Prim(最小全域木) (graph/mst/prim.hpp)

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• Last update: 2024-05-01 23:44:47+09:00
• Include: #include "graph/mst/prim.hpp"

概要

最小全域木(全域木のうち, その辺群の重みの総和が最小になる木)を求める. 既に到達した頂点集合からまだ到達していない頂点集合への辺のうち, コストが最小のものを選んでいくことによって, 最小全域木を構成している.

• prim(g): 連結な重み付きグラフ g の最小全域木を求める. cost には辺の重みの総和, edges にはそれを構成する辺が格納される.

計算量

• $O(E \log V)$

Depends on

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

Verified with

• test/verify/aoj-grl-2-a.test.cpp

Code

#pragma once

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

/**
 * @brief Prim(最小全域木)
 *
 */
template <typename T>
struct MinimumSpanningTree {
  T cost;
  Edges<T> edges;
};

template <typename T>
MinimumSpanningTree<T> prim(const Graph<T> &g) {
  T total = T();
  vector<int> used(g.size());
  vector<Edge<T> *> dist(g.size());
  using pi = pair<T, int>;
  priority_queue<pi, vector<pi>, greater<> > que;
  que.emplace(T(), 0);
  Edges<T> edges;
  while (!que.empty()) {
    auto p = que.top();
    que.pop();
    if (used[p.second]) continue;
    used[p.second] = true;
    total += p.first;
    if (dist[p.second]) edges.emplace_back(*dist[p.second]);
    for (auto &e : g[p.second]) {
      if (used[e.to] || (dist[e.to] && dist[e.to]->cost <= e.cost)) continue;
      que.emplace(e.cost, e.to);
    }
  }
  return {total, edges};
}

#line 2 "graph/mst/prim.hpp"

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

/**
 * @brief Graph Template(グラフテンプレート)
 */
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 4 "graph/mst/prim.hpp"

/**
 * @brief Prim(最小全域木)
 *
 */
template <typename T>
struct MinimumSpanningTree {
  T cost;
  Edges<T> edges;
};

template <typename T>
MinimumSpanningTree<T> prim(const Graph<T> &g) {
  T total = T();
  vector<int> used(g.size());
  vector<Edge<T> *> dist(g.size());
  using pi = pair<T, int>;
  priority_queue<pi, vector<pi>, greater<> > que;
  que.emplace(T(), 0);
  Edges<T> edges;
  while (!que.empty()) {
    auto p = que.top();
    que.pop();
    if (used[p.second]) continue;
    used[p.second] = true;
    total += p.first;
    if (dist[p.second]) edges.emplace_back(*dist[p.second]);
    for (auto &e : g[p.second]) {
      if (used[e.to] || (dist[e.to] && dist[e.to]->cost <= e.cost)) continue;
      que.emplace(e.cost, e.to);
    }
  }
  return {total, edges};
}

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