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