Extreme Vertex Set (graph/others/extreme-vertex-set.hpp)

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• Last update: 2024-07-16 02:45:20+09:00
• Include: #include "graph/others/extreme-vertex-set.hpp"

重み付き無向グラフ $G$ が与えられたとき、$G$ の Extreme Vertex Set（極点集合）をすべて列挙します。

extreme_vertex_set

template<typename T>
Graph<T> extreme_vertex_set(int n, const Edges<T> &es)

頂点数 $n$、辺が es からなる $G$ の Extreme Vertex Set を返します。

$2n - 1$ 頂点の根付きで表されます。根は $2n - 2$ です。頂点 $[0, n)$ は葉で、もとのグラフの頂点に対応します。それぞれの部分木が Extreme Vertex Set の候補に対応し、部分木からその親に生える辺の重みが、カットのコストです。

制約

• $G$ は頂点数 $n$ からなる非負重み付き単純無向グラフ

計算量

• $O(nm + n^2 \log n)$

Depends on

• Graph Template(グラフテンプレート) (graph/graph-template.hpp)

Required by

• Global Minimum Cut of Dynamic Star Augmented Graph (graph/flow/global-minimum-cut-of-dynamic-star-augmented-graph.hpp)

Verified with

• test/verify/yosupo-global-minimum-cut-of-dynamic-star-augmented-graph.test.cpp

Code

#include "../graph-template.hpp"

template <typename T>
Graph<T> extreme_vertex_set(int n, const Edges<T> &es) {
  for (auto &e : es) {
    assert(0 <= e.from and e.from < n);
    assert(0 <= e.to and e.to < n);
    assert(e.from != e.to);
    assert(0 <= e.cost);
  }
  using pi = pair<int, T>;
  Graph<T> res(2 * n - 1);
  vector<int> uf(n);
  vector<T> cur(2 * n - 1);
  iota(uf.begin(), uf.end(), 0);
  vector<bool> leaf(2 * n - 1);
  for (int i = 0; i < n; i++) {
    leaf[i] = true;
  }
  using qi = pair<T, int>;
  priority_queue<qi, vector<qi>, greater<> > que;
  for (int phase = 0; phase < n - 1; phase++) {
    Graph<T> g(2 * n - 1);
    vector<T> cost(2 * n - 1);
    for (auto e : es) {
      e.from = uf[e.from];
      e.to = uf[e.to];
      if (e.from != e.to) {
        cost[e.from] += e.cost;
        cost[e.to] += e.cost;
        g.add_edge(e.from, e.to, e.cost);
      }
    }
    for (int i = 0; i < 2 * n - 1; i++) {
      if (leaf[i]) {
        cur[i] = cost[i];
        que.emplace(cost[i], i);
      }
    }
    int x = -1, y = -1;
    while (not que.empty()) {
      auto [c, v] = que.top();
      que.pop();
      if (cur[v] == -1) {
        continue;
      }
      cur[v] = -1;
      y = x;
      x = v;
      for (auto &e : g[v]) {
        if (cur[e.to] != -1) {
          cur[e.to] -= e.cost;
          que.emplace(cur[e.to], e.to);
        }
      }
    }
    int z = n + phase;
    res.add_directed_edge(z, x, cost[x]);
    res.add_directed_edge(z, y, cost[y]);
    for (int i = 0; i < n; i++) {
      if (uf[i] == x or uf[i] == y) {
        uf[i] = z;
      }
    }
    leaf[x] = false;
    leaf[y] = false;
    leaf[z] = true;
  }
  return res;
}

#line 2 "graph/graph-template.hpp"

template <typename T = int>
struct Edge {
  int from, to;
  T cost;
  int idx;

  Edge() = default;

  Edge(int from, int to, T cost = 1, int idx = -1)
      : from(from), to(to), cost(cost), idx(idx) {}

  operator int() const { return to; }
};

template <typename T = int>
struct Graph {
  vector<vector<Edge<T> > > g;
  int es;

  Graph() = default;

  explicit Graph(int n) : g(n), es(0) {}

  size_t size() const { return g.size(); }

  void add_directed_edge(int from, int to, T cost = 1) {
    g[from].emplace_back(from, to, cost, es++);
  }

  void add_edge(int from, int to, T cost = 1) {
    g[from].emplace_back(from, to, cost, es);
    g[to].emplace_back(to, from, cost, es++);
  }

  void read(int M, int padding = -1, bool weighted = false,
            bool directed = false) {
    for (int i = 0; i < M; i++) {
      int a, b;
      cin >> a >> b;
      a += padding;
      b += padding;
      T c = T(1);
      if (weighted) cin >> c;
      if (directed)
        add_directed_edge(a, b, c);
      else
        add_edge(a, b, c);
    }
  }

  inline vector<Edge<T> > &operator[](const int &k) { return g[k]; }

  inline const vector<Edge<T> > &operator[](const int &k) const { return g[k]; }
};

template <typename T = int>
using Edges = vector<Edge<T> >;
#line 2 "graph/others/extreme-vertex-set.hpp"

template <typename T>
Graph<T> extreme_vertex_set(int n, const Edges<T> &es) {
  for (auto &e : es) {
    assert(0 <= e.from and e.from < n);
    assert(0 <= e.to and e.to < n);
    assert(e.from != e.to);
    assert(0 <= e.cost);
  }
  using pi = pair<int, T>;
  Graph<T> res(2 * n - 1);
  vector<int> uf(n);
  vector<T> cur(2 * n - 1);
  iota(uf.begin(), uf.end(), 0);
  vector<bool> leaf(2 * n - 1);
  for (int i = 0; i < n; i++) {
    leaf[i] = true;
  }
  using qi = pair<T, int>;
  priority_queue<qi, vector<qi>, greater<> > que;
  for (int phase = 0; phase < n - 1; phase++) {
    Graph<T> g(2 * n - 1);
    vector<T> cost(2 * n - 1);
    for (auto e : es) {
      e.from = uf[e.from];
      e.to = uf[e.to];
      if (e.from != e.to) {
        cost[e.from] += e.cost;
        cost[e.to] += e.cost;
        g.add_edge(e.from, e.to, e.cost);
      }
    }
    for (int i = 0; i < 2 * n - 1; i++) {
      if (leaf[i]) {
        cur[i] = cost[i];
        que.emplace(cost[i], i);
      }
    }
    int x = -1, y = -1;
    while (not que.empty()) {
      auto [c, v] = que.top();
      que.pop();
      if (cur[v] == -1) {
        continue;
      }
      cur[v] = -1;
      y = x;
      x = v;
      for (auto &e : g[v]) {
        if (cur[e.to] != -1) {
          cur[e.to] -= e.cost;
          que.emplace(cur[e.to], e.to);
        }
      }
    }
    int z = n + phase;
    res.add_directed_edge(z, x, cost[x]);
    res.add_directed_edge(z, y, cost[y]);
    for (int i = 0; i < n; i++) {
      if (uf[i] == x or uf[i] == y) {
        uf[i] = z;
      }
    }
    leaf[x] = false;
    leaf[y] = false;
    leaf[z] = true;
  }
  return res;
}

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