結果

問題 No.1559 Next Rational
ユーザー hitonanodehitonanode
提出日時 2021-07-03 12:49:33
言語 C++17
(gcc 12.3.0 + boost 1.83.0)
結果
AC  
実行時間 56 ms / 2,000 ms
コード長 15,066 bytes
コンパイル時間 1,685 ms
コンパイル使用メモリ 151,564 KB
実行使用メモリ 32,120 KB
最終ジャッジ日時 2024-06-30 04:39:29
合計ジャッジ時間 3,464 ms
ジャッジサーバーID
（参考情報） 		judge1 / judge4
このコードへのチャレンジ
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テストケース

テストケース表示
入力 結果 実行時間
実行使用メモリ
testcase_00 AC 55 ms
31,988 KB
testcase_01 AC 18 ms
10,488 KB
testcase_02 AC 56 ms
31,992 KB
testcase_03 AC 54 ms
31,992 KB
testcase_04 AC 55 ms
32,120 KB
testcase_05 AC 54 ms
31,996 KB
testcase_06 AC 55 ms
31,992 KB
testcase_07 AC 54 ms
31,996 KB
testcase_08 AC 56 ms
31,988 KB
testcase_09 AC 55 ms
31,992 KB
testcase_10 AC 55 ms
32,116 KB
testcase_11 AC 55 ms
31,988 KB
testcase_12 AC 2 ms
6,940 KB
testcase_13 AC 55 ms
31,988 KB
testcase_14 AC 56 ms
31,988 KB
testcase_15 AC 53 ms
31,992 KB
testcase_16 AC 31 ms
17,656 KB
testcase_17 AC 18 ms
10,484 KB
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

diff # 

#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 <numeric>
#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, typename V>
void ndarray(vector<T>& vec, const V& val, int len) { vec.assign(len, val); }
template <typename T, typename V, typename... Args> void ndarray(vector<T>& vec, const V& val, int len, Args... args) { vec.resize(len), for_each(begin(vec), end(vec), [&](T& v) { ndarray(v, val, args...); }); }
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; }
int floor_lg(long long x) { return x <= 0 ? -1 : 63 - __builtin_clzll(x); }
template <typename T1, typename T2> pair<T1, T2> operator+(const pair<T1, T2> &l, const pair<T1, T2> &r) { return make_pair(l.first + r.first, l.second + r.second); }
template <typename T1, typename T2> pair<T1, T2> operator-(const pair<T1, T2> &l, const pair<T1, T2> &r) { return make_pair(l.first - r.first, l.second - r.second); }
template <typename T> vector<T> sort_unique(vector<T> vec) { sort(vec.begin(), vec.end()), vec.erase(unique(vec.begin(), vec.end()), vec.end()); return vec; }
template <typename 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 <typename 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 <typename T> istream &operator>>(istream &is, vector<T> &vec) { for (auto &v : vec) is >> v; return is; }
template <typename T> ostream &operator<<(ostream &os, const vector<T> &vec) { os << '['; for (auto v : vec) os << v << ','; os << ']'; return os; }
template <typename T, size_t sz> ostream &operator<<(ostream &os, const array<T, sz> &arr) { os << '['; for (auto v : arr) os << v << ','; os << ']'; return os; }
#if __cplusplus >= 201703L
template <typename... T> istream &operator>>(istream &is, tuple<T...> &tpl) { std::apply([&is](auto &&... args) { ((is >> args), ...);}, tpl); return is; }
template <typename... T> ostream &operator<<(ostream &os, const tuple<T...> &tpl) { os << '('; std::apply([&os](auto &&... args) { ((os << args << ','), ...);}, tpl); return os << ')'; }
#endif
template <typename T> ostream &operator<<(ostream &os, const deque<T> &vec) { os << "deq["; for (auto v : vec) os << v << ','; os << ']'; return os; }
template <typename T> ostream &operator<<(ostream &os, const set<T> &vec) { os << '{'; for (auto v : vec) os << v << ','; os << '}'; return os; }
template <typename T, typename TH> ostream &operator<<(ostream &os, const unordered_set<T, TH> &vec) { os << '{'; for (auto v : vec) os << v << ','; os << '}'; return os; }
template <typename T> ostream &operator<<(ostream &os, const multiset<T> &vec) { os << '{'; for (auto v : vec) os << v << ','; os << '}'; return os; }
template <typename T> ostream &operator<<(ostream &os, const unordered_multiset<T> &vec) { os << '{'; for (auto v : vec) os << v << ','; os << '}'; return os; }
template <typename T1, typename T2> ostream &operator<<(ostream &os, const pair<T1, T2> &pa) { os << '(' << pa.first << ',' << pa.second << ')'; return os; }
template <typename TK, typename TV> ostream &operator<<(ostream &os, const map<TK, TV> &mp) { os << '{'; for (auto v : mp) os << v.first << "=>" << v.second << ','; os << '}'; return os; }
template <typename TK, typename TV, typename TH> ostream &operator<<(ostream &os, const 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) cerr << BRIGHT_CYAN << #x << COLOR_RESET << " = " << (x) << NORMAL_FAINT << " (L" << __LINE__ << ") " << __FILE__ << COLOR_RESET << endl
#define dbgif(cond, x) ((cond) ? cerr << BRIGHT_CYAN << #x << COLOR_RESET << " = " << (x) << NORMAL_FAINT << " (L" << __LINE__ << ") " << __FILE__ << COLOR_RESET << endl : cerr)
#else
#define dbg(x) (x)
#define dbgif(cond, x) 0
#endif

template <int md> struct ModInt {
#if __cplusplus >= 201402L
#define MDCONST constexpr
#else
#define MDCONST
#endif
    using lint = long long;
    MDCONST static int mod() { return md; }
    static int get_primitive_root() {
        static int primitive_root = 0;
        if (!primitive_root) {
            primitive_root = [&]() {
                std::set<int> fac;
                int v = md - 1;
                for (lint i = 2; i * i <= v; i++)
                    while (v % i == 0) fac.insert(i), v /= i;
                if (v > 1) fac.insert(v);
                for (int g = 1; g < md; g++) {
                    bool ok = true;
                    for (auto i : fac)
                        if (ModInt(g).pow((md - 1) / i) == 1) {
                            ok = false;
                            break;
                        }
                    if (ok) return g;
                }
                return -1;
            }();
        }
        return primitive_root;
    }
    int val;
    MDCONST ModInt() : val(0) {}
    MDCONST ModInt &_setval(lint v) { return val = (v >= md ? v - md : v), *this; }
    MDCONST ModInt(lint v) { _setval(v % md + md); }
    MDCONST explicit operator bool() const { return val != 0; }
    MDCONST ModInt operator+(const ModInt &x) const { return ModInt()._setval((lint)val + x.val); }
    MDCONST ModInt operator-(const ModInt &x) const { return ModInt()._setval((lint)val - x.val + md); }
    MDCONST ModInt operator*(const ModInt &x) const { return ModInt()._setval((lint)val * x.val % md); }
    MDCONST ModInt operator/(const ModInt &x) const { return ModInt()._setval((lint)val * x.inv() % md); }
    MDCONST ModInt operator-() const { return ModInt()._setval(md - val); }
    MDCONST ModInt &operator+=(const ModInt &x) { return *this = *this + x; }
    MDCONST ModInt &operator-=(const ModInt &x) { return *this = *this - x; }
    MDCONST ModInt &operator*=(const ModInt &x) { return *this = *this * x; }
    MDCONST ModInt &operator/=(const ModInt &x) { return *this = *this / x; }
    friend MDCONST ModInt operator+(lint a, const ModInt &x) { return ModInt()._setval(a % md + x.val); }
    friend MDCONST ModInt operator-(lint a, const ModInt &x) { return ModInt()._setval(a % md - x.val + md); }
    friend MDCONST ModInt operator*(lint a, const ModInt &x) { return ModInt()._setval(a % md * x.val % md); }
    friend MDCONST ModInt operator/(lint a, const ModInt &x) {
        return ModInt()._setval(a % md * x.inv() % md);
    }
    MDCONST bool operator==(const ModInt &x) const { return val == x.val; }
    MDCONST bool operator!=(const ModInt &x) const { return val != x.val; }
    MDCONST bool operator<(const ModInt &x) const { return val < x.val; } // To use std::map<ModInt, T>
    friend std::istream &operator>>(std::istream &is, ModInt &x) {
        lint t;
        return is >> t, x = ModInt(t), is;
    }
    MDCONST friend std::ostream &operator<<(std::ostream &os, const ModInt &x) { return os << x.val; }
    MDCONST ModInt pow(lint n) const {
        ModInt ans = 1, tmp = *this;
        while (n) {
            if (n & 1) ans *= tmp;
            tmp *= tmp, n >>= 1;
        }
        return ans;
    }

    static std::vector<ModInt> facs, facinvs, invs;
    MDCONST static void _precalculation(int N) {
        int l0 = facs.size();
        if (N > md) N = md;
        if (N <= l0) return;
        facs.resize(N), facinvs.resize(N), invs.resize(N);
        for (int i = l0; i < N; i++) facs[i] = facs[i - 1] * i;
        facinvs[N - 1] = facs.back().pow(md - 2);
        for (int i = N - 2; i >= l0; i--) facinvs[i] = facinvs[i + 1] * (i + 1);
        for (int i = N - 1; i >= l0; i--) invs[i] = facinvs[i] * facs[i - 1];
    }
    MDCONST lint inv() const {
        if (this->val < std::min(md >> 1, 1 << 21)) {
            while (this->val >= int(facs.size())) _precalculation(facs.size() * 2);
            return invs[this->val].val;
        } else {
            return this->pow(md - 2).val;
        }
    }
    MDCONST ModInt fac() const {
        while (this->val >= int(facs.size())) _precalculation(facs.size() * 2);
        return facs[this->val];
    }
    MDCONST ModInt facinv() const {
        while (this->val >= int(facs.size())) _precalculation(facs.size() * 2);
        return facinvs[this->val];
    }
    MDCONST ModInt doublefac() const {
        lint k = (this->val + 1) / 2;
        return (this->val & 1) ? ModInt(k * 2).fac() / (ModInt(2).pow(k) * ModInt(k).fac())
                               : ModInt(k).fac() * ModInt(2).pow(k);
    }
    MDCONST ModInt nCr(const ModInt &r) const {
        return (this->val < r.val) ? 0 : this->fac() * (*this - r).facinv() * r.facinv();
    }
    MDCONST ModInt nPr(const ModInt &r) const {
        return (this->val < r.val) ? 0 : this->fac() * (*this - r).facinv();
    }

    ModInt sqrt() const {
        if (val == 0) return 0;
        if (md == 2) return val;
        if (pow((md - 1) / 2) != 1) return 0;
        ModInt b = 1;
        while (b.pow((md - 1) / 2) == 1) b += 1;
        int e = 0, m = md - 1;
        while (m % 2 == 0) m >>= 1, e++;
        ModInt x = pow((m - 1) / 2), y = (*this) * x * x;
        x *= (*this);
        ModInt z = b.pow(m);
        while (y != 1) {
            int j = 0;
            ModInt t = y;
            while (t != 1) j++, t *= t;
            z = z.pow(1LL << (e - j - 1));
            x *= z, z *= z, y *= z;
            e = j;
        }
        return ModInt(std::min(x.val, md - x.val));
    }
};
template <int md> std::vector<ModInt<md>> ModInt<md>::facs = {1};
template <int md> std::vector<ModInt<md>> ModInt<md>::facinvs = {1};
template <int md> std::vector<ModInt<md>> ModInt<md>::invs = {0};
using mint = ModInt<1000000007>;

// Berlekamp–Massey algorithm
// https://en.wikipedia.org/wiki/Berlekamp%E2%80%93Massey_algorithm
// Complexity: O(N^2)
// input: S = sequence from field K
// return: L          = degree of minimal polynomial,
//         C_reversed = monic min. polynomial (size = L + 1, reversed order, C_reversed[0] = 1))
// Formula: convolve(S, C_reversed)[i] = 0 for i >= L
// Example:
// - [1, 2, 4, 8, 16]   -> (1, [1, -2])
// - [1, 1, 2, 3, 5, 8] -> (2, [1, -1, -1])
// - [0, 0, 0, 0, 1]    -> (5, [1, 0, 0, 0, 0, 998244352]) (mod 998244353)
// - []                 -> (0, [1])
// - [0, 0, 0]          -> (0, [1])
// - [-2]               -> (1, [1, 2])
template <typename Tfield> std::pair<int, std::vector<Tfield>> find_linear_recurrence(const std::vector<Tfield> &S) {
    int N = S.size();
    using poly = std::vector<Tfield>;
    poly C_reversed{1}, B{1};
    int L = 0, m = 1;
    Tfield b = 1;

    // adjust: C(x) <- C(x) - (d / b) x^m B(x)
    auto adjust = [](poly C, const poly &B, Tfield d, Tfield b, int m) -> poly {
        C.resize(std::max(C.size(), B.size() + m));
        Tfield a = d / b;
        for (unsigned i = 0; i < B.size(); i++) C[i + m] -= a * B[i];
        return C;
    };

    for (int n = 0; n < N; n++) {
        Tfield d = S[n];
        for (int i = 1; i <= L; i++) d += C_reversed[i] * S[n - i];

        if (d == 0)
            m++;
        else if (2 * L <= n) {
            poly T = C_reversed;
            C_reversed = adjust(C_reversed, B, d, b, m);
            L = n + 1 - L;
            B = T;
            b = d;
            m = 1;
        } else
            C_reversed = adjust(C_reversed, B, d, b, m++);
    }
    return std::make_pair(L, C_reversed);
}

// Calculate ^N \bmod f(x)$
// Known as `Kitamasa method`
// Input: f_reversed: monic, reversed (f_reversed[0] = 1)
// Complexity: (K^2 \log N)$ ($: deg. of $)
// Example: (4, [1, -1, -1]) -> [2, 3]
//          ( x^4 = (x^2 + x + 2)(x^2 - x - 1) + 3x + 2 )
// Reference: http://misawa.github.io/others/fast_kitamasa_method.html
//            http://sugarknri.hatenablog.com/entry/2017/11/18/233936
template <typename Tfield>
std::vector<Tfield> monomial_mod_polynomial(long long N, const std::vector<Tfield> &f_reversed) {
    assert(!f_reversed.empty() and f_reversed[0] == 1);
    int K = f_reversed.size() - 1;
    if (!K) return {};
    int D = 64 - __builtin_clzll(N);
    std::vector<Tfield> ret(K, 0);
    ret[0] = 1;
    auto self_conv = [](std::vector<Tfield> x) -> std::vector<Tfield> {
        int d = x.size();
        std::vector<Tfield> ret(d * 2 - 1);
        for (int i = 0; i < d; i++) {
            ret[i * 2] += x[i] * x[i];
            for (int j = 0; j < i; j++) ret[i + j] += x[i] * x[j] * 2;
        }
        return ret;
    };
    for (int d = D; d--;) {
        ret = self_conv(ret);
        for (int i = 2 * K - 2; i >= K; i--) {
            for (int j = 1; j <= K; j++) ret[i - j] -= ret[i] * f_reversed[j];
        }
        ret.resize(K);
        if ((N >> d) & 1) {
            std::vector<Tfield> c(K);
            c[0] = -ret[K - 1] * f_reversed[K];
            for (int i = 1; i < K; i++) { c[i] = ret[i - 1] - ret[K - 1] * f_reversed[K - i]; }
            ret = c;
        }
    }
    return ret;
}

// Guess k-th element of the sequence, assuming linear recurrence
// initial_elements: 0-ORIGIN
// Verify: abc198f https://atcoder.jp/contests/abc198/submissions/21837815
template <typename Tfield> Tfield guess_kth_term(const std::vector<Tfield> &initial_elements, long long k) {
    assert(k >= 0);
    if (k < static_cast<long long>(initial_elements.size())) return initial_elements[k];
    const auto f = find_linear_recurrence<Tfield>(initial_elements).second;
    const auto g = monomial_mod_polynomial<Tfield>(k, f);
    Tfield ret = 0;
    for (unsigned i = 0; i < g.size(); i++) ret += g[i] * initial_elements[i];
    return ret;
}


int main() {
    lint N;
    cin >> N;
    mint A, B, K;
    cin >> A >> B >> K;
    vector<mint> seq(1000);
    seq[0] = A, seq[1] = B;
    FOR(i, 2, seq.size()) seq[i] = (seq[i - 1] * seq[i - 1] + K) / seq[i - 2];
    cout << guess_kth_term(seq, N - 1) << '\n';
}
0