LowLink
(src/graph/low_link.hpp)

• View this file on GitHub
• Last update: 2024-11-09 01:34:39+09:00
• Include: #include "src/graph/low_link.hpp"

LowLink

無向グラフの橋や関節点を計算します．

橋とはグラフの辺であって，それを取り除くとグラフが非連結になるようなものです．
関節点とはグラフの頂点であって，それを取り除くとグラフが非連結になるようなものです．

コンストラクタ

LowLink<T> lowlink(Graph<T> g)

• $n$ 頂点 $m$ 辺の無向グラフ g に対して lowlink の初期化を行います．

計算量

• $O(n + m)$

メンバ変数

(1) vector<int> lowlink.ord
(2) vector<int> lowlink.low
(3) vector<int> lowlink.articulation
(4) vector<pair<int, int>> lowlink.bridge

• (1): 頂点 i がDFS木上で何番目に訪れたかを保持しています．
• (2): 頂点 i から後退辺を高々 $1$ 回用いて到達できる頂点 $v$ について， ord[v] の最小値を保持しています．
  後退辺は今回の場合，DFS木に含まれない辺と一致します．
• (3): g の関節点の集合を保持しています．
• (4): g の橋の集合を保持しています．

Depends on

• Graph (src/graph/graph_template.hpp)
• template (src/template/template.hpp)

Required by

• BiconnectedComponents (src/graph/biconnected_components.hpp)
• TwoEdgeConnectedComponents (src/graph/two_edge_connected_components.hpp)

Verified with

• verify/aizu_online_judge/grl/articulation_points.test.cpp
• verify/aizu_online_judge/grl/bridges.test.cpp
• verify/library_checker/graph/biconnected_components.test.cpp
• verify/library_checker/graph/two_edge_connected_components.test.cpp

Code

#pragma once
#include "../template/template.hpp"
#include "./graph_template.hpp"
template <typename T>
struct LowLink {
    vector<int> ord, low, articulation;
    vector<pair<int, int>> bridge;
    LowLink(const Graph<T>& g)
        : ord(g.size(), -1), low(g.size(), -1) {
        for(int i = 0, k = 0; i < g.size(); ++i) {
            if(ord[i] == -1) k = dfs(g, i, k, -1);
        }
    }

   private:
    int dfs(const Graph<T>& g, const int idx, int k, const int par) {
        low[idx] = (ord[idx] = k++);
        int cnt = 0;
        bool arti = false, second = false;
        for(const Edge<T>& e : g[idx]) {
            const int to = e.to;
            if(ord[to] == -1) {
                ++cnt;
                k = dfs(g, to, k, idx);
                low[idx] = min(low[idx], low[to]);
                arti |= (par != -1) and (low[to] >= ord[idx]);
                if(ord[idx] < low[to]) {
                    bridge.emplace_back(minmax(idx, to));
                }
            } else if(to != par or second) {
                low[idx] = min(low[idx], ord[to]);
            } else {
                second = true;
            }
        }
        arti |= (par == -1) and (cnt > 1);
        if(arti) articulation.emplace_back(idx);
        return k;
    }
};

#line 2 "src/template/template.hpp"
#include <bits/stdc++.h>
using namespace std;
using ll = long long;
using P = pair<long long, long long>;
#define rep(i, a, b) for(long long i = (a); i < (b); ++i)
#define rrep(i, a, b) for(long long i = (a); i >= (b); --i)
constexpr long long inf = 4e18;
struct SetupIO {
    SetupIO() {
        ios::sync_with_stdio(0);
        cin.tie(0);
        cout << fixed << setprecision(30);
    }
} setup_io;
#line 3 "src/graph/graph_template.hpp"
template <typename T>
struct Edge {
    int from, to;
    T cost;
    int idx;
    Edge()
        : from(-1), to(-1), cost(-1), idx(-1) {}
    Edge(const int from, const int to, const T& cost = 1, const int idx = -1)
        : from(from), to(to), cost(cost), idx(idx) {}
    operator int() const {
        return to;
    }
};
template <typename T>
struct Graph {
    Graph(const int N)
        : n(N), es(0), g(N) {}
    int size() const {
        return n;
    }
    int edge_size() const {
        return es;
    }
    void add_edge(const int from, const int to, const T& cost = 1) {
        assert(0 <= from and from < n);
        assert(0 <= to and to < n);
        g[from].emplace_back(from, to, cost, es);
        g[to].emplace_back(to, from, cost, es++);
    }
    void add_directed_edge(const int from, const int to, const T& cost = 1) {
        assert(0 <= from and from < n);
        assert(0 <= to and to < n);
        g[from].emplace_back(from, to, cost, es++);
    }
    inline vector<Edge<T>>& operator[](const int& k) {
        assert(0 <= k and k < n);
        return g[k];
    }
    inline const vector<Edge<T>>& operator[](const int& k) const {
        assert(0 <= k and k < n);
        return g[k];
    }

   private:
    int n, es;
    vector<vector<Edge<T>>> g;
};
template <typename T>
using Edges = vector<Edge<T>>;
#line 4 "src/graph/low_link.hpp"
template <typename T>
struct LowLink {
    vector<int> ord, low, articulation;
    vector<pair<int, int>> bridge;
    LowLink(const Graph<T>& g)
        : ord(g.size(), -1), low(g.size(), -1) {
        for(int i = 0, k = 0; i < g.size(); ++i) {
            if(ord[i] == -1) k = dfs(g, i, k, -1);
        }
    }

   private:
    int dfs(const Graph<T>& g, const int idx, int k, const int par) {
        low[idx] = (ord[idx] = k++);
        int cnt = 0;
        bool arti = false, second = false;
        for(const Edge<T>& e : g[idx]) {
            const int to = e.to;
            if(ord[to] == -1) {
                ++cnt;
                k = dfs(g, to, k, idx);
                low[idx] = min(low[idx], low[to]);
                arti |= (par != -1) and (low[to] >= ord[idx]);
                if(ord[idx] < low[to]) {
                    bridge.emplace_back(minmax(idx, to));
                }
            } else if(to != par or second) {
                low[idx] = min(low[idx], ord[to]);
            } else {
                second = true;
            }
        }
        arti |= (par == -1) and (cnt > 1);
        if(arti) articulation.emplace_back(idx);
        return k;
    }
};

Back to top page