(v);
    }

    int size(int u) {
        u = leader(u);
        return -data[u];
    }

  private:
    int n;
    std::vector<int> data;
};

}  // namespace internal

template <class T>
std::optional<std::pair<T, std::vector<int>>> directed_mst(int root,
                                                           const Graph<T> &g) {
    using Heap = skew_heap<T, Edge<T>, std::greater<T>>;
    int n = g.node_number();
    std::vector<int> used(2 * n, 0);
    std::vector<int> par_node(2 * n, -1);
    std::vector<int> best_id(2 * n, -1);
    std::vector<T> best_cost(2 * n);
    internal::union_find_not_merge_tech uf(2 * n);
    std::vector<Heap> heap(2 * n);
    int nxt = n;
    used[root] = 2;
    for (auto e : g.get_edges()) {
        heap[e.to].push(e.cost, e);
    }
    T sum = 0;
    for (auto s : std::views::iota(0, n)) {
        if (used[s]) continue;
        std::vector<int> path = {s};
        while (1) {
            int a = path.back();
            if (heap[a].empty()) return std::nullopt;
            auto [c, e] = heap[a].top();
            heap[a].pop();
            int v = uf.leader(e.from);
            if (a == v) continue;
            used[a] = 1;
            best_id[a] = e.id;
            best_cost[a] = c;
            sum += c;
            if (!used[v]) {
                path.emplace_back(v);
                continue;
            } else if (used[v] == 1) {
                int w = -1;
                int u = nxt++;
                while (w != v) {
                    w = path.back();
                    path.pop_back();
                    T sub = best_cost[w];
                    heap[w].add(-sub);
                    heap[u].meld(heap[w]);
                    par_node[w] = u;
                    uf.merge(u, w);
                    used[w] = 2;
                }
                path.emplace_back(u);
            } else
                break;
        }
        for (auto v : path) used[v] = 2;
    }
    std::vector<int> par(n, -1);
    std::vector<bool> done(nxt, false);
    done[root] = true;
    par[root] = root;
    T ret = 0;
    for (auto i : std::views::iota(0, nxt) | std::views::reverse) {
        if (done[i]) continue;
        int id = best_id[i];
        auto e = g.get_edge(id);
        par[e.to] = e.from;
        ret += e.cost;
        int x = e.to;
        while (x != -1 && !done[x]) {
            done[x] = true;
            x = par_node[x];
        }
    }
    assert(sum == ret);
    return std::pair<T, std::vector<int>>{sum, par};
}

}  // namespace ebi
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