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#include "graph/maximum_matching_size.hpp"
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。
#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) { 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) { 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 { return edges[i]; } std::vector<edge_type> get_edges() const { return edges; } const auto operator[](int i) const { return csr[i]; } auto operator[](int i) { 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