Kruskal(最小全域木) (graph/mst/kruskal.hpp)

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• Last update: 2024-05-01 23:44:47+09:00
• Include: #include "graph/mst/kruskal.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. Union-Findを用いて辺集合にある辺を加えて閉路を作らないか判定しながら, 辺を重みが小さい順に走査する.

• kruskal(edges, V): V 頂点の連結な重み付き辺集合 edges からなる重み付き連結グラフ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)
• Union Find (structure/union-find/union-find.hpp)

Verified with

• test/verify/aoj-grl-2-a-2.test.cpp
• test/verify/yosupo-manhattanmst.test.cpp

Code

#pragma once

#include "../../structure/union-find/union-find.hpp"
#include "../graph-template.hpp"

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

template <typename T>
MinimumSpanningTree<T> kruskal(Edges<T> &edges, int V) {
  sort(begin(edges), end(edges),
       [](const Edge<T> &a, const Edge<T> &b) { return a.cost < b.cost; });
  UnionFind tree(V);
  T total = T();
  Edges<T> es;
  for (auto &e : edges) {
    if (tree.unite(e.from, e.to)) {
      es.emplace_back(e);
      total += e.cost;
    }
  }
  return {total, es};
}

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

#line 2 "structure/union-find/union-find.hpp"

struct UnionFind {
  vector<int> data;

  UnionFind() = default;

  explicit UnionFind(size_t sz) : data(sz, -1) {}

  bool unite(int x, int y) {
    x = find(x), y = find(y);
    if (x == y) return false;
    if (data[x] > data[y]) swap(x, y);
    data[x] += data[y];
    data[y] = x;
    return true;
  }

  int find(int k) {
    if (data[k] < 0) return (k);
    return data[k] = find(data[k]);
  }

  int size(int k) { return -data[find(k)]; }

  bool same(int x, int y) { return find(x) == find(y); }

  vector<vector<int> > groups() {
    int n = (int)data.size();
    vector<vector<int> > ret(n);
    for (int i = 0; i < n; i++) {
      ret[find(i)].emplace_back(i);
    }
    ret.erase(remove_if(begin(ret), end(ret),
                        [&](const vector<int> &v) { return v.empty(); }),
              end(ret));
    return ret;
  }
};
#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 5 "graph/mst/kruskal.hpp"

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

template <typename T>
MinimumSpanningTree<T> kruskal(Edges<T> &edges, int V) {
  sort(begin(edges), end(edges),
       [](const Edge<T> &a, const Edge<T> &b) { return a.cost < b.cost; });
  UnionFind tree(V);
  T total = T();
  Edges<T> es;
  for (auto &e : edges) {
    if (tree.unite(e.from, e.to)) {
      es.emplace_back(e);
      total += e.cost;
    }
  }
  return {total, es};
}

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