Kruskal
(graph/kruskal.hpp)

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• Last update: 2026-08-13 01:41:40+09:00
• Include: #include "graph/kruskal.hpp"

Overview

Kruskal’s algorithm for a minimum spanning forest of an undirected weighted graph.

The algorithm sorts edges by cost and adds them one by one if they connect two different DSU components. This greedily builds a minimum-cost set of edges that connects each connected component.

Use it when you need a minimum spanning tree (connected graph) or minimum spanning forest (possibly disconnected graph). It is especially convenient when the input is an edge list or when $M \log M$ is acceptable.

Graph Orientation

Undirected only. Build the graph with add_edge. A directed edge does not represent the usual MST problem.

How to Use It

Build an undirected weighted graph with add_edge, then call kruskal(g). Although the function only looks at edge endpoints and costs, using add_directed_edge usually does not represent a normal MST problem.

The result contains these members:

Member	Type / Signature	Meaning
cost	T	Total cost of the selected forest.
edges	std::vector<Edge<T>>	Selected edges.
components	int	Number of connected components left after selecting edges.
is_spanning_tree	bool is_spanning_tree(int n) const	Returns whether the result is one spanning tree on n vertices.

If the graph is disconnected, the result is a minimum spanning forest and components will be greater than 1.

Functions

Function	Signature	Description	Complexity
kruskal	template <class T> MinimumSpanningForest<T> kruskal(const Graph<T>& g)	Returns total cost, selected edges, and component count.	$O(M \log M)$

Example

#include "graph/graph.hpp"
#include "graph/kruskal.hpp"
#include <iostream>

int main() {
    m1une::graph::Graph<long long> g(4);
    g.add_edge(0, 1, 1);
    g.add_edge(1, 2, 2);
    g.add_edge(2, 3, 3);
    g.add_edge(0, 3, 10);

    auto mst = m1une::graph::kruskal(g);
    std::cout << mst.cost << "\n";  // 6
}

Depends on

• DSU (Disjoint Set Union) (ds/dsu/dsu.hpp)
• Graph (graph/graph.hpp)

Required by

• Graph All (graph/all.hpp)
• Undirected Graph Algorithms (graph/undirected.hpp)

Verified with

• verify/graph/cow_game.test.cpp
• verify/graph/graph_algorithms.test.cpp
• verify/graph/minimum_spanning_tree.test.cpp
• verify/graph/range_edge_graph.test.cpp

Code

#ifndef M1UNE_GRAPH_KRUSKAL_HPP
#define M1UNE_GRAPH_KRUSKAL_HPP 1

#include <algorithm>
#include <vector>

#include "../ds/dsu/dsu.hpp"
#include "graph.hpp"

namespace m1une {
namespace graph {

template <class T>
struct MinimumSpanningForest {
    T cost;
    std::vector<Edge<T>> edges;
    int components;

    bool is_spanning_tree(int n) const {
        return components <= 1 && int(edges.size()) == std::max(0, n - 1);
    }
};

template <class T>
MinimumSpanningForest<T> kruskal(const Graph<T>& g) {
    int n = g.size();
    auto edges = g.edges();
    std::sort(edges.begin(), edges.end(), [](const auto& a, const auto& b) {
        return a.cost < b.cost;
    });

    m1une::ds::Dsu dsu(n);
    MinimumSpanningForest<T> result;
    result.cost = T(0);
    result.components = n;

    for (const auto& e : edges) {
        if (dsu.same(e.from, e.to)) continue;
        dsu.merge(e.from, e.to);
        result.cost += e.cost;
        result.edges.push_back(e);
        result.components--;
    }

    return result;
}

}  // namespace graph
}  // namespace m1une

#endif  // M1UNE_GRAPH_KRUSKAL_HPP

#line 1 "graph/kruskal.hpp"



#include <algorithm>
#include <vector>

#line 1 "ds/dsu/dsu.hpp"



#line 5 "ds/dsu/dsu.hpp"
#include <numeric>
#include <utility>
#line 8 "ds/dsu/dsu.hpp"

namespace m1une {
namespace ds {

struct Dsu {
   private:
    int _n;
    // parent_or_size[i] is the parent of i if it's >= 0.
    // If it's < 0, then i is a root and -parent_or_size[i] is the size of the group.
    std::vector<int> parent_or_size;

    // Returns {new leader, absorbed leader}. The absorbed leader is -1 when
    // both vertices already belong to the same component.
    std::pair<int, int> merge_leaders(int a, int b) {
        int x = leader(a), y = leader(b);
        if (x == y) return {x, -1};
        if (-parent_or_size[x] < -parent_or_size[y]) std::swap(x, y);
        parent_or_size[x] += parent_or_size[y];
        parent_or_size[y] = x;
        return {x, y};
    }

   public:
    Dsu() : _n(0) {}
    explicit Dsu(int n) : _n(n), parent_or_size(n, -1) {}

    // Merges the group containing 'a' with the group containing 'b'.
    // Returns the leader of the merged group.
    int merge(int a, int b) {
        return merge_leaders(a, b).first;
    }

    // Invokes callback(new_leader, absorbed_leader) after an actual merge.
    // Returns the leader of the merged group.
    template <class Callback>
    int merge(int a, int b, Callback&& callback) {
        std::pair<int, int> merged = merge_leaders(a, b);
        if (merged.second != -1) callback(merged.first, merged.second);
        return merged.first;
    }

    // Returns true if 'a' and 'b' belong to the same group.
    bool same(int a, int b) {
        return leader(a) == leader(b);
    }

    // Returns the leader (representative) of the group containing 'a'.
    int leader(int a) {
        if (parent_or_size[a] < 0) return a;
        // Path compression
        return parent_or_size[a] = leader(parent_or_size[a]);
    }

    // Returns the size of the group containing 'a'.
    int size(int a) {
        return -parent_or_size[leader(a)];
    }

    // Returns a list of all groups, where each group is a vector of its elements.
    std::vector<std::vector<int>> groups() {
        std::vector<int> leader_buf(_n), group_size(_n);
        for (int i = 0; i < _n; i++) {
            leader_buf[i] = leader(i);
            group_size[leader_buf[i]]++;
        }
        std::vector<std::vector<int>> result(_n);
        for (int i = 0; i < _n; i++) {
            result[i].reserve(group_size[i]);
        }
        for (int i = 0; i < _n; i++) {
            result[leader_buf[i]].push_back(i);
        }
        result.erase(std::remove_if(result.begin(), result.end(), [&](const std::vector<int>& v) { return v.empty(); }),
                     result.end());
        return result;
    }
};

}  // namespace ds
}  // namespace m1une


#line 1 "graph/graph.hpp"



#include <array>
#include <cassert>
#line 8 "graph/graph.hpp"

namespace m1une {
namespace graph {

template <class T = int>
struct Edge {
    using cost_type = T;

    int from;
    int to;
    T cost;
    int id;
    bool alive;

    Edge() : from(-1), to(-1), cost(T()), id(-1), alive(true) {}
    Edge(int from_, int to_, T cost_ = T(1), int id_ = -1, bool alive_ = true)
        : from(from_), to(to_), cost(cost_), id(id_), alive(alive_) {}

    int other(int v) const {
        assert(v == from || v == to);
        return from ^ to ^ v;
    }
};

template <class T = int>
struct Graph {
    using edge_type = Edge<T>;
    using cost_type = T;

   private:
    struct EdgePositions {
        std::array<std::pair<int, int>, 2> value{};
        int size = 0;

        void push_back(std::pair<int, int> position) {
            assert(size < 2);
            value[size++] = position;
        }
    };

    int _n;
    int _edge_count;
    std::vector<std::vector<edge_type>> _g;
    std::vector<EdgePositions> _edge_positions;

   public:
    Graph() : _n(0), _edge_count(0) {}
    explicit Graph(int n) : _n(n), _edge_count(0), _g(n) {
        assert(0 <= n);
    }

    int size() const {
        return _n;
    }

    bool empty() const {
        return _n == 0;
    }

    int edge_count() const {
        return _edge_count;
    }

    int add_vertex() {
        _g.emplace_back();
        return _n++;
    }

    int add_directed_edge(int from, int to, T cost = T(1)) {
        assert(0 <= from && from < _n);
        assert(0 <= to && to < _n);
        int id = _edge_count++;
        int idx = int(_g[from].size());
        _g[from].push_back(edge_type(from, to, cost, id));
        _edge_positions.emplace_back();
        _edge_positions.back().push_back({from, idx});
        return id;
    }

    int add_edge(int u, int v, T cost = T(1)) {
        assert(0 <= u && u < _n);
        assert(0 <= v && v < _n);
        int id = _edge_count++;
        int u_idx = int(_g[u].size());
        _g[u].push_back(edge_type(u, v, cost, id));
        int v_idx = int(_g[v].size());
        _g[v].push_back(edge_type(v, u, cost, id));
        _edge_positions.emplace_back();
        _edge_positions.back().push_back({u, u_idx});
        _edge_positions.back().push_back({v, v_idx});
        return id;
    }

    void set_edge_alive(int id, bool alive) {
        assert(0 <= id && id < _edge_count);
        for (int i = 0; i < _edge_positions[id].size; ++i) {
            auto [v, idx] = _edge_positions[id].value[i];
            _g[v][idx].alive = alive;
        }
    }

    void erase_edge(int id) {
        set_edge_alive(id, false);
    }

    void revive_edge(int id) {
        set_edge_alive(id, true);
    }

    bool is_edge_alive(int id) const {
        assert(0 <= id && id < _edge_count);
        assert(_edge_positions[id].size != 0);
        auto [v, idx] = _edge_positions[id].value[0];
        return _g[v][idx].alive;
    }

    const std::vector<edge_type>& operator[](int v) const {
        assert(0 <= v && v < _n);
        return _g[v];
    }

    std::vector<edge_type>& operator[](int v) {
        assert(0 <= v && v < _n);
        return _g[v];
    }

    const std::vector<std::vector<edge_type>>& adjacency() const {
        return _g;
    }

    std::vector<std::vector<edge_type>>& adjacency() {
        return _g;
    }

    std::vector<edge_type> edges(bool include_inactive = false) const {
        std::vector<edge_type> result;
        result.reserve(_edge_count);
        std::vector<char> used(_edge_count, false);
        for (int v = 0; v < _n; v++) {
            for (const auto& e : _g[v]) {
                if (!include_inactive && !e.alive) continue;
                if (0 <= e.id && e.id < _edge_count) {
                    if (used[e.id]) continue;
                    used[e.id] = true;
                }
                result.push_back(e);
            }
        }
        return result;
    }

    Graph reversed() const {
        Graph result(_n);
        result._edge_count = _edge_count;
        result._edge_positions.assign(_edge_count, {});
        for (int v = 0; v < _n; v++) {
            for (const auto& e : _g[v]) {
                int idx = int(result._g[e.to].size());
                result._g[e.to].push_back(edge_type(e.to, e.from, e.cost, e.id, e.alive));
                if (0 <= e.id && e.id < _edge_count) result._edge_positions[e.id].push_back({e.to, idx});
            }
        }
        return result;
    }
};

}  // namespace graph
}  // namespace m1une


#line 9 "graph/kruskal.hpp"

namespace m1une {
namespace graph {

template <class T>
struct MinimumSpanningForest {
    T cost;
    std::vector<Edge<T>> edges;
    int components;

    bool is_spanning_tree(int n) const {
        return components <= 1 && int(edges.size()) == std::max(0, n - 1);
    }
};

template <class T>
MinimumSpanningForest<T> kruskal(const Graph<T>& g) {
    int n = g.size();
    auto edges = g.edges();
    std::sort(edges.begin(), edges.end(), [](const auto& a, const auto& b) {
        return a.cost < b.cost;
    });

    m1une::ds::Dsu dsu(n);
    MinimumSpanningForest<T> result;
    result.cost = T(0);
    result.components = n;

    for (const auto& e : edges) {
        if (dsu.same(e.from, e.to)) continue;
        dsu.merge(e.from, e.to);
        result.cost += e.cost;
        result.edges.push_back(e);
        result.components--;
    }

    return result;
}

}  // namespace graph
}  // namespace m1une

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