グラフ/二部グラフ判定
(test/graph/is_bipartite.test.cpp)

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• Last update: 2023-07-21 12:14:05+09:00
• Problem: https://atcoder.jp/contests/arc099/tasks/arc099_c

Depends on

• union-find (include/emthrm/data_structure/union-find/union-find.hpp)
• 辺 (include/emthrm/graph/edge.hpp)
• 二部グラフ (bipartite graph) 判定 (include/emthrm/graph/is_bipartite.hpp)

Code

/*
 * @title グラフ/二部グラフ判定
 *
 * verification-helper: IGNORE
 * verification-helper: PROBLEM https://atcoder.jp/contests/arc099/tasks/arc099_c
 */

#include <algorithm>
#include <iostream>
#include <map>
#include <vector>

#include "emthrm/data_structure/union-find/union-find.hpp"
#include "emthrm/graph/edge.hpp"
#include "emthrm/graph/is_bipartite.hpp"

int main() {
  int n, m;
  std::cin >> n >> m;
  int ans = m;
  std::vector<std::vector<bool>> is_adjacent(n, std::vector<bool>(n, false));
  while (m--) {
    int a, b;
    std::cin >> a >> b;
    --a; --b;
    is_adjacent[a][b] = true;
    is_adjacent[b][a] = true;
  }
  emthrm::UnionFind union_find(n);
  std::vector<std::vector<emthrm::Edge<bool>>> graph(n);
  for (int i = 0; i < n; ++i) {
    for (int j = i + 1; j < n; ++j) {
      if (!is_adjacent[i][j]) {
        union_find.unite(i, j);
        graph[i].emplace_back(i, j);
        graph[j].emplace_back(j, i);
      }
    }
  }
  const std::vector<int> color = emthrm::is_bipartite(graph);
  if (color.empty()) {
    std::cout << "-1\n";
    return 0;
  }
  std::vector<bool> dp(n + 1, false);
  dp[0] = true;
  std::map<int, int> mp;
  for (int i = 0; i < n; ++i) {
    mp[union_find.root(i)] += color[i];
  }
  for (const auto& [root, size] : mp) {
    for (int i = n; i >= 0; --i) {
      if (dp[i]) {
        dp[i] = false;
        if (i + size <= n) dp[i + size] = true;
        if (i + union_find.size(root) - size <= n) {
          dp[i + union_find.size(root) - size] = true;
        }
      }
    }
  }
  for (int i = 0; i <= n; ++i) {
    if (dp[i]) ans = std::min(ans, i * (i - 1) / 2 + (n - i) * (n - i - 1) / 2);
  }
  std::cout << ans << '\n';
  return 0;
}

#line 1 "test/graph/is_bipartite.test.cpp"
/*
 * @title グラフ/二部グラフ判定
 *
 * verification-helper: IGNORE
 * verification-helper: PROBLEM https://atcoder.jp/contests/arc099/tasks/arc099_c
 */

#include <algorithm>
#include <iostream>
#include <map>
#include <vector>

#line 1 "include/emthrm/data_structure/union-find/union-find.hpp"



#include <utility>
#line 6 "include/emthrm/data_structure/union-find/union-find.hpp"

namespace emthrm {

struct UnionFind {
  explicit UnionFind(const int n) : data(n, -1) {}

  int root(const int ver) {
    return data[ver] < 0 ? ver : data[ver] = root(data[ver]);
  }

  bool unite(int u, int v) {
    u = root(u);
    v = root(v);
    if (u == v) return false;
    if (data[u] > data[v]) std::swap(u, v);
    data[u] += data[v];
    data[v] = u;
    return true;
  }

  bool is_same(const int u, const int v) { return root(u) == root(v); }

  int size(const int ver) { return -data[root(ver)]; }

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

}  // 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 1 "include/emthrm/graph/is_bipartite.hpp"



#include <ranges>
#line 6 "include/emthrm/graph/is_bipartite.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/is_bipartite.hpp"

namespace emthrm {

template <typename CostType>
std::vector<int> is_bipartite(
    const std::vector<std::vector<Edge<CostType>>>& graph) {
  const int n = graph.size();
  std::vector<int> color(n, -1);
  const auto dfs = [&graph, &color](auto dfs, const int ver, const int c)
      -> bool {
    color[ver] = c;
    for (const int e : graph[ver]
                     | std::views::transform(&Edge<CostType>::dst)) {
      if (color[e] == c || (color[e] == -1 && !dfs(dfs, e, c ^ 1))) {
        return false;
      }
    }
    return true;
  };
  for (int i = 0; i < n; ++i) {
    if (color[i] == -1 && !dfs(dfs, i, 0)) return std::vector<int>{};
  }
  return color;
}

}  // namespace emthrm


#line 16 "test/graph/is_bipartite.test.cpp"

int main() {
  int n, m;
  std::cin >> n >> m;
  int ans = m;
  std::vector<std::vector<bool>> is_adjacent(n, std::vector<bool>(n, false));
  while (m--) {
    int a, b;
    std::cin >> a >> b;
    --a; --b;
    is_adjacent[a][b] = true;
    is_adjacent[b][a] = true;
  }
  emthrm::UnionFind union_find(n);
  std::vector<std::vector<emthrm::Edge<bool>>> graph(n);
  for (int i = 0; i < n; ++i) {
    for (int j = i + 1; j < n; ++j) {
      if (!is_adjacent[i][j]) {
        union_find.unite(i, j);
        graph[i].emplace_back(i, j);
        graph[j].emplace_back(j, i);
      }
    }
  }
  const std::vector<int> color = emthrm::is_bipartite(graph);
  if (color.empty()) {
    std::cout << "-1\n";
    return 0;
  }
  std::vector<bool> dp(n + 1, false);
  dp[0] = true;
  std::map<int, int> mp;
  for (int i = 0; i < n; ++i) {
    mp[union_find.root(i)] += color[i];
  }
  for (const auto& [root, size] : mp) {
    for (int i = n; i >= 0; --i) {
      if (dp[i]) {
        dp[i] = false;
        if (i + size <= n) dp[i + size] = true;
        if (i + union_find.size(root) - size <= n) {
          dp[i + union_find.size(root) - size] = true;
        }
      }
    }
  }
  for (int i = 0; i <= n; ++i) {
    if (dp[i]) ans = std::min(ans, i * (i - 1) / 2 + (n - i) * (n - i - 1) / 2);
  }
  std::cout << ans << '\n';
  return 0;
}

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