cp-library

C++ Library for Competitive Programming

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:warning: グラフ/二重頂点連結成分分解
(test/graph/biconnected_component.test.cpp)

Depends on

Code

/*
 * @title グラフ/二重頂点連結成分分解
 *
 * verification-helper: IGNORE
 * verification-helper: PROBLEM https://atcoder.jp/contests/nadafes2022_day2/tasks/nadafes2022_day2_h
 */

#include <algorithm>
#include <cassert>
#include <iostream>
#include <iterator>
#include <set>
#include <utility>
#include <vector>

#include "emthrm/graph/biconnected_component.hpp"
#include "emthrm/graph/edge.hpp"

int main() {
  int n, m;
  std::cin >> n >> m;
  std::vector<std::vector<emthrm::Edge<bool>>> graph(n);
  while (m--) {
    int a, b;
    std::cin >> a >> b;
    --a; --b;
    graph[a].emplace_back(a, b);
    graph[b].emplace_back(b, a);
  }
  emthrm::BiconnectedComponent<bool, true> biconnected_component(graph);
  const int x = biconnected_component.articulation_points.size();
  const int y = biconnected_component.vertices.size();
  std::sort(biconnected_component.articulation_points.begin(),
            biconnected_component.articulation_points.end());
  std::vector<std::vector<int>> block_cut_tree(x + y);
  std::vector<int> weight(x + y, 0);
  for (int i = 0; i < n; ++i) {
    if (biconnected_component.id[i] == -1) {
      const int index =
          std::distance(biconnected_component.articulation_points.begin(),
                        std::lower_bound(
                            biconnected_component.articulation_points.begin(),
                            biconnected_component.articulation_points.end(),
                            i));
      for (const int block : biconnected_component.cutpoint[i]) {
        block_cut_tree[index].emplace_back(block + x);
        block_cut_tree[block + x].emplace_back(index);
      }
      ++weight[index];
    } else {
      ++weight[biconnected_component.id[i] + x];
    }
  }
  for (int i = 0; i < x + y; ++i) {
    std::sort(block_cut_tree[i].begin(), block_cut_tree[i].end());
    block_cut_tree[i].erase(
        std::unique(block_cut_tree[i].begin(), block_cut_tree[i].end()),
        block_cut_tree[i].end());
  }
  long long ans = static_cast<long long>(n) * (n - 1) / 2 * x;
  const auto dfs = [n, x, &block_cut_tree, &weight, &ans](
      auto dfs, const int par, const int ver) -> int {
    int subtree = weight[ver];
    if (ver < x) {
      for (const int e : block_cut_tree[ver]) {
        if (e != par) {
          const int child = dfs(dfs, ver, e);
          ans -= static_cast<long long>(child) * (child - 1) / 2 + child;
          subtree += child;
        }
      }
      ans -= static_cast<long long>(n - subtree) * (n - subtree - 1) / 2
             + (n - subtree);
    } else {
      for (const int e : block_cut_tree[ver]) {
        if (e != par) subtree += dfs(dfs, ver, e);
      }
    }
    return subtree;
  };
  assert(dfs(dfs, -1, 0) == n);
  std::cout << ans << '\n';
  return 0;
}
#line 1 "test/graph/biconnected_component.test.cpp"
/*
 * @title グラフ/二重頂点連結成分分解
 *
 * verification-helper: IGNORE
 * verification-helper: PROBLEM https://atcoder.jp/contests/nadafes2022_day2/tasks/nadafes2022_day2_h
 */

#include <algorithm>
#include <cassert>
#include <iostream>
#include <iterator>
#include <set>
#include <utility>
#include <vector>

#line 1 "include/emthrm/graph/biconnected_component.hpp"



// #include <algorithm>
#line 8 "include/emthrm/graph/biconnected_component.hpp"

#line 1 "include/emthrm/graph/edge.hpp"
/**
 * @title 辺
 */

#ifndef EMTHRM_GRAPH_EDGE_HPP_
#define EMTHRM_GRAPH_EDGE_HPP_

#include <compare>

namespace emthrm {

template <typename CostType>
struct Edge {
  CostType cost;
  int src, dst;

  explicit Edge(const int src, const int dst, const CostType cost = 0)
      : cost(cost), src(src), dst(dst) {}

  auto operator<=>(const Edge& x) const = default;
};

}  // namespace emthrm

#endif  // EMTHRM_GRAPH_EDGE_HPP_
#line 1 "include/emthrm/graph/lowlink.hpp"



#line 6 "include/emthrm/graph/lowlink.hpp"

#line 1 "include/emthrm/graph/edge.hpp"
/**
 * @title 辺
 */

#ifndef EMTHRM_GRAPH_EDGE_HPP_
#define EMTHRM_GRAPH_EDGE_HPP_

#include <compare>

namespace emthrm {

template <typename CostType>
struct Edge {
  CostType cost;
  int src, dst;

  explicit Edge(const int src, const int dst, const CostType cost = 0)
      : cost(cost), src(src), dst(dst) {}

  auto operator<=>(const Edge& x) const = default;
};

}  // namespace emthrm

#endif  // EMTHRM_GRAPH_EDGE_HPP_
#line 8 "include/emthrm/graph/lowlink.hpp"

namespace emthrm {

template <typename CostType>
struct Lowlink {
  std::vector<int> order, lowlink, articulation_points;
  std::vector<Edge<CostType>> bridges;
  const std::vector<std::vector<Edge<CostType>>> graph;

  explicit Lowlink(const std::vector<std::vector<Edge<CostType>>>& graph)
      : graph(graph) {
    const int n = graph.size();
    order.assign(n, -1);
    lowlink.resize(n);
    int t = 0;
    for (int i = 0; i < n; ++i) {
      if (order[i] == -1) dfs(-1, i, &t);
    }
  }

 private:
  void dfs(const int par, const int ver, int* t) {
    order[ver] = lowlink[ver] = (*t)++;
    int num = 0;
    bool is_articulation_point = false;
    for (const Edge<CostType>& e : graph[ver]) {
      if (order[e.dst] == -1) {
        ++num;
        dfs(ver, e.dst, t);
        lowlink[ver] = std::min(lowlink[ver], lowlink[e.dst]);
        if (order[ver] <= lowlink[e.dst]) {
          is_articulation_point = true;
          if (order[ver] < lowlink[e.dst]) {
            bridges.emplace_back(std::min(ver, e.dst), std::max(ver, e.dst),
                                 e.cost);
          }
        }
      } else if (e.dst != par) {
        lowlink[ver] = std::min(lowlink[ver], order[e.dst]);
      }
    }
    if ((par == -1 && num >= 2) || (par != -1 && is_articulation_point)) {
      articulation_points.emplace_back(ver);
    }
  }
};

}  // namespace emthrm


#line 11 "include/emthrm/graph/biconnected_component.hpp"

namespace emthrm {

template <typename CostType, bool IS_FULL_VER = false>
struct BiconnectedComponent : Lowlink<CostType> {
  std::vector<int> id;
  std::vector<std::vector<int>> vertices, cutpoint;
  std::vector<std::vector<Edge<CostType>>> block;

  explicit BiconnectedComponent(
      const std::vector<std::vector<Edge<CostType>>>& graph)
      : Lowlink<CostType>(graph) {
    const int n = graph.size();
    id.assign(n, -2);
    if constexpr (IS_FULL_VER) {
      cutpoint.resize(n);
      is_articulation_point.assign(n, false);
      for (const int articulation_point : this->articulation_points) {
        is_articulation_point[articulation_point] = true;
      }
    }
    for (int i = 0; i < n; ++i) {
      if (id[i] == -2) dfs(-1, i);
    }
    // const int m = vertices.size();
    // for (int i = 0; i < m; ++i) {
    //   std::sort(block[i].begin(), block[i].end());
    // }
    // if constexpr (IS_FULL_VER) {
    //   for (int i = 0; i < m; ++i) {
    //     std::sort(vertices[i].begin(), vertices[i].end());
    //   }
    //   for (int i = 0; i < n; ++i) {
    //     std::sort(cutpoint[i].begin(), cutpoint[i].end());
    //   }
    // }
  }

 private:
  std::vector<bool> is_articulation_point;
  std::vector<Edge<CostType>> tmp;

  void dfs(const int par, const int ver) {
    id[ver] = -1;
    for (const Edge<CostType>& e : this->graph[ver]) {
      if (e.dst == par) continue;
      int src = ver, dst = e.dst;
      if (src > dst) std::swap(src, dst);
      if (id[e.dst] == -2 || this->order[e.dst] < this->order[ver]) {
        tmp.emplace_back(src, dst, e.cost);
      }
      if (id[e.dst] == -2) {
        dfs(ver, e.dst);
        if (this->lowlink[e.dst] >= this->order[ver]) {
          const int idx = block.size();
          block.emplace_back();
          std::set<int> st;
          while (true) {
            const Edge<CostType> edge = tmp.back();
            tmp.pop_back();
            block.back().emplace_back(edge);
            if constexpr (IS_FULL_VER) {
              st.emplace(edge.src);
              st.emplace(edge.dst);
            }
            if (edge.src == src && edge.dst == dst) break;
          }
          if constexpr (IS_FULL_VER) {
            vertices.emplace_back();
            for (const int el : st) {
              vertices.back().emplace_back(el);
              if (is_articulation_point[el]) {
                cutpoint[el].emplace_back(idx);
              } else {
                id[el] = idx;
              }
            }
          }
        }
      }
    }
  }
};

}  // namespace emthrm


#line 1 "include/emthrm/graph/edge.hpp"
/**
 * @title 辺
 */

#ifndef EMTHRM_GRAPH_EDGE_HPP_
#define EMTHRM_GRAPH_EDGE_HPP_

#include <compare>

namespace emthrm {

template <typename CostType>
struct Edge {
  CostType cost;
  int src, dst;

  explicit Edge(const int src, const int dst, const CostType cost = 0)
      : cost(cost), src(src), dst(dst) {}

  auto operator<=>(const Edge& x) const = default;
};

}  // namespace emthrm

#endif  // EMTHRM_GRAPH_EDGE_HPP_
#line 18 "test/graph/biconnected_component.test.cpp"

int main() {
  int n, m;
  std::cin >> n >> m;
  std::vector<std::vector<emthrm::Edge<bool>>> graph(n);
  while (m--) {
    int a, b;
    std::cin >> a >> b;
    --a; --b;
    graph[a].emplace_back(a, b);
    graph[b].emplace_back(b, a);
  }
  emthrm::BiconnectedComponent<bool, true> biconnected_component(graph);
  const int x = biconnected_component.articulation_points.size();
  const int y = biconnected_component.vertices.size();
  std::sort(biconnected_component.articulation_points.begin(),
            biconnected_component.articulation_points.end());
  std::vector<std::vector<int>> block_cut_tree(x + y);
  std::vector<int> weight(x + y, 0);
  for (int i = 0; i < n; ++i) {
    if (biconnected_component.id[i] == -1) {
      const int index =
          std::distance(biconnected_component.articulation_points.begin(),
                        std::lower_bound(
                            biconnected_component.articulation_points.begin(),
                            biconnected_component.articulation_points.end(),
                            i));
      for (const int block : biconnected_component.cutpoint[i]) {
        block_cut_tree[index].emplace_back(block + x);
        block_cut_tree[block + x].emplace_back(index);
      }
      ++weight[index];
    } else {
      ++weight[biconnected_component.id[i] + x];
    }
  }
  for (int i = 0; i < x + y; ++i) {
    std::sort(block_cut_tree[i].begin(), block_cut_tree[i].end());
    block_cut_tree[i].erase(
        std::unique(block_cut_tree[i].begin(), block_cut_tree[i].end()),
        block_cut_tree[i].end());
  }
  long long ans = static_cast<long long>(n) * (n - 1) / 2 * x;
  const auto dfs = [n, x, &block_cut_tree, &weight, &ans](
      auto dfs, const int par, const int ver) -> int {
    int subtree = weight[ver];
    if (ver < x) {
      for (const int e : block_cut_tree[ver]) {
        if (e != par) {
          const int child = dfs(dfs, ver, e);
          ans -= static_cast<long long>(child) * (child - 1) / 2 + child;
          subtree += child;
        }
      }
      ans -= static_cast<long long>(n - subtree) * (n - subtree - 1) / 2
             + (n - subtree);
    } else {
      for (const int e : block_cut_tree[ver]) {
        if (e != par) subtree += dfs(dfs, ver, e);
      }
    }
    return subtree;
  };
  assert(dfs(dfs, -1, 0) == n);
  std::cout << ans << '\n';
  return 0;
}
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