Library

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:heavy_check_mark: Block Cut Tree
(tree/block_cut_tree.hpp)

説明

二重連結成分分解をし、二重連結成分と関節点を結ぶ木（非連結グラフなら森）を作る。 $O(N + M)$ 頂点番号の割り当ては、関節点は何個目の関節点か、二重連結成分は関節点の個数 + 何個目の二重連結成分かになっている。

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Code

#pragma once

#include <cassert>

#include "../graph/biconnected_components.hpp"

namespace ebi {

template <class T> struct block_cut_tree : biconnected_components<T> {
  public:
    block_cut_tree(const Graph<T> &g)
        : biconnected_components<T>(g), rev(this->n, -1) {
        int cnt = 0;
        for (auto v : this->_articulation) {
            rev[v] = cnt++;
        }
        int sz = cnt + this->bc.size();
        tree.resize(sz);
        std::vector<int> last(this->n, -1);
        for (int i = cnt; i < sz; i++) {
            for (auto e : this->bc[i - cnt]) {
                for (auto v : {e.first, e.second}) {
                    if (rev[v] != -1 && rev[v] < cnt) {
                        if (std::exchange(last[v], i) != i) {
                            tree[i].emplace_back(rev[v]);
                            tree[rev[v]].emplace_back(i);
                        }
                    } else {
                        rev[v] = i;
                    }
                }
            }
        }
        groups.resize(sz);
        for (int i = 0; i < this->n; i++) {
            if (rev[i] < 0) {
                rev[i] = sz++;
                groups.emplace_back();
                tree.emplace_back();
            }
            groups[rev[i]].emplace_back(i);
        }
    }

    std::vector<std::vector<int>> bcc() {
        int cnt = this->_articulation.size();
        int sz = groups.size() - cnt;
        std::vector _bcc(sz, std::vector<int>());
        for (int i = 0; i < sz; i++) {
            _bcc[i] = groups[cnt + i];
            for (auto nv : tree[cnt + i]) {
                assert(0 <= nv && nv < cnt);
                assert(groups[nv].size() == 1);
                _bcc[i].emplace_back(groups[nv][0]);
            }
        }
        return _bcc;
    }

  private:
    std::vector<int> rev;
    std::vector<std::vector<int>> tree;
    std::vector<std::vector<int>> groups;
};

}  // namespace ebi
#line 2 "tree/block_cut_tree.hpp"

#include <cassert>

#line 2 "graph/biconnected_components.hpp"

#line 2 "graph/low_link.hpp"

#include <algorithm>
#include <utility>
#include <vector>

#line 2 "graph/base.hpp"

#line 4 "graph/base.hpp"
#include <iostream>
#include <ranges>
#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) {
        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) {
        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 {
        return edges[i];
    }

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

    const auto operator[](int i) const {
        return csr[i];
    }
    auto operator[](int i) {
        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 8 "graph/low_link.hpp"

namespace ebi {

template <class T> struct low_link {
  private:
    void dfs(int v, int par = -1) {
        static int k = 0;
        low[v] = ord[v] = k++;
        int cnt = 0;
        bool is_arti = false, is_second = false;
        for (auto e : g[v]) {
            int nv = e.to;
            if (ord[nv] == -1) {
                cnt++;
                dfs(nv, v);
                low[v] = std::min(low[v], low[nv]);
                is_arti |= (par != -1) && (low[nv] >= ord[v]);
                if (ord[v] < low[nv]) {
                    _bridge.emplace_back(std::minmax(v, nv));
                }
            } else if (nv != par || is_second) {
                low[v] = std::min(low[v], ord[nv]);
            } else {
                is_second = true;
            }
        }
        is_arti |= par == -1 && cnt > 1;
        if (is_arti) _articulation.emplace_back(v);
    }

  public:
    low_link(const Graph<T> &g) : n(g.size()), g(g), ord(n, -1), low(n) {
        for (int i = 0; i < n; i++) {
            if (ord[i] == -1) dfs(i);
        }
    }

    std::vector<int> articulation() const {
        return _articulation;
    }

    std::vector<std::pair<int, int>> bridge() const {
        return _bridge;
    }

  protected:
    int n;
    Graph<T> g;
    std::vector<int> ord, low, _articulation;
    std::vector<std::pair<int, int>> _bridge;
};

}  // namespace ebi
#line 4 "graph/biconnected_components.hpp"

namespace ebi {

template <class T> struct biconnected_components : low_link<T> {
  private:
    void dfs(int v, int par = -1) {
        used[v] = true;
        for (auto e : this->g[v]) {
            int nv = e.to;
            if (nv == par) continue;
            if (!used[nv] || this->ord[nv] < this->ord[v]) {
                tmp.emplace_back(std::minmax(v, nv));
            }
            if (!used[nv]) {
                dfs(nv, v);
                if (this->low[nv] >= this->ord[v]) {
                    bc.emplace_back();
                    while (true) {
                        auto e = tmp.back();
                        bc.back().emplace_back(e);
                        tmp.pop_back();
                        if (e.first == std::min(v, nv) &&
                            e.second == std::max(v, nv)) {
                            break;
                        }
                    }
                }
            }
        }
    }

  public:
    biconnected_components(const Graph<T> &g)
        : low_link<T>(g), used(this->n, false) {
        for (int i = 0; i < this->n; i++) {
            if (!used[i]) dfs(i);
        }
    }

  protected:
    std::vector<bool> used;
    std::vector<std::vector<std::pair<int, int>>> bc;
    std::vector<std::pair<int, int>> tmp;
};

}  // namespace ebi
#line 6 "tree/block_cut_tree.hpp"

namespace ebi {

template <class T> struct block_cut_tree : biconnected_components<T> {
  public:
    block_cut_tree(const Graph<T> &g)
        : biconnected_components<T>(g), rev(this->n, -1) {
        int cnt = 0;
        for (auto v : this->_articulation) {
            rev[v] = cnt++;
        }
        int sz = cnt + this->bc.size();
        tree.resize(sz);
        std::vector<int> last(this->n, -1);
        for (int i = cnt; i < sz; i++) {
            for (auto e : this->bc[i - cnt]) {
                for (auto v : {e.first, e.second}) {
                    if (rev[v] != -1 && rev[v] < cnt) {
                        if (std::exchange(last[v], i) != i) {
                            tree[i].emplace_back(rev[v]);
                            tree[rev[v]].emplace_back(i);
                        }
                    } else {
                        rev[v] = i;
                    }
                }
            }
        }
        groups.resize(sz);
        for (int i = 0; i < this->n; i++) {
            if (rev[i] < 0) {
                rev[i] = sz++;
                groups.emplace_back();
                tree.emplace_back();
            }
            groups[rev[i]].emplace_back(i);
        }
    }

    std::vector<std::vector<int>> bcc() {
        int cnt = this->_articulation.size();
        int sz = groups.size() - cnt;
        std::vector _bcc(sz, std::vector<int>());
        for (int i = 0; i < sz; i++) {
            _bcc[i] = groups[cnt + i];
            for (auto nv : tree[cnt + i]) {
                assert(0 <= nv && nv < cnt);
                assert(groups[nv].size() == 1);
                _bcc[i].emplace_back(groups[nv][0]);
            }
        }
        return _bcc;
    }

  private:
    std::vector<int> rev;
    std::vector<std::vector<int>> tree;
    std::vector<std::vector<int>> groups;
};

}  // namespace ebi
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