minimum spanning tree / 最小全域木 (Kruskal's method)
(graph/kruskal.hpp)

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• Last update: 2021-08-30 04:35:37+09:00
• Include: #include "graph/kruskal.hpp"

Depends on

• Union-Find Tree (data_structure/union_find_tree.hpp)
• utils/macros.hpp

Verified with

• graph/kruskal.aoj.test.cpp

Code

#pragma once
#include "../data_structure/union_find_tree.hpp"
#include "../utils/macros.hpp"
#include <algorithm>
#include <numeric>
#include <tuple>
#include <vector>

/**
 * @brief minimum spanning tree / 最小全域木 (Kruskal's method)
 * @note $O(E \log E)$
 * @note it becomes a forest if the given graph is not connected
 * @return a list of indices of edges
 */
template <typename T>
std::vector<int> compute_minimum_spanning_tree(int n, std::vector<std::tuple<int, int, T> > edges) {
    std::vector<int> order(edges.size());
    std::iota(ALL(order), 0);
    std::sort(ALL(order), [&](int i, int j) {
        return std::make_pair(std::get<2>(edges[i]), i) < std::make_pair(std::get<2>(edges[j]), j);
    });
    std::vector<int> tree;
    union_find_tree uft(n);
    for (int i : order) {
        int x = std::get<0>(edges[i]);
        int y = std::get<1>(edges[i]);
        if (not uft.is_same(x, y)) {
            uft.unite_trees(x, y);
            tree.push_back(i);
        }
    }
    return tree;
}

#line 2 "data_structure/union_find_tree.hpp"
#include <vector>

/**
 * @brief Union-Find Tree
 * @docs data_structure/union_find_tree.md
 * @note union-by-size + path-compression
 */
struct union_find_tree {
    std::vector<int> data;
    union_find_tree() = default;
    explicit union_find_tree(std::size_t n) : data(n, -1) {}
    bool is_root(int i) { return data[i] < 0; }
    int find_root(int i) { return is_root(i) ? i : (data[i] = find_root(data[i])); }
    int tree_size(int i) { return - data[find_root(i)]; }
    int unite_trees(int i, int j) {
        i = find_root(i); j = find_root(j);
        if (i != j) {
            if (tree_size(i) < tree_size(j)) std::swap(i, j);
            data[i] += data[j];
            data[j] = i;
        }
        return i;
    }
    bool is_same(int i, int j) { return find_root(i) == find_root(j); }
};
#line 2 "utils/macros.hpp"
#define REP(i, n) for (int i = 0; (i) < (int)(n); ++ (i))
#define REP3(i, m, n) for (int i = (m); (i) < (int)(n); ++ (i))
#define REP_R(i, n) for (int i = (int)(n) - 1; (i) >= 0; -- (i))
#define REP3R(i, m, n) for (int i = (int)(n) - 1; (i) >= (int)(m); -- (i))
#define ALL(x) std::begin(x), std::end(x)
#line 4 "graph/kruskal.hpp"
#include <algorithm>
#include <numeric>
#include <tuple>
#line 8 "graph/kruskal.hpp"

/**
 * @brief minimum spanning tree / 最小全域木 (Kruskal's method)
 * @note $O(E \log E)$
 * @note it becomes a forest if the given graph is not connected
 * @return a list of indices of edges
 */
template <typename T>
std::vector<int> compute_minimum_spanning_tree(int n, std::vector<std::tuple<int, int, T> > edges) {
    std::vector<int> order(edges.size());
    std::iota(ALL(order), 0);
    std::sort(ALL(order), [&](int i, int j) {
        return std::make_pair(std::get<2>(edges[i]), i) < std::make_pair(std::get<2>(edges[j]), j);
    });
    std::vector<int> tree;
    union_find_tree uft(n);
    for (int i : order) {
        int x = std::get<0>(edges[i]);
        int y = std::get<1>(edges[i]);
        if (not uft.is_same(x, y)) {
            uft.unite_trees(x, y);
            tree.push_back(i);
        }
    }
    return tree;
}

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