Low Link (関節点・橋)
(graph/low_link.hpp)

• View this file on GitHub
• Last update: 2024-01-25 10:46:02+09:00
• Include: #include "graph/low_link.hpp"

• LowLink llink(G); で作成
• llink.articulations に関節点が、llink.bridges に橋となる辺が含まれる

Depends on

• graph/graph_template.hpp

Verified with

• verify/aoj_grl/aoj_grl_3_a.test.cpp
• verify/aoj_grl/aoj_grl_3_b.test.cpp

Code

#pragma once

#include "graph/graph_template.hpp"

template <class T> struct LowLink {
    int n;
    std::vector<int> ord, low;
    std::vector<int> articulations;
    std::vector<int> roots;
    std::vector<std::pair<int, int>> bridges;  // edges {u, v} (u < v)
    std::vector<std::vector<int>> dfs_tree;

    LowLink(const Graph<T>& g) : n(int(g.size())) {
        ord.assign(n, -1);
        low.assign(n, -1);
        dfs_tree.resize(n);
        int ord_id = 0;
        auto dfs = [&](auto f, int cur, int par) -> void {
            low[cur] = ord[cur] = ord_id++;
            bool is_articulation = false;
            for (auto& e : g[cur]) {
                if (ord[e.to] == -1) {
                    // DFS 木上の辺に対する処理
                    f(f, e.to, cur);
                    dfs_tree[cur].push_back(e.to);
                    // e が DFS 木に含まれているので後退辺をすでに通った low[e.to] を使って更新して良い
                    low[cur] = std::min(low[cur], low[e.to]);
                    is_articulation |= (par != -1) and (ord[cur] <= low[e.to]);
                    if (ord[cur] < low[e.to]) {
                        bridges.emplace_back(std::minmax(cur, e.to));
                    }
                } else if (e.to != par) {
                    // 後退辺に対する処理
                    // Todo: multiple edges
                    low[cur] = std::min(low[cur], ord[e.to]);
                }
            }
            is_articulation |= par == -1 and int(dfs_tree[cur].size()) > 1;
            if (is_articulation) articulations.push_back(cur);
            return;
        };
        for (int i = 0; i < n; i++) {
            if (ord[i] == -1) {
                roots.push_back(i);
                dfs(dfs, i, -1);
            }
        }
    }
    // 連結成分数
    int count_components() { return int(roots.size()); }
    // 頂点 x を取り除くともともと 1 つだった連結成分がいくつになるか
    int count_components_remove(int x) {
        if (std::binary_search(roots.begin(), roots.end(), x)) {
            int c = int(dfs_tree[x].size());
            return c;
        } else {
            int c = 0;
            for (auto& e : dfs_tree[x]) {
                if (ord[x] <= low[e]) c++;
            }
            // 親の分で +1
            return c + 1;
        }
    }
};

#line 2 "graph/low_link.hpp"

#line 2 "graph/graph_template.hpp"

#include <vector>
template <class T> struct Edge {
    int from, to;
    T cost;
    int id;

    Edge() = default;
    Edge(int from, int to, T cost = 1, int id = -1) : from(from), to(to), cost(cost), id(id) {}

    friend std::ostream &operator<<(std::ostream &os, const Edge<T> &e) {
        // output format: "{ id : from -> to, cost }"
        return os << "{ " << e.id << " : " << e.from << " -> " << e.to << ", " << e.cost << " }";
    }
};

template <class T> using Edges = std::vector<Edge<T>>;
template <class T> using Graph = std::vector<std::vector<Edge<T>>>;
#line 4 "graph/low_link.hpp"

template <class T> struct LowLink {
    int n;
    std::vector<int> ord, low;
    std::vector<int> articulations;
    std::vector<int> roots;
    std::vector<std::pair<int, int>> bridges;  // edges {u, v} (u < v)
    std::vector<std::vector<int>> dfs_tree;

    LowLink(const Graph<T>& g) : n(int(g.size())) {
        ord.assign(n, -1);
        low.assign(n, -1);
        dfs_tree.resize(n);
        int ord_id = 0;
        auto dfs = [&](auto f, int cur, int par) -> void {
            low[cur] = ord[cur] = ord_id++;
            bool is_articulation = false;
            for (auto& e : g[cur]) {
                if (ord[e.to] == -1) {
                    // DFS 木上の辺に対する処理
                    f(f, e.to, cur);
                    dfs_tree[cur].push_back(e.to);
                    // e が DFS 木に含まれているので後退辺をすでに通った low[e.to] を使って更新して良い
                    low[cur] = std::min(low[cur], low[e.to]);
                    is_articulation |= (par != -1) and (ord[cur] <= low[e.to]);
                    if (ord[cur] < low[e.to]) {
                        bridges.emplace_back(std::minmax(cur, e.to));
                    }
                } else if (e.to != par) {
                    // 後退辺に対する処理
                    // Todo: multiple edges
                    low[cur] = std::min(low[cur], ord[e.to]);
                }
            }
            is_articulation |= par == -1 and int(dfs_tree[cur].size()) > 1;
            if (is_articulation) articulations.push_back(cur);
            return;
        };
        for (int i = 0; i < n; i++) {
            if (ord[i] == -1) {
                roots.push_back(i);
                dfs(dfs, i, -1);
            }
        }
    }
    // 連結成分数
    int count_components() { return int(roots.size()); }
    // 頂点 x を取り除くともともと 1 つだった連結成分がいくつになるか
    int count_components_remove(int x) {
        if (std::binary_search(roots.begin(), roots.end(), x)) {
            int c = int(dfs_tree[x].size());
            return c;
        } else {
            int c = 0;
            for (auto& e : dfs_tree[x]) {
                if (ord[x] <= low[e]) c++;
            }
            // 親の分で +1
            return c + 1;
        }
    }
};

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