Library

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:warning: Maximum Matching Size
(graph/maximum_matching_size.hpp)

説明

グラフを与えて、最大マッチングのサイズを返す。頂点数を $N$ として $O(N^3)$

アルゴリズムとしてはtutte行列を用いている。

Depends on

Code

#pragma once

#include "../graph/base.hpp"
#include "../matrix/base.hpp"
#include "../matrix/gauss_jordan.hpp"
#include "../modint/modint61.hpp"
#include "../utility/random_number_generator.hpp"

namespace ebi {

template <class T> int maximum_matching_size(const Graph<T> &g) {
    static random_number_generator rng;
    using mint = modint61;
    int n = g.node_number();
    matrix<mint> tutte(n, n, 0);
    for (auto e : g.get_edges()) {
        mint x = rng.get<std::uint64_t>(0, mint::mod());
        tutte[e.from][e.to] += x;
        tutte[e.to][e.from] -= x;
    }
    return tutte.rank() / 2;
}

}  // namespace ebi
#line 2 "graph/maximum_matching_size.hpp"

#line 2 "graph/base.hpp"

#include <cassert>
#include <iostream>
#include <ranges>
#include <vector>

#line 2 "data_structure/simple_csr.hpp"

#line 4 "data_structure/simple_csr.hpp"
#include <utility>
#line 6 "data_structure/simple_csr.hpp"

namespace ebi {

template <class E> struct simple_csr {
    simple_csr() = default;

    simple_csr(int n, const std::vector<std::pair<int, E>>& elements)
        : start(n + 1, 0), elist(elements.size()) {
        for (auto e : elements) {
            start[e.first + 1]++;
        }
        for (auto i : std::views::iota(0, n)) {
            start[i + 1] += start[i];
        }
        auto counter = start;
        for (auto [i, e] : elements) {
            elist[counter[i]++] = e;
        }
    }

    simple_csr(const std::vector<std::vector<E>>& es)
        : start(es.size() + 1, 0) {
        int n = es.size();
        for (auto i : std::views::iota(0, n)) {
            start[i + 1] = (int)es[i].size() + start[i];
        }
        elist.resize(start.back());
        for (auto i : std::views::iota(0, n)) {
            std::copy(es[i].begin(), es[i].end(), elist.begin() + start[i]);
        }
    }

    int size() const {
        return (int)start.size() - 1;
    }

    const auto operator[](int i) const {
        return std::ranges::subrange(elist.begin() + start[i],
                                     elist.begin() + start[i + 1]);
    }
    auto operator[](int i) {
        return std::ranges::subrange(elist.begin() + start[i],
                                     elist.begin() + start[i + 1]);
    }

    const auto operator()(int i, int l, int r) const {
        return std::ranges::subrange(elist.begin() + start[i] + l,
                                     elist.begin() + start[i + 1] + r);
    }
    auto operator()(int i, int l, int r) {
        return std::ranges::subrange(elist.begin() + start[i] + l,
                                     elist.begin() + start[i + 1] + r);
    }

  private:
    std::vector<int> start;
    std::vector<E> elist;
};

}  // namespace ebi
#line 9 "graph/base.hpp"

namespace ebi {

template <class T> struct Edge {
    int from, to;
    T cost;
    int id;
};

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

    Graph(int n_) : n(n_) {}

    Graph() = default;

    void add_edge(int u, int v, cost_type c) {
        assert(!prepared && u < n && v < n);
        buff.emplace_back(u, edge_type{u, v, c, m});
        edges.emplace_back(edge_type{u, v, c, m++});
    }

    void add_undirected_edge(int u, int v, cost_type c) {
        assert(!prepared && u < n && v < n);
        buff.emplace_back(u, edge_type{u, v, c, m});
        buff.emplace_back(v, edge_type{v, u, c, m});
        edges.emplace_back(edge_type{u, v, c, m});
        m++;
    }

    void read_tree(int offset = 1, bool is_weighted = false) {
        read_graph(n - 1, offset, false, is_weighted);
    }

    void read_parents(int offset = 1) {
        for (auto i : std::views::iota(1, n)) {
            int p;
            std::cin >> p;
            p -= offset;
            add_undirected_edge(p, i, 1);
        }
        build();
    }

    void read_graph(int e, int offset = 1, bool is_directed = false,
                    bool is_weighted = false) {
        for (int i = 0; i < e; i++) {
            int u, v;
            std::cin >> u >> v;
            u -= offset;
            v -= offset;
            if (is_weighted) {
                cost_type c;
                std::cin >> c;
                if (is_directed) {
                    add_edge(u, v, c);
                } else {
                    add_undirected_edge(u, v, c);
                }
            } else {
                if (is_directed) {
                    add_edge(u, v, 1);
                } else {
                    add_undirected_edge(u, v, 1);
                }
            }
        }
        build();
    }

    void build() {
        assert(!prepared);
        csr = simple_csr<edge_type>(n, buff);
        buff.clear();
        prepared = true;
    }

    int size() const {
        return n;
    }

    int node_number() const {
        return n;
    }

    int edge_number() const {
        return m;
    }

    edge_type get_edge(int i) const {
        assert(prepared);
        return edges[i];
    }

    std::vector<edge_type> get_edges() const {
        assert(prepared);
        return edges;
    }

    const auto operator[](int i) const {
        assert(prepared);
        return csr[i];
    }
    auto operator[](int i) {
        assert(prepared);
        return csr[i];
    }

  private:
    int n, m = 0;

    std::vector<std::pair<int, edge_type>> buff;

    std::vector<edge_type> edges;
    simple_csr<edge_type> csr;
    bool prepared = false;
};

}  // namespace ebi
#line 2 "matrix/base.hpp"

#include <algorithm>
#line 8 "matrix/base.hpp"

namespace ebi {

template <class T> struct matrix;

template <class T> matrix<T> identify_matrix(int n) {
    matrix<T> a(n, n);
    for (int i = 0; i < n; i++) {
        a[i][i] = 1;
    }
    return a;
}

template <class T> struct matrix {
  private:
    using Self = matrix<T>;

  public:
    matrix(int n_, int m_, T init_val = 0)
        : n(n_), m(m_), data(n * m, init_val) {}

    matrix(const std::vector<std::vector<T>> &a)
        : n((int)a.size()), m((int)a[0].size()) {
        data = std::vector(n * m);
        for (int i = 0; i < n; i++) {
            std::copy(a[i].begin(), a[i].end(), data.begin() + i * m);
        }
    }

    Self operator+(Self &rhs) const noexcept {
        return Self(*this) += rhs;
    }

    Self operator-(Self &rhs) const noexcept {
        return Self(*this) -= rhs;
    }

    Self operator*(Self &rhs) const noexcept {
        return Self(*this) *= rhs;
    }

    Self operator/(Self &rhs) const noexcept {
        return Self(*this) /= rhs;
    }

    friend Self operator*(const T &lhs, const Self &rhs) {
        return Self(rhs) *= lhs;
    }

    friend Self operator*(const Self &lhs, const T &rhs) {
        return Self(lhs) *= rhs;
    }

    std::vector<T> operator*(const std::vector<T> &rhs) noexcept {
        assert(m == (int)rhs.size());
        std::vector<T> res(n, 0);
        for (int i = 0; i < n; i++) {
            for (int j = 0; j < m; j++) {
                res[i] += (*this)[i][j] * rhs[j];
            }
        }
        return res;
    }

    Self &operator+=(Self &rhs) noexcept {
        assert(this->size() == rhs.size());
        for (int i = 0; i < n; ++i) {
            for (int j = 0; j < m; ++j) {
                (*this)[i][j] += rhs[i][j];
            }
        }
        return *this;
    }

    Self &operator-=(Self &rhs) noexcept {
        assert(this->size() == rhs.size());
        for (int i = 0; i < n; ++i) {
            for (int j = 0; j < m; ++j) {
                (*this)[i][j] -= rhs[i][j];
            }
        }
        return *this;
    }

    Self &operator*=(Self &rhs) noexcept {
        int h = n, w = rhs.column_size();
        assert(m == rhs.row_size());
        Self ret(h, w);
        for (int i = 0; i < h; ++i) {
            for (int k = 0; k < m; ++k) {
                for (int j = 0; j < w; ++j) {
                    ret[i][j] += (*this)[i][k] * rhs[k][j];
                }
            }
        }
        return *this = ret;
    }

    Self &operator/=(const Self &rhs) noexcept {
        auto ret = rhs.inv();
        assert(ret);
        return *this *= ret.value();
    }

    Self &operator*=(const T &rhs) noexcept {
        for (int i = 0; i < n; i++) {
            for (int j = 0; j < m; j++) {
                (*this)[i][j] *= rhs;
            }
        }
        return *this;
    }

    const auto operator[](int i) const {
        return std::ranges::subrange(data.begin() + i * m,
                                     data.begin() + (i + 1) * m);
    }

    auto operator[](int i) {
        return std::ranges::subrange(data.begin() + i * m,
                                     data.begin() + (i + 1) * m);
    }

    void swap(int i, int j) {
        std::swap_ranges(data.begin() + i * m, data.begin() + (i + 1) * m,
                         data.begin() + j * m);
    }

    int rank() const;

    Self transposition() const {
        Self res(m, n);
        for (int i = 0; i < n; i++) {
            for (int j = 0; j < m; j++) {
                res[j][i] = (*this)[i][j];
            }
        }
        return res;
    }

    std::optional<Self> inv() const {
        assert(row_size() == column_size());
        Self a = *this;
        Self b = identify_matrix<T>(n);
        for (int r = 0; r < n; r++) {
            for (int i = r; i < n; i++) {
                if (a[i][r] != 0) {
                    a.swap(r, i);
                    b.swap(r, i);
                    break;
                }
            }
            if (a[r][r] == 0) return std::nullopt;
            T x = a[r][r].inv();
            for (int j = 0; j < n; j++) {
                if (r < j) a[r][j] *= x;
                b[r][j] *= x;
            }
            for (int i = 0; i < n; i++) {
                if (i == r) continue;
                for (int j = 0; j < n; j++) {
                    if (r < j) a[i][j] -= a[i][r] * a[r][j];
                    b[i][j] -= a[i][r] * b[r][j];
                }
            }
        }
        return b;
    }

    Self pow(long long k) const {
        assert(row_size() == column_size() && k >= 0);
        Self res = identify_matrix<T>(row_size());
        Self x = *this;
        while (k) {
            if (k & 1) res *= x;
            x *= x;
            k >>= 1;
        }
        return res;
    }

    int row_size() const {
        return n;
    }

    int column_size() const {
        return m;
    }

    std::pair<int, int> size() const {
        return {n, m};
    }

  private:
    int n, m;
    std::vector<T> data;
};

template <class T> T det(matrix<T> a) {
    assert(a.row_size() == a.column_size());
    T d = 1;
    int n = a.row_size();
    for (int r = 0; r < n; r++) {
        if (a[r][r] == 0) {
            for (int i = r + 1; i < n; i++) {
                if (a[i][r] != 0) {
                    a.swap(r, i);
                    d = -d;
                }
            }
        }
        if (a[r][r] == 0) return 0;
        d *= a[r][r];
        T inv = a[r][r].inv();
        for (int i = r + 1; i < n; i++) {
            T x = a[i][r] * inv;
            for (int j = r; j < n; j++) {
                a[i][j] -= x * a[r][j];
            }
        }
    }
    return d;
}

template <class T> std::istream &operator>>(std::istream &os, matrix<T> &a) {
    for (int i = 0; i < a.row_size(); i++) {
        for (int j = 0; j < a.column_size(); j++) {
            os >> a[i][j];
        }
    }
    return os;
}

template <class T>
std::ostream &operator<<(std::ostream &os, const matrix<T> &a) {
    for (int i = 0; i < a.row_size(); i++) {
        for (int j = 0; j < a.column_size(); j++) {
            os << a[i][j];
            if (j < a.column_size() - 1) os << ' ';
        }
        if (i < a.row_size() - 1) os << '\n';
    }
    return os;
}

}  // namespace ebi
#line 2 "matrix/gauss_jordan.hpp"

#line 4 "matrix/gauss_jordan.hpp"

namespace ebi {

template <class T> int find_pivot(const matrix<T> &a, int r, int w) {
    for (int i = r; i < a.row_size(); i++) {
        if (a[i][w] != 0) return i;
    }
    return -1;
}

template <class T> int gauss_jordan(matrix<T> &a) {
    int h = a.row_size(), w = a.column_size();
    int rank = 0;
    for (int j = 0; j < w; j++) {
        int pivot = find_pivot(a, rank, j);
        if (pivot == -1) continue;
        a.swap(rank, pivot);
        T inv = T(1) / a[rank][j];
        for (int k = j; k < w; k++) {
            a[rank][k] *= inv;
        }
        for (int i = 0; i < h; i++) {
            if (i != rank && a[i][j] != 0) {
                T x = a[i][j];
                for (int k = j; k < w; k++) {
                    a[i][k] -= a[rank][k] * x;
                }
            }
        }
        rank++;
    }
    return rank;
}

template <class T> int matrix<T>::rank() const {
    matrix<T> a = *this;
    return gauss_jordan(a);
}

template <class T> int gauss_jordan(matrix<T> &a, std::vector<T> &b) {
    int h = a.row_size(), w = a.column_size();
    assert(h == (int)b.size());
    int rank = 0;
    for (int j = 0; j < w; j++) {
        int pivot = find_pivot(a, rank, j);
        if (pivot == -1) continue;
        a.swap(rank, pivot);
        std::swap(b[rank], b[pivot]);
        T inv = T(1) / a[rank][j];
        for (int k = j; k < w; k++) {
            a[rank][k] *= inv;
        }
        b[rank] *= inv;
        for (int i = 0; i < h; i++) {
            if (i != rank && a[i][j] != 0) {
                T x = a[i][j];
                for (int k = j; k < w; k++) {
                    a[i][k] -= a[rank][k] * x;
                }
                b[i] -= b[rank] * x;
            }
        }
        rank++;
    }
    return rank;
}

template <class T>
std::optional<std::vector<std::vector<T>>> solve_linear_equations(
    matrix<T> a, std::vector<T> b) {
    assert(a.row_size() == (int)b.size());
    int rank = gauss_jordan(a, b);
    int h = a.row_size(), w = a.column_size();
    for (int i = rank; i < h; i++) {
        if (b[i] != 0) return std::nullopt;
    }
    std::vector res(1, std::vector<T>(w, 0));
    std::vector<int> pivot(w, -1);
    {
        int p = 0;
        for (int i = 0; i < rank; i++) {
            while (a[i][p] == 0) p++;
            res[0][p] = b[i];
            pivot[p] = i;
        }
    }
    for (int j = 0; j < w; j++) {
        if (pivot[j] == -1) {
            std::vector<T> x(w, 0);
            x[j] = -1;
            for (int i = 0; i < j; i++) {
                if (pivot[i] != -1) x[i] = a[pivot[i]][j];
            }
            res.emplace_back(x);
        }
    }
    return res;
}

}  // namespace ebi
#line 2 "modint/modint61.hpp"

#line 4 "modint/modint61.hpp"
#include <cstdint>
#line 6 "modint/modint61.hpp"

#line 2 "modint/base.hpp"

#include <concepts>
#line 6 "modint/base.hpp"

namespace ebi {

template <class T>
concept Modint = requires(T a, T b) {
    a + b;
    a - b;
    a * b;
    a / b;
    a.inv();
    a.val();
    a.pow(std::declval<long long>());
    T::mod();
};

template <Modint mint> std::istream &operator>>(std::istream &os, mint &a) {
    long long x;
    os >> x;
    a = x;
    return os;
}

template <Modint mint>
std::ostream &operator<<(std::ostream &os, const mint &a) {
    return os << a.val();
}

}  // namespace ebi
#line 8 "modint/modint61.hpp"

namespace ebi {

struct modint61 {
  private:
    using mint = modint61;
    using u64 = std::uint64_t;
    constexpr static u64 m = (1ull << 61) - 1;
    constexpr static u64 MASK31 = (1ull << 31) - 1;
    constexpr static u64 MASK30 = (1ull << 30) - 1;

  public:
    constexpr static u64 mod() {
        return m;
    }

    constexpr modint61() : _v(0) {}

    constexpr modint61(long long v) {
        v %= (long long)umod();
        if (v < 0) v += (long long)umod();
        _v = u64(v);
    }

    constexpr u64 val() const {
        return _v;
    }

    constexpr u64 value() const {
        return val();
    }

    constexpr mint &operator++() {
        _v++;
        if (_v == umod()) _v = 0;
        return *this;
    }

    constexpr mint &operator--() {
        if (_v == 0) _v = umod();
        _v--;
        return *this;
    }

    constexpr mint &operator+=(const mint &rhs) {
        _v += rhs._v;
        _v = safe_mod(_v);
        return *this;
    }

    constexpr mint &operator-=(const mint &rhs) {
        if (_v < rhs._v) _v += umod();
        assert(_v >= rhs._v);
        _v -= rhs._v;
        return *this;
    }

    constexpr mint &operator*=(const mint &rhs) {
        u64 au = _v >> 31, ad = _v & MASK31;
        u64 bu = rhs._v >> 31, bd = rhs._v & MASK31;
        u64 mid = ad * bu + au * bd;
        u64 midu = mid >> 30;
        u64 midd = mid & MASK30;
        _v = (au * bu * 2 + midu + (midd << 31) + ad * bd);
        _v = safe_mod(_v);
        return *this;
    }

    constexpr mint &operator/=(const mint &rhs) {
        return *this *= rhs.inv();
    }

    constexpr mint pow(long long n) const {
        assert(0 <= n);
        mint x = *this, res = 1;
        while (n) {
            if (n & 1) res *= x;
            x *= x;
            n >>= 1;
        }
        return res;
    }

    constexpr mint inv() const {
        assert(_v);
        return pow(umod() - 2);
    }

    friend mint operator+(const mint &lhs, const mint &rhs) {
        return mint(lhs) += rhs;
    }
    friend mint operator-(const mint &lhs, const mint &rhs) {
        return mint(lhs) -= rhs;
    }
    friend mint operator*(const mint &lhs, const mint &rhs) {
        return mint(lhs) *= rhs;
    }
    friend mint operator/(const mint &lhs, const mint &rhs) {
        return mint(lhs) /= rhs;
    }
    friend bool operator==(const mint &lhs, const mint &rhs) {
        return lhs.val() == rhs.val();
    }
    friend bool operator!=(const mint &lhs, const mint &rhs) {
        return !(lhs == rhs);
    }
    friend bool operator<(const mint &lhs, const mint &rhs) {
        return lhs._v < rhs._v;
    }
    friend bool operator>(const mint &lhs, const mint &rhs) {
        return rhs < lhs;
    }

  private:
    u64 _v = 0;

    constexpr static u64 umod() {
        return m;
    }

    constexpr u64 safe_mod(const u64 &a) {
        u64 au = a >> 61;
        u64 ad = a & umod();
        u64 res = au + ad;
        if (res >= umod()) res -= umod();
        return res;
    }
};

}  // namespace ebi
#line 2 "utility/random_number_generator.hpp"

#line 5 "utility/random_number_generator.hpp"
#include <numeric>

#include <random>

#line 8 "utility/random_number_generator.hpp"

namespace ebi {

struct random_number_generator {
    random_number_generator(int seed = -1) {
        if (seed < 0) seed = rnd();
        mt.seed(seed);
    }

    void set_seed(int seed) {
        mt.seed(seed);
    }

    template <class T> T get(T a, T b) {
        std::uniform_int_distribution<T> dist(a, b - 1);
        return dist(mt);
    }

    std::vector<int> get_permutation(int n) {
        std::vector<int> p(n);
        std::iota(p.begin(), p.end(), 0);
        std::shuffle(p.begin(), p.end(), mt);
        return p;
    }

  private:
    std::mt19937_64 mt;
    std::random_device rnd;
};

}  // namespace ebi
#line 8 "graph/maximum_matching_size.hpp"

namespace ebi {

template <class T> int maximum_matching_size(const Graph<T> &g) {
    static random_number_generator rng;
    using mint = modint61;
    int n = g.node_number();
    matrix<mint> tutte(n, n, 0);
    for (auto e : g.get_edges()) {
        mint x = rng.get<std::uint64_t>(0, mint::mod());
        tutte[e.from][e.to] += x;
        tutte[e.to][e.from] -= x;
    }
    return tutte.rank() / 2;
}

}  // namespace ebi
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