#1111097 (C++23) No.3231 2×2行列相似判定〜hard〜

提出ソース
結果

問題	No.3231 2×2行列相似判定〜hard〜
ユーザー	hitonanode
提出日時	2025-08-08 22:04:48
言語	C++23 (gcc 13.3.0 + boost 1.87.0)
結果	AC
実行時間	2 ms / 2,000 ms
コード長	14,254 bytes
コンパイル時間	3,312 ms
コンパイル使用メモリ	243,000 KB
実行使用メモリ	7,720 KB
最終ジャッジ日時	2025-08-08 22:04:54
合計ジャッジ時間	4,662 ms
ジャッジサーバーID （参考情報）	judge3 / judge1
このコードへのチャレンジ
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ファイルパターン	結果
sample	AC * 3
other	AC * 37
権限があれば一括ダウンロードができます
ソースコード

raw source code
// https://judge.yosupo.jp/submission/183810
#include <algorithm>
#include <array>
#include <bitset>
#include <cassert>
#include <chrono>
#include <cmath>
#include <complex>
#include <deque>
#include <forward_list>
#include <fstream>
#include <functional>
#include <iomanip>
#include <ios>
#include <iostream>
#include <limits>
#include <list>
#include <map>
#include <memory>
#include <numeric>
#include <optional>
#include <queue>
#include <random>
#include <set>
#include <sstream>
#include <stack>
#include <string>
#include <tuple>
#include <type_traits>
#include <unordered_map>
#include <unordered_set>
#include <utility>
#include <vector>
using namespace std;
using lint = long long;
using pint = pair<int, int>;
using plint = pair<lint, lint>;
struct fast_ios { fast_ios(){ cin.tie(nullptr), ios::sync_with_stdio(false), cout << fixed << setprecision(20); }; } fast_ios_;
#define ALL(x) (x).begin(), (x).end()
#define FOR(i, begin, end) for(int i=(begin),i##_end_=(end);i<i##_end_;i++)
#define IFOR(i, begin, end) for(int i=(end)-1,i##_begin_=(begin);i>=i##_begin_;i--)
#define REP(i, n) FOR(i,0,n)
#define IREP(i, n) IFOR(i,0,n)
template <typename T> bool chmax(T &m, const T q) { return m < q ? (m = q, true) : false; }
template <typename T> bool chmin(T &m, const T q) { return m > q ? (m = q, true) : false; }
const std::vector<std::pair<int, int>> grid_dxs{{1, 0}, {-1, 0}, {0, 1}, {0, -1}};
int floor_lg(long long x) { return x <= 0 ? -1 : 63 - __builtin_clzll(x); }
template <class T1, class T2> T1 floor_div(T1 num, T2 den) { return (num > 0 ? num / den : -((-num + den - 1) / den)); }
template <class T1, class T2> std::pair<T1, T2> operator+(const std::pair<T1, T2> &l, const std::pair<T1, T2> &r) { return std::make_pair(l.first + r.first, l.second + r.second); }
template <class T1, class T2> std::pair<T1, T2> operator-(const std::pair<T1, T2> &l, const std::pair<T1, T2> &r) { return std::make_pair(l.first - r.first, l.second - r.second); }
template <class T> std::vector<T> sort_unique(std::vector<T> vec) { sort(vec.begin(), vec.end()), vec.erase(unique(vec.begin(), vec.end()), vec.end()); return vec; }
template <class T> int arglb(const std::vector<T> &v, const T &x) { return std::distance(v.begin(), std::lower_bound(v.begin(), v.end(), x)); }
template <class T> int argub(const std::vector<T> &v, const T &x) { return std::distance(v.begin(), std::upper_bound(v.begin(), v.end(), x)); }
template <class IStream, class T> IStream &operator>>(IStream &is, std::vector<T> &vec) { for (auto &v : vec) is >> v; return is; }

template <class OStream, class T> OStream &operator<<(OStream &os, const std::vector<T> &vec);
template <class OStream, class T, size_t sz> OStream &operator<<(OStream &os, const std::array<T, sz> &arr);
template <class OStream, class T, class TH> OStream &operator<<(OStream &os, const std::unordered_set<T, TH> &vec);
template <class OStream, class T, class U> OStream &operator<<(OStream &os, const pair<T, U> &pa);
template <class OStream, class T> OStream &operator<<(OStream &os, const std::deque<T> &vec);
template <class OStream, class T> OStream &operator<<(OStream &os, const std::set<T> &vec);
template <class OStream, class T> OStream &operator<<(OStream &os, const std::multiset<T> &vec);
template <class OStream, class T> OStream &operator<<(OStream &os, const std::unordered_multiset<T> &vec);
template <class OStream, class T, class U> OStream &operator<<(OStream &os, const std::pair<T, U> &pa);
template <class OStream, class TK, class TV> OStream &operator<<(OStream &os, const std::map<TK, TV> &mp);
template <class OStream, class TK, class TV, class TH> OStream &operator<<(OStream &os, const std::unordered_map<TK, TV, TH> &mp);
template <class OStream, class... T> OStream &operator<<(OStream &os, const std::tuple<T...> &tpl);

template <class OStream, class T> OStream &operator<<(OStream &os, const std::vector<T> &vec) { os << '['; for (auto v : vec) os << v << ','; os << ']'; return os; }
template <class OStream, class T, size_t sz> OStream &operator<<(OStream &os, const std::array<T, sz> &arr) { os << '['; for (auto v : arr) os << v << ','; os << ']'; return os; }
template <class... T> std::istream &operator>>(std::istream &is, std::tuple<T...> &tpl) { std::apply([&is](auto &&... args) { ((is >> args), ...);}, tpl); return is; }
template <class OStream, class... T> OStream &operator<<(OStream &os, const std::tuple<T...> &tpl) { os << '('; std::apply([&os](auto &&... args) { ((os << args << ','), ...);}, tpl); return os << ')'; }
template <class OStream, class T, class TH> OStream &operator<<(OStream &os, const std::unordered_set<T, TH> &vec) { os << '{'; for (auto v : vec) os << v << ','; os << '}'; return os; }
template <class OStream, class T> OStream &operator<<(OStream &os, const std::deque<T> &vec) { os << "deq["; for (auto v : vec) os << v << ','; os << ']'; return os; }
template <class OStream, class T> OStream &operator<<(OStream &os, const std::set<T> &vec) { os << '{'; for (auto v : vec) os << v << ','; os << '}'; return os; }
template <class OStream, class T> OStream &operator<<(OStream &os, const std::multiset<T> &vec) { os << '{'; for (auto v : vec) os << v << ','; os << '}'; return os; }
template <class OStream, class T> OStream &operator<<(OStream &os, const std::unordered_multiset<T> &vec) { os << '{'; for (auto v : vec) os << v << ','; os << '}'; return os; }
template <class OStream, class T, class U> OStream &operator<<(OStream &os, const std::pair<T, U> &pa) { return os << '(' << pa.first << ',' << pa.second << ')'; }
template <class OStream, class TK, class TV> OStream &operator<<(OStream &os, const std::map<TK, TV> &mp) { os << '{'; for (auto v : mp) os << v.first << "=>" << v.second << ','; os << '}'; return os; }
template <class OStream, class TK, class TV, class TH> OStream &operator<<(OStream &os, const std::unordered_map<TK, TV, TH> &mp) { os << '{'; for (auto v : mp) os << v.first << "=>" << v.second << ','; os << '}'; return os; }
#ifdef HITONANODE_LOCAL
const string COLOR_RESET = "\033[0m", BRIGHT_GREEN = "\033[1;32m", BRIGHT_RED = "\033[1;31m", BRIGHT_CYAN = "\033[1;36m", NORMAL_CROSSED = "\033[0;9;37m", RED_BACKGROUND = "\033[1;41m", NORMAL_FAINT = "\033[0;2m";
#define dbg(x) std::cerr << BRIGHT_CYAN << #x << COLOR_RESET << " = " << (x) << NORMAL_FAINT << " (L" << __LINE__ << ") " << __FILE__ << COLOR_RESET << std::endl
#define dbgif(cond, x) ((cond) ? std::cerr << BRIGHT_CYAN << #x << COLOR_RESET << " = " << (x) << NORMAL_FAINT << " (L" << __LINE__ << ") " << __FILE__ << COLOR_RESET << std::endl : std::cerr)
#else
#define dbg(x) ((void)0)
#define dbgif(cond, x) ((void)0)
#endif


// #include <bits/stdc++.h>

using u32 = unsigned int;
using u64 = unsigned long long;

constexpr unsigned int P = 1000000007, P2 = P * 2;

struct Fp {
	u32 v;
	Fp() : v() {}
	Fp(u32 v) : v(v) {}

	Fp operator+(const Fp& rhs) const {
		u32 a = v + rhs.v;
		if (a >= P) a -= P;
		return a;
	}
	Fp& operator+=(const Fp& rhs) { if ((v += rhs.v) >= P) v -= P; return *this; }
	Fp operator-() const { return v ? P - v : v; }
	Fp operator-(const Fp& rhs) const {
		u32 a = v - rhs.v;
		if (int(a) < 0) a += P;
		return a;
	}
	Fp& operator-=(const Fp& rhs) { if (int(v -= rhs.v) < 0) v += P; return *this; }
	Fp operator*(const Fp& rhs) const { return u32(u64(v) * rhs.v % P); }
	Fp& operator*=(const Fp& rhs) { v = u64(v) * rhs.v % P; return *this; }
	Fp inv() const;
	Fp operator/(const Fp& rhs) const { return *this * rhs.inv(); }
	Fp& operator/=(const Fp& rhs) { return *this *= rhs.inv(); }
	operator u32() const { return v; }

	Fp quo2() const { return (v & 1) ? ((v + P) >> 1) : (v >> 1); }
};

Fp mpow(const Fp& a, unsigned k) {
	if (k == 0) return 1u;
	Fp ret = mpow(a * a, k >> 1);
	if (k & 1) ret *= a;
	return ret;
}
Fp Fp::inv() const { return mpow(*this, P - 2); }

std::mt19937 rng(std::chrono::steady_clock::now().time_since_epoch().count());
std::uniform_int_distribution<int> uid(0, P - 1);

using Poly = std::vector<Fp>;
using Vec = std::vector<Fp>;
using Matrix = std::vector<Vec>;

const int _ = 505;

int N;

Matrix transpose(const Matrix& mat) {
	Matrix ret(N, Vec(N));
	for (int i = 0; i != N; ++i)
		for (int j = 0; j != N; ++j)
			ret[i][j] = mat[j][i];
	return ret;
}

Matrix operator*(const Matrix& lhs, Matrix rhs) {
	for (int i = 0; i != N; ++i)
		for (int j = 0; j != i; ++j)
			std::swap(rhs[i][j], rhs[j][i]);
	Matrix ret(N, Vec(N));
	for (int i = 0; i != N; ++i)
		for (int j = 0; j != N; ++j)
			for (int k = 0; k != N; ++k)
				ret[i][j] += lhs[i][k] * rhs[j][k];
	return ret;
}

Vec operator+(const Vec& lhs, const Vec& rhs) {
	Vec ret(N);
	for (int i = 0; i != N; ++i) ret[i] = lhs[i] + rhs[i];
	return ret;
}
Vec operator-(const Vec& lhs, const Vec& rhs) {
	Vec ret(N);
	for (int i = 0; i != N; ++i) ret[i] = lhs[i] - rhs[i];
	return ret;
}
Vec operator*(const Vec& lhs, const Fp& rhs) {
	Vec ret(N);
	for (int i = 0; i != N; ++i) ret[i] = lhs[i] * rhs;
	return ret;
}

struct Basis {
	std::vector<Vec> vectors;
	Matrix reduced, coefficients;
	Basis() : reduced(N), coefficients(N) {}
	Basis(const Basis& basis) : vectors(basis.vectors), reduced(basis.reduced), coefficients(basis.coefficients) {}

	Poly insert(Vec vec) {
		int id = vectors.size();
		Vec coefficient(N);

		vectors.push_back(vec);
		for (int i = 0; i != N; ++i) {
			if (vec[i] == 0) continue;
			if (!reduced[i].empty()) {
				Fp c = vec[i];
				vec = vec - reduced[i] * c;
				coefficient = coefficient - coefficients[i] * c;
			} else {
				Fp nv = vec[i].inv();
				coefficient[id] = 1;
				reduced[i] = vec * nv;
				coefficients[i] = coefficient * nv;
				return Poly();
			}
		}
		coefficient.resize(id + 1);
		coefficient[id] = 1;
		return coefficient;
	}

	Matrix inv() {
		for (int i = N - 1; i; --i)
			for (int j = 0; j != i; ++j) {
				Fp c = reduced[j][i];
				reduced[j] = reduced[j] - reduced[i] * c;
				coefficients[j] = coefficients[j] - coefficients[i] * c;
			}
		return coefficients;
	}
};

Vec operator*(const Matrix& mat, const Vec& vec) {
	Vec ret(N);
	for (int i = 0; i != N; ++i)
		for (int j = 0; j != N; ++j)
			ret[i] += mat[i][j] * vec[j];
	return ret;
}

Poly div(Poly a, Poly b) {
	int n = a.size() - 1, m = b.size() - 1;
	Poly ret(n - m + 1);
	for (int i = n; i >= m; --i) {
		ret[i - m] = a[i];
		Fp c = -a[i];
		for (int j = 0; j <= m; ++j) {
			a[i - m + j] += c * b[j];
		}
	}
	for (int i = 0; i != m; ++i) assert(a[i] == 0);
	return ret;
}

Poly mod(Poly a, Poly b) {
	int n = b.size() - 1;
	if (a.size() < n) a.resize(n);
	int m = a.size() - 1;
	for (int i = m; i >= n; --i)
		for (int j = 1; j <= n; ++j)
			a[i - j] -= a[i] * b[n - j];
	a.resize(n);
	return a;
}

Poly powmod(Poly modulo, u64 K) {
	int n = modulo.size() - 1;
	if (K == 0) {
		Poly ret(n); ret[0] = 1;
		return ret;
	}
	Poly half = powmod(modulo, K >> 1);
	Poly ret(n * 2 - 1);
	for (int i = 0; i != n; ++i)
		for (int j = 0; j != n; ++j)
			ret[i + j] += half[i] * half[j];
	if (K & 1) ret.insert(ret.begin(), 0);
	return mod(ret, modulo);
}

Matrix Calc(Matrix a, int N, u64 K) {
    std::vector<Poly> elementaryDivisors;

    Basis basis = Basis();
    while (basis.vectors.size() < N) {
        // std::cerr << basis.vectors.size() << '\n';

        Vec initVector(N);
        for (int i = 0; i != N; ++i) initVector[i].v = uid(rng);
        Vec iterVector = initVector;
        Basis test = basis;
        Poly coefficient;
        while (true) {
            coefficient = test.insert(iterVector);
            if (!coefficient.empty()) break;
            iterVector = a * iterVector;
        }

        Poly minimalPolynomial(coefficient.begin() + basis.vectors.size(), coefficient.end());
        int minPolyDegree = minimalPolynomial.size() - 1;

        int pre = 0;
        for (int i = 0; i != elementaryDivisors.size(); ++i) {
            int degree = elementaryDivisors[i].size() - 1;
            if (degree <= minPolyDegree) {
                pre += degree;
                continue;
            }
            // std::cerr << degree << ' ' << minPolyDegree << '\n';
            // std::cerr << "COEFF ";
            // for (int j = 0; j != degree; ++j)
            // 	std::cerr << coefficient[pre + j].v << ' ';
            Poly res = div(Poly(coefficient.begin() + pre, coefficient.begin() + pre + degree),
                           minimalPolynomial);
            // std::cerr << "SIZ " << res.size() << '\n';
            // std::cerr << "RES:      " << res[0] << '\n';
            for (int j = 0; j != degree - minPolyDegree; ++j)
                initVector = initVector + basis.vectors[pre + j] * res[j];
            pre += degree;
        }

        elementaryDivisors.push_back(minimalPolynomial);
        for (int rep = 0; rep != minPolyDegree; ++rep) {
            basis.insert(initVector);
            initVector = a * initVector;
        }
    }

    Matrix b = transpose(basis.vectors);
    Matrix inv = transpose(basis.inv());

    Matrix canonicalFormPower(N, Vec(N));
    int pre = 0;
    for (Poly poly : elementaryDivisors) {
        int degree = poly.size() - 1;
        Poly res = powmod(poly, K);
        for (int i = 0; i != degree; ++i) {
            for (int j = 0; j != degree; ++j) canonicalFormPower[pre + j][pre + i] = res[j];
            res.insert(res.begin(), 0);
            res = mod(res, poly);
        }
        pre += degree;
    }
    return canonicalFormPower;
}

int main() {
    N = 2;
    std::ios::sync_with_stdio(false); std::cin.tie(nullptr); std::cout.tie(nullptr);

    Matrix A(2, Vec(2));
    Matrix B(2, Vec(2));
    REP(i, 2) REP(j, 2) { cin >> A[i][j].v; }
    REP(i, 2) REP(j, 2) { cin >> B[i][j].v; }
    dbg(A);
    dbg(B);
    auto ca = Calc(A, 2, 1);
    dbg(ca);
    auto cb = Calc(B, 2, 1);
    dbg(cb);

    if (ca == cb) {
        puts("Yes");
    } else {
        puts("No");
    }
	// std::ios::sync_with_stdio(false); std::cin.tie(nullptr); std::cout.tie(nullptr);

	// u64 K;
	// std::cin >> N >> K;
	
	// Matrix a(N, Vec(N));
	// for (int i = 0; i != N; ++i)
	// 	for (int j = 0; j != N; ++j)
	// 		std::cin >> a[i][j].v;
	

	// Matrix power = b * canonicalFormPower * inv;
	// for (int i = 0; i != N; ++i) {
	// 	for (int j = 0; j != N; ++j)
	// 		std::cout << power[i][j].v << " \n"[j == N - 1];
	// }

	// return 0;
}
yukicoder

結果

ソースコード
