Count Four Cycles
(graph/count_four_cycles.hpp)

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

Overview

This header counts subgraphs isomorphic to the four-edge cycle $C_4$. It can return either the total number of four-cycles or, for every graph edge id, the number of four-cycles containing that particular edge.

The implementation compresses parallel edges into multiplicities, orders vertices by degree, and counts pairs of oriented length-two paths. This avoids enumerating vertex quadruples.

Graph Behavior

Every active edge is interpreted as undirected, including an edge inserted with add_directed_edge. Inactive edges are ignored and receive a per-edge count of zero. Parallel edges are supported and remain distinct choices: replacing one side of a four-cycle by k parallel edges creates k different four-edge subgraphs. Self-loops are not supported.

The per-edge result has graph.edge_count() entries indexed by original edge id. Each count refers to one particular edge copy, not its whole parallel-edge group.

Functions

Function	Exact signature	Description	Complexity
`count_four_cycles_per_edge`	`template <class T> std::vector<long long> count_four_cycles_per_edge(const Graph<T>& graph)`	Returns the number of four-cycles containing every edge id.	$O(N + M\sqrt M)$ time and $O(N + M)$ memory
`count_four_cycles`	`template <class T> long long count_four_cycles(const Graph<T>& graph)`	Returns the total number of four-cycles.	$O(N + M\sqrt M)$ time and $O(N + M)$ memory

M denotes the number of active edges, including parallel copies. Counts must fit in long long.

Example

#include "graph/count_four_cycles.hpp"
#include "graph/graph.hpp"

#include <cassert>

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

    assert(m1une::graph::count_four_cycles(graph) == 1);
    auto per_edge = m1une::graph::count_four_cycles_per_edge(graph);
    assert(per_edge[4] == 0);  // The diagonal is not in the four-cycle.
}

Depends on

Graph (graph/graph.hpp)

Required by

Verified with

Code

#ifndef M1UNE_GRAPH_COUNT_FOUR_CYCLES_HPP
#define M1UNE_GRAPH_COUNT_FOUR_CYCLES_HPP 1

#include <algorithm>
#include <cassert>
#include <tuple>
#include <utility>
#include <vector>

#include "graph.hpp"

namespace m1une {
namespace graph {

namespace four_cycle_detail {

// Counts C4s containing one particular copy of each edge in a simple graph
// whose edge weights represent parallel-edge multiplicities.
inline std::vector<long long> count_simple_per_edge(
    int vertex_count,
    std::vector<int> first,
    std::vector<int> second,
    const std::vector<long long>& multiplicity
) {
    const int edge_count = int(first.size());
    assert(second.size() == first.size());
    assert(multiplicity.size() == first.size());

    std::vector<int> degree(vertex_count, 0);
    for (int edge = 0; edge < edge_count; edge++) {
        degree[first[edge]]++;
        degree[second[edge]]++;
    }

    int maximum_degree = 0;
    for (int value : degree) maximum_degree = std::max(maximum_degree, value);
    std::vector<int> degree_start(maximum_degree + 2, 0);
    for (int value : degree) degree_start[value + 1]++;
    for (int value = 0; value <= maximum_degree; value++) {
        degree_start[value + 1] += degree_start[value];
    }
    std::vector<int> cursor = degree_start;
    std::vector<int> order(vertex_count);
    for (int vertex = 0; vertex < vertex_count; vertex++) {
        order[cursor[degree[vertex]]++] = vertex;
    }
    std::vector<int> rank(vertex_count);
    for (int i = 0; i < vertex_count; i++) rank[order[i]] = i;
    for (int edge = 0; edge < edge_count; edge++) {
        first[edge] = rank[first[edge]];
        second[edge] = rank[second[edge]];
        if (first[edge] < second[edge]) {
            std::swap(first[edge], second[edge]);
        }
    }

    std::vector<int> start(vertex_count + 1, 0);
    for (int vertex = 0; vertex < vertex_count; vertex++) {
        start[vertex + 1] = start[vertex] + degree[order[vertex]];
    }
    std::vector<int> end = start;
    std::vector<int> edge_at(2 * edge_count);
    std::vector<int> to(2 * edge_count);
    for (int edge = 0; edge < edge_count; edge++) {
        int position = end[first[edge]]++;
        edge_at[position] = edge;
        to[position] = second[edge];
    }

    std::vector<int> downward_end = end;
    for (int vertex = 0; vertex < vertex_count; vertex++) {
        for (int i = start[vertex]; i < downward_end[vertex]; i++) {
            int edge = edge_at[i];
            int neighbor = to[i];
            int position = end[neighbor]++;
            edge_at[position] = edge;
            to[position] = vertex;
        }
    }

    std::vector<long long> path_count(vertex_count, 0);
    std::vector<long long> result(edge_count, 0);
    for (int vertex = vertex_count - 1; vertex >= 0; vertex--) {
        for (int i = start[vertex]; i < end[vertex]; i++) {
            int first_edge = edge_at[i];
            int middle = to[i];
            end[middle]--;
            for (int j = start[middle]; j < end[middle]; j++) {
                int second_edge = edge_at[j];
                int opposite = to[j];
                path_count[opposite] +=
                    multiplicity[first_edge] * multiplicity[second_edge];
            }
        }

        for (int i = start[vertex]; i < end[vertex]; i++) {
            int first_edge = edge_at[i];
            int middle = to[i];
            for (int j = start[middle]; j < end[middle]; j++) {
                int second_edge = edge_at[j];
                int opposite = to[j];
                long long other_paths =
                    path_count[opposite] -
                    multiplicity[first_edge] * multiplicity[second_edge];
                result[first_edge] +=
                    other_paths * multiplicity[second_edge];
                result[second_edge] +=
                    other_paths * multiplicity[first_edge];
            }
        }

        for (int i = start[vertex]; i < end[vertex]; i++) {
            int middle = to[i];
            for (int j = start[middle]; j < end[middle]; j++) {
                path_count[to[j]] = 0;
            }
        }
    }
    return result;
}

}  // namespace four_cycle_detail

// Returns, for every graph edge id, the number of C4 subgraphs containing it.
// Parallel active edges are distinct choices; inactive edges receive zero.
template <class T>
std::vector<long long> count_four_cycles_per_edge(const Graph<T>& graph) {
    struct ActiveEdge {
        int first;
        int second;
        int id;
    };

    std::vector<ActiveEdge> active_edges;
    active_edges.reserve(graph.edge_count());
    for (const Edge<T>& edge : graph.edges()) {
        assert(edge.from != edge.to);
        assert(0 <= edge.id && edge.id < graph.edge_count());
        if (edge.from == edge.to) continue;
        active_edges.push_back(ActiveEdge{
            std::min(edge.from, edge.to),
            std::max(edge.from, edge.to),
            edge.id
        });
    }
    std::sort(
        active_edges.begin(),
        active_edges.end(),
        [](const ActiveEdge& left, const ActiveEdge& right) {
            return std::tie(left.first, left.second) <
                   std::tie(right.first, right.second);
        }
    );

    std::vector<int> first;
    std::vector<int> second;
    std::vector<long long> multiplicity;
    std::vector<int> group_of_edge(graph.edge_count(), -1);
    first.reserve(active_edges.size());
    second.reserve(active_edges.size());
    multiplicity.reserve(active_edges.size());
    for (const ActiveEdge& edge : active_edges) {
        if (first.empty() || first.back() != edge.first ||
            second.back() != edge.second) {
            first.push_back(edge.first);
            second.push_back(edge.second);
            multiplicity.push_back(0);
        }
        multiplicity.back()++;
        group_of_edge[edge.id] = int(first.size()) - 1;
    }

    std::vector<long long> simple_result =
        four_cycle_detail::count_simple_per_edge(
            graph.size(),
            std::move(first),
            std::move(second),
            multiplicity
        );
    std::vector<long long> result(graph.edge_count(), 0);
    for (const ActiveEdge& edge : active_edges) {
        result[edge.id] = simple_result[group_of_edge[edge.id]];
    }
    return result;
}

template <class T>
long long count_four_cycles(const Graph<T>& graph) {
    std::vector<long long> per_edge = count_four_cycles_per_edge(graph);
    long long incidence_count = 0;
    for (long long count : per_edge) incidence_count += count;
    assert(incidence_count % 4 == 0);
    return incidence_count / 4;
}

}  // namespace graph
}  // namespace m1une

#endif  // M1UNE_GRAPH_COUNT_FOUR_CYCLES_HPP

#line 1 "graph/count_four_cycles.hpp"



#include <algorithm>
#include <cassert>
#include <tuple>
#include <utility>
#include <vector>

#line 1 "graph/graph.hpp"



#include <array>
#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 11 "graph/count_four_cycles.hpp"

namespace m1une {
namespace graph {

namespace four_cycle_detail {

// Counts C4s containing one particular copy of each edge in a simple graph
// whose edge weights represent parallel-edge multiplicities.
inline std::vector<long long> count_simple_per_edge(
    int vertex_count,
    std::vector<int> first,
    std::vector<int> second,
    const std::vector<long long>& multiplicity
) {
    const int edge_count = int(first.size());
    assert(second.size() == first.size());
    assert(multiplicity.size() == first.size());

    std::vector<int> degree(vertex_count, 0);
    for (int edge = 0; edge < edge_count; edge++) {
        degree[first[edge]]++;
        degree[second[edge]]++;
    }

    int maximum_degree = 0;
    for (int value : degree) maximum_degree = std::max(maximum_degree, value);
    std::vector<int> degree_start(maximum_degree + 2, 0);
    for (int value : degree) degree_start[value + 1]++;
    for (int value = 0; value <= maximum_degree; value++) {
        degree_start[value + 1] += degree_start[value];
    }
    std::vector<int> cursor = degree_start;
    std::vector<int> order(vertex_count);
    for (int vertex = 0; vertex < vertex_count; vertex++) {
        order[cursor[degree[vertex]]++] = vertex;
    }
    std::vector<int> rank(vertex_count);
    for (int i = 0; i < vertex_count; i++) rank[order[i]] = i;
    for (int edge = 0; edge < edge_count; edge++) {
        first[edge] = rank[first[edge]];
        second[edge] = rank[second[edge]];
        if (first[edge] < second[edge]) {
            std::swap(first[edge], second[edge]);
        }
    }

    std::vector<int> start(vertex_count + 1, 0);
    for (int vertex = 0; vertex < vertex_count; vertex++) {
        start[vertex + 1] = start[vertex] + degree[order[vertex]];
    }
    std::vector<int> end = start;
    std::vector<int> edge_at(2 * edge_count);
    std::vector<int> to(2 * edge_count);
    for (int edge = 0; edge < edge_count; edge++) {
        int position = end[first[edge]]++;
        edge_at[position] = edge;
        to[position] = second[edge];
    }

    std::vector<int> downward_end = end;
    for (int vertex = 0; vertex < vertex_count; vertex++) {
        for (int i = start[vertex]; i < downward_end[vertex]; i++) {
            int edge = edge_at[i];
            int neighbor = to[i];
            int position = end[neighbor]++;
            edge_at[position] = edge;
            to[position] = vertex;
        }
    }

    std::vector<long long> path_count(vertex_count, 0);
    std::vector<long long> result(edge_count, 0);
    for (int vertex = vertex_count - 1; vertex >= 0; vertex--) {
        for (int i = start[vertex]; i < end[vertex]; i++) {
            int first_edge = edge_at[i];
            int middle = to[i];
            end[middle]--;
            for (int j = start[middle]; j < end[middle]; j++) {
                int second_edge = edge_at[j];
                int opposite = to[j];
                path_count[opposite] +=
                    multiplicity[first_edge] * multiplicity[second_edge];
            }
        }

        for (int i = start[vertex]; i < end[vertex]; i++) {
            int first_edge = edge_at[i];
            int middle = to[i];
            for (int j = start[middle]; j < end[middle]; j++) {
                int second_edge = edge_at[j];
                int opposite = to[j];
                long long other_paths =
                    path_count[opposite] -
                    multiplicity[first_edge] * multiplicity[second_edge];
                result[first_edge] +=
                    other_paths * multiplicity[second_edge];
                result[second_edge] +=
                    other_paths * multiplicity[first_edge];
            }
        }

        for (int i = start[vertex]; i < end[vertex]; i++) {
            int middle = to[i];
            for (int j = start[middle]; j < end[middle]; j++) {
                path_count[to[j]] = 0;
            }
        }
    }
    return result;
}

}  // namespace four_cycle_detail

// Returns, for every graph edge id, the number of C4 subgraphs containing it.
// Parallel active edges are distinct choices; inactive edges receive zero.
template <class T>
std::vector<long long> count_four_cycles_per_edge(const Graph<T>& graph) {
    struct ActiveEdge {
        int first;
        int second;
        int id;
    };

    std::vector<ActiveEdge> active_edges;
    active_edges.reserve(graph.edge_count());
    for (const Edge<T>& edge : graph.edges()) {
        assert(edge.from != edge.to);
        assert(0 <= edge.id && edge.id < graph.edge_count());
        if (edge.from == edge.to) continue;
        active_edges.push_back(ActiveEdge{
            std::min(edge.from, edge.to),
            std::max(edge.from, edge.to),
            edge.id
        });
    }
    std::sort(
        active_edges.begin(),
        active_edges.end(),
        [](const ActiveEdge& left, const ActiveEdge& right) {
            return std::tie(left.first, left.second) <
                   std::tie(right.first, right.second);
        }
    );

    std::vector<int> first;
    std::vector<int> second;
    std::vector<long long> multiplicity;
    std::vector<int> group_of_edge(graph.edge_count(), -1);
    first.reserve(active_edges.size());
    second.reserve(active_edges.size());
    multiplicity.reserve(active_edges.size());
    for (const ActiveEdge& edge : active_edges) {
        if (first.empty() || first.back() != edge.first ||
            second.back() != edge.second) {
            first.push_back(edge.first);
            second.push_back(edge.second);
            multiplicity.push_back(0);
        }
        multiplicity.back()++;
        group_of_edge[edge.id] = int(first.size()) - 1;
    }

    std::vector<long long> simple_result =
        four_cycle_detail::count_simple_per_edge(
            graph.size(),
            std::move(first),
            std::move(second),
            multiplicity
        );
    std::vector<long long> result(graph.edge_count(), 0);
    for (const ActiveEdge& edge : active_edges) {
        result[edge.id] = simple_result[group_of_edge[edge.id]];
    }
    return result;
}

template <class T>
long long count_four_cycles(const Graph<T>& graph) {
    std::vector<long long> per_edge = count_four_cycles_per_edge(graph);
    long long incidence_count = 0;
    for (long long count : per_edge) incidence_count += count;
    assert(incidence_count % 4 == 0);
    return incidence_count / 4;
}

}  // namespace graph
}  // namespace m1une

m1une's library

This documentation is automatically generated by online-judge-tools/verification-helper

Count Four Cycles
(graph/count_four_cycles.hpp)

Overview

Graph Behavior

Functions

Example

Depends on

Required by

Verified with

Code

Count Four Cycles (graph/count_four_cycles.hpp)

Overview

Graph Behavior

Functions

Example

Depends on

Required by

Verified with

Code

Count Four Cycles
(graph/count_four_cycles.hpp)